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.
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2
國立嘉義大學財務金融系
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.
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3
國立嘉義大學財務金融系
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 risktaking behaviors.
– Devers et al. (2007)
– O’Connell and Teo (2009)
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4
國立嘉義大學財務金融系
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)
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5
國立嘉義大學財務金融系
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.
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6
國立嘉義大學財務金融系
Reciprocal Reinsurance
 Cai et al. (2013, JRI) designed the optimal
reinsurance treaty f that maximize
 the joint survival probability
 and the joint profitable probability.
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7
國立嘉義大學財務金融系
Loss, Premium, Wealth, Utility
 Loss X and Premium P
I  E[ I ( ~
x )]
x  E[x~ ]
P  h(I )
h()  0
h(0)  0
 Insured’s and Insurer’s final wealth
~
~  w  P  I (~
W  W0  P  ~
x  I (~
x ) and w
x)
0
 Objective of the optimal reinsurance
~
~)], λ  weight
E[U (W )  V (w
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8
國立嘉義大學財務金融系
S-shaped Loss Aversion Utility
 Insured’s loss aversion utility
 u1 (W  Wˆ )
if W  Wˆ

U (W )  
0
if W  Wˆ
  u (Wˆ  W ) if W  Wˆ
2

u1()  0  u1()
u2 ()  0  u2()
 Insurer’s loss aversion utility
if
 v1 ( w  wˆ )

V ( w)  
0
if
  v ( wˆ  w) if
2

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9
w  wˆ
w  wˆ
w  wˆ
v1()  0  v1()
v2 ()  0  v2()
國立嘉義大學財務金融系
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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國立嘉義大學財務金融系
Optimal indemnity schedule for RAU-RAU

~
~
Maximize E[U (W )   V ( w)]  0 [U (W )   V ( w)] f ( x)dx
0 I ( x )  x
W  W0  P  x  I ( x) and w  w0  P  I ( x)
with
 By calculus of variations, the Hamiltonian
Maximize H  {U (W )  V ( w)} f ( x)
0 I ( x )  x
 {U (W0  P  x  I ( x))  V ( w0  P  I ( x))} f ( x)
FOC: H / I  [U (W )  V (w)] f ( x)  0  I ( x)  Iˆ( x)
SOC :  2 H / I 2  [U (W )  V (w)] f ( x)  0
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國立嘉義大學財務金融系
Optimal indemnity schedule for RAU-RAU
 Proposition 1 for RAU-RAU:
min{Iˆ( x), x} if
I ( x)  
ˆ( x), 0} if
max{
I

*
0  Iˆ( x) 
ARAU
1
ARAU  ARAV
ARAU  U (W ) / U (W )
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Iˆ(0)  0
Iˆ(0)  0
12
~
ARAR (WR )  V  / V 
國立嘉義大學財務金融系
Unconstrained and Constrained Optimal Insurance
 Unconstrained
optimal reinsurance
 Optimal insurance
I (x)
Iˆ( x)
2
1.9
1.8
Iˆ( x)
xˆ
1.7
45 line
2.25
x
1.6
2.2
1.5
2.15
1.4
2.1
0
0.1
1.3
2.05
2
Iˆ(0) 1.950
1.9
0.4
0.5
0.6
0.7
0.8
0.9
x
dˆ
1.8
0.5
0.3
I (x)
Iˆ(0) 1.850
Iˆ(0) 1.750
0.2
x
xˆ
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Iˆ( x)
0.9
2
1.9
1.8
1.7
0
dˆ
x
1.6
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1.5
國立嘉義大學財務金融系
Optimal indemnity schedule for RAU-LAU
~
~
~)]  E[U (W
~  wˆ )1~   v ( wˆ  w
~)1 }]
Maximize E[U (W )   V ( w
)  {v1 ( w
w  wˆ
2
w  wˆ
0 I ( x )  x

