Chapter 12: Risk Aversion and Insurance

A. The Insurance Industry

• Most individuals are risk-averse• Insurance is the reassignment of a risk in exchange for a

premium, whereby the more risk-averse entity (the insured) assigns to (indemnify against) another (the insurer) risk in exchange for a cash payment (premium).

• Insurance companies pool their risks among many insured clients, enabling them to mitigate risks through diversification.

• Most insurers employ actuaries who analyze and calculate risks and fair premiums.

Early Insurance Industry History

• 3rd millennium BCE in China with distributing their risks (insurance) and 2nd millennium Babylon paying risk premiums to lenders to forgive debts should they lose cargo.

• Maritime insurance in 14th century Genoa. • Building and fire insurance grew in usage after the Great Fire of London of 1666• Lloyd’s of London was a partially mutualized insurance marketplace located in a

1680s coffee house• Life insurance date to 1706 London, when the Amicable Society for a Perpetual

Assurance Office was founded.• The Philadelphia Contributionship was founded by Benjamin Franklin in 1752• The early growth of the life insurance industry was impeded by the moral

complications related to associating monetary values to life and the inability of women to inherit.

• Early insurance was state regulated in part due to the 1868 Supreme Court decision in Paul v. Virginia.

Insurance Regulation

• The McCarran–Ferguson Act of 1945– partially exempts insurance companies from

federal anti-trust legislation– allows states to regulate insurance

• The Gramm-Leach-Bliley Act of 1999 overturned certain provisions of Glass-Steagall legislation, permitting banks, insurers and securities firms to affiliate and cross-sell each other’s products.

Primary Types of Insurance Policies

1. Life Insurance: Pays at the death of the policy holder or at the policy’s maturity. Types of life insurance policies include, but are not limited to:

a. Term life insurance: Pays a specified death benefit over a set term.

b. Whole life insurance: Pays a death benefit and builds cash value based on a set schedule for the life of the insured.

c. Universal life insurance: Pays a death benefit and earns a fixed interest rate on the cash value in the policy for the life of the insured. Premiums and benefits can be flexible.

d. Variable universal life insurance: Pays a death benefit and ties the policy cash value to an investment such as a stock market index.

2. Casualty: Pays in the event of a specified damage to a given asset.

3. Health: Provides for payment for designated treatments or care related to health, sickness or physical injury.

4. Automobile and Other Vehicle: Pays in the event of a designated event causing specified types of damage.

B. Measuring Risk Aversion and Calculating Insurance Premiums

Figure 1: Expected Utility of Wealth and Utility of Expected Wealth

w

U

E[w] wMIN wMAX

UMAX

U(E[w])

UMIN

E[Uw]

Risk Aversion in the Small• Let W be initial wealth and z be a normally distributed risk with E[z] =

0 and E[z2] = σ2:

E[U( W + z )] = U(W - π)

With Risk z Without Risk z

• Expand both sides in a Taylor Series polynomial:

E[U(W) + zU’(W) + ½ z2U”(W) + …] = U(W) - πU’(W) + .....

• Since E[z] = 0, E[z]U’(W) can be dropped from the equality and σ2 = E[z2]. Because z is normally distributed, and risks are presumed to be small, higher order terms can be dropped as well:

E[U(W)] + ½ σ2 U”(W) = U(W) - πU'(W)

Risk Aversion and Premiums

E[U(W)] + ½σ2 U”(W) = U(W) - πU'(W)

 • Now, we solve for the risk premium as follows:

 

 • When used in this context, -U”(W)/U’(W) is referred to as the Arrow-Pratt Absolute Risk

Aversion Coefficient (ARA), which indicates an investor's aversion to a small risk.• A given investor A will accept a particular gamble that is unacceptable to another

investor B if his ARA is smaller at their current wealth levels:

 

• In this scenario, Investor B is said to be more risk averse than is Investor A; his utility of wealth function exhibits more concavity than does the utility curve for Investor A.

Illustration: CARA Utility

Suppose an investor has the following CARA utility of wealth function:

U(W) = 1 – e-.5W

The investor’s Absolute Risk Aversion (ARA) coefficient is calculated as follows:

U’(W) = .5e-.5W ; U”(W) = -.25e-.5W

ARA = = = .5

Suppose that the investor faces a risk σ2 = 100. We calculate the Arrow-Pratt premium that she would be willing to indemnify this risk as follows:

C. Annuities• A large number of life insurance payouts arte

in the form of annuities.• Many of these annuities are deferred.

