Wenyen Hsu 1

Agency Cost and Bonus Policy of Participating Policies

Wenyen Hsu

Feng Chia University

Email: wyhsu@fcu.edu.tw

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Table of Contents

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

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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.

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Policy value according rG

B(t)

P(t)Age

Face value

The Features of Participating Policies

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The Features of Participating Policies

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

)})1(

)1((,max{)(

tP

tBrtr Gp

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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.

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The Features of Participating Policies

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

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Questions

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?

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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.

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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.

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Literature Review

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

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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

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RAROC

RAROC: Risk adjusted return on capital

CaR: Capital at Risk

RAROC focuses on the left tail.

CaR

emiumPrRiskRAROC

_

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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.

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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.

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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.

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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.

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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.

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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.

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Assets Liabilities

Risky Asset

Zero Coupon Bond

Policy Reserve

Bonus Reserve

)(tV

)(tA

)(tC

)(tP

)(tB

)(tV )(tV

Simulation

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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).

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Asset Side

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:

ttt dWdtrbadr ][

tA dZtdWdttA

tdA 21)([)(

)(

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Liability Side

The Liability Side of Balance Sheet policyholder interest rate in t

Value of policy in year t

)})1(

)1((,max{)(

tP

tBrtr Gp

1))(1( tpt PtrP

t

ipt irPP

10 ))(1(

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Simulation

)})0(

)0((,max{)1( P

Brr Gp

)0())1(1()1( PrP p

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

Determine

)1()1()1( PAB

)})1(

)1((,max{)2( P

Brr Gp

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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%

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40 50

α =0

α =0.25

α =0.5

α =0.75

α =1

VaR

$

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α β VaR(95) ROR

0 61% 38.042 86.55

0.25 64% 32.198 84.56

0.5 71% 23.797 73.60

0.75 73% 20.302 56.04

1 75% 18.636 37.76

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%

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Efficient Frontiers with Various Participating Levels - Policyholders

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 5 10 15 20 25

α =0

α =0.25

α =0.5

α =0.75

α =1

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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

0

5

10

15

20

10 15 20 25 30

rg=4%

rg=3%

VaR

$

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Efficient Frontiers with Different Guaranteed Rates

αrG=4% rG=3%

β VaR(95) ROR% β VaR(95)

ROR%

0.75 73% 20.302 56.04 78% 17.889 66.31

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

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Efficient Frontiers with Different ex-ante Default Risks

0

5

10

15

20

10 15 20 25 30

5%

10%

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Efficient Frontiers with Different ex-ante Default Risks

α

Prob=0.05 Prob=0.10

β VaR(95) ROR% β VaR(90)

ROR%

0.75 73% 20.302 56.04 77% 14.961 77.97

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

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Conclusions

The frontier present the investment opportunity sets for insurers.

The risk premium decreases with higher α. Therefore, insurers are likely to become more c

onservative with higher α since the payoff of additional risk decreases.

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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.

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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.

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Thank You for Listening!