Actuarial and financial risks in life insurance, pensions and household finance

Regis, Luca (Ed.) Book — Published Version Actuarial and financial risks in life insurance, pensions and household finance Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Regis, Luca (Ed.) (2018) : Actuarial and financial risks in life insurance, pensions and household finance, ISBN 978-3-03842-729-2, MDPI, Basel, https://doi.org/10.3390/books978-3-03842-729-2 This Version is available at: https://hdl.handle.net/10419/182437 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ Actuarial and Financial Risks in Life Insurance, Pensions and Household Finance Luca Regis www.mdpi.com/journal/risks Edited by Printed Edition of the Special Issue Published in Risks Actuarial and Financial Risks in Life Insurance, Pensions and Household Finance Special Issue Editor Luca Regis MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Books MDPI Special Issue Editor Luca Regis University of Siena Siena Editorial Office MDPI AG St. Alban-Anlage 66 Basel, Switzerland This edition is a reprint of the Special Issue published online in the open access journal Risks (ISSN 2227-9091) from 2016–2017 (available at: http://www.mdpi.com/journal/risks/special_issues/household_finance). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: Lastname, F.M.; Lastname, F.M. Article title. Journal Name. Year. Article number, page range. First Edition 2018 ISBN 978-3-03842-730-8 (Pbk) ISBN 978-3-03842-729-2 (PDF) Articles in this volume are Open Access and distributed under the Creative Commons Attribution license (CC BY), which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book taken as a whole is © 2018 MDPI, Basel, Switzerland, distributed under the terms and conditions of the Creative Commons license CC BY-NC-ND (http://creativecommons.org/licenses/by-nc-nd/4.0/). Books MDPI Table of Contents About the Special Issue Editor ...................................... v Luca Regis Special Issue Actuarial and Financial Risks in Life Insurance, Pensions and Household Finance doi: 10.3390/risks5040063 ....................................... 1 Kevin Dowd, David Blake and Andrew J. G. Cairns The Myth of Methuselah and the Uncertainty of Death: The Mortality Fan Charts doi: 10.3390/risks4030021 ....................................... 3 Gabriella Piscopo and Marina Resta Applying Spectral Biclustering to Mortality Data doi: 10.3390/risks5020024 ....................................... 10 Carlo Maccheroni and Samuel Nocito Backtesting the Lee–Carter and the Cairns–Blake–Dowd Stochastic Mortality Models on Italian Death Rates doi: 10.3390/risks5030034 ....................................... 23 Pierre Devolder and S´ebastien de Valeriola Minimum Protection in DC Funding Pension Plans and Margrabe Options doi: 10.3390/risks5010005 ....................................... 46 Catherine Donnelly A Discussion of a Risk-Sharing Pension Plan doi: 10.3390/risks5010012 ....................................... 60 Thomas G. Koch The Shifting Shape of Risk: Endogenous Market Failure for Insurance doi: 10.3390/risks5010009 ....................................... 80 Pierre Devolder and Adrien Leb`egue Compositions of Conditional Risk Measures and Solvency Capital doi: 10.3390/risks4040049 ....................................... 93 Yuguang Fan , Philip S. Griffin, Ross Maller, Alex Szimayer and Tiandong Wang The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation doi: 10.3390/risks5010003 .......................................114 Jan Natolski and Ralf Werner Mathematical Analysis of Replication by Cash Flow Matching doi: 10.3390/risks5010013 .......................................141 iii Books MDPI Books MDPI About the Special Issue Editor Luca Regis, Assistant Professor at the Department of Economics and Statistics of the University of Siena, has research interests that lie at the interface between financial and actuarial mathematics, and include corporate finance as well. In particular, he has recently been focusing on the modeling and management of longevity risk and on the capital and ownership structure choices of business groups. v Books MDPI Books MDPI risks Editorial Special Issue “Actuarial and Financial Risks in Life Insurance, Pensions and Household Finance” Luca Regis Department of Economics and Statistics, University of Siena, Siena 53100, Italy; [email protected] Received: 16 October 2017; Accepted: 28 November 2017; Published: 5 December 2017 The aim of the Special Issue is to address some of the main challenges individuals and companies face in managing financial and actuarial risks, when dealing with their investment/retirement or business-related decisions. We have received a large number of submissions, and ultimately published the nine high quality contributions that compose this issue. The papers address a variety of important issues, ranging from mortality modeling to risk management. The paper by Kevin Dowd, David Blake and Andrew Cairns (Dowd et al. 2016), introduces mortality fan charts as a novel instrument to visualize the most likely forecasts of human mortality rates, together with their uncertainty. Their application to UK mortality data suggests that there are clear limits to mortality improvements: living as long as Methusalah, who according to the Bible reached the age of 969, is pure utopia. Pierre Devolder and Adrièn Lebegue (Devolder and Lebègue 2016), in their contribution, study compositions of time-consistent dynamic risk measures, a crucial issue insurance companies have to tackle when computing their economic capital. Yuguang Fan, Philip S. Griffin, Ross Maller, Alexander Szimayer and Tiandong Wang (Fan et al. 2017) carry out an extensive simulation-based study of the effects of two reinsurance policies, namely the Largest Claim and the Excess of Loss, on the ruin probability, ruin time and value of an insurance company under the classical compound Poisson risk model. Their exercise should help guide insurers in the design of their most appropriate reinsurance strategy. The contribution by Pierre Devolder and Sébastien de Valeriola (Devolder and de Valeriola 2017) examines the two options that a new Belgian law offers to employers about the types of guarantees that the pension plans they offer to their employers should embed. Two different methodologies compare the alternatives. The paper highlights that the reform will most likely have the effect of changing the investment choices of pension plan funding vehicles. Thomas Koch (Koch 2017) addresses theoretically the issue of adverse selection in insurance markets, and applies the theoretical framework to better understand the equilibrium in the market for insurance against medical risks in the U.S. The paper suggests that changes in insurance prices were the most important determinant of the changes in the demand for medical insurance. The article by Catherine Donnelly (Donnelly 2017) deals with risk-sharing pension plans, that adjust the investment strategy and benefits to stabilize the funding ratio. The paper compares the theoretical performance of this type of plan vis-à-vis defined contribution plans, comparing the degrees of stability of the benefits provided to plan members. The paper by Jan Natolski and Ralf Werner (Natolski and Werner 2017) gives a proper mathematical formulation to the replicating portfolio approach that insurance companies commonly utilize to compute risk capital under the Solvency II framework. Gabriella Piscopo and Marina Resta (Piscopo and Resta 2017) propose an application of spectral bi-clustering to human mortality datasets. This technique can be a useful tool to guide mortality model selection, because, for instance, it may help identify the presence of cohort effects. Finally, the paper by Carlo Maccheroni and Samuel Nocito (Maccheroni and Nocito 2017) provides a backtesting analysis of the performance of the two most known stochastic mortality Risks 2017,5, 63 1 www.mdpi.com/journal/risks Books MDPI Risks 2016,4,21 way of visually representing that uncertainty: put simply, the wider the fan charts, the more uncertainty there is. They are also quantitatively plausible, in so far as they are based on an underlying mortality model that is known to provide a good fit to the data, and because these data reflect recent mortality improvements. Our fan charts suggest three main findings. First and foremost, despite recent mortality improvements, the fan chart projections suggest that no-one will live to the very extreme old ages that have sometimes been suggested in the more ‘optimistic’ literature on likely future longevity. According to these results, the idea that anyone will live to the ages reported for any of the unnaturally long-lived Patriarchs in Genesis—let alone Methusalah—is a pipe dream. Second, the fan charts clearly show that future mortality rates are highly uncertain and become more uncertain as the forecast horizon increases. They also show that future mortality rates for older cohorts are more uncertain than those for younger cohorts, other things being equal. Mortality rates are difficult to predict, especially for older cohorts, and this is so even though the mortality model provides a close fit to recent historical data: one is reminded of the old quip that forecasting is a difficult business, especially forecasting the future. Finally, the fan charts show that allowing for parameter uncertainty has a very major impact in widening the dispersion of estimated fan charts: as a rough rule of thumb, we found that allowing for uncertain parameters nearly doubles the dispersion of our fan charts. Failing to allow for parameter uncertainty therefore leads to fan chart forecasts that are far too narrow and that have lower bounds with a significant upwards bias. If we wish to reliably forecast the uncertainty in future mortality, we must therefore take account of the uncertainty in our parameter estimates as well. 6 These findings have obvious and disturbing implications for those providing services to the elderly such as health and long-term care or pensions. If people are living longer than previously anticipated, then provision needs to be made for more services for the elderly and pensioners will be drawing pensions for longer than expected. Someone then has to bear the resulting higher costs. In addition, since future mortality is very uncertain, the fan charts also suggest that the healthcare system, pension funds, life companies and, indeed, the state itself, are all heavily exposed to longevity risk, and their exposure to this risk needs to be managed. Author Contributions: KD and AJGC wrote the underlying software; all authors contributed to writing the paper. Conflicts of Interest: The authors declare no conflict of interest. Appendix A Mortality Model The mortality fan charts discussed in this paper are based on an underlying mortality model set out by Cairns et al. [ 12 ]. Let q(t,x) be the realized mortality rate in year t+ 1 (that is, from time tto time t+ 1) of a cohort aged xat time 0. We assume that q(t,x) is governed by the following two-factor Perks stochastic process: qpt,xq“ exp rA1pt`1q`A2pt`1qpt`xqs 1`exp rA1pt`1q`A2pt`1qpt`xqs (A1) where A 1 (t+1) and A 2 (t+1) are themselves stochastic processes that are measurable at time t+1 (see Perks [ 18 ], Benjamin and Pollard [ 19 ]). Now let A(t)=(A 1 (t), A 2 (t)) and assume that A(t)isa random walk with drift: Apt`1q“Aptq`μ`CZpt`1q(A2) 6 Of course, in interpreting the fan chart forecasts, we also need to be on our guard against possible biases in the model. (1) The mortality forecasts have a possible upward bias, in so far as they do not take account of future improvements to medical science (e.g., miracle cures of major illnesses) that we cannot predict; (2) On the other hand, the forecasts have a possible downward bias in that they ignore important factors such as the impact of obesity that threaten to increase future mortality but have not yet fed through into the mortality data on which our model is calibrated. Readers who have strong views on these issues might wish to take them into account in interpreting the fan charts. 8 Books MDPI Risks 2016,4,21 where μis a constant 2 ˆ1 vector of drift parameters, Cis a constant 2 ˆ2 lower triangular Choleski square root matrix of the covariance matrix V, and Z(t)isa2 ˆ 1 vector of independent standard normal variables. Cairns et al. (2006) show that this model provides a good fit to UK Government Actuary’s Department (GAD) data for English and Welsh males over 1961–2002. For each set of parameter values, we simulated 10,000 paths of A 1 (t+1) and A 2 (t+1), and then used these in Equation (A1) to obtain 10,000 simulated paths of q(t,x) over a chosen horizon. For each given t, the quantiles of q(t,x) were then obtained from the relevant order statistics of our ‘sample’ of q(t,x) values. These quantiles give us the bounds of the fan chart intervals. References 1. Vaupel, J.; Carey, J.; Christensen, K.; Johnson, T.; Yashin, A.; Holm, V.; Iachine, I.; Kannisto, V.; Khazaeli, A.; Liedo, P.; et al. Biodemographic Trajectories of Longevity. Science 1998,280, 855–860. [CrossRef] [PubMed] 2. Tuljapurkar, S.; Li, N.; Boe, C. A Universal Pattern of Mortality Decline in the G7 Countries. Nature 2000 ,405, 789–792. [CrossRef] [PubMed] 3. Oeppen, J.; Vaupel, J.W. Broken Limits of Life Expectancy. Science 2002 ,296, 1029–1031. [CrossRef] [PubMed] 4. Tuljapurkar, S. Future Mortality: A Bumpy Road to Shangri-La? Sci. Aging Knowl. Environ. 2005 ,2005. [CrossRef] [PubMed] 5. Olshansky, S.J.; Carnes, B.A.; Cassel, C. In Search of Methuselah: Estimating the Upper Limits to Human Longevity. Science 1990,250, 634–640. [CrossRef] [PubMed] 6. Olshansky, S.J.; Carnes, B.A.; Désesquelles, A. Prospects for Human Longevity. Science 2001 ,291, 1491–1492. [CrossRef] [PubMed] 7. Olshansky, S.J.; Passaro, D.; Hershow, R.; Layden, J.; Carnes, B.A.; Brody, J.; Hayflick, L.; Butler, R.N.; Allison, D.B.; Ludwig, D.S. A Potential Decline in Life Expectancy in the United States in the 21st Century. N. Engl. J. Med. 2005,352, 1103–1110. [CrossRef] [PubMed] 8. Mizuno, T.; Shu, I.-W.; Makimura, H.; Mobbs, C. Obesity Over the Life Course. Sci. Aging Knowl. Environ. 2004,2004. [CrossRef] 9. Loladze, I. Rising Atmospheric CO 2 and Human Nutrition: Toward Globally Imbalanced Plant Stoichiometry? Trends Ecol. Evolut. 2002,17, 457–461. [CrossRef] 10. De Grey, A.D.N.J. Extrapolaholics Anonymous: Why Demographers’ Rejections of a Huge Rise in Cohort Life Expectancy in This Century are Overconfident. Ann. N. Y. Acad. Sci. 2006,1067, 83–93. [CrossRef] [PubMed] 11. Sandars, N.K. The Epic of Gilgamesh: An English Version with an Introduction, 3rd ed.; Penguin: Harmondsworth, Middlesex, UK, 1972. 12. Cairns, A.J.G.; Blake, D.; Dowd, K. A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration. J. Risk Insur. 2006,73, 687–718. [CrossRef] 13. Bank of England. Inflation Report; Bank of England: London, UK, 1996. 14. King, M.A. What Fates Impose: Facing up to Uncertainty?; The British Academy: London, UK, 2004. 15. Blake, D.; Cairns, A.J.G.; Dowd, K. Longevity Risk and the Grim Reaper’s Toxic Tail: The Survivor Fan Charts. Insur. Math. Econ. 2008,42, 1062–1066. [CrossRef] 16. Dowd, K.; Blake, D.; Cairns, A.J.G. Facing up to Uncertain Life Expectancy: The Longevity Fan Charts. Demography 2010,47, 67–78. [CrossRef] [PubMed] 17. Li, J.S.H.; Ng, A.C.Y.; Chan, W.S. Stochastic Life Table Forecasting: A Time-Simultaneous Fan Chart Application. Math. Comput. Simul. 2013,93, 98–107. [CrossRef] 18. Perks, W. On Some Experiments in the Graduation of Mortality Statistics. J. Inst. Actuar. 1932,63, 12–57. 19. Benjamin, B.; Pollard, J.H. The Analysis of Mortality and Other Actuarial Statistics, 3rd ed.; Institute of Actuaries: London, UK, 1993. © 2016 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 9 Books MDPI risks Article Applying Spectral Biclustering to Mortality Data Gabriella Piscopo * and Marina Resta DIEC, University of Genova, 16126 Genova, Italy; [email protected] *Correspondence: [email protected]; Tel.: +39-010-2095008 Academic Editor: Luca Regis Received: 6 October 2016; Accepted: 29 March 2017; Published: 4 April 2017 Abstract: We apply spectral biclustering to mortality datasets in order to capture three relevant aspects: the period, the age and the cohort effects, as their knowledge is a key factor in understanding actuarial liabilities of private life insurance companies, pension funds as well as national pension systems. While standard techniques generally fail to capture the cohort effect, on the contrary, biclustering methods seem particularly suitable for this aim. We run an exploratory analysis on the mortality data of Italy, with ages representing genes, and years as conditions: by comparison between conventional hierarchical clustering and spectral biclustering, we observe that the latter offers more meaningful results. Keywords: mortality data; biclustering; cohort effect 1. Introduction During the last decade, the analysis of mortality datasets has been deepened by actuaries and statisticians. The growing attention devoted to mortality investigation is due to the financial impact on actuarial liabilities of private life insurance companies, pension funds and national pension systems: as a matter of fact, monthly pension payments are generally based on remaining life expectancy at retirement; the accurate modeling and projection of mortality rates and life expectancy are therefore of growing interest to researchers. In this respect, it makes sense to monitor how much longevity improvements are sizeable: it has been noted that, in the past century survival, probabilities have increased for each age group, even though with some undeniable differences. As mortality forecasts have become increasingly important, various parametric models have been proposed to describe mortality patterns and produce predictions: in the actuarial literature and practice, the well-known Lee-Carter (LC) model [ 1 ] extrapolates the trends of mortality and describes the secular change in longevity as a function of three determinants: the overall time trend, the age component, and the extent of change over time by age. Let us introduce the matrix M = {log m x,t }of log-mortality data for the age xat time tin more detail: xis an integer number in the range [0, 120], and tan integer representing the year of observation (to make an example: m 58,2001 is the log-mortality rate recorded in 2001 for individuals 58 years old), so that M has 121 rows and a number of columns depending on the overall number of years for which log-mortality data are available. The model seeks to summarize an age-period surface of log–mortality rates in terms of the vectors a and b along the age dimension, and k along the time dimension: log m x,t =a x +b x k t + εx,t , for every x ∈ [0, 120], and for every t, with restrictions such that the “b”s are normalized to sum to one, the “k”s sum to zero, and the “a”s are average log rates. The vector a can be interpreted as an average age profile, tracking mortality changes over time; the vector b determines how much each age group changes when ktchanges. Finally, the error term εx,t reflects age-period effects eventually not captured by the model. In the basic model, the fit to historical data is made through the Singular Value Decomposition (SVD) [ 2 ] of M, and then the time-varying parameter is modeled and forecasted as an ARIMA process using standard Box-Jenkins methodology. Risks 2017,5, 24 10 www.mdpi.com/journal/risks Books MDPI Risks 2017,5,24 As per [ 3 ], the diffusion of the LC model is mainly due to its capability to generate realistic life expectancy forecasts; moreover, many developments of the LC method have been suggested (see [ 4 , 5 ]). Furthermore, in the last decade, several authors have proposed approaches to mortality fitting based on smoothing procedures [ 6 , 7 ]; finally, [ 8 ] shows a version of the Lee–Carter methodology, the so-called Functional Demographic Model, based on the combination of smoothing techniques and functional data analysis. More recently, authors have focused attention on mortality patterns of particular generations: increased focus on the size of pension-scheme deficits, in fact, led actuaries to know more about the possible trends in future life span [ 9 ]: to such aim, some studies compare the relative influence of both the year of birth and the year of observation, and [ 10 ] introduced a variation on the LC model in order to capture the so-called cohort effect, which occurs when a particular generation exhibits patterns of mortality improvements different to those of previous generations. The LC can model the age-period effects; nevertheless, it does not consider the cohort effects, as the analysis of UK data confirms [ 11 ]. As a matter of fact, the LC age-period model does not always fit empirical data well (see, for example, [ 12 ]), while incorporating the cohort effect into the model sensitively improves the results of the fitting, as well as the comprehension of the mortality evolution. Finally, in the most recent years, a different approach is being experimented to consider the instances and issues highlighted in previous rows; it is based on clustering techniques that are employed to improve the estimation of mortality rates and the cohort effect [ 13 ]; in addition, a few number of contributions applied fuzzy logic techniques in mortality projection [ 14 , 15 ]. In particular, [ 15 ] applied fuzzy logic to mortality datasets of different countries, capturing the main time effect of the Lee-Carter model for each country. With respect to the cited literature, our work fits into different research streams, as it extends the application of biclustering techniques, already widely employed to classify gene expression matrices, to the problem of exploring mortality patterns, and it has contact points with the actuarial research strand focused on pattern recognition towards the path already opened by [ 13 ]. While, in general, the aim of clustering in genomics is to find gene similarities across all given conditions, in the mortality context, we may assume that the genes are represented by ages and the conditions by years. In the physical world, as with mortality datasets, however, a group of genes often shows similar behavior across just some of the conditions and different properties when looking at others; in these cases, the simple clustering approach is unsuitable for describing the specific patterns in the data. To overcome this limitation, biclustering approaches have been introduced to capture specific patterns in the data even if genes behave similarly over just a subset of conditions ([ 16 – 18 ], see also: [ 19 ] for a survey). Considering this, as well as the peculiar structure of mortality datasets, we follow the recent actuarial literature on pattern recognition and enhance the clustering approach with the novel introduction of biclustering to analyze mortality data. In practice, while clusters correspond to disjoint strips in the data matrix, biclusters correspond to arbitrary subsets of rows and columns. To this extent, the notion of bicluster gives rise to a more flexible computational framework. A bicluster can be defined as a submatrix spanned by a set of genes and a set of samples or equivalently, in our case, as the corresponding age and year subsets. Given a mortality matrix, we can therefore characterize the features it embodies by a collection of biclusters, each representing a different type of joint behaviour of a set of ages in a corresponding set of years, without any a priori constraints on the organization of biclusters. The biclustering problem consists then in finding a set of significant biclusters in the data matrix. Keeping in mind what was stated in previous rows, using biclustering on mortality data should make it possible to identify structural changes in the mortality trend, avoiding the estimation of terms specifically designed to incorporate cohort effects, but rather letting the features of data emerge from the analysis in quite a natural and intuitive way, as we will explain in the section of the paper devoted to describe the methodology. This, in turn, might help to develop more tailored models of mortality forecasting, providing a possibility to compare cohort effects among different countries or groups. 11 Books MDPI Risks 2017,5,24 The paper is organized as follows: Section 2 briefly recalls the notions of biclustering and spectral biclustering; Section 3 applies spectral biclustering on a mortality dataset: some numerical results and comments are presented; and final remarks are offered in Section 4. 2. Methodological Aspects of Spectral Biclustering 2.1. Biclustering The goal of clustering is to partition the elements into sets (clusters), while trying to both optimize groups homogeneity (i.e., elements of a cluster should be highly similar to each other), and group separation, which is to say: elements from different clusters should have low similarity to each other. Clustering may be a very powerful tool to automatically detect relevant sub–groups when one does not have prior knowledge about the hidden structure of these data. However, an important issue related to clustering techniques was firstly investigated in the past decade while identifying local patterns in gene expression data: it was highlighted that cluster analysis makes several a priori assumptions [ 16 ] that may not be perfectly adequate in all circumstances. As a matter of fact, clustering can be applied to either genes or samples, implicitly directing the analysis to a peculiar aspect of the system under study; in addition, clustering algorithms usually seek a disjoint cover of the set of elements, requiring that no gene or sample belongs to more than one cluster. The notion of bicluster then gave rise to a more flexible computational framework. Given a gene expression matrix, we can characterize the biological phenomena that it embodies by a collection of biclusters, each representing a different type of joint behavior of a set of genes in a corresponding set of samples. Moreover, as clustering can be separately applied to either the rows or the columns of the data matrix; biclustering, on the other hand, performs clustering in these two dimensions simultaneously. This, in turn, means that while clustering derives a global model, biclustering produces a local model [ 19 ]: likewise, in genomics, this aspect is of importance within the mortality context because of the interest of actuaries for age-specific mortality patterns. As already mentioned in Section 1, in fact, scholars are intended to capture three relevant aspects: the period, the age and the cohort effects, as their knowledge is a key factor in understanding actuarial liabilities of private life insurance companies, pension funds and national pension systems. 2.2. Spectral Biclustering Following the work of [ 16 ], biclustering has quickly become popular in analyzing gene expression data and various biclustering algorithms have been proposed [ 17 – 26 ]. Since each algorithm focuses on identification of different bicluster patterns, it is a very challenging task to thoroughly evaluate these algorithms. The rationale inside the spectral biclustering method [ 27 ] is that checkerboard structures may emerge in matrices of expression data once they have been properly arranged using a linear algebra approach, by means of the Singular Value Decomposition (SVD). Using SVD, the data matrix D of dimensions N × M can be decomposed asD=U Λ V T , where Λ is a diagonal matrix with decreasing non-negative entries, and U and V are orthonormal column matrices with dimensions N × min(N,M) and M × min(N,M), respectively. If the data matrix has a block diagonal structure (with all elements outside the blocks equal to zero), then each block can be associated with a bicluster. Specifically, if the data matrix is of the form: D=⎡ ⎢ ⎢ ⎢ ⎢ ⎣ D10...0 0D2... 0 . . .. . ..... . . 0 0 ... Dk ⎤ ⎥ ⎥ ⎥ ⎥ ⎦(1) where D i (i=1,...,k) are arbitrary matrices, then, for each D i , there will be a singular vector pair (u i ,v i ) such that a nonzero component of u i corresponds to rows occupied by D i , and a nonzero component of vicorresponds to columns occupied by Di. 12 Books MDPI Risks 2017,5,24 In a less idealized case, when the elements outside the diagonal blocks are not necessarily zeros but the diagonal blocks still contain dominating values, the SVD can reveal the biclusters as dominating components in the singular vector pairs. Box 1 illustrates the algorithmic details of the procedure, as reported in [27]. Box 1. Spectral biclustering: the algorithm. U: conditions; V: genes DN×M: gene expression matrix. Compute R = diag(D·1M) and C = diag(1MTD) Compute SVD of R−1/2 DC −1/2 Discard the pair of eigenvectors corresponding to the largest eigenvalue For each pair of eigenvectors u,vof R−1DC−1DTand C−1DTR−1D with the same eigenvalue do: Apply K-means to check the fit of u and v to stepwise vectors Report the block structure for the p couples u,vwith the best stepwise fit. In our case, the choice of the biclustering method was suggested by the analogies existing between spectral biclustering and the Singular Value Decomposition on which the Lee–Carter method [1]—LCm—is based. Furthermore, the starting point of the introduction of biclustering method in mortality context is to design a clustering approach suitable for the kind of data we are working on. Most biclustering algorithms, in fact, are apt to binary data only, and, therefore, they are not appropriate for mortality rates. Instead, spectral biclustering appears suitable for death rates matrix because it finds the checkerboard structure in data through the extraction of significant eigenvectors using the SVD associated to some normalization steps; this approach is also in line with the most used approach for fitting mortality data, the LCm, based on the SVD of the mortality matrix. Moreover, the normalization step reduces the effects of random differences in experimental conditions and levels of genes, and it is useful to highlight biclusters, if a structure exists. Finally, as a common practice in biclustering analysis is transforming the data by taking logarithms, we believe the log-normalization is appropriate in a mortality context because two or more ages are experiencing the same evolution if their mortality profiles are constantly proportional, and the log-transformation is not affected by an additive constant. 