The Portfolio Selection Problem

Suppose one has the positive amountw to be invested among n differ-ent securities. If the amount a is invested in security i (i = 1, ..., n) then, after one period, that investment returns aXi, where Xi is a non-negative random variable. In other words, if we let Ri be the the rate of return from investment i, then

a= aXi

1+ Ri

or Ri = Xi− 1.

The Portfolio Selection Problem 175

Figure 9.4: An Exponential Utility Function

Ifwi is invested in each security i = 1, ..., n, then the end-of-period wealth is

W =

n i=1

wiXi.

The vectorw1, ...,wnis called a portfolio. The problem of determining the portfolio that maximizes the expected utility of one’s end-of-period wealth can be expressed mathematically as follows:

choose w1, ...,wn satisfying wi ≥ 0, i = 1, ..., n,

n i=1

wi = w, to

maximize E[U(W )],

where U is the investor’s utility function for the end-of-period wealth.

To make the preceding problem more tractable, we shall make the as-sumption that the end-of-period wealth W can be thought of as being a normal random variable. Provided that one invests in many securities that are not too highly correlated, this would appear to be, by the central

176 Valuing by Expected Utility

limit theorem, a reasonable approximation. (It would also be exactly true if the Xi, i = 1, ..., n, have what is known as a multivariate normal distribution.)

Suppose now that the investor has an exponential utility function U(x) = 1 − e^−bx, b > 0,

and so the utility function is concave. If Z is a normal random variable, then e^Z is lognormal and has expected value

E[e^Z]= exp{E[Z] + Var(Z )/2}.

Hence, as−bW is normal with mean −bE[W] and variance b²Var(W ), it follows that

E[U(W )] = 1 − E[e^−bW]= 1 − exp{−bE[W] + b²Var(W )/2}.

Therefore, the investor’s expected utility will be maximized by choos-ing a portfolio that

maximizes E[W ]− b Var(W )/2.

Observe how this implies that, if two portfolios give rise to random end-of-period wealths W₁and W₂such that W₁has a larger mean and a smaller variance than does W₂, then the first portfolio results in a larger expected utility than does the second. That is,

E[W₁]≥ E[W2] & Var(W1) ≤ Var(W2)

⇒ E[U(W1)] ≥ E[U(W2)]. (9.1) In fact, provided that all end-of-period fortunes are normal random vari-ables, (9.1) remains valid even when the utility function is not expo-nential, provided that it is a nondecreasing and concave function. Con-sequently, if one investment portfolio offers a risk-averse investor an expected return that is at least as large as that offered by a second in-vestment portfolio and with a variance that is no greater than that of the second portfolio, then the investor would prefer the first portfolio.

Let us now compute, for a given portfolio, the mean and variance of W. With security i’s rate of return Ri = Xi − 1, let

ri = E[Ri], vi²= Var(Ri).

The Portfolio Selection Problem 177

Example 9.3a An important case which results in W having a normal distribution is the case where R₁, . . . , Rnhas a multivariate normal dis-tribution, defined as follows.

Definition Let Z₁, . . . , Zm be independent standard normal random variables. If for some constantsμi, i = 1, . . . , n and ai j, i = 1, . . . , n,

178 Valuing by Expected Utility Because any linear combinationn

i=1wiXiis also a linear combination of the independent normal random variables Z₁, . . . , Zm, it follows that

n

i=1wiXi is a normal random variable.

Example 9.3b Suppose you are thinking about investing your fortune of 100 in two securities whose rates of return have the following ex-pected values and standard deviations:

r₁= .15, v1= .20; r2= .18, v2= .25.

If the correlation between the rates of return isρ = −.4, find the opti-mal portfolio when employing the utility function

U(x) = 1 − e^−.005x.

Solution. Ifw1= y and w2 = 100 − y, then from Equation (9.2) we obtain

E[W ]= 100 + .15y + .18(100 − y) = 118 − .03y.

Also, since c(1, 2) = ρv1v2= −.02, Equation (9.3) gives

Var(W ) = y²(.04) + (100 − y)²(.0625) − 2y(100 − y)(.02)

= .1425y²− 16.5y + 625.

We should therefore choose y to maximize

118− .03y − .005(.1425y²− 16.5y + 625)/2 or, equivalently, to maximize

.01125y − .0007125y²/2.

Simple calculus shows that this will be maximized when y = .01125

.0007125 = 15.789.

That is, the maximal expected utility of the end-of-period wealth is ob-tained by investing 15.789 in investment 1 and 84.211 in investment 2.

Substituting the value y = 15.789 into the previous equations gives

The Portfolio Selection Problem 179 E[W ]= 117.526 and Var(W ) = 400.006, with the maximal expected utility being

1− exp{−.005(117.526 + .005(400.006)/2)} = .4416.

This can be contrasted with the expected utility of .3904 obtained when all 100 is invested in security 1 or the expected utility of .4413 when all 100 is invested in security 2.

Example 9.3c Suppose only two securities are under consideration, both with normally distributed returns that have same expected rate of return. Then, since every portfolio will yield the same expected value, it follows that the best portfolio for any concave utility function is the one whose end-of-period wealth has minimal variance. Ifαw is invested in security 1 and(1 − α)w is invested in security 2, then with c = c(1, 2) we have

Var(W ) = α²w²v₁²+ (1 − α)²w²v²₂+ 2α(1 − α)w²c

= w²[α²v²₁+ (1 − α)²v²₂+ 2cα(1 − α)].

