Lecture 29Insurance Markets

10 December 2013

Ins Mkt

Logistics

1. Online course evaluations available until Friday December 13,2013.

2. Final Exam: 5:05–7:05 pm Sunday December 15, 2013. SocialScience 5208.

Ins Mkt

Insurance

I Insurance is a mechanism by which to smooth consumptionover uncertain states of nature.

I In U.S. formal insurance market exists. We are required bystate law to have liability insurance to drive a car.

I Dane country (and elsewhere in WI) experienced a severedrought during the summer of 2012.

I Many farmers lost their crop. Approximately, one–third hadcrop insurance, typically the largest farms.

I In this example, The “uncertain state” is weather and itsinfluence on the corn crop. Assume for simplicity that weatheris the only source of variability of the corn crop. If theweather is Good production is high, if Bad production is low.

I Critical. States of Nature makes reference to events beyondthe control of the farmer.

Ins Mkt

Moral Hazard

I Recognize the challenge of moral hazard to creation ofinsurance. Hope by this time, the threat of moral hazard toinsurance market.

I Underwriter (of insurance) must know the risk probabilities.

I Events insured must be verifiable.

I Insure risks beyond the control or influence of the insuredparty.

Ins Mkt

Structure of Insurance

I Many different types of insurance. Property, casualty(liability), health, and life insurance.

I Life insurance. Purchase policy, pay fee, F . Beneficiary ofpolicy receives payment upon death of the insured.

I Insured event must be verifiable (death certificate).

I Payment excludes suicides (the extreme form of moralhazard).

Ins Mkt

Casualty Insurance

I Liability insurance. Automobile insurance, pays off if ownerfound “liable” for accident.

I Pays medical care of self and injured parties.

I Pays for repair (replacement) of vehicle(s).

I Pay insurance premium for coverage over period of time(year).

I Premiums depend on characteristics of insured.

Ins Mkt

Pricing of Insurance Policy

I (Idealized) Life Insurance.

I Let T represent person’s age (lifetime).

I The mortality density function is f (t) which represents theprobability of death, at exact age t.

I Then the probability that an individual survives to age x isPr(T > x) = 1− F (x). This is frequently called the survivorfunction (the probability of surviving to age x).Let the Pr(T > τ) = S(τ) is the probability that the personsurvives to at least age τ .

Ins Mkt

Mortality: Exponential Random Variable

I Let f (t) = λe−λt, pdf, for exponential r.v..

I Memoryless process, probability of death is constant.

I Used because of simple properties.

I What is the value of a life insurance that pays B upon death,given that mortality process is characterized by exponentialdistribution with parameters λ?

Ins Mkt

Life Insurance Example

Let interest rate equal r :

E [V ] =

∫ ∞0

Bf (t) exp(−rt) dt

=B

∫ ∞0

λ exp(−λt) exp(−rt) dt

=Bλ

∫ ∞0

exp (−[λ+ r ]t]) dt

=λ

r + λB

Ins Mkt

Numerical Example

I If λ = .0125, life expectancy at birth = 80 years.

I r = .03

I V = B .0125.0125+.03 = .0125

.04125

I V = .303B

Ins Mkt

Impersonal Market

I Markets as we know are impersonal (though especially hard toremember in the job market).

I The insurance company writes policies for a large number ofcustomers.

I The insurance company works off the law of large numbers: in“large” samples, the sample mean is “close” to the populationmean.

I Pooling Insurance company engages in pooling (or averaging)which is a form of smoothing.

I Ideally, the customers should be dispersed over a largegeographical region. Why?

Ins Mkt

LLN

As a mathematical theorem, the Law of Large Numbers hasconditions on the nature of the population and the samplingprocess.

These conditions relate to similarity (homogeneity) of thepopulation units, and possible dependence of sampled observations.

Think of a large of population of similar farms. Same size, samefarming skills, same quality of land. Homogenous farms all verymuch a like.