3. Discussion Case The structural description of mortality table constitutes the base to understand mortality patterns and to derive an opportune mortality model to fit and project historical data; our research aims to identify every demographic feature in mortality data, capturing both the so-called period and age effects: the former is related to the evolution of life expectancy during the years, the latter to the differences of mortality across ages. Recently, demographers and actuaries have focused their attention on a third effect, the so-called cohort effect, looking for generations that show particular mortality patterns. Without an ex-ante investigation of the structure of data, the demographical models are often worsened with cohort parameters to capture generational effect, while the interaction of age–time effects in the data structure could be misunderstood and, for example, wrongly attributed to the cohort effect. For this reason, a preliminary investigation of data becomes useful to design a parsimonious mortality model. While the microarray technology is a central tool in biological studies, it is an unexplored field in demographic context, where the identification of individuals with similar mortality patterns represents a key step in the analysis of life tables. However, traditional clustering algorithms, like the hierarchical one, organize the mortality matrix into submatrices assuming that all genes behave similarly in groups of conditions, but it would be unreasonable in the mortality context. The biclustering methods are useful to find meaningful checkerboard patterns in matrices where data represent marker genes expressed under a particular set of conditions; they permit simultaneously clustering genes and conditions. In the mortality datasets, the genes are represented by ages and the 13 Books MDPI Risks 2017,5,24 conditions by years: we observe how the mortality for individuals aged differently has changed under different conditions, i.e., as time changes. The aim of biclustering is therefore to highlight groups of mortality rates in which ages are clustered together if they exhibit similar patterns across years and, likewise, years are clustered together if they include ages whose mortality has shown a similar evolution. In this way, it is possible to discover homogeneous submatrices that are relevant for understanding mortality evolution in social and actuarial studies. 3.1. Demographical Settings and Notational Conventions In the context of mortality data, demographers and actuaries work on life tables. A life table is a finite decreasing sequence l 0 ,l 1 , ... , l ω where l x refers to an integer age xand represents the estimated number of people alive at xin a given population composed by l 0 individuals aged 0 at inception; note that ω commonly indicates the so-called extreme age, representing the age at which it occurs: l ω = 0: for sake of convenience, we can assume ω = 120. A cohort table is obtained if the sequence l 0 , l 1 , ... , l ω is the longitudinal observation of the actual numbers of individuals alive at ages 1, 2, . . . , ω out of a given initial cohort of l 0 newborns. If we consider an existing population and observe the frequency of death at different ages in a given period, for example, one year, then we obtain the period table. Finally, M = {log m ij } is usually employed to denote the matrix of death rates of a particular population, where the data are organized in rows by age, and in columns by years, so that m i,j is the mortality rate at the age i(i=0,...,ω) in the year j. 3.2. Empirical Results In our case study, we considered the death rates of the whole (male and female) Italian population; the data were downloaded from the Human Mortality Database 1 (HMD). The genes are represented by ages collected between 40 and 60, and the conditions by the years between 1950 and 2006. We focus on the ages in the range [40, 60] because those ages are the most important in pensions and actuarial product design. We applied to data both the hierarchical Euclidean clustering (whose algorithm is provided in Appendix A, at the end of the paper) and the spectral biclustering, already explained in Box 1 of Section 2, and visualized the results through heatmaps. The first step of our analysis consists in grouping the data through a Euclidean hierarchical cluster method; we applied the algorithm to the columns and rows of death rates matrix separately, and Figures 1 and 2 show the corresponding heat maps. Clustering performs data in one dimension at a time, so each gene in a given gene cluster is defined using all the conditions, and, similarly, each condition in a condition cluster is characterized by the activity of all the genes that belong to it. From a graphical point of view, this produces vertical regions that appear in Figures 1 and 2, where the period effect and the age effect become visible: moving from yellow regions to red regions, the mortality decreases during years and increases across ages. We also highlight the presence of the cohort effect, as some generations have experienced specific mortality patterns differently from others. If the cohort effect is present, a kind of “ladder” appears in the heatmap: genes aged xin t,x+1int+ 1, and so on are grouped in the same cluster and are represented with the same color shade in the heatmap. 1http://www.mortality.org. 14 Books MDPI Risks 2017,5,24 ȱ Figure 1. Clustering: the period effect. ȱ Figure 2. Clustering: the age effect. In more detail, in Figure 1, we may observe a diagonal line, corresponding to the individuals aged 40 in 1958, i.e., the generation born in 1918; such graphical evidence suggests that this group has experienced a particular reduction in mortality during the years. However, it is not reasonable that groups of genes behave similarly under all conditions; it could occur, for example, that group of ages show similar features just during some years, and they cluster differently during other years. To capture this detailed structure, we introduce the biclustering as a useful tool to find meaningful checkerboard patterns. 15 Books MDPI Risks 2017,5,24 In the second step of our application, we have therefore randomly applied the spectral biclustering algorithm to the matrix of mortality rates, normalized through the log-transformation, thus obtaining 20 clusters: main results are visually summarized in Figures 3–7. Above all, the heat map of Figure 3 shows not only vertical regions, but also rectangular areas produced by the interaction between age and period effects. Figure 3. Biclustering on the mortality dataset. Moreover, Figures 4 and 5 highlight the composition of each bicluster, summarized in Table 1: four groups of ages (40–47, 48–53, 54–57, 58–60) are defined, each of them characterized by significant patterns across particular years. For example, the first bicluster contains ages 58–60 during the years 1987–1993; the same ages act differently during the years 1970–1986 and are grouped in the second bicluster, and so on. Figure 4. Biclusters 1–10. 16 Books MDPI Risks 2017,5,24 Figure 5. Biclusters 11–20. Table 1. The composition of biclusters. Cluster Age Years 1 58–60 1987–1993 2 58–60 1970–1986 3 58–60 1952, 1956, 1957, 1962 4 58–60 1994–2006 5 58–60 1950–1969 except for 1952, 1956, 1957, 1962 6 48–53 1987–1993 7 48–53 1970–1986 8 48–53 1952, 1956, 1957, 1962 9 48–53 1994–2006 10 48–53 1950–1969 except for 1952, 1956, 1957, 1962 11 40–47 1987–1993 12 40–47 1970–1986 13 40–47 1952, 1956, 1957, 1962 14 40–47 1994–2006 15 40–47 1950–1969 except for 1952, 1956, 1957, 1962 16 54–57 1987–1993 17 54–57 1970–1986 18 54–57 1952, 1956, 1957, 1962 19 54–57 1994–2006 20 54–57 1950–1969 except for 1952, 1956, 1957, 1962 In order to provide the reader with additional interpretative keys, in Figures 6 and 7, we show parallel coordinate graphs for mortality levels. 17 Books MDPI Risks 2017,5,34 computations (ANIA 2014). Therefore, we chose to compare the original formulation of the LC model to the original CBD since they also represent the two most used parametric families of mortality models. On the one hand, the Lee–Carter model has sparked a deep methodological revolution in the field of demographic forecast, particularly in mortality. The mortality model has been used together with a similar fertility model and deterministic migration assumptions to generate stochastic forecasts about the population and its components. These stochastic population forecasts, in turn, have been used as the key component of stochastic projections of the finances of the US Social Security system. The stochastic forecast avoids some of the problems inherent to using the classic scenario method for representing forecast uncertainty (Lee 2000). Then, in concurrence with the main demographic applications, the LC model suggested: • an important research front on problems related to the parameter estimations (Booth et al. 2006), with many applications also in the actuarial and economics literature (Loisel and Serant 2007); and • extension of the forecasting analysis with disaggregated projections on demographic subsets to maintain consistency at the aggregate level (Lee and Miller 2001; Li and Lee 2005; Li 2010). On the other hand, the Cairns–Blake–Dowd model, even if more recent in its formulation than the LC model, has played an important role in forecasting mortality at higher ages (i.e., ages starting at 60 and over). The mortality model made great contributions for pension funds, life-insurance companies and private annuity providers in general. It is mainly used for pricing longevity bonds as suggested also by the authors in the first formulation of the model (Cairns et al. 2006). The second aim of this work is to analyse the medium-length forecast with respect to the short term, observing potential differences in the parameter estimations (Mavros et al. 2014) 3 accordingly with changes in the starting point of the database. Chan et al. (2014) have also studied the new-data-invariant property on the quality of the CBD mortality index. For this purpose, we introduced a new backtesting approach named the jumping fixed-length horizon, which makes short-run projections of five years, “jumping forward” in the historical database by five-year-steps. Considerations of the backtesting results do not imply a conclusive evaluation of the models, since we perform the analysis exclusively for the range of ages 57 to 90. The choice for the interval of ages was motivated by the fact that, in Italy, Ragioneria dello Stato computes the so-called transformation coefficients for pension annuities, starting from age 57. Moreover, since the CBD model is recommended as a good predictor of mortality at higher ages, we chose this interval of ages to make a more prudent and accurate comparison between the models. Furthermore, we decided to take into consideration only death probabilities qx,t among all of the other possible biometric functions. We used4death probabilities qx,tprovided by ISTAT spanning the period 1975–2014. Then, over the designated horizon of historical data, we select the “lookback” and the “lookforward” windows 5 , respectively, for the parameter estimation and forecast. In particular, the length of the forecast window will be different for each of the three backtesting approaches proposed by the work: •fixed horizon backtests: lookback and lookforward windows of 20 years; • jumping fixed-length horizon backtests: lookback window of 20 years and lookforward window of 5 years (short-term projections); and • rolling fixed-length horizon backtests: lookback window of fixed-length (20 years) and a contracting lookforward window from 20 to 2 years of projections. The paper is organized as follows. Section 2 briefly presents the models and the adopted terminology, Section 3 shows the historical Italian mortality data, and Section 4 and subsections explain methodology and the backtesting results obtained by the different approaches. Section 5 provides conclusions. 3Particularly for the case of Cairns–Blake–Dowd model. 4Data downloaded on June 2016. Source: http://demo.istat.it/tvm2016/index.php?lingua=eng. 5For the sake of simplicity, we decided to adopt the same terminology used by Dowd et al. (2010a). 24 Books MDPI Risks 2017,5,34 2. Model Specifications 2.1. The Lee–Carter Model We took into consideration the original formulation of Lee and Carter (1992), represented by the following model equation: mx,t=eαx+βxkt+εx,t, (1) where mx,tis the central rate of mortality at age xand at time t, and it is given by the formula: mx,t=dx,t Lx,t, with 6dx,t representing the number of deaths that occurred between x and x+ 1, and Lx,t called the age units living in x, which is simply the average number of individuals alive between xand x+1. For simplicity, the model was implemented by adopting its logarithm transformation: ln mx,t=αx+βxkt+εx,t, with the following parameter interpretations: •ktis the time index representing the level of mortality at time t; •αxrepresents the average trend of mortality on the time horizon at age x; •βx represents a measure of the sensitivity in movement from the parameter kt . In particular, βxdescribes the relative speed of mortality changes, at each age, when ktchanges; and •εx,t is the homoskedastic error term, which incorporates historical trends not considered by the model. It is assumed to be εx,t∼N(0, σ2 ε). Appendix A illustrates the method adopted for the estimation and projection of the parameters. 2.2. The Cairns–Blake–Dowd Model We considered the original formulation of the model provided by Cairns et al. (2006) with the following model equation: ln qx,t px,t=k(1) t+k(2) t(x−¯ x)+εx,t, (2) where •k(1) tand k(2) tare two stochastic processes and represent the two time indexes of the model; •qx,t and px,t represent, respectively, the death and the survival probability, at time t for an individual aged x; •ln qx,t px,t=ln (φx)=logit qx,t is the logit transformation of qx,t , with φx representing the mortality odds; •¯ xis the mean age of the considered interval of ages; and •εx,t is the error term that encloses the historical trend that the model does not express. All of the error terms are i.i.d following the Normal distribution with mean 0 and variance σ2 ε. The model is fully identified, so it does not require additional constraints. Moreover, the time index k(1) t is the intercept of the model. It affects every age in the same way, and it represents the level of mortality at time t . More precisely, if it declines over time, it means that 6The variables dx,tand Lx,tare the common biometric functions as described in the life tables. 25 Books MDPI Risks 2017,5,34 the mortality rate has been decreasing over time at all ages. The time index k(2) t represents the slope of the model: every age is differently affected by this parameter. For instance, if during the fitting period, the mortality improvements have been greater at lower ages than at higher ages, the slope period term k(2) t would be increasing over time. In such a case, the plot of the logit of death probabilities against age would become steeper as it shifts downwards over time (Pitacco et al. 2009). Appendix B illustrates the estimation and projection methods involved. 3. Case Study: Italian Mortality Data from 1975 to 2014 The application of the presented models requires the use of the death probabilities time series for extrapolating mortality forecast. As already mentioned, we use data provided by ISTAT because these data are commonly used by private insurance companies and public pension providers. The range of ages is 57 ≤x≤ 90. In particular, we chose the upper limit for taking into consideration the ISTAT graduation method of ending the life table (Istat 2001). The calculation of the probabilities of dying for ages over 95 is performed by extrapolating the qx,tgraduated values following the Thatcher et al. (1998) model7: qx,t=ϑeγx 1+ϑeγx;x≥95. (3) This kind of graduation could affect the backtesting results, comparing realized data with forecasts obtained by applying the LC (1) and the CBD (2) models, since they offer a different mortality pattern at old ages. For the ages from 5 to 94, ISTAT uses a moving average of crude rates with the length of seven values. Moreover, we selected the time period from 1975 to 2014 because, from in the mid-seventies in Italy, the successful fight against cardiovascular diseases began. More recently, efforts against tumors, which are still the main cause of death, have been launched. These successes have contributed to an extraordinary acceleration of growth in life expectancy, especially at higher ages: e.g., from 1975 to 2014, life expectancy at 60 years has seen an average increase of about four hours each day, both for men and women. In the male case, this phenomena extraordinarily occurred. Previously, life expectancy at birth had registered the first significant increase due to the control of infant and child mortality, while during the years under review, it has also benefited from the control of adult age mortality. Currently, the probability of reaching an old age for a young adult is really high: for a 30 year old, the probability of reaching the age of 60 is almost 94% for males and 96.4% for females. However, it remains difficult to reach the threshold of 90 years, especially for men. Table 1 accurately shows 8 how this probability changed starting from age 50. Moreover, it shows how the difference in probability between genders became greater as the age increased. This process is known as the rectangularization and shift forward of the survival curves. Its measure can be derived from the entropy of a life table (Equation (4)). It was introduced by Keyfitz and Caswell (2005) and it is referred to in this paper as tHK,ξ with ξ the age by which the survival curve is built, and t the year of the period life table at which the entropy is computed (in our case t= 1975, 1976, ... 2014). Then, tHK,ξ=−∑j(ln lj)lj ∑jlj , (4) 7 In Equation (3) ϑ and γ are parameters that need to be estimated. In general, those parameters are estimated by applying Ordinary Least Squares (OLS) on the logit transformation of Equation (3). 8 Even though the backtesting analysis will be focused on the interval of ages 57–90, here we decided to provide information also on ages lower than x= 57. In this way, we are able to present a more accurate Italian demographic scenario for the period observed. 26 Books MDPI Risks 2017,5,34 where lj is the probability of surviving from age 9ξ ( ξ= 0, 1, ..., w ; lξ =1 ∀ξ )toage j ( j=ξ+ 1, ξ+ 2, . .. w ). The entropy index becomes smaller whenever the survivorship curve lj moves towards a rectangular form; in this limit case, tHK,ξ=0. Table 1. Proportion of persons aged 30 and expected to be alive at selected ages. Italian Period Life Tables Ages 1975 1980 1985 1990 1995 2000 2005 2010 2014 Male 50 0.9438 0.9483 0.9554 0.9583 0.9591 0.9662 0.9722 0.9755 0.9777 60 0.8406 0.8487 0.8646 0.8839 0.8951 0.90962 0.9242 0.9324 0.9376 70 0.6292 0.6409 0.6691 0.7081 0.7351 0.7732 0.8060 0.8257 0.8385 80 0.3014 0.3161 0.3539 0.4029 0.4406 0.4936 0.5434 0.5912 0.6188 90 0.0464 0.0527 0.0682 0.0954 0.1170 0.1396 0.1648 0.1996 0.2250 95 0.0080 0.0096 0.0140 0.0235 0.0318 0.0401 0.0491 0.0595 0.0743 Female 50 0.9703 0.9739 0.9769 0.9785 0.9796 0.9822 0.9850 0.9865 0.9871 60 0.9194 0.9290 0.9364 0.9427 0.9473 0.9525 0.9585 0.9620 0.9639 70 0.8009 0.8168 0.8337 0.8546 0.8681 0.8828 0.8972 0.9053 0.9087 80 0.5070 0.5403 0.5814 0.6249 0.6576 0.69561 0.7322 0.7540 0.7674 90 0.1154 0.1433 0.1629 0.2141 0.2547 0.2860 0.3297 0.3653 0.3878 95 0.0226 0.0326 0.0380 0.0626 0.0830 0.1030 0.1259 0.1420 0.1654 Figure 1 shows how the trend of the rectangularization process has changed according to ages (i.e., from ξ= 50 to ξ= 65, 75). Regarding women, this process was already in place before 1975. In particular, starting from ages 50 and 65, it is continued with a substantially linear continuity. In the case of men, the rectangularization process begins to escalate smoothly after 1984. However, the following trend shows a deep reduction of mortality, from which is derived an attenuation of the inequality between sexes even though it has not disappeared. In Figure 1, tHk,ξ shows that the mortality improvement in the elderly population has taken place at different rates over time, particularly with a faster steep decline for both sexes after 1993. Figure 1. Italian life tables 1975–2014: males and females entropy (tHK,ξ). The differentiation of the pace in reducing mortality of both sexes starting from adult age up to those who are old is confirmed by the results of the Kullback and Leibler (1951) divergence: 9The starting point for the final age interval is denoted by w. 27 Books MDPI Risks 2017,5,34 tDKL,ξ(hz,gz)= w−ξ ∑ z=0 hzlnhz gz, (5) where hz and gz are the probability distributions of the “time until death” random variable Zξ for a person aged ξ , respectively, for males and females. Equation (5) measures the “difference” between these two probability distributions, which, in our case, is taken as the reference model gz . The choice is motived not only by the fact that mortality is significantly lower for women than for men, but also because the continuous decline of female mortality in the reporting period occurred much more regularly (Maccheroni 2014). The divergence in mortality between genders mortality has different characteristics depending on the considered age group. Figure 2 shows that the divergence in mortality between sexes presents different characteristics, depending on the observed age. In particular, until 1981, the divergence gradually increased on the full range of ages. At a later time, differentials in mortality between sexes decrease whenever x is lower than 60, while they progressively increase at higher ages. These diverging trends make the application of the models interesting, especially for the comparison of results. Needless to say, the mortality forecast will be more accurate for women than men because women experienced a death risk reduction process with greater regularity than men. Figure 2. Kullback–Leibler divergence with respect to Zξat selected ages. 4. Backtesting Analysis In this section, we introduce the three different backtesting frameworks, and we present the related forecast results. • The fixed horizon backtest uses a fixed twenty-year historical “lookback” interval, 1975 ≤t≤ 1994 , and a fixed “lookforward” horizon, 1995 ≤t≤2014 (20 years). • The jumping fixed-length horizon backtests make short run projections of five years 10 and keep fixed the length of the “lookback” horizon (20 years), but make jumps of five years ahead to cover the “lookforward” interval, 1995 ≤t≤ 2014. This analysis is divided into four groups of estimations and forecast, described in Table 2. • Finally, the rolling fixed-length horizon backtests keep fixed the length of the “lookback” horizon (i.e., 20 years) and let it roll ahead year by year. The projections are made over the remaining horizon, keeping fixed the last year of the projection at t= 2014. This analysis is divided into nineteen groups of estimations and forecast, described in Table 3. 10 We made a forecast of each year in the short-run projection window (5 years). 28 Books MDPI Risks 2017,5,34 Table 2. Jumping fixed-length horizon backtests data horizon. Lookback Horizon Lookforward Horizon 1975–1994 1995–1999 1980–1999 2000–2004 1985–2004 2005–2009 1990–2009 2010–2014 Table 3. Rolling fixed-length horizon backtests data horizon. Lookback Lookforward Lookback Lookforward 1975–1994 1995–2014 (20) 1985–2004 2005-2014 (10) 1976–1995 1996–2014 (19) 1986–2005 2006–2014 (9) 1977–1996 1997–2014 (18) 1987–2006 2007–2014 (8) 1978–1997 1998–2014 (17) 1988–2007 2008–2014 (7) 1979–1998 1999–2014 (16) 1989–2008 2009–2014 (6) 1980–1999 2000–2014 (15) 1990–2009 2010–2014 (5) 1981–2000 2001–2014 (14) 1991–2010 2011–2014 (4) 1982–2001 2002–2014 (13) 1992–2011 2012–2014 (3) 1983–2002 2003–2014 (12) 1993–2012 2013–2014 (2) 1984–2003 2004–2014 (11) The numbers in parentheses show the length of the “lookforward” horizon. Moreover, they indicate the position of the year 2014 over the related projection interval. This will be particularly useful for the analysis of results that will be presented in Section 4.3. Before going in depth about the backtesting analysis, we check for the estimation quality of the models over the historical “lookback” interval, 1975 ≤t≤ 1994. For this purpose, we use the index Λ2 x, a form of R2that particularly fits our case (Draper and Smith 2014), described as follows: Λ2 x=1− 1 n∑t(qO x,t−qft x,t)2−1 n∑t(qO x,t−qft x,t)2 1 n∑t(qO x,t)2−1 n∑t(qO x,t)2, where qft x,t is the fitted value for the qx,t and n is the total number of considered years (i.e., n= 20). The index provides the proportion of the temporal variance explained by the model for all 57 ≤x≤ 90. Figure 3 shows that both models fit the observed data generally well. Particularly in the case of males, the share of the “explained variance” at any age is always greater than 88%, while, for females, in the case of LC, it falls to 85% at x= 63. However, such a decrease takes place within a very limited age range between 61 and 65 years. Figure 3. Proportion of temporal variance explained by the models: 1975–1994. 29 Books MDPI Risks 2017,5,34 More specifically, by the analysis of the “explained variance” for both models, we see that the irregular path of the curves may be influenced by a cohort effect before the age x= 80. This effect is diagonally observable on the graphs in Figure 4 for those individuals aged 57–59 in 1975 and 76–78 in 1994, respectively. These are the generations born during the First World War (1915-1918) who, in the course of their lives, have experienced higher mortality at the same ages than the previous and next cohorts (Maccheroni 2016). For ages older than x= 75, the differences between the models are sharply evident. In particular, LC overestimates qO x,tand CBD underestimates (Figure 4). Figure 4. Residuals qO x,t−qP x,tby age x: 1975–1994. Analysis of the projection results that will be presented in the next section shows that the described cohort effect has an impact on the forecast quality of the models in two ways. • Both models slightly suffer the cohort effect for both populations over the projection horizon (1995–2014) for the same cohort aged 77–79 in 1995 that is no longer observed from 2006. In particular, both models show an underestimated forecast for such birth cohorts on both sexes with observed values above the upper limit of the confidence interval for some ages of the cohort. This occurred particularly for males. • The observed male qx,t for individuals aged 57–59 in 1995 and 76–78 in 2014, respectively, are often under the lower extreme of the forecast confidence interval. It seems that models have replicated the cohort effect over an homologous cohort in 1995, but since the male mortality evolution has changed consistently from 1975–1994 to 1995–2014, the two homologous cohorts (i.e., 57–59 in 1975 and 57–59 in 1995) showed different trends that lead to forecast errors. This scenario does not occur for females, since women experienced a more ordinary mortality evolution. Therefore, the homologous cohorts are similar, so the bias is not observable. For these reasons, the results obtained with the three backtesting approaches need to be evaluated, taking into consideration the analysed cohort effect and its related impact on the forecast. In particular, forecasts seem to suffer the cohort effect as long as the data used for the estimation of the parameters take into account years from 1975 to 1985. After 1985, the cohort effect is small compared to the overall sample; therefore, projections do not suffer greatly from it. 30 Books MDPI Risks 2017,5,34 4.1. Fixed Horizon Backtest (1995–2014) The first backtesting analysis takes into account a forecast horizon that is demographically considered a medium-term projection horizon. The comparisons proposed are among the most likely values of qP x,t prediction, which is the projected central value derived by the model on which we constructed the 95% confidence interval and those observed qO x,t ; comparisons between the central value and extremes of the confidence interval occur only for the ages 65 and 85. These are the ages that in the demographic literature mark the entrance in the range of so-called “young-old” and “oldest-old”. Unfortunately, due to space limitations, it was not possible to present the comparison to the age of 75, which divides the old from the “young-old” (Vaupel 2010). The qO x,t can present a strong temporal variability due to the observed cohort effect and to the so-called “period effect”, which is the time condition that affects mortality via a variety of factors. Among these, the best known is the climatic effect that can, for instance, cause a rise in mortality at old ages during a very hot summer (e.g., an episode occurred in Italy in 2003), or epidemiological effects that arise from flu in winter in low-mortality countries. Needless to say, the impact of those factors is stronger on the most vulnerable people. For this reason, a rise in mortality due to those factors is generally followed by a decrease in mortality, since those who remained alive have a lower frailty level. These mortality shocks can affect short-term forecasts rather than medium-term ones, since the latter are usually more capable of capturing changes in environmental and socio-economic conditions and people’s lifestyles. From an applicative point of view, particularly focused on the insurance and social security sector, we were interested in analysing the performance of the models on assessing the risk of death at various ages. It is from this point of view that we are going to develop our analysis. For this purpose, we make a brief assessment of forecast errors that was performed using as an index the Root Mean Squared Errors (RMSE), defined as follows: RMSE =√MSE, with MSE =1 υ∑ x ∑ t (qO x,t−qP x,t)2, where the mean squared errors (MSE) are equal to the sum of squared errors adjusted for the residual degrees of freedom υ . Moreover, qO x,t and qP x,t are, respectively, the death probabilities observed and forecast (projected). We use the root of the adjusted SSE to take into account the difference in the number of free parameters between the models. Table 4 shows RMSE for the first and the second backtesting approach that will be presented in the following section. Moreover, it takes into account exclusively the central value of the confidence interval as the most relevant for pension policy-makers and annuity providers (Whitehouse 2007). Table 4 shows how the LC model proposes a more accurate forecast with respect to the CBD model for the period 1995–2014 for females; it is more difficult to judge the models’ performances for males given the small difference between the RMSE results. These predictions are produced on the extrapolated parameters kt (Appendix Equations (A7) and (A9)), but the result is made more flexible by the stochastic component of the models that allows building of the forecast confidence interval. One cause of error can arise from the fact that the central value of the projection may be shifted with respect to the observed data, even though it does not differ from the observed trend recorded over the projection horizon. Figure 5 provides a graphic explanation of the phenomenon. In particular, for individuals aged 65, the male forecasts 1995–2014 are above the mortality trend observed for the same period, with divergent paths for the LC model. In the female case (age 65), only the CBD model shows divergences. However, these deviations may be instead very low, as in the case of the LC model for females aged 65, or in the case of both models for both sexes aged 85 (Figure 5). Moreover, the bias due to the continuing fluctuations of the risk of death over time has to be taken in consideration. 