 0 [U (W ) {v1 ( w  wˆ )1w~  wˆ   v2 ( wˆ  w)1w wˆ } ] f ( x)dx
W  W0  P  x  I ( x) and w  w0  P  I ( x)
with
Panel A
Panel B
Utility
Panel C
Utility
v1 (w  wˆ )
Utility
v1 (w  wˆ )
v1 (w  wˆ )
U (W )
U (W )
0
0
0
  v2 (w  wˆ )
IˆbR
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U (W )
  v2 (w  wˆ )
I (x)
IˆbR
14
  v2 (w  wˆ )
I (x)
IˆbR
I (x)
國立嘉義大學財務金融系
Optimal indemnity schedule for RAU-LAU
Panel A
Ut ilit y
ˆ)
v1 (w  w
U (W )
0
ˆ)
  v2 (w  w
Iˆ
R
b
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15
I (x)
國立嘉義大學財務金融系
Optimal indemnity schedule for RAU-LAU
Panel B
Ut ilit y
ˆ)
v1 (w  w
U (W )
0
ˆ)
  v2 (w  w
Iˆ
R
b
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16
I (x)
國立嘉義大學財務金融系
Optimal indemnity schedule for RAU-LAU
Panel C
Ut ilit y
ˆ)
v1 (w  w
0
U (W )
ˆ)
  v2 (w  w
Iˆ
R
b
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17
I (x)
國立嘉義大學財務金融系
Optimal indemnity schedule for RAU-LAU
Maximize H  {U (W )  [v1 ( w  wˆ )1w wˆ   v2 ( wˆ  w)1w wˆ ]} f ( x)
0 I ( x )  x
with
W  W0  P  x  I ( x) and w  w0  P  I ( x)
 H  {U (W )  v1 ( w  wˆ )} f ( x)

if
H / I  {U (W ) v1 ( w  wˆ )} f ( x)
 2
2

H
/

I
 {U (W ) v1( w  wˆ )} f ( x)  0

w  wˆ
 H  {U (W )  v2 ( wˆ  w)} f ( x)

if
H / I  {U (W )  v2 ( wˆ  w)} f ( x)
 2
2

H
/

I
 {U (W )   v2( wˆ  w)} f ( x)

w  wˆ
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國立嘉義大學財務金融系
Optimal indemnity schedule for RAU-LAU
I * ( x)  min{max{Iˆ( x), 0}, x} for large λβ
H ( x)
H ( x)
ˆ
H
ˆ
H
Iˆ
IˆbR
I (x)
 x 1x xˆ  min{Iˆ( x), x} 1x xˆ
2
2

I * ( x)   x 1x xˆ0
x

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19
Iˆ IˆbR Iˆ1
if
if
if
Iˆ2
I (x)
0  Iˆ  Iˆ1
Iˆ  0  Iˆ1 for small λβ
Iˆ  Iˆ  0
1
國立嘉義大學財務金融系
Optimal indemnity schedule for RAU-LAU
 Panel A. I * ( x)  min{max{Iˆ( x), 0}, x} for large λβ
I (x)
I (x)
Iˆ( x)
2
1.9
1.8
dˆ
xˆ
1.7
1.6
Iˆ( x)
2
x
1.9
1.5
1.8
1.4
1.3
0
0.1
0.2
0.3
0.4
xˆ
0.5
x
0.6
0.7
0.8
0.9
1.7
x
dˆ
0
1.6
1.5
1.4
1.3
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20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
國立嘉義大學財務金融系
Optimal indemnity schedule for RAU-LAU
 Panel B. for small λβ
 x 1x xˆ  min{Iˆ( x), x} 1x xˆ
2
2

I * ( x)   x 1x xˆ0
x

I (x)
if
0  Iˆ  Iˆ1
Iˆ  0  Iˆ
if
Iˆ  Iˆ1  0
if
I (x)
1
I (x)
x
xˆ2
2
1.9
Iˆ( x)
1.8
xˆ
xˆ0
1.7
x
1.6
1.5
1.4
1.3
0
0.1
0.2
0.3
0.4
xˆ
0.5
0.6
0.7
0.8
xˆ2
0.9
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x
0
x
xˆ0
21
x
0
國立嘉義大學財務金融系
Optimal indemnity schedule for LAU-RAU
~
~ ˆ
~ ˆ
~)]  E[u (W
~  wˆ )]
~
Maximize E[U (W )   V ( w

