Collective Defined Contribution Plans – Backtesting Based on German Capital Market Data 1950 - 2022

Forschung am ivwKöln Band 4/2022 Collective Defined Contribution Plans – Backtesting Based on German Capital Market Data 1950 - 2022 Oskar Goecke Collective Defined Contribution Plans – Backtesting Based on German Capital Market Data 1950 - 2022 Oskar Goecke TH Köln (University of Applied Sciences), Institute for Insurance Studies, Claudiusstrasse 1, D50678 Köln, Germany [email protected] Abstract Using historical capital market data for Germany (1950-2022) we analyze and compare (individual) defined contribution (IDC-) and collective defined contribution (CDC) pension plans. To this end we define simple asset liability management rules that govern a CDC pension plan and compare these to IDC-plans with the same asset allovation. Our main result is, that the CDC pension plans allow for a significant improvement of the risk return profile compared to individual pension plans. Hereby we consider different risk measures. This empirical study affirms the theoretical results based on stochastic CDC-models. 1. Introduction Pension systems worldwide consist of a combination of Pay-as-You-Go (PAYG) systems and capital-funded systems. From the macroeconomic viewpoint a PAYG system is linked to the production factor labor while a capital-funded systems is linked to the production factor capital. Both systems have their merits and drawbacks. One argument in favor of capital funding is the fact that in most developed countries we observe an aging working population, that puts pressure on PAYG-Systems. However, a pension fund cannot directly invest into the 2 04.11.2022 09:46 production factor capital in the notion of national income accounting.1 In particular an investment into corporate bonds only indirectly secures a share in capital stock of an economy since part of the profit generated by the company, namely a risk premium, goes to the shareholders. Similar considerations apply to government bonds. Thus - at least in theory - only a 100% real investment (stocks and real estate) would insure a full participation in the production factor capital. But a 100% equity investment means an unacceptable risk for most savers. So, capital funded pension systems are confronted with the risk-return dilemma: The capital can be invested in risky asset (such as equities) with high returns or can be invested safely (e.g. in government bonds) with significant lower returns. The equity risk premium (ERP) is a well-established key figure to gauge the average extra return if one invests in equities rather than in bonds. However, an individual saver who puts aside part of her/ his labor income during working life may face a stock market crash just at the moment she/ he wants to retire or to buy an annuity. The problem is that an individual saver (the same applies to an age cohort of savers) cannot realize the average return on equities. Collective defined contribution (CDC) pension systems try to overcome this problem by some kind of collective agreement among generations of savers aiming to redistribute “deviations” from the average. 2 The idea of intergenerational risk transfer with respect to capital market risks is old since classical with-profit life insurance or endowment policies can be regarded as an implementation of some kind of intergenerational (capital market) risk transfer.3 The last years research into intergenerational risk transfer arrangements has helped to better understand CDC pension schemes. Applying stochastic models (including stochastic simulation techniques) one can prove a positive welfare effect (e.g. Gordon/ Varian 1988, 1 Only a small fraction of the national capital (machinery, real estate, patents, intellectual property, …) are traded on capital (including real estate) markets. 2 Cf. Gollier (2008) 3 Cf. Goecke (2003) 3 04.11.2022 09:46 Gullién/ Jørgensen/ Nielsen 2006, Gollier 2008, Hoevenaars 2008 etc.) of CDC-arrangement or we can derive optimal ALM-strategies (Chen/ Kanagawa/ Zhang (2021)). However, to the best knowledge of the author, there are only few empirical studies with respect to CDC plans (Wesbroom/ Arends/ Turnock/ Harding 2015 with UK-Data). We want to fill this “empirical gap” by a backtesting based on German capital market data. Even though we are working with real world capital market data it should be clear, that the following is not a study of real CDC-pension systems; it is rather a study about how a CDC-pension system would have performed if as proper CDC-system had been installed in the past decades. The author is well aware of the fact that the following study only refers to the German market and that historic market data may not be representative for what might happen in future. However, one should keep in mind that even the most elaborated stochastic model is equally limited with respect to the predictive power – the future is only a realization of one of millions of possible future random walks!4 The CDC-model underlying our backtesting is inspired by the