Risk Aversion Expected Utility And The St Petersburg Paradox

We digress here to examine the rationale behind our contention that investors are risk averse. Recognition of risk aversion as central in investment decisions goes back at least to 1738. Daniel Bernoulli, one of a famous Swiss family of distinguished mathematicians, spent the years 1725 through 1733 in St. Petersburg, where he analyzed the following coin-toss game. To enter the game one pays an entry fee. Thereafter, a coin is tossed until the first head appears. The number of tails, denoted by n, that appears until the first head is tossed is used to compute the payoff, $R, to the participant, as

The probability of no tails before the first head (n = 0) is xh and the corresponding payoff is 20 = $1. The probability of one tail and then heads (n = 1) is xh X xh with payoff 21 = $2, the probability of two tails and then heads (n = 2) is xh X xh X i/2, and so forth. The following table illustrates the probabilities and payoffs for various outcomes:

Tails

Probability

Payoff = $R(n)

Probability x Payoff

0

1/2

$1

$1/2

1

1/4

$2

$1/2

2

Va

$4

$1/2

3

Vie

$8

$1/2

n

(1/2)n+1

$2n

$1/2

The expected payoff is therefore q

CHAPTER 6 Risk and Risk Aversion

The evaluation of this game is called the "St. Petersburg Paradox." Although the expected payoff is infinite, participants obviously will be willing to purchase tickets to play the game only at a finite, and possibly quite modest, entry fee.

Bernoulli resolved the paradox by noting that investors do not assign the same value per dollar to all payoffs. Specifically, the greater their wealth, the less their "appreciation" for each extra dollar. We can make this insight mathematically precise by assigning a welfare or utility value to any level of investor wealth. Our utility function should increase as wealth is higher, but each extra dollar of wealth should increase utility by progressively smaller amounts.4 (Modern economists would say that investors exhibit "decreasing marginal utility" from an additional payoff dollar.) One particular function that assigns a subjective value to the investor from a payoff of $R, which has a smaller value per dollar the greater the payoff, is the function ln(R) where ln is the natural logarithm function. If this function measures utility values of wealth, the subjective utility value of the game is indeed finite, equal to .693.5 The certain wealth level necessary to yield this utility value is $2.00, because ln(2.00) = .693. Hence the certainty equivalent value of the risky payoff is $2.00, which is the maximum amount that this investor will pay to play the game.

Von Neumann and Morgenstern adapted this approach to investment theory in a complete axiomatic system in 1946. Avoiding unnecessary technical detail, we restrict ourselves here to an intuitive exposition of the rationale for risk aversion.

Imagine two individuals who are identical twins, except that one of them is less fortunate than the other. Peter has only $1,000 to his name while Paul has a net worth of $200,000. How many hours of work would each twin be willing to offer to earn one extra dollar? It is likely that Peter (the poor twin) has more essential uses for the extra money than does Paul. Therefore, Peter will offer more hours. In other words, Peter derives a greater personal welfare or assigns a greater "utility" value to the 1,001st dollar than Paul does to the 200,001st. Figure 6B.1 depicts graphically the relationship between the wealth and the utility value of wealth that is consistent with this notion of decreasing marginal utility.

Individuals have different rates of decrease in their marginal utility of wealth. What is constant is the principle that the per-dollar increment to utility decreases with wealth. Functions that exhibit the property of decreasing per-unit value as the number of units grows are called concave. A simple example is the log function, familiar from high school mathematics. Of course, a log function will not fit all investors, but it is consistent with the risk aversion that we assume for all investors.

Now consider the following simple prospect:

$100,000

$150,000

$80,000

4 This utility is similar in spirit to the one that assigns a satisfaction level to portfolios with given risk and return attributes. However, the utility function here refers not to investors' satisfaction with alternative portfolio choices but only to the subjective welfare they derive from different levels of wealth.

5 If we substitute the "utility" value, ln(R), for the dollar payoff, R, to obtain an expected utility value of the game (rather than expected dollar value), we have, calling V(R) the expected utility,

Figure 6B.1 Utility of wealth with a log utility function.

Figure 6B.1 Utility of wealth with a log utility function.

Log Utility Function Hedging

This is a fair game in that the expected profit is zero. Suppose, however, that the curve in Figure 6B.1 represents the investor's utility value of wealth, assuming a log utility function. Figure 6B.2 shows this curve with numerical values marked.

Figure 6B.2 shows that the loss in utility from losing $50,000 exceeds the gain from winning $50,000. Consider the gain first. With probability p = .5, wealth goes from $100,000 to $150,000. Using the log utility function, utility goes from ln(100,000) = 11.51 to ln(150,000) = 11.92, the distance G on the graph. This gain is G = 11.92 - 11.51 = .41. In expected utility terms, then, the gain is pG = .5 X .41 = .21.

