How Does the Life Settlement Market Aect the Primary Life Insurance - - PowerPoint PPT Presentation

How Does the Life Settlement Market A¤ect the Primary Life Insurance Market?

Hanming Fang Edward Kung

Duke University

June 19, 2008

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Introduction

A life settlement is a …nancial transaction in which a life insurance policyholder sells the policy to a third party for more cash than the cash value o¤ered by the original insurer. The third party is then responsible for maintaining premium payments to the insurance company, but is also entitled to the bene…ts at the time of the insured’s death. The life settlements industry is young but growing rapidly: from a few billion dollars in the late 1990s to about $12-15 billion in 2007. Some projections see the industry growing to more than $100 billion in the next decade.

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The life settlements industry is interesting as it relates to a large IO literature on secondary markets for durable goods, with one key distinction: in life insurance, secondary market transactions directly a¤ect the primary market seller’s payo¤s. The life settlements industry is also interesting in that its emergence has triggered controversies over its e¤ect on consumer welfare: see Doherty and Singer (2002), the Deloitte Report (2005) Despite this, life settlements have received little serious attention from economists, except for a short paper by Daily, Hendel and Lizzeri (2008) on which we base our analysis.

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We study the e¤ects of the secondary market for life insurance on the structure of primary market life insurance contracts and consumer welfare. Our model builds o¤ of Hendel and Lizzeri (HL, 2003) and Daily, Hendel and Lizzeri (DHL, 2008). HL showed that dynamic life insurance contracts can partially insure consumers against reclassi…cation risk via front-loading.

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We show that in the presence of a secondary market, the nature and extent of this dynamic insurance is a¤ected in a qualitatively signi…cant (and potentially severe) manner. We also show that within our framework, the presence of a secondary market is welfare reducing. Finally, we analyze how the choice of cash surrender values can evolve to partially mitigate the e¤ect of the secondary market.

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Economic Intuition

Two main features of life insurance contracts underline the opportunity for a life settlements market.

1

Most life insurance contracts have premiums that stay …xed over the course of the contract (front-loading).

2

Life insurance contracts typically have zero (as in the case of Term policies) or low and non-health-contingent cash surrender values (as in the case of Whole Life policies).

The gap between the cash surrender value and the actuarial value of the contract provides an opportunity for gains of trade between policyholders with impaired health and the life settlements companies.

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To understand the mechanism by which the secondary market may a¤ect primary market contracts, consider a policyholder who no longer needs his policy. Absent a secondary market, the only option available to this consumer is to allow his policy to lapse by failing to pay the premium or surrendering the policy for a very low surrender value. To the extent that the lapsed policy is actuarially favorable to the policyholder (due to front-loading), the life insurance company can pocket the value of the lapsed policy as a pro…t. If the primary market is competitive, lapsation pro…ts can be passed on to consumers through lower pricing of premiums.

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In the presence of a settlement market, policies that otherwise would have lapsed or been surrendered are now sold to life settlements …rms. Life insurance companies continue to receive premium payments but are also liable for paying the death bene…ts. By denying the insurance companies the returns on lapsed or surrendered policies, the secondary market may make it more costly to provide life insurance on the primary market. These costs may have to be passed on to consumers through higher premiums. As a result, the structure of the contracts chosen in equilibrium may change, and consumers may ultimately be made worse o¤.

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Baseline (HL) Model: Health, Income and Bequests

A perfectly competitive primary market for life insurance operates in 2 periods. For now, assume there is no secondary market. In the …rst period, the policyholder has probability of death p1. In the second period, the policyholder has a new probability of death p2 which is a random variable drawn from (). The realization of p2 is not known in period 1 but is symmetrically learned (and thus common knowledge) at the beginning of period 2.

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In period 1 the policyholder earns an income stream y g and in period 2 he earns an income stream y + g. In each period, the policyholder cares about two sources of utility. His own, given by u(), and his dependent’s, given by v(), should he die. Both u and v are strictly increasing and concave in their arguments. In period 2, there is an exogenous probability, given by (1 q), that the policyholder no longer has a bequest motive. In that case, the policyholder only cares about his

• wn utility u().

y, g, p1, and q are all common knowledge.