continuous time (c.t.) model presented in Goecke (2013) and we want to check whether the theoretical results derived for the c.t. model remain valid for real capital markets. To this end firstly, we have to formalize a discrete time (d.t.) version and secondly, we have to select proper capital market data. To differentiate from CDC-models we will refer to (individual) defined contribution (IDC-) models, where the saver participates in a normal investment fund arrangement, where the individual pension capital is one-to-one linked to the market value of assets of that particular investment fund. 4 We do not discuss the rather philosophical question whether stochastic models build on the mathematical concept of probability are in principle appropriate to model economic scenarios; the fundamental problem is that the mathematical concept of probability implicitly presumes that uncertain events (like throwing dices) can be repeated arbitrarily often and that economic scenarios are not the result of throwing dices but mainly result of human decisions, partly driven by emotions and irrationalities. 4 04.11.2022 09:46 Capital Market Data for Backtesting The c.t. model in Goecke (2013) presumes a stylized capital market with a risk-free asset with a constant interest rate and a risky asset driven by a geometric Brownian motion with constant drift and volatility. It is assumed that the pension assets are a mix of the risk-free and the risky asset and that the equity ratio (i.e. the relative part of risky assets) can be adjusted continuously. If we want to check the theoretical time continuous model with real capital market data, we must find a suitable proxy for a risk-free asset and a risky asset. For our backtesting we assume that for a pension manager an investment into top rated government bonds is regarded as a risk-free investment and that an investment into a well-diversified portfolio of equities is regarded as a risky investment with a positive extra return. We take the German government bond index REXP as a proxy of a risk-free bond portfolio and the German equity index DAX a proxy of a broadly diversified equity portfolio. Clearly, an investment into a REXP-Portfolio is not risk-free in the strict sense since from month to month we observe unexpected gains or losses from volatile market interest rates leveraged by the duration of the portfolio. Consistent with our pragmatic definition of a riskfree asset, we will substitute the constant risk-free interest rate of the time continuous model by the current yield of outstanding government bonds is(t), which we interpret as the expected return of a bond investment. The time continuous model implicitly presumes a zero-inflation economy. We therefore adjust our real capital market data to eliminate inflation. To this end we take a suitable consumer price index CPI(t) and calculate price adjusted index values REXPp and DAXp. On the basis is(t) we define µ s(t), the price adjusted expected (log-)return (based on information up to time t) for the month following t of a risk-free investment. Details are explained in Appendix. 5 04.11.2022 09:46 2. A simple CDC model for backtesting 2.1. Model description The CDC model for our backtesting is a discrete time (d.t.) adoption of the continuous time (c.t.) model presented in Goecke (2013). The discrete time interval is ∆ = 1/12 i.e. one month. We consider a pension fund with assets and liabilities. The assets at time t – denoted P(t) – are invested in a mixture of equities (represented by DAXp) and bonds (represented by REXPp). The liability side of the balance sheet of our pension fund consists of two parts: the total of all individual accounts – denoted by V(t) – and a capital reserve – denoted by R(t) – which serves as a buffer. We have P(t) = R(t) + V(t); we assume that always P(t) > 0 and V(t) > 0, but we allow for a negative reserve R(t) < 0 in case of an underfunding of the pension fund. Figure 1. Balance sheet of a CDC pension funds We assume that at the beginning of each month a new generation enters working life and starts paying contributions into the pension fund. At the same time the generation of freshly retired leaves the pension fund taking with them their individual assets, i.e. their share of V(t). We thus do not consider lifetime annuity payments to retirees, instead we assume that all retirees receive a one-off payment upon retirement. We implicitly assume that all savers live to see their retirement. 