31 Books MDPI Risks 2017,5,34 The confidence interval provided by the two models takes into account this stochastic component of mortality (Figures 7 and 9), although this may occur with different levels of precision (Figure 10). Figures 6 and 8 show the overall error dynamic highlighted by the ratio between the projected qP x,tand the observed qO x,t. Figure 5. LC and CBD models: comparisons between observed and forecast mortality trends. As far as men are concerned, the LC model initially overestimates qO x,t from ages 57 to 80 (approximately), with persistence across years. In particular, the overestimation errors become sharply evident as the projection is extended to the last year of the forecast horizon. Figure 6 multi-dimensionally shows the ratio between projected and the real death probabilities. The described LC performance trend is also graphically reported by the Figure 7, comparing projections at ages 65 and 85 to the observed data. The overestimation starts decreasing from age 80, pointing out that the divergence between qO x,t and qP x,t is really close to zero. However, for high ages at the extreme of the interval, LC forecasts systematically underestimate qO x,t. Figure 6. Lee–Carter Fixed Horizon Backtest: qP x,t qO x,t ratio. As for women, the divergence between qO x,t and qP x,t is sharply smaller than for men. This is particularly evident in Figure 6, which shows that the forecast initially underestimates real data converging at the age 65 and then starts overestimating for a wide span of ages. Furthermore, the last 32 Books MDPI Risks 2017,5,34 part of the age range is again characterized by an underestimation path. However, the overestimation experienced at higher ages is smaller than the one observed in the male case. Figure 7. LC Fixed Horizon Backtest forecast: comparison between observed death rates and the corresponding 95% confidence interval of the forecast based on the time series 1975–1994. The CBD forecast greatly overestimates the male mortality historical evolution, particularly for the central and last years of projection. The error is evident in the full range of ages, although it becomes smaller at the age 80, after which forecasts start underestimating qO x,t with an increasing magnitude until the last age and the last projection year (i.e., x=90 and t=2014) (Figure 8). When we look at the female case, the accuracy of the CBD forecast is worse. In this case, in fact, we can notice a wide and systematic underestimation on approximately all of the first half of the age range for almost the totality of the forecast horizon. In particular, the forecast error reduces around the age 68, then it starts overestimating until x= 85, after which it underestimates again. However, at x= 85, the forecast is relatively accurate, with values of qO x,t all inside the confidence interval (Figure 9). Figure 8. Cairns–Blake–Dowd Fixed Horizon Backtest: qP x,t qO x,t ratio. 33 Books MDPI Risks 2017,5,34 Figure 18. CBD Rolling Fixed-Length Horizon Backtests (age 85): convergence to real data (2014). Figure 19. LC Rolling Fixed-Length Horizon Backtests (age 65): convergence to real data (2014). Figure 20. LC Rolling Fixed-Length Horizon Backtests (age 85): convergence to real data (2014). 5. Conclusions The main aims of this paper are to scrutinize the forecast for both sexes proposed by the original formulation of the models, given the wide use of LC at the national level, and to analyse the long-term forecast with respect to the short term, observing qualitative differences in the estimation of the parameter accordingly to changes in the starting point of the database. Regarding the former, we find that, basically, neither model was able to capture the shock in terms of improvements on the male mortality trend, with greater biases for ages lower than x= 75, which were those more affected by the improvement. In this sense, CBD forecasts for those ages are more biased than LC projections in terms of overestimations. The limited capacity of the models to predict male mortality is evident in all of the three backtesting frameworks. Table 4 numerically summarizes the difference in terms of performances between sexes for the first two backtesting approaches. In addition, the analysis of the forecast for the year 2014 that we provided with the third 40 Books MDPI Risks 2017,5,34 approach confirms this result. Moreover, women’s forecasts are widely more accurate than men’s, with small biases observed both in the short and the medium-term. However, in the female case, CBD projections showed particularly deep and systematic underestimations with respect to ages lower than 75. From the comparison between the short-term and the medium-term forecast, we find that changes in the starting point of the database widely affect the estimation of the LC parameters, particularly for βx with observable impacts on the projections. The female forecasts are more influenced by those changes in βx . The CBD model satisfies the “new-data-invariant” property for the estimation of the parameter k(1) t , while k(2) t presents persistent changes for the same year as the dataset slides forward. This aspect is more evident in males than in females. In particular, the adjustment of the parameter k(2) t (i.e., x−¯ x ) affects mortality forecasts with weights of the opposite sign at the extremes of the considered age range. The weight is greater the larger the age range. This structural characteristic of the model, albeit simultaneously with k(1) t , results in a systematic underestimation of the qO x,t for ages lower than ¯ x that gradually decrease as x moves towards ¯ x . Moreover, mortality forecasts around ¯ x are almost exclusively explained by k(1) t , since ( x−¯ x ) is really close to 0 in that case. On the contrary, as x gets closer to the upper limit of the age range, the weight of ( x−¯ x ) on mortality forecasts changes with the opposite sign, with resulting overestimation of the qO x,t . For these reasons, the risk in terms of application of the models is conspicuous because it could potentially affect both the mortality risk and the longevity risk. Taking into consideration the variability of both the parameters βx (LC) and k(2) t (CBD), it is difficult to judge a priori what these two rigidities penalize more in the mortality forecast. As far as the CBD model is concerned, we find that projections are not reliable for describing mortality at ages before x= 75. For this reason, LC projections are preferable for describing Italian mortality in this particular framework of years and ages. However, CBD forecasts showed a more restrained variability of the forecast error at higher ages with respect to LC. This result and the fact that usually the CBD confidence interval at higher ages is wider (i.e., LC is nested in CBD) than LC ones provide a more accurate theoretical robustness to the CBD for ages greater than x=75. We would like to make clear that we examined the models in their original form, so we cannot rule out the possibility that some extensions of the models might resolve these issues on Italian data (1975–2014). In particular, we expect that the results of both models may be improved with the adoption of the model extensions, including a cohort component, in order to reduce the bias caused by the cohort effect of those born during the First World War. Moreover, the CBD extension, including the quadratic term of the age component, may solve the weighting issue of the model over the considered interval of ages on this data. In conclusion, the results seem to be relevant for private and public Italian annuity providers that use LC forecasts as demographic bases. From this perspective, the choice between the two models may vary in accordance with the purpose of the use of the model (e.g., the age and the sex of the insured). Even though we limited our analysis to the study of the forecast qx,t , we can infer that a backtesting analysis of annuity prices, based on the forecast obtained by the original formulations of the models, would show evidence of a distortion caused by the forecast error on the money’s worth of an annuity and on reserves. Acknowledgments: The authors want to thank the Center for Research on Pensions and Welfare Policies (CeRP) for supporting this study. It is one of the CeRP’s contributions to the hackUniTO research project about aging, promoted by the University of Turin. Author Contributions: The authors contributed equally to this work. Conflicts of Interest: The authors declare no conflict of interest. 41 Books MDPI Risks 2017,5,34 Appendix A. Lee–Carter Estimation and Projection Appendix A.1. Parameter Estimations The parameter estimation was computed with respect to the Ordinary Least Square (OLS) estimation method in accordance with the original approach suggested by the authors. The following constraints were used to find a unique solution for the parameters: xm ∑ x=x1 βx=1and tn ∑ t=t1 kt=0. (A1) To obtain the estimation for the variable ˆ αx , it was necessary to compute the partial derivative of the equation LS(α , β , k)=∑x∑t(ln (mx,t)−αx−βxkt)2 , with respect to αx . Then, as a first order condition, we get: ˆ αx=1 tn−t1+1∑ t ln (mx,t), (A2) where the denominator simply represents the number of years considered in the dataset, and x=x1 , ..., xm is the considered range of ages. As is expressed by the Equation (A2), the estimation for the first parameter αx was given by the average of the logarithms of the central rate of mortality over time t . Furthermore, the estimations of ˆ βx and ˆ kt for the parameters βx and kt were obtained by adopting the Singular Value Decomposition of the matrix A of elements (ln mxi,tj−αxi) , with i as age index and jas time index (years considered in the data). At this point, the estimated parameters were recalibrated so the differences between the actual and the estimated total deaths in each year were zero. This implies that the recalibrated ˆ k∗ t solves the equation15: ∑ x dx,t=∑ x e(ˆ α∗ x+ˆ β∗ xˆ k∗ t)Lx,t. (A3) Finally, the estimated parameters were adjusted to satisfy the constraint at (A1) for the parameter ˆ k∗ t. Then: a∗ x=ˆ αx+ˆ βx¯ k, (A4) β∗ x=ˆ βxxm−x1+1 ∑ j=1 ˆ β1j, (A5) k∗ t=( ˆ k∗ t−¯ k)xm−x1+1 ∑ j=1 ˆ β1j, (A6) where ¯ k=1 tn−t1+1∑tn t=1ˆ k∗ t is the arithmetic average of ˆ k∗ t with respect to time t , and ∑xm−x1+1 j=1ˆ β1j is simply the sum of all the estimated ˆ β , which sum to 1. The fitted model is then used to estimate the median and the 95% prediction interval. 15 Equation (A3) has no explicit solution, so it has to be solved numerically. 42 Books MDPI Risks 2017,5,34 Appendix A.2. Parameter Projection We projected the estimated 16 parameters k∗ t of the Lee–Carter model using a Random Walk with Drift equation: kt=kt−1+d+ηtwith ηx,t∼N(0, 1)aandaE(ηs,ηt)=0, (A7) where the drift dis estimated by the formula: ˆ d=(k∗ 2−k∗ 1)+(k∗ 3−k∗ 2)+... +(k∗ T−k∗ T−1) tn−t1 =(k∗ T−k∗ 1) tn−t1 , with k∗ T and k∗ 1 , respectively, given by the first and the last elements of the vector k∗ t=[k∗ 1 , ..., k∗ T] . The drift is simply the arithmetic mean of the differenced series of estimated parameters. After having solved Equation (A7) of the RWD model, we describe the projection of the parameter ktat time T+Δtas follows: ˆ kT+Δt=k∗ T+(Δt)ˆ d+√Δtηt. At this point, it was possible to get the equation for the projection of the central rates of mortality as follows: ˆ mx,T+Δt=ea∗ x+b∗ xˆ kT+Δt. Finally, we transformed the central mortality rates into probabilities by adopting the Reed and Merrell (1939) method. The relation is expressed by the equation: nqx,t=1−e−n(mx,t)−n30.008(mx,t)2. Appendix B. Cairns–Blake–Dowd Estimation and Projection According to the original formulation of the model proposed by Cairns et al. (2006), Equation (2) is the result of the logit transformation of the following model equation: qx,t=ek(1) t+k(2) t(x−¯ x) 1+ek(1) t+k(2) t(x−¯ x). (A8) Fitted values for the stochastic processes k(1) t and k(2) t were obtained using least squares applied to the Equation (A8). The fitted model is then used to estimate the median and the 95% prediction interval. The parameters vector  kt=k(1) t , k(2) t has been projected by considering the following equation of a two-dimensional random walk with drift:  kt+1= kt+μ+CN(t+1), (A9) where •μ is a constant 2 × 1 vector of drifts, computed as the arithmetic mean of the differenced series of estimated parameters; •C is a constant 2 × 2 upper triangular matrix, derived by the unique Cholesky decomposition of the variance–covariance matrix V=CCof the parameters vector kt+1; and •N(t+1)is a two-dimensional standard normal random variable. 16 We used MATLAB (R2010b, The MathWorks, Inc., Natick, Massachusetts 01760 USA) for estimation and forecast. 43 Books MDPI Risks 2017,5,34 The adopted forecast method treats the estimated parameters as if they were the true parameter values (parameters certainty). In particular, the presented projections were computed 17 considering parameter certainty based on 5000 simulation trials. References ANIA. 2014. Le Basi Demografiche Per Rendite Vitalizie a 1900–2020 e a62. Technical report. Roma: Associazione Nazionale fra le Imprese Assicuratrici (ANIA). Avraam, Demetris, Joao Pedro de Magalhaes, and Bakhtier Vasiev. 2013. A mathematical model of mortality dynamics across the lifespan combining heterogeneity and stochastic effects. Experimental Gerontology 48: 801–11. Booth, Heather, Rob Hyndman, Leonie Tickle, and Piet De Jong. 2006. Lee-carter mortality forecasting: A multi-country comparison of variants and extensions. Demographic Research 15: 289–310. Cairns, Andrew J. G., David Blake, and Kevin Dowd. 2006. A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance 73: 687–718. Cairns, Andrew J. G., David Blake, Kevin Dowd, Guy D. Coughlan, David Epstein, Alen Ong, and Igor Balevich. 2009. A quantitative comparison of stochastic mortality models using data from england and wales and the united states. North American Actuarial Journal 13: 1–35. Chan, Wai-Sum, Johnny Siu-Hang Li, and Jackie Li. 2014. The cbd mortality indexes: Modeling and applications. North American Actuarial Journal 18: 38–58. Dowd, Kevin, Andrew J. G. Cairns, David Blake, Guy D Coughlan, David Epstein, and Marwa Khalaf-Allah. 2010a. Backtesting stochastic mortality models: An ex post evaluation of multiperiod-ahead density forecasts. North American Actuarial Journal 14: 281–98. Draper, Norman R., and Harry Smith. 2014. Applied Regression Analysis. New York: John Wiley & Sons. Istat. 2001. Tavole di mortalità della popolazione italiana per provincia e regione di residenza. anno 1998. Roma: Servizio Popolazione Istruzione e Cultura. Istat. 2008. Previsioni demografiche. 1 gennaio 2007-1 gennaio 2051. Nota informativa, Popolazione. Technical report. Roma: Istat. Istat. 2016. Indicatori Demografici: Stime Per L’anno 2015. Technical report. Roma: Istat. Keyfitz, Nathan, and Hal Caswell. 2005. Applied Mathematical Demography. New York: Springer, vol. 47. Kullback, Solomon, and Richard A. Leibler. 1951. On information and sufficiency. The Annals of Mathematical Statistics 22: 79–86. Lee, Ronald. 2000. The lee-carter method for forecasting mortality, with various extensions and applications. North American Actuarial Journal 4: 80–91. Lee, Ronald, and Timothy Miller. 2001. Evaluating the performance of the lee-carter method for forecasting mortality. Demography 38: 537–49. Lee, Ronald D., and Lawrence R Carter. 1992. Modeling and forecasting US mortality. Journal of the American Statistical Association 87: 659–71. Li, Jackie. 2010. Projections of new zealand mortality using the lee-carter model and its augmented common factor extension. New Zealand Population Review 36: 27–53. Li, Nan, and Ronald Lee. 2005. Coherent mortality forecasts for a group of populations: An extension of the lee-carter method. Demography 42: 575–94. Li, Ting, and James Anderson. 2013. Shaping human mortality patterns through intrinsic and extrinsic vitality processes. Demographic Research 28: 341–72. Li, Ting, and James J Anderson. 2009. The vitality model: A way to understand population survival and demographic heterogeneity. Theoretical Population Biology 76: 118–31. Loisel, Stéphane, and Daniel Serant. 2007. In the core of longevity risk: Hidden dependence in stochastic mortality models and cut-offs in prices of longevity swaps. Cahier de Recherche de l’ISFA WP2044. Working Paper. Available online: http://isfaserveur.univ-lyon1.fr/ stephane.loisel/Loisel-Serant-ISFA-WP2044.pdf 17 We used MATLAB for estimation and forecast. 44 Books MDPI Risks 2017,5,34 Maccheroni, Carlo. 2014. Diverging tendencies by age in sex differentials in mortality in italy. South East Journal of Political Science (SEEJPS) 2: 42–58. Maccheroni, Carlo. 2016. The Actuarial Aging of Italian Veterans of World War I Born 1889-1906 and a Comparison to the Cohorts Born During the Years Immediately Following. Technical report. Torino: Department of Economics and Statistics (WP36), University of Torino. Mavros, George, Andrew J.G. Cairns, Torsten Kleinow, and George Streftaris. 2014. A Parsimonious Approach to Stochastic Mortality Modelling with Dependent Residuals. Technical report. Edinburgh: Citeseer. Pitacco, Ermanno, Michel Denuit, Steven Haberman, and Annamaria Olivieri. 2009. Modelling Longevity Dynamics for Pensions and Annuity Business. London: Oxford University Press. Reed, Lowell Jacob, and Margaret Merrell. 1939. A short method for constructing an abridged life table. American Journal of Epidemiology 30: 33–62. Thatcher, A. Roger, Väinö Kannisto, and James W. Vaupel. 1998. The force of mortality at ages 80 to 120. Odense Monographs on Population Aging 5: 104-20. Vaupel, James W. 2010. Biodemography of human ageing. Nature 464: 536–42. Whitehouse, Edward. 2007. Life-expectancy risk and pensions: Who bears the burden? OECD Social, Employment, and Migration Working Papers.(60): 46 pp. Working Paper. Retrieved from: http://www.oecd.org/social/soc/39469901.pdf c 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 45 Books MDPI Article Minimum Protection in DC Funding Pension Plans and Margrabe Options Pierre Devolder * and Sébastien de Valeriola * Institute of Statistic, Biostatistic and Actuarial Science (ISBA), Université catholique de Louvain (UCL), Voie du Roman Pays, 20, 1348 Louvain-la-Neuve, Belgium *Correspondence: [email protected] (P.D.); [email protected] (S.d.V.) Academic Editor: Luca Regis Received: 14 November 2016; Accepted: 10 January 2017; Published: 18 January 2017 Abstract: The regulation on the Belgian occupational pension schemes has been recently changed. The new law allows for employers to choose between two different types of guarantees to offer to their affiliates. In this paper, we address the question arising naturally: which of the two guarantees is the best one? In order to answer that question, we set up a stochastic model and use financial pricing tools to compare the methods. More specifically, we link the pension liabilities to a portfolio of financial assets and compute the price of exchange options through the Margrabe formula. Keywords: pensions; Defined Contributions; guaranteed rate; option theory; Margrabe formula 1. Introduction In most European countries, the sustainability of the pension system has become a major concern. The past few decades have therefore seen an evolution of the pension plan designs (as remarked e.g., by the European Commission, [ 1 ]). On one hand, more and more plans are financed through funding by pension funds or group/life insurance contracts (as opposed to plans financed through pay-as-you-go mechanisms). On the other hand, Defined Contribution (DC) schemes have tended to overtake Defined Benefit (DB) schemes. This new pension plan model (funding and DC) addresses—at least partially—the sustainability problem. However, it generates another issue, which is related to the adequacy of the benefits: in such a plan, the affiliates bear all of the financial risk. To protect the affiliates and make these plans politically acceptable, many European countries have enacted legislations. Among various types of guarantees (presented e.g., in [ 2 ]), some of them impose on pension sponsors ensuring a minimum financial performance on plan contributions. The Swiss system is one of them, and an early and interesting example. Dating back to 1985, it imposes a minimal guaranteed rate that can be revised every other year by the Federal Council, which takes into account financial markets to do so. Accordingly, the rate was progressively lowered from 4% (its value between 1985 and 2002) to its current level of 1.25%. In 2003, Belgian authorities implemented the Law on Complementary Pension (LCP), obliging sponsors to guarantee a minimum rate of 3.25% on the employer’s contributions and of 3.75% on employees’ contributions. Since then, these values have proven to be unsustainable in the context of a decreasing interest rates market. A reform act 1 has therefore been passed by the government (applicable since the beginning of 2016), that transforms the fixed minimum guaranteed rate into a variable rate, linked to the yield rate 1 “Loi du 18 décembre 2015 visant à garantir la pérennité et le caractère social des pensions complémentaires et visant à renforcer le caractère complémentaire par rapport aux pensions de retraite”, published in the Moniteur Belge on 24 December 2015. Risks 2017,5, 5 46 www.mdpi.com/journal/risks Books MDPI Risks 2017,5,5 of the 10 year Belgian governmental bond. As it is fluctuating, one needs to indicate how the guarantee is applied to the previously paid contributions. Two different computation approaches are described in the new legal text, designated as the horizontal and vertical methods. In the case of newly created plans, sponsors are allowed to choose between them. On the contrary, in the case of already existing plans, the method that has to be used depends on the funding vehicle of the plan. The existence of two computation methods raises the natural question of their comparison. In this paper, we address it by setting a stochastic framework for the interest rates and considering the evolution of the pension liability over the years. In order to compare the amounts obtained from one initial contribution using the horizontal and vertical methods, we embrace two distinct approaches. On one hand, we simply compute the expectations of the corresponding stochastic processes and determine which method leads to the best results. This approach can be associated with the affiliates’ point of view, as it estimates how much the affiliates will earn from the initial contribution. On the other hand, we take the pension sponsor’s point of view and compute the price of both pension guarantees. The topic of pension guarantee pricing has been much studied in the literature (starting with the seminal papers of Pennacchi and Fischer [ 3 , 4 ] and going on after that with [5,6] , among many others). More specifically, we follow an Asset and Liability Management-driven methodology (as done for example in [ 7 – 9 ]): we suppose that the sponsor has an asset investment portfolio in front of its pension liabilities, and we try to determine which method is preferable for him, taking into account its investment preferences. In order to do so, we consider options allowing for exchanging the asset portfolio for the horizontal (respectively vertical) liability and compute their prices: the best method is the one associated to the cheapest option. This methodology has already been used, for example in [ 10 ]. The pricing of these exchange options is achieved using the Margrabe formula (see [11]). More precisely, we use a generalization of this formula to a stochastic interest rates framework obtained (in the general case, where assets and rates are not independent) by Bernard and Cui [12] (see also [13]). The paper is organized as follows: in Section 2, we present the details of the reform act, such as the definition of the guaranteed rate, and describe the horizontal and vertical computation methods. Then, we set up a stochastic framework and perform the first comparison of the two methods in Section 3. Section 4 is devoted to our second comparison approach, using option prices and the Margrabe formula. Finally, we comment on the results and conclude in Section 5. 2. Detailed Design of the Reform Act The new LCP links the guaranteed rate to a 24-month moving average of the 10-year OLO (i.e., Belgian governmental bond) yield rate . A cap and a floor is applied to this average, and the result is multiplied by a constant: rguaranteed(t0)=max 1.75%; min π1 24 23 months ∑ t=0 months rOLO 10(t0−t); 3.75%. The value of π is defined as 65%, but the law states that, under some circumstances (which are linked to the evolution of the maximum rate of long-term insurance contracts), the National Bank of Belgium can decide to raise this percentage to 75% in 2018 and to 85% in 2020. Figure 1 shows the evolution of these rates between January 1991 and September 2015. 47 Books MDPI Risks 2017,5,5 Figure 1. Different rates appearing in the definition of the reference rate. Let us now present the two computation methods, which are summarized in Figure 2. The vertical method is used when the vehicle of an existing pension scheme is a pension fund (or a pure unit linked product sold by an insurance company), or when the sponsor of a new pension scheme (i.e., created before 1 January 2015) chooses so. In this case, the guaranteed rate of year t is applied to the whole amount of cotisations already paid by the affiliated up to year t. The vertical liability is then similar to the one produced by a savings product where the contributions are deposited, whose interest rate is the guaranteed rate, susceptible to be different from year to year. ݎ ଵ ݎ ଶ ݎ் ݎ ଶ ݎ ଵ ݎ் Horizontal method Vertical method … … t=1 t=2 t=T t=1 t=2 t=T … … t=1 t=2 t=T … Instants of payment Instants of payment t=1 t=2 t=T … Instants of valuation Instants of valuation Figure 2. The two different methods of guaranteed rate application. The horizontal method is used when the financing vehicle is a life insurance contract with interest rate guarantee or when the sponsor of a new pension scheme chooses so. Remark that, in this case, the guarantee level offered by the insurer can be different from the guarantee level that the sponsor must legally ensure to its affiliates. The guaranteed rate of year t is then applied to the new cotisations paid in year t , the cotisations already paid in years t− 1 being accounted for with the guaranteed 48 Books MDPI Risks 2017,5,5 rate of year t− 1, etc. The resulting horizontal liability is similar to the one produced by a standard life insurance product. Let us first consider a very simple example to illustrate the two methods: assume that an affiliate successively pays two contributions of 1 Euro in 2016 and 2017, and retires in 2018. In Table 1, we consider two scenarios for the evolution of the guaranteed interest rate. Table 1. Simple example illustrating how the horizontal and vertical methods are applied to the plan contributions. Scenario 1 Scenario 2 2016 guaranteed rate 2.5% 2.5% 2017 guaranteed rate 3.5% 2% 2018 liability (horizontal) 1.0252+1.035 =2.086 1.0252+1.02 =2.071 2018 liability (vertical) 1.025 ·1.035 +1.035 =2.096 1.025 ·1.02 +1.02 =2.066 Highest liability vertical horizontal Although very simple, this small toy example provides some intuition about the relation between the two computation methods and the market evolution. If the interest rates are increasing, the vertical method leads to a larger amount. If, on the contrary, the interest rates are decreasing, the horizontal liability is larger. 3. Direct Comparison In this section, we set up a stochastic framework for the interest rates and use it to compare the level of the two guarantees in absolute terms. 3.1. Interest Rate Framework We assume that the short rate rt is modelled with a Vasicek one-factor model, i.e., that in the risk-neutral world drt=k(θ−rt)dt+σdWr t, (1) where k> 0, σ> 0 and θ are real constants and Wr is a standard Brownian motion. It is well-known (see e.g., [14]) that the solution of this stochastic differential equation is rt=r0e−kt +θ(1−e−kt)+σt 0e−k(t−s)dWr s, (2) and that the K-years zero-coupon bond yield is given by rK t=C(K)+D(K)rt K=A(K)+B(K)rt, (3) where A(K)=C(K) K, B(K)=D(K) K, C(K)=σ2 2k2−θ(D(K)−K)+σ2 4kD2(K), D(K)=1 k1−e−kK. (4) 49 Books MDPI Risks 2017,5,5 4.3. Comparison of Margrabe Option Prices We are now able to compare the two liabilities using the ALM methodology explained supra. On one hand, the price of the option giving the right at time T to exchange the asset portfolio for the vertical liability is equal to pv 0=exp ˜ mv(T)+1 2˜ s2 v(T)Φ˜ mv(T)+1 2˜ s2 v(T)+1 2s2 v(T)+s2 a(T)−2cv,a(T) s2 v(T)+s2 a(T)−2cv,a(T) −Φ˜ mv(T)+1 2˜ s2 v(T)−1 2s2 v(T)+s2 a(T)−2cv,a(T) s2 v(T)+s2 a(T)−2cv,a(T). On the other hand, the price of the option giving the right at time T to exchange the asset portfolio for the horizontal liability, which reduces to a standard put option on the asset portfolio with the strike being equal to the final value of the horizontal liability, is equal to ph 0=−Φ⎛ ⎝log Lh TP(0, T)−1 2s2 a(T) sa(T)⎞ ⎠+Lh TP(0, T)Φ⎛ ⎝log Lh TP(0, T)+1 2s2 a(T) sa(T)⎞ ⎠. We can now compute these option prices for a set of various investment portfolios, using the parameters of Tables 2 and 3. Table 4 gathers the results of the comparison. Among the example portfolios that we have considered, the “Typical insurer portfolio” is meant to mimic the investment habits of Belgian insurers. The prices of the two options are given in this case in Table 5. Table 3. Supplementary parameters used in the computations of the ALM comparison. θKημ 1.34% 10 25% 5% Table 4. Results of the ALM comparison for some example portfolios. The cheapest liability is the horizontal one when H is displayed, and the vertical one when V is displayed. Composition Cheapest Liability Description xyzρ=−1ρ=−0.5 ρ=0ρ=0.5 ρ=1 Only stocks 100% 0% 0% H H V V V Only bonds 0% 100% 0% H H H H H Only cash 0% 0% 100% V V V V V Stocks and bonds 50% 50% 0% H H V V V Stocks and cash 50% 0% 50% H H V V V Bonds and cash 0% 50% 50% V V V V V Equal repartition 33% 33% 33% H H V V V Typical insurer portfolio 10% 80% 10% H H V V V Table 5. Prices of the options giving the right to exchange the “Typical insurer” asset portfolio (as defined in Table 4) for the horizontal (resp. vertical) liability. ρ=−1ρ=−0.5 ρ=0ρ=0.5 ρ=1 ph 00.0044 0.0077 0.0112 0.0148 0.0183 pv 00.0108 0.0099 0.0090 0.0082 0.0073 ph 0−pv 0−0.0065 −0.0023 0.0022 0.0066 0.0111 Let us first consider portfolios without any stock. In the case of the “only bonds” one, the horizontal liability is cheaper. This result is not really surprising, as this method looks like the way 56 Books MDPI Risks 2017,5,5 bonds produce money. On the contrary, in the case of the “only cash” portfolio, the vertical option price is smaller. Again, this conclusion is not unexpected because this method is similar to a savings product. When stocks are included in the portfolio, the hierarchy between the methods depends on the correlation between stocks and rates. If ρ> 0, the stock and the cash are similar in some sense. The vertical method is thus the cheapest one, as in the case of portfolios without stock. If, on the contrary, ρ< 0, the asset portfolio is very different from cash, and the horizontal method is preferred. The funding vehicle of the pension plan in question is therefore an important feature to consider when comparing the two computation methods. The nature of this institution has indeed a strong influence on its investment habits: insurers tend to invest a lot in bonds, while pension funds often prefer stocks. 