Sampling process such that individual elements are independent ofone another. Knowing that production on farm i is Yi provides noinformation about production on farm j , Yj .

Ins Mkt

Homogeneity and Independence

I Simplest (and first) LLN developed for homogenous andindependent samples. {Y1,Y2, . . . ,Yn}

I Yet LLN exist for heterogeneous populations with elementsthat exhibit some dependence. Yet, there is a limit on theamount of dependence allowable.

I In the application of insurance, dependence, and particularlypositive dependence is what matters most.

I If farms are from the same community (e.g., Waunakee,Verona) farm production likely to be dependent, becausefarms will experience much the same weather.

Ins Mkt

LLN

Let the probability of crop damage in the population be θ. And forn policy holders, denote the proportion of policy holdersexperiencing crop damage by Pn.

Then for sufficiently large n, Pn ≈ θ.

For any c > 0, and ε > 0, there exists an n, such that

Pr (|Pn − θ| > c) < ε.

That is, with enough (similar) policy holders the insurance candiversify sufficiently to effectively eliminate its risk.

Ins Mkt

A caveat

True in the perfect world of mathematics, where the LLN is truly astatement on infinitely large samples.

So the LLN does not say that in every large sample (sayn = 10, 000) Pn and θ will be close. Rather LLN says thateventually (some n) the sample quantity will converge to thepopulation value.

Ins Mkt

Weather as Natural Disaster

Think Hurricane Sandy.

Many extreme weather events affect everyone within region.Positive Dependence.

To protect capital reserves, the insurance company must limitexposure to risks. Hence, company may limit the number ofpolicies written in a region. (Risk management)

Crop–Hail Insurance one of the first types of crop insuranceavailable (in 1820s in France) because hail concentrated in narrowisolated bands.

Summer 2011 — Waunakee Hail Storm. Our neighborhood (1–2mile radius). Other side of town not affected.

Ins Mkt

Spread risk over group

See that insurance spreads risk over group.

Want the insured event to be beyond the individual’s control.

Must be able to verify outcome.

Insurance company needs to limit the amount of positiveassociation of outcomes among policy holders. (Insurers andReinsurers)

Ins Mkt

Mutual Insurance

Same principles of insurance apply in developing country context.May not have institutions established that will support marketsolution.

Ray’s example at beginning of chapter of two farmers Asaf andSharif. Described as identical identical (same crop, same amount ofland =⇒ homogenous) but each subject to uncertainty of harvest.

Can establish insurance scheme between themselves. (Mutualinsurance, no share holders, only policy holders.)

Ins Mkt

Asaf and Sharif

Insurance will be an agreement that if one harvest is bad and theother is good then the one with the good harvest will give incometo the other.

Insurance is pooling to lower risk.

Standard way to measure risk is by the variance or dispersion inoutcomes.

Denote Asaf’s harvest by Qa and Sharif’s harvest by Qs . These arerandom variables, and yield the bountiful harvest with probability pand the low harvest with probability 1− p.

The two point probability distribution is the second simplestrandom variable. I want to make a basic point, and it is moredirect to do that using the notation of a general or generic randomvariable.

Ins Mkt

Pooling

Let Qa and Qs be random variables with pdf f and cdf F . Themean of the distribution is E [Qa] = E [Qs ] = µ, and the variance isVar(Qa) = Var(Qs) = σ2.

Qa, Qs follow the same probability distribution but are distinctrealizations of this random (i.e., uncertain) agricultural process.

Let Z = Qa + Qb be the pooled agricultural output. Basically theinsurance agreement is one of sharing output.

Asaf and Sharif are identical so natural outcome is to divide totaloutput equally.

Thus to see the potential advantage of sharing want to compareQa or Qs versus Z/2.