W
)
1


u
(
W
 W ) 1W~ Wˆ  V ( w
1
2
W  wˆ
0 I ( x) x

 0 {u1 (W  Wˆ )1W Wˆ   u2 (Wˆ  W )1W Wˆ   V ( w)} f ( x)dx
W  W0  P  x  I ( x) and w  w0  P  I ( x)
with
Panel A
Panel B
Panel C
Utility
Utility
Utility
u1 (W  Wˆ )
u1 (W  Wˆ )
0
0
V (W )
u1 (W  Wˆ )
0
V (W )
V (W )
  u2 (Wˆ  W )
  u2 (Wˆ  W )
IˆbI
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  u2 (Wˆ  W )
IˆbI
I (x)
22
I (x)
IˆbI
I (x)
國立嘉義大學財務金融系
Optimal indemnity schedule for LAU-RAU
 Panel A. I * ( x)  min{max{Iˆ( x), 0}, x} for small λ/α
I (x)
Iˆ( x)
2
1.9
1.8
H (x )
xˆ
1.7
Hˆ
x
1.6
1.5
1.4
0
0.1
1.3
0.2
0.3
0.4
xˆ
0.5
x
0.6
0.7
0.8
0.9
I (x)
IˆbI
Iˆ
I (x)
dˆ
Iˆ( x)
2
1.9
1.8
1.7
0
dˆ
x
1.6
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1.5
國立嘉義大學財務金融系
Optimal indemnity schedule for LAU-RAU
min{Iˆ( x), x} 1x xˆ
0

*
I ( x)  min{I ( x), x}
0

 Panel B.
for large λ/α
Iˆ1  0  Iˆ2  Iˆ
Iˆ1  Iˆ2  0  Iˆ
Iˆ  0 or Iˆ  0
if
if
if
1
I (x)
H (x )
Iˆ( x)
2
Hˆ
1.9
xˆ
1.8
1.7
x
1.6
1.5
xˆ 0
Iˆ1
1.3
0
I (x)
Iˆ2 IˆbI Iˆ
1.4
I (x)
0.1
xˆ 0
0.2
0.3
0.4
0.5
xˆ
0.6
0.7
0.8
0.9
x
I (x)
Iˆ( x)
2
1.9
1.8
xˆ
1.7
x
1.6
1.5
1.4
1.3
0
0.1
0.2
0.3
0.4
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xˆ
0.5
x
0.6
0.7
0.8
0
0.9
24
x
國立嘉義大學財務金融系
Optimal indemnity schedule for LAU-LAU
~
~)]
Maximize E[U (W )   V ( w
0 I ( x )  x
with W  W0  P  x  I ( x) and w  w0  P  I ( x)
~
 E[u1 (W  Wˆ )1W Wˆ   u2 (Wˆ  W )1W Wˆ
~  wˆ )1  v ( wˆ  w
~ )1 }]
 {v ( w
1

