REGS 2013: Variable Annuity Guaranteed Minimum Benefits

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1 Department of Mathematics University of Illinois, Urbana-Champaign REGS 2013: Variable Annuity Guaranteed Minimum Benefits By: Vanessa Rivera Quiñones Professor Ruhuan Feng September 30, 2013 The author acknowledges support from National Science Foundation grant DMS "EMSW21-MCTP: Research Experience for Graduate Students". 1

2 2 1. INTRODUCTION A variable annuity is an insurance contract in which the insurer agrees to make periodic payments to the policyholder at some future date. It is purchased by a single payment or various payments which are then put in subaccounts and invested in various funds depending on the investment objective of the policyholder. A guaranteed minimum maturity benefit (GMMB) guarantees the policyholder a specific amount at the time of the maturity of the contract. This helps protect the insured against adverse economic scenarios. The purpose of this project was to explore the insurance liability of the guarantee benefit of the average of a pool of contracts and to compare with previous results obtained in the work for Klein (2012) and Feng and Volkmer (2013). In addition, we wished to extend these results to two types of risk measures, the Value at Risk(VaR) and the Conditional Tail Expectation(CTE), while improving the numerical implementation presented in their work. The α-quantile risk measure, also referred to as Value at Risk (VaR), is used to assess the overall risks of insurance liabilities. This measure is defined as, V α := inf{v: P[L (i) L (i) > V α ]}, 0 α 1. which is the interpreted as the minimum capital required to ensure that there are sufficient funds to cover the future liability with the probability of at least α. We also considered the Conditional Tail Expectation (CTE), which is calculated as CTE α := E[L (i) L (i) > V α ] where α is the confidence level. It can be defined as the capital required to cover the average liablity given that it exceeds an α-percentile. 2. COMPUTING RISK MEASURES Numerical PDE methods have been used for pricing and hedging in finance and actuarial sciences literature. Feng (2013) provides two propositions that describe how these methods apply to the computation of the quantile risk measure V α and the conditional tail expectation risk measure CTE α which are discussed below Individual Model. Define the tail distribution of the net liability L as P(y) : P(L > y). As a survival probability, P is a decreasing function on y. We want to determine V α such that for a given level α, 1 α = P(V α ). For the purpose of risk management, we are only concerned with the risk measure for the percentile level α such that V α > 0. So, suppose V α > 0 then P is determined by P(V α ) = T p x u(0, z 0 ) where z 0 := (1 + m e )ert m e and T p x denotes the probability that the policy holder at age x at the time of valuation survives T periods.the function satisfies the backward parabolic PDE with terminal condition u t + σ 2 2 (z q(t))2 2 y = 0, t 0, z R (2.1) z2 where for 0 t T, u(t, z) = I {z<k} q(t) := m e (e(t t) 1), := µ r m + σ 2 2, K = e rt G V α F 0. (2.2) and m e represents the rider charges, m is the insurance fee, r is the constant risk-free force of interest, F 0 is the sub-account value at time t = 0, and G is the guaranteed amount. We can compute the