• A series of cash flows CFt is valued as follows:

 

(1)

(1.a)  

Geometric Expansions and Annuities

A geometric expansion is an algebraic procedure used to simplify a geometric series. Suppose we wished to solve the following finite geometric series for , where CF is a constant and PVA is the present value of an annuity:

 

(2)

 

First, divide both sides of the equation by (1+k):

 

(3)

Next, to eliminate repetitive terms, subtract (3) from (2) and simplify:

 

(4)

 

(5) ===

Growing Annuities

Illustration:CF1 = 100 k = .10g = .04 n = 5

 

t CFt PV[CFt] ∑ PV[CFt]

1 100 90.90909 90.90909 

2 104 85.95041 176.8595 

3 108.16 81.26221 258.1217 

4 112.4864 76.82972 334.9514 

5 116.9859 72.63901 407.5905 

Deferred Annuities

• Illustration:• CF1 = 100 k = .10• s = 5 n = 5 

Rothschild and Stiglitz:Adverse Selection and Screening

• Assume a population comprised of 10% high risk drivers and 90% low risk drivers, all with wealth of $40,000.

• Each high risk driver has a 75% probability of an accident, with a claim of $20,000.

• Each low risk driver has only a 25% probability of an accident with a claim of $20,000.

• The expected claim for each driver is $6,000 = $20,000 × (.10×.75 + .90×.25). Suppose that π = $6,000.

• The insurance company knows the proportion of high- and low-risk drivers, but cannot distinguish the two types of drivers.

• Drivers know whether they are high risk or low risk.• Each driver has a log utility function U = lnW.

Adverse Selection and Screening Illustration, cont.

• Uninsured high risk drivers have expected utility of .75×ln(40,000-20,000) +.25×ln (40,000) = 10.07677.

• Uninsured low risk drivers have expected utility of .25×ln(40,000-20,000) +.75×ln(40,000) = 10.42335.

• Again, the insurance company offers policies for $6,000. • Insured high risk drivers have expected utility level equal

to .75×ln(40,000-6,000-20,000+20,000) +.25×ln(40,000-6,000) = 10.43412, and will purchase the policy.

• Insured low risk drivers have expected utility levels equal to .25×ln(40,000-6,000-20,000+20,000) +.75× ln(20,000-6,000) = 9.768638. Low risk drivers will not purchase the policy.

• This is a standard case of adverse selection.

Adverse Selection, Cont.

• The insurance company cannot sell policies to low-risk drivers at the pooling premium of $6,000. 

• The expected loss on each policy sold to high-risk drivers would be .75×20,000 – 6,000 = 9,000. 

• The insurance company can break even if it charges a premium greater than $15,000, where only the high risk drivers would purchase insurance. 

• How can the insurance company capture the low risk driver market?

Adverse Selection, Cont.• Consider a policy with  a deductible equal to D and a premium equal to P. This second policy will be targeted towards the 

low risk drivers. • If a low risk driver purchases the policy with the deductible, his expected utility will be .25×ln(40,000-P-D) 

+.75×ln(40,000-P).• The premium be determined as follows: P = .25×(20,000-D) =5,000-.25×D. • To induce low risk drivers to accept this policy, their expected utility levels would have to be at least: 

10.42335 = .25×ln(40,000 – P - 20,000 + 20,000 - D) +.75×ln(40,000 – P)10.42335 = .25×ln(35,000  - .75D) +.75×ln(35,000 + .25D)

• Next, we determine what level of D would be required to exclude high risk drivers. To induce high risk drivers to reject this policy, their expected utility levels would have to be less than with full insurance at a premium of $15,000:

 10.12663 = .75×ln(40,000 – P - 20,000 + 20,000 - D) +.25×(40,000 – P)

10.12663 = .75×ln(35,000-.75×D) +.25×ln(35,000+.25D) • Thus, for this example, a particularly high deductible of $18,051 (found by substitution) is needed to screen out the high 

risk driver. The high-risk driver prefers the full insurance policy at a premium of $15,000 to this high deductible policy. • The premium for this high deductible policy is P = $5,000 - .25D = $487.25 (based on low-risk drivers).• In this scenario, the low risk driver prefers the partial insurance policy with the high deductible of $18,051 for a premium 

of $487.25. The high risk driver prefers the full insurance for a premium of $15,000.

Lesson:

• Insurance deductibles serve two primary purposes:

1. Deductibles mitigate the adverse selection problem by making low-premium policies less attractive to high risk policy-holders.

2. Deductibles mitigate the moral hazard by forcing policy-holders to participate in loses.