Modeling cycle dependence in credit insurance

Caja, Anisa; Planchet, Frédéric Article Modeling cycle dependence in credit insurance Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Caja, Anisa; Planchet, Frédéric (2014) : Modeling cycle dependence in credit insurance, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 2, Iss. 1, pp. 74-88, https://doi.org/10.3390/risks2010074 This Version is available at: https://hdl.handle.net/10419/103615 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. 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. http://creativecommons.org/licenses/by/3.0/ Risks 2014,2, 74-88; doi:10.3390/risks2010074 OPEN ACCESS risks ISSN 2227-9091 www.mdpi.com/journal/risks Article Modeling Cycle Dependence in Credit Insurance Anisa Caja * and Frédéric Planchet Université Claude Bernard Lyon 1, ISFA, 69007, France; E-Mail: [email protected] *Author to whom correspondence should be addressed; E-Mail: [email protected]. Received: 31 December 2013; in revised form: 12 February 2014 / Accepted: 20 February 2014 / Published: 14 March 2014 Abstract: Business and credit cycles have an impact on credit insurance, as they do on other businesses. Nevertheless, in credit insurance, the impact of the systemic risk is even more important and can lead to major losses during a crisis. Because of this, the insurer surveils and manages policies almost continuously. The management actions it takes limit the consequences of a downturning cycle. However, the traditional modeling of economic capital does not take into account this important feature of credit insurance. This paper proposes a model aiming to estimate future losses of a credit insurance portfolio, while taking into account the insurer’s management actions. The model considers the capacity of the credit insurer to take on less risk in the case of a cycle downturn, but also the inverse, in the case of a cycle upturn; so, losses are predicted with a more dynamic perspective. According to our results, the economic capital is over-estimated when not considering the management actions of the insurer. Keywords: credit insurance; cycles; regime-switching Markov chain; rating transition matrix; multi-factor Merton model; economic capital 1. Introduction Credit insurance is concerned with business and credit cycle fluctuations, but in an unusual way. Indeed, it is extremely impacted by the defaults at the beginning of a crisis, but afterwards, the influence of the cycle downturn lowers the impact in a relatively short amount of time. Risks 2014,275 Commercial businesses need a credit insurer if they think their clients will not be able to pay their invoices on time or risk becoming insolvent shortly. Therefore, they obtain an insurance policy that guarantees them that a part of or the whole invoice amount will be reimbursed by the insurer in the case that the client could not pay. Every company may obtain an insurance policy to insure against the default payment of its clients, so small businesses and very big companies are insured. Thus, the credit insurer takes on the risk of insolvency or protracted default.1This makes the credit insurer very sensitive to credit and, eventually, business cycles. While the type of risk the insurer bears is still a credit risk, it is quite different from the credit risk of financial markets. First, because the source of risk is very diversified, defaulting firms can come from very different sectors: they can be small around-the-corner businesses or big multinational corporations. Second, the risk is not quoted in a market; there is no bid or sale. Therefore, the credit insurer should manage its portfolio, along with the cycles and its risk aversion, and thus, management actions will not be the same during this period. We will try to take this into account. Many papers aim to find out whether business and credit cycles coincide, and it seems there is no final answer to this question (see, for example, [1,2] and the references therein). Both cycles seem to follow some macro-economic variables, and GDP variations seem to explain, at least partly, credit cycle movements. Anyway, the difference between these two cycles will not be important in what follows. We can work with either of them, as long as we distinguish upturns from downturns; that in downturn periods there are (significantly) more defaults than in upturn periods and that the hypothesis that the cycles are a Markov process are realistic. For us, the important issue will be being able to distinguish the point at which the number of defaults changes significantly, i.e., at which the default regime switches. When the cycle goes down, the number of insolvent firms or those not being able to pay their invoices increases dramatically. Hence, the insurer should reimburse huge amounts of money, and it risks becoming insolvent itself. This would probably be the case if credit insurers could not limit their losses. They can do this by diminishing their exposure towards firms (clients of the insured business) whose creditworthiness is decreasing. In this case, the insurer warns the insured firm , whose granted amount is decreased. Its client may not be solvent anymore, and it should itself reduce the amount of commercial