6 04.11.2022 09:46 We assume that the sum of contribution of all active workers just matches the total of money paid out to the retirees. By this assumption we rule out dynamic effects from a growing or shrinking population. We do not rebuild the start-up phase of a CDC system; instead we start with an initial balance sheet (P(t0), V(t0), R(t0)) in t0 = 0 representing01.01.1950. However, what we do is we simulate more or less favorable starting conditions by setting the initial reserve ratio at different levels. As pointed out, we use price adjusted data. Thus, 1 Euro contribution payed by a generation of active workers has the same real value as 1 Euro paid out to the same generation decades later. Notation and Definitions Our d.t. model is based on monthly data, i.e. our time unit ∆ represents one month. The time index t represents the first day of the months within the backtesting period (01.01.1950 to 01.07.2022). Generally, we identify the end of a month with the beginning the following month. For example, if we consider a 480-month saving plan starting at the beginning of Jan. 1960 and falling due at the end of December 1999, we say that the saving plan starts at t = 120∆ and ends at t = 600∆. At time t, the beginning of month [t, t+∆[, the pension manager determines the equity ratio β (t) ≥ 0. Within the time interval [t, t +∆[ we follow a buy-and-hold strategy, i.e. with respect to the performance of P(t) we assume that ( ) () () ()() 1 () () () () pp pp DAX t REXP t Pt tt P t DAX t REXP t ββ +∆ +∆ +∆ = +− for t ≥ 0. (Eq 1) For t ≥ ∆ we define () ( ): ln () P Pt tPt µ  = −∆  . 7 04.11.2022 09:46 Furthermore, at time t ≥ 0 the pension manager determines the declaration η (t), namely the (log-) interest rate for the month [t, t+∆[, by which the individual accounts are updated, i.e. ( ) ( ) ()exp ()Vt Vt t η +∆ = or () ( ) ln () Vt tVt η +∆  =  . Note that µ P(t) can be observed at time t and refers to the investment period [t-∆, t [, while η (t) refers to [t, t + ∆[. β (t) and η (t) are determined at time t based on the information available up to time t . Thus β (t), η (t) and µ P(t) can be regarded as t-adapted processes. We define () () ( ) : ln ln 1 () () Pt Rt tVt Pt ρ    = =−−       and call it reserve ratio at time t. We prefer this log-definition instead of R(t)/P(t) because it simplifies the notation. For example, using this definition we get the following simple recursion for the reserve ratio: ( ) () ( ) () P t tt t ρ ρµ η +∆ = + +∆ − . (Eq 2) The fact that η (t) is determined at time t while µ P(t+∆) cannot be observed before t+∆ implies that the ALM-manager cannot guarantee a positive reserve ratio. The ALM-strategy proposed in Goecke (2013) and analyzed in Chen e.a. (2021) is characterized by the following three rules: Rule 1: The pension manager tries to keep the reserve ratio ρ (t) close to a given strategic reserve ratio ρ s ≥ 0; we call ˆ(): () s tt ρ ρρ = − the reserve gap. Rule 2: The pension manager follows a given strategic equity ratio β s ∈[0, 1]. However, we allow for monthly adjustments of β (t) as a reaction of a positive or negative reserve gap. 8 04.11.2022 09:46 Rule 3: The declaration η (t) should basically follow the expected rate of return of the pension assets, in particular a high equity ratio should result in a higher declaration. However, because of Rule 1, the actual declaration is adjusted according to the reserve gap. These general rules must be specified. Again, we follow the approach in Goecke (2013) and define an asset management rule (AM-rule) and liability management rule (LM-rule) depending on parameters α ∆ ≥ 0 and θ ∆ ≥ 0 : AM-Rule: ( ) ˆ () () () s ss tt t β βαρ ρ βαρ ∆∆ =+ −=+ with side constraint 0 ≤ β (t) ≤ 1, 5 LM-Rule: ( ) ˆ () () () () () ee P sP tt t t t ηµθρρµθρ ∆∆ = + −= + , where () e P t µ denotes the expected rate of return of the assets based on the information up to time t . We define ( ) 22 1 2 (): () () () e Ps M t t t ERP t µ µ β βσ = +∆ − , where µ s(t) denotes the expected return of a risk-free investment, ERP is the equity risk premium and σ M is an estimate of the average market volatility. We take ERP = 5% and σ M = 20% - see Appendix 1 for details. We should keep in mind that the calibration of ERP and σ M only has effect on the expected return on assets; the actual performance of the pension portfolio P(t) is not affected by the choice of η (t).. However, if we overestimate ERP, then the declaration η (t) tends to be too optimistic and the probability on a negative reserve gap will increase. Given ERP and σ M the asset liability management is fully determined by only four parameters, namely β s, ρ s, α ∆ and θ ∆. 5 We could allow for β (t) > 100%, which corresponds to a debt-financed investment in equities. However, in the context of pension management a debtfinanced speculative investment makes little sense apart from the fact that this kind of investment might be prohibited by supervisory authorities. 15 04.11.2022 09:46 1216.64, which is 57.24% lower that the final capital of the saving plan maturing 12 months before (2845.21). The average over all maturing dates is ICImean = 11.52%. For a β = 50% equity portfolio we get ICImax = 30.86% and ICImean = 6.06%, and for β = 0% we get ICImax = 18.24% and ICImean = 2.95%. We see that even for a presumed secure investment in German government bonds within only 12 months the pension capital of two generations of savers can differ by more than 18%. The reason for this is, that between June 2021 and June 2022 the interest rates and inflation rates increased substantially. As pointed out, we believe that to assess the risk of a certain pension vehicle (and hence the acceptance by consumers) it is not enough to evaluate the final capital but also the whole accumulation period. The ups and downs during the accumulation period definitely can stress the saver – in other words we think that a saving plan with heavy ups and downs are perceived to be riskier than a saving plan with a continuously increasing accrued capital. To illustrate this, we look at the 40-years saving plan that starts on 01.04.1963 and ends on 31.03.2003 – cf. Figure 3. Figure 3. Accrued capital of a saving plan (1.4.1963-31.3.2003) for β = 0%, 50%, 100%. 