Now consider the possibility of coming up on the short end of the prospect. In that case, wealth goes from $100,000 to $50,000. The loss in utility, the distance L on the graph, is L = ln(100,000) - ln(50,000) = 11.51 - 10.82 = .69. Thus the loss in expected utility terms is (1 - p)L = .5 X .69 = .35, which exceeds the gain in expected utility from the possibility of winning the game.

We compute the expected utility from the risky prospect:

11.37

If the prospect is rejected, the utility value of the (sure) $100,000 is ln(100,000) = 11.51, greater than that of the fair game (11.37). Hence the risk-averse investor will reject the fair game.

Using a specific investor utility function (such as the log utility function) allows us to compute the certainty equivalent value of the risky prospect to a given investor. This is the amount that, if received with certainty, she would consider equally attractive as the risky prospect.

If log utility describes the investor's preferences toward wealth outcomes, then Figure 6B.2 can also tell us what is, for her, the dollar value of the prospect. We ask, "What sure

CHAPTER 6 Risk and Risk Aversion

Figure 6B.2 Fair games and expected utility.

CHAPTER 6 Risk and Risk Aversion

Figure 6B.2 Fair games and expected utility.

Expected Utility Risk Premium Certain

level of wealth has a utility value of 11.37 (which equals the expected utility from the prospect)?" Ahorizontal line drawn at the level 11.37 intersects the utility curve at the level of wealth WCE. This means that ln(WcE) = 11.37

which implies that

$86,681.87

WCE is therefore the certainty equivalent of the prospect. The distance Y in Figure 6B.2 is the penalty, or the downward adjustment, to the expected profit that is attributable to the risk of the prospect.

Y = E(W) - WCE = $100,000 - $86,681.87 = $13,318.13

This investor views $86,681.87 for certain as being equal in utility value as $100,000 at risk. Therefore, she would be indifferent between the two.

Suppose the utility function is U(W = VW.

a. What is the utility level at wealth levels $50,000 and $150,000?

b. What is expected utility if p still equals .5?

c. What is the certainty equivalent of the risky prospect?

d. Does this utility function also display risk aversion?

e. Does this utility function display more or less risk aversion than the log utility function?

Does revealed behavior of investors demonstrate risk aversion? Looking at prices and past rates of return in financial markets, we can answer with a resounding "yes." With remarkable consistency, riskier bonds are sold at lower prices than are safer ones with otherwise similar characteristics. Riskier stocks also have provided higher average rates of

CONCEPT CHECK ^ QUESTION B.1

PART II Portfolio Theory return over long periods of time than less risky assets such as T-bills. For example, over the 1926 to 1999 period, the average rate of return on the S&P 500 portfolio exceeded the T-bill return by about 9% per year.

It is abundantly clear from financial data that the average, or representative, investor exhibits substantial risk aversion. For readers who recognize that financial assets are priced to compensate for risk by providing a risk premium and at the same time feel the urge for some gambling, we have a constructive recommendation: Direct your gambling impulse to investment in financial markets. As Von Neumann once said, "The stock market is a casino with the odds in your favor." A small risk-seeking investment may provide all the excitement you want with a positive expected return to boot!

PROBLEMS: APPENDIX B

Suppose that your wealth is $250,000. You buy a $200,000 house and invest the remainder in a risk-free asset paying an annual interest rate of 6%. There is a probability of .001 that your house will burn to the ground and its value will be reduced to zero. With a log utility of end-of-year wealth, how much would you be willing to pay for insurance (at the beginning of the year)? (Assume that if the house does not burn down, its end-of-year value still will be $200,000.)

If the cost of insuring your house is $1 per $1,000 of value, what will be the certainty equivalent of your end-of-year wealth if you insure your house at:

b. Its full value.

c. 1V2 times its value.

SOLUTIONS

B.1. a.

U(W) = VW

TO CONCEPT

U(50,000) = V50,000 = 223.61

CHECKS

U(150,000) = 387.30

b.

E(U) = (.5 X 223.61) + (.5 X 387.30) = 305.45

c.

We must find WCE that has utility level 305.45. Therefore VWCE = 305.45

WCE = 305.452 = $93,301

d.

Yes. The certainty equivalent of the risky venture is less than the expected outcome of $100,000.

e.

The certainty equivalent of the risky venture to this investor is greater than it was for the log utility investor considered in the text. Hence this utility function displays less risk aversion.

Lessons From The Intelligent Investor

Lessons From The Intelligent Investor

If you're like a lot of people watching the recession unfold, you have likely started to look at your finances under a microscope. Perhaps you have started saving the annual savings rate by people has started to recover a bit.

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