Life Insurance Settlement and the Monopolistic Insurance Market

Life Insurance Settlement and the Monopolistic Insurance Market
Jimin Hong
Business School
Seoul National University
1 Gwanak Ro, Gwanak Gu, Seoul, 151-916, Korea
[email protected]
S. Hun Seog1
Business School
Seoul National University
1 Gwanak Ro, Gwanak Gu, Seoul, 151-916, Korea
[email protected]
1
Seog is grateful for the financial support from the Management Research Institute of the
Seoul National University.
Life Insurance Settlement and the Monopolistic Insurance Market
2014.6.
Abstract: We analyze the effects of life insurance settlement on the insurer’s profit and
the consumer and social welfare. We consider a one-period model in which the insurance
market is monopolistic and the settlement market is competitive. Policyholders face
heterogeneous liquidity risks in addition to mortality risks. Liquidity risks are introduced to
address the case in which policyholders need urgent cash, leading them to surrender or settle
the policies. It is assumed that the insurer cannot observe the liquidity risks nor discriminate
among policyholders based on liquidity risk. It is further assumed that no costs are incurred
in policy surrender or settlement. We find that introduction of the life settlement market
lowers monopolistic rent and insurer’s profit, and raises the insurance premium. The effects
on consumer welfare and social welfare are mixed. Consumer welfare increases only when
demand increases sufficiently. This finding implies that social welfare, as measured by the
sum of consumer welfare and the insurer's profit, can increase if the increase in consumer
welfare is greater than the decrease in insurer's profit. This finding contrasts with the
existing literature, in which the settlement market lowers consumer and social welfare.
Keywords: life insurance, settlement market, consumer welfare, monopoly, social welfare
Life Insurance Settlement and the Monopolistic Insurance Market
I. Introduction
Life settlement is a transaction such that a policyholder sells her insurance policy to a third
party investor, a so-called settlement provider. As a result of the transaction, the
policyholder receives the settlement price, while the investor becomes the beneficiary of the
life insurance. Life settlement has been used by policyholders as a means of securing cash.
Without the settlement market, the policyholder can surrender the policy to receive cash,
called the surrender value. Therefore, the life settlement market provides the policyholder
with an alternative to policy surrender. Given that surrender value is generally lower than
the actuarial value of the insurance contract, a policyholder may opt to settle the policy rather
than surrendering it when the settlement price is higher than the surrender value.
Life settlement also benefits financial investors in several ways. First, since the
settlement price is lower than the actuarial value, investors may earn profits. Profits can be
greater when policyholders settle policies when there is an urgent need for cash. Second,
policy settlement can provide investors with investment opportunities that are not correlated
with existing portfolios. The inclusion of life settlement in investment portfolios will lower
portfolio risks.
According to Gatzert (2010), settlement transactions are allowed in Germany, the U.K. and
the U.S. For example, terminally-ill AIDS patients are allowed to sell their policies in the
settlement market in the U.S. Conning Research & Consulting (2008) reported that the size
of the U.S. settlement market was 120 billion dollars in 2007 and that it may reach one
trillion dollars in 2016.
Several economic studies address life settlement. Doherty and Singer (2002) argue that
the settlement market may enhance consumer welfare, since it can reduce the monopsony
power of the insurer. They observe that, without the settlement market, the policyholder
would surrender the policy to the insurer, where the insurer plays a role similar to a
monopsonistic firm. The introduction of a settlement market effectively increases
competition among buyers, which lowers the monopsony rent of the insurer.
Based on Hendel and Lizzeri (2003), Daily, Hendel and Lizzeri (2008, DHL hereafter),
consider a dynamic insurance contract in a competitive insurance market. The insurance
contract is a one-sided commitment which means only the insurer’s commitment is binding.
They assume that the policyholder’s income is growing and that the insurer can observe the
policyholder’s health risk (symmetric learning). DHL focused on the zero cash surrender
value (CSV) and the “front-loaded” contracts, in which policyholders pay front-loaded
premiums. Front loading allows the policyholder to avoid reclassification risk in the
following period. When a policyholder surrenders her policy, the insurer can get the
surrender profit from the prepaid front loaded premium. DHL argues that the introduction
of a settlement market lowers consumer welfare, because the premium increases and the
policyholder is unable to hedge the reclassification risk. They, however, conjecture that the
settlement market may improve consumer welfare if the income stream is reversed.
Fang and Kung (2008, FK hereafter) extend the discrete model of DHL into the continuous
model. Unlike DHL, FK allow a non-zero CSV whose value is dependent on the
policyholder’s health condition. FK's findings are in line with DHL. FK shows that the
optimal CSV is equal to zero and that the settlement deteriorates consumer welfare because
consumers lose the hedging opportunity as well.
On the other hand, Hong and Seog (2012) analyze the case in which the income stream is
reverse in the DHL model. They refute the conjecture of DHL. That is, even if the income
stream decreases from high to low, the introduction of a settlement market still lowers
consumer welfare because policyholders lose the opportunity to hedge the reclassification
risk.
Seog and Hong (2012) investigate the effects of a settlement market on the monopolistic
insurance market, where death benefits as well as the CSV and premiums are determined
endogenously. These authors find that the monopolistic insurer can fully extract the
consumer surplus through adjustment of premiums, and the policyholder can fully hedge the
risk. Unlike DHL and FK, the CSV can be positive. The CSV is less than or equal to the
actuarial value of insurance. When a settlement market is introduced, social welfare tends
to decrease, as the monopoly rent of the insurer decreases, while consumer welfare remains
zero.
Gatzert, Hoermann and Schmeiser (2008) argue that the introduction of a settlement
market may worsen the insurer’s profit using a simulation based on actuarial assumptions.
The profit is reduced because the insurer needs to pay a higher benefit instead of a lower
surrender value. The profit reduction also leads to a rise in premiums.
Along a similar line of thinking, this paper investigates the effects of the settlement market
on the design of insurance contracts, consumer welfare and social welfare. We consider a
monopolistic insurer and a competitive settlement market. The focus of this paper is on the
liquidity risk of policyholders, which reflects the fact that the settlement market has been
often used by policyholders who need cash for medical treatment or urgent care. Facing
liquidity needs, policyholders should decide whether to surrender or settle the policy. We
assume that policyholders face heterogeneous liquidity risks and that the insurer cannot
observe the liquidity risks. We further assume that the insurer cannot offer contracts that
discriminate against policyholders based on the liquidity risks. This assumption reflects the
observation that insurers are often disallowed from discriminating against policyholders
based on non-insurance risks. Policyholders are homogeneous except for their liquidity
risks.
We find that the introduction of the settlement market lowers monopolistic rent of the
insurer, while increasing insurance premiums. As a result, the insurer's profit decreases.
The effects on welfare are mixed, depending on the distribution of liquidity risks and the
utility shapes of policyholders. Consumer and social welfare can be improved when the
changes in insurance contracts attract a sufficient number of potential policyholders. This
finding contrasts with published studies that claim that the settlement market deteriorates
welfare.
The remainder of the paper is proceeds as follows. Section II provides the model
description. Section III investigates the basic model with no settlement market. Section
IV studies the model with a settlement market. Sections V and VI study the effects of the
settlement market on the insurance contract, and on the consumer and social welfare,
respectively. Sections VII and VIII deal with the example and compare it with the
competitive insurance market. Section IX discusses the implications of the findings.
Section X set forth the conclusions.
II. Model description
We consider a monopolistic insurance market in a one period model. Time is denoted by
t = 0 and t = 1. A potential policyholder purchases life insurance in t = 0, and the death
event occurs with probability p1 in t = 1. The insurance premium is denoted by Q and the
death benefit of insurance is fixed as D. The premium is composed of pure premium and
the monopolistic rent which is denoted by R. We assume that policyholders are
homogeneous except for liquidity risk. Liquidity risk is measured by the probability that the
policyholder needs urgent cash, which is denoted by q which is distributed on [0, 1]. It is
assumed that the event of cash need occurs immediately prior to death event in t = 1. When
the policyholder needs cash, the policyholder has to surrender the policy to the insurer and
receive surrender value S, if there is no settlement market. However, the policyholder can
choose between surrender and settlement, if a settlement market exists.
The population of potential policyholders is distributed over the liquidity risk. The
population density function (p.d.f.) and the cumulative density function (c.d.f.) of q are
denoted by f(q) and F(q), respectively. The monopolistic insurer cannot observe the
liquidity risk nor offer contracts that discriminate among policyholders based on their
liquidity risk. The discount factor is denoted by ρ. The time line of the model is depicted
in Figure 1.
Suppose that there is no settlement market. When the insurer sells insurance to a
policyholder with liquidity risk q, the pure premium becomes qS + (1 – q)p1D. With rent
R, premium Q can be denoted as follows.
Q   qS   (1  q) p1D  R
(1)
Let us denote the endowment income of policyholders as W at t = 0. In t = 1, the
policyholder experiences a liquidity crisis, so the income flow would be W - y with
probability q or the income would be W1 with probability 1- q.
We suppose that the
policyholder considers two sources of utility. If the policyholder is alive and her
consumption is W, then the utility is denoted as u(W ) . However, if the policyholder is
dead and a dependent spends W, then the utility becomes v(W ) . This assumption is in
accordance with DHL and FK. The policyholder incorporates the dependent’s consumption
into her expected utility, reflecting a bequest motive. Utility functions are strictly concave
and twice differentiable. That is, u '(W )  0, u ''(W )  0 and v '(W )  0, v ''(W )  0 .
The policyholder’s expected utility with no insurance can be written as
u(W )   qu(W  y)   (1  q)(1  p1 )u(W1 ) . The expected utility with insurance is as
follows:
u(W  Q)   qu(W  y  S )   (1  q) p1v(D)   (1  p1 )(1  q)u(W1 )
The difference between the two expected utilities with and without insurance is called the
net benefit of the policyholder, or NB(q). NB(q) can be expressed as follows.
NB(q)  u(W  Q)   qu(W  y  S )   (1  q) p1v(D)   (1  p1 )(1  q)u(W1 )
[u(W )   qu(W  y)   (1  q)(1  p1 )u(W1 )]
We suppose that u(W  y  S )  [ p1v( D)  u(W  y)] . That is, the utility to hold the
insurance is higher than the utility to lapse the insurance. This assumption allows us to
avoid the case that the insured chooses the surrender strategically to obtain the cash when
surrender value is high.
III.
The basic model : no settlement market
1. Demand given target liquidity risk
The monopolistic insurer determines the premium and the surrender value to maximize the
(expected) profit. Since policyholders have different liquidity risks and the insurer cannot
offer a contract conditional on liquidity risk, policyholders may have different preferences
over the insurance contract terms. Only those who obtain nonnegative net benefits will
purchase insurance. Let us refer to the marginal policyholder (liquidity risk) with zero net
benefit as a target policyholder (liquidity risk). Technically, a target policyholder’s net
benefit is as follows.
NB(q)  u(W  Q)   qu(W  y  S )   (1  q) p1v(D)   (1  p1 )(1  q)u(W1 )
(2)
[u(W )   qu(W  y)   (1  q)(1  p1 )u(W1 )]  0
Now, Lemma 1 is obtained.
Lemma 1. Potential policyholders with lower liquidity risk than the target liquidity risk
purchase insurance.
Proof. See the Appendix.//
In this case, potential policyholders with lower liquidity risk than the target obtain positive
net benefits such that they prefer to purchase insurance. This result also implies that the
insurer cannot fully extract rents from policyholder, allowing policyholders to enjoy positive
net benefits.
2. Surrender value and target liquidity risk
The insurer will determine optimal premium Q and optimal surrender value S for profit
maximization. From (1) and (2), the problem is equivalent to finding optimal surrender
value S* and target liquidity risk q* in    ( R(S , q), S , q) . Note that rent R, and thus
premium Q, follow from (2).
The problem for target risk q’, R’,S’ and Q’ can be written as:
Max  ( R '( S ', q '), S ', q ')
S ', q ', R
 Q ' F (q ')   S '  qf (q)dq  p1D  (1  q) f (q)dq
q'
q'
0
0
 [  S '  p1D][q ' F (q ')   qf (q)dq] R ' F (q ')
q'
0
s.t.
u(W  Q)   q ' u(W  y  S ')   (1  q ') p1v(D)   (1  q ')(1  p1 )u(W1 )
[u(W )   q ' u(W  y)   (1  q ')(1  p1 )u(W1 )]  0
where Q '   q ' S   (1  q ') p1D  R '
(3)
Note that monopolistic profit is composed of the difference between the surrender value
and the fair value of the death benefit and rent. If the fair value is greater than the surrender
value, the difference term is negative. That is, the insurer paid a higher benefit than the
benefit for which it set the price.
Now, we obtain Proposition 1 by solving (3).
Proposition 1. The optimal surrender value S *, R * and q * satisfy (4) and (5).
q'
u '(W  Q)  qf (q)dq  qF (q)u '(W  y  S )
(4)
0
Rf (q) 
[ p1v( D)  u (W  y)  u (W  y  S )]
u '(W  Q ')
F (q )
(5)
where Q '   q ' S   (1  q ') p1D  R '
Proof. See the Appendix. //
The LHS of (4) measures the sum of the policyholders’ net benefit decrease following the
premium increase due to the increase in S. Since the insurer needs to compensate the target
policyholder for the net benefit decrease, the LHS also indicates the marginal cost for the
insurer. The RHS of (4) measures the sum of the net benefits increase following the
surrender value increase. It indicates the marginal revenue of the insurer. As a result, (4)
requires marginal revenue to equal marginal cost at the optimal surrender value.
On the other hand, (5) indicates the condition that in optimum, revenue change is equal to
cost change following the premium decrease due to the increase in q, and the increase in q
itself. That is, the marginal revenue of the insurer is equivalent to the marginal cost to the
insurer. We prove a detailed proof process in the Appendix.
3. Consumer welfare and social welfare
Let us define consumer welfare (CW) as the sum of the net benefit of all consumers.
Given S*, R*, and q*, CW can be expressed as:
CW  u (W  Q*) F (q*)  u (W  y  S *)  qf (q)dq   p1v( D)  (1  q) f (q)dq