16 04.11.2022 09:46 We see that the final pension capital is more or less the same for all three levels of the equity ratio.12 However, investing into REXPp ( β = 0) results in a more or less continuously increasing accrued capital, while a DAXp ( β = 100%) investment makes any reasonable projection of the final pension capital impossible. The volatility of the monthly returns13 during the investment period is 3.87% ( β = 0%), 9.23% ( β = 50%) and 17.77% ( β = 100%). The risk indicators MDD and MLD are illustrated in Figure 4. Figure 4 depicts the accumulation period for saving plans maturing end of June 2022 for β = 0% and β = 100%. The MDD of the β = 100% saving plan is 67.92%, since between end of Febr. 2000 and end of March 2003 the accrued capital fell from 931.87 to 298.95 - by 67.92%. The MDD for β = 0% is 16.71%, respectively. The MLD for β = 100% and 0% is (respectively) 202 and 121 months. The MLD for β = 0% is illustrated in Figure 4: the accrued capital end of May 2012 is 728.05 and 121 months later (end of June 2022) it is still lower, namely 725.14. The extreme high loss duration for a “safe” investment vehicle is caused by low (price adjusted) interest rates during most of the saving period, the increasing interest rates combined with high and inflation in 2022. Still true is that a risk-averse person paid 121 contributions while at the same time the purchasing power of her of his pension capital did not grow. 12 The rate of return is 4.48% (for β = 0%), 4.81% (for β = 50%) and 4.18% (for β =100%) 13 Annualized standard deviation of 480 monthly (log-) returns 17 04.11.2022 09:46 Figure 4. Accumulation period of a saving plan (1.7.1982-30.6.2022) for β = 0% and 100%. 3. Results 3.1.1. Lump sum investment Before evaluating the saving plans we illustrate the effect of the intergenerational risk transformation of CDC-plans. To this end evaluate a lump-sum investment for the total backtesting period (01.01.1950 to 30.06.2022) for different investment vehicles:  Constant mix portfolio with a share of β invested into DAXp and (1β ) invested into REXPp portfolio for β ∈ {0%, 10%, …, 100%} – abbreviated CM( β ).  Investing into a CDC-Portfolio with θ ∆ = 2%, α ∆ = 0, ˆ(0) 0 ρ = and with an underlying constant mix portfolio with β ∈ {0%, 10%, …, 100%} - abbreviated CDC( θ ∆= 2%, β )  Investing into a CDC-Portfolio with θ ∆ ∈ {0.5%, 1%, 2%, 5%, 10%, 20%, 40%, 60% 80%, 100%}, α ∆ = 0, ˆ(0) 0 ρ = with β = 100% - abbreviated CDC( β =100%, θ ∆). 18 04.11.2022 09:46 For each portfolio we evaluate the rate of return and the annualized volatility. The annualized volatility (Vola) is calculated as () () 12 ln :0 Pt Pt StDev t T +∆   ⋅ ≤ ≤ −∆     for constant mix portfolio investment and as () () 12 ln :0 Vt Vt StDev t T +∆   ⋅ ≤ ≤ −∆     for investment into V(t). To start with, Figure 5 just illustrates the interplay of µ P(t), η (t), and the reserve gap ˆ() () s tt ρρρ = − . In this example the underlying portfolio is a DAXp investment ( β =100%) and the parameters for the CDC-model are ( ) ˆ , , , (0) (100%, 2%, 0, 0) s βθα ρ ∆∆ = . The smoothing effect of the intergenerational risk transfer is obvious; the annualized volatility of the µ P(t)-paths is 17.83% and 2.05% for the η (t)-paths – a reduction of more than 88%! This return smoothing is only possible if we allow for a volatile reserve gap. Since for β s = 100% and α ∆ =0 ˆˆ () () () () 0.03 () e Ps tt tt t η µ θρ µ θρ ∆∆ = + = + ∆+ we get as a rough estimation ( ) ( ) ˆ () ()StdDev t StdDev t ηθ ρ ∆ ≈ , if we neglect the impact of the µ s(t)-term. Actually, in this example we have ( ) ()StdDev t η = 0.59% and ( ) ˆ()StdDev t θρ ∆ = 0.545%. The mean reverting character of the ALM-rules insure that the reserve ratio always returns to the strategic level. This effect is even more pronounced in practice since equity bear markets have always be followed by bull markets and vice versa. 19 04.11.2022 09:46 Figure 5. ( µ P(t), η (t), ρ (t) - ρ s) for 0 ≤ t ≤ T For the three types of investment vehicles explained above (CM( β ), CDC( θ ∆ = 2%, β ), CDC( β = 100%, θ ∆)) we calculate risk-return profiles, where risk is measured as the volatility and the return as the (annualized) rate of return over the back-testing period. The volatility-return profile for CM( β ) confirms the risk-return dilemma, i.e. that an increasing (expected) return comes along with increasing risk (here: volatility). As pointed out the investment in a pure REXPp portfolio is not risk free with respect to volatility. However, volatility – as defined here - might overstate the investment risk, if we want to evaluate long-term investments. DAXp and REXPp are not fully correlated, so there is a diversification effect between both. The risk-return profile for CDC( θ ∆= 2%, β ) shows the risk-mitigating effect of buffering capital market variations. There is also a risk-mitigating effect for a pure REXPp -portfolio ( β = 0) since the collective reserve buffers short term interest rate fluctuations. 