5. Conclusions The two comparison methodologies give different points of view of the two computation methods. The first one confirms what was trivially suggested by the intuition: the horizontal liability is larger than the vertical liability in the case of a decreasing rates market, and vice versa. The second methodology yields more interesting conclusions, as it connects the hierarchy between the two methods to the investment profile of the institution granting the guarantee. We have seen that two methods represent two different philosophies: the horizontal one is closer to the insurers’ asset management habits, while the vertical one is closer to pension funds’ investment preferences. For this reason, the reform act will possibly have consequences regarding the investment strategies of the Belgian pensions plans’ funding vehicles. We have chosen to consider a valuation framework in order to be consistent with the IAS philosophy, and thus with the funding vehicles’ interests. It should, however, be noted that other methodologies would make sense. For example, it is possible to compute, instead of exchange option prices, risk measures (such as VaR or TVaR) applied to the gap between the asset portfolio value and the liability value. Political circumstances have led Belgian authorities to let the choice for new pensions schemes, but, as the results generated by the two methods are different, a whole set of legal questions arise (out of scope here). Among them, we can mention discrimination problems (possible arbitrage by the sponsor, see e.g., [15–17]). Our work leaves some questions open for further research, mainly in two directions. On one hand, we could consider the exact definition of the guaranteed rate, i.e., take into account the cap and the floor as defined in the legal text. In order to compute options prices similar to the ones that we have obtained with closed formulas, it would then be necessary to use numerical methods. On the other hand, we could consider more complex models for the interest rate and the stock price, including a more complete dependence structure between the two risk factors. Acknowledgments: We would like to thank the anonymous referees for their useful remarks. This work has been done in the context of the Chaire d’excellence sur les pensions of the Université catholique de Louvain. Author Contributions: Both authors have contributed equally to this paper. Conflicts of Interest: The authors declare no conflict of interest. 57 Books MDPI Risks 2017,5,5 Appendix A. Proof of Proposition 1 Proof. First, we rewrite the sum of R’s consecutive values: 1 π T−1 ∑ t=0 Rt= T−1 ∑ t=0 r10 t−2+r10 t−1+r10 t 3 = T−1 ∑ t=0 A+Brt−2+A+Brt−1+A+Brt 3 =AT +B1 3r−2+2 3r−1+r0+ T−3 ∑ t=1 rt+2 3rT−2+1 3rT−1 =α+B T−1 ∑ t=1 λ(t)rt, where we denoted the sum of the deterministic terms by α and introduced the notation λ(t) defined in Formula (7). Using the expression for rtgiven by Equation (2), 1 π T−1 ∑ t=0 Rt=α+B(r0−θ) T−1 ∑ t=1 λ(t)e−kt +Bθ T−1 ∑ t=1 λ(t)+Bσ T−1 ∑ t=1 λ(t)e−kt t 0eksdWr s =α+B(r0−θ)Λ(1)+Bθ(T−2)+Bσ T−1 ∑ t=1 Λ(t)t t−1eksdWr s, where we have used the notation Λ(t)defined in Formula (8), as T−1 ∑ t=1 λ(t)e−kt t 0# =λ(1)e−k1 0#+λ(2)e−k22 0#+···+λ(T−1)e−k(T−1)T−1 0# =λ(1)e−k+λ(2)e−k2+···+λ(T−1)e−k(T−1)1 0# +λ(2)e−k2+···+λ(T−1)e−k(T−1)2 1# +... +λ(T−1)e−k(T−1)T−1 T−2#, where # denotes any integrand. These notations seem cumbersome, but allow handling the sum of independent stochastic integrals, which is much easier for the following computations than dependent stochastic integrals. The mv(T) parameter is straightforwardly obtained from the expression obtained for the sum of R’s values. For the volatility, we compute s2 v(T)=E⎡ ⎣πBσ T−1 ∑ t=1 Λ(t)t t−1eksdWr s2⎤ ⎦ =π2B2σ2T−1 ∑ t=1 Λ2(t)t t−1e2ksds =π2B2σ2 2k T−1 ∑ t=1 Λ2(t)e2kt −e2k(t−1), which completes the proof of Proposition 1. 58 Books MDPI Risks 2017,5,5 References 1. European Commission. An Agenda for Adequate, Safe and Sustainable Pensions; White Paper COM(2012)55; European Commission: Brussels, Belgium; Luxembourg, 2012; Volume 55. 2. Antolín, P.; Payet, S.; Whitehouse, E.; Yermo, J. The Role of Guarantees in Defined Contribution Pensions; OECD Working Papers on Finance, Insurance and Private Pensions; OECD Publishing: Paris, France, 2011; Volume 11. 3. Pennacchi, G. The Value of Guarantees on Pension Fund Returns. J. Risk Insur. 1999,66, 219–237. 4. Fischer, K. Pricing Pension Fund Guarantees: A Discrete Martingale Approach. Can. J. Adm. Sci. 1999 ,16, 256–266. 5. Lachance, M.E.; Mitchell, O.; Smetters, K. Guaranteeing Defined Contribution Pensions: The Option to Buy Back a Defined Benefit Promise. J. Risk Insur. 2003,70, 1–16. 6. Yang, S.; Yueh, M.L.; Tang, C.H. Valuation of the interest rate guarantee embedded in defined contribution pension plans. Insur.: Math. Econ. 2008,42, 920–934. 7. Deelstra, G.; Grasselli, M.; Koehl, P. Optimal investment strategies in the presence of a minimum guarantee. Insur.: Math. Econ. 2003,33, 189–207. 8. Consiglio, A.; Saunders, D.; Zenios, S. Asset and liability management for insurance products with minimum guarantees: The UK case. J. Bank. Financ. 2006,30, 645–667. 9. Consiglio, A.; Cocco, F.; Zenios, S. Asset and liability modelling for participating policies with guarantees. Eur. J. Oper. Res. 2008,186, 380–404. 10. Consiglio, A.; Tumminello, M.; Zenios, S. Designing and pricing guarantee options in defined contribution pension plans. Insur.: Math. Econ. 2015,65, 267–279. 11. Margrabe, W. The Value of an Option to Exchange One Asset for Another. J. Financ. 1978,33, 177–186. 12. Bernard, C.; Cui, Z. A Note on Exchange Options under Stochastic Interest Rates. Available online: https://ssrn.com/abstract=1626020 (accessed on 16 June 2010). 13. Huerlimann, W. Option pricing in the multidimensional Black-Scholes market with Vasicek interest rates. Math. Financ. Lett. 2013,2, 1–18. 14. Brigo, D.; Mercurio, F. Interest Rate Models—Theory and Practice. With Smile, Inflation and Credit; Springer: Berlin, Germany, 2006. 15. Flohimont, V. Gelijkheid in de Pensioenregelingen voor Ambtenaren, Werknemers en Zelfstandigen; Die Keure: Bruges, Belgium, 2012. 16. Flohimont, V. Comparaison et comparabilité dans la jurisprudence de la Cour constitutionnelle: Rigueur ou jeu de hasard? Revue belge de droit constitutionnel 2008,3, 217–235. 17. Ellis, E.; Watson, P. The principle of equality as applied to pensions. In EU Anti-Discrimination Law; Oxford University Press: Oxford, UK, 2012; pp. 195–208. c 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 59 Books MDPI risks Article A Discussion of a Risk-Sharing Pension Plan Catherine Donnelly Department of Actuarial Mathematics and Statistics, and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK; [email protected]; Tel.: +44-131-451-3251 Academic Editor: Luca Regis Received: 2 October 2016; Accepted: 27 January 2017; Published: 14 February 2017 Abstract: I show that risk-sharing pension plans can reduce some of the shortcomings of defined benefit and defined contributions plans. The risk-sharing pension plan presented aims to improve the stability of benefits paid to generations of members, while allowing them to enjoy the expected advantages of a risky investment strategy. The plan does this by adjusting the investment strategy and benefits in response to a changing funding level, motivated by the with-profits contract proposed by Goecke (2013). He suggests a mean-reverting log reserve (or funding) ratio, where mean reversion occurs through adjustments to the investment strategy and declared bonuses. To measure the robustness of the plan to human factors, I introduce a measurement of disappointment, where disappointment is high when there are many consecutive years over which benefit payments are declining. Another measure introduced is devastation, where devastation occurs when benefit payments are zero. The motivation is that members of a pension plan who are easily disappointed or likely to get no benefit, are more likely to exit the plan. I find that the risk-sharing plan offers more disappointment than a defined contribution plan, but it eliminates the devastation possible in a plan that tries to accumulate contributions at a steadily increasing rate. The proposed risk-sharing plan can give a narrower range of benefits than in a defined contribution plan. Thus it can offer a stable benefit to members without the risk of running out of money. Keywords: defined contribution; defined benefit; risk-sharing; pension 1. Introduction Risk-sharing pension plans offer a middle ground between the extremes of defined benefit (DB) and defined contribution (DC) pension plans. They don’t promise a certain level of pension at retirement like DB plans, but neither do they promise something that they may not deliver. They don’t give members control over their own investment strategy like DC plans, but neither do they require members to be financial experts. Instead, they try to pay a stable pension to the members. They try to give members an idea of what their pension may be in retirement, but an aim or target amount rather than a guarantee. And they try to actively maintain the sustainability of the plan to adverse investment returns, by adjusting one or more of the investment strategy, benefits and contributions to be paid. I discuss a model risk-sharing pension plan that adjusts only the investment strategy and retirement benefits. It is closer to a DC plan than a DB plan. In the risk-sharing plan, the retirement benefits are the accumulated value of the contributions. The plan reduces retirement benefit volatility to the plan members, paying a more stable benefit to its members than under a DC plan. It does not require additional contributions to make good a funding deficit. Instead, the benefits paid and investment strategy adjust so that the funding level is self-correcting. Adjusting the investment strategy means adjusting the level of investment risk taken by the plan. Moreover, the structure of the plan is highly flexible. By changing the parameters of the plan, the amount of benefit volatility experienced by different generations of members can be adjusted. Risks 2017,5, 12 60 www.mdpi.com/journal/risks Books MDPI Risks 2017,5,12 Although DB plans are currently in decline, they have some very appealing features for their members. A retirement “defined benefit” income or lump sum is promised to the plan members without regard to the achieved investment return. This is very attractive to the membership since they are insulated from the ups and downs of investment returns. However, the adverse consequence is that a DB plan can require large cash injections from its sponsoring employer. If the sponsor does not make the cash injections, then the plan may have to wind up and secure whatever fraction of the promised benefits it can for its members. In recent years, poor equity returns and low interest rates have resulted in many DB plans being underfunded [ 1 ]. As a result, many employers are closing down their DB plans and replacing them by DC plans [2]. In contrast, DC plans are in the ascendancy in countries such as the UK and USA. They have some very appealing features for the employer who sets them up on behalf of their employees. DC plans require no unexpected contributions to make good a funding deficit. This means that the employer has no uncertainty about their contribution rate to a DC plan, unlike in a DB plan. Typically, members and the employer pay in contributions which are invested in the financial market. The accumulated value of their contributions, which varies directly with the investment return achieved, is the value of the benefit paid to a member at their retirement date. There are no guarantees as to the amount of the retirement benefit, unless a financial guarantee is purchased. Thus members are exposed to the full volatility of their investment choices. If returns are much lower than expected, members may question the value of saving for retirement and not save enough for their retirement. Moreover, in DC plans the member must choose the investment strategy, although many people are financially illiterate [ 3 ] and do not consult a financial advisor ([Figure 325], [4]). They are other types of pension plans, such as collective DC (CDC) plans. They are a type of risk-sharing plan, which can adjust the benefits paid to members in response to achieved investment returns. If annuities are paid as a benefit from a CDC plan, then the benefits can also be adjusted in response to mortality experience. The Dutch hybrid pension plans [ 5 ] can be considered as the latter type of CDC plan. Plans with inter-generational risk-sharing, such as CDC plans and other risk-sharing plans, have been shown to be welfare-improving, compared to the individually optimal lifecycle strategy [ 6 , 7 ]. This is when measuring welfare using an expected utility-of-consumption framework, and maximizing utility across all generations. Inter-generational risk-sharing plans have higher and smoother consumption patterns, averaged across generations, than in a pure DC plan. More generally, risk-sharing in pension plans is discussed in Pugh and Yermo [ 8 ] and Blommestein et al. [ 9 ], both in terms of types of risk and who bears each risk. For example, in both DB and cash-balance plans the sponsoring employers bear the investment risk, whereas the members bear this risk in DC plans and in the plan analyzed here. Another example are conditional indexation plans: they share investment risk between the sponsor and the members, and the members’ basic benefit is guaranteed by the sponsor but benefit increases are not. Closely related to the idea of risk-sharing plans are participating policies, which are contracts issued by insurance companies. Participating policies typically increase contributions annually at the risk-free rate, and thus provide a minimum guarantee of the benefit paid at retirement. Briys and De Varenne [ 10 ] study a model participating policy, focusing on the maturity value of the policy. Other studies including the minimum guarantee are Barbarin and Devolder [ 11 ], Graf et al. [ 12 ], Grosen and Jørgensen [ 13 ], Hieber et al. [ 14 ]. Guillén et al. [ 15 ] analyze a policy that was launched by a Danish insurance company. The policy smooths investment returns over the contract duration, but does not guarantee a minimum investment return. In these new contracts, there is no direct or indirect inter-generational investment risk-sharing. Baumann and Müller [ 16 ] propose a pension plan in which benefits are increased by the risk-free rate plus or minus a proportion of the funding level above or below a desired funding level. They find the static investment strategy that maximizes the discounted expected utility of employees averaged over all generations. They find that their proposed plan increases the risk tolerance and the expected utility of the retirement benefits. Analyses of different 61 Books MDPI Risks 2017,5,12 surplus appropriation schemes in participating policies are provided by Bohnert and Gatzert [ 17 ] and Zemp [18]. The risk-sharing pension plan that I study changes explicitly both the accumulation of contributions and the investment strategy in response to changes in a notional funding level. Thus it is not like a Dutch-style plan, nor a typically considered CDC plan in the literature, in which there is no explicit adjustment to the investment strategy. It is an adaptation of the scheme proposed by Goecke [ 19 ], which he proposed in the context of a with-profits contract. Unlike many with-profit contracts, there are no investment guarantees in Goecke’s scheme, and hence there is no need for an insurance company to financially underwrite the proposed contract. Thus Goecke’s scheme could be considered as a type of CDC plan, since the policyholders collectively bear all of the risk. However, Goecke [ 19 ] considers only a single generation of members, all of whom enter the contract at the same time with the same amount of money, and obtain the terminal value of the contract at the same time. In this paper, I analyze how the plan performs over time when there are regular withdrawals from its assets, which is not generally a consideration for with-profits contracts and was thus, unsurprisingly, not considered in Goecke [19]. The risk-sharing plan promises to increase the accumulated value of contributions each year in line with the expected return on assets, adjusted for the plan being over- or under-funded relative to a target funding level. Although the plan uses a notional funding level in its operation, benefits are not guaranteed. While the increase in the accumulation of contributions varies each year, the idea is that the annual variation in the accumulation increase is kept small. The notional funding level is simply a mechanism to determine the investment strategy and accumulation on the contributions at a selected time. The target funding level is fixed and its value helps to determine the spread of investment returns across generations. The plan invests its money in line with a long-term strategic investment strategy, which would be chosen by the trustees or plan managers. However, if the plan is over-funded, then the investment risk of its strategy is increased (and correspondingly its expected investment return is increased). The opposite happens if the plan is under-funded: the investment risk is decreased. The two adjustments, to the accumulated contributions and the investment strategy, act to encourage the funding level back towards the target funding level. I examine the robustness of the proposed risk-sharing pension plan from the viewpoint of individual members. I believe this is more appropriate for private pension plans than a social-planning viewpoint. Plans can fail because of what happens to one generation, regardless if they are expected to be a success averaged across all generations. I focus on the actual retirement benefit payments made to members, as this is the ultimate purpose of the plan, rather than, for example, maximizing the expected utility derived from the payments. To do this, I examine the stability of the retirement benefit payments between generations of members and look at their distribution. Furthermore, in risk-sharing plans, it is inevitable that benefits have to be cut for some generations. However, the benefit cuts should not be too large between generations. A plan structure which means there is a high chance of members getting nothing is unlikely to succeed long-term in the marketplace. Similarly if the benefits decline year-on-year. For this reason, I calculate the probability of the plan’s assets being exhausted before all benefits have been paid, which I call devastation, and the probabilities of benefits declining year-on-year between generations, which I call disappointment. I investigate (i) how changing the parameterization of the risk-sharing plan changes the range of benefits across generations; (ii) the retirement benefit stability across generations of members; (iii) disappointment, namely the frequency of runs of declining accumulation of contributions; and (iv) devastation, namely the chance of the plan exhausting its assets before all members have received their benefit. I find that a suitably parameterized risk-sharing plan can offer a stable retirement benefit across generations. With a suitable choice of plan-specific parameters, it has a zero chance of running out 62 Books MDPI Risks 2017,5,12 of money before all members have retired, for the chosen financial market model. The downside is that members must expect that retirement benefits may decrease; there is no guaranteed minimum retirement benefit. While this is also seen in the DC plan, and many people are in DC plans, the risk-sharing plan members may compare themselves with other plan members and feel that they should get more each year. 2. Risk-Sharing Plan The risk-sharing plan has the aim of paying a reasonable retirement benefit to its members, while maintaining the financial security of the non-retired members. Financial security is measured through the funding level of the plan, namely the ratio of the plan’s assets to its liabilities. The most attractive aspect is that each plan can be adjusted to the needs of the beneficiaries and sponsors, if any. A target funding level is fixed and the plan’s managers seek to bring the funding level back to the chosen target, by adjusting the accumulated value of the members’ contributions and the investment strategy. The motivation is to maintain a cushion of assets as financial security for the non-retired plan members. This is done through applying the scheme of Goecke [ 19 ], and adapting it for use as an inter-generational pension plan. Rather than pension plans that pay benefits, Goecke [ 19 ] studies with-profit contracts and introduces a mechanism for the smoothing of capital market returns in them. This involves changing the bonus declaration and the investment strategy in response to changes in the reserve ratio. A highly attractive property is that the log reserve ratio is mean-reverting to a fixed target log reserve ratio. As Goecke [ 19 ] studies with-profits contracts, which are held to the same maturity date, there is naturally no consideration of payments out. However, this must be allowed for in the risk-sharing plan as a pension plan is constantly paying out benefits to its retired members. Furthermore, I do not allow short-selling of any asset, which is a realistic assumption for a pension plan since the plan manager must act as a “prudent investor” [20]. 2.1. Investment Strategy Assume that time is measured in years. In the financial market, the plan can invest in a risky stock and a risk-free bond. The annual effective return over the time period [n−1, n) on the risky stock is represented by the random variable Rn>− 1, for n= 1, 2, ... . For the risk-free bond, its annual effective return over the time period [n−1, n) is the constant r>− 1. The plan’s investment strategy at time nis represented by the proportion πn of assets invested in the risky stock, for each n=0, 1, 2, . . .. The proportion is determined through several elements. One of the elements is a long-term strategic investment of sLT in the risky stock. This can be decided by reference to the risk attitude of a typical member or, more realistically, the risk attitude of the trustees of the plan, a small group of people who run the plan in the interests of the members. An approach to deciding an investment strategy for a long-term investor like a pension fund is given in de Jong [21]. Another element of the strategy is positive or negative at time n, according to the funding level Fn of the plan. The funding level is defined as the value of assets divided by the value of the liabilities, and it is formally defined in Section 2.3. If the plan is over-funded relative to a fixed target funding level F≥ 1, then it invests more in the risky stock. The idea is that the plan can risk losing money by following a more risky investment strategy. The reverse is true when the plan is under-funded. I impose the realistic investment constraints that the plan cannot short-sell the stock and neither can it invest more than 100% of its asset value in the stock. Taking these constraints into account, the total proportion of assets which are invested in the risky stock at time n, for investment over the period [n,n+1)is πn:=max$0, min$1, sLT +aFn−F%%, 63 Books MDPI Risks 2017,5,12 in which a≥ 0 is a constant called the investment risk adjustment that is chosen when the plan is set up, and Fn is the plan’s notional funding level at time n. I state in Section 2.3 how the funding level is calculated. The higher the value of the constant a, the more sensitive is the strategy to over- and under-funding and hence the more volatile is the proportion invested in the risky stock. 2.2. Accumulation of Contributions The members’ contributions are increased each year using a rate calculated via a pre-specified formula. Once each member’s retirement benefit, i.e., the accumulated value of their contributions, is paid to them as a lump sum on their retirement date, there is no more plan liability to that member. The annual accumulation factor (AAF) granted on each member’s contributions at time n(i.e., for accumulation from time n−1 to time n)is 1+r+πn−1(E(Rn)−r)+βFn−−F, (1) for n= 1, 2, ... . The funding level Fn− is the funding level just prior to any payment made at time n and it is defined in Section 2.3. The AAF adjustment β≥ 0 is chosen when the plan is set-up. There are two components to the AAF. The first is a stable return, r+πn−1(E(Rn)−r) , equal to the expected return on the assets for the current investment strategy. The second component is a return, βFn−−F , that is positive or negative according to the funding level of the plan relative to the target. The AAF is higher when the plan is over-funded relative to the target funding level, i.e., when Fn−>F , as the plan can afford to grant higher accumulation factors. The higher value of the accumulated contributions acts to decrease the funding level when the plan is over-funded. The reverse happens when the plan is under-funded. The AAF adjustment β decrees the extent to which the AAF responds to situations of over- and under-funding; the higher the value of β , the greater the sensitivity. Notice that if β= 0 then the AAF is not affected by deviations in the notional funding level from the target. Here the funding level just prior to any payment out that is made at time nis used to calculate the AAF. If the funding level at time nwas used then the AAF would have to be solved for implicitly, which makes the accumulation calculation more opaque to the members. Let B(k) n denote the value of the accumulated contributions of generation kat time n, for n= 1, ... , k . It is calculated from the previous year’s accumulated value, B(k) n−1 , and any new contributions as B(k) n=B(k) n−11+r+πn−1(E(Rn)−r)+βFn−−F+C(k) n, in which C(k) n≥ 0 represents the new contribution made at time n= 1, ... , k− 1 by generation k= 2, ...N . As generation 1 retires at time 1, the only contribution that generation 1 makes is B(1) 0 at time 0. The retirement benefit depends directly on the investment strategy of the risk-sharing plan. This is not quite the same as in a DC plan, in which the contributions are increased at the rate of investment return on the assets. For example, a DC plan has its contributions increasing at the annual rate r+πn−1(Rn−r) , i.e., with the actual return on the DC plan’s assets rather than with the expected return, as in the risk-sharing plans considered here. 2.3. Funding Level The funding level of the risk-sharing plan is encouraged to move towards the target funding level. The AAF changes in response to deviations in the funding level, in order to bring the funding level back to the target value. Simultaneously, the investment strategy adjusts its investment risk-taking, lowering it if the plan is under-funded and otherwise raising it. 64 Books MDPI Risks 2017,5,12 The funding level of the risk-sharing plan at time nis Fn:=An Ln, where An is the value of the assets at time n. The value Ln at time n, is a notional liability, since the plan does not guarantee to pay any benefit. The plan simply accumulates contributions. However, I regard the accumulated contributions as a notional deferred benefit. The notional liability is calculated assuming that the accumulated contributions at time nare increased at every subsequent year by the current expected return on the long-term investment strategy and discounted by the same expected return. In valuing the notional liabilities like this, the idea is that the retirement benefit is intended, but not guaranteed, to be the accumulated contributions that would be obtained from following the strategic long-term investment strategy and receiving the expected return on that strategy. Thus for N starting generations in the plan, of whom generations n+1, . . . , Nwill retire after time n, Ln= N ∑ k=n+1B(k) n+C(k) n, i.e., it is a notional liability to pay the current value of accumulated contributions to each currently non-retired generation, including their new contributions. The notional liability at time n− includes the generation who are about to retire at time n, but accumulates their contributions at the expected, long-term strategic return over [n−1, n)rather than the actual return, i.e., Ln−:=1+r+sLT (E(Rn)−r)N ∑ k=n B(k) n−1. Accumulating the previous’s years contributions using the expected return on the long-term strategy is for two reasons: (i) to reflect the investment aim of the plan to invest in line with the long-term strategy; and (ii) to avoid the implicit equation involving Ln− (through Fn− ) that would arise if we used the AAF shown in (1). 3. Illustrations 3.1. Model Parameterization For simplicity, I assume that the risky stock’s annual effective random returns R1 , R2 , ... are independent, lognormally distributed random variables each with location parameter μ and scale parameter σ> 0. The continuously-compounded annual risk-free rate is denoted by the constant r.To parameterize the model, I used long-term returns. However, the risk-free bond return is “normalized” to 0% p.a. The reason for the normalization is to make it clear when members get a higher benefit from investing in the plan rather than investing all their money in the risk-free bond. The choice of the risk-free rate of 0% per annum means that, for example, a member who obtains a retirement benefit higher in value than their total contributions has done better than investing their contributions entirely in the risk-free asset. This corresponds to an average AAF which is greater than one. The annual return on the S&P500 index over the years 1956–1999 is about 12% with an annual volatility of about 15%, based on the values in (Table 2.1, Hardy [ 22 ]). Over the same time period, annual 10-year US Treasury bond returns were around 7% 1 . This gives the S&P500 annual return to be about 5% above the 10-year US Treasury bond return. 1 Based on the average annual 10-year US Treasury bond returns obtained from https://fred.stlouisfed.org/series/ IRLTLT01USA156N over the years 1956–1999. 65 Books MDPI Risks 2017,5,12 Figure 5. The annualized average accumulation factor for each generation in the model, for the more realistic membership, as both the investment risk adjustment (a) and the AAF adjustment ( β ) are varied. The target funding level is F= 100%. In this membership profile, the kth generation contributes $ (40 −k+1) at time 0, and contributes a further $1 per annum at the end of each year that they are not retired. (a) 75% quantile, F=100%; (b) Median, F=100%; (c) 25% quantile, F=100%. The widest range of possible retirement benefits is with the DC plan. The earliest generations of the DC plan experience a huge range of possible benefits, although this declines with the investment period. Increasing the target funding level to F= 120% for the risk-sharing plans, Figure 6 shows the same quantiles. Since the risk-sharing plans are only 100% funded at time 0, they attempt to reach the 72 Books MDPI Risks 2017,5,12 target funding level through reducing the AAFs (via the adjustment β ) and reducing the investment in the risky stock (via the adjustment a). This is observed in the figures, as the average AAF is below one (shown in Figure 6 as negative values, since the figure displays the value of the average AAF minus one) for the earliest retiring generations, and becomes greater than one (shown in Figure 6 as positive values) for the later generations as the target funding level is attained. The average AAF remains more stable (i.e., has a lower inter-quartile range) under the risk-sharing plans than the DC plan, for generations up to, roughly, the 30th. Figure 6. The annualized average accumulation factor for each generation in the model, for the more realistic membership, as both the investment risk adjustment (a) and the AAF adjustment ( β ) are varied. The target funding level is F= 120%. In this membership profile, the kth generation contributes $ (40 −k+1) at time 0, and contributes a further $1 per annum at the end of each year that they are not retired. (a) 75% quantile, F=120%; (b) Median, F=120%; (c) 25% quantile, F=120%. 