q*
q*
0
0

 u(W ) F (q*)  u(W  y)  qf (q)dq
q*
0
(6)
We transform (6) using (2):
CW  [q * F (q*)   qf (q)dq][ p1v( D)  u (W  y )  u (W  y  S *)]
q*
0
(7)
From (7), we know that the consumer welfare comprises the difference in utility between
holding the insurance and surrender the insurance. The insurer can extract the utility
difference as  qf (q)dq . However, policyholders enjoy the utility difference as q * F (q*) ,
q*
0
because the insurer cannot discriminate among policyholders depending on liquidity risk q .
Similarly, let us define social welfare (SW) as the sum of the consumer welfare and the
insurer’s profit. Thus,;
SW  CW  
 [q * F (q*)   qf (q)dq][ p1v( D)  u (W  y)  u (W  y  S *)  ( p1D  S *)]
q*
R * F (q*)
IV.
0
(8)
The model with a settlement market
1. Demand given target liquidity risk
Suppose that a competitive settlement market exists. We assume that the market is
perfect, so there is no transaction cost and the investors (settlement providers) are risk neutral.
As a result, a policyholder can sell her policy at an actuarially fair price, p1 D , in the
settlement market. Our task is to seek the equilibrium outcome for the insurance contract
when the settlement market exists. For notational clarity, we add script s to indicate the
existence of the settlement market: for example, liquidity risk qs, surrender value Ss and so on.
Applying the same logic as Lemma 1, we easily obtain the result that potential policyholders
with lower liquidity risks than the target risk prefer to purchase insurance.
2. Surrender value
First, note that if S* as determined in (5) is greater than or equal to p1 D , then the
settlement market cannot exist because policyholders opt to surrender rather than settle.
Therefore, if the settlement market is to be meaningful, S* should be lower than p1 D .
The condition that settlement investors cannot enter the market is written as Lemma 2.
Lemma 2. The settlement market cannot exist, if the following condition is satisfied
u '(W  Q*)
q * F (q*)
 q*
u '(W  y  p1 D )
 qf (q)dq
0
where Q*   p1D  R *
Proof.
(9)
By Proposition 1, S*  p1D when (9) is satisfied. //
Lemma 2 indicates that if condition (9) is satisfied, then the marginal benefit is still greater
than the marginal cost at p1 D . As a result, the optimal surrender value is greater than the
settlement price.
Based on this observation, we hereafter focus on the case in which S* < p1 D . Now,
suppose that the settlement market exists. If the insurer sets the surrender value below p1 D ,
then no policyholder will surrender the policy. Once the policy is settled, the insurance
contract is alive and its actuarial value is p1 D . Therefore, the insurer’s profit is the same as
the case in which no policyholders choose surrender.
On the other hand, the insurer can set the surrender value higher than the settlement price
to block the entry of settlement investors strategically. However, the following lemma
shows that the optimal surrender value should be equal to the settlement price p1 D .
Lemma 3. Suppose a competitive settlement market.
less than the settlement price.
Then, the optimal surrender value is
Proof. See the Appendix.//
When no settlement market exists, a low surrender value is needed to smooth the income
stream, which allows the insurer to extract a rent. The introduction of a settlement market
restricts the insurer’s extraction. To cope with investors, the insurer has to increase the
surrender value up to the settlement price.
3. Target liquidity risk
By Lemma 3, the premium can be expressed as follows.
Qs   p1D  Rs
(10)
As in Section III, the net benefit of the policyholder with target risk qs equals zero.
obtain the following expression.
We
NBqs  u(W  Qs )   qu(W  y  p1D)   (1  qs ) p1v(D)   (1  p1 )(1  qs )u(W1 )
[u(W )   qsu(W  y)   (1  p1 )(1  qs )u(W1 )]  0
(11)
We also have the following result from (11) and (2).
u(W  Qs )   qsu(W  y  p1D)  qs p1v(D)  qsu(W  y)
 u(W  Q*)   q * u(W  y  S*)  q * p1v(D)  q * u(W  y)
(12)
As in Proposition 1, the optimal target risk qs* solves the following problem.
Max  s ( Rs (qs ), qs )
qs
 Rs F (qs )
s.t. u(W  Qs )   qsu(W  y  p1D)   (1  qs ) p1v(D)   (1  qs )(1  p1 )u(W1 )
[u(W )   qsu(W  y)   (1  p1 )(1  qs )u(W1 )]  0
where Qs   p1D  Rs
(13)
Proposition 2. The optimal target liquidity risk qs * and Rs * are determined as follows.
Rs f (qs ) 
[ p1v( D)  u (W  y )  u (W  y  p1D)]
u '(W  Qs )
Proof. See the Appendix.//
F ( qs )
(14)
Note that (14) is the same as (5) with the additional constraint that the surrender value is
equal to p1 D . The interpretation of (14) is similar to (5). The insurer determines qs *
where marginal revenue equals marginal cost. As a result, the existence of a settlement
market leads the insurer to choose ( Ss *  p1D, qs *, Rs * ), instead of the original optimal
contract ( S*, q*, R * ).
4. Consumer welfare and social welfare
Now, consumer welfare CWs and social welfare SWs can be expressed as follows:
CWs  u(W  Qs *) F (qs *)  u (W  y  p1D)  qf (q)dq   p1v( D)  (1  q) f (q)dq

qs *
qs *
0
0

 u(W ) F (qs *)  u(W )  (1  q) f (q)dq
qs *
0
 [qs * F (qs *)  
qs *
SWs  [qs * F (qs *)  
qs *
0
0
qf (q)dq][ p1v( D)  u(W  y)  u(W  y  p1D)]
(15)
qf (q)dq][ p1v( D)  u (W  y)  u (W  y  p1D)]
 Rs * F (qs *)
(16)
As in the basic model, the policyholder with liquidity risk lower than target risk qs* enjoys
the positive net benefit. This implies that consumer welfare is positive.
V.
Effects of the life settlement market
Let us analyze the effects of the settlement market on the insurance contract. By
comparing cases with and without the settlement market, we obtain the following results.
Proposition 3. The monopolistic insurer's profit is lower when the settlement market exists.
Proof. The existence of the settlement market effectively imposes additional constraint (S =
p1D) on the insurer. That is, the insurer should solve the profit maximization problem under
an additional constraint. Therefore, the profit should be smaller when a settlement market
exists. We have:
   ( R(S (q*), q*), S (q*), q*)   s   s ( Rs (qs *), qs *)
(17)
Equality holds only when S* equals p1D. //
Comparative statics analyses are performed to investigate the effects of the settlement
market on the insurance contract.
To examine the effects on the contract, we take the total
differentiation of the first order conditions when there is no settlement market.
With Proposition 3, we can show that the rent is lower when the settlement market exists,
which implies that the monopsony power of the insurer is reduced. We also find that the
premium is higher. These observations are summarized in Proposition 4.
Proposition 4. Suppose that the settlement market exists. Comparing with the case with no
settlement market, we have the following results.
(1) The premium is higher.
(2) The insurer’s rent per premium is lower.
(3) The target liquidity risk can be increased even though the premium is higher.
Proof. See the Appendix.//
From Propositions 3 and 4, we know that the insurer’s profit and monopoly power are
lowered when the settlement market is introduced. This can be interpreted as resulting from
competition between the insurer and the settlement market. While the rent is lower, the
premium is always increased. This is because the insurer adjusts the premium to reflect the
higher surrender value as well as to compensate for the loss. When the premium increases,
the income gap between t=0 and t=1 is higher, so the net benefit of a consumer who has low
liquidity risk is decreased. However, the increase in the surrender value leads the
improvement in net benefit for the consumer who has high liquidity risk. As a result, even
though the premium increases, more policyholders can buy insurance. Demand can
decrease as well when the premium is too high.
The sign of demand change depends on
the utility shape and the wealth level of the policyholders.
These factors affect the insurer’s profit. In this case, the population distribution is also
important since it affects the revenue of the insurer. Since the insurer selects the target risk
to maximize its profit, the target risk can be greater or smaller according to the premium
change compared with the case where there is no settlement market.
VI.
Welfare comparison
By comparing welfare with and without the settlement market, we have proposition 5.
Proposition 5. Suppose that the settlement market exists. Comparing with the case with no
settlement market,
(1) consumer welfare is higher if the the following condition is satisfied:
[qs * F (qs *)  
qs *
0
q*
qf (q)dq]
[q * F (q*)   qf (q)dq]