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Baseline Model: Timing, Commitment, and Contracts

Life insurance contracts are of the form h(Q1; F1); f(Q2(p); F2(p)) : p 2 [0; 1]gi. Assume one-sided commitment: insurance companies can commit to the contract terms in the second period, but the policyholder may renege on the contract by lapsing his premium payment.

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Period 1 Timing: Consumer earns y g and chooses a contract h(Q1; F1); f(Q2(p); F2(p)) : p 2 [0; 1]gi. He then pays the …rst period premium Q1 and consumes the remaining y g Q1 With probability p1 he dies and his dependents receive F1. Period 2 Timing: Consumer and life insurance companies learn p2. With probability 1 q the policyholder loses his bequest motive. Consumer earns y + g. He then decides to either stay with his policy, lapse his policy and repurchase insurance on the spot market, or lapse his policy and remain uninsured. If he stays with his contract or repurchases, he pays the premium Q2 and consumes the remaining y + g Q2: With probability p2 he dies and his depends receive the face value F2 of his standing insurance contract, if any.

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Baseline Model: Equilibrium Characterization

In a competitive equilibrium, the insurance company will o¤er a contract h(Q1; F1); f(Q2(p); F2(p)) : p 2 [0; 1]gi that maximizes consumer welfare u(y g Q1) + p1v(F1)+ (1 p1) Z fq [u(y + g Q2(p)) + pv(F2(p))] + (1 q)u(y + g)g d(p) (1) subject to the following constraints: Q1 p1F1 + (1 p1)q Z fQ2(p) pF2(p)g d(p) = 0 (2) 8p : Q2(p) pF2(p) 0; (3) Constraint (2) is a zero pro…t condition re‡ecting perfect competition in the primary market. Constraint (3) is a no-lapsation constraint re‡ecting one-sided commitment.

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Solving the above problem we obtain the following equations characterizing the equilibrium contract: u0(y g Q1) = v0(F1) = (4) 8p : u0(y + g Q2(p)) = v0(F2(p)) = + (p) (1 p1)q(p) (5) where > 0 is the Lagrange multiplier for the zero pro…t constraint (2) and (p) 0 is the Lagrange multiplier for the no-lapsation constraint (3). Conditions (4) and (5) imply that full event insurance is obtained in period 1, and in every health state of period 2. These equations give a one-to-one correspondence between premia and face values in the equilibrium contract.

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Consider again the no-lapsation constraint (3). If (3) binds, i.e. Q2(p) = pF2(p), then we say the premium is actuarially fair relative to face value F2(p). Let

• QF I

2 (p); F F I 2

(p)

• be de…ned by:

QF I

2 (p) pF F I 2

(p) = u0(y + g QF I

2 (p))

= v0(F F I

2

(p)) For convenience, we will simply refer to

• QF I

2 (p); F F I 2

(p)

• as the actuarially fair

premium and face value for health state p. If (3) does not bind, i.e. Q2(p) < pF2(p), then we say that the premium is actuarially favorable (or actuarially unfair from the perspective of the …rm)

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Proposition (Hendel and Lizzeri 2003)

In the absence of a secondary market, equilibrium life insurance contracts will satisfy the following:

• 1. All policyholders obtain full event insurance in period 1 and in all health states of

period 2.

• 2. In the second period, there is a threshold health state p such that for p p the

premiums are actuarially fair and for p > p the premiums are actuarially favorable. For p > p the premium is constant and given by Q2(p) = Q1 + 2g

• 3. For g su¢ciently small, p < 1, i.e. reclassi…cation risk insurance is provided for

policyholders with low income growth.

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• 6

p Q2 (p) Q2 (p) QF I

2

(p) Q1 + 2g p

Figure: The Equilibrium Second-Period Premium as a Function of Health State p: The Case Without Life Settlement Market.

