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Chapter 13 NAME
Risky Assets
Introduction. Here you will solve the problems of consumers who wish
to divide their wealth optimally between a risky asset and a safe asset.
The expected rate of return on a portfolio is just a weighted average of
the rate of return on the safe asset and the expected rate of return on
the risky asset, where the weights are the fractions of the consumer’s
wealth held in each. The standard deviation of the portfolio return is
just the standard deviation of the return on the risky asset times the
fraction of the consumer’s wealth held in the risky asset. Sometimes
you will look at the problem of a consumer who has preferences over
the expected return and the risk of her portfolio and who faces a budget
constraint. Since a consumer can always put all of her wealth in the
safe asset, one point on this budget constraint will be the combination
of the safe rate of return and no risk (zero standard deviation). Now
as the consumer puts xpercent of her wealth into the risky asset, she
gains on that amount the difference between the expected rate of return
for the risky asset and the rate of return on the safe asset. But she also
absorbs some risk. So the slope of the budget line will be the difference
between the two returns divided by the standard deviation of the portfolio
that has xpercent of the consumer’s wealth invested in the risky asset.
You can then apply the usual indifference curve–budget line analysis to
find the consumer’s optimal choice of risk and expected return given her
preferences. (Remember that if the standard deviation is plotted on the
horizontal axis and if less risk is preferred to more, the better bundles will
lie to the northwest.) You will also be asked to apply the result from the
Capital Asset Pricing Model that the expected rate of return on any asset
is equal to the sum of the risk-free rate of return plus the risk adjustment.
Remember too that the expected rate of return on an asset is its expected
change in price divided by its current price.
13.1 (3) Ms. Lynch has a choice of two assets: The first is a risk-free
asset that offers a rate of return of rf, and the second is a risky asset (a
china shop that caters to large mammals) that has an expected rate of
return of rmand a standard deviation of σm.
(a) If xis the percent of wealth Ms. Lynch invests in the risky asset,
what is the equation for the expected rate of return on the portfolio?
176 RISKY ASSETS (Ch. 13)
(b) By solving the second equation above for xand substituting the result
into the first equation, derive an expression for the rate of return on the
(c) Suppose that Ms. Lynch can borrow money at the interest rate rf
and invest it in the risky asset. If rm= 20, rf= 10, and σm= 10, what
will be Ms. Lynch’s expected return if she borrows an amount equal to
100% of her initial wealth and invests it in the risky asset? (Hint: This
(d) Suppose that Ms. Lynch can borrow or lend at the risk-free rate. If
rfis 10%, rmis 20%, and σmis 10%, what is the formula for the “budget
graph below.
0102030
40
10
20
30
Standard deviation
Expected return
40
Budget line
U=0
U=5
U=10
(e) Which of the following risky assets would Ms. Lynch prefer to her
present risky asset, assuming she can only invest in one risky asset at a
time and that she can invest a fraction of her wealth in whichever risky
asset she chooses? Write the word “better,” “worse,” or “same” after
each of the assets.
NAME 177
(f) Suppose Ms. Lynch’s utility function has the form u(rx,σ
x)=rx−2σx.
How much of her portfolio will she invest in the original risky asset?
(You might want to graph a few of Ms. Lynch’s indifference curves be-
fore answering; e.g., graph the combinations of rxand σxthat imply
13.2 (3) Fenner Smith is contemplating dividing his portfolio between
two assets, a risky asset that has an expected return of 30% and a standard
deviation of 10%, and a safe asset that has an expected return of 10%
and a standard deviation of 0%.
(a) If Mr. Smith invests xpercent of his wealth in the risky asset, what
(b) If Mr. Smith invests xpercent of his wealth in the risky asset, what
(c) Solve the above two equations for the expected return on Mr. Smith’s
(d) Plot this “budget line” on the graph below.
0 5 10 15 20
10
20
30
Standard deviation
Expected return
40
Budget line
Indifference
curves
Optimal choice
178 RISKY ASSETS (Ch. 13)
(e) If Mr. Smith’s utility function is u(rx,σ
x)=min{rx,30 −2σx},then
One of the equations is the budget constraint.)
(f) Plot Mr. Smith’s optimal choice and an indifference curve through it
in the graph.
(g) What fraction of his wealth should Mr. Smith invest in the risky asset?
13.3 (2) Assuming that the Capital Asset Pricing Model is valid, com-
plete the following table. In this table p0is the current price of asset i
and Ep1is the expected price of asset iin the next period.
rfrmriβip0Ep1
13.4 (2) Farmer Alf Alpha has a pasture located on a sandy hill. The
return to him from this pasture is a random variable depending on how
much rain there is. In rainy years the yield is good; in dry years the yield
is poor. The market value of this pasture is $5,000. The expected return
from this pasture is $500 with a standard deviation of $100. Every inch
of rain above average means an extra $100 in profit and every inch of rain
below average means another $100 less profit than average. Farmer Alf
has another $5,000 that he wants to invest in a second pasture. There are
two possible pastures that he can buy.
(a) One is located on low land that never floods. This pasture yields
an expected return of $500 per year no matter what the weather is like.
What is Alf Alpha’s expected rate of return on his total investment if he
NAME 179
(b) Another pasture that he could buy is located on the very edge of the
river. This gives very good yields in dry years but in wet years it floods.
This pasture also costs $5,000. The expected return from this pasture is
$500 and the standard deviation is $100. Every inch of rain below average
means an extra $100 in profit and every inch of rain above average means
another $100 less profit than average. If Alf buys this pasture and keeps
his original pasture on the sandy hill, what is his expected rate of return
on his total investment? 10%.What is the standard deviation of the
rate of return on his total investment in this case? 0%.
(c) If Alf is a risk averter, which of these two pastures should he buy
