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Baylor University
Hankamer School of Business
Department of Finance, Insurance & Real Estate
Risk Management
Dr. Garven
Problem Set 4
Name:
SOLUTIONS
√
1. Consider a risk averse investor with utility function U (W ) = W who is deciding how much
of her initial wealth (W 0 ) to invest in a bond and how much to invest in a stock. The
current prices of the bond and stock are B 0 and S 0 respectively. Although neither security
pays dividends or interest, Investor A expects to receive income from selling these securities
at their end-of-period prices, which are B 1 for the bond and S 1 for the stock. Since the bond
is riskless, its end-of-period price is known with certainty to be B 1 = B 0 (1+r ), where r is the
riskless rate of interest. The price of the stock at t = 1 can be high or low; i.e., it will be
S 0 (1+s) with probability .6 and it will be S 0 (1-s) with probability .4. Furthermore, assume
that W 0 = $100, r = .05, and s = .3.
A. How much of the investor’s initial wealth should be invested in the stock, and how much in
the bond?
SOLUTION: We will define the proportion of initial wealth to be invested in the stock as α,
and the proportion of initial wealth to be invested in the bond is (1-α). Expected utility is
E (U (W )) = .6(αW (1 + s) + (1 − α)W (1 + r)).5 + .4(αW (1 − s) + (1 − α)W (1 + r)).5
= .6(α100(1.3) + (1 − α)100(1.05)).5 + .4(α100(.70) + (1 − α)100(1.05)).5
=.6(α130 + (1 − α)105).5 + .4(α70 + (1 − α)105).5
=.6(105 + 25α).5 + .4(105 − 35α).5
Since our square root utility function is itself a function of α; i.e., U (W ) = W (α).5 , this means
that we must apply the chain rule in order to compute the first order condition and solve for
dU dW
dW
dU
α; i.e.,
=
= .5W (α)−.5
. Therefore, the first order condition is:
dα
dW dα
dα
∂E(U (W ))
= .3(105 + 25α)−.5 (25) − .2(105 − 35α)−.5 (35) = 0.
∂α
Next, we solve for α:
.3(105 + 25α)−.5 (25) = .2(105 − 35α)−.5 (35) ⇒ 7.5(105 + 25α)−.5 = 7(105 − 35α)−.5
7.5(105 − 35α).5 = 7(105 + 25α).5 ⇒ 56.25(105 − 35α) = 49(105 + 25α)
5, 906.25 − 1, 968.75α = 5, 145 + 1, 225α
3, 193.75α = 761.25
α = 761.25/3, 193.75 = 0.2384.
Since α=0.2384, Investor A should invest $23.84 in the stock, and $76.16 in the bond.
B. What will be the investor’s expected wealth and standard deviation of wealth at t = 1 from
this investment strategy?
SOLUTION: In order to determine expected wealth and standard deviation of wealth, we must
first calculate the expected value of the investor’s stock investment at time 1 and add to this
the future value of her bond investment:
• E(s) =
n
P
ps ss = .6(.3) − .4(.3) = .06 ⇒ E(S1 ) = S0 (1 + E(s))= $23.84(1.06) = $25.27;
s=1
• B 1 = B 0 (1+r ) = $76.16(1.05) = $79.97; and
• E(W1 ) = E(S) + B1 = $25.27 + $79.97 = $105.24.
Since bonds are riskless, the only source of risk in the investor’s portfolio is from her stock
investment. Therefore, the standard deviation of wealth must be equal to the value of her
stock investment multiplied by the standard deviation of the stock return; i.e., σW = S0 σs =
23.84σs . Solving for σs , we find that
r
n
P
q
ps (ss − E(s)) = .6(30 − 6)2 + .4(−30 − 6)2
σs =
s=1
p
√
√
= .6(576)+.4(1296) = 345.6 + 518.4 = 864 = 29.39%.
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Therefore, σW = 23.84(.2939) = $7.01.
C. Suppose that this investor starts out with initial wealth of $200 rather than $100. In this case,
what proportion of her initial wealth should be invested in the stock, and how much in the
bond?
SOLUTION: Since the investor has a square root utility function, we know that she has
decreasing absolute risk aversion and constant relative risk aversion. Constant relative risk
aversion implies that as the investor grows richer, the proportion of her wealth that she exposes
to risk remains the same. Thus, the ”rich” and ”poor” versions of this investor will allocate
23.84% of her wealth to the stock and 76.16% of her wealth to the bond.
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