73 Books MDPI Risks 2017,5,12 The shift in the median quantile, between Figures 5b and 6b, is interesting. For the higher target funding level F= 120%, the median average AAF has an increasing trend as time increases for the risk-sharing plans. This is the opposite trend to the plans with a target funding level F= 100%. The conclusion is that the target funding level can be used to shift the investment gains from the latest retiring generations to the earliest retiring generations. The reason for this shift is that the plan attempts to build up a cushion of assets, to reach the target funding level F= 120%. This cushion comes at the expense of the earlier generations, who pay for it and hence who are more likely to receive a lower average AAF than the later generations. The figures demonstrate the flexibility of the risk-sharing plan; the plan parameters allow us to adjust the average AAF quantiles (and hence the amount of retirement benefit) across the generations. 3.4.4. Analyzing Benefit Stability Here I analyze the stability of the average AAFs across generations for each plan. Rather than looking at means, I try to capture the spread of the possible average AAFs through an inter-quantile range. Setting for generation n=1, 2, . . . , 40, IQRn:=95% average AAF quantile of nth generation −5% average AAF quantile of nth generation, Next I calculate the range of {IQRn}n=1,...,40 for each plan, namely IQR instability :=max n=1,2,...,40{IQRn}−min n=1,2,...,40{IQRn}. A plan with low instability should have an average AAF whose 5% to 95% quantile range does not change much between generations. However, this does not tell us whether the values in the range change between generations; a plan can have zero instability but generation 1 faces an average AAF in the range [0%, 3%] , generation 2 in the range [1%, 4%] , and so on. The disparity between the values of the average AAF is also important, so that a plan does not give highly negative increases to the earlier generations and highly positive increases to the later generations, or vice versa. To capture the latter, define Quantile inequity :=max n=1,2,...,40{95% average AAF quantile of nth generation} −min n=1,2,...,40{5% average AAF quantile of nth generation}. The lower the quantile inequity, the lower is the overall range of the average AAFs across all generations. This means that a plan with a low quantile inequity does not have the earlier generations benefiting at the expense of the later generations, or vice versa. A narrower measure is the median inequity, which looks at the range of the median average AAF across all generations. Median inequity :=max n=1,2,...,40{50% average AAF quantile of nth generation} −min n=1,2,...,40{50% average AAF quantile of nth generation}. Table 2 shows the values of instability and inequity of the plans, for different target funding levels. Note that the values for the DC plan do not depend on the target funding level. 74 Books MDPI Risks 2017,5,12 Table 2. Instability and inequity, across the generations for different target funding levels. Plan Type aβTarget Funding Level F=100% IQR Instability Qu. Inequity Median Inequity Risk-sharing 0.2 0.2 9.4% 10.3% 3.8% Risk-sharing 0.2 0.4 15.7% 15.7% 3.7% Risk-sharing 0.4 0.2 9.7% 10.5% 3.8% DC N/A N/A 40.7% 40.7% 3.3% Plan Type aβTarget Funding Level F=120% IQR Instability Qu. Inequity Median Inequity Risk-sharing 0.2 0.2 9.5% 12.1% 4.4% Risk-sharing 0.2 0.4 14.9% 19.4% 8.9% Risk-sharing 0.4 0.2 9.5% 12.2% 4.6% DC N/A N/A 40.7% 40.7% 3.3% The risk-sharing plans have significantly lower values of IQR instability and quantile inequity than the DC plan. Thus, in terms of having a low range of possible average AAFs faced by different generations between the 5% and 95% quantiles, the risk-sharing plans perform well. The lowest measures are seen in the two plans with β= 0.2. Looking further down the table, the IQR instability measures decline for the risk-sharing plans as the target funding level increases, but this is at the cost of the quantile inequity increasing. Thus while the 5%–95% inter-quantile range declines over all generations, the range of possible values that the average AAF can take increases. Interestingly, the DC plan has the lowest median average AAF inequity. For the risk-sharing plans, the median inequity rises as the target funding level rises. This is the due to the plans starting with a funding level of 100%, and therefore being under-funded relative to the higher target funding levels considered. Earlier generations are granted lower AAFs as the plan attempts to reach the target funding level. Later generations gain from the lower payments out from the plan’s assets and the accumulated investment returns. 3.4.5. Disappointment Although reducing the accumulated contributions is a integral part of the considered risk-sharing plans, perhaps some plan structures result in a higher chance of reductions in the accumulated value of contributions than others. If reductions occur too often, then it is possible that members will exit the plan. They will compare what they are getting with the recently retired members, and will be unhappy if they get a lower retirement benefit than them [23–26]. I call this member disappointment. Member disappointment is related to disappointment aversion, which was introduced by Gul [ 27 ]. Ang et al. [ 28 ] develop a portfolio choice framework under disappointment aversion preferences, using a dynamic CRRA problem. Fielding and Stracca [29] analyze both loss and disappointment aversion, and find that stocks may disappoint in the long term. Here I introduce a way of trying to measure retiring members’ disappointment if their average AAF is less than the average AAF for the previous year’s retiring generation. To measure member disappointment over two consecutive years, in each simulation I count the number of runs of exactly two consecutive years in which the average AAF is strictly declining. Doing this across all simulations, I calculate the probability of having two consecutive years in which the average AAF is strictly declining. Using the same approach, I calculate the probability of having three consecutive years in which the average AAF is strictly declining. I repeat this for four consecutive years, five consecutive years and so on. The resulting probability can be used to see the frequency of runs of declining average AAFs. Figures 7 and 8 show the probability of disappointment for the plans, i.e., the probability of a fixed number of years of declining average AAF, for a target funding levels of 100% and 120%, respectively. 75 Books MDPI Risks 2017,5,12 Members of the plans with the higher target funding level F= 120% (Figure 8) are less likely to suffer disappointment. Thus the higher target funding level gives more security to the members. In all figures, it is observed that members of the risk-sharing plans are more likely to suffer declining average AAFs than DC plan participants. Therefore, communication of this possibility to members is essential for risk-sharing plans. Figure 7. Disappointment for the more realistic membership when the target funding level F= 100%. The probability of a declining average AAF plotted against the number of consecutive years that the decline occurs over. Both the investment risk adjustment (a) and the AAF adjustment (β) are varied. Figure 8. Disappointment for the more realistic membership when the target funding level F= 120%. The probability of a declining average AAF plotted against the number of consecutive years that the decline occurs over. Both the investment risk adjustment (a) and the AAF adjustment (β) are varied. Figures 9 and 10 show the probability of disappointment for the plans, but for the AAF instead of the average of the AAFs. The message is the same, albeit the disappointment measure is very close to that of the DC plan. Disappointment is reduced by increasing the target funding level. This suggests that members who are easily disappointed should be in a plan with a higher target funding level. 76 Books MDPI Risks 2017,5,12 Figure 9. Disappointment for the more realistic membership when the target funding level F= 100%. The probability of a declining AAF plotted against the number of consecutive years that the decline occurs over. Both the investment risk adjustment (a) and the AAF adjustment (β) are varied. Figure 10. Disappointment for the more realistic membership when the target funding level F= 120%. The probability of a declining AAF plotted against the number of consecutive years that the decline occurs over. Both the investment risk adjustment (a) and the AAF adjustment (β) are varied. 4. Conclusions The risk-sharing plan presented enables its members to receive more stable increases on their contributions than under a DC plan. However, the downside is that members may face year-on-year reductions in the accumulated value of their contributions, which I call disappointment. There is a higher chance of this than under a DC plan, for the chosen financial market model and membership profile. The possibility of members’ accumulated contributions declining over time would need to be well-communicated to them. However, as the average accumulation factors on the contributions made in the risk-sharing plan are more stable than under a DC plan, a risk-sharing plan may allow members to plan better for their retirement since they have more certainty about the amount of their retirement benefit. 77 Books MDPI Risks 2017,5,12 The chance of year-on-year reductions in the accumulated value of contributions can be reduced by increasing the target funding level. This suggests that members who wish to reduce the chance of reductions should be in a plan with a higher target funding level. However, increasing the target funding level means that earlier generations pay for a cushion of assets from which they may not benefit. Instead, the later retiring generations are more likely to benefit from the cushion, in the form of higher accumulation factors granted on their contributions. The choice of the plan parameters—which determine by how much to adjust the investment strategy, the accumulation factors and the target funding level—are critical to the sustainability of the plan, and the degree of risk-sharing between generations. Based on the studied plans, I draw the following conclusions. The accumulation factor granted on contributions is most sensitive to the parameter that adjusts the investment strategy. The chance of the plan running out of money is most sensitive to the parameter that adjusts the accumulation factors. The target funding level can be used to shift the level of accumulation factors between generations; a higher target funding level means that earlier generations are granted lower accumulation factors than later generations, and vice versa. The choice of the plan parameters requires a careful study of the expected membership profile, allowing for a financial market model or models. A preliminary analysis by the author, using the membership profiles analyzed in the paper, suggests that the conclusions continue to hold when the financial model is changed. While this paper analyzes a selection of the plan parameters, any implementation in practice would require further analysis to choose appropriate values for the expected plan membership. Acknowledgments: The author gratefully acknowledges the financial support of the 2013 Individual Grant received from The Actuarial Foundation. The author thanks the reviewers of the paper, who made valuable comments which helped to greatly improve the paper. Conflicts of Interest: The author declares no conflict of interest. References 1. Ramaswamy, S. The Sustainability of Pension Schemes; Technical Report; Working Papers No. 368; Bank for International Settlements: Basel, Switzerland, 2012. 2. Franzen, D. Managing Investment Risk in Defined Benefit Pension Funds; Technical Report; Organisation for Economic Co-Operation and Development; OECD Working Papers on Insurance and Private Pensions; Number 38; OECD Publishing: Paris, France, 2010. 3. Lusardi, A.; Mitchell, O. Financial literacy around the world: An overview. J. Pension Econ. Financ. 2011 ,10, 497–508. [CrossRef] 4. Greenwald & Associates. 2015 Risks and Process of Retirement Survey; Technical Report; Society of Actuaries: Schaumburg, IL, USA, January 2016. 5. Ponds, E.; Riel, B. Sharing risk: The Netherlands’ new approach to pensions. J. Pension Econ. Financ. 2009 ,8, 91–105. [CrossRef] 6. Cui, J.; de Jong, F.; Ponds, E. Intergenerational risk sharing within funded pension schemes. J. Pension Econ. Financ. 2011,10, 1–29. [CrossRef] 7. Gollier, C. Intergenerational risk-sharing and risk-taking of a pension fund. J. Public Econ. 2008 ,92, 1463–1485. [CrossRef] 8. Pugh, C.; Yermo, J. Funding Regulations and Risk-Sharing; Technical Report; Organisation for Economic Co-operation and Development; OECD Working Papers on Insurance and Private Pensions; Number 17; OECD Publishing: Paris, France, 2008. 9. Blommestein, H.; Janssen, P.; Kortleve, N.; Yermo, J. Evaluating the Design of Private Pension Plans: Costs and Benefits of Risk-Sharing; Technical Report; Organisation for Economic Co-operation and Development; OECD Working Papers on Insurance and Private Pensions; Number 34; OECD Publishing: Paris, France, 2009. 10. Briys, E.; de Varenne, F. On the risk of insurance liabilities: Debunking some common pitfalls. J. Risk Insur. 1997,64, 673–694. [CrossRef] 11. Barbarin, J.; Devolder, P. Risk measure and fair valuation of an investment guarantee in life insurance. Insur. Math. Econ. 2005,37, 297–323. [CrossRef] 78 Books MDPI Risks 2017,5,12 12. Graf, S.; Kling, A.; Ruß, J. Risk analysis and valuation of life insurance contracts: Combining actuarial and financial approaches. Insur. Math. Econ. 2011,49, 115–125. [CrossRef] 13. Grosen, A.; Jørgensen, P. Fair valuation of life insurance liabilities: The impact of interest rate guarantees, surrender options, and bonus policies. Insur. Math. Econ. 2000,26, 37–57. [CrossRef] 14. Hieber, P.; Korn, R.; Scherer, M. Analyzing the effect of low interest rates on the surplus participation of life insurance policies with different annual interest rate guarantees. Eur. Actuar. J. 2015,5, 11–28. [CrossRef] 15. Guillén, M.; Jørgensen, P.; Nielsen, J. Return smoothing mechanism in life and pension insurance. Insur. Math. Econ. 2006,38, 229–252. [CrossRef] 16. Baumann, R.; Müller, H. Pension funds as institutions for intertemporal risk transfer. Insur. Math. Econ. 2008 , 42, 1000–1012. [CrossRef] 17. Bohnert, A.; Gatzert, N. Analyzing surplus appropriation schemes in participating life insurance from the insurer’s and the policyholder’s perspective. Insur. Math. Econ. 2012,50, 64–78. [CrossRef] 18. Zemp, A. Risk comparison of different bonus distribution approaches in participating life insurance. Insur. Math. Econ. 2011,49, 249–264. [CrossRef] 19. Goecke, O. Pension saving schemes with return smoothing mechanisms. Insur. Math. Econ. 2013 ,53, 678–689. [CrossRef] 20. Galer, R. “Prudent Person Rule” Standard for the Investment of Pension Fund Assets; Technical Report; Organisation for Economic Co-Operation and Development; Financial Market Trends; Number 83; OECD Publishing: Paris, France, November 2002. 21. De Jong, F. Pension fund investments and the valuation of liabilities under conditional indexation. Insur. Math. Econ. 2008,42, 1–13. [CrossRef] 22. Hardy, M. Investment Guarantees: Modeling and Risk Management for Equity-Linked Life Insurance; Wiley Finance: Hoboken, NJ, USA, 2003. 23. Boyce, C.; Brown, G.; Moore, S. Money and happiness: Rank of income, not income, affects life satisfaction. Psychol. Sci. 2010,21, 471–475. [CrossRef] [PubMed] 24. Clark, A.; Senik, C. Who compares to whom? The anatomy of income comparisons in Europe. Econ. J. 2010 , 120, 573–594. [CrossRef] 25. Luttmer, E. Neighbors as negatives: Relative earnings and well-being. Q. J. Econ. 2005 ,120, 963–1002. [CrossRef] 26. Wolbring, T.; Keuschnigg, M.; Negele, E. Needs, comparisons, and adaptation: The importance of relative income for life satisfaction. Eur. Sociol. Rev. 2013,29, 86–104. [CrossRef] 27. Gul, F. A theory of disappointment aversion. Econometrica 1991,59, 667–686. [CrossRef] 28. Ang, A.; Bekaert, G.; Liu, J. Why stocks may disappoint. J. Financ. Econ. 2005,76, 471–508. [CrossRef] 29. Fielding, D.; Stracca, L. Myopic loss aversion, disappointment aversion, and the equity premium puzzle. J. Econ. Behav. Organ. 2007,64, 250–268. [CrossRef] © 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 79 Books MDPI Article The Shifting Shape of Risk: Endogenous Market Failure for Insurance Thomas G. Koch Bureau of Economics, Federal Trade Commission, Washington, DC 20580, USA; [email protected]; Tel.: +1-512-809-8014 Academic Editor: Luca Regis Received: 6 July 2016; Accepted: 9 December 2016; Published: 27 January 2017 Abstract: This article considers an economy where risk is insurable, but selection determines the pool of individuals who take it up. First, we demonstrate that the comparative statics of these economies do not necessarily depend on its marginal selection (adverse versus favorable), but rather other characteristics. We then use repeated cross-sections of medical expenditures in the U.S. to understand the role of changes in the medical risk distribution on the fraction of Americans without medical insurance. We find that both the level and the shape of the distribution of risk are important in determining the equilibrium quantity of insurance. Symmetric changes in risk (e.g., shifts in the price of medical care) better explain the shifting insurance rate over time. Asymmetric changes (e.g., associated with a shifting age distribution) are not as important. Keywords: risk sharing; medical insurance; adverse selection JEL Classification: D3; I1 1. Introduction This article considers how the price for insurance and the number of people insured varies when the risk and other characteristics of the population change. We study these relationships, known as comparative statics, for insurance markets that suffer from either adverse or advantageous selection, and then consider the specific case of medical insurance markets. The comparative statics of medical insurance markets is of interest, as these markets leave millions of Americans uninsured against medical risk. Eighty percent of the private medical insurance in the US is employer-provided 1 , and employers are not allowed to discriminate according to relevant demographic and medical information when they price insurance provided to their employees. 2 Policies, such as the income-tax deductability of health insurance, or a “Cadillac tax” on expensive health plans, should induce changes in prices and quantity. I demonstrate that increases in the cost of insurance, such as the “Cadillac tax”, may actually increase the amount of insurance, rather than decrease it. The actual price and quantity effects of policies designed to shift either the supply or demand curves in insurance markets may have counter-intuitive effects. For example, many aspects of the Affordable Care Act (ACA), such as the Cadillac tax or ACA plan subsidies, that are designed to decrease or increase the amount of insurance, respectively, may have opposite effects. In these models, the amount of risk left uninsured due to selection depends upon the distribution of risks—for example, what is the ratio of high-cost to low-cost individuals, and how different in average costs are high- and low-cost individuals. We use repeated cross-sections of medical expenditure 1This number is based upon the author’s calculations using the MEPS, which is described below. 2 This is why a menu of contracts is unlikely to solve the adverse selection problem here: any separate pooling they could induce would rely on indirect, and possibly manipulable, correlates with expected cost. Risks 2017,5, 9 80 www.mdpi.com/journal/risks Books MDPI Risks 2017,5,9 data to infer such ratios, and use them in a model of insurance markets with asymmetric information. I also find that changes to the distribution of medical risk faced by consumers in recent years should have lead to more expensive medical insurance. It should also have lead to more people with medical insurance. The price of employer-provided medical insurance to cover a single employee has grown five percent a year from 1996 to 2005. The price of family coverage has grown 5.7 percent per year over the same period. 3 At the same time, more Americans have gone without medical insurance. Among the non-poor and non-elderly, the percentage of individuals without private insurance rose 22 percent (18 percent to 22) from 1996 to 2004.4 One explanation for higher insurance prices is the well-documented growth in average medical spending (Swartz [1] ; Freudenheim [2] ). As medical costs grow, the supply curve for medical insurance shifts up, and induces movement along the demand curve. This new equilibrium price is higher, while equilibrium quantity is lower. However, changes in the distribution of medical expenditures (e.g., mean, median, variance, etc.) are the result of changes in the distribution of medical risk. In this case, the demand curve also shifts up. The new equilibrium price is larger, but whether the new equilibrium quantity is larger or smaller is unclear. This article attempts to disentangle these changes. This study complements the findings of Gruber and Levy [3] , which documented similar (though differently measured) changes to the distribution of medical risk. We use changes in the distribution of medical expenditures in the US to infer changes in the distribution of medical risk. We then take these measured differences in the distribution of medical risk into a quantitative model of insurance choice with asymmetric information. We find that measured changes in the distribution of medical risk should have made insurance more expensive and more prevalent. While the former trend is consistent with the data, the latter is not. This work can help shed light on recent research that considers alternative risk-adjustment factors; that is, what kinds of information could be used to price insurance that would limit the amount of market failure due to adverse selection. Both [ 4 , 5 ] find evidence that age-based health insurance prices may drive some of those age-specific markets to collapse. Here, we identify some of the characteristics of risk pools that make them sustainable. The lessons from this exercise go beyond the sectoral concerns of medical insurance. If the source of uninsurance in other markets is asymmetric information, then this reminds us that distributions matter. If there is an across-the-board increase in risk, then insurance markets likely would provide more insurance, instead of less, as the demand curve shifts out further than the supply curve because of risk aversion. Furthermore, it introduces the notion that comparative statics do not depend upon the nature of the selection (adverse vs. advantageous). Instead, the comparative statics depend upon the stability of the equilibrium. For example, if wage volatility goes up, it matters whose wage volatility goes up. This also relates to the market failure as discussed in recent work on insurance against aggregate, catastrophic risk [ 6 – 8 ]. When a natural disaster strikes, it may test the solvency of the risk pool, leading to partial insurance. The risks under study here have idiosyncratic realizations, as opposed to the widespread loss of property due to an earthquake or hurricane. For example, we do not consider how insurance markets might deal with a flu pandemic. Likewise, the tax subsidy of health insurance should mitigate adverse selection, and the indirect nature of the contract (typically via employers, instead of directly to consumers) may also influence the extent of adverse selection. 3 The level changes from 1996 to 2005 for these two prices is 1992 to 3227, and 4954 to 8675, as reported by the MEPS-IC. These calculations adjust for changes in the price level, as measured by the Urban CPI. 4 These numbers are calculated from the MEPS-HC, which collects information from a nationally-representative sample of households. 81 Books MDPI Risks 2017,5,9 MEPS, starting in 1996, going to 2004, in order to infer each year’s distribution of medical risk—each year’s Gamma distribution, its (αt , βt) pair. We will later use these values to infer the amount of private insurance possible in the asymmetric-information model. We restrict the sample to the non-elderly and the non-poor in order to avoid complications due to Medicare and Medicaid. Veterans are also removed, due to access to VA care. In all three cases, publicly-provided insurance can crowd out private insurance. This restriction also avoids potential confusion about private insurance among the elderly (over 65). Unfortunately, the data do not present a perfect candidate for the insurable (i.e., covered by insurance) realizations of medical risk. One potential measure is the total charges for an individual over a year. Because medical insurance contracts typically have co-payments or deductibles, this is an imprecise measure of the realized insurable medical risk. This specification of risk has an internally-coherent and tractable way to back out the insurable fraction of total risk. If a fraction ρ−1< 1 of an exponential risk is insurable, parameterized by λ , then this fraction of a risk is also an exponential risk. This new risk is parameterized by λ ρ . If this fraction ρ is common across agents, then all of the exponential risks are scaled by ρ . A common ρ might come from the fact that these co-insurance rates are used to solve a hidden-action problem, and type privacy does not allow for discrimination. Finally, since β is a scale parameter for the Gamma distribution, the new distribution of insurable risk is also a Gamma distribution, with parameters αand β ρ. The distribution of insurable risk can also be inferred from the distribution of medical expenditures paid by private insurance companies for the privately insured. The first moment of the conditional distribution, the average realized risk of the insured, can be found by integrating the expected realized risk over the types who choose insurance; i.e., E(' mxi|ιi=1)=λm 0t−1tαe−t β βα+1Γ(λm β,α+1)dt = Γ(λm β,α) βΓ(λm β,α+1). The average square of realized risk of the insured (i.e., the second non-central conditional moment) is found similarly: E(' mx2 i|ιi=1)=λm 02t−2tαe−t β βα+1Γ(λm β,α+1)dt =2Γ(λm β,α−1) β2Γ(λm β,α+1). These are two unique moments that identify the two parameters of the risk distribution, (α,β). That said, the amount paid for treatment by a private insurance company is unlikely to be the amount billed to the uninsured. Insurance companies buy a large volume of medical goods and services throughout a year, and frequently bargain with with medical providers over the cost of treatment. This bargaining power is not available to the individual uninsured patient. The resolution here is the same as for charges—assume a common proportional mark-up from the expenditures paid for by private medical insurance companies. This new marked-up distribution of risk is also a Gamma distribution, with parameters α and β ρ . The balance of this article will use both methods—marking up from private medical expenditures, and marking down from total charges. Both series tell the same story. It may be useful to characterize changes in α and β . As the discussions of mark-up and mark-down above suggest, changes in β are easier to interpret. Since β is a scale parameter, a decrease in β scales up the medical risk faced by every agent an equal amount. 