XII. RISK-SPREADING VIA FINANCIAL INTERMEDIATION: LIFE INSURANCE

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1 XII. RIS-SPREADIG VIA FIACIAL ITERMEDIATIO: LIFE ISURACE As discussed briefly at the end of Section V, financial assets can be traded directly in the capital markets or indirectly through financial intermediaries. In general, "standardized" securities are traded in markets (e.g., government bonds, wheat futures, shares of IBM) while "custom" contracts (e.g., individual mortgage, personal loan, or insurance) are handled through financial intermediaries. In this section, the classical case of pure life insurance is examined to show how efficient risksharing can be achieved using a combination of financial intermediation and the capital market. The Life Insurance Company Suppose that there are people in the economy each with wealth (per capita) W. Hence, national wealth W W. Suppose further that each person purchases a one-year term life insurance policy which pays $c in the event of death and we define q c/w to be the amount of insurance coverage purchased by each person as a fraction of his wealth. Let y i be a random variable describing the death of the i th person where y i 0 if person i survives the year and y i 1 if person i dies during the year. Assume that the mortality tables are such that E( yi ) ρ, the same for all people, i 1,,..., and Var( y ) V which is also the same for all people. Hence, ρ is the expected number of deaths per person (0 < ρ <1). Let Y y be the random variable for deaths of all people and it is equal to the number of deaths in the economy. If the death of one person is independent of another (a crucial but reasonable assumption), then i i1 i E[Y ] ρ ; Var(Y ) V 41

2 Robert C. Merton If a single competitive insurance company writes all the policies, then the analysis will determine the: Premium per policy charged, P The amount of equity capital required by the company to do business,. The required (expected) return on the equity by investors in the insurance company. Premiums are received at the beginning of the year in the amount, P. Benefits are paid at the end of the year in the (random variable) amount C c Y. Hence, E[ C ] C cρ Var( C ) σ c V Suppose that investors are mean-variance imizers and that the conditions for the Capital Asset Pricing Model (Section XI) hold. If R 1 + rate of interest, then the return per dollar invested in equity of the insurance company is E[ Z Var[ Z ] Z ] σ Z σ R[( P Z R( P + )- C, and )+ c V ] - cρ In equilibrium, supply must equal demand, and so, for the equity of the insurance company to be held, we have that Z must satisfy the basic equilibrium condition for the CAPM: supply Demand for asset; or wz fraction of the market portfolio W where W Z is also the fraction in optimal combination of risky assets given by: 4

3 Finance Theory w vz j Z j1 Z n ( - R) ( A-RC) j Suppose (as is reasonable) that Z is independent of the returns on all other assets (i.e., Cov ( Z, Z j ) 0 for Z Z j ), then n Cov Cov c V c V ( Z, ZM ) wj ( Z, Z j) wz Var ( Z ) ( )( ). j1 W W From Section XI, we have from the Security Market Line that Substituting for Z, we have that Z - R rs Cov ( Z,Z R[ P + ] - cρ - R r M ) r S c V W cv RP - c ρ r S or substituting for c qw W q WV RP - qρw r S S c V W or The (equilibrium) premium per policy can be written as qwρ 1 cρ 1 P + r S q WV r S cqv and therefore, R R R R r S cqv Z R +, the (equilibrium) expected return on equity in the insurance company. 43

4 Robert C. Merton ote: In this formal analysis, we have not taken into account the ed liability feature of the equity of the insurance company which leads us to the last question to be answered: What is the appropriate value for? To answer this question, one must go back and ask what service is the financial intermediary to provide to the customer? What does he want? The customer wants a certain payment, c, in the event of death. ow, if the total number of deaths is such that the (ex-post) benefits required to be paid, c, is larger than the company's total assets, R( P + ), then by ed liability on R equity, the customer will not receive the full benefits promised, but only ( P + )< c. y Obviously, the larger is the less likely is default. If is "too small", then the probability of default is higher, and the customer in purchasing the policy does not get the simple security he wanted which pays $c for sure in the event of his death, but rather has the more complicated security which pays $c in the event of his death, conditional on the company being solvent, pays the R (r.v.) amount ( P + ) y in the event of death, conditional on the company not being solvent. Clearly, to assess the probabilities about possible payoffs, the customer would have to know the amount of capital the firm has; the nature and quantity of policies written for other customers; the probability of these customers dying; etc. In short, nearly everything about the company that the management knows, the customer would have to know and analyze. Essentially, the customer takes a (partial) equity position in the company. Since one purpose of the financial intermediary is to the amount of information required by customers to make a decision and because the service wanted is basically life insurance, the equity capital should be large enough to (virtually) eliminate the chance of default. In doing so, the separation between customer and equityholder (or general liability or debtholder) is made as large as possible. Define: reserves as the amount of assets required to be held by the insurance company to ensure that payment will be made to customers with some probability. Let ( P ) R reserves requires R + So, given the premium, there is a one-to-one correspondence between reserves and capital (equity). Clearly, the amount of reserves required to ensure with absolute certainty that all customers will be paid in every state of the world would come by 44