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An Invariance Result

We now consider the case when u() = v(). The …rst order conditions (4) and (5) imply that the following are satis…ed in equilbrium: 8p p : Q2(p) = p(y + g) 1 + p ; F2(p) = y + g 1 + p 8p p : Q2(p) = Q1 + 2g; F2(p) = y g Q1 p = Q1 + 2g y g Q1 F1 = y g Q1 Note that every contract term has been written in terms of Q1. To obtain for Q1 one must simply solve the zero pro…t condition. The resulting solution for Q1 will not depend on the shape of the utility function u().

Proposition (Invariance of contract structure to utility parameters)

In the absence of a secondary market, when u() = v(), the equilibrium contract does not depend on any other features of the utility function.

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Introducing the Secondary Market

Absent the secondary market, the policyholder’s only possible period 2 actions were to remain with the contract terms speci…ed from period 1, to lapse the contract and repurchase on the spot market, or to lapse the contract and remain uninsured. With the secondary market, the policyholder has an extra option available to him. Suppose the policyholder is in health state p with contract terms (Q2(p); F2(p)). He may now sell his contract on the settlements market for a return of [pF2(p) Q2(p)] V2(p), where:

parameterizes the level of competition or e¢ciency of the settlements market.

As before he may still repurchase insurance on the spot market or remain insured.

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Claim

In our framework, a policyholder will sell his contract to the settlement market if and

• nly if he loses his bequest motive.

Proof.

A policyholder with a bequest motive will not sell because the best he could do is re-obtain the same contract terms. A policyholder without a bequest motive will sell because as long as > 0, he will always get a positive return as compared to simply lapsing.

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Secondary Market: Characterizing the Equilibrium

In equilibrium, contracts h(Qs

1; F s 1 ); f(Qs 2(p); F s 2 (p)) : p 2 [0; 1]gi will be chosen to

maximize u(y g Qs

1) + p1v(F s 1 )+

(1 p1) Z fq [u(y + g Qs

2(p)) + pv(F s 2 (p))] + (1 q)u(y + g + V s 2 (p))g d(p)

(6) subject to the constraints: Qs

1 p1F s 1 + (1 p1)

Z fQs

2(p) pF s 2 (p)g d(p) = 0

(7) 8p : Qs

2(p) pF s 2 (p) 0;

(8)

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There are two key di¤erences between the problem with a secondary market and the problem without one:

1

Consumers who lose their bequest motives in period 2 now consume an extra V s

2 (p)

as compared to without the secondary market;

2

The life insurance companies can no longer avoid liability for contracts lapsed by consumers in the no bequest state. As such, the q term drops from the zero pro…t condition (7).

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The following equations must hold in the equilibrium contract: u0(y g Qs

1) = v0(F s 1 ) =

(9) u0(y + g Qs

2(p)) = v0(F s 2 (p))

(10) qu0(y + g Qs

2(p)) + (1 q)u0(y + g + V s 2 (p)) = +

(p) (1 p1)(p) (11) Full event insurance is again obtained in both periods and in all period 2 health states. However, the …rst order condition characterizing the period 2 premiums has changed. There are some qualitative similarities between the equilibrium contract with and without a secondary market:

1

Consumers still obtain full event insurance in period 1 and in all health states of period 2.

2

There is still a threshold state ps such that for p ps the second period premium actuarially fair and for p > ps the second period premium is actuarially favorable.

Point 2 can be proved easily from the …rst order conditions (9)-(11).

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There are some important qualitative di¤erences between the equilibrium contract with and without a secondary market.

Proposition (Premium discounts vs. ‡at premiums)

For p > p, the second period premium Qs

2(p) is increasing in p (strict if q < 1).

Thus, reclassi…cation risk insurance no longer takes the form of ‡at premiums, but rather premium discounts.

Proof.

The equations that need to be satis…ed for all p > p are the following: qu0(y + g Qs

2(p)) + (1 q)u0(y + g + V s 2 (p))

= u0(y g Qs

1)

V s

2 (p)

= pF s

2 (p) Qs 2(p)

u0(y + g Qs

2(p))

= v0(F s

2 (p))

Simply taking total derivatives for each of these equation we can solve for dQs

2=dp:

dQs

2

dp = F s

2 (p) qu00(y+gQs

2(p))

2(1q)u00(y+g+V s

2 (p)) + 1 + p

u00(y+gQs

2(p))

v00(F s

2 (p))

which we can see is positive.

