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Optimal Reciprocal Insurance Contract for Loss Aversion Preference Hung-Hsi Huang 黃黃黃 National Chiayi University Ching-Ping Wang 黃黃黃 National Kaohsiung University of Applied Sciences

Purpose and Abstract

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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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Page 1: Purpose and Abstract

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

Page 2: Purpose and Abstract

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

Page 3: Purpose and Abstract

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

Page 4: Purpose and Abstract

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

Page 5: Purpose and Abstract

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

Page 6: Purpose and Abstract

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

Page 7: Purpose and Abstract

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

Page 8: Purpose and Abstract

Loss, Premium, Wealth, Utility

Loss X and Premium P

Insured’s and Insurer’s final wealth

Objective of the optimal reinsurance

www.ncyu.edu.tw/fin 國立嘉義大學財務金融系8

]~[xx E )]~([ xII E

)(IhP 0)( h 0)0( h

)~(~ and )~(~~00 xIPwwxIxPWW

weight λ)],~()~

([ wVWU E

Page 9: Purpose and Abstract

S-shaped Loss Aversion Utility

Insured’s loss aversion utility

Insurer’s loss aversion utility

www.ncyu.edu.tw/fin 國立嘉義大學財務金融系9

ˆif

ˆif

ˆif

)ˆ(

0

)ˆ(

)(

2

1

WW

WW

WW

WWu

WWu

WU

ww

ww

ww

wwv

wwv

wV

ˆif

ˆif

ˆif

)ˆ(

0

)ˆ(

)(

2

1

)(0)( 11 uu

)(0)( 22 uu

)(0)( 11 vv

)(0)( 22 vv

Page 10: Purpose and Abstract

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

www.ncyu.edu.tw/fin 國立嘉義大學財務金融系10

Page 11: Purpose and Abstract

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

Page 12: Purpose and Abstract

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 /)~

(

Page 13: Purpose and Abstract

Unconstrained and Constrained Optimal Insurance

Unconstrained optimal reinsurance

Optimal insurance

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Page 14: Purpose and Abstract

Optimal indemnity schedule for RAU-LAU

www.ncyu.edu.tw/fin 國立嘉義大學財務金融系14

)(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

Page 15: Purpose and Abstract

Optimal indemnity schedule for RAU-LAU

www.ncyu.edu.tw/fin 國立嘉義大學財務金融系15

Panel A

Page 16: Purpose and Abstract

Optimal indemnity schedule for RAU-LAU

www.ncyu.edu.tw/fin 國立嘉義大學財務金融系16

Panel B

Page 17: Purpose and Abstract

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

Page 18: Purpose and Abstract

Optimal indemnity schedule for RAU-LAU

www.ncyu.edu.tw/fin 國立嘉義大學財務金融系18

)(and)( with

)(}])ˆ()ˆ([)({Maximize

00

ˆ2ˆ1)(0

xIPwwxIxPWW

xfwwvwwvWUH wwwwxxI

11

ww

xfwwvWUIH

xfwwvWUIH

xfwwvWUH

ˆif

0)()}ˆ()({/

)()}ˆ()({/

)(})ˆ()({

122

1

1

ww

xfwwvWUIH

xfwwvWUIH

xfwwvWUH

ˆif

)()}ˆ()({/

)()}ˆ()({/

)(})ˆ()({

222

2

2

Page 19: Purpose and Abstract

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

1ˆˆ

*

0

22

IIx

IIx

IIxxIx

xI xx

xxxx

1

11

Page 20: Purpose and Abstract

Optimal indemnity schedule for RAU-LAU

Panel A. for large λβ

www.ncyu.edu.tw/fin 國立嘉義大學財務金融系20

}},0),(ˆmin{max{)(* xxIxI

Page 21: Purpose and Abstract

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

1ˆˆ

*

0

22

IIx

IIx

IIxxIx

xI xx

xxxx

1

11

Page 22: Purpose and Abstract

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

Page 23: Purpose and Abstract

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

Page 24: Purpose and Abstract

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

Page 25: Purpose and Abstract

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

Page 26: Purpose and Abstract

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

Page 27: Purpose and Abstract

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

Page 28: Purpose and Abstract

Optimal indemnity schedule for LAU-LAU

www.ncyu.edu.tw/fin 國立嘉義大學財務金融系28

Panel C

Page 29: Purpose and Abstract

Optimal indemnity schedule for LAU-LAU

Panel A. for small λ

Panel B. for large λ

www.ncyu.edu.tw/fin 國立嘉義大學財務金融系29

xxI )(*

0)(* xI

Page 30: Purpose and Abstract

Optimal indemnity schedule for LAU-LAU

Panel C.for small λ

www.ncyu.edu.tw/fin 國立嘉義大學財務金融系30

0ˆˆif

ˆ0ˆif

ˆˆ0if}),(ˆmin{

)(

1

1ˆˆ

*

0

22

IIx

IIx

IIxxIx

xI xx

xxxx

1

11

Page 31: Purpose and Abstract

Optimal indemnity schedule for LAU-LAU

Panel D.for large λ

www.ncyu.edu.tw/fin 國立嘉義大學財務金融系31

0ˆor0ˆif0

ˆ0ˆˆif}),(min{

ˆˆ0ˆif}),(ˆmin{

)(

1

21

21ˆ

*0

II

IIIxxI

IIIxxI

xIxx1

Page 32: Purpose and Abstract

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

Page 33: Purpose and Abstract

www.ncyu.edu.tw/fin 國立嘉義大學財務金融系33

Conclusions and Further Works