Ins Mkt

Potential Advantage of Pooling

Need to calculate the mean and variance of Z/2 and compare toQ (drop subscript).Mean of Z is

E [Z/2] =1

2(E [Qa] + E [Qs ]) =

1

2(µ+ µ) = µ

The variance of Z/2 is

Var(Z/2) = Var

[1

2(Qa + Qs)

]=

1

4(Var(Qa) + Var(Qs) + 2Cov(Qa,Qs)

=2σ2 + 2Cov(Qa,Qs)

4

=σ2 + Cov(Qa,Qs)

2

Ins Mkt

Covariance

I The last term Cov is covariance, which is how Qa,Qs co–vary.

I Covariance measures the association of two random variables.

I If the variables move together (one large the other likely islarge) then Covariance is positive.

I If the variables move opposite one another (one large theother likely small) then Covariance is negative.

I If the variables are independent, have no association, thenCovariance is zero.

Ins Mkt

Pooling if . . .

I When does pooling reduce the variance of the harvest?

I Tantamount to asking: Var(Z/2) < σ2

I Substituting the expression for Var(Z/2) in the inequality

Var(Z/2) < σ2

σ2 + Cov(Qa,Qs)

2< σ2

σ2 + Cov(Qa,Qs) < 2σ2

Cov(Qa,Qs) < σ2.

Ins Mkt

Pooling

If Qa and Qa are independent, Cov(Qa,Qs) = 0.

Qa,Qb can have some positive association. Indeed, as long asQa,Qb are not perfectly correlated (=1), the variation of theshared variance will be less than the individual harvest.

Express condition in terms of correlation coefficient ρ,

For arbitrary random variables, X1,X2,

Cov(X1,X2) = ρσ1σ2

ρ =Cov(X1,X2)

σ1σ2

Ins Mkt

Pooling

Now use the fact that Qa, Qs come from the same agriculturalprocess so their means and variances are the same so

Cov(Qa,Qs) = ρσσ = ρσ2

So our condition becomes

Cov(Qa,Qs) < σ2

ρσ2 < σ2

ρ < 1.

Ins Mkt

Insurance (Continuation) Part II

Review:

I Insurance is a mechanism by which to smooth consumptionover uncertain states of nature.

I Provision of insurance increases utility for risk averse agents.

I Provision of insurance provide incentive to shirk (less effort).

I Insurance must balance risk aversion and work incentives.

I Key component: information.

Ins Mkt

Voluntary Behavior

All contracts, whether formal or informal are voluntarily entered.Can not force someone to accept insurance.

Adverse Selection — healthy individuals (e.g., young) may wantnot want to enroll.

If the healthy opt out and only those who most need insuranceparticipate the insurance plan may not be financially viable.

Ins Mkt

Must check incentives

For the given structure of information (who knows what andwhen), must make sure that all parties of the insurance contractare no worse off and some are better off.

This means:

1. Insurance companies break even; premiums and benefits arefinancial viable.

2. Risk averse individual is better off with insurance thanwithout.

3. Insurance contract written so that risk averse individual doesnot shirk.

Illustrate these ideas through an example developed from onepresented on page 604 of the textbook.

Ins Mkt

Enforcement

I Developing countries, insurance informal contracts, socialnorms encourage reciprocity and punish deviations throughsocial sanctions.

I Once again must investigate incentives of individuals toreciprocate and to deviate.

I Assumption: rational individuals take option that yieldshighest payoff. (Payoff in utility)

I Enforcement: through social sanctions.

I Equation (15.6) summary of enforcement forces to maintaininsurance contract.

Ins Mkt

Insurance and Credit Blurred Boundary

I Key feature of insurance: lack of memory on history.

I Basing insurance payouts to history of past claims, a schemeceases to be one of pure insurance, and acquires somecharacteristics of credit.

I Such history dependent insurance contracts permits a greaterdegree of consumption smoothing.

I Such arrangements: part insurance, part credit are common indeveloping countries.

I Social norms and family members perform many of thefunctions done by markets in developed countries.

Ins Mkt