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Suppose that you must choose one of two possible investments, each of which can result in any of n consequences, denoted C₁, ..., Cn. Suppose

Valuing Investments by Expected Utility 167 that if the first investment is chosen then consequence i will result with probability pi (i = 1, ..., n), whereas if the second one is cho-sen then consequence i will result with probability qi (i = 1, ..., n), wheren

i=1pi =n

i=1qi = 1. The following approach can be used to determine which investment to choose.

We begin by assigning numerical values to the different consequences as follows. First, identify the least and the most desirable consequence, call them c and C respectively; give the consequence c the value 0 and give C the value 1. Now consider any of the other n− 2 consequences, say Ci. To value this consequence, imagine that you are given the choice between either receiving Ci or taking part in a random experiment that earns you either consequence C with probability u or consequence c with probability 1− u. Clearly your choice will depend on the value of u. If u = 1 then the experiment is certain to result in consequence C;

since C is the most desirable consequence, you will clearly prefer the experiment to receiving Ci. On the other hand, if u = 0 then the ex-periment will result in the least desirable consequence, namely c, and so in this case you will clearly prefer the consequence Ci to the ex-periment. Now, as u decreases from 1 down to 0, it seems reasonable that your choice will at some point switch from the experiment to the certain return of Ci, and at that critical switch point you will be indif-ferent between the two alternatives. Take that indifference probability u as the value of the consequence Ci. In other words, the value of Ci is that probability u such that you are indifferent between either receiving the consequence Ci or taking part in an experiment that returns conse-quence C with probability u or conseconse-quence c with probability 1− u.

We call this indifference probability the utility of the consequence Ci, and we designate it as u(Ci).

In order to determine which investment is superior, we must eval-uate each one. Consider the first one, which results in consequence Ci with probability pi (i = 1, ..., n). We can think of the result of this investment as being determined by a two-stage experiment. In the first stage, one of the values 1, ..., n is chosen according to the prob-abilities p₁, ..., pn; if value i is chosen, you receive consequence Ci. However, since Ci is equivalent to obtaining consequence C with prob-ability u(Ci) or consequence c with probability 1−u(Ci), it follows that the result of the two-stage experiment is equivalent to an experiment in

168 Valuing by Expected Utility

which either consequence C or c is obtained, with C being obtained with probability

n i=1

piu(Ci).

Similarly, the result of choosing the second investment is equivalent to taking part in an experiment in which either consequence C or c is ob-tained, with C being obtained with probability

n i=1

qiu(Ci).

Since C is preferable to c, it follows that the first investment is prefer-able to the second if

n i=1

piu(Ci) >

n i=1

qiu(Ci).

In other words, the value of an investment can be measured by the ex-pected value of the utility of its consequence, and the investment with the largest expected utility is most preferable.

In many investments, the consequences correspond to the investor re-ceiving a certain amount of money. In this case, we let the dollar amount represent the consequence; thus, u(x) is the investor’s utility of receiving the amount x. We call u(x) a utility function. Thus, if an investor must choose between two investments, of which the first returns an amount X and the second an amount Y, then the investor should choose the first if

E[u(X )] > E[u(Y )]

and the second if the inequality is reversed, where u is the utility func-tion of that investor. Because the possible monetary returns from an investment often constitute an infinite set, it is convenient to drop the requirement that u(x) be between 0 and 1.

Whereas an investor’s utility function is specific to that investor, a general property usually assumed of utility functions is that u(x) is a nondecreasing function of x. In addition, a common (but not universal) feature for most investors is that, if they expect to receive x, then the

Valuing Investments by Expected Utility 169

Figure 9.2: A Concave Function

extra utility gained if they are given an additional amount is nonin-creasing in x; that is, for fixed  > 0, their utility function satisfies

u(x + ) − u(x) is nonincreasing in x.

A utility function that satisfies this condition is called concave. It can be shown that the condition of concavity is equivalent to

u^(x) ≤ 0.

That is, a function is concave if and only if its second derivative is non-positive. Figure 9.2 gives the curve of a concave function; such a curve always has the property that the line segment connecting any two of its points always lies below the curve.

An investor with a concave utility function is said to be risk-averse.

This terminology is used because of the following, known as Jensen’s inequality, which states that if u is a concave function then, for any ran-dom variable X,

E[u(X )] ≤ u(E[X]).

Hence, letting X be the return from an investment, it follows from Jensen’s inequality that any investor with a concave utility function would prefer the certain return of E[X ] to receiving a random return with this mean.

170 Valuing by Expected Utility We now give a proof of

Jensen’s Inequality If U is concave then E[U(X)] ≤ U(E[X])

Proof of Jensen’s Inequality. The Taylor series formula with remainder of U(x) expanded about μ = E[X] gives, for some value of τ between x andμ, that

U(x) = U(μ) + U^(μ)(x − μ) + U^(τ)(x − μ)²/2 But U being concave implies that U^≤ 0, showing that

U(x) ≤ U(μ) + U^(μ)(x − μ) Consequently,

U(X) ≤ U(μ) + U^(μ)(X − μ) Now take expectations of both sides to obtain the result:

E[U(X)] ≤ U(μ) + U^(μ)E[X − μ] = U(μ) An investor with a linear utility function

u(x) = a + bx, b > 0,

is said to be risk-neutral or risk-indifferent. For such a utility function, E[u(X )] = a + bE[X]

and so it follows that a risk-neutral investor will value an investment only through its expected return.

