PART 2 Ch. 7
7.9 Measuring Systematic Risk: Describe and justify what the value of the beta of a U.S.
Treasury bill should be.
Solution:
Since the beta of any asset is the slope of the line of best fit for the plot of an asset against
that of the market return, then we can use that logic to help us understand the beta of a T-
bill. If we purchased a T-bill five years ago and held the same T-bill through each of the
last 60 months then the return for each of those 60 months would be exactly the same.
Therefore, the vertical axis coordinates of each of the monthly returns would have the
same value and therefore the slope (beta) of the line of best fit would be zero. The
meaning of a beta of zero means that our T-bill has no systematic risk. That is logical
given that we know that a T-bill has no risk at all since it is a riskless asset.
7.10 Measuring Systematic Risk: If the expected rate of return for the market is not much
greater than the risk-free rate of return, what is the general level of compensation for
bearing systematic risk?
Solution:
Such a situation suggests that return compensation for investing in an asset is determined
largely by the risk-free return than by the market’s compensation for bearing systematic
risk. This means that the price for bearing systematic risk is very low. This may be
caused by a very low perceived level of risk in the market or by an abundance of funds in
the market seeking to be invested in risky assets.
7.11 CAPM: Describe the Capital Asset Pricing Model (CAPM) and what it tells us.
Solution:
The CAPM is a model that describes the relation between systematic risk and the
expected return. The model tells us that the expected return on an asset with no
systematic risk equals the risk free rate. As systematic risk increases, the expected return
increases linearly with beta. The CAPM is written as E(Ri) = Rrf + β i(E(Rm) – Rrf) .
7.12 The Security Market Line: If the expected return on the market is 10 percent and the
risk-free rate is 4 percent, what is the expected return for a stock with a beta equal to 1.5?
What is the market risk premium for the set of circumstances described?
Solution:
Following the CAPM prediction:
(Rcs) = Rrf + β (E(RM) – Rrf) = 0.04 + 1.5(0.1 - 0.04) = 0.13
The market risk premium is (E(RM) – Rrf) = 0.06
Intermediate
7.13 Expected Returns: You are thinking about purchasing a soft drink machine and placing
it in a business office. You know that there is a 5 percent probability that someone who
walks by the machine will make a purchase from the machine, and you know that the
profit on each soft drink sold is $0.10. If you know that 1,000 people per day pass by the
machine and you require a complete return of your investment in one year, then what is
the maximum price that you are willing to pay for the soft drink machine? Assume 250
working days in a year, and ignore taxes.
Solution:
E(Profit) = 1,000 x 0.05 x $.10 x 250 days = $1,250
Therefore, the most you are willing to pay for the machine in $1,250.
7.14 Interpreting the Variance and Standard Deviation: The distribution of grades in an
introductory finance class is normally distributed, with an expected grade of 75. If the
standard deviation of grades is 7, in what range would you expect 90 percent of the
grades to fall?
Solution:
90% is 1.645 standard deviations from the mean =>
75 – 1.645(7) = 63.485
75 + 1.645(7) = 86.515
7.15 Calculating the Variance and Standard Deviation: You recently invested in real estate
with the intention of selling the property one year from today. You have modeled the
returns on that investment based on three economic scenarios. You believe that if the
economy stays healthy, then your investment will generate a 30 percent return. However,
if the economy softens as predicted, the return will be 10 percent, while the return will be
-25 percent if the economy slips into a recession. If the probabilities of the healthy, soft,
and recessionary states are 0.4, 0.5, and 0.1, respectively, then what are the expected
return and the standard deviation for your investment?
Solution:
E(Return) = (0.4)(0.3) + (0.5) (0.1) + (0.1) (-.25)= 0.145
σ2Return = (0.4)(0.3 - 0.145)2 + (0.5) (0.1 - 0.145)2 + (0.1) (-.25 - 0.145)2 =
0.02623 =>
σReturn = (0.02623)1/2 = 0.16194
7.16 Calculating the Variance and Standard Deviation: You are considering investing in a
stock, and you are aware that the return on that investment is particularly sensitive to how
the economy is performing. Your analysis suggests that four states of the economy can
affect your investment. Using the table of returns and probabilities below, find the
expected return and the standard deviation of the return on your investment.
Solution:
E(Return) = 0.1(0.25) + (0.4) (0.15) + (0.3) (0.1) + (0.2) (-.05)= 0.105
σ2Return = 0.1(0.25 - 0.105)2 + (0.4) (0.15 - 0.105)2 + (0.3) (0.1 - 0.105)2 +
(0.2) (-.05 - 0.105)2 = 0.00773 =>
σReturn = (0.00773)1/2 = 0.08789
7.17 Calculating the Variance and Standard Deviation: You would like to invest in gold,
and you are aware that the returns on such an investment can be quite volatile. Use the
table of states, probabilities, and returns below to determine the expected return on your
gold investment.