20 04.11.2022 09:46 The rates of return for CDC( θ ∆= 2%, β ) are slightly higher than the rates of return for CM( β ) due to the fact that the final reserve gaps are slightly negative. The risk-return profile for CDC( β =100%, θ ∆) shows that θ ∆ is the crucial ALM-parameter to manage the smoothing effect. We do not include the case θ ∆ = 0 because in this case the reserve gap is just a “random” walk and not mean reverting. Figure 6. Return-volatility-profile of different IDC-/ CDC-investment vehicles Const-Mix-Portfolio CDC( θ ∆ = 2%, β ) CDC( β = 100%, θ ∆) β rate of return Vola β rate of return Vola θ ∆ rate of return Vola 0% 3.07% 3.84% 0% 3.18% 0.75% 0.5% 7.03% 1.01% 10% 3.61% 3.91% 10% 3.73% 0.78% 1% 7.10% 1.39% 20% 4.13% 4.73% 20% 4.25% 0.85% 2% 7.19% 2.05% 30% 4.61% 5.99% 30% 4.73% 0.95% 5% 7.28% 3.43% 40% 5.07% 7.48% 40% 5.18% 1.08% 10% 7.32% 4.98% 50% 5.49% 9.08% 50% 5.60% 1.22% 20% 7.32% 7.00% 60% 5.87% 10.76% 60% 5.99% 1.37% 40% 7.27% 9.84% 70% 6.23% 12.49% 70% 6.34% 1.53% 60% 7.25% 12.37% 80% 6.54% 14.24% 80% 6.66% 1.70% 80% 7.24% 14.94% 90% 6.83% 16.03% 90% 6.94% 1.87% 100% 7.25% 17.77% 100% 7.07% 17.83% 100% 7.19% 2.05% Table 1: Data underlying Figure 6 21 04.11.2022 09:46 Figure 7 and Table 2 analyze the behavior of the reserve gap. For CDC( θ ∆ = 2%, β ) the mean of the reserve gap is close to zero asserting the meanreversion character of the reserve process. Figure 7 also shows that the distribution of the reserve ratio is roughly symmetric. This is not the case if θ ∆ falls below 2%, because then θ ∆ is to low to force the reserve gap back to zero. Figure 7. Evaluation of { } ˆ( ), 0,...,tt T ρ = for different CDC-variants. 22 04.11.2022 09:46 min max 10%- quantile 90%- quantile median mean StdDev β CDC( θ ∆ = 2%, β ) 0% -16.30% 13.22% -7.08% 8.44% 0.96% 1.11% 6.10% 10% -17.56% 15.57% -7.89% 9.19% 1.16% 2.06% 6.61% 20% -20.85% 19.45% -9.59% 11.38% 1.40% 1.77% 8.00% 30% -24.51% 28.98% -11.42% 14.60% 1.67% 2.02% 9.91% 40% -28.26% 39.34% -13.47% 18.21% 1.96% 2.13% 12.10% 50% -33.92% 49.72% -15.71% 22.00% 2.29% 2.30% 14.45% 60% -42.75% 60.13% -17.76% 25.82% 2.64% 2.20% 16.90% 70% -51.79% 70.57% -20.06% 29.21% 3.02% 2.44% 19.42% 80% -61.04% 81.04% -21.98% 33.21% 3.43% 2.34% 21.99% 90% -70.52% 91.54% -24.08% 37.82% 3.86% 2.50% 24.60% 100% -80.24% 102.08% -25.97% 41.99% 4.31% 2.57% 27.25% θ ∆ CDC( β = 100%, θ ∆) 0.5% -68.06% 148.90% -25.83% 64.45% 14.81% 9.96% 37.03% 1% -74.87% 128.83% -31.54% 46.95% 7.94% 4.34% 33.14% 2% -80.24% 102.08% -25.97% 41.99% 4.31% 2.57% 27.25% 5% -74.13% 66.37% -20.31% 26.60% 1.86% 1.04% 19.41% 10% -63.83% 49.03% -15.91% 18.37% 0.95% 1.10% 14.33% 20% -50.69% 35.23% -10.78% 12.24% 0.46% 0.88% 10.12% 40% -39.39% 23.85% -7.89% 8.36% 0.22% 0.43% 7.12% 60% -33.52% 18.79% -6.29% 6.83% 0.14% 0.61% 5.97% 80% -30.65% 17.65% -5.86% 6.13% 0.10% 0.51% 5.41% 100% -29.84% 19.15% -5.40% 5.81% 0.08% 0.29% 5.14% Table 2: Data underlying Figure 7 3.1.2. Analyzing 40-years saving plans Figure 8 depicts the final capital of 391 (overlapping) 480-months IDC-saving plans with constant equity ratio β = 0%, 50% and 100% - shown as solid lines. The corresponding final capitals for CDC-pension plans with θ ∆ = 2% and β = 0%, 50% and 100% are shown as broken lines. The difference between the solid and the broken line indicates the degree of intergenerational transfer of assets. 23 04.11.2022 09:46 Figure 8. Final capital of 40-year saving plans for different IDC-/ CDC-variants; the xaxis shows the maturity date The heavy ups and downs for IDC-saving plans with β = 100% indicate that even for a longterm saving plan it is nearly impossible to predict the outcome. As pointed out we want to evaluate long-term saving processes under the aspect of intergenerational fairness, having in mind, that a capital funded system should enable a fair participation in the economic capital. To this end we introduced the risk measure intergenerational imbalance (IGI) – in relation to the rate of return, – cf. the risk-return profiles in Figure 9. There is some resemblance to the risk-return profiles in Figure 6. Note that the range of rates of return for IDC-saving plans in Table 3 (from 3.97% to 6.66%) smaller than the range in Table 1 (from 3.07% to 7.07%). This is due to the fact that in Table 3 we analyze the rates of return of 480-months saving periods and that the rate of return of each of these is already a weighted average. The intergenerational risk transfer is particularly strong with respect to the maximal IGI. Here we observe that increasing β (from 0% to 60%) comes along with a decreasing of the maximal IGI. With respect to the CDC( β = 100%, θ ∆) saving plans we notice for θ ∆ = 0.5% an 24 04.11.2022 09:46 abnormal risk-return profile. Again, this is just the consequence that the mean reverting effect is to low. Figure 9. Average (top) and maximal (bottom) intergenerational imbalance (IGI) for IDCand CDC-saving plans. 