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• 6

p Qs

2 (p)

Qs

2 (p)

QF I

2

(p) ps

Figure: The Equilibrium Second-Period Premium as a Function of Health State p: The Case With Life Settlement Market.

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Welfare E¤ects of the Secondary Market

We now prove a result that shows a potentially stark e¤ect of the secondary market

• n the provision of reclassi…cation risk insurance by the primary market.

Proposition (Potential for unraveling)

Fix u(), v(), y and (). There exists a threshold ^ q such that if q < ^ q then for any g; the set of health states for which constraint (8) does not bind is empty. That is, the equilibrium contract is simply the set of short term spot market contracts, regardless of the consumer’s income growth.

Proof.

If there is a p such that (8) does not bind, then the following must be satis…ed: q

• u0(y + g Qs

2(p)) u0(y + g + V s 2 (p))

• = u0(y g Qs

1) u0(y + g + V s 2 (p))

The proof amounts to showing that the RHS is bounded below and the LHS is bounded above by quantities that do not depend on p and g (we omit the details here).

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The unraveling results proves that welfare must be reduced by the secondary market for very low q. We can, in fact, show that within our framework the seocndary market reduces consumer welfare unambiguously.

Proposition (Welfare e¤ect of the secondary market)

Consumer welfare is reduced by the presence of a life settlement market.

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Proof.

Let Cs = h(Qs

1; F s 1 ); f(Qs 2(p); F s 2 (p)) : p 2 [0; 1]gi be any contract feasible in the

presence of a secondary market. Let C = h(Q1; F s

1 ); f(Qs 2(p); F s 2 (p)) : p 2 [0; 1]gi be the same contract, except with

the …rst period premium lowered so that the contract is feasible without a secondary market. From the zero pro…t conditions, we see that Qs

1 Q1 = (1 p1)(1 q)

Z V s

2 (p)d(p) > 0:

Let W s(Cs) denote the expected consumer welfare associated with contract Cs in a world with a secondary market. Let W(C) denote the expected consumer welfare associated with contract C in a world without a secondary market.

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Proof.

(continued) W(C) W s(Cs) = u(y g Q1) u(y g Qs

1)

(1 p1)(1 q) Z [u(y + g + V s

2 (p)) u(y + g)] d(p)

In the case with a secondary market, consumers without bequest motives in the second period get an extra V s

2 (p) to their consumption.

The tradeo¤ is that they must pay a higher …rst period premium (more front-loading). It can be shown that W(C) W s(Cs) is non-negative and strictly positive if Qs

1 Q1 is strictly positive (i.e. there is some reclassi…cation risk insurance under

Cs). (Intuition: risk aversion makes the transfer from the low income …rst period to the high income second period undesirable.) Since C is just a candidate contract, the equilibrium contract when there is no secondary market cannot make the consumer worse o¤.

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Failure of the Invariance Result

Let us again consider the case when u() = v(). In this case, for p > p, we have: dQs

2

dp = F s

2 (p) qu00(y+gQs

2(p))

2(1q)u00(y+g+V s

2 (p)) + 1 + p

Note that the term

qu00(y+gQs

2(p))

2(1q)u00(y+g+V s

2 (p)) depends on the shape of the utility

function. Thus the invariance result fails to hold in the presence of a secondary market.

Proposition (Failure of the invariance result)

In the presence of a secondary market, the equilibrium contract terms will depend on the shape of the utility function, even when u() = v(). To the extent that we believe u() = v() is a reasonable assumption, the invariance result could be a useful basis for empirical tests of the e¤ects of the secondary market.

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Numerical Example

We now consider a numerical example with CARA utility functions with coe¢cient

• f absolute risk aversion r:

u(x) = v(x) = 1 exp(rx) and uniformly distributed health states: v U[0; 1] The numerical example was computed directly from the optimization problem using AMPL.