[ p1v( D)  u (W  y )  u (W  y  S *)]
[ p1v( D)  u (W  y )  u (W  y  p1D)]
(18)
0
(2) social welfare can be higher when the increase in consumer welfare is greater than the
decrease in the insurer’s profit.
Proof. See the Appendix.//
From Proposition 5, it is evident that the introduction of settlement market can increase
consumer welfare. Consumer welfare is improved when the target risk sufficiently
increases, even if the premium is increased. This result is illustrated in Figure 2.
On the other hand, if qs* is smaller than q*, then consumer welfare can decrease. This
result is depicted in Figure 3.
The effect on social welfare is not clear. While the insurer’s profit always decreases,
consumer welfare can increase or decrease. If consumer welfare decreases, then social
welfare decreases as well. However, if consumer welfare increases enough to offset the
profit decrease, then it is possible that social welfare increases.
VII.
Numerical example
Let us suppose that the population of policyholders over liquidity risk has uniform
distribution with [0,1]. In addition, we assume that the utility function is
u(W )  1000exp(aW ) and v(W )  1,000[1  exp(aW )] . We impose the initial value as
W  20, D  12, a  0.3, y  20, p1  0.2 and find the optimal contract.
We first identify that there are cases where a settlement market can exist and cannot exist
when parameters a, y and p1 change2. These results are illustrated in Figures 4, 5, and 6,
and Table 1,2, and 3.
These results can be explained intuitively. At first, if the policyholders are more risk
averse, they gain more benefit when the premium is lower due to the lower surrender value,
because the income at t = 0 and t = 1 would be smooth. The insurer can extract this rent
imposing low surrender value. Second, when the income shock is higher, people would like
to get the higher surrender value to smooth the income between t = 0 and t = 1. Finally, as
death probability increases, the policyholder seeks lower surrender value and premium to
increase the utility at t = 0, that the policyholder is certainly alive. As a result, these results
are attributed that the insurer would like to increase the net benefit it extracts.
Next, we show that consumer and social welfare can be enhanced as in this numerical
example. In addition, we infer that consumer welfare would be decreased as risk aversion
increases when a settlement market is introduced. This is because the target policyholders
prefer low surrender value as in Figure 4. The increase in surrender value caused by
introducing the settlement market lowers the increase in consumer welfare and furthermore
decreases consumer welfare. Similarly, we anticipate that consumer welfare declines as the
income shock and death probability increase. We can elaborate this conjecture by
comparing welfare and demand when the settlement market can exists.
We also observe that there exists a case where social welfare is deteriorated even though
the market is deep. In this example, the target liquidity risk, demand, increases and the
premium is higher, so the market size is larger, but welfare is lower.
This is contrary to the
general belief that as the market deepens, welfare is enhanced.
These results are depicted
in Figure 7,8, and 9, and Table 4,5, and 6.
Demand always increases in the above example. However, there are other examples in
which the demand decreases and consumer welfare also decreases, or where consumer
welfare decreases even if demand increases by changing the death benefit. The premium is
higher when the death benefit increases, so optimal surrender value decreases to smooth
intertemporal consumption. In addition, we already see that if the optimal surrender value
without a settlement market is higher, then the net benefit difference with a settlement market
decreases in (7) and (15). As a result, as the death benefit increases, consumer welfare tends
to worsen. Demand decreases following the difference in net benefit becoming larger.
Policyholders cannot buy the insurance because the premium is too high. These results are
shown in Figure 10 and Table 7.
Moreover, we show how the result depends on the population distribution.
We can
observe that if the density function of population is an increasing function, then consumer
2
We used MATLAB R2014a and ran 1,000 times optimization in each case.
welfare also increases more. Conversely, if the density function is decreasing function, then
the consumer welfare decreases. Demand is higher in all cases. However, change in
demand is greater when the density function is decreasing.
The changes are depicted in
Table 8. Hence, we obtain that with sufficient increased demand, consumer welfare and
social welfare increase following the population distribution after a settlement market is
introduced.
VIII.
Comparison with competitive insurance market
Now, we consider a competitive insurance market. We suppose that q is distributed on
[0,1] and the mean value of q is  . All other assumptions are identical to that of
monopolistic insurance market. In this case, the insurer’s expected profit should be zero.
In a competitive equilibrium without a settlement market, premium and surrender value must
maximize consumer welfare. In addition, insurers sell the insurance to all consumers to
maximizing consumer welfare. The problem can be written as:
Max CW  u (W  Q )  u (W  y  S ')   p1 (1   )v ( D )   (1  p1 )(1   )u (W  y )
Q,S
 u (W )  u (W  y )   (1  p1 )(1   )u (W  y )
s.t. Q   S '    p1D(1   )
S 0
(19)
We obtain optimal surrender value by solving (19). When a settlement market does not
exist, optimal surrender value is zero for consumption smoothing when income shock y is
lower than premium. However, if y is higher than premium, positive surrender value is
possible.
Now, we suppose that the optimal surrender value without settlement market is lower than
p1 D . If the surrender value is higher than p1 D , then settlement investors have not
incentive to enter the market. In this case, welfare does not change. If a settlement market
is introduced, surrender value is increased at p1 D and consumer welfare is reduced. This is
because the premium increases when surrender value increases, so policyholders cannot
smooth the income between t=0 and t=1. This result is summarized in Lemma 4.
Lemma 4. In a competitive insurance market, we obtain following results.
(1) Without a settlement market, optimal surrender value is equal to zero when income shock
is higher than the premium. Otherwise, surrender value has a positive value.
(2) When a settlement market is introduced, consumer welfare is deteriorated.
Proof. See the Appendix.//
We know that the non-discriminatory contract design based on liquidity risk derives the
consumer welfare improvement in monopolistic insurance markets in contrast with
competitive insurance markets.
IX.
Discussion
Our research distinguishes itself from earlier studies in several respects. First, we
consider the heterogeneous liquidity risk of policyholders. Policyholders may surrender or
settle their policies for cash. The liquidity risk consideration reflects the fact that the
settlement market has been introduced to help terminally-ill people pay for medical treatment.
Our approach differs from Seog and Hong (2012) because they consider only homogeneous
liquidity risk. Note that, in DHL and FK, the main reason for surrender is disappearance of
the bequest motive.
Second, we focus on a monopolistic insurer, unlike DHL, FK and Hong and Seog (2012)
who all consider competitive insurance markets. Focusing on a monopolistic insurer
enhance our understanding of the settlement market. The effects of the settlement market
differ significantly from the competitive case, as shown in our analysis. In the competitive
insurance market, consumer welfare and social welfare are never increased by the