31 04.11.2022 09:46 ratio. Note that for θ ∆ = 1 we have ˆ () () () e P t tt ηµ ρ = + and thus the volatility of ( ) ˆ()t ρ is very close to the volatility of ( ) ()t η . 4.1.2. Asset adjustment parameter α ∆ The following Proposition is the motivation for our LM-Rule ( ) (): () ss tt β βαρ ρ ∆ =+− and it helps to calibrate the parameters β s and α ∆ . Suppose at time t we want to control the “ruin probability” that the reserve ρ (t+∆) falls behind a given minimum level ρ min , then the following holds: Proposition 15 If we assume that ( ) () () ( ) distr e P P Mt t t t t WW µ µ βσ +∆ +∆ − − = ,16 we get ( ) min min ˆ (1 ) ( ) ˆˆ ( ) () () s s M t tt t ρ ρ θρ ρ ρ ρρ βσ ∆  − +− +∆ ≤ − =Φ −   ∆   and ( ) ( ) min ( ) () () () ss tttt ρ ρ ρ ε β βαρ ρ ∆ +∆ ≤ ≤ ⇔ ≤ + − , (Eq 4) with min 1 : s s M u ε ρρ βσ − − = ∆ , 1 1 : Mu ε θ ασ ∆ ∆ − − =∆ and 1 1 : (1 )u ε ε − − =Φ− , the 1ε quantile of the standard normal distribution. Here (Wt) denotes a (discrete) standard Wiener process. ◊ Remark: The assumption ( ) ( ) () () distr e P P Mt t t t t WW µ µ βσ +∆ +∆ − − = is daring for several reasons. Firstly, even for β (t) = 0 ( ) () e PP tt µµ +∆ − bears an interest rate risk, secondly, the monthly 15 The proof is straight forward and left to the reader. 16 “ distr = ” stands for „has the same distribution as” 32 04.11.2022 09:46 returns of a pure equity portfolio are only approximately normal-distributed. Thirdly, given ρ (t) the distribution of ( ) () e PP tt µµ +∆ − is not independent of the t-history – in other words, real world capital markets are not efficient.17 Example: For σ M = 20%, ρ s − ρ min = 20%, ε = 1% and θ ∆ = 2% we get u1ε = 2.3263 , min 1 148.91% s s Mu ε ρρ βσ − − = = ∆ , 1 17.2965 M u ε θ ασ ∆ ∆ − − = = ∆ . This means that – at least in theory - even a debt financed equity investment ( β > 100%) would be allowed, if we require that in only 1 of 100 months the reserve gap falls below - 20%. As can be seen from Figure 15 the “ruin probabilities” (here: the proportion of case where the reserve gap falls below – 20%) are much higher than the Proposition above suggests. The reason is that log-returns of equity markets are not normally distributed.18 17 For example, “irrational exuberances” - analyzed by Nobel laureate Robert Shiller (Shiller 2014) - strongly indicate that equity markets are not efficient. 18 If we take σ M = 40% instead of σ M = 20% in the example above the ruin probabilities according Eq 4 approximately correspond to the backtesting results in Figure 15. 33 04.11.2022 09:46 Figure 15: Probability that ˆ()t ρ falls below -20% for different α ∆-levels with underlying ALM parameters θ ∆ = 2% and β s = 100%/ 75%/ 50% (red/ orange/ blue column). Backtesting results in Figure 15 show that the LM-parameter α ∆ is a decisive instrument to manage the reserve gap. Figure 15 also indicates that high α ∆-levels could be counterproductive at least for β s = 75% or β s = 50%. 5. Conclusion and outlook Our analysis shows that collective defined contribution (CDC) schemes are more than just a nice idea. Based on the theoretical model presented in (Goecke, 2013) we could prove that the risk return profile for CDC plans is much better than that for DC plan. Even in a worst case scenario CDC plans perform better than individual DC plans. Clearly, there is no guarantee that an excellent performance observed in the past will recur in future. But it should be stressed that our analysis is based on observations of 721/2 years. Within this time span, we have observed extreme situations such as the oil price crisis 1973, stock market crashes and 34 04.11.2022 09:46 bubbles, high inflation rates, and extreme short and long term interest rates.19 The strength of the CDC plan is that it is self adjusting due to resilience factors in the ALM rules. With respect to the fundamental objective to ensure a fair participation in production factor capital, a CDC plan is superior to a pension plan with an interest rate guarantee. Looking back into the economic history, we should realize that long term interest rate guarantees either are worthless or are unbearable for the warrantor. Pension systems must be adjustable to the economic reality! It has been criticized that a CDC plan is just a zero-sum game: the advantage for one generation of savers is the disadvantage for the other. This view totally ignores the principle of risk sharing and insurance. A CDC scheme is an “insurance” contract where the saver pays a premium (in form of contributions into the collective reserve) and receives benefits (in form of payments out of the collective reserve). In A Theory of Justice John Rawls introduced the concept of fairness under the veil of ignorance.20 