**8. Conclusions and outlook**

26 Risk Management – Current Issues and Challenges

[27], [28].

**Cohort size**

**(x,n)=(30,20)**

**(x,n)=(40,20)**

**Cohort size**

**(x,n)=(30,20)**

**(x,n)=(40,20)**

that the chosen results for the normal distribution are only approximate, especially for small cohort sizes. In this respect, we think that the displayed figures are most likely lower bounds due to the fact that often a normal approximation rather underestimates than overestimates risk. A more detailed analysis of this point is left as open issue for further investigation (however, the use of the gamma approximation makes no big difference). On the other hand, solvency capital ratios of cohort sizes exceeding 10'000 policyholders and late contract times tend more and more to the lower limiting bound as expected from the central limit theorem. Fig. 2 visualizes these findings. In virtue of the made confidence level calibration, the VaR & CVaR solvency capital ratios are of the same order of magnitude. Finally, the considered example points out to another difficulty. Though almost negligible in absolute value, we note that the standard solvency capital ratios change their signs repeatedly over the time axis. In this respect, one can ask whether fixed transition shifts are the "crucial scenarios". As a response to this "biometric worst- and best-case scenarios" are proposed in Christiansen

100 53.815% 26.466% 17.602% 13.150% 10.483% 8.714% 4.628% 2.525% 1.464% 500 24.067% 11.836% 7.872% 5.881% 4.688% 3.897% 2.070% 1.129% 0.655% 1'000 17.018% 8.369% 5.566% 4.158% 3.315% 2.756% 1.463% 0.798% 0.463% 10'000 5.381% 2.647% 1.760% 1.315% 1.048% 0.871% 0.463% 0.252% 0.146% 100'000 1.702% 0.837% 0.557% 0.416% 0.332% 0.276% 0.146% 0.080% 0.046%

100 91.895% 45.158% 30.663% 23.298% 18.838% 15.846% 8.844% 5.223% 3.176% 500 41.097% 20.195% 13.713% 10.419% 8.425% 7.087% 3.955% 2.336% 1.421% 1'000 29.060% 14.280% 9.696% 7.368% 5.957% 5.011% 2.797% 1.652% 1.004% 10'000 9.190% 4.516% 3.066% 2.330% 1.884% 1.585% 0.884% 0.522% 0.318% 100'000 2.906% 1.428% 0.970% 0.737% 0.596% 0.501% 0.280% 0.165% 0.100%

100 138.7% 68.9% 45.7% 34.1% 27.1% 22.5% 11.8% 6.4% 3.8% 500 62.0% 31.1% 20.6% 15.3% 12.2% 10.1% 5.2% 2.8% 1.7% 1'000 43.9% 22.1% 14.6% 10.9% 8.6% 7.1% 3.7% 2.0% 1.2% 10'000 13.9% 7.3% 4.8% 3.5% 2.8% 2.3% 1.1% 0.6% 0.4% 100'000 4.4% 2.6% 1.7% 1.2% 0.9% 0.7% 0.3% 0.1% 0.1% **Limiting QIS5 ratio SCRt/VZt** 0.0% 0.5% 0.3% 0.1% 0.1% 0.0% -0.1% -0.1% 0.0% **QIS5 TC ratio = (SCRt + RMt)/VZt** -0.3% 0.3% 0.1% 0.0% 0.0% -0.1% -0.1% -0.1% 0.0%

100 235.6% 117.2% 79.3% 60.1% 48.5% 40.7% 22.7% 13.4% 8.1% 500 105.2% 52.6% 35.5% 26.9% 21.7% 18.2% 10.1% 6.0% 3.6% 1'000 74.3% 37.4% 25.2% 19.0% 15.3% 12.8% 7.1% 4.2% 2.5% 10'000 23.3% 12.1% 8.1% 6.0% 4.8% 4.0% 2.2% 1.3% 0.8% 100'000 7.1% 4.1% 2.7% 1.9% 1.5% 1.2% 0.7% 0.4% 0.2% **Limiting QIS5 ratio SCRt/VZt** -0.3% 0.4% 0.2% 0.0% 0.0% -0.1% -0.1% 0.0% 0.0% **QIS5 TC ratio = (SCRt + RMt)/VZt** -0.7% 0.2% 0.0% -0.1% -0.1% -0.1% -0.1% 0.0% 0.0%

**0 1 2 3 4 5 10 15 18**

**0 1 2 3 4 5 10 15 18**

**Contract Time**

**Contract Time**

**Table 2.** Coefficients of variation of the shifted prospective insurance loss

**Table 3.** VaR solvency capital ratios for the endowment with waiver of premium

Let us summarize the present work. We have derived a general solvency rule for the prospective liability, which has resulted in two simple liability VaR & CVaR target capital requirements. The proposed approach has been applied to determine the biometric solvency risk capital for a portfolio of general traditional life contracts within the Markov chain model of life insurance. Our main actuarial tools have been the backward recursive actuarial reserve formula and the theorem of Hattendorff. Based on this we have determined the conditional mean and variance of a portfolio's prospective liability risk and have used a gamma approximation to obtain the liability VaR & CVaR solvency capital. Since our first formulas include only the process risk and do not take into account the possibility of systematic risk, we have proposed either to shift the biometric transition probabilities, or apply a stochastic model, which allows for random biometric transition probabilities.