exchanges with the client. On the other hand, when the insurer thinks that economic conditions are more favorable, it takes on more risk and increases the guarantees. See [3] for an introduction to risk mitigation in credit insurance. The large number of firms defaulting at the same time at the beginning of a crisis impacts the credit insurer quite considerably (if the insurer has not correctly predicted the starting point of the crisis). However, once the number of defaults increases and the consequences of the crisis are observed, the credit insurer can lower its exposures towards riskier firms; there is a trade-off between the immediate loss of money and subsequent credibility loss (which would mean a loss of future wealth) and the diminishing of its losses. To resume, the insurer has the power to increase or decrease the risk it is bearing, which is more difficult for banks, for example. 1There is a default if the payment period lasts longer than was initially agreed upon by the insured and its client. Risks 2014,276 For all these reasons, it is important for a credit insurer to predict the state of the economy and for the actuarial teams to compute a risk capital, which depends on the state of the economy. The insurer computes its economic capital for a one-year period. In the current modeling, its capability to take actions on the exposures and portfolio is ignored, and this actually leads to an overestimation of its reserves. The reason for this is that the consequences of lowering its exposures in hard times are bigger and faster than those of increasing the risk appetite in prosperous periods. Our paper presents a way to adjust to cycles and to introduce lowering or increasing exposures. In the first section, we will present the reference model, where all parameters are fixed during the period. In the second section, we will introduce a two-period modeling, which allows one to adjust to the cycle. In the last section, we apply the model to a credit insurance portfolio and compare the values of economic capital under the two models. 2. Reference Single-Period Modeling The model we will introduce in this section is the cornerstone of the two-period model we propose. Let us start by describing this model, so that the new steps we want to introduce and their usefulness may be clearer later on. 2.1. Default Modeling Defaults are currently modeled with a multi-factor Merton model. The model is Merton-like (see [4–7]), since one client, which we will henceforth call the buyer, will default if a latent value, called the ability to pay and notated as Z, falls below a certain threshold, d. In the true Merton model, the latent value is the value of the assets of the firm, and the firm defaults if its assets value fall below the amount of its liabilities. The probability of default for buyer nis then: pn=P(Zn≤dn) The parameter estimated here is not the default threshold, because we are working with a latent variable, but rather, the default probabilities, pn. In our case, we will assume that they are given and are exogenous to our main concern. The default modeling is a multi-factor one, because the latent variable, Zn, is modeled as the sum of systemic risk and buyer individual risk. See [8–12] for further information on one-factor and multi-factor models and the associated copulae. In our case, we have: Zn=%ntwnR+p1−%2 nεn where: •Ris the systemic risk vector following a multivariate Gaussian distribution, N(0,Σ), with Σthe covariance matrix. In our case, it is a (105,105) type of matrix; •εnis the idiosyncratic (individual) risk of buyer nand follows a standard Gaussian distribution, N(0,1); •εnand Rare independent; •wnis the vector of weights of buyer nfor risk factors in R; Risks 2014,277 •%ndescribes the correlation of buyer nto systemic risk (economy); the bigger it is, the higher the correlation to systemic risk, thus the higher the correlation between firms. In the multi-factor model, the parameters one needs to estimate are the covariance matrix, Σ,%n, and the weights, wn. In practice, the estimation of Σand %nseems to be the hardest part, but it will not be the object of our paper. We should also notice that since it is easier to manipulate standard Gaussian variables, and Znwith the above definition is not a standard one, we will use instead the following definition of Z: Zn=%n twnR ktwnMk+p1−%2 nεn where Mis such that Σ=tMM (after the Cholesky decomposition of Σ). 2.2. Loss Modeling A defaulting buyer will produce a loss. This loss will be equal to the insured amount, which is defined as the minimum value among the invoice amount and the exposure the insurer has on the buyer.2 Loss n= min (Invoice Amountn;Exposuren) Since the insured amount is not known until the default occurs, it is modeled through UGDs (Usage Given Default), defined with the following formula: UGD =Insured amountn Exposuren UGD nis another parameter the insurer should model and estimate. Given UGD , the insurer estimates the loss from buyer nto be equal to: Loss n=UGDn×Exposuren in case buyer ndefaults. 