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Agency Cost and Bonus Policy of Participating Policies

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Agency Cost and Bonus Policy of Participating Policies

Wenyen Hsu

Feng Chia University

Email: [email protected]

- The features of participating policies
- Literature Review
- Approach of the Paper
- Simulation Results
- Conclusions

- 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.

The Features of Participating Policies

Face value

B(t)

Policy value according rG

P(t)

Age

- In mathematic form,
- rp(t) policyholder interest rate in t
- rG guaranteed interest rate
- B(t) policyholder reserve in t
- P(t) policyholder reserve in t
- γ target buffer ratio
- α distribution ratio

- Therefore, the interest rate guarantee implies a floor of the credited rate.
- The dividend mechanism is an option element of the contract.

- Options embedded in a participating policy
- Bonus option
- Guaranteed rate
- Insolvency put option from insurer

- 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?

- 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.

- 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.

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

- 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

- Introducing risk capital

- RAROC: Risk adjusted return on capital
- CaR: Capital at Risk

- 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.

- 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.

- May be not!

- 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.

- 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.

- 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.

- 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.

Assets

Liabilities

Risky Asset

Policy Reserve

Zero Coupon Bond

Bonus Reserve

Simulation

- 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).

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

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

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

Efficient Frontiers with Various Participating Levels - Insurer

γ= 0, rG=0.04,P(0)=100, r(0)=4%, β= 0.49~0.99, ρ=-0.1, T=5, VaR=95%

$

VaR

Efficient Frontiers with Various Participating Levels - Insurer

γ= 0, rG=0.04,P(0)=100, r(0)=4%, β= 0.49~0.99, ρ=-0.1, T=5, VaR=95%

Efficient Frontiers with Different Guaranteed Rates

γ= 0, P(0)=100, r(0)=4%, β= 0.49~0.99, ρ=-0.1, T=5, VaR=95%

$

VaR

γ= 0, P(0)=100, r(0)=4%, β= 0.49~0.99, ρ=-0.1, T=5, VaR=95%

γ= 0, rG=0.04,P(0)=100, r(0)=4%, β= 0.49~0.99, ρ=-0.1, T=5

- 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.

- 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.

- 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.

Thank You for Listening!