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Q2 vs. p

y=2, g=0.3, beta=1, r=1, q=0.7

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.6 0.7 0.8 0.9 1

baseline secondary market fair premium

Figure: The Relationship Between Q2 and Qs

2 with p:

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pstar vs. q

y=2, g=0.3, beta=1, r=1

0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

baseline secondary market

Figure: The Relationship Between p and ps with the Probability of Retaining Bequest Motive q:

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Q1 vs. q

y=2, g=0.3, beta=1, r=1

0.28 0.29 0.3 0.31 0.32 0.33 0.2 0.4 0.6 0.8 1

baseline secondary market

Figure: How Does the Secondary Market A¤ect the First Period Premium, for Varying Values of q?

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pstar vs. g

y=2, beta=1, r=1, q=0.7

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6

baseline secondary market

Figure: The Relationship Between ps and p and g:

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Q1 vs. g

y=2, beta=1, r=1, q=0.7

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 0.6

baseline secondary market

Figure: The Relationship Between Q1 and Qs

1 and g:

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pstar vs. beta

y=2, g=0.3, r=1, q=0.7

0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0.2 0.4 0.6 0.8 1

secondary market

Figure: The Relationship Between ps and with the Secondary Market.

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Q1 vs. beta

y=2, g=0.3, r=1, q=0.7

0.2832 0.2834 0.2836 0.2838 0.284 0.2842 0.2844 0.2846 0.2 0.4 0.6 0.8 1

secondary market

Figure: The Relationship Between Qs

1 and with the Secondary Market.

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pstar vs. r

y=2, g=0.3, beta=1, q=0.7

0.6 0.7 0.8 0.9 1 1 1.5 2 2.5 3 3.5 4

baseline secondary market

Figure: The Relationship Between p and ps with the Absolute Risk Aversion Parameter r:

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Q1 vs. r

y=2, g=0.3, beta=1, q=0.7

0.28 0.285 0.29 0.295 0.3 0.305 0.31 0.315 0.32 1 1.5 2 2.5 3 3.5 4

baseline secondary market

Figure: The Relationship Between Q1 and Q

1 and the Absolute Risk Aversion Parameter r:

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Welfare (baseline) - Welfare (secmkt) vs. q

y=2, g=0.3, r=1

0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120 0.2 0.4 0.6 0.8 1

beta=1 beta=0.7 beta=0.5

Figure: The Welfare E¤ects of the Life Settlement Market, for Varying Levels of :

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Endogenously Chosen CSVs

The primary market could potentially respond by o¤ering contracts with endogenously chosen cash surrender values. We show that by specifying optimally chosen health-contingent surrender values, the insurance companies can partially mitigate the e¤ects of the secondary market. There are legal di¢culties and regulations involved with health-contingent surrender values, however, so we also study optimal non-health-contingent surrender values. We …nd that endogenously chosen surrender values are useless if they are restricted to be independent of health.

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Health-Contingent CSVs Without a Secondary Market

As a baseline we show that in the absence of a secondary market, the option to include health contingent surrender values will not be exercised. The contract h(Q1; F1); f(Q2(p); F2(p); S (p)) : p 2 [0; 1]gi will be chosen to maximize: u(y g Q1) + p1v(F1) + (12) (1 p1) Z fq [u(y + g Q2(p)) + pv(F2(p))] + (1 q)u(y + g + S(p))g d(p) subject to the constraints Q1 p1F1 + (1 p1) Z fq [Q2(p) pF2(p)] (1 q)S(p)g d(p) = 0 (13) 8p : Q2(p) pF2(p) 0 (14) 8p : S(p) 0 (15)

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At the equilibrium, the following two equations must be satis…ed: u0(y g Q1) =

• u0(y + g + S(p))

= + (p) (1 p1)(1 q)(p) where 0 is the Lagrange multiplier for the non-negativity constraint (15). If the surrender value were not zero for some p, i.e. S(p) > 0, then (p) = 0 and thus u0(y + g + S(p)) = u0(y g Q1): This is impossible, so in equilibrium we have S(p) = 0 for all p.