Following Rawls’ theory a transgenerational contract between savers will be regarded as fair, if the savers were to agree upon this contract provided they did not know in advance which generation they belong to – i.e. under the veil of ignorance. From this perspective a CDC plan is fair. However, it is also clear that a saver might feel unfairly treated if she or he was obliged to enter a heavily underfunded CDC scheme. CDC schemes are not an all-purpose answer to the pension challenge in an aging society. There are apparent limitations. For example a CDC plan can only work if the participants cannot withdraw money at will. Contractual compliance is essential for intergenerational risk transfer. That does not mean that CDC plans are Ponzi plans. Whenever the flow of new 19 The 1-month money market rates ranged between 13.33% (Sept. 1973) and – 0.6% (Dec. 2021). 20 Cf. (Rawls 1991), in particular Section 24 (pp. 118ff.) and Section 44 (pp. 251ff.) 35 04.11.2022 09:46 entrants stops, a CDC plan can be converted into a simple DC plan by setting the target reserve ratio to zero. Our backtesting analysis considered only the accumulation phase. We assumed that the wealth at retirement is paid out and reinvested outside the fund. An apparent extension of the model is to include a decumulation phase and then analyze the pension risk, e.g. the volatility of pensions in payment. It would be equally interesting to analyze population dynamics (growing or shrinking generations, winding off) in a CDC scheme. We hope at least, that our analysis has added one more argument in favour of CDC plans as a good alternative to DC and DB plans. 36 04.11.2022 09:46 Appendix: Data Description and Calibration of ERP and σ M We take REXP as a proxy for a portfolio of German government bonds and DAX as a proxy for a well-diversified portfolio of German equities. REXP and DAX are both performance indices calibrated such that REXP(end of year 1987) = 100 and DAX(end of year 1987) = 1000. The index REXP is calculated on the basis of a portfolio 30 different fictious German government bonds with coupon rates 6%, 7.5% and 9% and maturities between 1 and 10 years.21 The average Macauly duration is 5.02, 4,81 and 4.62 for interest rates 0%, 3% and 6%, respectively. Clearly, an investment into a REXP-Portfolio (or in German government bonds) is not risk-free in the strict sense since from month to month a pension asset manager will experience unexpected gains or losses from volatile market interest rates leveraged by the duration of the portfolio. Consistent to our pragmatic definition of a risk-free asset, we will substitute the constant risk-free interest rate of the c.t. model by the current yield of outstanding government bonds (described in detail below), which we interpret as the expected return of a REXP-investment. DAX is a weighted performance index comprising the 40 biggest German joint stock companies.22 End of month index values for REXP and DAX are provided by Deutsche Bundesbank from end of Jan. 1967 (REXP) and end of Dec. 1987 (DAX) onwards.23 The end of month index values (based on the last fixing on the last trading date of the particular month) is considered 21 Deutsche Bundesbank (2021); Statistische Fachreihe Kapitalmarktkennzahlen, Januar 2022, p. 17f. 22 Cf. Deutsche Börse AG (2022). 23 Time series BBK01.WU3141 and BBK01.WU046A 37 04.11.2022 09:46 to be identical with the index value of the first trading date of the following month just before the first fixing. Remark on notation: We use the time index t representing any point in time between t0 = 0 (representing the beginning of the first trading day of Jan. 1950) and T = 870∆ (representing the beginning of the first trading day of July 2022). Since we have a discrete time (d.t.) model based on monthly data, we sometimes write t = 01.mm.yyyy 24 instead of t = k ∆ 25 just to make transparent what date t is referring to. For REXPand DAX-index values not published by Deutsche Bundesbank, we use backward projections from different sources: Backward Projection for DAX(t) (t0 ≤ t ≤ 01.01.1988) We use the backward projection of Gielen (1994),26 adjusted such that the index value matches DAX(01.01.1988) = 1000. Backward Projection for REXP(t) (t0 ≤ t ≤ 01.01.1967) For 01.01.1950 ≤ t ≤ 01.01.1967 we use the following formula for backward projection ( ) 1 12 1( ) () ( ) 1 () 1 () D s s s it REXP t REXP t i t it −+ +∆  = +∆ ⋅ + ⋅ +  , (Eq A1) with REXP(01.02.1967) = 21.24,27 is(t) = risk free interest rate observed at time t and D = 4.80.28 We interpret is(t) to be the yield of an investment into German government bonds held to maturity invested at time t. However, the actual return on investment, calculated at time 24 The first of a month is identified with the beginning of the first trading date of this month. 25 k counts the number of months after 01.01.1950, i.e. k = 12·(yyyy-1950) + mm-1 26 Gielen, Gregor (1994): Können Aktienkurse noch steigen? Langfristige Trendanalyse des deutschen Aktienmarktes, Gabler-Verlag, Wiesbaden 1994 27 REXP index value for end of Jan. 1967 according to Deutsche Bundesbank data. 28 4.8 equals Macauly-Duration of REXP-portfolio for an interest rate of 3.19%. 