introduction of the settlement market. However, our analysis shows that welfares can
increase. Our result depends not only on the monopoly condition but also on nondiscriminatory contract design based on liquidity risk.
Third, we consider a population distribution, while DHL, FK and Hong and Seog consider
the representative consumer. Different potential policyholders may have different
willingness to pay for an insurance contract.
X.
Conclusion
We analyze the effects of life insurance settlement on insurer’s profit and consumer and
social welfare. We consider a one-period model in which the insurance market is
monopolistic and the settlement market is competitive. Policyholders face heterogeneous
liquidity risk in addition to mortality risk. Liquidity risk is introduced to address the case in
which policyholders need urgent cash, leading them to surrender or settle policies. It is
assumed that the insurer cannot observe liquidity risk nor discriminate among policyholders
based on liquidity risk. It is further assumed that no costs are incurred in policy surrender or
settlement. We find that the introduction of a life settlement market lowers monopolistic
rent and profit. In addition, premium increases. The effects on consumer welfare and
social welfare are mixed. Consumer welfare increases only when demand increases
sufficiently. This finding implies that social welfare, as measured by the sum of consumer
welfare and insurer’s profit, can increase if the increase in consumer welfare is greater than
the decrease in insurer's profit. This finding contrasts with the existing literature, in which
the settlement market lowers social welfare.
Numerical Example.
W0 = 20, D = 12, a = 0.3, y = 10, p1  0.2 , u(W )  1000exp(aW ) , v(W )  1,000[1  exp(aW )] , q ~ U [0,1]
Table 1. Surrender Value when risk aversion changes
a
S
Settlement price
0.2
4.9384
2.4
0.22
4.7570
2.4
0.24
3.5290
2.4
0.26
2.9784
2.4
0.28
2.4666
2.4
0.3
1.9923
2.4
0.32
1.5532
2.4
0.34
1.1464
2.4
0.36
0.7689
2.4
0.38
0.4179
2.4
0.4
0.0908
2.4
Table 2. Surrender Value when income shock changes
a
S
Settlement price
8
0.2601
2.4
8.2
0.4349
2.4
8.4
0.6094
2.4
8.6
0.7836
2.4
8.8
0.9576
2.4
9
1.1312
2.4
9.2
1.3044
2.4
9.4
1.4772
2.4
9.6
1.6495
2.4
9.8
1.8212
2.4
10
1.9923
2.4
10.2
2.1627
2.4
10.4
2.3322
2.4
10.6
2.5008
2.4
10.8
2.6683
2.4
11
2.8345
2.4
0.2
1.9923
2.4
0.21
1.8814
2.52
0.22
1.7746
2.64
0.23
1.6716
2.76
0.24
1.5722
2.88
0.25
1.4762
3
0.26
1.3833
3.12
0.27
1.2933
3.24
0.28
1.2061
3.36
0.29
1.1215
3.48
0.3
1.0393
3.6
Table 3. Surrender Value when death probability changes
a
S
Settlement price
0.15
2.6211
1.8
0.16
2.4842
1.92
0.17
2.3533
2.04
0.18
2.2279
2.16
0.19
2.1077
2.28
Table 4.Change in welfare and demand when risk aversion changes
a
0.29
0.30
0.31
0.32
0.33
0.34
0.35
0.36
0.37
0.38
0.39
0.40
S*
P1D
q*
qs
π
πs
CW
CWs
SW
SWs
△q
△CW
△SW
2.2249
1.9923
1.7686
1.5532
1.3460
1.1464
0.9542
0.7689
0.5903
0.4179
0.2515
0.0908
2.4
2.4
2.4
2.4
2.4
2.4
2.4
2.4
2.4
2.4
2.4
2.4
0.8200
0.8172
0.8153
0.8141
0.8136
0.8136
0.8141
0.8149
0.8161
0.8175
0.8192
0.8210
0.8237
0.8251
0.8266
0.8282
0.8299
0.8317
0.8335
0.8353
0.8372
0.8391
0.8410
0.8430
6.4822
6.6602
6.8381
7.0158
7.1929
7.3692
7.5445
7.7186
7.8914
8.0628
8.2327
8.4009
6.4805
6.6510
6.8161
6.9761
7.1313
7.2820
7.4283
7.5705
7.7089
7.8435
7.9747
8.1025
54.7352
55.8057
56.8372
57.8324
58.7933
59.7219
60.6195
61.4878
62.3279
63.1410
63.9282
64.6905
54.7553
55.8474
56.8924
57.8924
58.8496
59.7661
60.6441
61.4858
62.2932
63.0682
63.8127
64.5285
61.2174
62.4659
63.6754
64.8482
65.9862
67.0911
68.1640
69.2064
70.2193
71.2038
72.1608
73.0914
61.2358
62.4984
63.7085
64.8685
65.9809
67.0481
68.0724
69.0563
70.0020
70.9117
71.7873
72.6310
0.0037
0.0079
0.0113
0.0141
0.0163
0.0180
0.0194
0.0204
0.0211
0.0216
0.0219
0.0220
0.0201
0.0417
0.0552
0.0600
0.0562
0.0442
0.0246
-0.0020
-0.0347
-0.0728
-0.1155
-0.1620
0.0184
0.0325
0.0331
0.0203
-0.0054
-0.0430
-0.0916
-0.1501
-0.2173
-0.2921
-0.3735
-0.4604
Table 5. Change in welfare and demand when income shock changes
y
S*
P1D
q*
qs
π
πs
CW
CWs
SW
SWs
△q
△CW
△SW
8
8.2
8.4
8.6
8.8
9
9.2
9.4
9.6
9.8
10
10.2
10.4
0.2601
0.4349
0.6094
0.7836
0.9576
1.1312
1.3044
1.4772
1.6495
1.8212
1.9923
2.1627
2.3322
2.4
2.4
2.4
2.4
2.4
2.4
2.4
2.4
2.4
2.4
2.4
2.4
2.4
0.7534
0.7573
0.7615
0.7663
0.7715
0.7774
0.7838
0.7909
0.7988
0.8076
0.8172
0.8279
0.8398
0.7724
0.7761
0.7801
0.7844
0.7890
0.7940
0.7993
0.8051
0.8112
0.8179
0.8251
0.8329
0.8413
6.4065
6.4126
6.4221
6.4353
6.4524
6.4739
6.5000
6.5312
6.5679
6.6107
6.6602
6.7170
6.7820
6.2262
6.2562
6.2885
6.3231
6.3602
6.4001
6.4431
6.4893
6.5392
6.5929
6.6510
6.7137
6.7817
53.0438
53.1682
53.3188
53.4981
53.7084
53.9526
54.2338
54.5556
54.9217
55.3368
55.8057
56.3342
56.9288
52.2804
52.5328
52.8035
53.0939
53.4059
53.7411
54.1018
54.4901
54.9085
55.3599
55.8474
56.3746
56.9453
59.4503
59.5808
59.7409
59.9334
60.1608
60.4265
60.7338
61.0867
61.4896
61.9475
62.4659
63.0512
63.7108
58.5066
58.7890
59.0919
59.4170
59.7661
60.1413
60.5449
60.9794
61.4476
61.9528
62.4984
63.0883
63.7270
0.0190
0.0189
0.0186
0.0181
0.0175
0.0166
0.0155
0.0141
0.0124
0.0104
0.0079
0.0050
0.0015
-0.7634
-0.6354
-0.5154
-0.4041
-0.3025
-0.2115
-0.1320
-0.0655
-0.0132
0.0231
0.0417
0.0404
0.0165
-0.9437
-0.7918
-0.6490
-0.5164
-0.3948
-0.2852
-0.1890
-0.1074
-0.0420
0.0053
0.0325
0.0371
0.0163
Table 6. Change in welfare and demand when death probability changes
P1
S*
P1D
q*
qs
π
πs
CW
CWs
SW
SWs
△q
△CW
△SW
0.183
0.192
0.201
0.21
0.219
0.228
0.237
0.246
0.255
0.264
0.273
0.282
0.291
0.3
2.1913
2.0843
1.9810
1.8814
1.7851
1.6919
1.6017
1.5142
1.4294
1.3470
1.2669
1.1890
1.1132
1.0393
2.196
2.304
2.412
2.52
2.628
2.736
2.844
2.952
3.06
3.168
3.276
3.384
3.492
3.6
0.8272
0.8216
0.8167
0.8124
0.8085
0.8051
0.8020
0.7992
0.7968
0.7945
0.7925
0.7906
0.7889
0.7874
0.8274
0.8262
0.8250
0.8237
0.8223
0.8209
0.8194
0.8179
0.8163
0.8147
0.8131
0.8115
0.8098
0.8081
6.6089
6.6363
6.6631
6.6892
6.7144
6.7387
6.7619
6.7840
6.8051
6.8250
6.8438
6.8615
6.8781
6.8936
6.6089
6.6335
6.6529
6.6674
6.6776
6.6837
6.6862
6.6852
6.6812
6.6742
6.6646
6.6525
6.6381
6.6216
51.1633
53.6155
56.0801
58.5563
61.0433
63.5402
66.0466
68.5618
71.0854
73.6167
76.1554
78.7011
81.2534
83.8119
51.1644
53.6512
56.1210
58.5732
61.0077
63.4241
65.8223
68.2019
70.5628
72.9048
75.2278
77.5315
79.8158
82.0806
57.7722
60.2518
62.7433
65.2455
67.7577
70.2789
72.8085
75.3459
77.8904
80.4417
82.9992
85.5626
88.1315
90.7056
57.7732
60.2848
62.7738
65.2406
67.6853
70.1078
72.5084
74.8871
77.2440
79.5791
81.8924
84.1840
86.4539
88.7022
1.6546
1.6478
1.6417
1.6360
1.6308
1.6259
1.6214
1.6171
1.6131
1.6093
1.6056
1.6021
1.5987
1.5955
0.0011
0.0357
0.0408
0.0169
-0.0356
-0.1161
-0.2244
-0.3600
-0.5225
-0.7119
-0.9276
-1.1696
-1.4376
-1.7314
0.0011
0.0330
0.0306
-0.0049
-0.0724
-0.1711
-0.3001
-0.4588
-0.6464
-0.8626
-1.1068
-1.3786
-1.6776
-2.0034
Table 7. Change in optimal Surrender value without settlement market, consumer welfare, social welfare and demand when death benefit
changes
D
11
13
15
17
19
21
23
25
27
29
31
33
35
S*
2.0206
1.9697
1.9354
1.9100
1.8892
1.8708
1.8534
1.8365
1.8196
1.8025
1.7851
1.7672
1.7488
P1D
2.2
2.6
3
3.4
3.8
4.2
4.6
5
5.4
5.8
6.2
6.6
7
q*
0.8191
0.8156
0.8129
0.8105
0.8082
0.8060
0.8038
0.8016
0.7993
0.7970
0.7946
0.7921
0.7895
qs
0.8228
0.8270
0.8295
0.8304
0.8296
0.8271
0.8231
0.8174
0.8101
0.8012
0.7908
0.7788
0.7652
π
6.7372
6.5574
6.4075
6.2291
6.0468
5.8626
5.6773
5.4915
5.3055
5.1195
4.9335
4.7477
4.5621
πs
6.7353
6.5574
6.3490
6.1194
5.8744
5.6177
5.3521
5.0798
4.8028
4.5226
4.2409
3.9590
3.6781
CW
55.3615
56.1044
56.4049
56.4597
56.3769
56.2158
56.0084
55.7722
55.5166
55.2466
54.9646
54.6716
54.3681
CWs
55.3983
56.1052
56.1957
55.8811
55.2789
54.4555
53.4492
52.2830
50.9712
49.5235
47.9469
46.2469
44.4280
SW
110.7597
112.2096
112.6005
112.3407
