Download 1 Some Implications of Risk Aversion for Portfolio Choice Consider a

Some Implications of Risk Aversion for Portfolio Choice
Consider a simple portfolio choice problem. An economic agent has beginning-of-period
real wealth W0 that is to be invested in either or both of two assets. The first asset has a
(gross) rate of return Rf that is known with certainty. (Rf is often referred to as the riskfree rate of interest). The second asset is risky and has a (gross) rate of return R1 that is a
random variable. (Note that both Rf and R1 are real rates of return).
Let α denote the proportion of W0 invested in the risky asset and (1-α) be the proportion
invested in the risk-free asset. At the end of the time period the return on the risky asset
will be realized and the agent's portfolio will deliver real wealth:
(1)
W1 = [(1- α)Rf + αR1] W0 .
At the end of the period the agent consumes the entire amount W1 and enjoys utility
denoted U(W1). Observe that from the perspective of the beginning of period, neither W1
nor U(W1) is known with certainty. The agent's portfolio choice problem is to choose the
value of α that maximizes E [U(W1)] , subject to the portfolio constraint given in
Equation (1). This problem may be more simply stated as:
Max E [ U ( [ (1- α)Rf + αR1] W0 ) ]
α
I will impose no restrictions on the value of α here; the agent is permitted to undertake
"short sales" of either security so that α may be less than zero (when the agent sells the
risky asset short and invests the proceeds in the risk-free asset) or greater that 1.0 (when
the agent short sells the risk-free asset and invests the proceeds in the risky asset).
The solution to the problem is rather simple: We must set the derivative of
E [ U ( [(1- α)Rf + αR1]W0 ) ] with respect to α equal to zero and solve for the value of α
that satisfies this condition; i.e., we must solve the following for the value of α:
(2)
E [ U' ( [ (1- α)Rf + αR1] W0 ) (R1 -Rf ) W0 ] = 0 .
Digression:
Some students are bewildered the first time they encounter an expression like the
RHS of Equation (2), which is the derivative of an expected value of a utility
function. The bewilderment arises because such students mistakenly interpret the
expectation operator E[ ] as a function and inappropriately try to apply the chain
dE[U (W 1)]
.
rule of elementary calculus to derive the value of an expression like
dα
E[ ] is an operation, not a function, so the chain rule is not appropriate.
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What follows is a long-hand demonstration of how to take the total differential of
the expected value of a utility function.
We wish to find how the value of E [U(W1)] changes in response to small changes
in W1. Let dW1 be a small change in W1. Then the change in E [U(W1)] is given
by
dE [U(W1)] = E [U(W1 + dW1 )] - E [U(W1)] .
(3)
Observe that the RHS of Equation (3) is the difference between two expected
values. Since the expected value operator is a linear operator, we can re-write
Equation (3) as
(3)'
dE [U(W1)] = E [ U(W1 + dW1 ) - U(W1)].
The utility function U( ) is indeed a function, so we know from elementary
calculus that lim U(W1 + dW1) = U(W1 ) + U' ( W1) dW1 .
dW1 →0
It follows then that
lim dE [U(W1)] = E [U(W1 ) + U' ( W1) dW1 ] - E [U(W1)] ,
dW1 →0
and Equation (3)' becomes
(4)
dE [U(W1)] = E [ U' ( W1) dW1 ] .
One may be tempted to convert this into a derivative by dividing both sides of
Equation (4) by dW1 but, in general, this can not be done because dW1 is a random
variable and can not be disassociated from the expectations operation on the RHS.
No matter. We can readily apply Equation (4) to our portfolio choice problem by
recognizing that the agent's decision to change the value of α causes W1 to change
in the following way:
(5)
dW1 = ( R1 - Rf ) W0 dα .
Substituting Equation (5) into Equation (4) yields
(6)
dE [U(W1)] = E [ U' ( W1) ( R1 - Rf ) W0 dα ] .
Because the value of α is chosen by the agent at the start of the time period,
neither α nor dα is a random variable. Consequently, Equation (6) may be
written as
(7)
dE [U(W1)] = E [ U' ( W1) ( R1 - Rf ) W0 ] dα .
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Now divide both sides of Equation (7) by dα and set the result equal to zero and
we have the first order condition for the maximization of expected utility given
earlier as Equation (2).
End Digression………………………………………………………
Let us denote by the symbol α* the value of α that solves Equation (2) and maximizes
the agent's expected utility. If the agent is risk averse, we can deduce the following about
α* . (For students who are interested in such things, proofs are provided in an Appendix).
α* > 0 if and only if E [R1] > Rf
α* = 0 if and only if E [R1] = Rf
α* < 0 if and only if E [R1] < Rf
.
In order to make further inferences about α*, we need to know more about the form of
the agent's utility function.
Constant Relative Risk Aversion (CRRA)
Suppose the agent has a CRRA utility function. A class of utility functions that have this
property takes the form U (W) = (1/(1-γ)) W 1-γ for γ > 1 and U (W) = ln(W) for γ = 1.