38 04.11.2022 09:46 t+ ∆, depends also on the interest rate is(t+∆) and the duration of the bond portfolio. We stipulate that the market value at time t+1 of a German government bond investment of 100€ at time t is ( ) 1/12 1 () 100€ 1 ( ) 1( ) D s s s it it it +  ⋅+ ⋅  ++∆  . This is the motivation for (Eq A1). Risk free Interest Rates: is(t) (t0 ≤ t ≤ T):  01/1950 ≤ t ≤ 01/1953: Mean value of Central Bank lombard and discount rate29  02/1953 ≤ t ≤ 01/1954: Current yield of outstanding public bonds (5.0% coupon)30  02/1954 ≤ t ≤ 02/1956: Current yield of outstanding public bonds (5.5% coupon)31  03/1956 ≤ t ≤ 03/1960: Current yield of outstanding public covered bonds32  04/1960 ≤ t ≤ 07/2022: Current yield of outstanding German government bonds.33 Note that e.g. is(t) is the average of observed yields of the foregoing month; is(t) can be observed time t. Money-Market Index MMI(t) We calculate a Money Market index based on the following money market rates iMM(t) published by Deutsche Bundesbank:  01/1950 ≤ t ≤ 01/1960: day-to-day money (average of minimum and maximum)34  02/1960 ≤ t ≤ 01/1999: 1-month-money (monthly average)35  02/1999 ≤ t ≤ T: 1-month-EURIBOR (monthly average).36 We set MMI(t = 01/1950):= 100 and define recursively ( ) 1 12 ( ) () 1 () MM MMI t MMI t i t+∆ = ⋅ + . 29 Deutsche Bundesbank, Time Series BBK01.SU112 and BBK01.SU113 30 Calculated from published market values for covered bonds; Wirtschaft und Statistik, Montsberichte 06/1953 (p. 256*), 12/1953 (p. 676*) and 12/1954 (p. 652*) https://www.statistischebibliothek.de/mir/receive/DESerie_mods_00000012. 31 Evaluation of Deutsche Bundesbank Monthly Reports April – Dec. 1956 32 Deutsche Bundesbank, Time Series BBSIS.M.I.UMR.RD.EUR.MFISX.B.A150.A.R.A.A._Z._Z.A 33 Deutsche Bundesbank, Time Series BBSIS.M.I.UMR.RD.EUR.S1311.B.A604.A.R.A.A._Z._Z.A 34 Deutsche Bundesbank, Time Series BBK01.SU0102 and BBK01.SU0103 35 Deutsche Bundesbank, Time Series BBK01.SU0104 36 Deutsche Bundesbank, Time Series BBK01.SU0310 39 04.11.2022 09:46 Implicitly we hereby assume that at time t we can invest in a 1-month money market paper with interest rate iMM(t). Consumer Price Index CPI(t) for t = 01.01.1949 to t = 01.07.2022: CPI(t) is the concatenation of the following two time series:  01.01.1949 ≤ t ≤ 01.01.1991: Consumer Price Index for West Germany 37  01.02.1991 ≤ t ≤ 01.07.2022: Consumer Price Index38. Figure A1: Price Index, Performance of a DAX-, REXPand a Money Marketinvestment, normalized at 100 for t0 =1.1.1950, log-scaled Figure A1 illustrates the performance of a DAX-, REXPand MMI-investment starting with an initial capital of 100 on 01.01.1950. To illustrate the reverse projection for REXP in Figure A1 we have added a forward projection of REXP (from 01/1967 onwards) using formula (Eq 37 Statistisches Bundesamt (2021), Preise, Verbraucherpreise für Deutschland, Lange Reihen ab 1948, „Früheres Bundesgebiet, Preisindex für die Lebenshaltung, 4-Personen-Haushalte von Arbeitern und Angestellten mit mittlerem Einkommen, index basis = 100 (average for year 1995). This time series has been rescaled such that the index value for 01/1991 equals 86.9; the rescaled index value are rounded to two decimal points. 38 Statistisches Bundesamt, Fachserie 17, Reihe 7 (time series 61111-0002), index basis = 100 (average for year 2015) 40 04.11.2022 09:46 A1) – see dotted line. Obviously REXP can well be approximated by using only one interest rate (namely is(t) as described above) and a suitable estimation of the average (Macauly-) duration. Price adjusted indices REXPp(t), DAXp(t) and MMIp(t) for t = 01.01.1950 to 01.07.2022: 0 () () () () p CPI t REXP t REXP t CPI t = , 0 () () () () p CPI t DAX t DAX t CPI t = , 0 () () () () p CPI t MMI t MMI t CPI t = with t0 = 01.01.1950 ≤ t ≤ 01.07.2022 Price adjusted interest rates µ s(t) and µ s(t) for t = 01.01.1950 to 01.07.2022: ( ) ( 12 ) 1 () ln 1 () 12 ( ) ss CPI t t it CPI t µ −∆  = +   We use the log-interest rate on monthly basis to simplify the notation. We interpret µ s(t) as the expected real return of a risk-free investment at time t for the following month [t, t+ ∆] on the basis of the observed yield is(t) and the experienced depreciation of the foregoing year.39 µ s(t) can be calculated on the basis of information up to time t. Figure A2: Performance of a price-adjusted DAX-, REXPand Money Marketinvestment of 100 (log-scaled). 39 We prefer a price adjustment on a yearly basis to eliminate seasonal effect of the price index. Publikationsreihe „Forschung am ivwKöln“ Die Veröffentlichungen der Online-Publikationsreihe "Forschung am ivwKöln" (ISSN: 2192-8479) werden üblicherweise über Cologne Open Science (Publikationsserver der TH Köln) veröffentlicht. Die Publikationen werden hierdurch über nationale und internationale Bibliothekskataloge, Suchmaschinen sowie andere Nachweisinstrumente erschlossen. 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