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Optimal Reciprocal Insurance Contract for Loss Aversion Preference Hung-Hsi Huang 黃鴻禧 National Chiayi University Ching-Ping Wang 汪青萍 National Kaohsiung University of Applied Sciences. Purpose and Abstract. - PowerPoint PPT Presentation
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Optimal Reciprocal Insurance Contract for Loss Aversion
Preference
Hung-Hsi Huang 黃鴻禧 National Chiayi University
Ching-Ping Wang 汪青萍 National Kaohsiung University of Applied Sciences
Purpose and Abstract
The reciprocal insurance contract is defined by maximizing the weighted expected wealth utility of the insured and the insurer.
For fitting the gap of the optimal insurance field, this study develops the reciprocal optimal insurance under the four situations:– risk-averse insured versus risk-averse insurer– risk-averse insured versus loss-averse insurer– loss-averse insured versus risk-averse insurer– loss-averse insured versus loss-averse insurer.
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系2
Motivation
Kahneman and Tversky (1979) states that investors are characterized by a loss-averse utility preference, in which individuals are much more sensitive to losses than to gains.
Wang and Huang (2012) and Sung et al. (2011) have investigated the optimal insurance contract for maximizing a risk-averse insured’s objective against a risk-neutral insurer.
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系3
Loss Aversion Behavior Evidence
Benartzi and Thaler (1995) found that the equity premium is consistent with the loss aversion utility.
Hwang and Satchell (2010) demonstrated that investors in financial markets are more loss averse than assumed in the literature.
In addition to individual loss aversion, several scholars have drawn on loss aversion to explain executive behaviors or institution risk-taking behaviors. – Devers et al. (2007)– O’Connell and Teo (2009)
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系4
Optimal Insurance Studies
Raviv (1979, AER) is the pioneer who uses the optimal control theory in deriving the optimal insurance contract.
Extension
– Uninsurable asset: Gollier (1996, JRI)
– VaR (value-at-risk) constraint: Wang et al. (2005, GRIR), Huang (2006, GRIR), Zhou and Wu (2009, GRIR)
– Expected loss constraint: Zhou and Wu (2008, IME)
– Loss limit: Zhou et al. (2010, IME)
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系5
Optimal Insurance for Prospect Theory
Wang and Huang (2012) developed an optimal insurance for loss aversion insured.– The representative optimal insurance form is
the truncated deductible insurance. – When losses exceed a critical level, the insured
retains all losses and adopts a particular deductible otherwise.
Sung et al. (2011) studied the optimal insurance policy with convex probability distortions.– Under a fixed premium rate, the results showed
that either an insurance layer or a stop-loss insurance is an optimal insurance policy.
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系6
Reciprocal Reinsurance
Cai et al. (2013, JRI) designed the optimal reinsurance treaty f that maximize
the joint survival probability
and the joint profitable probability.
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系7
Loss, Premium, Wealth, Utility
Loss X and Premium P
Insured’s and Insurer’s final wealth
Objective of the optimal reinsurance
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]~[xx E )]~([ xII E
)(IhP 0)( h 0)0( h
)~(~ and )~(~~00 xIPwwxIxPWW
weight λ)],~()~
([ wVWU E
S-shaped Loss Aversion Utility
Insured’s loss aversion utility
Insurer’s loss aversion utility
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ˆif
ˆif
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0
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)(0)( 22 uu
)(0)( 11 vv
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The Optimal Reciprocal Insurance Form
Optimal indemnity schedule for RAU-RAU
Optimal indemnity schedule for RAU-LAU
Optimal indemnity schedule for LAU-RAU
Optimal indemnity schedule for LAU-LAU
RAU = Risk Aversion Utility
LAU = Loss Aversion Utility
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Optimal indemnity schedule for RAU-RAU
By calculus of variations, the Hamiltonian
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系11