111.6558
110.6713
109.4576
108.0552
106.4878
104.7701
102.9115
100.9185
98.7961
SWs
166.1580
168.3148
168.7962
168.2218
166.9347
165.1268
162.9068
160.3381
157.4590
154.2936
150.8584
147.1655
143.2242
△q
0.0037
0.0114
0.0167
0.0199
0.0214
0.0211
0.0192
0.0158
0.0108
0.0042
-0.0038
-0.0133
-0.0243
△CW
0.0368
0.0007
-0.2092
-0.5786
-1.0980
-1.7602
-2.5591
-3.4892
-4.5455
-5.7231
-7.0176
-8.4247
-9.9401
△SW
0.0350
0.0007
-0.2677
-0.6883
-1.2705
-2.0051
-2.8843
-3.9009
-5.0482
-6.3199
-7.7102
-9.2134
-10.824
Table 8. Change in welfare and demand following the population density function
f(q)=2q
f(q)=I(0,1)
f(q)=-2q+2
S*
CW
CWs
SW
SWs
q*
qs
△CW
△SW
△q
2.2921
1.9923
0.7503
44.6336
55.8057
45.4786
44.7817
55.8474
43.9963
50.7151
62.4659
60.1875
50.6959
62.4984
58.6704
0.9331
0.8172
0.5585
0.9356
0.8251
0.5761
0.1481
0.0417
-1.4823
-0.0193
0.0325
-1.5171
0.0025
0.0079
0.0176
Figure 1. Time line of model
t=1
t=0
Insurance is purchased
and premium Q paid.
Liquidity needs occur
Surrender value S is paid with pr. q
or settlement occurs with price p1 D
A loss occurs and insurance
benefit D paid with pr. p1
Figure 2. Case that consumer welfare can be improved
Net
benefit
NB
NBs
q*
non-existence of settlement market
qs
q
existence of settlement market
Figure 3. Case that consumer welfare is deteriorated
Net
Benefit
NB
NBs
qs
q*
q
non-existence of settlement market
existence of settlement market
Figure 4. Change of surrender value when risk aversion a changes
5
S
settlement market cannot exist
4
3
settlement market
can exist
2
1
a
0
0.224
0.244
0.264
0.284
0.304
0.324
0.344
0.364
0.384
Figure 5. Change of surrender value when income shock y changes
3
S
settlement market
cannot exist
2.5
2
settlement market
can exist
1.5
1
0.5
y
0
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10 10.25 10.5 10.75 11
Figure 6. Change of surrender value when death probability p1 changes
3.00
S
settlement market
2.50
cannot exist
settlement market
2.00
can exist
1.50
1.00
0.50
p1
0.00
0.15
0.17
0.19
0.21
0.23
0.25
0.27
0.29
Figure 7. Change in consumer welfare, social welfare and demand when risk aversion a
changes
△CW
0.10
0.05
a
0.00
-0.05
0.29
0.3
0.31
0.32
0.33
0.34
0.35
0.36
0.37
0.38
0.39
0.4
-0.10
-0.15
-0.20
△q
0.0250
0.0200
0.0150
0.0100
0.0050
a
0.0000
0.29
0.3
0.31
0.32
0.33
0.34
0.35
0.36
0.37
0.38
0.39
△SW
0.10
a
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
0.29
0.3
0.31
0.32
0.33
0.34
0.35
0.36
0.37
0.38
0.39
Figure 8. Change in consumer welfare, social welfare and demand when income shock y
changes
△CW
0.2
y
0.0
-0.2
8
8.2
8.4
8.6
8.8
9
9.2
9.4
9.6
9.8
10 10.2 10.4
-0.4
-0.6
-0.8
-1.0
△q
0.02
0.02
0.01
0.01
y
0.00
8
8.2
8.4
8.6
8.8
9
9.2
9.4
9.6
9.8
10 10.2 10.4
△SW
0.2
y
0.0
8
-0.2
-0.4
-0.6
-0.8
-1.0
8.2
8.4
8.6
8.8
9
9.2
9.4
9.6
9.8
10 10.2 10.4
Figure 9. Change in consumer welfare, social welfare and demand when death probability
p1 changes
△CW
0.5
P1
0
0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29
-0.5
-1
-1.5
-2
△q
0.025
0.020
0.015
0.010
0.005
P1
0.000
0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29
△SW
0.5
P1
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29
Figure 10. Change in optimal surrender value without settlement market, consumer welfare,
social welfare and demand when death benefit changes
S
2.10
2.00
1.90
1.80
1.70
D
1.60
11
14
17
20
23
26
29
32
35
△CW
2
D
0
-2
-4
-6
-8
-10
-12
11
14
17
20
23
26
29
32
35
△q
0.03
0.02
0.01
0.00
-0.01
11
14
17
20
23
26
29
32
35
-0.02
-0.03
△SW
2.0
D
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
-12.0
11
14
17
20
23
26
29
32
35
References
Daily, Glenn, Igal Hendel and Alessandro Lizzeri (2008), Does the Secondary Life Insurance
Market Threaten Dynamic Insurance?, American Economic Review 98 (2): 151-156.
Doherty, N. A. and Singer, H. J. (2003), The Benefits of a Secondary Market for Life
Insurance Policies, Real Property, Probate and Trust Journal : 449-478..
Fang, Hanming and Edward Kung (2008), How Does the Life Settlement Affect the Primary
Life Insurance Market?, Working Paper, Department of Economics of Duke University.
Gatzert, Nadine (2010), The Secondary Market for Life Insurance in the United Kingdom,
Germany, and the United States: Comparison and Overview, Risk Management and
Insurance Review 13(2): 279–301.
Gatzert, Nadine, Gudrun Hoermann and Hato Schmeiser (2008), The Impact of the
Secondary Market on Life Insurers' Surrender Profits, Working Papers on Risk
Management and Insurance 54, University of St. Gallen.
Hendel, Igal and Alessandro Lizzeri (2003), The Role of Commitment in Dynamic Contracts:
Evidence from Life Insurance, Quarterly Journal of Economics 118 (1): 299-327.
Hong, Jimin and S. Hun Seog (2012), A Study of the Effect on the Life Insurance Market
of Life Settlement, Korean Insurance Journal 95 : 23-50.
Seog, S. Hun and Jimin Hong (2012), A Study on the Introduction of Life Settlement based
in Monopoly Insurance Market, The Journal of Risk Management 23 (2) : 1-34.
Appendix
1. Proof of Lemma 1.
Lemma 1. Potential policyholders with lower liquidity risk than the target liquidity risk
purchase insurance.
Proof. If net benefit is greater than 0, potential policyholders with liquidity risk q ' will
buy insurance contracts. Thus we have
0  NB(q)  NB(q ')
 u(W  Q)   qu(W  y  S )   (1  q) p1v(D)  u(W )  qu(W  y)
 u(W  Q)   q ' u(W  y  S )   (1  q ') p1v( D)  u(W )  q ' u(W  y)
(A.1)
From this relation, we obtain the following.
(q ' q) [u(W  y  S )  p1v( D)  u(W  y)]  0
(A.2)
In (A.2), u(W  y  S )  [ p1v( D)  u(W  y)] , because of the assumption that without
liquidity risk, the utility of keeping the insurance contract is higher than that of surrender.
Therefore, q  q ' must be satisfied for equation (A.2). //
2. Proof of Proposition 1.
Proposition 1. The optimal surrender value S *, R * and q * are satisfying (4) and (5).
q'
u '(W  Q)  qf (q)dq  qF (q)u '(W  y  S )
(4)
0
Rf (q) 
[ p1v( D)  u (W  y)  u (W  y  S )]
u '(W  Q ')
F (q )
(5)
where Q '   q ' S   (1  q ') p1D  R '
Proof. We can find the optimal S *, R * and q * using the Lagrangian optimization, where
λ is the Lagrange multiplier.
L  [  S   p1D][q ' F (q ')  0 qf (q)dq]  RF (q ')
q'
 (q ')[u(W  Q ')   q ' u(W  y  S )   (1  q ') p1v(D)  u(W )  q ' u(W  y)]
(A.3)
The first order conditions for an optimum with respect to S, R, and λ are as follows.
LS  [q ' F (q ')   qf (q)dq]  (q ')  q '[u '(W  Q ')  u '(W  y  S )]  0
q'
0
LR  F (q ')  (q ')u '(W  Q ')  0
Lq  [  S   p1D]F (q ')  Rf (q ')
(A.4)
(A.5)
 (q ')[(  S   p1D)u '(W  Q ')  u(W  y  S )   p1v(D)  u(W  y)]  0
L  u(W  Q ')   q ' u(W  y  S )   (1  q ') p1v( D)  u(W )  q ' u(W  y)  0
(A.7)
(A.6)
By (A.5) we have,
 (q ') 
F (q ')
u '(W  Q ')
(A.8)
Let us plug (A.8) into (A.4) and (A.6). Now, the first order conditions can be simplified as
q'
u '(W  Q)  qf (q)dq  qF (q)u '(W  y  S )
0
and
Rf (q) 
[ p1v( D)  u (W  y)  u (W  y  S )]
u '(W  Q ')
F (q) //
In equilibrium, the premium is Q*   q * S *   (1  q*) p1D  R * , so the marginal revenue
with respect to q is as follows.
[  S *  p1D]F (q*)  [  q * S *   (1  q*) p1D  R*] f (q*)
(A.9)
In (A.9), [  S *  p1D]F (q*) is the marginal revenue from existing policyholders, and
[  q * S *   (1  q*) p1D  R*] f (q*) is the marginal revenue from new policyholders.
Meanwhile, marginal cost when q increases and  is the shadow price is as follows.
[  q * S *   (1  q*) p1D] f (q*)
[(  S *  p1D)u '(W  Q*)   p1v(D)  u(W  y)  u(W  y  S*)]
(A.10)
In (A.10), the insurer needs to pay [  q * S *  (1  q*) p1D] f (q*) more when q increases
because of the increase in surrender or death benefit for new consumers. This is the cost for
the insurer. On the other hand, the insurer extracts the net benefit from policyholders with
liquidity risk q* until the net benefit is equal to zero. If q increases, the net benefit the
insurer can extract changes as below:
dNB(q*) Q * dNB(q*) NB(q*)