3 3 conditional tail expectation risk measure in a similar way. is determined by CTE α := E[L (i) L (i) > V α ] = 1 1 α E[LI {L>V α }] CTE α := T p x 1 α [V α + F 0 u(0, z 0 )], where u satisfies the PDE in (2.1) with the parameters in (2.2) and the terminal condition u(t, z) = (K z) Average Model. PDE methods may also be used to compute measures in the average model. Similarily to the individual model, consider the tail distribution of the net liability, P (V α ) = T p x u(0, z 0) The function u satisfies the backward parabolic PDE with terminal condition where for 0 t T u t + σ 2 2 (z q (t)) 2 2 y = 0, t 0, z R (2.3) z2 u(t, z) = I z<k q (t) := m e T t e(t s) 2 T sp x σ ds, := µ r m + Tpx 2, K = T p x e rt G V α. (2.4) Tp x F 0 The constant z 0 is given by z 0 := q (0) + et.we can compute the conditional tail expectation risk measure in a similar way. is determined by CTE α := E[L L > V α ] = 1 1 α E[L I {L >V α }] CTE α := T p x 1 α [V α + F 0 u(0, z 0 )], where u satisfies the PDE in (2.3) with the parameters in (2.4) and the terminal condition u(t, z) = (K z) +. Life-tables are used to determine the distribution of t p x, which is the probability that (x) survives at least to age (x + t) to reduce the amount of computation. However, life-tables only allow us to compute t p x for integer years and therefore assumptions need to be made to address the periods in between years. We can use two fractional age assumptions to reduce the computational time that entails computing q (t) and account for these non-integer periods. These are described in detail in the work presenter by Feng (2013). By using these fractional age assumptions (Dickson et al., 2009, Chapter 3), we obtain that for 0 t T q (t) = m T t 1 e p(k, k + 1) + p( T t, T t) Tp x k=0 where. is the floor function and for 0 s 1, p(k, k + s) := k+s k eu up x du = k p x ek s 0 et tp x+k dt.

4 Uniform Distribution of Deaths. For 0 s 1, it is assumed that s q x = sq x. Then, under this assumption [ es p(k, k + s) = k p ek 1 ses es ] 1 x q x+k + q x+k Constant Force of Mortality. For 0 s 1, it is assumed that s p x = (p x ) s. Then, under this assumption p(k, k + s) = k p x ek (e p x+k ) s 1 + lnp x+k. 3. SIMULATIONS IN MATLAB In this section the probability both risk measures were computed for a set of parameters. To obtain the probability of survival for the individuals Table 1 obtained from Feng and Volkamer (2012) was used. TABLE 1. Predicted mortality rates of a male at the age of 65 x q x k kp TABLE 2. Set of parameters Parameter Value µ 0.09 σ 0.5 r 0.04 m 0.01 T 10 m e G 1 F SIMULATION After obtaining numerical results for the average and individual model a Maple simulation code provided by Professor Feng was translated into Matlab and 10,000 simulations were performed using the parameters in Table 2. The empirical distributions for the net liabilities obtained from the simulation for the average and individual model can be observed in Figure 1 and Figure 2. In addition, the tail probability and conditional tail expectation, for α = {0.20, 0.40} for both models are expressed in the Table 3. TABLE 3. Risk measures in the average mode and individual model Risk Measure Average model Individual model V 0.20 V 0.40 V 0.20 V 0.40 Tail Probability Conditional Tail Expectation CONCLUSION Through this project, it was shown that the risk measures, Value at Risk (VaR) and the Conditional Tail Expectation (CTE), can be computed using numerical methods and are comparable to those in the literature. In addition, the time of computation was significantly reduced and the previous results were extended into non-discrete time periods by using fractional age assumptions.

5 5 (A) Average Model (B) Individual Model FIGURE 1. Empirical distributions of net liabilities FIGURE 2. Comparison of empirical distributions of net liability in the average model and individual model 6. REFERENCES Dickson, D. C. M., Hardy, M. R., and Waters, H. R. (2009). Actuarial mathematics for life contingent risks. International Series on Actuarial Science. Cambridge University Press, Cambridge. Feng, R. and Volkmer, H. (2012). Analytical calculation of risk measures for variable annuity guaranteed benefits. Insurance Math. Econom., 51(3): Feng,R.(2013) Computations of risk measues for guaranteed minimum benefit by a Numerical PDE Methos. Preprint. Schwaerzler, R. (2012). The method of control variates applied to the estimation of valueat-risk for variable annuities. Master s thesis, University of Wisconsin-Milwaukee. Klein, S. (2012) Determination of the loss distribution for variable annuity guarantee benefits using partial differential equation, Master s Thesis, University of Wisconsin-Milwaukee.