 0
w wˆ
2
w wˆ
{[u1 (W  Wˆ )1W Wˆ   u2 (Wˆ  W )1W Wˆ
[v1 ( w  wˆ )1w wˆ  v2 ( wˆ  w)1w wˆ ]} f ( x)dx
Maximize H  {[u1 (W  Wˆ )1W Wˆ   u2 (Wˆ  W )1W Wˆ
0 R ( x )  x
[v1 ( w  wˆ )1w wˆ  v2 (wˆ  w)1w wˆ ]} f ( x)
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25
國立嘉義大學財務金融系
Optimal indemnity schedule for LAU-LAU
Panel A
Ut ilit y
ˆ)
u1 (W  W
ˆ)
v1 (w  w
0
ˆ W )
  u2 (W
ˆ)
  v2 (w  w
Iˆb
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26
I (x)
國立嘉義大學財務金融系
Optimal indemnity schedule for LAU-LAU
Panel B
Ut ilit y
ˆ)
u1 (W  W
ˆ)
v1 (w  w
0
ˆ)
  v2 (w  w
ˆ W )
  u2 (W
IˆbI
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27
IˆbR
I (x)
國立嘉義大學財務金融系
Optimal indemnity schedule for LAU-LAU
Panel C
Ut ilit y
u1 (W  Wˆ )
ˆ)
v1 (w  w
0
ˆ W )
  u2 (W
IˆbR
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ˆ)
  v2 (w  w
IˆbI
28
I (x)
國立嘉義大學財務金融系
Optimal indemnity schedule for LAU-LAU
*
 Panel A. I ( x)  x for small λ
H ( x)
I (x)
ˆ
H
I (x)
Iˆb
0
x
 Panel B. I * ( x)  0 for large λ
H ( x)
I (x)
ˆ
H
Iˆb
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I (x)
29
0
x
國立嘉義大學財務金融系
Optimal indemnity schedule for LAU-LAU
 Panel C.
for small λ
 x 1x xˆ  min{Iˆ( x), x} 1x xˆ
2
2

I * ( x)   x 1x xˆ0
x

H ( x)
if
0  Iˆ  Iˆ1
Iˆ  0  Iˆ
if
Iˆ  Iˆ1  0
if
1
H ( x)
ˆ
H
ˆ
H
Iˆ IˆbR Iˆ1 Iˆ2
I (x)
IˆbI
IˆbI Iˆ IˆbR Iˆ1
I (x)
I (x)
Iˆ2
I (x)
I (x)
x
xˆ2
2
1.9
xˆ0
Iˆ( x)
1.8
xˆ
1.7
x
1.6
1.5
1.4
1.3
0
0.1
0.2
0.3
0.4
xˆ
0.5
0.6
0.7
0.8
xˆ2
x
0
x
xˆ0
0
0.9
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30
x
國立嘉義大學財務金融系
Optimal indemnity schedule for LAU-LAU
min{Iˆ( x), x} 1x xˆ
0

I * ( x)  min{I ( x), x}
0

 Panel D.
for large λ
H ( x)
if
if
if
Iˆ1  0  Iˆ2  Iˆ
Iˆ1  Iˆ2  0  Iˆ
Iˆ  0 or Iˆ  0
1
H ( x)
ˆ
H
ˆ
H
I (x)
Iˆ1 Iˆ2 IˆbI Iˆ
IˆbR
Iˆ( x)
2
Iˆ( x)
2
1.9
I (x)
I (x)
I (x)
I (x)
xˆ
Iˆ2 IˆbI Iˆ IˆbR
Iˆ1
1.9
1.8
1.7
1.8
x
1.6
1.5
xˆ 0
xˆ
1.7
1.4
1.3
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x
1.6
0.9
1.5
1.4
0
xˆ 0
xˆ
www.ncyu.edu.tw/fin
x
1.3
0
0.1
0.2
0.3
31
0.4
xˆ
0.5
x
0.6
0.7
0.8
0
x
0.9
國立嘉義大學財務金融系
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.

~
~
Maximize E[U (W )   V ( w)]  0 [U (W )   V ( w)] f ( x)dx
P
subject to W  W0  P  x  I ( x;  P ) and w  w0  P  I ( x;  P )
h( I )  P, I  E[ I ( ~
x ;  )]
P
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32
國立嘉義大學財務金融系
Conclusions and Further Works
This study has developed the reciprocal
optimal insurance under the four
situations:
RAU-RAU, RAU-LAU, LAU-RAU, LAU-LAU.
The further works should further present
the result intuitions. Moreover, the results
will be compared with the previous works,
especially Raviv (1979), Wang and Huang
(2012) and Sung et al. (2011).
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33
國立嘉義大學財務金融系