dq *
q * dQ *
q *
 (  S *  p1D)u '(W  Q*)  [  p1v( D)  u(W  y)  u(W  y  S*)]
(A.11)
The insurer should give up the increase in net benefit following the premium decrease
caused by an increase in q from existing policyholders by as much as
(  S *  p1D)u '(W  Q*) .
On the contrary, the insurer can obtain the net benefit of as much as
[  p1v( D)  u(W  y)  u(W  y  S*)] from consumers because of an increase in q.
As a result, for the insurer,
{( S *  p1D)u '(W  Q*)  [ p1v(D)  u(W  y)  u(W  y  S*)]}
is the additional cost for the increase in q.
Total marginal cost for q is as follows.
[  q * S *  (1  q*) p1D] f (q*)
[  p1v( D)  u (W  y )  u (W  y  S *)]F (q*)
[(  S *  p1D)]F (q*) 
u '(W  Q*)
(A.12)
(A .13) should be satisfied such that marginal cost equals marginal revenue to maximize
insurer’s profit:
[  S *  p1D]F (q*)  [  q * S *   (1  q*) p1D  R*] f (q*)  [  q * S *   (1  q*) p1D] f (q*)
[(  S *   p1D)]F (q*) 
[  p1v( D)  u (W  y )  u (W  y  S *)]F (q*)
u '(W  Q*)
(A.13)
That is rearranged as below and it is equivalent to (A.9)
R * f (q*) 
[  p1v( D)  u (W  y )  u (W  y  S *)]
F (q*)
u '(W  Q*)
That is, optimal q is determined at the point that marginal cost equals marginal revenue of
q.//
3. Proof of Lemma 3.
Lemma 3. Suppose a competitive settlement market.
less than the settlement price.
Then, the optimal surrender value is
Proof. Let us suppose that the premium Q ' is as follows, given the optimal target liquidity
risk q ' , when the settlement market exists.
Q '   q ' S   (1  q ') p1D  R (A.14)
Thus the profit maximization problem can be stated as follows.
Max  ( R( S ), S )
S
 Q ' F (q)   S  qf (q)dq  p1D  (1  q) f (q)dq
q'
q'
0
0
 [  S   p1D][q ' F (q ')   q ' f (q ')dq] RF (q ')
q'
0
s.t.
u(W  Q ')   q ' u(W  y  S )   (1  q ') p1v( D)   (1  q ')(1  p1)u(W  y)
[u (W 
)  q ' u (W
 y)  ( 1 q ' )1 ( 1p 
u ) W( y ) ] 0
S  p1 D
(A.15)
The Lagrangian becomes
L  [  S   p1D][q ' F (q ')   qf (q)dq]  RF (q ')
q'
0
 (q ')[u(W  Q ')   q ' u(W  y  S )   (1  q ') p1v(D)  u(W )  q ' u(W  y)]
[ p1D  S ]
where   0
(A.16)
For S ' , the first order condition is
LS '  [q ' F (q ')   qf (q)dq]
q'
0
 (q ')  q '[u '(W  Q ')  u '(w  y  S ')]    0 (A.17)
We define that LS  LS (q*, S*)  0 in (A.4). If we assume that optimal q * and S *
are unique, then, for q ' , LS (q ', S ')  0 . This is because q* and S* do not satisfy the
additional constraint S '  p1D . Thus, for q ' and S ' , LS ' should be negative and
LS '  LS (q ', S ')    0 .
As a result,  should be positive to satisfy (A.17) and we have p1D  S '  0 by
complementary slackness condition [ p1D  S ']  0 . //
4. Proof of Proposition 2.
Proposition 2. The optimal target liquidity risk qs * and Rs * are determined as follows.
Rs f (qs ) 
[ p1v( D)  u (W  y )  u (W  y  p1D)]
u '(W  Qs )
F ( qs )
(14)
Proof. The Lagrangian is written as
Ls  Rs F (qs )
 (qs )[u(W  Qs )   qsu(W  y  p1D)   (1  qs ) p1v(D)  u(W )  qsu(W  y)]
(A.18)
The first order condition is obtained as follows.
Lqs  Rs f (qs )   (qs )[ u(W  y  p1D)   p1v( D)  u (W  y)]  0
LRs  F (qs )  (qs )u '(W  Qs )  0
(A.19)
(A.20)
Ls  u (W  Qs )   qsu (W  y  p1D)   (1  qs )v( D)  u (W )   qsu (W  y )  0
(A.21)
From (A.19) to (A.21), the optimal qs* and Rs* should satisfy the following condition.
Rs f (qs ) 
[ p1v( D)  u (W  y )  u (W  y  p1D)]
u '(W  Qs )
F ( qs )
(A.22)//
5. Proof of Proposition 4
Proposition 4. Suppose that the settlement market exists. Comparing with the case with no
settlement market, we have the following results.
(1) The premium is higher.
(2) The insurer’s rent per premium is lower.
(3) The target liquidity risk can be increased even though the premium is higher.
Proof.
(1) We know that if dq* < 0 then dQ* > 0. Let us assume q*  qs * and Q*  Qs * . Then,
by (12), we obtain following relation (A.23) and it is a contradiction, so in case of q*  qs * ,
Q*  Qs * is true.
u(W  Q*)   q * u(W  y  S*)   (1  q*) p1v( D)  q * u(W  y)
 u(W  Q*)   p1v(D)  q *[ p1v(D)  u(W  y)  u(W  y  S*)]
 u(W  Q*)   p1v( D)   qs [ p1v( D)  u(W  y)  u(W  y  S*)]
 u(W  Q*)   qs * u(W  y  S*)   (1  qs *) p1v(D)  qs * u(W  y)
 u(W  Qs *)   qs * u(W  y  p1D)   (1  qs *) p1v(D)  qs * u(W  y)
 u(W  Qs *)   qs * u(W  y  p1D)   (1  qs *) p1v(D)  qs * u(W  y)
(A.23)
dq *
dQ *
is increased following relation
 0 , the sign of
dS *
dS *
Qs * Q*  (  p1D  Rs *)  (  q * S *   (1  q*) p1D  R*)
Meanwhile, for the case of
 ( Rs * R*)  q *(  p1D   S*)
 dQ*  dR * q * dS *
(A.24)
Frrom (A.23) and (A.24), we know that the premium is always increased.
(2) (a) f '(q*)  0
From the first order condition (5), we obtain the following relation.
DdS*  EdR *  Fdq *
(A.25)
where
D
 q * u ''(W  Q*) [ p1v( D)  u(W  y)  u(W  y  S*)]
u '(W  Q*)2

