wenyen hsu1 agency cost and bonus policy of participating policies wenyen hsu feng chia university...
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![Page 1: Wenyen Hsu1 Agency Cost and Bonus Policy of Participating Policies Wenyen Hsu Feng Chia University Email: wyhsu@fcu.edu.tw](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649ecb5503460f94bd9507/html5/thumbnails/1.jpg)
Wenyen Hsu 1
Agency Cost and Bonus Policy of Participating Policies
Wenyen Hsu
Feng Chia University
Email: [email protected]
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Wenyen Hsu 2
Table of Contents
The features of participating policies Literature Review Approach of the Paper Simulation Results Conclusions
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Wenyen Hsu 3
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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Wenyen Hsu 4
Policy value according rG
B(t)
P(t)Age
Face value
The Features of Participating Policies
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Wenyen Hsu 5
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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Wenyen Hsu 6
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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Wenyen Hsu 7
The Features of Participating Policies
Options embedded in a participating policy Bonus option Guaranteed rate Insolvency put option from insurer
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Wenyen Hsu 8
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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Wenyen Hsu 9
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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Wenyen Hsu 10
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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Wenyen Hsu 11
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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Wenyen Hsu 12
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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Wenyen Hsu 13
RAROC
RAROC: Risk adjusted return on capital
CaR: Capital at Risk
RAROC focuses on the left tail.
CaR
emiumPrRiskRAROC
_
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Wenyen Hsu 14
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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Wenyen Hsu 15
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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Wenyen Hsu 16
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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Wenyen Hsu 17
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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Wenyen Hsu 18
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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Wenyen Hsu 19
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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Wenyen Hsu 20
Assets Liabilities
Risky Asset
Zero Coupon Bond
Policy Reserve
Bonus Reserve
)(tV
)(tA
)(tC
)(tP
)(tB
)(tV )(tV
Simulation
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Wenyen Hsu 21
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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Wenyen Hsu 22
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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Wenyen Hsu 23
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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Wenyen Hsu 24
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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Wenyen Hsu 25
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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Wenyen Hsu 26
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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Wenyen Hsu 27
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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Wenyen Hsu 28
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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Wenyen Hsu 29
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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Wenyen Hsu 30
Efficient Frontiers with Different ex-ante Default Risks
0
5
10
15
20
10 15 20 25 30
5%
10%
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Wenyen Hsu 31
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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Wenyen Hsu 32
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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Wenyen Hsu 33
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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Wenyen Hsu 34
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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Wenyen Hsu 35
Thank You for Listening!