3 3. A New Modeling Approach-Taking into Account the Actions of Management The current modeling is single period, which means that the whole parameters as well as the considered variables are defined over one period. The ability to pay is the one-period ability to pay. If there is a default, there is one default in the period. Since we do not know exactly when it occurs, we assume that all defaults occur at the end of the period. In practice, the period we work on is a year, since reserves are computed on a one-year basis. 2The maximum amount of money the insurer guarantees the insured in case of the default of his client, the buyer, n. 3This will not be the final loss, because other contract clauses, such as reinsurance or deductibles, for example, will be taken into account. We will not consider those clauses in this paper. Risks 2014,278 We want to introduce into the modeling the possibility for the insurer to manage exposures during the year. Indeed, it can lower exposures for buyers whose creditworthiness decreases and even cancel them, and this has an impact on its reserves.4 Therefore, thanks to this type of management, the reserves of the credit insurer should be lower than those estimated with the one-period model, and we want to take this into consideration. For this, we will introduce a half-period step into the model, which gives us a two-period model. This two-period model can easily be transposed into a multi-period model. Nevertheless, in credit insurance, working on a semester basis is coherent and sufficient, since it corresponds to an average reaction time. This means that six months after the beginning of a crisis, the big management actions have already taken place, and so, exposures on very risky firms have been cut or canceled. Therefore, if they are to change substantially, the portfolio features are to change every six months. Thus, the reserves estimated with this model should be more realistic. 3.1. Two-Period Modeling We will describe here the general idea of the model and deduce afterwards the parameters to be estimated. The idea here is to divide the single period into two sub-periods. 3.1.1. First Period The parameters are predicted for the first period. The default probabilities are the parameters we are more interested in, for now. They withhold information about the phase of the cycle in which we are during the period. At the end of the period, we compute the number of insolvencies, and given a criterion applied to this number, we determine if it is more probable for the first period to be in a high or a low cycle phase. We will give more details about this criterion in the section below. We then compute the losses related to insolvency defaults and protracted defaults. The creditworthiness of buyers may change at the end of the period, and their grades may change, consequently; whereas, in the one-period model, buyers could change the rating class only at the end of the period. 3.1.2. Second Period At the beginning of the second period, we thus have a new portfolio, since the buyers might have changed their grading class, and some of them have defaulted, so they have exited the portfolio. The exposures might also have changed from the beginning of the first period if the creditworthiness is lower. In the single-period model, this was not possible. 4This former case is modeled through the change in the UGD, whereas the latter is taken into account in the default probabilities. The reason for this is that in the case of the default of the buyer, when the guarantee has been canceled, the insurer pays nothing; this is as though the default never occurred from the insurer’s point of view. Risks 2014,279 The cycle phase of the second period may be high or low;5this depends on the a posteriori phase of the cycle in the first period. The dependence is modeled by a Markov chain, i.e., the probabilities of transition between high or low cycle phases. Examples in the literature are many: for example [13], where they work with business cycles. The losses of the second period are then added to those of the first. 3.1.3. Parameters The parameters we need to estimate for the recession and expansion periods are then the following, for low or high cycle phases: •grade transition matrices, with insolvency probability defaults in the last column, notated as Pland Ph; •a vector of protracted default probabilities for each grade, πland πh; •a vector with UGDs for each grade, UGDland UGDh; •a vector (or matrix) of coefficients indicating the variation of exposures, cland ch. The parameters for the first period should be predicted, especially the insolvency default probabilities. The grade transition matrix should be given, too. The estimation of those parameters will not be the object of this article, but finding consistent estimators for those parameters would be of great interest in a future work. A consistent estimation approach would be a hidden Markov chain (or regime-switching Markov chain; see [13–15]). Some other papers that present estimation techniques for conditional (on business, credit cycles or other) factors are [1,16–20]. 