For any function in this class, U'(W) = W -γ and U''(W) = -γW -γ-1 and the Pratt-Arrow
measure of relative risk aversion is -W (U'' / U' ) = γ . The larger the value of γ , the
more risk averse is an agent with a CRRA utility function.
Using the expression above for U'(W), First Order Condition (2) becomes
(8)
E [ [ (1- α*)Rf + α*R1] -γ (R1 -Rf ) W01-γ ] = 0 .
Now because W0 is known with certainty, we can divide both sides of Equation (8) by
W01-γ with the result
(9)
E [ [ (1- α*)Rf + α*R1] -γ (R1 -Rf ) ] = 0 .
The important implication of Equation (9) is that the optimal allocation of the portfolio
between risky and risk-free securities is independent of the initial value of wealth for any
agent with CRRA preferences. α* does not depend upon W0 but does depend upon the
value of the relative risk aversion parameter γ. Using a Taylor's Series expansion, we
can derive an approximate solution for α* from Equation (9):
(10)
α* = (1/γ) [(Rf (µ1-Rf ) / (σ12 + (µ1-Rf )2] ,
where µ1 = E[R1] and σ12 = Var[R1].
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Observe that α* depends inversely upon the relative risk aversion parameter γ.
Comparing two agents with differing CRRA utility functions, the more risk averse agent
has the larger value for γ and will wish to invest a smaller share of invested wealth in the
risky security. In other words, the more risk averse agent will choose a lower value for
α*. An numerical example will help to illustrate this.
Suppose the (net) risk-free rate of interest is 2%, which implies Rf = 1 + 0.02 = 1.02.
Suppose also that the (net) rate of return on the risky security is a normally distributed
random variable with a mean of 3% and a standard deviation of 5%, which implies
µ1 = 1.03 and σ12 = 0.0025. Then approximate Solution (10) implies α* = 3.923 (1/γ).
An agent with a coefficient of relative risk aversion of γ = 1 will want to hold the risky
asset in an amount that is 3.923 times his initial wealth W0. To do so, this agent will have
to short sell the risk-free asset in an amount equal to 2.923 times W0. (If the risk-free
asset is debt, this will amount to borrowing an amount 2.923W0 and using the borrowed
funds to purchase the risky asset). In contrast, an agent with a coefficient of relative risk
aversion of γ = 10 will choose α* = 0.392 and invest a little more than 39% of initial
wealth in the risky security and the remaining 61 % in the risk-free asset.
[We will see Equation (10) again, latter in the course.]
We could determine empirically whether real world agents have CRRA preferences by
monitoring a group of investors over time and observing whether they change the fraction
of their portfolios invested in risky assets as their wealth fluctuates. Friend and Bloom
conducted just such a study in 1975 using data supplied by the US Internal Revenue
Service. They found that actual investors do behave as if they have CRRA preferences
with values of γ between 1 and 4. Subsequent empirical investigations have drawn
similar conclusions.
Appendix
Recall from elementary statistics that the expected value of the product of any two
random variables X and Y may be expressed as
E[XY] = E[X] E[Y] + Cov[X,Y] .
Recall also that the Covariance between any two random variables is a measure of the
association between their respective outcomes. Cov[X,Y] > 0 when "high" outcomes for Y
occur most frequently in association with "high" outcomes for X. Cov[X,Y] < 0 when
"high" outcomes for Y occur most frequently in association with "low" outcomes for X.
Observe that the necessary and sufficient condition for maximizing expected utility in the
portfolio choice problem appearing as Equation (2) is the expected value of the product
of two random variables and may be written as
(2)'
E[XY] = E[X] E[Y] + Cov[X,Y] = 0 ,
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where X = U'(W1) and Y = (R1 -Rf ). [And W1 = (1-α)Rf + αR1 )].
Now consider how the Covariance between these two variables depends upon the sign of
α.
If α > 0, a "high" outcome for R1 generates a "high" outcome for both Y and W1.
But a "high" value for W1 generates a "low" value for X = U' (W1) because of
diminishing marginal utility. Therefore, Cov[X,Y] < 0 whenever α > 0. This
means that any solution of Equation (2)' that yields an optimal value for α that is
greater than zero will also have a value for E[X] E[Y] that is greater than zero. X
is a marginal utility so E[X] is always greater than zero; consequently, positive
values for α occur whenever E[Y] = E[R1] - Rf > 0.
If α = 0, the variable X is non-random and Cov[X,Y] = 0. A solution of Equation
(2)' that has α = 0 will, therefore, only occur when E[Y] = E[R1] - Rf = 0.
If α < 0, a "high" outcome for R1 generates a "high" outcome for Y and a "low" outcome forW1.
But a "low" value for W1 generates a "high" value for X = U' (W1) because of
diminishing marginal utility. Therefore, Cov[X,Y] > 0 whenever α < 0. This
means that any solution of Equation (2)' that yields an optimal value for α that is
less than zero will also have a value for E[X] E[Y] that is less than zero. X is a
marginal utility so E[X] is always greater than zero; consequently, negative values
for α occur whenever E[Y] = E[R1] - Rf < 0.
.
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