5 Finance Theory requiring that assets be large enough to payoff everyone in the event that everyone dies. I.e., C c or R imum required reserves c R( P + ) 1 r S cqv c( - ρ ) -. However, for c large and ρ reasonably small, the amount of R R capital required to meet the imum reserves could be prohibitively large. Further, if ρ is small, there is a very small chance that everyone will die (especially since the events of death are independent) and one would expect that for large, there would be some diversifying effects. Hence, it may not be necessary for the company to hold the imum reserves while still performing the essential service required. Suppose that instead it was required that reserves be such that the probability of default is less than some assigned level,, p i.e., Prob{ C R } p, capital as ( p ). ote: (0). Prob{ C > R } n 1 > and define the associated required R ( p ) s i)c > R ( ) + cρ + r cqv } Prob { s X > ( ) + Prob{( y p rcqv V c or 1 y -ρ i and E( ) 0; Var( ) 1. where X X X V For large, X will be distributed approximately standard normal (Gaussian). Hence, R ( p ) rs qv R ( p ) r S qv Prob { X >( ) + } 1- Φ[( ) + ] p where Φ [ ] is the Vc Vc cumulative density function for the normal distribution. For this distribution, there is a one-to-one correspondence between p and the number of standard deviations to the right of the mean. I.e., let µ number of standard deviations, then 45

6 Robert C. Merton µ(p ) p (1 chance in 10,000) (1 chance in 5,000) Hence, R ( p ) r S qv µ ( p )( ) + or Vc Vc ( p ) ( ) ( ) - qv p rs R µ for large. Thus, for a given p (or µ ( p )), we have an expression for the required equity capital for large. 1 Asymptotic Results for large ( ) ( > p > 0) ( p ) ( Vc R ) µ ( p ) cρ ( ) ; ( ) R; ( ) R P Z Z R lim σ ( p µ >0. ) ( Required Equity Total Premiums ) ( p ( P ) ) 0; lim ( p ( W ) ) 0; Suppose Required Equity Maximum Equity Suppose ρ.005; ( p ( c $30,000 ) ) 0 p ,000 V.005 R 1 [ µ ( p ) 4] 46

7 Finance Theory 10,000 (1x10 4 ) 90,000 (9x10 4 ) 1,000,000 (1x10 6 ) one million 9,000,000 (9x10 6 ) 1x10 8 one hundred million P (0) (.00004) (.00004) $750,000 $99,50,000 $600, $6,750,000 $75,000,000 $,693,50,000 ($.7 billion) $30,000,000,000 ($30 billion) P $1,800, $6,000, $675,000,000 $70 billion $18,000, $7,500,000,000 ($7.5 billion) $3 trillion (3x10 1 ) $60,000,000 (60 million) (6x10 7 ).0080 Thus, we observe a characteristic property of (many) financial intermediaries: namely, that net worth is a small fraction of total assets (in the example, less than 1%); further total (potential) liabilities are many orders of magnitude larger than total assets or reserves. Of course, sales and other operating expenses would have to be added to the premium and other assets (buildings, etc.) have been excluded. The benefits of the financial intermediary in this case are obvious: if each insurance policy were written by one person for one other person (and if and q 1), then the minimum premium for a $30,000 policy would be $5 versus $75 charged by the company. Further, the reserves required would be $30,000 (or 1 ) per policy versus $.60 per policy for the intermediary! (50,000 times as much!) ote despite the tremendous diversifying power of many policies, if there were no equity capital market to raise the funds, it is doubtful if such an organization could occur without substantially higher premiums. Suppose one (wealthy) individual provided all the capital ($60 million): (at the derived rates with R 1) the expected rate of return is zero and there is a.16 probability of one standard deviation to the right which translates into a $15 million loss! Few r S 47

8 Robert C. Merton risk-averse utility imizers would accept such an investment. But, by diversifying the risk by issuing equity in the capital market and if individuals hold well-diversified equity portfolios (as they should), then the loss would be around 15 per investor which is trivial for an investor with initial wealth of $30,000. Thus, through the combined use of the capital market and the financial intermediary, the individual investor can get the service or asset he wants (virtually no-default life insurance) to eliminate a substantial non-systematic risk, at minimum cost. 48