3.1.4. Mathematical Computation of the Losses Let Nbe the total number of buyers in the portfolio. Let Lnbe the exposure of buyer nat the beginning of the period. Let Gnbe the grade of buyer nat the beginning of the period, Gn∈ {1, ..., J}, where Jis the number of grading classes. Losses of the first period. The total loss at the end of the first period will be: Loss1= N X n=1 Ln×UGD(Gn)×[Zn≤d(Gn)] where [.]denotes the Iverson bracket. Remark 1. In the formula, we have UGDGn, because the UGDs are estimated by the grades and the probabilities of default, too. Practically, we compute d(Gn)for each grade as a standard Gaussian quantile d(Gn) = Φ−1(pGn), where Φdenotes the cumulative distribution function of a standard normal distribution, as does Zn. 5We will assume here that the high ( low) cycle phase is the period associated with a lower (higher) insolvency rate. Risks 2014,280 Changes in the portfolio during the first period. Let def be the number of defaults in the first period, def =PN n=1 [Zn≤d(Gn)]. If the default rate def N=PN n=1 [Zn≤d(Gn)] Nsatisfies a certain criterion, then we suppose the state of the first period was El; otherwise, we suppose it was Eh. We will develop this in 3.2. Concerning transitions between grades, the same principles as for defaults apply: default thresholds are computed using the transition matrices, P, and the assumption on Zbeing a standard Gaussian. The probability for a buyer, n, to go from Gn=ito j∈ {1, ..., J}is the following: P(di,j+1 ≤Z≤dij) = Φ (dij)−Φ (di,j+1) = pij with di,j+1 ≤dij. The thresholds, dij, of going from Grade ito Grade jare computed: ∀j∈ {J, ..., 1}, dij = Φ−1 J−1 X k=j pik +pi!(1) where J= 10 in our case. At the end of the first period, the structure of the portfolio would have changed: buyers having defaulted have exited the portfolio, and the others may have changed grades, which implies that their characteristics, such as, for example, the default probabilities, change for the second period. Losses in the second period. The transition matrix on the second period will be given conditionally on the cycle phase in the first period. P2=   Ph,with probability P(E2=h|E) Pl,with probability P(E2=l|E) The protracted default probabilities will be randomly chosen, conditional on the first period, so that: π2=   πh,with probability P(E2=h|E) πl,with probability P(E2=l|E) We compute a new default threshold with Formula (1) for high and low cycle phases, and then, we just have to choose between the two. The losses of the second period are given by the formula: Loss2= N−def X n=1 L2 n×UGD(E, G2 n)×Z2 n≤d2(G2 n) where L2 nis the exposure of buyer nat the beginning of the second period. It is estimated to be equal to cn×Ln, where cnis a coefficient indicating if the exposure of grade nhas fallen or increased. Risks 2014,281 The estimation of the coefficients, cn, is highly important. They can be estimated using the history of claims declarations and seeing how the exposure of buyers who have defaulted has evolved during the year before the default. We would expect that in high cycles, the exposures go up, and when the cycle is low, the exposures fall. Ideally, we could estimate a matrix, C, of coefficients indicating the evolution of the exposure for buyers going from one grade to another. However, in order to have more observations and a more robust estimation, we choose to give a vector of coefficients, indicating the evolution of exposures of buyers relative to their grade at the beginning of the period. The total loss of the year would then be Loss =Loss1+Loss2. 3.2. Hypothesis If the default rate at the end of the period is “high enough”, namely PN n=1 [Zn≤d(Gn)] N>def?, then Elis more probable than Eh. Otherwise, Ehis more probable than El. Proposition 1. The default rate for a given R, notated as def N, is a good estimator of the mean of conditional probabilities of all buyers in the portfolio 1 NPN n=1 pn|R. Proof. We will use Kolmogorov’s theorem; see the Appendix. We may apply Kolmogorov’s theorem to the “sequence” of random variables representing conditional default, i.e., Xn= [Zn≤dn|R] = n≤dn−%nR √1−%2 n|Rfor n= 1, ..., N. Indeed, those variables are independent (n)for n= 1, ..., N. We choose an=nfor n= 1, ..., +∞. The first hypothesis is satisfied. E([Zn≤dn|R])2=V([Zn≤dn|R]) + (E([Zn≤dn|R]))2(2) =pn|R(1 −pn|R) + pn|R2=pn|R<1(3) The second hypothesis is satisfied, since we have: +∞ X n=1 V([Zn≤dn|R]) n2= +∞ X n=1 pn|R(1 −pn|R) n2< +∞ X n=1 1 4n2<+∞(4) So PN n=1 [Zn≤dn|R]−EPN n=1 [Zn≤dn|R] N−→ 0a.s. Then, using the following: 1 NE N X n=1 [Zn≤dn]|R!= N X n=1 P(Zn≤dn|R)(5) =1 N N X n=1 pn|R(6) 1 N N X n=1 [Zn≤dn|R]−1 N N X n=1 pn|R−→ 0a.s. (7) Risks 2014,288 18. Jafry, Y.; Schuermann, T. Measurement, Estimation and Comparison of Credit Migration Matrices. J. Bank. Financ. 2004,28, 2603–2639. 19. Nickell, P.; Perraudin, W.; Varotto, S. Stability of Rating Transitions. J. Bank. Financ. 2000, 24, 203–227. 20. Lando, D.; Skødeberg, T. Analyzing Rating Transitions and Rating Drift with Continuous Observations. J. Bank. Financ. 2002,26, 423–444. Appendix A. Kolmogorov’s Theorem Theorem 2. Kolmogorov Let (Xn)n⩾1be a sequence of independent random variables, such that: •for all n⩾1,E(X2 n)<+∞ •there exists a sequence, (an)n⩾1, of positive numbers that grows to +∞, such that: +∞ X n=1 V(Xn) a2 n <+∞ Then: PN n=1 Xn−EPN n=1 Xn an−→N→+∞0 If a−1 nE N X n=1 Xn!−→ m, then PN n=1 Xk an−→ m. c 2014 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 license (http://creativecommons.org/licenses/by/3.0/).