88 Books MDPI Risks 2017,5,9 The parameter α is best characterized by a distribution’s kurtosis. A distribution’s kurtosis is the ratio of its fourth moment to the square of its second moment (variance). As discussed in Balanda and MacGillivray [15] , kurtosis is synonymous with “peakedness” and “tail weight.” Two distributions may have the same variance, but kurtosis measures whether that variance is due to a large amount of moderate dispersion or a moderate amount of large dispersion. Flatter distributions have lower kurtosis. Changes in kurtosis are central to the ability of insurance markets to spread risk across types. The variance of risk types may increase. If this increase is because many types became a little more different, then it is unlikely that many will newly refuse to buy the same insurance. However, if a few of the agents became very different from the rest, then cross-type risk sharing becomes less likely. The kurtosis for a gamma distribution is 6 α . Thus, as α decreases, the risk types are becoming more and more disparate, and market failure becomes more pervasive. The sickest members of the population are becoming relatively more sick than the rest of the population. We will refer to this effect as the kurtosis effect—as α increases, risk types become more similar, and insurance becomes more widespread. When κis large, the kurtosis effect dominates. However, the calibrated value of κ is only $5600. This reflects the many ways in which uninsured individuals can seek and receive free or uncompensated care. Hospitals reported providing an average of $20 billion per year in uncompensated and charity care (American Hospital Association [16] ). The charity portion of care reflects the not-for-profit and humanitarian missions of many hospitals. Uncompensated care is typically provided in emergency rooms, where hospitals are required to provide care independent of insurance status or ability to pay. In either case, this provides an upper bound risk faced by an uninsured individual, and it has consequences for an agent’s willingness to pay for insurance (i.e., in distribution, the demand curve for insurance). The calibrated value for CARA risk aversion, r=0.0018, is in the range of risk preferences estimated by Cohen and Einav [17]. The subsidy rate, s , has important implications for the calibration. Because ninety percent of the privately insured get medical insurance from their employer, and the cost of that insurance is not taxed as wage income would be, the subsidy rate is the marginal tax rate faced by the individual. Marginal tax rates are important, because they provide extra incentive to have insurance, and increase the insurance rate. This model corresponds to a competitive labor market where workers are compensated with their productivity, and only their compensation can come in the form of taxed wages or untaxed medical insurance. Saez [18] provides the average marginal tax rates for the US. These calculations stop at the year 2000. The marginal tax rate for the year 2001 and later will be the average value for the years 1996 to 2000. When α decreases, the marginal agent faces a larger risk, and her willingness to pay for insurance increases. However, when that agent faces only a fraction of the realized risk (i.e., κ is small), the marginal agent’s willingness to pay for insurance does not increase more than the increase in the price of insurance. Here, the price of insurance is increasing because everyone who has insurance is also becoming more expensive. The kurtosis effect is dominated by the price effect of decreasing α. Recall the claim in Swartz [1] that increases in the average cost of care led to the increased price of insurance, and therefore fewer people had medical insurance. If the increase in average medical expenditure is due to a common increase in medical risk ( β goes down), then the insurance rate may increase. As everyone’s medical risk grows, so does their certainty equivalent—how much above and beyond a fair price each agent is willing to pay for insurance. Larger certainty equivalents allow for more risk sharing across types. If the increase in medical expenditure is the result of a change in the shape of the distribution of medical risk, then the outcome is ambiguous—either the price effect or the kurtosis effect may dominate. As described in Table 2, the changes inferred from 1996 to 2004 are a mixture of the two. Over this period, we might derive these changes as coming from two sources: an increasing price of medical goods and services (falling β ); and an increasing incidence of costly medical conditions (falling α ). This 89 Books MDPI Risks 2017,5,9 latter trend is consistent with the aging American population and the increased incidence of obesity, each associated with costly medical conditions. The mark-up rate for a given year is exactly identified as the ratio of the price of insurance and the average medical expenditure paid for insurance on behalf of the insured, both from the data. The mark-down rate from total charges for a particular year is found by matching the model’s price of insurance to that found in the data. These estimates are presented in Table 2. The price of insurance used here is the average price of single-coverage employer-provided medical (hospital and doctor) insurance in the US, as reported in the MEPS. Since the risk is reported on the individual level, the price of the corresponding coverage is considered as well. As mentioned in the introduction, the price of single-coverage and family-coverage employer-provided insurance grew at similar rates. Over this period, family coverage (covering two or more individuals) was steadily two-and-a-half times as expensive as single coverage. The implicit assumption is that there are constant returns to scale when it comes to family insurance plans versus single coverage, and there is no differential selection across them. Table 2. Parameter values that vary by year; see text for explanation of subsidy rate for years since 2000. Year Charges Expenditures Subsidy αyear βyear ρyear αyear βyear ρyear 1996 1.091 5.30 ×10−40.9706 1.0453 0.0010 1.8046 0.2475 1997 1.107 5.17 ×10−40.968 1.1557 0.0011 2.0931 0.2533 1998 1.087 4.70 ×10−40.9079 1.1645 0.0011 2.1757 0.2556 1999 1.128 4.57 ×10−40.988 1.1479 0.0010 2.1994 0.2584 2000 1.156 3.95 ×10−41.0028 1.1203 0.0010 2.2978 0.2613 2001 1.132 3.51 ×10−40.9145 1.166 0.0008 2.118 0.2522 * 2002 1.141 3.21 ×10−40.9293 1.1838 0.0008 2.2149 0.2522 * 2003 1.049 3.01 ×10−40.84 1.0547 0.0008 1.9889 0.2522 * 2004 1.067 2.72 ×10−40.8126 1.1096 0.0007 2.0232 0.2522 * Figure 4a,b plot the insurance rate predicted by the model against the rate observed in the data. The two trends are within four percentage points of one another until 2001, the year of significant tax reform legislation. Since this tax reform decreased the marginal tax rates faced by households, this may explain why the model predicts increasing insurance rates, while the data finds insurance rates dropping slightly since 2000. (a)(b) Figure 4. Predicted trends are consistent across methods. 90 Books MDPI Risks 2017,5,9 Figure 5a,b assess the relative importance of changes in the level of risk ( β ) and the shape of risk ( α ). They each plot two counterfactual insurance rates—the first, if the shape of the distribution of risk stayed the same, but the level of risk changed as described in Table 2 (i.e., α1996 for all years, βt for all t ); the second, if the level of risk did not change, but the shape of the distribution of risk changed as described in the same table (i.e., β1996 for all years, αtfor all t). (a)(b) Figure 5. Changes in levels dominate changes in shape. Changes common to the medical risk of all agents are driving the insurance rate trends predicted by the model. Inferred changes in the shape in the distribution of medical risk (i.e., changes in α ) play a very small roll in the predicted trends. Thus, in this period, changes due to the shifting age distribution or obesity have very little explanatory power. Instead, the model suggests a focus on changes common to all agents, such as moving prices for medical goods and services. 5. Conclusions A competitive insurance market’s capacity to provide insurance in spite of asymmetric information depends upon the distribution of risk. When an economy faces greater average risk, the amount of uninsured risk may vary, but depends crucially upon what kind of change took place. An observed increase in average medical costs could be due to either a common increase in the risk faced by all; or, it could reflect the change in risk faced by some agents. In the former case, there is more opportunity for risk sharing across types. In the latter, uninsurance is more frequent because the risk types are more disparate. Changes in the cross-sectional distribution of medical risk suggest that there have been changes to both the level and the shape in the distribution of medical risk. The changes in the medical insurance rate predicted here are due primarily to the former. This emphasizes the role of changing medical prices for the rate of and price for insurance. It meanwhile suggests that changes due to shifting demographics, such as age and obesity, have had little effect on the number of people with medical insurance. Conflicts of Interest: The views expressed in this article are those of the author. They do not necessarily represent those of the Federal Trade Commission, and the work was not prepared as part of the author’s work for the FTC. References 1. Swartz, K. Reinsuring Health: Why More Middle-Class People Are Uninsured and What Government Can Do; Russell Sage Foundation Publications: New York, NY, USA, 2006. 91 Books MDPI Risks 2017,5,9 2. Freudenheim, M. Cost of Health Insurance Rises Again, but at a Slightly Slower Rate; New York Times: New York, NY, USA, 2007. 3. Gruber, J.; Levy, H. The Evolution of Medical Spending Risk. J. Econ. Perspect. 2009,23, 25–48. 4. Handel, B.R.; Hendel, I.; Whinston, M.D. Equilibria in Health Exchanges: Adverse Selection vs. Reclassification Risk. Econometrica 2015,83, 1261–1313. 5. Koch, T.G. One pool to insure them all? Age, risk and the price (s) of medical insurance. Int. J. Ind. Organ. 2014b,35, 1–11. 6. Charpentier, A.; le Maux, B. Natural catastrophe insurance: How should the government intervene? J. Public Econ. 2014,115, 1–17. 7. Cummins, J.D. Should the government provide insurance for catastrophes. Fed. Reserve Bank St. Louis Rev. 2006,88, 337–379. 8. Kousky, C.; Cooke, R. Explaining the failure to insure catastrophic risks. Geneva Pap. Risk Insur. Issues Pract. 2012,37, 206–227. 9. Jovanovic, B. Favorable Selection with Asymmetric Information. Q. J. Econ. 1982,97, 535–39. 10. Selten, R. Re-examination of the perfectness concept for equilibrium points in extensive games. Int. J. Game Theory 1975,4, 25–55. 11. Kohlberg, E.; Mertens, J.-F. On the Strategic Stability of Equilibria. Econometrica 1986,54, 1003–1037. 12. Koch, T.G. Bankruptcy, medical insurance, and a law with unintended consequences. Health Econ. 2014 ,23, 1326–1339. 13. Harris, C.M. The Pareto Distribution as a Queue Service Discipline. Oper. Res. 1968,16, 307–313. 14. Stanton, M.W. Employer-Sponsored Health Insurance: Trends in Cost and Access; Research in Action 17; Agency for Healthcare Research and Quality: Rockville, MD, USA, 2004. 15. Balanda, K.P.; MacGillivray, H.L. Kurtosis: A Critical Review. Am. Stat. 1988,42, 111–119. 16. American Hospital Association. Uncompensated Hospital Care Cost Fact Sheet; American Hospital Association: Washington, DC, USA, 2014. 17. Cohen, A.; Einav, L. Estimating Risk Preferences from Deductible Choice. Am. Econ. Rev. 2007,97, 745–788. 18. Saez, E. Reported Incomes and Marginal Tax Rates, 1960–2000: Evidence and Policy Implications; NBER Working Papers 10273; National Bureau of Economic Research, Inc.: Cambridge, MA, USA, 2004. c  2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 92 Books MDPI risks Article Compositions of Conditional Risk Measures and Solvency Capital Pierre Devolder and Adrien Lebègue * Institut de Statistique, Biostatistique et Sciences Actuarielles, Université catholique de Louvain, Voie du Roman Pays 20 bte L1.04.01, B-1348 Louvain-la-Neuve, Belgium; [email protected] *Correspondence: [email protected] Academic Editor: Luca Regis Received: 14 November 2016; Accepted: 9 December 2016; Published: 16 December 2016 Abstract: In this paper, we consider compositions of conditional risk measures in order to obtain time-consistent dynamic risk measures and determine the solvency capital of a life insurer selling pension liabilities or a pension fund with a single cash-flow at maturity. We first recall the notion of conditional, dynamic and time-consistent risk measures. We link the latter with its iterated property, which gives us a way to construct time-consistent dynamic risk measures from a backward iteration scheme with the composition of conditional risk measures. We then consider particular cases with the conditional version of the value at risk, tail value at risk and conditional expectation measures. We finally give an application of these measures with the determination of the solvency capital of a pension liability, which offers a fixed guaranteed rate without any intermediate cash-flow. We assume that the company is fully hedged against the mortality and underwriting risks. Keywords: dynamic risk measures; time consistency; iterated risk measures; pension liability; solvency capital 1. Introduction The determination of the economic capital of a life insurer or pension fund has become of paramount importance over the past decade due to the recent financial crisis in the banking sector. The management and measurement of risk is of great interest, as can be seen from the recent Basel II and III regulatory frameworks for banks and Solvency II for the European insurance sector. These new regulations are designed to be risk sensitive. The Solvency II regulation recognizes that the insurer faces different kinds of risks, like market and longevity risks, while the previous regulation (Solvency I) is factor based. These regulations use risk measures as introduced in [ 1 – 3 ] in order to quantify the riskiness of financial positions and to provide a criterion to determine their acceptability. For instance, the new Solvency II framework requires a value at risk (VaR) measure to determine the solvency capital requirement, while the Swiss Solvency Test uses the tail value at risk (TVaR) measure for its target capital. These measures consider a real-valued random variable describing a future financial value and measure its risk at the beginning of the time period. Under the Solvency II and the Swiss Solvency Test regulations, the length of the time period is equal to one year, i.e., these frameworks consider the evolution of their own funds of the insurance undertaking over one year. However, life insurers and pension funds hold also long-term products, such as pension liabilities, which are the topic of this paper. In this paper, we consider pension liabilities, and we study risk measures that consider the long-term characteristic of these products. We focus here on the market risk, especially the equity risk. We also assume that the company is fully hedged against the mortality and underwriting risks. An important drawback of the classic static risk measures is that they do not take into account the information disclosed through time. These measures only consider the end-points of the time period. Risks 2016,4, 49 93 www.mdpi.com/journal/risks Books MDPI Risks 2016,4,49 If we deal with liabilities with a maturity of one year, then these risk measures are adapted as we work with a time horizon equal to the accounting horizon. Because we consider pension liabilities with a long-term horizon, this information could be meaningful in the computation of the solvency capital through time, especially on a yearly basis, as is the case for accounting purposes. That is why we consider dynamic risk measures as studied in [ 3 – 6 ]. The information is modeled by a filtration, and this filtration is incorporated in the computation of the capital each year. A first example of a dynamic risk measure could be the recalculated version of the static risk measures. We could for instance recalculate the solvency capital each year according to a static risk measure by incorporating the information available. However, this approach only incorporates past information, and for well-known measures, such as the VaR or TVaR measures, this could lead to time inconsistency, i.e., a position that was acceptable could become undesirable in any scenario. Therefore, we require a particularly useful property for dynamic risk measures, which is called time consistency. This property tells us that if we prefer a position tomorrow in almost every state of the world, then we already prefer it today. It is well known that the recalculated versions of the VaR and TVaR measures are not time consistent (see [7,8]). It has been proven that a time-consistent dynamic risk measure is closely linked to a backward iteration scheme [ 9 ]. We then consider this property to construct time-consistent risk measures through the iteration of conditional risk measures. This approach has been considered for the determination of the solvency capital of life insurance products in [ 10 – 12 ]. Nevertheless, it appears that the solvency capital obtained can be very expensive if we do not take care of the confidence level of each conditional VaR and TVaR measures involved in the iteration scheme. This is linked to a result obtained in [ 13 ], which tells us that, under certain hypotheses, if we consider a great number of iterations, the risk measure obtained is close to either the conditional expectation of the risk or its essential supremum. In order to overcome this difficulty, we consider the iterations of different conditional risk measures with a yearly time step fitting the accounting point of view. We also build these measures in such a way that they are coherent with the Solvency II or Swiss Solvency Test frameworks, meaning that for the last year of the product, the measures we introduce here correspond to the one used in these frameworks. This paper then introduces a new way to compute the solvency capital in life insurance, in order to take into account simultaneously the following three constraints •to be in line with Solvency II on a one-year horizon; •to be time consistent; •to incorporate in the risk measurement the maturity of the product. The paper is organized as follows. We set the mathematical framework in Section 2. Then, we recall the definition of conditional and dynamic risk measures in Section 3, of time consistency and iterated risk measures in Section 4 and present some compositions of conditional risk measures in order to determine the solvency capital in Section 5. We then compute the solvency capital in Section 6 and finally give a numerical illustration in Section 7. 2. Framework Throughout the paper, we consider a complete atomless probability space (Ω , F , P) . We write N∗=N\{ 0 } .Wefix T∈N∗ seen as the maturity of a pension liability and define T={ 0, ... , T− 1 } as the set of intermediate dates. We also consider a filtration: F=(Ft)t∈T∪{T}, such that F0={∅ , Ω} and FT=F . All inequalities and equalities between random variables (r.v.’s) are meant to hold P -almost surely ( P -a.s.) if not stated otherwise. Let t∈T∪{T} . We define L1(Ω , Ft , P) as the space of all real-valued Ft -measurable r.v.’s X , such that E[|X|]<+∞ , 94 Books MDPI Risks 2016,4,49 where two r.v.’s are identified if they coincide a.s. We consider an equivalence class X∈L1(Ω , Ft , P) as a r.v., and we understand X as a financial amount at date t of date t money. We also identify L1(Ω,F0,P)with R. 3. Conditional and Dynamic Risk Measures We consider dynamic risk measures in order to determine the solvency capital of a pension liability. The solvency capital is the amount the company has to put aside, in addition to the initial value of its portfolio, in order to be solvent. The idea behind dynamic risk measures is to consider the amount of information available through time. In our setting, this amount of information is modeled by the filtration F. Let t∈T . We consider a reference instrument ιt,T∈L1(Ω , FT , P) , as introduced in [ 1 ] in the static case. The amount ιt,T is the value at time T of one unit of currency invested in this reference instrument between time t and T . The solvency capital computed at date t is then invested in this financial instrument ιt,T from date t to the maturity T . We assume that ιt,T(ω)> 0 for all ω∈Ω . This reference instrument could be a risk-free zero-coupon bond, and the value at time Twould be: ιt,T=1 P(t,T), where P(t , T) is the price of a risk-free zero-coupon bond at time t with maturity T and paying one unit of currency. We first recall the definition of a conditional risk measure. In the following, we understand X∈L1(Ω , FT , P) as a profit r.v., which means that if X(ω)< 0, then the company faces a loss at maturity Tunder scenario ω∈Ω; otherwise, it makes a profit. Definition 1. A conditional risk measure ρton L1(Ω,FT,P)with a reference instrument ιt,Tis a function: ρt:L 1(Ω,FT,P)−→ L1(Ω,Ft,P), such that it satisfies the following properties, for X,Y∈L1(Ω,FT,P), •(monotonicity) X ≤Y implies ρt(X)≥ρt(Y); •(conditional cash invariance) for all mt∈L1(Ω,Ft,P), ρt(X+mtιt,T)=ρt(X)−mt; •(normalization) ρt(0)=0. We can also define convex and coherent conditional risk measures as in the classic static setting. Definition 2. A conditional risk measure ρt on L1(Ω , FT , P) is called convex if for X , Y∈L1(Ω , FT , P) and λ∈L1(Ω,Ft,P),0≤λ≤1, ρt(λX+(1−λ)Y)≤λρt(X)+(1−λ)ρt(Y). Furthermore, ρt is called coherent if it is convex and, for X∈L1(Ω , FT , P) and λ∈L1(Ω , Ft , P) , λ≥ 0, ρt(λX)=λρt(X). A dynamic risk measure is simply a sequence of conditional risk measures. Definition 3. A dynamic risk measure: ρ=(ρt)t∈T 95 Books MDPI Risks 2016,4,49 on L1(Ω , FT , P) is a sequence of conditional risk measures ρt on L1(Ω , FT , P) (with the reference instrument ιt,T), for t ∈T. It is called convex (resp. coherent)ifρtis convex (resp. coherent) for all t ∈T. Remark 1. We refer the reader to [3–6] for a detailed study of this kind of measure. For X∈L1(Ω , FT , P) seen as a final net worth at date T of a life insurer or pension fund and t∈T , the amount ρt(X) (which is an r.v.) can be seen as a solvency capital at date t of date t money for the final net worth X, which is invested in the reference instrument ιt,Tover the period [t,T], i.e., ρt(X+ρt(X)ιt,T)=0, by the conditional cash invariance property, where we recall that this equality holds in the P -a.s. sense. 4. Time Consistency and Iterated Risk Measures A time-consistent risk measure tells us that if we prefer a position tomorrow, in almost all states of nature, then we already prefer it today (see [4,5], for instance). Definition 4. A dynamic risk measure ρon L1(Ω,FT,P)is called time consistent if: ρt+1(X)≤ρt+1(Y), implies: ρt(X)≤ρt(Y), for all t ∈T,t=T−1and X,Y∈L1(Ω,FT,P). Example 1. In order to understand why this property is important, in particular in the field of a pension liability, we consider two r.v.’s X and Y described in Figure 1. While in Section 2, we consider an atomless probability space, we set here for the purpose of the example a discrete probability space (Ω,F,P). Ω {ω3,ω4} 5−5ω4 0.99 −150 25 ω3 0.01 0.99 {ω1,ω2} −10 −50 ω2 0.99 −250 −500 ω1 XYΩ 0.01 0.01 Figure 1. Distributions of Xand Y(Example 1). The set of states of nature Ωis given by: Ω={ω1,ω2,ω3,ω4}, with the σ-algebra Fgiven by: F=P(Ω), 96 Books MDPI Risks 2016,4,49 and a probability measure Pgiven by: P[{ω1}]=0.01 ×0.01 =0.0001 P[{ω2}]=0.01 ×0.99 =0.0099 P[{ω3}]=0.99 ×0.01 =0.0099 P[{ω4}]=0.99 ×0.99 =0.9801. We consider two periods, i.e., we set T =2. We then define a filtration F=(Ft)t∈{0,1,2}with: F0={∅,Ω} F1=σ({ω1,ω2},{ω3,ω4}) F2=F. This filtration means that at time t= 1, we know if we are at node {ω1 , ω2} or {ω3 , ω4} , i.e., at time t=1, we know that two scenarios are possible, either ω1and ω2or ω3and ω4. We also compute that: P[{ω1}|{ω1,ω2}]=0.01 P[{ω2}|{ω1,ω2}]=0.99 P[{ω3}|{ω3,ω4}]=0.01 P[{ω4}|{ω3,ω4}]=0.99. We see X and Y as two possible outcomes (strategies) for our life insurer or pension fund at time T= 2. For instance, if we follow the strategy of X , we see that at time t= 1, if we are at node {ω1 , ω2} , the probability of facing a loss of 250 is 1%. We emphasize that when X or Y are positive, the company faces a profit, while it faces a loss when X or Y are negative. That is why the amount of the loss is 250, while X takes as a value − 250. We assume here that ι0,2 =ι1,2 =1, so that we neglect the time value of money for this example. We consider the VaR measure with a level of 99% in order to make a choice between X and Y . We first compute the VaR measure with a confidence level of 99% at time t =0(see [1,3]), VaR0.99 0(X)=−inf {x∈R:P[X≤x]>0.01} =10 , and: VaR0.99 0(Y)=5, which means that at time t =0, we prefer Y over X, because: VaR0.99 0(X)>VaR0.99 0(Y). However, at time t =1, we compute that, if we are at node {ω1,ω2}, VaR0.99 1(X){ω1,ω2}=−inf {x∈R:P[X≤x|{ω1,ω2}]>0.01} =10 , and: VaR0.99 1(Y){ω1,ω2}=50 ; 97 Books MDPI Risks 2016,4,49 for all t∈[0, T]. As the term structure is flat and known with certainty, the price at time t∈[ 0, T] of a zero-coupon bond paying one unit of currency at maturity s∈[0, T],t≤s, is given by: P(t,s)=e−r(s−t). 6.3. Final Net Worth The final net worth (at maturity T ) of the pension liability is simply given by the difference between its final assets and liabilities, i.e., given by the r.v.: FNWT=AT−LT. For ω∈Ω, the company faces a loss at maturity under scenario ωif: FNWT(ω)<0, otherwise, it is solvent. 6.4. Solvency Capital We define the solvency capital as the amount the company has to put aside, in addition to the initial value of its portfolio, in order to be solvent at maturity according to a particular measure. This amount at time t∈Tis invested in a product ιt,Tover the period [t,T]. We consider here: ιt,T=1 P(t,T), which means that the solvency capital at time t is invested in a zero-coupon bond between dates t and T . This choice for the reference instrument can be considered as prudent. For instance, we could have decided to invest the solvency capital in the portfolio of assets, i.e., ιt,T=AT At, which is clearly riskier. This approach has been considered in [ 12 ] where it is observed that the solvency capital computed according to this investment strategy is higher than the prudent approach of a risk-free zero-coupon bond. The solvency capital given by the iterated VaR and TVaR measures (see Definitions 8 and 11) is computed as follows. Proposition 3. Let α∈(0, 1)T. We compute that: IVaRα t(FNWT) =LTP(t,T)−P(t,T)Atexp θμ +(1−θ)r−θ2σ2 2(T−t)+θσ T−t ∑ i=1 Φ−1(1−αi), (2) and: ITVaRα t(FNWT)=LTP(t,T)−P(t,T)Atexp [(θμ +(1−θ)r)(T−t)] T−t ∏ i=1ΦΦ−1(1−αi)−θσ 1−αi, 104 Books MDPI Risks 2016,4,49 for t ∈T, where Φdenotes the cumulative distribution function of a standard normal r.v., i.e., for y ∈R, Φ(y)= 1 √2πy −∞e−x2 2dx, and Φ−1its inverse. Proof. See Appendix B. Remark 10. For both the iterated VaR and TVaR measures, we observe that the solvency capital is a decreasing function of μ , while it is increasing with σ . Furthermore, for the iterated VaR, the first derivative with respect to μis a decreasing function, meaning that the speed of decrease will increase with μ, while the first derivate with respect to σis an increasing function. We also consider the solvency capital given by the expected VaR and TVaR measures (see Definitions 13 and 14). Proposition 4. Let α∈(0, 1). We compute that: EVaRα t(FNWT)=LTP(t,T)−P(t,T)Atexp (θμ +(1−θ)r)(T−t)−θ2σ2 2+θσΦ−1(1−α), and: ETVaRα t(FNWT)=LTP(t,T)−P(t,T)Atexp [(θμ +(1−θ)r)(T−t)]ΦΦ−1(1−α)−θσ 1−α, for t ∈T. Proof. Similar to the proof of Proposition 3. If we compare Propositions 3 and 4, we see that the iterated VaR (resp. TVaR) considers a sequence of bad events, while the expected VaR (resp. TVaR) only considers one bad event affecting the last year of the product (the first iteration). The expected VaR (resp. TVaR) incorporates a security margin in the first iteration, for the last year of the product. However, the iterated VaR (resp. TVaR) can incorporate a security margin in each iteration, according to the confidence level considered at each step. For instance, we could consider a linearly-decreasing margin when the number of iterations increases (see examples below). We could then consider the expected VaR or TVaR as a lower bound for the solvency capital. Finally, we compare this solvency capital with that obtained by means of the recalculated risk measures. Proposition 5. Let α∈(0, 1). We compute that: VaRα 0(FNWT)=LTP(0, T)−P(0, T)A0exp θμ +(1−θ)r−θ2σ2 2T+θσ√TΦ−1(1−α), and: TVaRα 0(FNWT)=LTP(0, T)−P(0, T)A0exp [(θμ +(1−θ)r)T] ΦΦ−1(1−α)−θσ√T 1−α. Proof. We only consider one step in the proof of Proposition 3. 105 Books MDPI Risks 2016,4,49 7. Numerical Illustration In this section, we consider different values for α∈(0, 1)T and α∈( 0, 1 ) in order to compute the solvency capital. We only compare the initial values of the solvency capital, i.e., t=0. We assume that T∈{ 1, ... ,45 } , θ= 40%, so that 40% of the portfolio is invested in the stock, that the guaranteed interest rate is equal to the risk free rate, i.e., rG=r , and that π0= 1000 e . Concerning the financial market, the stock has been calibrated on daily log returns of the Belgian BEL20 index from 2 April 1991 to 31 December 2013 by means of the maximum likelihood estimation (MLE) method (see Table 1). We also set the constant interest rate to 1.5%. Table 1. Parameters of the GBMobtained by the MLE method with the corresponding standard errors between brackets. μσ 0.05564 (0.03827) 0.18415 (0.00172) In order to be consistent with either a Solvency II or a Swiss Solvency Test approach, we will consider: α(SII) =0.995 , when working with VaR measures and: α(SST) =0.99 , for TVaR measures. We first define the confidence levels in the case of the recalculated measures of Proposition 5. We follow the maturity approach as introduced in [10] and set: α1=α(SII)T, and: α2=α(SST)T. We also consider four definitions for the vector α∈(0, 1)T for the VaR and TVaR cases. We write for the j-th definition, j∈{1, ···,4}, α(SII),j=α(SII),j ii∈{1,...,T}, when dealing with VaR measures and: α(SST),j=α(SST),j ii∈{1,...,T}, for TVaR measures. We give an illustration of these definitions in Figure 4 for the SII case. The first one is the constant case, which is also studied in [12], α(SII),1 i=α(SII) , and: α(SST),1 i=α(SST) , for i∈{1, . . . , T}. For the three others definitions, we consider an integer K> 0. We see this parameter as the maturity after which we consider that a product is a long-term product, for instance K=8 years. 106 Books MDPI Risks 2016,4,49 t αT−t 115 30 45 0.5 0.75 0.995 α(SII),1 α(SII),2 α(SII),3 α(SII),4 Figure 4. Different choices for the confidence vector, for a maturity T= 45 and a long-term threshold K=8. The second definition follows a linear increase when the time to maturity is decreasing, i.e., when idecreases, α(SII),2 i=α(SII) −α(SII)−0.5 K−1(i−1)if i<K 0.5 if i≥K, and: α(SST),2 i=α(SST) −α(SST)−1 K−1(i−1)if i<K 0ifi≥K, for i∈{ 1, ... , T} . We see that the confidence parameter decreases linearly towards either 50% or 0% within K years and afterwards stays constant. We chose 50% for the measures based on the VaR in order to obtain either the mean of the solvency capital if its distribution is symmetric or its median if not. With 0% for the measures based on the TVaR, we obtain directly the mean of the solvency capital. These choices are linked with the EVaR and ETVaR measures where the decrease is instantaneous ( K= 1). In other words, for a time to maturity greater than K years, the parameter stays constant while it starts increasing towards either 99.5% or 99% when the time to maturity decreases. Now, instead of considering a linear increase in terms of time to maturity, we could consider a square or an exponential increase, which is the aim of the two others definitions. We then have , for the square increase, α(SII),3 i=⎧ ⎪ ⎨ ⎪ ⎩ 1−0.5−α(SII) (K−1)2i2−2Ki +K2+0.5 (K−1)2 0.5−α(SII) if i<K 0.5 if i≥K , α(SST),3 i=⎧ ⎪ ⎨ ⎪ ⎩ 1+α(SST) (K−1)2i2−2Ki +K2−100%(K−1)2 α(SST) if i<K 0ifi≥K , and for the exponential increase, α(SII),4 i=⎧ ⎨ ⎩ 0.5 +α(SII)−0.50.5−s(SII) α(SII)−0.5 i−1 K−1if i<K 0.5 if i≥K , α(SST),4 i=⎧ ⎨ ⎩ α(SST)1−s(SST) α(SST) i−1 K−1if i<K 0ifi≥K , 107 Books MDPI Risks 2016,4,49 for t∈T , where 0 <s(SII) <α(SII) and 0 <s(SST) <α(SST) are some adjustment parameter for the smoothness around K, typically s(SII) close to α(SII) and s(SST) close to α(SST). We now study the level of the initial solvency capital according to the previous definitions of the confidence vectors. In Figure 5, we consider the solvency capital at time t= 0 given by Propositions 3, 4 and 5, for T∈{ 1, ... ,45 } and K= 8. We only consider the positive part. We observe that for the constant case, i.e., α(SII),1 and α(SST),1 , the level of the solvency capital significantly increases with the maturity of the product. As already mentioned, it has been observed in [ 11 , 12 ]. The level of the solvency capital converges towards 100% of the initial value of the portfolio, i.e., towards the present value of the liability, which is here its supremum. T SC / A0 115 30 45 0 0.5 1 IVaR0with α(SII),1 IVaR0with α(SII),2 IVaR0with α(SII),3 IVaR0with α(SII),4 EVaR0with α(SII) VaR0with α1 Figure 5. Computation of the solvency capital in the proportion of the initial value of the portfolio, for maturities T∈{1, . . . , 45}and a long-term threshold K=8. However, for the three others definitions of the confidence level, we remark that the solvency capital level increases for short maturities, while it decreases for long maturities. This decrease in the level is higher when the decrease of the confidence level increases as well, as expected. For instance, the solvency capital under the exponential function α(SII),4 declines more quickly than the solvency capital under the square function α(SII),3 . We also observe a similar increasing and decreasing shape with the recalculated VaR and TVaR measures. This shape is very similar to the one given by the maturity approach of [ 10 ]. However, the main advantage is that the measure that we consider now is time consistent. Concerning the EVaR measures, we see that we could consider it as the lower bound for the solvency capital level. This solvency capital is simply the expectation of the solvency capital needed for the last year of the product. It is the best estimate of the capital needed for the last year and does not include any security margin, except for the last year. The difficulty we face now is the determination of a reasonable or relevant confidence level function. We could obtain any shape of the curve of the solvency capital with a particular choice of the confidence function. A way of solving this would be to consider an easy understandable function, such as the linear increase, e.g., starting K years before the maturity, we increase the confidence level from a fixed amount each year towards a classic level, such as the Solvency II or the Swiss Solvency Test levels. Finally, according to Remark 10, we also observe in Figure 6 that the solvency capital increases when μdecreases and σincreases, while it decreases when μincreases and σdecreases. 