Optimal Multi-Period Asset Allocation for Life Insurance Policies Hong-Chih Huang Associate Professor, Department of Risk Management and Insurance, National Chengchi University, Taiwan Yung-Tsung Lee Ph.D. Student, Department of Risk Management and Insurance, National Chengchi University, Taiwan

Purpose  employ a multi-assets model and investigate the multi-period optimal asset allocation on life insurance reserves. for a general portfolio of life insurance policy

Literature Review  Marceau and Gaillardetz (1999)  Huang and Cairns (2006).

Contributions  provide a good contribution on solving multi-period asset allocation problems of the application of life insurance policies.  find that the optimal investment strategy will be very different under different durations of policy portfolios.

The liability model

Multi-asset return model

2-2. Multi-asset model

Moments of Loss Functions

Three Cases of Policy Portfolios  All cases have 10 endowments policies, with the same term 10 years and the same sum assumed 1.  Case A: 10 new policies at the valuation date Case B: 10 policies with different uniform maturity dates Case C: The maturity date is selected randomly.

Three Cases of Policy Portfolios  The maturity dates of case C are as follows: Maturity dates Policy amounts

Optimal asset allocation  Single-Period Rebalance  Multi-Period Rebalance

Single-Period Rebalance Mean – variance plot: case B

Mean – variance plot: case B  An efficient frontier can be found at the left part of the plot.  Insurance company can minimize variance of loss under a contour line of mean; or minimize mean under a contour line of variance.

Objective Function

Optimal Asset Allocation Single-period rebalance Casecashlong bondstock A B C

Optimal Asset Allocation Multi-period rebalance case A and case B

Optimal Asset Allocation Multi-period rebalance case A and case B (with short constraints )

4-2. Multi-period rebalance - case A and case B  The holding pattern of riskless/risky asset are totally different between case A and case B, regardless of a short constrain exist or not.  Under case A, the proportion of cash is increasing and the proportion of risky assets is decreasing; whereas an opposite pattern arise under case B.

Optimal Asset Allocation Multi-period rebalance-case C

Multi-period rebalance- case C  Due to the randomness of the maturity dates of the policies, the optimal investment strategy appears a saw-toothed variation, whereas the pattern is similar with case B (the uniform case).  The optimal asset allocation with short constrain under case C is almost the same as the without constrain one, so we display the result of without constrain only.

Sensitivity Analysis of k The optimal asset allocations of multi-period rebalance, k=0.5, 1 and2

Sensitivity Analysis of asset model High excess mean: The optimal asset allocations under case B

Sensitivity Analysis of asset model High variance: The optimal asset allocations under case B

The Case of large sample  We examine the optimal asset allocations under 4 policy portfolios.  These 4 portfolios have a same statistic property: the maturity dates of a same portfolio has p-value under chi- square goodness of fit test.  The null hypothesis is that the maturity dates are selected form a discrete uniform distribution.  Thus, these 4 portfolios are unlike uniformly distributed in a statistical sense.

The Case of large sample The optimal asset allocation of the 4 special portfolios

The Case of large sample The optimal asset allocation of a specific portfolio

6. Conclusion  This paper successfully derives the formulae of the first and second moments of loss functions based on a multi-assets return model.  With these formulae, we can analyze the portfolio problems and obtain optimal investment strategies.  Under single-period rule, we found an efficient frontier in the mean-variance plot. This efficient frontier can be found under an arbitrary policies portfolio.

6. Conclusion  In multi-period case, we found that the optimal asset allocation can vary enormously under different policy portfolios. A. “Top-Down” strategy for a single policy B. “Down-Top” strategy for a portfolio with numbers of policies