Proposition (Irrelevance of surrender values in the absence of a secondary market)

In the absence of a secondary market, the option to include surrender values in long term contracts will not be used. Equilibrium contracts and outcomes will be the same regardless of whether surrender values are allowed.

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Health-Contingent CSVs with a Secondary Market

In the presence of a secondary market, the contract h(Q1; F1); f(Q2(p); F2(p); S (p)) : p 2 [0; 1]gi will be chosen to maximize: u(y g Q1) + p1v(F1) + (16) (1 p1) Z fq [u(y + g Q2(p)) + pv(F2(p))] + (1 q)u(y + g + S(p))g d(p) subject to the constraints Q1 p1F1 + (1 p1) Z fq [Q2(p) pF2(p)] (1 q)S(p)g d(p) = 0 (17) 8p : Q2(p) pF2(p) 0 (18) 8p : S(p) [pF2(p) Q2(p)] (19) Constraint (19) says that for each health state, the surrender value must be greater than what could be obtained on the settlement market.

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The …rst order conditions for an optimum are: u0(y g Q1) = v0(F1) = u0(y + g Q2(p)) = + (p) (1 p1)q(p) (p) (1 p1)q(p) v0(F2(p)) = + (p) (1 p1)q(p) (p) (1 p1)q(p) u0(y + g + S(p)) = + (p) (1 p1)(1 q)(p) Again, it is easy to see that constraint (19) must bind for every p. Thus, in equilibrium the surrender value will be chosen to exactly equal the amount

• btainable from the settlements market, S(p) = V2(p).

In essence, the primary market insurance companies must now compete against secondary market …rms for the repurchase of actuarially favorable life insurance contracts.

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After substituting S(p) = V2(p) into the …rst order conditions, we …nd that the following must be satis…ed in equilibrium for every p: qu0(y + g Q2(p)) + (1 q)u0(y + g + V2(p)) = (q + (1 q)) + (p) (1 p1)(p) Compare this to the …rst order condition that characterizes second period premiums for the case without surender values: qu0(y + g Q2(p)) + (1 q)u0(y + g + V2(p)) = + (p) (1 p1)(p) Thus, we see that there are similarities between the equilibrium contracts in the secondary market, no surrender values case and the secondary market with surrender values case.

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In the presence of a secondary market, the equilibrium contracts with and without surrender values are qualitatively similar in the following ways:

1

Full event insurance is obtained in period 1 and in every health state of period 2.

2

There is a threshold p such that the second period premium is fair for p p and favorable for p > p. The threshold may di¤er depending on whether surrender values are allowed or not.

3

The second period premiums are increasing with p for p > p, although the exact shape of the premium pro…le may di¤er.

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Cross-Regime Welfare Comparisons

Let us categorize the four regimes we have studied so far:

Regime A. no secondary market with cash surrender values Regime B. no secondary market without cash surrender values Regime C. secondary market with endogenous cash surrender values Regime D. secondary market without cash surrender values.

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We have already shown that the welfare is equivalent between regime A and regime B. It is easy to see that the welfare is higher in regime C than regime D. Any outcome feasible in regime D can be obtained at a lower …rst period premium in regime C. It can also be shown that the welfare is higher in regime B than regime C. That is, welfare is higher without a secondary market regardless of whether or not cash surrender values are endogenously chosen. (The proof is similar to above).

Proposition (Cross regime welfare comparison)

In terms of consumer welfare, the four regimes can be ranked as follows: A = B C D. Endogenously chosen surrender values can be used to partially mitigate but not eliminate the welfare loss associated with the secondary market.

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Non-Health Contingent CSVs

Legal issues and regulation may make it di¢cult or impossible for insurance companies to specify health contingent CSVs. Indeed, life insurance contracts typically have no or very low and non-health contingent cash surrender values. Due to the earlier irrelevance result, we know that there is no impact in restricting CSVs to be non-health contingent in the absence of a secondary market. We now study equilibrium contracts with non-health contingent surrender values in the presence of a secondary market.