108 Books MDPI Risks 2016,4,49 T SC / A0 115 30 45 0 0.1 0.2 0.3 0.4 μand σ μ×1.1 and σ×0 μ×0.9 and σ×1 Figure 6. Computation of the solvency capital in the proportion of the initial value of the portfolio according to the iterated VaR measure with α(SII),2 and with different values for μ and σ , for maturities T∈{1,...,45}and a long-term threshold K=8. 8. Conclusions In this paper, we presented different constructions of time-consistent dynamic risk measures in order to determine the solvency capital of a life insurer or pension fund by taking into account the information disclosed through time. These dynamic risk measures are built such that they are coherent with the Solvency II or Swiss Solvency Test regulatory frameworks in the last year of the product. We saw that according to the choice of the conditional risk measures considered for the iteration scheme and, in particular, according to the choice of the confidence levels for each iteration, different shapes for the solvency capital can be obtained. Several confidence levels that seem convenient have been proposed in order to benefit from the long-term characteristic of these products. We also introduced a time-consistent measure where the first iteration is a VaR or TVaR measure, and the subsequent iterations make use of the conditional expectation. This measure is a kind of best estimate of the capital needed for the last year of the product and is particularly intuitive and elegant. Finally, an important step forward will be to consider the case of multiple cash flows and to include more risks, such as the interest rate and mortality risks. Acknowledgments: We would like to thank the referees of an earlier version of this paper for useful remarks and suggestions. We also acknowledge the financial support from the Pensionvaluation and solvency AGInsurance Chair. Author Contributions: Both authors have contributed equally to this paper. Conflicts of Interest: The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript; nor in the decision to publish the results. Appendix A. Link between Time-Consistent and Iterated Measures We recall here the equivalence between time-consistent and iterated risk measures (see [4]). Theorem A1. Let ρbe a dynamic risk measure on L1(Ω,FT,P). The following conditions are equivalent: (a) ρis time consistent; (b) if: ρt+1(X)=ρt+1(Y), then: ρt(X)=ρt(Y), for all t ∈T,t=T−1and X,Y∈L1(Ω,FT,P); (c) ρis iterated. 109 Books MDPI Risks 2016,4,49 Proof. We first prove (a) implies (b). Let t∈T,t=T−1 and X,Y∈L1(Ω,FT,P). If: ρt+1(X)=ρt+1(Y), then by time consistency, we have either: ρt(X)≤ρt(Y), or: ρt(X)≥ρt(Y), and the first conclusion holds. We now prove (b) implies (c). Let t∈T , t=T− 1 and X∈L1(Ω , FT , P) . By conditional cash invariance and normalization of the conditional risk measure, we compute that: ρt+1(−ρt+1(X)ιt+1,T)=ρt+1(X), and by (b) with: Y=−ρt+1(X)ιt+1,T, ρis iterated. Finally, we see that (c) implies (a). Let t∈T,t=T−1 and X,Y∈L1(Ω,FT,P). If: ρt+1(X)≤ρt+1(Y), then by the assumption of recursiveness and the monotonicity of the conditional risk measure, we have: ρt(X)=ρt(−ρt+1(X)ιt+1,T) ≤ρt(−ρt+1(Y)ιt+1,T) =ρt(Y), and the proof is complete. Appendix B. Proof of Proposition 3 The demonstration requires the following lemma; see ([ 17 ] [Lemma A.108]), for a proof of this result. Lemma B1. Let X , Y∈L1(Ω , FT , P) and G⊆F T be a σ -algebra, such that X is independent of G and Y is G-measurable. Then, for every B(R2)-measurable bounded (or non-negative) function: h:R2−→ R, we have that: E[h(X,Y)|G]=g(Y), where B(R2)is the Borel σ-algebra on R2and: g(y)=E[h(X,y)], for y ∈R. 110 Books MDPI Risks 2016,4,49 We only consider the case of the IVaR measure, and we start the proof with t=T− 1. We find that: IVaRα T−1(FNWT)=VaRα1 T−1(AT−LT) =VaRα1 T−1(AT)+LTP(T−1, T), by conditional cash invariance. Let V∈L1(Ω,FT−1,P). We compute that, by Equation (1), P[ATP(T−1, T)≤V|FT−1] =PAT−1 AT AT−1 P(T−1, T)≤VFT−1 =PAT−1exp θμ +(1−θ)r−θ2σ2 2+θσ(WT−WT−1)P(T−1, T)≤VFT−1 =PZ≤1 θσ ln V−ln AT−1−ln P(T−1, T)−θμ +(1−θ)r−θ2σ2 2FT−1, with Z a standard normal r.v. independent of FT−1 and the r.v. V must satisfy the condition V> 0. According to Lemma B1 with X=Z, Y=1 θσ ln V−ln AT−1−ln P(T−1, T)−θμ +(1−θ)r−θ2σ2 2, and: h(x,y)=χ{x≤y}, for x,y∈R, where χstands for the indicator function, we find that: P[ATP(T−1, T)≤V|FT−1] =Φ1 θσ ln V−ln AT−1−ln P(T−1, T)−θμ +(1−θ)r−θ2σ2 2. Due to the continuous and strictly increasing properties of the cumulative distribution function Φ , it is invertible and: P[ATP(T−1, T)≤V|FT−1]>α1, is equivalent to: 1 θσ ln V−ln AT−1−ln P(T−1, T)−θμ +(1−θ)r−θ2σ2 2>Φ−1(α1). We find that V∈L1(Ω,FT−1,P)must also satisfy: V>P(T−1, T)AT−1exp θμ +(1−θ)r−θ2σ2 2+θσΦ−1(α1), and the case t=T−1 follows. 111 Books MDPI Risks 2016,4,49 We assume now that the result is true for t∈T , t= 0, and we show that it is also true for t− 1. We have that, by Equation (2), IVaRα t−1(FNWT) =VaRαT−t+1 t−1−IVaRα t(FNWT)1 P(t,T) =VaRαT−t+1 t−1Atexp θμ +(1−θ)r−θ2σ2 2(T−t)+θσ T−t ∑ i=1 Φ−1(αi)+LTP(t−1, T). Let V∈L1(Ω,Ft−1,P). We then find that, PAtexp θμ +(1−θ)r−θ2σ2 2(T−t)+θσ T−t ∑ i=1 Φ−1(αi)P(t−1, T)≤VFt−1 =PZ≤1 θσ ln V−ln At−1−ln P(t−1, T)−θμ +(1−θ)r−θ2σ2 2(T−(t−1)) −θσ T−t ∑ i=1 Φ−1(αi)Ft−1, where again Z is a standard normal r.v. independent of Ft−1 , and the r.v. V∈L1(Ω , Ft−1 , P) must satisfy the condition V> 0. According to Lemma B1 and the properties of the cumulative distribution function Φ, the first result follows from the same reasoning as the first part of the proof. The second part of the result follows from the same reasoning adapted to the TVaR measure. References 1. Artzner, P.; Delbaen, F.; Eber, J.M.; Heath, D. Coherent measures of risk. Math. Financ. 1999,9, 203–228. 2. Frittelli, M.; Rosazza Gianin, E. Putting order in risk measures. J. Bank. Financ. 2002,26, 1473–1486. 3. Föllmer, H.; Schied, A. Stochastic Finance: An Introduction in Discrete Time, 3rd ed.; De Gruyter Graduate; Walter de Gruyter: Berlin, Germany, 2011. 4. Acciaio, B.; Penner, I. Dynamic Risk Measures. In Advanced Mathematical Methods for Finance; Di Nunno, G., Øksendal, B., Eds.; Springer: Berlin/Heidelberg, Germany, 2011; Chapter 1, pp. 1–34. 5. Detlefsen, K.; Scandolo, G. Conditional and dynamic convex risk measures. Financ. Stoch. 2005 ,9, 539–561. 6. Pflug, G.C.; Römisch, W. Modeling, Measuring and Managing Risk; World Scientific Publishing Co. Pte. Ltd.: Toh Tuck Link, Singapore, 2007. 7. Artzner, P.; Delbaen, F.; Eber, J.M.; Heath, D.; Ku, H. Coherent multiperiod risk adjusted values and Bellman’s principle. Ann. Oper. Res. 2007,152, 5–22. 8. Cheridito, P.; Stadje, M. Time-inconsistency of VaR and time-consistent alternatives. Financ. Res. Lett. 2009 , 6, 40–46. 9. Cheridito, P.; Kupper, M. Composition of time-consistent dynamic monetary risk measures in discrete time. Int. J. Theor. Appl. Financ. 2011,14, 137–162. 10. Devolder, P. Revised version of: Solvency requirement for a long-term guarantee: Risk measures versus probability of ruin. Eur. Actuar. J. 2011,1, 199–214. 11. Hardy, M.R.; Wirch, J.L. The iterated CTE: A dynamic risk measure. N. Am. Actuar. J. 2004,8, 62–75. 12. Devolder, P.; Lebègue, A. Iterated VaR or CTE measures: A false good idea? Scand. Actuar. J. 2016 , 1–32, doi:10.1080/03461238.2015.1126343. 13. Kupper, M.; Schachermayer, W. Representation results for law invariant time consistent functions. Math. Financ. Econ. 2009,2, 189–210. 14. Devolder, P.; Lebègue, A. Risk measures versus ruin theory for the calculation of solvency capital for long-term life insurances. Depend. Model. 2016, in press. 112 Books MDPI Risks 2016,4,49 15. Black, F.; Scholes, M.S. The pricing of options and corporate liabilities. J. Political Econ. 1973,81, 637–654. 16. Merton, R.C. Theory of rational option pricing. Bell J. Econ. Manag. Sci. 1973,4, 141–183. 17. Pascucci, A. PDE and Martingale Methods in Option Pricing; Bocconi & Springer Series; Springer: Milan, Italy, 2011; Volume 2. c 2016 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 113 Books MDPI Risks 2017,5,3 and * τu,T=∑n3 i=1τu,T,i n3 . (15) For each of the n3paths of Type (c) we have τu,T,i≤τR u,T,i, implying of course that * τu,T≤* τR u,T. For the simulations in the next section we need to make choices for the parameters T , θ , μ , λ and u . We discuss these choices in more detail in Section 5.2, but for the present purposes, we set them as follows: expectation of claims distributions μ = 2; claim arrival rate is λ= 1; safety loading θ= 0.1. Initial reserve takes values u= 10, 30, 50, 70, 100 and time spans are T= 100, 500, 1000. Each time unit is 0.01 year = 3.65 days. 4. Results In this section, we report on simulations for the classical compound Poisson risk model in which the claim surplus process takes the form specified in (1) and the reinsured process is as in (3) . We inspected the impact of EOL and LCR reinsurances in the three different tail regimes by varying the claim size distributions. In all examples, we chose the claim arrival rate as λ= 1 and the safety loading as θ= 0.1. For a variety of combinations of initial capital u and follow-up time T , we recorded the estimated original and the reinsured ruin probabilities, and the estimated ruin times. 4.1. Largest Claim Reinsurance For the case of LCR we denote the claims surplus process in (3) by CM and the ruin time in (13) by τM u . We chose a Pareto ( 1, 2 ) distribution for the simulations in the heavy-tailed case. This choice of parameters parallels that of [ 12 ], who calculated the ultimate ruin probabilities for these particular Cramér-Lundberg risk models. So we can benchmark our results against theirs to check on the accuracy of our simulations. The results are summarised in Table 1. Comparing Columns 3 and 4 in Table 1, we see that the estimated ruin probability * P(τu<T) drops substantially to * P(τM u<T) after reinsurance. Correspondingly, significant increases in the expected conditional lifetime of the company with reinsurance are observed (compare Columns 5 and 6). Column 7 gives the percentage change in the conditional ruin times due to reinsurance. As expected, the effect tends to diminish when u is increased, but remains substantial even for u= 100. The probabilities in Columns 3 and 4 of Table 1, and in similar tables below, are estimated correct to 2 decimal places (standard error less than 10 −2 ). Numbers in the T=∞rows in Table 1 are calculated from Algorithm III in [12]. We next investigate the impact of reinsurance on the Cramér-Lundberg model with light or medium-heavy tailed claim distributions. The specific examples chosen are Gamma ( 2, 1 ) (light tailed) and IG ( 2, 1.5 ) (medium-heavy tailed). For consistency, we chose the expectations of the claims distributions to be μ = 2 (the same as in the Pareto case), and all other parameters (claim arrival rate λ=1, safety loading θ=0.1, initial reserves uand time spans T) also the same. 120 Books MDPI Risks 2017,5,3 Table 1. LCR reinsurance for Pareto ( 1, 2 ) distributed claims. The safety loading is θ= 0.1, expected claim size is μ = 2, claim arrival rate is λ= 1, and each time unit is 0.01 years. Simulations are done with N= 100, 000 sample paths. The T=∞ case refers to the results obtained from Algorithm III in [12]. uT* P(τu<T)* P(τM u<T)* τu,T* τM u,T% Changes 10 100 0.43 0.14 19.06 37.03 93.34 500 0.53 0.20 38.14 85.45 124.02 1000 0.55 0.21 44.75 104.35 133.19 ∞0.56 ±0.03 - - - - 30 100 0.14 0.02 35.78 58.97 64.84 500 0.26 0.06 90.65 164.25 81.19 1000 0.28 0.06 113.18 214.20 89.25 ∞0.32 ±0.02 - - - - 50 100 0.06 0.00 44.64 66.40 55.74 500 0.14 0.02 129.20 215.24 66.59 1000 0.17 0.03 172.73 303.81 75.89 ∞0.20 ±0.02 - - - - 70 100 0.03 0.00 45.66 73.67 61.37 500 0.09 0.01 157.07 263.60 67.82 1000 0.11 0.01 221.55 380.55 71.77 ∞0.14 ±0.02 - - - - 100 100 0.01 0.00 37.32 75.64 102.71 500 0.05 0.00 180.25 300.30 66.60 1000 0.06 0.00 258.22 450.58 74.50 ∞0.081 ±0.017 - - - - Graphical illustrations are in Figures 3 and 4. Relatively smaller claim sizes occur in these two cases (compare the vertical scales of these two plots with that of Figure 1), and as a result the impact of reinsurance is not as dramatic as it is for the heavy-tailed cases. A similar conclusion can be drawn from the numerical results in Tables 2 and 3. In both the Gamma and Inverse Gaussian cases, improvements in ruin probabilities after reinsurance are significant, especially for u small, but proportionally not as substantial as for the Pareto. Figure 3. Sample paths of the insurance risk processes without reinsurance (black line), with LCR reinsurance (red line), and with excess of loss (EOL) reinsurance (green line), for a Gamma ( 2, 1 ) claim distribution. The safety loading is θ= 0.1, expected claim size is μ = 2, claim arrival rate is λ= 1, and each time unit is 0.01 years. L for the EOL scheme is chosen so that the expected values of the LCR and EOL claim distributions are equal at maturity time T=5000. 121 Books MDPI Risks 2017,5,3 Figure 4. Sample paths of the insurance risk processes without reinsurance (black line), with LCR reinsurance (red line), and with EOL reinsurance (green line), for an IG ( 2, 1.5 ) claim distribution. The safety loading is θ= 0.1, expected claim size is μ = 2, claim arrival rate is λ= 1, and each time unit is 0.01 years. L for the EOL scheme is chosen so that the expected values of the LCR and EOL claim distributions are equal at maturity time T=5000. Table 2. LCR reinsurance for Gamma ( 2, 1 ) distributed claims. The safety loading is θ= 0.1, expected claim size is μ = 2, claim arrival rate is λ= 1, and each time unit is 0.01 years. Simulations are done with N=100, 000 sample paths. uT * P(τu<T)* P(τM u<T)* τu,T* τM u,T% Changes 10 100 0.43 0.25 22.22 34.91 57.11 500 0.49 0.32 44.35 70.81 61.18 1000 0.50 0.32 47.79 77.57 63.39 30 100 0.08 0.04 47.95 59.49 24.08 500 0.14 0.08 115.46 145.61 26.11 1000 0.15 0.09 135.16 170.45 26.11 50 100 0.01 0.00 62.13 72.35 16.45 500 0.04 0.02 177.90 208.29 17.08 1000 0.04 0.02 214.04 254.08 18.71 70 100 0.00 0.00 68.07 79.12 16.24 500 0.01 0.01 233.81 264.30 13.04 1000 0.01 0.01 292.09 327.81 12.23 100 100 0.00 0.00 - - - 500 0.00 0.00 315.52 347.89 10.26 1000 0.00 0.00 393.28 438.94 11.61 122 Books MDPI Risks 2017,5,3 Table 3. LCR reinsurance for IG ( 2, 1.5 ) distributed claims. The safety loading is θ= 0.1, expected claim size is μ = 2, claim arrival rate is λ= 1, and each time unit is 0.01 years. Simulations are done with N=100, 000 sample paths. uT * P(τu<T)* P(τM u<T)* τu,T* τM u,T% Changes 10 100 0.51 0.24 18.63 35.88 92.61 500 0.59 0.33 40.69 83.27 104.67 1000 0.60 0.34 47.97 100.35 109.18 30 100 0.16 0.05 38.97 56.63 45.33 500 0.27 0.13 101.28 153.44 51.51 1000 0.28 0.14 122.80 189.76 54.53 50 100 0.04 0.01 52.30 68.41 30.81 500 0.12 0.05 150.57 205.16 36.25 1000 0.13 0.06 197.59 272.25 37.79 70 100 0.01 0.00 61.72 78.22 26.73 500 0.05 0.02 197.71 253.26 28.10 1000 0.06 0.03 261.72 339.91 29.88 100 100 0.00 0.00 54.87 64.62 17.77 500 0.01 0.00 250.91 303.45 20.94 1000 0.02 0.01 368.16 455.16 23.63 4.2. Excess of Loss Reinsurance In this section we examine the EOL reinsurance scheme. We denote the corresponding claims surplus process by CL and the ruin time by τL u . Under this treaty, the reinsurer pays the total amount of any claim in excess of some pre-determined retention level L . For the results in the present section, in order to afford some degree of comparability with the LCR scheme, we chose Lsuch that E(CL T)=E(CM T), For any t>0 we have E(CM t)=E(Ct)−Emax 1≤i≤Nt ξi, and E(CL t)=E(Ct)−ENt ∑ i=1 (ξi−L)1{ξi>L}, so for comparability we need to solve the equation Emax 1≤i≤Nt ξi=ENt ∑ i=1 (ξi−L)1{ξi>L}(16) for L=L(t). The left-hand side of (16) is equal to Emax 1≤i≤Nt ξi= ∞ ∑ n=0 Emax 1≤i≤nξie−λt(λt)n n! = ∞ ∑ n=0∞ 0Pmax 1≤i≤nξi>xdx×e−λt(λt)n n! =∞ 0dx ∞ ∑ n=0 (1−Fn(x)) ×e−λt(λt)n n! =∞ 0(1−e−λtF(x))dx, 123 Books MDPI Risks 2017,5,3 where F(x)=1−F(x)is the tail of the distribution of the ξi. The right-hand side of (16) is equal to ENt ∑ i=1 (ξi−L)1{ξi>L}= ∞ ∑ n=0 e−λt(λt)n n!nE((ξ1−L)1{ξ1>L})=λt∞ L(x−L)dF(x). Choosing t=Tand λ=1, Lis required to solve ∞ L(x−L)dF(x)=∞ LF(x)dx=1 T∞ 0(1−e−TF(x))dx. (17) This is easily done in the R package, which we used for the simulations also. Once having selected L in this way, we used the same approach as before to estimate ruin probabilities and ruin times. The results are displayed in Tables 4–6. In these tables we abuse notation slightly and continue to use * τu,T as the estimated conditional ruin time for the plain risk process, noting, however, that in the present case the conditioning is on the event τL u≤Tand not on τM u≤Tas in Tables 1–3. This is the reason for the differing values of * τu,Tin Tables 4–6 as opposed to Tables 1–3. Tables 4–6 contain an extra column “No Effect” as compared to Tables 1–3. The extra column records the proportion of paths for which ruin occurs but the ruin times are the same for the original sample path Ct as for the reinsured path CL t . In these cases the reinsurance scheme does not avoid ruin. There are two ways in which this can happen. One is that ruin occurs but reinsurance is not invoked at all; that is, there was no claim larger than L before ruin. The second scenario is that even though reinsurance was invoked at some time or times before ruin, nevertheless the jump causing ruin has magnitude less than L. There is no saving effect from the EOL scheme in these cases. Table 4. EOL reinsurance for Pareto ( 2, 1 ) distributed claims. The safety loading is θ= 0.1, expected claim size is μ = 2, claim arrival rate is λ= 1, and each time unit is 0.01 years. Simulations are done with N= 100, 000 sample paths. Retention level L is the solution to (17) .For T= 100, L(T)=5.64; for T=500, L(T)=12.62; for T=1000, L(T)=17.84. uT * P(τu<T)* P(τL u<T)* τu,T* τL u,T% Changes No Effect 10 100 0.43 0.20 15.88 21.36 34.55 0.08 500 0.53 0.39 31.50 36.55 16.03 0.30 1000 0.55 0.44 38.93 43.54 11.84 0.38 30 100 0.14 0.01 33.48 49.71 48.47 0.00 500 0.26 0.08 71.37 92.83 30.08 0.03 1000 0.28 0.12 87.85 106.86 21.63 0.07 50 100 0.06 0.00 45.22 67.26 48.76 0.00 500 0.14 0.02 107.64 145.85 35.50 0.00 1000 0.17 0.03 134.88 172.85 28.15 0.01 70 100 0.03 0.00 70.52 79.83 13.20 0.00 500 0.09 0.00 144.90 195.31 34.79 0.00 1000 0.11 0.01 182.04 243.62 33.82 0.00 100 100 0.01 0.00 - - - 0.00 500 0.05 0.00 181.23 281.15 55.13 0.00 1000 0.06 0.00 235.18 318.21 35.31 0.00 124 Books MDPI Risks 2017,5,3 Table 5. EOL reinsurance for Gamma ( 2, 1 ) distributed claims. The safety loading is θ= 0.1, expected claim size is μ = 2, claim arrival rate is λ= 1, and each time unit is 0.01 years. Simulations are done with N= 100, 000 sample paths. Retention level L is the solution to (17) .For T= 100, L(T)=4.49; for T=500, L(T)=6.10; for T=1000, L(T)=6.79. uT * P(τu<T)* P(τL u<T)* τu,T* τL u,T% Changes No Effect 10 100 0.43 0.32 20.02 24.34 21.55 0.13 500 0.49 0.45 40.67 44.52 9.45 0.32 1000 0.50 0.47 45.80 48.49 5.85 0.38 30 100 0.08 0.03 44.45 53.24 19.77 0.00 500 0.14 0.11 106.87 116.82 9.31 0.04 1000 0.15 0.13 127.08 135.13 6.33 0.07 50 100 0.01 0.00 57.85 68.12 17.74 0.00 500 0.04 0.03 165.39 181.42 9.69 0.01 1000 0.04 0.03 200.86 214.19 6.64 0.01 70 100 0.00 0.00 66.66 80.33 20.52 0.00 500 0.01 0.01 219.09 239.44 9.29 0.00 1000 0.01 0.01 279.28 294.17 5.33 0.00 100 100 0.00 0.00 - - - 0.00 500 0.00 0.00 306.48 334.20 9.04 0.00 1000 0.00 0.00 385.12 410.96 6.71 0.00 Table 6. EOL reinsurance for IG ( 2, 1.5 ) distributed claims. The safety loading is θ= 0.1, expected claim size is μ = 2, claim arrival rate is λ= 1, and each time unit is 0.01 years. Simulations are done with N= 100, 000 sample paths. Retention level L is the solution to (17) . For T= 100, L(T)= 6.89; for T=500, L(T)=11.27; for T=1000, L(T)=13.39. uT * P(τu<T)* P(τL u<T)* τu,T* τL u,T% Changes No Effect 10 100 0.51 0.33 15.99 21.11 32.05 0.15 500 0.59 0.52 35.92 41.51 15.55 0.39 1000 0.60 0.56 44.17 48.64 10.10 0.47 30 100 0.16 0.04 35.16 47.97 36.44 0.00 500 0.27 0.18 89.10 106.78 19.85 0.07 1000 0.28 0.22 109.79 124.45 13.35 0.12 50 100 0.04 0.00 46.48 63.63 36.89 0.00 500 0.12 0.06 133.06 159.35 19.76 0.01 1000 0.13 0.09 175.11 199.22 13.77 0.03 70 100 0.01 0.00 55.19 71.45 29.44 0.00 500 0.05 0.02 177.18 211.36 19.29 0.00 1000 0.06 0.03 233.06 267.53 14.79 0.01 100 100 0.00 0.00 - - - 0.00 500 0.01 0.00 227.50 272.16 19.63 0.00 1000 0.02 0.00 333.10 384.93 15.56 0.00 Improvements under EOL reinsurance are more substantial when the claims have a heavier tailed distribution (the Pareto ( 1, 2 ) case) as opposed to the medium-heavy and light tailed cases, where decreases in ruin probabilities and increases in conditional ruin times are comparatively minor. Comparing the results in Tables 4–6 to those in Tables 1–3 correspondingly, we see that when u≤ 30 the LCR treaty gives larger percentage improvements in the ruin probabilities over all three tail regimes, but this superiority diminishes as u grows. The same is true of the conditional lifetimes. The EOL method appears to perform markedly better than no reinsurance only when there are heavy tailed claims, whereas the LCR treaty shows consistent improvements over all three classes of claim distributions. 125 Books MDPI Risks 2017,5,3 4.3. Comparisons Across Distributions The simulations also allow us to make interesting comparisons across distributions, that is, between the Pareto, Inverse Gaussian and Gamma distributed cases. Intuitively our initial expectation might be that heavier tailed claims distributions would tend to lead to higher ruin probabilities than lighter tailed ones. Seemingly perplexing at first, then, might be that the ruin probability with or without reinsurance is, for small reserve levels ( u≤ 30), larger for Inverse Gaussian claims than for Pareto-type claims, despite the fact that the Inverse Gaussian has much lighter tails than the power law distributions. This is true for both LCR (compare Columns 3 and 4 in Table 1 with Columns 3 and 4 in Table 3) and for EOL (compare Columns 3 and 4 in Table 4 with Columns 3 and 4 in Table 6), to varying degrees. The explanation for this is that in general ruin probabilities and are not closely correlated with “heaviness” of tails, at least for moderate values of u . Ruin can occur by the accumulation of many small or medium sized jumps as well as by occasional huge jumps. When the claim size distribution follows Pareto ( 1, 2 ) , we see in Figure 1 that most claims have relatively small sizes, roughly in the range 1 to 4. Eventually, though, as in Figure 1, a huge claim (having magnitude near 60 in the figure), will arrive. Thus, in a heavy tailed situation, the ruinous jump is very likely to be due to the largest claim. However ruin may occur by the accumulation of many smaller jumps. In Figure 4, for the Inverse Gaussian, we see this effect; there are many small and moderate sized claims which can accumulate to give ruin. The effect tends to be more noticeable when the initial reserve is small. Figure 5 plots the tails of the three distributions used in the simulations. The tail of the Pareto ( 1, 2 ) is undoubtedly much bigger than for the other two distributions (not obvious in this figure, but apparent if the x –axis is extended further to the right). Correspondingly, there is less probability mass at small and medium sized claims than for the Gamma and Inverse Gaussian. The Gamma distribution has distinctly higher probability mass around relatively small (<5) claim sizes. In the medium size range (5–15), the Inverse Gaussian provides many substantial claims whose sum can contribute to ruin for a small reserve, more so than for a heavy tailed distribution. Figure 5. Tails of the three claim distributions involved in our simulations. The table entries for * τu,T or * τR u,T (with R=M or L ) are expected ruin times conditional on ruin occurring by time T for the reinsured processes, and consequently are not particularly meaningful across distributions. The percentage changes however are of some interest. In this case improvements due to reinsurance are greater for the Pareto than for the Inverse Gaussian, as evidenced by the values of the percentage-wise increases (Column 7) in all tables. 126 Books MDPI Risks 2017,5,3 5. Cost of Reinsurance Reinsurance treaties are undertaken to reduce risk, but there is a cost attached. In the present section we employ a dividend discount model to determine the available means by which the company is able to pay for reinsurance without reducing the firm’s value, and how this affects risk as measured by the standard deviation of the company value. 5.1. Reinsurance Premium and Dividend Adjustment We assume the company’s current value is given by its future potential dividend stream, discounted to present value. Let ρ be the time value of money and assume that dividends are paid at constant rate d until the cedant’s default, if this occurs. Then the claim surplus process in (1) must be modified to reflect the dividend payment: Yt:=Ct+dt = Nt ∑ i=1 ξi−(c−d)t,t>0. (18) The insurance company will require a specified safety loading θ to be in effect after the dividend is paid, so the net profit condition (2) is modified to c−d=(1+θ)λμ. (19) In (18) , Y does not depend directly on c and d , only on c−d through the value of θ . Since our main interest is in the cost of reinsurance, we will take c and d as given. In practice their values will be dependent on policyholders’ willingness to pay and the choice of safety loading θ. Note also that the values of cin (1) and (18) must differ if d>0 and the same safety loading is used in both cases. The ruin time of the company is now given by τu=inf{t:Yt>u} , for an initial capital level u>0, and the cumulative dividend income by Iu=dτu 0e−ρtdt=d ρ(1−e−ρτu). (20) The company is subsequently valued at Vu=E(Iu)=d ρ1−E(e−ρτu;τu<∞). (21) Now suppose a reinsurance scheme is incorporated, for which the cedant pays the reinsurer a premium which is constant in time at rate r . As a result of the consequent change in risk profile of the insurer, policyholders may be willing to pay an increased premium c∗≥c , while shareholders will accept a reduced dividend d∗≤d. The reinsured claim surplus process is then given by Y∗ t= Nt ∑ i=1 ξi−(c∗−r−d∗)t−Rt,t>0, (22) where the nondecreasing process R represents the reduction in claims due to reinsurance. This is given by (4) in the case of an EOL treaty, and by (5) for the LCR treaty. The reinsured claim surplus process has ruin time τ∗ u=inf{t:Y∗ t>u} , and the dividend income Equation (20) and the valuation Equation (21) are then modified by replacing dand τuwith d∗and τ∗ urespectively. Thus V∗ u=E(I∗ u)=Ed∗τ∗ u 0e−ρtdt=d∗ ρ1−E(e−ρτ∗ u;τ∗ u<∞). (23) 127 Books MDPI Risks 2017,5,3 Since the aim of reinsurance is to prevent, or at least delay ruin, it is natural to require that τ∗ u≥τu for all u> 0. For the LCR and EOL reinsurance schemes, this can only be guaranteed if c∗−r−d∗≥ c−d , and so we make this assumption. Thus for a given new premium rate c∗ and dividend rate d∗ , the largest reinsurance premium the cedant would consider paying is r=c∗−c+d−d∗ . When this condition holds, (22) becomes Y∗ t= Nt ∑ i=1 ξi−(c−d)t−Rt:=Y t, (24) which does not depend on c∗or on d∗, and the valuation Equation (23) becomes V∗ u=d∗ ρ1−E(e−ρτ u;τ u<∞), (25) where τ u=inf{t:Y t>u}does not depend on c∗or on d∗. In particular, reducing d∗reduces V∗ u. Adopting a “utility indifference” rationale ([ 13 , 14 ]) whereby the reinsurance contract is beneficial for the cedant if its utility with reinsurance exceeds that without, and “utility” is taken to be the net present value of dividend income received, acceptable reinsurance contracts must satisfy V∗ u≥Vu .So to find the maximal reinsurance premium rmax that the cedant is willing to pay for a reinsurance treaty R, we should maximize r=c∗−c+d−d∗over all d∗∈[0, d]for which V∗ u≥Vu. Since V∗ u is increasing in d∗ , it follows immediately from (21) and (25) that the maximizing value of d∗is given by d∗ max(u)=d1−E(e−ρτu;τu<∞) 1−E(e−ρτ u;τ u<∞), (26) and the corresponding maximal reinsurance premium by rmax(u)=c∗−c+d−d∗ max(u)=c∗−c+dE(e−ρτu;τu<∞)−E(e−ρτ u;τ u<∞) 1−E(e−ρτ u;τ u<∞). (27) One interesting aspect of (27) is that the factor vθ(u):=E(e−ρτu;τu<∞)−E(e−ρτ u;τ u<∞) 1−E(e−ρτ u;τ u<∞)∈(0, 1)(28) depends on u and θ only, and not on d , and so represents the proportion of the dividend that may be used to pay the reinsurance premium for a given safety loading. Thus if the reinsurer demands a premium which does not exceed dvθ(u) , then, without reducing the value of the firm, the premium can be paid for entirely with a reduction in dividend. However if the insurance premium is in excess of dvθ(u) , then the insurance company will be forced to turn to policyholders to pay part of the cost if a reduction in the value of the firm is to be avoided. The calculation of dmax , rmax and vθ(u) amounts to the evaluation of the Laplace transforms of τu and τ u , where τ u represents the ruin time under whichever type of reinsurance is being considered. For LCR, τ u=τM u , and for EOL, τ u=τL u as specified in Sections 4.1 and 4.2. Currently there are no known theoretical results for the Laplace transform of τM u , and it would be of interest and useful to derive them 2. 2 Indeed, from a theoretical perspective, very little appears to be known about the effects of trimming on an insurance risk process and the subsequent ruin quantities. A series of approximate premium calculations for LCR treaties has been made in the literature; see, for example, [15,16], and [17–20], and their references. 