F (q*)
 u '(W  y  S *)
F (q*)  0
u '(W  Q*)
[ p1v( D)  u (W  y )  u (W  y  S *)]u ''(W  Q*)
E   f (q*) 
F (q*)  0
u '(W  Q*)2
[ p1v( D)  u (W  y )  u (W  y  S *)]
F   R * f '(q*) 
f (q*)
u '(W  Q*)
[ p1v( D)  u (W  y )  u (W  y  S *)][  p1D   S *]u ''(W  Q*)

F (q*)
u '(W  Q*)2
If we assume that f '(q*)  0 , then the sign of F in (A.25) is positive. By (A.25), we obtain
D > 0, E < 0, and F > 0. From this information, let us investigate the sign of dR* and dq*,
when dS* > 0. We can know that the case of dS* > 0, dR* > 0 and dq* < 0 is contradicted
by (A.25) because the LHS of (A.25) is positive but the RHS is negative.
In addition, from (17),
  [  S *   p1D][q * F (q*)   qf (q)dq] R * F (q*)   s  Rs * F (qs *) should be satisfied.
q*
0
This implies R * F (q*)  Rs * F (qs *) , because  S   p1D by Lemma 2. If dq is positive
then F(qs* ) > F(q* ), so Rs* should be decreased. Therefore, we can exclude the case in
which dq* > 0 and dR* > 0.
As a result, only two cases are possible: (i) dR* < 0, dq* > 0 and (ii) dR* < 0, dq* < 0.
That is, the rent should decrease.
(b) f '(q*)  0
Let us compare (5) and (14).
If dS* > 0 then, [  p1v( D)  u(W  y)  u(W  y  S*)]
 [ p1v( D)  u(W  y)  u(W  y  p1D)] , and u '(W  Qs *)  u '(W  Q*) because
F (q)
is an increasing function in q. As a
Qs *  Q * by (1) of Proposition 4. In addition,
f (q )
result, even if f '(q*)  0 , R should be decreased. As a result, the insurer’s monopolistic
power weakens with the introduction of a settlement market. //
(3) By (2), we know that dq* can be positive. //
6. Proof of Proposition 5
Proposition 5. Suppose that the settlement market exists. Comparing with the case with no
settlement market,
(1) consumer welfare is higher if the the following condition is satisfied:
[qs * F (qs *)  
qs *
0
q*
qf (q)dq]
[q * F (q*)   qf (q)dq]