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The equilibrium contract h(Q1; F1); f(Q2(p); F2(p); ) : p 2 [0; 1]g ; Si must be chosen to maximize: u(y g Q1) + p1v(F1) + (1 p1)q Z 1 [u(y + g Q2(p)) + pv(F2(p))] d(p) +(1 p1)(1 q) Z

SV2(p)

u(y + g + S)d(p) +(1 p1)(1 q) Z

S<V2(p)

u(y + g + V2(p))d(p) (20) subject to the constraints: V2(p)

• S; 8p

(21) S

• (22)

Q1 p1F1 = (1 p1)q Z 1 V2(p)d(p) + (1 p1)(1 q) Z

SV2(p)

Sd(p) +(1 p1)(1 q) Z

S<V2(p)

V2(p)d(p) (23)

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Supposing that the optimal V2() is continuous and monotonically increasing in p, then for every S there exists a p0(S) such that V2(p) S if and only if p p0(S). So we can write the optimization problem as maximizing: u(y g Q1) + p1v(F1) + (1 p1)q Z 1 [u(y + g Q2(p)) + pv(F2(p))] d(p) +(1 p1)(1 q) Z p0(S) u(y + g + S)d(p) +(1 p1)(1 q) Z 1

p0(S)

u(y + g + V2(p))d(p) (24) subject to the constraints V2(p)

• S; 8p

(25) S

• (26)

Q1 p1F1 = (1 p1)q Z 1 V2(p)d(p) + (1 p1)(1 q) Z p0(S) Sd(p) +(1 p1)(1 q) Z 1

p0(S)

V2(p)d(p) (27)

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Using the Leibniz integral rule, the derivative of the Langrangian with respect to S is then: @L @S = (1 p1)(1 q)u0(y + g + S)(p0(S)) + Z 1 (p)dp +

• Term A

z }| { (1 p1)(1 q)(p0(S)) +

Term B

z }| { (1 p1)(1 q)(1 )V2(p0(S)) (p0(S)) V 0

2(p0(S))

where (p) 0, 0, and > 0 are the Lagrange multipliers with respect to constraints (25), (26), and (27), respectively.

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Now we argue that the derivative of the Lagrangian is strictly negative when S deviates from 0 to a small " > 0. lim

s!"=0+

@L @S = (1 p1)(1 q)

• u0(y + g)
• (p0(0)) +

Z 1 (p)dp The …rst order conditions with respect to Q1 is the same as in all previous cases: u0(y g Q1) = so we have that u0(y + g) < 0. Furthermore, (p) 0 for all p, so lims!"=0+ @L

@S < 0.

Thus, the optimal cash surrender value S = 0.

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A B A

p p V2(p) ß ßV2(p)

Figure: The E¤ect of Increasing S by " > 0 on the Firm’s Pro…ts in the Second Period.

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Proposition (Irrelevance of non-health contingent cash surrender values)

When the primary insurers are restricted to o¤er only non-health contingent cash surrender values, they will choose S=0 in equilibrium. An important consequence is that when surrender values are restricted to be non-health contingent, they lose their usefulness as a device to mitigate the welfare loss associated with the settlement market.

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Conclusion

We have examined in detail the e¤ect of the life settlement market on the structure

• f the long term contracts o¤ered by the primary insurance market, and on

consumer welfare. Of most importance, we have shown that the secondary market a¤ects the nature by which reclassi…cation risk insurance is delivered. In particular, the secondary market tends to lower the amount of reclassi…cation risk insurance obtained in equilibrium and as a result reduces consumer welfare. We also showed that this welfare loss can be partially mitigated with optimally chosen health-contingent surrender values, but that none of the welfare loss can be eliminated if surrender values are restricted to be non-health contingent. There are a number of directions for further research, on both theoretical and empirical fronts. Theoretical: Alternative reasons for lapsation of long term contracts (income shocks) Empirical: Does the evidence support our model’s theoretical predictions?

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