A commonly assumed utility function is the log utility function u(x) = log(x);

see Figure 9.3. Because log(x) is a concave function, an investor with a log utility function is risk-averse. This is a particularly important utility function because it can be mathematically proven in a variety of situa-tions that an investor faced with an infinite sequence of investments can maximize long-term rate of return by adopting a log utility function and then maximizing the expected utility in each period.

Valuing Investments by Expected Utility 171

Figure 9.3: A Log Utility Function

To understand why this is true, suppose that the result of each invest-ment is to multiply the investor’s wealth by a random amount X. That is, if Wndenotes the investor’s wealth after the nth investment and if Xn

is the nth multiplication factor, then

Wn = XnWn−1, n ≥ 1.

With W₀ denoting the investor’s initial wealth, the preceding implies that

Wn = XnWn−1

= XnXn−1Wn−2

= XnXn−1Xn−2Wn−3

...

= XnXn−1· · · X1W₀.

If we let Rndenote the rate of return (per investment) from the n invest-ments, then

Wn

(1 + Rn)ⁿ = W0

172 Valuing by Expected Utility or

(1 + Rn)ⁿ = Wn

W₀ = X1· · · Xn. Taking logarithms yields that

log(1 + Rn) =

n

i=1log(Xi)

n .

Now, if the Xi are independent with a common probability distribution, then it follows from a probability theorem known as the strong law of large numbers that the average of the values log(Xi), i = 1, ..., n, con-verges to E[log(Xi)] as n grows larger and larger. Consequently,

log(1 + Rn) → E[log(X )] as n → ∞.

Therefore, if one has some choice as to the investment – that is, some choice as to the probabilities of the multiplying factors Xi – then the long-run rate of return is maximized by choosing the investment that yields the largest value of E[log(X )].

Moreover, because Wn= W0X₁· · · Xn, it follows that log(Wn) = log(W0) +

n i=1

log(Xi).

Hence,

E[log(Wn)] = log(W0) + nE[log(X)]

which shows that maximizing E[log(X)] is equivalent to maximizing the expectation of the log of the final wealth.

The following example shows how much a log utility investor should invest in a favorable gamble.

Example 9.2a An investor with capital x can invest any amount be-tween 0 and x; if y is invested then y is either won or lost, with respective probabilities p and 1− p. If p > 1/2, how much should be invested by an investor having a log utility function?

Solution. Suppose the amountαx is invested, where 0 ≤ α ≤ 1. Then the investor’s final fortune, call it X, will be either x + αx or x − αx

Valuing Investments by Expected Utility 173 with respective probabilities p and 1− p. Hence, the expected utility of this final fortune is

p log((1 + α)x) + (1 − p) log((1 − α)x)

= p log(1 + α) + p log(x) + (1 − p) log(1 − α) + (1 − p) log(x)

= log(x) + p log(1 + α) + (1 − p) log(1 − α).

To find the optimal value ofα, we differentiate p log(1 + α) + (1 − p) log(1 − α) to obtain

d

dα( p log(1 + α) + (1 − p) log(1 − α)) = p

1+ α −1− p 1− α. Setting this equal to zero yields

p− αp = 1 − p + α − αp or α = 2p − 1.

Hence, the investor should always invest 100(2p − 1) percent of her present fortune. For instance, if the probability of winning is .6 then the investor should invest 20% of her fortune; if it is .7, she should invest 40%. (When p≤ 1/2, it is easy to verify that the optimal amount to in-vest is 0.)

Our next example adds a time factor to the previous one.

Example 9.2b Suppose in Example 9.2a that, whereas the investment αx must be immediately paid, the payoff of 2αx (if it occurs) does not take place until after one period has elapsed. Suppose further that what-ever amount is not invested can be put in a bank to earn interest at a rate of r per period. Now, how much should be invested?

Solution. An investor who investsαx and puts the remaining (1 − α)x in the bank will, after one period, have (1 + r)(1 − α)x in the bank, and the investment will be worth either 2αx (with probability p) or 0 (with probability 1− p). Hence, the expected value of the utility of his

174 Valuing by Expected Utility fortune is

p log((1 + r)(1 − α)x + 2αx) + (1 − p) log((1 + r)(1 − α)x)

= log(x) + p log(1 + r + α − αr)

+ (1 − p) log(1 + r) + (1 − p) log(1 − α).

Hence, once again the optimal fraction of one’s fortune to invest does not depend on the amount of that fortune. Differentiating the previous equation yields

d

dα(expected utility) = p(1 − r)

1+ r + α − αr −1− p 1− α.

Setting this equal to zero and solving yields that the optimal value ofα is given by

α = p(1 − r) − (1 − p)(1 + r)

1− r = 2 p− 1 − r

1− r .

For instance, if p = .6 and r = .05 then, although the expected rate of return on the investment is 20% (whereas the bank pays only 5%), the optimal fraction of money to be invested is

α = .15

.95 ≈ .158.

That is, the investor should invest approximately 15.8% of his capital and put the remainder in the bank.

Another commonly used utility function is the exponential utility func-tion

u(x) = 1 − e^−bx, b > 0.

The exponential is also a risk-averse utility function (see Figure 9.4).