Thus, the optimal portfolio is obtained by choosing the value ofα that minimizesα²v₁²+ (1 − α)²v²₂+ 2cα(1 − α). Differentiating this quan-tity and setting the derivative equal to zero yields

2αv₁²− 2(1 − α)v²₂+ 2c − 4cα = 0.

Solving forα gives the optimal fraction to invest in security 1:

α = v²₂− c v₁²+ v²₂− 2c.

For instance, suppose the standard deviations of the rate of returns are v1= .20 and v2= .30, and that the correlation between the two rates of return isρ = .30. Then, as c = ρv1v2= .018, we obtain that the opti-mal fraction of one’s investment capital to be used to purchase security 1 is

α = .09 − .018

.04 + .09 − .036 = 72/94 ≈ .766.

That is, 76.6% of one’s capital should be used to purchase security 1 and 23.4% to purchase security 2.

180 Valuing by Expected Utility

If the rates of returns are independent, then c = 0 and the optimal fraction to invest in security 1 is

α = v²₂

v²₁+ v²₂ = 1/v₁² 1/v²₁+ 1/v²₂.

In this case, the optimal percentage of capital to invest in a security is determined by a weighted average, where the weight given to a security is inversely proportional to the variance of its rate of return. This result also remains true when there are n securities whose rates of return are uncorrelated and have equal means. Under these conditions, the optimal fraction of one’s capital to invest in security i is

1/v_i²

n j=11/vj²

.

Determining a portfolio that maximizes the expected utility of one’s end-of-period wealth can be computationally quite demanding. Often a reasonable approximation can be obtained when the utility function U(x) satisfies the condition that its second derivative is a nondecreasing function – that is, when

U^(x) is nondecreasing in x. (9.4) It is easily checked that the utility functions

U(x) = x^a, 0 < a < 1, U(x) = 1 − e^−bx, b > 0, U(x) = log(x)

all satisfy the condition of Equation (9.4).

We can approximate U(W ) by using the first three terms of its Taylor series expansion about the pointμ = E[W]. That is, we use the approx-imation

U(W ) ≈ U(μ) + U^(μ)(W − μ) + U^(μ)(W − μ)²/2.

Taking expectations gives that

E[U(W )] ≈ U(μ) + U^(μ)E[W − μ] + U^(μ)E[(W − μ)²]/2

= U(μ) + U^(μ)v²/2,

The Portfolio Selection Problem 181 wherev²= Var(W ) and where we have used that

E[W − μ] = E[W] − μ = μ − μ = 0.

Therefore, a reasonable approximation to the optimal portfolio is given by the portfolio that maximizes

U(E[W]) + U^(E[W]) Var(W )/2. (9.5) If U is a nondecreasing, concave function that also satisfies condition (9.4), then expression (9.5) will have the desired property of being both increasing in E[W ] and decreasing in Var(W ).

Utility functions of the form U(x) = x^a or U(x) = log(x) have the property that there is a vector

α^∗₁, ..., α^∗n, αi^∗≥ 0,

n i=1

α^∗i = 1,

such that the optimal portfolio under a specified one of these utility func-tions iswα^∗₁, ...,wα^∗nfor every initial wealthw. That is, for these utility functions, the optimal proportion of one’s wealthw that should be in-vested in security i does not depend onw. To verify this, note that

W = w

n i=1

αiXi

for any portfoliowα1, ...,wαn. Hence, if U(x) = x^athen E[U(W )] = E[W^a]

= E

w^a

n

i=1

αiXi

a

= w^aE

n

i=1

αiXi

a

and so the optimalαi(i = 1, ..., n) do not depend on w. (The argument for U(x) = log(x) is left as an exercise.)

An important feature of the approximation criterion (9.5) is that, when U(x) = x^a(0 < a < 1), the portfolio that maximizes (9.5) also has the property that the percentage of wealth it invests in each security does

182 Valuing by Expected Utility

not depend onw. This follows since equations (9.2) and (9.3) show that, for the portfoliowi = αiw (i = 1, ..., n),

E[W ]= wA, Var(W ) = w²B, where

A= 1 +

n i=1

αiri,

B=

n i=1

α²ivi²+

n i=1



j=i

αiαjc(i, j).

Thus, since

U^(x) = a(a − 1)x^a−2, we see that

U(E[W]) + U^(E[W]) Var(W )/2

= w^aA^a+ a(a − 1)w^a−2A^a−2w²B/2

= w^a[ A^a+ a(a − 1)A^a−2B/2].

Therefore, the investment percentages that maximize (9.5) do not de-pend onw.

Example 9.3d Let us reconsider Example 9.3b, this time using the utility function

U(x) =√ x. Then, withα1= α and α2= 1 − α we have

A= 1 + .15α + .18(1 − α),

B= .04α²+ .0625(1 − α)²− 2(.02)α(1 − α), and we must choose the value ofα that maximizes

f(α) = A^1/2− A^−3/2B/8.

The solution can be obtained by setting the derivative equal to zero and then solving this equation numerically.

In document Fnancial Mathematics (Page 192-200)