[ p1v( D)  u (W  y )  u (W  y  S *)]
[ p1v( D)  u (W  y )  u (W  y  p1D)]
(18)
0
(2) social welfare can be higher when the increase in consumer welfare is greater than the
decrease in the insurer’s profit.
Proof.
(1) First, we obtain (A.26) by (7) and (15).
CWs  CW
 [qs * F (qs *)  
qs *
0
q*
qf (q)dq][ p1v( D)  u(W  y)  u(W  y  p1D)]
[q * F (q*)   qf (q)dq][ p1v( D)  u (W  y)  u (W  y  S *)]
0
In addition, when q*  qs , (A.27) should be satisfied.
1 qs *
1 q*
qf
(
q
)
dq

F
(
q
*)

qf (q )dq
qs * 0
q * 0
1 q*
1 q*
1 qs *
 F (qs *)  F (q*) 
qf (q)dq   qf (q )dq 
qf (q)dq

0
0
q*
qs
qs * q*
F (qs *) 
(A.26)
qs *
1
1 qs *
1 qs *
[qs * F (q*)  qs * F (qs *)   qf (q)dq] 
qf (q)dq 
qf (q)dq

q*
qs *
qs * 0
qs * 0
q*
q*
1
(A.27)
 [q *  qf (q)dq   qf (q)dq]  0
qs *
qs *
q*

As a result, the condition for CWs  CW  0 is as follows.
[qs * F (qs *)  
qs *
0
q*
qf (q)dq]
[q * F (q*)   qf (q)dq]

[ p1v( D)  u (W  y )  u (W  y  S *)]
[ p1v( D)  u (W  y )  u (W  y  p1D)]
0
(2) The sign of the difference in social welfare is not clear:
SWs  SW
 [qs * F (qs *)   qf (q)dq][ p1v( D)  u(W  y)  u(W  y  p1D)]
qs *
0
q*
[q * F (q*)   qf (q)dq][ p1v( D)  u (W  y)  u (W  y  S *)  ( p1D  S *)]
0
 Rs * F (qs *) R * F (q*)
(A.28)
However, if consumer welfare is sufficiently increased, social welfare can also be increased.//
7. Proof of lemma 4
Lemma 4. In a competitive insurance market, we obtain following results.
(1) Without a settlement market, optimal surrender value is equal to zero when income shock
is higher than the premium. Otherwise, surrender value has a positive value.
(2) When a settlement market is introduced, consumer welfare is deteriorated.
Proof.
(1) In a competitive insurance market, q=1 because insurers sell life insurance to all
consumers. We use Lagrangian maximization where  and  are Lagrange multiplier.
L  u(W  Q)  u(W  y  S ')   p1v( D)(1   )  u(W )  u(W  y)(1  )
(A.29)
[Q   S '   p
1 D( 1   ) ] S
  0 should be satisfied for the complementary slackness condition.
first order conditions are as follows.
Ls  u '(W  y  S ')      0
LQ  u '(W  Q)    0
Rearranging (A.30) and (A.31), we obtain
(A.30)
(A.31)
For S and Q, the

(A.32)

This condition means that if y  Q , then S > 0. On the other hand, S > 0 when y  Q .
u '(W  y  S ')  u '(W  Q) 
(2) If the settlement market is introduced, the insurer faces the additional constraint that
S  p1D . As a result, consumer welfare decreases. //

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