128 Books MDPI Risks 2017,5,3 In general the Laplace transforms need to be approximated by some means. We did this by using the simulations to directly estimate Ee−ρ(τu∧T)for large T, and then observing that 0≤Ee−ρ(τu∧T)−E(e−ρτu;τu<∞)≤e−ρT. (29) This applies equally well to τ u . As a check on this, and to decide on the number of simulations needed for sufficient accuracy, we also used Proposition A1 in the Appendix, which shows that E(e−ρτu;τu<∞)=P(Yeρ>u)(30) where Yt=sup0≤s≤tYs and eρ is an independent exponential random variable with mean 1 /ρ . The right hand side of (30) can be estimated by simulating the paths of Y . (30) also holds if Y and τu are replaced by Y and τ u , so the Laplace transform of τ u can be estimated by the same means. Then dmax,rmax and vθ(u)can be evaluated by d∗ max(u)=dP(Yeρ≤u) P(Y eρ≤u)rmax(u)=c∗−c+d⎛ ⎝1−P(Yeρ≤u) P(Y eρ≤u)⎞ ⎠. (31) and vθ(u):=1−P(Yeρ≤u) P(Y eρ≤u). (32) 5.2. Choice of Parameters Below we report on simulations for some of the derived quantities in the present section. We want to give reasonably realistic simulation scenarios, so we have to make a credible choice of parameter values. There seems to be little guidance in the literature for doing this. In the end, the values we decided on are loosely based on some given in [21,22] together with some pragmatic considerations. To start with, the initial reserve level u is only determined up to a scale constant. It can be thought of as units of $10 k, or $1 m, etc., as convenient. The mean claim size μ is then to be taken relative to u . The time unit we set to be 0.01 years = 3.65 days, so values of T= 100, 500, 1000, as designated in Section 3.3 and in the finite horizon scenarios considered in Section 5.5, correspond to 1 year, 5 years, 10 years. The time value of money is set at ρ= 0.0005. Taken together with the time unit specified, this corresponds to a discount rate of 5% p.a. To approximate the infinite time horizon we take T= 13800 in (29) so that the error of the asymptotic approximation to (30) is bounded by e−13800ρ≈10−3. Safety loadings are taken to be θ= 0, 0.025, 0.05, 0.075, 0.1. The expected claim size μ= 2 and claims rate of λ= 1 are again as designated in Section 3.3. Thus claims accumulate on average an amount of 2 units per unit time length. This again is taken relative to u . The rate of premium inflow c and the dividend rate d need not be specified because as shown in Section 5.1, only the difference c−d=( 1 +θ)λμ is relevant for the computations in the present section, and this is fixed by our choice of θ,λand μ. How to decide on the value of L for the EOL reinsurance is also problematic. Again we could find little guidance in the literature 3 . We want to maintain comparability between the LCR and EOL schemes as far as possible. The values of L used in Tables 4–6 (finite horizon cases) were chosen so 3 The work of [ 23 ] suggests that one common principle in choosing L is to keep it at “a level at which claims become very infrequent”. 129 Books MDPI Risks 2017,5,3 Nevertheless, the classification is a useful way of specifying a range of tail behaviours on which to base simulation investigations. The work of [ 33 ] gives a detailed analysis of the classical Norwegian Fire Claims data set, comparing a number of distributions for goodness of fit and using them to calculate value-at-risk and related measures. Of the six probability models considered some are heavy tailed, such as the GPD (generalised Pareto), others are lighter-tailed (the Weibull-Pareto). They argue “it is certainly tempting to conclude that simpler distributions, such as GPD and FT (foldedt ) are preferred for the task of measuring tail risk” but “lead to substantially different risk evaluations”. This underlines the value of investigations like ours for understanding the behaviour of the risk process across a variety of tail regimens. The work of [ 33 ] further stress the need for formal statistical analysis for measuring and pricing tail risk. In any case, as we discussed in Section 4.1, the behaviour of ruin probabilities and ruin times for finite u is not necessarily closely correlated with tail heaviness, however defined. These characteristics can be strongly influenced by the distribution of small and medium claim sizes. In this context we refer to discussions in [ 34 , 35 ] where asymptotic analyses of path properties of the process are given for convolution equivalent distributions, and related to the ruin prospects of the company. On the other hand, of course, in any scenario, reinsurance in either of the ways we have defined it increases the lifetime of the company. 6.4. Lévy Insurance Risk Models The LCR model can be extended in various directions. Insofar as our analysis is restricted to the classical compound Poisson risk process, it can be generalised to a broader class of processes, the “general Lévy insurance risk models”. See for example [ 7 – 10 , 36 , 37 ] , where these models and some subclasses of them are considered in this context. 7. Summary We considered two types of reinsurance, EOL and LCR, and investigated the pros and cons of each by simulations. We took as outcomes the extent of increases in ruin times and decreases in ruin probabilities as a result of reinsurance. Using a dividend discount model, we also investigated the amount of the dividend available to pay for reinsurance and the consequent effect on the standard deviation of the company value. We found in Section 4.2 that the EOL method performs markedly better than no reinsurance in terms of lower ruin probability and longer ruin times mainly when there are heavy tailed claims, whereas the LCR treaty shows consistent improvements over all three classes of claim distributions. Regarding payment for reinsurance, we saw in Figures 6 and 8 that for a Pareto claim distribution a greater proportion of the dividend is available to pay the reinsurance premium than for Inverse Gaussian, which is greater again than for a Gamma claim distribution. Over a finite time horizon, with equal expected aggregate claims, LCR is at least as effective as EOL in averting ruin (Tables 1–6). When they are equally effective, the proportion of the dividend available to pay for reinsurance is comparable. When LCR is more effective, then the proportion is greater for LCR than for EOL (Figure 5). LCR and EOL both reduce risk considerably as compared with no reinsurance, in a variety of situations, as measured by the standard deviation of the dividend income. Acknowledgments: This work was partially supported by Simons Foundation Grant #226863, ARC Grant DP1092502, DFG grant SZ/321/2-1 and the ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS). 136 Books MDPI Risks 2017,5,3 Author Contributions: All authors contributed equally to the development of the theory and the application in this paper as it evolved. The initial inspiration arose during discussions among Y.F., P.G., R.M., T.W. at a conference on Lévy processes and their applications held at Kioloa in February 2014. Later A.S. initiated the ideas in Section 5 which were then further developed by all parties. Y.F. and T.W. were also responsible for calculations, simulations and figures. Conflicts of Interest: The authors declare no conflict of interest. Appendix A. Laplace Transforms Here we state some useful results concerning Laplace transforms of passage times. We used the formulae in this section for checking the asymptotic values in Section 5.1. Simulating the Laplace transforms: The Laplace transforms Ee−ρ(τu∧T) etc. for finite or infinite times Tcan be simulated using the following formulae. Proposition A1. For any ρ>0, T>0and u ≥0, Ee−ρ(τu∧T)=e−ρT+P(Yeρ>u,eρ≤T), (A1) while for ρ>0and u ≥0, E(e−ρτu;τu<∞)=P(Yeρ>u). (A2) The same results hold if Y and τuare replaced by Yand τ u. Proof. For (A1), we have Ee−ρ(τu∧T)=E(e−ρτu;τu≤T)+e−ρTP(τu>T) =[0,T]e−ρtP(τu∈dt)+e−ρTP(τu>T) =e−ρTP(τu≤T)+[0,T]ρe−ρtP(τu≤t)dt+e−ρTP(τu>T) =e−ρT+[0,T]ρe−ρtP(Yt>u)dt =e−ρT+P(Yeρ>u,eρ≤T). A check of the calculation shows this also holds if Y and τu are replaced by Y and τ u . Letting T→∞then proves (A2) in both cases. Although we did not use it in the simulation exercises, the asymptotic dividend d∗ max can be estimated similarly using the formula P(Yeρ≤u,eρ≤s)≤P(Yeρ≤u)≤P(Yeρ≤u,eρ≤s)+e−ρs(A3) for P(Yeρ≤u). (Take slarge enough that e−ρsis negligible.) Explicit Formula for Exponentially Distributed Claims: The Laplace transform of τu has been well studied in the literature; see for example [ 37 – 41 ]. Explicit, or even semi-explicit, formulae are rarely available. The simplest instance of an explicit formula is when claims are exponentially distributed with mean 1/δ. Then by Proposition 4.1.2 of [6], E(e−ρτu;τu<∞)=e−νu1−ν δ, 137 Books MDPI Risks 2017,5,3 where νis given by ν=(c−d)δ−λ−ρ+((c−d)δ−λ−ρ)2+4(c−d)ρδ 2(c−d). Setting ρ=0 gives the probability of ultimate ruin as P(τu<∞)=exp(−θδu(1+θ)−1) 1+θ, where c−d=(1+θ)λδ−1. An Upper Bound for P(τM u<∞) :With the notation in (13) , assume the Cramér case, so that (6) is satisfied for some ν0> 0. Assume that Ct is defined on a filtered probability space (Ω , Ft , F , P) , and let P∗be the exponentially tilted probability measure given by dP∗:=eν0CtdPon Ft. Then dP=e−ν0CtdP∗ and dP=e−ν0CτM udP∗ on Ft∩{τM u<∞}. It follows from Corollary 3.11 of [37] that eν0uP(τM u<∞)=eν0uE∗exp −ν0CτM u =E∗exp −ν0CM τM u−u+ZM u where CM τM u−u≥0 is the overshoot for the trimmed process over level uand, ZM u=sup 0<s≤NτM u ξi≥0. So we get eν0uP(τM u<∞)≤E∗exp −ν0ZM u. Assuming ξ1 has unbounded support, then ZM u→∞ almost surely as u→∞ , so eν0uP(τM u<∞)→0 , whereas eν0uP(τu<∞)→c> 0. This shows that the probability of eventual ruin is much smaller when trimming and suggests a way of quantifying this effect via the overshoot of the trimmed process. References 1. Böcker, K. and Klüppelberg, C. Multivariate models for operational risk. Quant. Finance 2010,10, 855–869. 2. Embrechts, P.; Klüppelberg, C.; Mikosch, T. 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Benktander, G. Largest claims reinsurance (LCR). A quick method to calculate LCR-risk rates from excess of loss risk rates. Astin Bull. 1978,10, 54–58. 16. Berglund, R.M. A note on the net premium for a generalized largest claims reinsurance cover. Astin Bull. 1998,28, 153–162. 17. Kremer, E. Rating of largest claims and ECOMOR reinsurance treaties for large portfolios. Astin Bull. 1982 ,13, 47–56. 18. Kremer, E. Distribution-free upper bounds on the premiums of the LCR and ECOMOR treaties. Insur. Math. Econom. 1983,2, 209–213. 19. Kremer, E. The asymptotic efficiency of largest claims reinsurance treaties. Astin Bull. 1990,20, 11–22. 20. Kremer, E. Largest claims reinsurance premiums under possible claims dependence. Astin Bull. 1998 ,28, 257–267. 21. Grandell, J. Aspects of Risk Theory; Springer Series in Statistics; Springer: New York, NY, USA, 1991. 22. Wikstad, N. Exemplification of ruin probabilities. Astin Bull. 1971,6, 147–152. 23. Bradshaw, A.J.; Bride, M.; English, A.B.; Hindley, D.J.; Maher, G.P.M. Reinsurance and Retentions—A London Market Actuaries’ Group Paper; Casualty Actuarial Society: Arlington, VA, USA, 1991; Volume I. 24. Beveridge, C.J.; Dickson, D.C.M.; Wu, X. Optimal dividends under reinsurance. Bulletin de l’Association Suisse des Actuaires 2008,2, 149–166. 25. De Finetti, B. Su un’impostazion alternativa dell teoria collecttiva del rischio. Trans. Internat. Congr. Actuar. 1957,2, 433–443. 26. Dickson, D.C.M.; Waters, H.R. Some optimal dividends problems. Astin Bull. 2004,34, 49–74. 27. Gerber, H.U.; Shiu, E.S.W. Optimal dividends: Analysis with Brownian motion. N. Am. Actuar. J. 2004 ,8, 1–20. 28. Ladoucette, S.A.; Teugels, J.L. Reinsurance of large claims. J. Comput. Appl. Math. 2006,186, 163–190. 29. Thépaut, A. Une nouvelle forme de réassurance: Le traité d’excédent du coût moyen relatif (ECOMOR). Bull. Trim. Inst. Actu. Fr. 1950,49, 273–343. 30. Teugels, J.L. 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Am. Actuar. J. 2006,10, 196–216. 37. Kyprianou, A. Introductory Lectures on Fluctuations of Lévy Processes with Applications; Springer: Berlin, Germany, 2006. 38. Dickson, D.C.M.; Willmot, G.E. The density of the time to ruin in the classical Poisson risk model. Astin Bull. 2005,35, 45–60. 39. Elghribi, M.; Haouala, E. Laplace transform of the time of ruin for a perturbed risk process driven by a subordinator. IAENG Int. J. Appl. Math. 2009,39, 221–230. 40. Lima, F.D.P.; Garcia, J.M.A.; Egídio dos Reis, A.D. Fourier/Laplace transforms and ruin probabilities. Astin Bull. 2002,32, 91–105. 41. Percheskii, E.A.; Rogozin, B.A. On the joint distribution of random variables associated with fluctuations of a process with independent increments. Theory Probab. Appl. 1969,14, 410–423. © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 140 Books MDPI risks Article Mathematical Analysis of Replication by Cash Flow Matching Jan Natolski and Ralf Werner * University of Augsburg, Universitätsstraße 14, 86159 Augsburg, Germany; [email protected] *Correspondence: [email protected]; Tel.: +49-821-598-2220 Academic Editor: Luca Regis Received: 17 August 2016; Accepted: 24 February 2017; Published: 28 February 2017 Abstract: The replicating portfolio approach is a well-established approach carried out by many life insurance companies within their Solvency II framework for the computation of risk capital. In this note, we elaborate on one specific formulation of a replicating portfolio problem. In contrast to the two most popular replication approaches, it does not yield an analytic solution (if, at all, a solution exists and is unique). Further, although convex, the objective function seems to be non-smooth, and hence a numerical solution might thus be much more demanding than for the two most popular formulations. Especially for the second reason, this formulation did not (yet) receive much attention in practical applications, in contrast to the other two formulations. In the following, we will demonstrate that the (potential) non-smoothness can be avoided due to an equivalent reformulation as a linear second order cone program (SOCP). This allows for a numerical solution by efficient second order methods like interior point methods or similar. We also show that—under weak assumptions—existence and uniqueness of the optimal solution can be guaranteed. We additionally prove that—under a further similarly weak condition—the fair value of the replicating portfolio equals the fair value of liabilities. Based on these insights, we argue that this unloved stepmother child within the replication problem family indeed represents an equally good formulation for practical purposes. Keywords: life insurance; replicating portfolio; market consistent valuation; cash flow matching; fair value; stochastic Fermat–Torricelli problem 1. Introduction Market-consistent valuation has gained increasing importance in the risk management of life insurance policies (e.g., Bauer et al. [1] ). However, many life insurance contracts have features which are too complicated to undertake analytical analysis. These include participating contracts, surrender options, and guaranteed interest rates, to name a few. They leave only little structure in the payoff profile, thus making analytical risk neutral valuation almost impossible. Therefore, one has to resort to some type of Monte Carlo method for market-consistent valuation of life insurance policies. As a result, Monte Carlo methods of various types have attracted a lot of attention in recent research. Although Monte Carlo methods work quite well for one specific valuation, they prove to be rather inefficient for repeated application in risk capital computation. For instance, the computational inefficiency of nested Monte Carlo methods for risk capital computation (cf. below and Figure 1 for a brief description, or see Bauer et al. [1] for more details) has led to an investigation of alternative methods. Andreatta and Corradin [2] , Baione et al. [3] , Glasserman and Yu [4] , Stentoft [5] , and several other authors apply or analyze the well-known least squares Monte Carlo approach, which was originally introduced by Longstaff and Schwartz [6] and Tsitsiklis and van Roy [7] to price American options. Bergmann [8] also mentions the stochastic mesh method of Broadie and Glasserman [9] , although no application to life insurance policies is known to our best knowledge. Risks 2017,5, 13 141 www.mdpi.com/journal/risks Books MDPI Risks 2017,5,13 In hindsight, Pelsser [10] probably first suggested valuation of with-profits guaranteed annuity options—which are typical life insurance products—via static replicating portfolios. Although static hedging seems overly simplistic—especially compared to more advanced dynamic hedging—a remarkably good fit of the behaviour of annuity options was obtained by a static portfolio of vanilla swaptions. In general, by constructing such a static replicating portfolio, one tries to form a static portfolio of a finite number of selected financial instruments which are easy to price, such that they generate cash-flows which approximate cash-flows on the liability side at each point in time and in each scenario. If this approximation is accurate, one obtains a good estimate of the market value of liabilities from the fair value of the replicating portfolio. Further, the market consistent embedded value (MCEV) can be obtained accordingly by subtracting this value from the value of the insurer’s asset holdings. One could of course determine the value of liabilities directly by Monte Carlo simulations: generate risk-neutral sample paths of liability cash-flows and compute the mean of present values of the discounted cash-flows (e.g., Grosen and Jørgensen [11] ). However, as the Solvency II capital requirement demands the computation of the 99.5% value at risk (see EIOPA [12] , page 7) of the MCEV distribution in one year under the real world measure, it is not sufficient to compute the MCEV today, but its distribution in the future. This requires the aforementioned nested approach; see Figure 1. Dsϭс Y ;Ͳ>)ϭͿ 7 /ŶŶĞƌƐĐĞŶĂƌŝŽƐ KƵƚĞƌƐĐĞŶĂƌŝŽƐ 2QH S HULRGRIHFRQRPLFULV N  W  W  ZĞĂůǁŽƌůĚ ZŝƐŬŶĞƵƚƌĂů Ͳ> Figure 1. Replicating portfolio approach: Each node at t= 1 and t=T represents a state of the portfolio and liabilities. The target is to estimate the 99.5% quantile of the market consistent embedded value (MCEV) distribution at t=1. As static replication is much more simplistic than dynamic replication, it has been an open question until quite recently, as to whether the approach of replicating portfolio can actually work. Although replication seems to resemble traditional immunization approaches, one major difference stands out: on the one hand, immunization works well in one period (as sensitivities to risk factors have been immunized), but leaves the portfolio un-immunized in the next period, which requires re-immunization (this is the main idea of dynamic delta hedging). On the other hand, replication matches cash flows, so there is hope that although one is not fully immunized in the first period, the immunization is still reasonable in the next period, as future cash flows are still replicated well. Very recent results have shown that replication indeed works under quite general setups, with efficiency comparable to the least squares Monte Carlo approach. The first theoretical foundations for replicating portfolios have been given in a series of papers by Beutner et al. [13] , Pelsser and Schweizer [14] , 142 Books MDPI Risks 2017,5,13 and Beutner et al. [15] . They analyze the asymptotic behaviour of matching the terminal value of liabilities (instead of matching cash flows; see below for more details) as the number of scenarios and the number of replicating assets grows at specific relative speeds. Recently, Cambou and Filipovic [16] proved that the matching of terminal values indeed has a mathematical foundation in the sense that a good match of the terminal values is strongly intertwined with a good approximation of the risk measure of the future MCEV (i.e., the resulting risk capital). Simultaneously, Natolski and Werner [17] demonstrated that this foundation holds for any form of matching problem in Natolski and Werner [18] (and many more), especially including all cash flow matching and all terminal value matching problems considered here. In both Cambou and Filipovic [16] and Natolski and Werner [18] ,itis shown that it is possible to change from the real world measure in the first period to the risk neutral measure while maintaining the strong link between objective function and error in risk capital. These results provide the basis for the practically most relevant replication setup 1 where only the risk neutral measure is considered (see Section 2 for the mathematical setup). The replicating portfolio approach represents a well-established approach carried out by many life insurance companies within their Solvency II framework for the computation of market risk capital. It is very specific to life insurance companies, and to our best knowledge so far, not used outside life insurance. We note that it could in theory also be used in banks for improving risk figure computation for hard-to-price portfolios2. Of course, only considering financial instruments might fall short of matching actuarial risks like mortality or lapse risk. This is one of the well-known weaknesses of replicating portfolios, and usually leads to a remaining mismatch. However, as replicating portfolios are usually only applied within the market risk computation in Solvency II, this only represents a minor issue of replicating portfolios in practice. As already briefly indicated above, there exist different choices for the specific matching problem. In Natolski and Werner [18] , some of these criteria were already investigated in more detail. The focus there was put on the two criteria which are most popular in the insurance industry: • Squared cash flow matching: this was probably the first formulation considered in the context of replicating portfolios. Here, the difference of cash flows at the same time point is measured by the squared L2 -norm, and the sum of these is used as matching criterion. This problem is called (RPSCF) in the following. • Terminal value matching: as an alternative to the cash flow matching criterion, terminal value matching—called ( RP ˜ TV ) below—was introduced by Oechslin et al. [20] . The main idea is that it is expected that a good approximation of the risk capital can already be obtained if the value of the cash flows is matched sufficiently well, while the timing of the cash flows should not be of any importance. Usually, cash flows are aggregated to the terminal time point, leading to the consideration of terminal values. This point of view was mathematically supported in Natolski and Werner [17]. These two (and other) choices might differ in important properties: • Depending on the choice of the fitting criterion, similarly good fits of the fitting criterion lead to differing good or even only reasonably good approximations of the risk capital figure. In summary, terminal value matching provides better bounds than cash flow matching. Further, squared cash flow matching provides worse bounds than cash flow matching (to be defined below). Details on the exact relationships are provided in Natolski and Werner [17]. 1 In Natolski and Werner [18] it is argued that working with the risk neutral measure is feasible, while it is better to keep the mix of measures as indicated in Figure 1. 2 A recent paper by Broadie et al. [19] gives some first arguments in this direction. Although the paper does not explicitly treat replicating portfolio, but focuses on least squares Monte Carlo methods, the idea can also be extended to replicating portfolios. The reason for this is that replication and least squares Monte Carlo are strongly interconnected, as has been shown by Glasserman and Yu [4]. 143 Books MDPI Risks 2017,5,13 • Depending on the choice of the fitting criterion, the solution of the corresponding optimization problems might pose difficulties. Although all formulations usually lead to convex problems, not all problems might be strictly convex, and thus may have non-unique optimal solutions. Further, non-smooth problems might be much harder to solve than smooth problems. Finally, some fitting criteria (for instance, those based on L1 -norms instead of L2 -norms) lead to an increase of the problem dimension with increasing number of scenarios, while others do not face this issue. • It is expected by practitioners that the fair value of the replicating portfolio also matches the fair value of the liabilities. This has already been proven for the two most practically relevant criteria, but this question remains open for other criteria. The analysis of the replication problems in Natolski and Werner [18] focused on the two most popular matching criteria, as these are a) quite easy to analyze and b) are strongly connected. As has been shown in Natolski and Werner [18] , the terminal value match can be seen as a squared cash flow match plus an additional dynamic trading strategy moving cash flows optimally in time. From a theoretical perspective, both choices provide a good link between matching criterion and approximation of risk capital (see Natolski and Werner [17] ), although terminal value matching dominates squared cash flow matching. Further, both problems lead to strictly convex quadratic optimization problems possessing a unique optimal solution (under rather weak assumptions). From a practical perspective, however, both problems show some unwanted behaviour. Extensive numerical tests (Daul and Vidal [21] ) have shown that the out-of-sample performance of terminal value matching is worse than the corresponding performance of (squared) cash flow matching. However, the latter problem reacts more sensitively to the scaling and the conditioning of the data than terminal value matching due to increased dimensions of relevant covariance matrices. The bad out-of-sample performance of terminal value matching can be linked to the property that it actually already includes a dynamic cash flow distribution strategy. This indicates that an unwanted overfitting takes place; i.e., there are too many optimization variables given only a few number of scenarios. As pointed out, it is not feasible to provide more scenarios, as these are computationally quite expensive. Therefore, the consideration of more enhanced replication—for example, by a more dynamic replication strategy—is clearly not advocated. For the above reasons, it might make sense to look for alternative criteria which have the potential to successfully address these issues. For this purpose, we consider a modified cash flow matching problem—called ( RPCF )—where the penalization of cash flows does not use the squared L2 -norm, but just the L2 -norm itself. This problem has already been briefly considered in Natolski and Werner [18], but a full analysis was out of the scope of that paper. Compared to the other two problems ( RPSCF ) and ( RP ˜ TV ), it has a more complicated objective function, since the objective function cannot be expressed as a convex quadratic function. Thus, it is impossible to obtain an explicit expression for the solution of the problem, and results on the existence and uniqueness of a solution are not straightforward. Further, the objective function of ( RPCF ) might be non-smooth at certain points. We believe that it is for this reason that problem formulation ( RPCF ) did not yet get as much attention as the other two formulations, both from practitioners and academics. In the following, we will argue that on the contrary, there are good reasons to believe that ( RPCF ) is indeed at least as preferable as (RPSCF)or(RP ˜ TV). To be able to do so, we first provide the detailed proofs for the existence and uniqueness of the solution under weak assumptions. Further, we show that despite the similarity to the famous Fermat–Torricelli problem, it matches the fair value of the replicating portfolio and the fair value of liabilities under an additional rather weak assumption. We also prove that the objective function is uniformly convex in practical problem instances (satisfying an additional assumption), a useful feature for optimization and statistical properties. Finally, we demonstrate that ( RPCF ) can be equivalently cast as a second-order cone problem, which allows its unique solution to be obtained by efficient second-order methods like interior point methods (without the additional assumption satisfied in practical problems). Ultimately, we provide a discussion as to why problem (RPCF) should be preferred over problems (RPSCF) and (RP ˜ TV), and why not. 144 Books MDPI Risks 2017,5,13 The rest of the paper is organized as follows. In Section 2, we introduce the mathematical setup for the financial market following Natolski and Werner [17] , and we recall the cash flow matching problem of the Fermat–Torricelli type. The main Section 3 states and proves the aforementioned properties of the replication problem, and Section 4 concludes. 2. The Mathematical Setup As mentioned in the introduction, this setup is taken from Natolski and Werner [17] .Wefix a finite time horizon T∈N and set T:={t=1, 2, . . . , T} , T0:={0}∪T . Let Ω,F,(Ft)t∈T0,Q be a filtered probability space with risk-neutral3measure and numéraire (Nt)t∈T0, where F0is assumed to be complete and trivial and FT=F. Denote by 1. ˜ CF t=˜ CF i,ti=1,...,m∈L 2(Q)m , t∈T , the discounted 4 financial cash flows of all assets at time t , where 1 ≤i≤mand mdenotes the number of available financial assets, 2. ˜ AF:=∑t∈T ˜ CF tthe vector of discounted terminal5financial cash flows, 3. ˜ CL t∈L 2(Q),t∈Tthe discounted liability cash flows, and by 4. ˜ AL:=∑t∈T ˜ CL t, the discounted terminal value of liability cash flows. The final cash flow ˜ CF T is equal to the corresponding asset value. The interpretation is that all assets are sold at the time horizon T . Assets are bought and sold at time t before the time t cash flows take place. The following three replication problems were considered in Natolski and Werner [18] , where the first problem ( RPCF ), cash flow matching, is in the focus of this exposition. In this context, a portfolio α represents the units of the financial instruments which have to be bought or sold. inf α∈Rm T ∑ t=1EQ˜ CL t−α˜ CF t2F01 2 .(RPCF) inf α∈RmT ∑ t=1 EQ˜ CL t−α˜ CF t2F01 2 .(RPSCF) inf α∈RmEQ˜ AL−α˜ AF2F01 2 .(RP ˜ TV) The first formulation penalizes the deviation of cash flows by the L2 -norm, whereas the second formulation penalizes the cash flows by the squared L2 -norm. Otherwise, the formulations are the same. The third formulation (where the sum has moved inside the square inside the expected value) represents the penalized squared deviation of the terminal values. Remark 1. It has to be noted that there is no budget constraint in all three optimization problems, as the budget is not restricted, but allowed to vary freely. In Section 3.4, we will characterize the optimal budget, which is equal to the fair value of the replicating portfolio. It is shown there that the optimal budget equals the fair value of the liabilities under rather weak assumptions. 3 All results of this exposition remain true with one obvious exception: if instead of the risk neutral measure the real world measure is chosen. Naturally, the result on the fair value in Section 3.4 crucially depends on the property that a risk neutral measure is chosen. 4We use tilded variables to express the fact that the variable is discounted. 5˜ AF i represents the pathwise discounted terminal value of all cash flows of asset i , i.e., the cash flow at time t is invested in the numeraire until time T, and the aggregated value of all these is discounted by the terminal numeraire value to time 0. 145 Books MDPI