Wenyen Hsu1 Agency Cost and Bonus Policy of Participating Policies Wenyen Hsu Feng Chia University

Wenyen Hsu2 Table of Contents The features of participating policies Literature Review Approach of the Paper Simulation Results Conclusions

Wenyen Hsu3 The Features of Participating Policies Policyholders share the surplus accumulated by the insurer because of deviations of actual from assumed experience. Mortality rate Interest rate Expense ratio The assumptions are relatively conservative.

Wenyen Hsu4 Policy value according r G B(t) P(t) Age Face value The Features of Participating Policies

Wenyen Hsu5 The Features of Participating Policies In mathematic form, r p (t) policyholder interest rate in t r G guaranteed interest rate B(t) policyholder reserve in t P(t) policyholder reserve in t γ target buffer ratio α distribution ratio

Wenyen Hsu6 The Features of Participating Policies Therefore, the interest rate guarantee implies a floor of the credited rate. The dividend mechanism is an option element of the contract.

Wenyen Hsu7 The Features of Participating Policies Options embedded in a participating policy Bonus option Guaranteed rate Insolvency put option from insurer

Wenyen Hsu8 Questions Does the fact that policyholders share the upside potential while insurers retain all the downside risk alter the investment incentives of insurers? How these options interact with each other?

Wenyen Hsu9 Literature Review Grosen and Jorgensen (2000) Propose a formula for credited interest rate and argue the participating policies consist a risk free bond element and an option element Assume insurer invests in risky assets and simulate the value of participating policies in terms of the policyholders under various combined of α, γ and asset risk.

Wenyen Hsu10 Literature Review However, the paper assumes Only bond investment Value of a policy does not depend only on the demand side, supply side’s behavior also matters. Do not incorporate capital.

Wenyen Hsu11 Literature Review Iwaki and Yumae (2004) Incorporate the supply side’s decision. Add capital in the model Find the efficient frontier for insurer

Wenyen Hsu12 Approach of the Paper Want to improve theory by Introducing risk capital Risk Adjusted Return on Capital (RAROC) Incentive effect of participating policies on insurer’s investment decisions Participating levels Guaranteed rates Default risks

Wenyen Hsu13 RAROC RAROC: Risk adjusted return on capital CaR: Capital at Risk RAROC focuses on the left tail.

Wenyen Hsu14 Incentive Problems The features of participating policies A combination of interest rate guarantee and an option element The value of the option depends on the risk of asset portfolio More volatile assets lead to higher value of the option for policyholders and more capital for stockholders.

Wenyen Hsu15 Incentive Problems Would the insurer increase the stock assets to enhance the value of option? May be not! Most of returns would accrue to policyholders but stockholders bear the risk. Such incentive problem becomes more severe as the share (α) of the return to policyholders increases.

Wenyen Hsu16 Incentive Problems Since insurers share return with policyholders but retain all the downside risk. The payoff of the policies to insurers is asymmetric. Therefore, this paper uses the RAROC, instead of the Sharpe Index.

Wenyen Hsu17 Hypotheses Holding probability of default constant, There exists an one-to-one relationship between participating ratio and risk-return for policyholders. Higher guaranteed rates lead to more aggressive investment policies. Higher ex-ante default risks lead to more conservative investment policies.

Wenyen Hsu18 Simulation Assumptions and constraints Insurers operate in a perfect financial markets Expense charges, lapses and mortality are ignored. The insurer offers only a participating policy, expiring at time T, T>0.

Wenyen Hsu19 Simulation At time t=0, the policyholder pays a single premium for a 5-year, with minimum guaranteed benefit participating policy. The dividend is credited each year.

Wenyen Hsu20 AssetsLiabilities Risky Asset Zero Coupon Bond Policy Reserve Bonus Reserve Simulation

Wenyen Hsu21 Asset Side Two assets a risky asset A(t) and a zero coupon bond C(t). Asset allocation factor β, denotes the proportion of the initial zero coupon bond C(0), i.e. C(0) = βV(0).

Wenyen Hsu22 Asset Side By Vasicek (1997) model, the dynamics of risk free interest rate r t follows the stochastic differential equation: The portfolio of the risky asset A(t) is assumed to follow the stochastic process:

Wenyen Hsu23 Liability Side The Liability Side of Balance Sheet policyholder interest rate in t Value of policy in year t

Wenyen Hsu24 Simulation Valuation of Participating Policy – Grosen and Jørgensen (2000) Determine Simulate A(1) Calculate Determine

Wenyen Hsu25 Efficient Frontiers with Various Participating Levels - Insurer γ= 0, r G =0.04, P(0)=100, r(0)=4%, β= 0.49~0.99, ρ=-0.1, T=5, VaR=95% VaR $

Wenyen Hsu26 α βVaR(95)ROR 0 61% % % % % Efficient Frontiers with Various Participating Levels - Insurer γ= 0, r G =0.04, P(0)=100, r(0)=4%, β= 0.49~0.99, ρ=-0.1, T=5, VaR=95%

Wenyen Hsu27 Efficient Frontiers with Various Participating Levels - Policyholders

Wenyen Hsu28 γ= 0, P(0)=100, r(0)=4%, β= 0.49~0.99, ρ=-0.1, T=5, VaR=95% Efficient Frontiers with Different Guaranteed Rates VaR $

Wenyen Hsu29 Efficient Frontiers with Different Guaranteed Rates α r G =4%r G =3% βVaR(95)ROR%βVaR(95)ROR% % % γ= 0, P(0)=100, r(0)=4%, β= 0.49~0.99, ρ=-0.1, T=5, VaR=95%

Wenyen Hsu30 Efficient Frontiers with Different ex-ante Default Risks

Wenyen Hsu31 Efficient Frontiers with Different ex-ante Default Risks α Prob=0.05Prob=0.10 βVaR(95)ROR%βVaR(90)ROR% % % γ= 0, r G =0.04, P(0)=100, r(0)=4%, β= 0.49~0.99, ρ=-0.1, T=5

Wenyen Hsu32 Conclusions The frontier present the investment opportunity sets for insurers. The risk premium decreases with higher α. Therefore, insurers are likely to become more conservative with higher α since the payoff of additional risk decreases.

Wenyen Hsu33 Conclusions If the slope of frontier measures the risk premium, the risk premium decreases with higher α. Therefore, insurers are likely to become more conservative with higher α since the payoff of additional risk decreases.

Wenyen Hsu34 Conclusions There exists an one-to-one relationship between participating ratio and risk-return for policyholders. Higher guaranteed rates lead to more aggressive investment policies. Higher ex-ante default risks lead to more conservative investment policies.

Wenyen Hsu35 Thank You for Listening!