)(and)( with
)()]()([)]~()~
([Maximize
00
0)(0
xIPwwxIxPWW
dxxfwVWUwVWUExxI
)())}(())(({
)()}()({Maximize
00
)(0
xfxIPwVxIxPWU
xfwVWUHxxI
)(ˆ)(0)()]()([/ :FOC xIxIxfwVWUIH
0)()]()([/ :SOC 22 xfwVWUIH
Optimal indemnity schedule for RAU-RAU
Proposition 1 for RAU-RAU:
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系12
0)0(ˆif}0),(ˆmax{
0)0(ˆif}),(ˆmin{)(*
IxI
IxxIxI
1)(ˆ0
VU
U
ARAARA
ARAxI
)(/)( WUWUARAU VVWARA RR /)~
(
Unconstrained and Constrained Optimal Insurance
Unconstrained optimal reinsurance
Optimal insurance
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Optimal indemnity schedule for RAU-LAU
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)(and)( with
)(]})ˆ()ˆ({)([
}])~ˆ()ˆ~({)~
([)]~()~
([Maximize
00
0 ˆ2ˆ~1
ˆ2ˆ~1)(0
xIPwwxIxPWW
dxxfwwvwwvWU
wwvwwvWUEwVWUE
wwww
wwwwxxI
11
11
Panel A Panel B Panel C
Optimal indemnity schedule for RAU-LAU
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Panel A
Optimal indemnity schedule for RAU-LAU
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Panel B
Optimal indemnity schedule for RAU-LAU
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系17
Panel C
Optimal indemnity schedule for RAU-LAU
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)(and)( with
)(}])ˆ()ˆ([)({Maximize
00
ˆ2ˆ1)(0
xIPwwxIxPWW
xfwwvwwvWUH wwwwxxI
11
ww
xfwwvWUIH
xfwwvWUIH
xfwwvWUH
ˆif
0)()}ˆ()({/
)()}ˆ()({/
)(})ˆ()({
122
1
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xfwwvWUH
ˆif
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222
2
2
Optimal indemnity schedule for RAU-LAU
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系19
λβ largefor }},0),(ˆmin{max{)(* xxIxI
λβ smallfor
0ˆˆif
ˆ0ˆif
ˆˆ0if}),(ˆmin{
)(
1
1ˆ
1ˆˆ
*
0
22
IIx
IIx
IIxxIx
xI xx
xxxx
1
11
Optimal indemnity schedule for RAU-LAU
Panel A. for large λβ
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}},0),(ˆmin{max{)(* xxIxI
Optimal indemnity schedule for RAU-LAU
Panel B. for small λβ
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系21
0ˆˆif
ˆ0ˆif
ˆˆ0if}),(ˆmin{
)(
1
1ˆ
1ˆˆ
*
0
22
IIx
IIx
IIxxIx
xI xx
xxxx
1
11
Optimal indemnity schedule for LAU-RAU
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系22
)(and)( with
)()}()ˆ()ˆ({
)]ˆ~()ˆ~()ˆ~
([)]~()~
([Maximize
00
0 ˆ2ˆ1
ˆ~2ˆ~1
)(0
xIPwwxIxPWW
dxxfwVWWuWWu
wwVWWuWWuEwVWUE
WWWW
WWwWxxI
11
11
Panel A Panel B Panel C
Optimal indemnity schedule for LAU-RAU
Panel A. for small λ/α
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系23
}},0),(ˆmin{max{)(* xxIxI
Optimal indemnity schedule for LAU-RAU
Panel B. for large λ/α
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系24
0ˆor0ˆif0
ˆ0ˆˆif}),(min{
ˆˆ0ˆif}),(ˆmin{
)(
1
21
21ˆ
*0
II
IIIxxI
IIIxxI
xIxx1
Optimal indemnity schedule for LAU-LAU
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系25
)( and )(with
)]~()~
([Maximize
00
)(0
xIPwwxIxPWW
wVWUExxI
}])~ˆ()ˆ~({
)ˆ()ˆ~([
ˆ2ˆ1
ˆ2ˆ1
wwww
WWWW
wwvwwv
WWuWWuE
11
11
0ˆ2ˆ1
ˆ2ˆ1
)(}])ˆ()ˆ([
)ˆ()ˆ([{
dxxfwwvwwv
WWuWWu
wwww
WWWW
11
11
)(}])ˆ()ˆ([
)ˆ()ˆ([{ Maximize
ˆ2ˆ1
ˆ2ˆ1)(0
xfwwvwwv
WWuWWuH
wwww
WWWWxxR
11
11
Optimal indemnity schedule for LAU-LAU
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系26
Panel A
Optimal indemnity schedule for LAU-LAU
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系27
Panel B
Optimal indemnity schedule for LAU-LAU
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Panel C
Optimal indemnity schedule for LAU-LAU
Panel A. for small λ
Panel B. for large λ
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xxI )(*
0)(* xI
Optimal indemnity schedule for LAU-LAU
Panel C.for small λ
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0ˆˆif
ˆ0ˆif
ˆˆ0if}),(ˆmin{
)(
1
1ˆ
1ˆˆ
*
0
22
IIx
IIx
IIxxIx
xI xx
xxxx
1
11
Optimal indemnity schedule for LAU-LAU
Panel D.for large λ
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0ˆor0ˆif0
ˆ0ˆˆif}),(min{
ˆˆ0ˆif}),(ˆmin{
)(
1
21
21ˆ
*0
II
IIIxxI
IIIxxI
xIxx1
Optimal Premium and Coverage Level
For step 1, Section 3 derives the optimal indemnity schedule being a function of premium P.
Subsequently, this section aims to determine the optimal premium and the coverage level.
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系32
)];~([,)(
);(and);( subject to
)()]()([)]~()~
([Maximize
00
0
P
PP
P
xIEIPIh
xIPwwxIxPWW
dxxfwVWUwVWUE
www.ncyu.edu.tw/fin 國立嘉義大學財務金融系33
Conclusions and Further Works