Similarly to [1], Section 8, the adopted general methodology is in agreement with several known facts as (i) the process risk is negligible for portfolios with increasing size and has a small impact on medium to large insurers (ii) all else equal, process risk will increase (decrease) with higher (lower) coefficients of variation (aggregated effect of both decrement rates and sums at risk). Another interesting observation has been made at the end of Section 6.3 that the model with shifted biometric transitions can be

viewed as a sub-model of the model with Poisson-Gamma time dependent biometric transitions.

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 29

[7] Milbrodt, H., Helbig, M. Mathematische Methoden der Personenversicherung. Berlin: De

[8] Wolthuis, H. Life Insurance Mathematics (The Markovian Model). Brüssel: CAIRE;

[9] Norberg, R. Thiele differential equation, Springer Online Reference Works.

[12] Norberg, R. Life insurance mathematics, In: Teugels, J., Sundt, B., editors. Encyclopedia

[13] Bühlmann, H. A probabilistic approach to long-term insurance (typically life insurance).

[14] Gerber, H.U. An Introduction to Mathematical Risk Theory. Huebner Foundation

[15] Gerber, H.U; Leung, B.P.K., Shiu, E.S.W. Indicator function and Hattendorff theorem.

[16] Hattendorff, K. Das Risiko bei der Lebensversicherung. Maisius' Rundschau der

[17] Kremer, E. Stochastic Life-Insurance Mathematics. Lecture Notes No. 5 – Special Issue.

[18] Patatriandafylou, A., Waters, H.R. Martingales in life insurance. Scandinavian Actuarial

[19] Bowers, N.L.; Gerber, H.U.; Hickman, J.C.; Jones, D.A., Nesbitt, C.J. Actuarial

[20] Frasier, W. M. Second to Die Joint Life Cash Values and Reserves. 1978; The Actuary:

[22] Margus, P. Generalized Frasier claim rates under survivorship life insurance

[23] Olivieri, A., Pitacco, E. Stochastic mortality: the impact on target capital.ASTIN Bulletin.

[24] Christiansen, M.; Denuit, M., Lazar, D. The Solvency II square-root formula for systematic biometric risk. Insurance: Math. and Economics. 2012; 50(2): 257-265. [25] Chuard, C. (2008). Zur Geschichte der Sterbetafeln. Vortrag an der GV der Schweiz.

[26] Saxer, W. Versicherungsmathematik I. Series of Comprehensive Studies in

[27] Christiansen, M. A sensitivity analysis of typical life insurance contracts with respect to

Kammer der Pensionskassenexperten, Bern. 2008 April 23. Available from: www.chuard.com/new/d/wpcontent/uploads/2009/03/hist\_sterbetafeln.pdf

the technical basis. Insurance: Math. and Economics. 2008; 42(2): 787-796.

Mathematics, vol. 79 (Reprint 1979). Berlin: Springer; 1955.

Hamburg: Verein Förderung angew. Math. Statistik & Risikotheorie; 1979.

[10] Gerber, H.U. Lebensversicherungsmathematik. Berlin: Springer; 1986. [11] Gerber, H.U. Life Insurance Mathematics. 3rd ed. Berlin: Springer; 1997.

Trans. 20th Int. Congress of Actuaries. 1976; vol. 4: 267-276.

Monograph, vol. 8. Homewood, Illinois: R.D. Irwin; 1979.

Gruyter; 1999.

2001.Available from:

http://eom.springer.de/T/t130080.htm.

of Actuarial Science. New York: J. Wiley; 2004.

North Amer. Actuarial J. 2003; 7(1): 38-47.

Mathematics. Itascal, IL: Society of Actuaries; 1986.

policies.North Amer. Actuarial J. 2002; 6(2): 76-94.

[21] Second to Die. Letters. 1978 June; The Actuary: 3.

Versicherungen. 1868; 18: 169-183.

Journal. 1984; 2: 210-230.

2009; 39(2): 541-563.

4.

1994.

Moreover, a detailed analysis for a single cohort of identical endowment contracts with waiver of premium by disability has been undertaken in Section 7. Besides a complete Markov chain specification, which seems to be missing in the literature, the numerical illustration has shown, as expected, that the cohort size is a main driving factor of process risk. Due to the statistical law of large numbers, the larger the cohort size the less solvency capital is actually required. In contrast to the life annuity "longevity risk" study in [1], the stochastic approach penalizes almost all insurers (except the very large ones) because the current standard approach prescribes almost negligible solvency capital ratios and does not measure explicitly the process risk effects.

The interested actuary might challenge the proposed approach with alternatives from other regulatory environments than Solvency II. Moreover, it is important to point out that a lot of technical issues remain to be settled properly. They are not only regulatory specific but also related to the complex mathematics of related software products and go beyond the Markov chain model. Today's life insurance contracts include many embedded options and are henceforth even more complex. A challenging issue is the definition of capital requirements for unit-linked contracts without and with guarantee and variable annuities with guaranteed minimum benefits (so-called variable GMXB annuities).