Solution
E(Return) = 0.1(0.4) + (0.2) (0.3) + (0.3) (0.15) + (0.2) (0.02) +
(0.2) (-0.12) = 0.125
σ2Return = 0.1(0.4 - 0.125)2 + (0.2) (0.3 - 0.125)2 + (0.3) (0.15 - 0.125)2 +
(0.2) (0.02 - 0.125)2 + (0.2) (-.12 - 0.125)2
= 0.02809 =>
σReturn = (0.02809)1/2 = 0.16759
7.18 Single Asset Portfolios: Using the information from Problems 78.15, 78.16, and 78.17
calculate each coefficient of variation.
Solution:
Coefficient of Variation = σReturn / E(return)
Problem 15: 0.16194/0.145 = 1.11684 (using the exact values rather than the printed)
Problem 16: 0.08789/0.105 = 0.837047 (using the exact values rather than the printed)
Problem 17: 0.16759/0.125 = 1.34069 (using the exact values rather than the printed)
7.19 Portfolios with More Than One Asset: You are analyzing a two-stock combination
portfolio made up of Utility stock and Commodity stock. Utility has a standard deviation
of 40 percent, and Commodity has a standard deviation of 30 percent. However, you do
not know the exact covariance in the returns of the two stocks. You would like to plot the
variance of the portfolio for each of three cases—covariance of 0.12, 0, and -0.12—in
order to understand how the variance of such a portfolio would react. Do the calculation
for each of the extreme cases (0.12 and –0.12), assuming an equal proportion of each
stock in your portfolio.
Solution:
Part 1, σ12 = 0.12:
(0.5)2 (0.4)2 + (0.5)2 (0.3)2 + 2(0.5)(0.5)(0.12)= 0.1225
Part 2, ρ = 0.0:
(0.5)2 (0.4)2 + (0.5)2 (0.3)2 + 2(0.5)(0.5)(0.0) = 0.0625
Part 3, σ12 = -0.12:
(0.5)2 (0.4)2 + (0.5)2 (0.3)2 + 2(0.5)(0.5)(-0.12) = 0.0025
7.20 Portfolios with More Than One Asset: Given the returns and probabilities for the three
possible states below, calculate the covariance between the returns of Stock A and Stock B.
For convenience, assume that the expected returns of Stock A and Stock B are 11.75
percent and 18 percent, respectively.
Solution:
7.21 Compensation for Bearing Systematic Risk: You have constructed a diversified
portfolio of stocks such that there is no nonsystematic risk. Explain why the expected
return of that portfolio should be greater than the expected return of a risk-free security.
Solution: Your portfolio contains no non-systematic risk but it does in fact contain systematic
risk. Therefore, the market should compensate the holder of this portfolio for the
systematic risk that the investor bears. The risk-free security has no risk and therefore
requires no compensation for risk bearing. The expected return of the portfolio should
therefore be greater than the return of the risk-free security.
7.22 Compensation for Bearing Systematic Risk: Write out the equation for the covariance
in the returns of two assets, Asset 1 and Asset 2. Using that equation, explain the easiest
way for the two asset returns to have a covariance of zero.
Solution:
We know that all state probabilities must be greater than zero and thus the source of a
zero covariance cannot be from the state probabilities. The easiest way for the entire
probability weighted sum to equal zero is then for one of the assets, say Number 1(2),
to have a value in all states j that is equal to the expected return of Number 1(2).
Another way of saying that is for one of the assets to have a constant return in all
states. If that occurs then the second term in the equation will always be equal to zero
causing the sum, or covariance, to be zero.
7.23 Compensation for Bearing Systematic Risk: Evaluate the following statement: By
fully diversifying a portfolio, such as by buying every asset in the market, we can
completely eliminate all types of risk, thereby creating a synthetic Treasury bill.
Solution:
The statement is false. Even if we could afford such a portfolio and thus completely
diversify our portfolio, we would only be eliminating non-systematic risk. The
systematic risk generated by the portfolio would remain. Otherwise, the expected rate of
return on the market portfolio would be equal to the risk-free rate of return. We know
that to be a false statement.
7.24 The Capital Asset Pricing Model: You know that the beta of your portfolio is equal to
1, but you do not know the risk-free rate of return or the market risk premium. You do
know that the expected return on the market is 8 percent. What is the expected return on
your portfolio?
Solution:
Following the CAPM prediction:
(Rcs) = Rrf + β (E(RM) – Rrf) = Rrf + E(RM) – Rrf = E(RM) = 0.08
Advanced
7.25 Portfolios with More Than One Asset: You are going to purchase two stocks to form
the initial holdings in your portfolio. Iron stock has an expected return of 15 percent,
while Copper stock has an expected return of 20 percent. If you plan to invest 30 percent
of your funds in Iron and the remainder in Copper, then what will be the expected return
of your portfolio? What if you invest 70 percent of your funds in Iron stock?
Solution:
Part 1: E(Rport) = (0.3)(0.15) + (0.7)(0.2) = 0.185
Part 2: E(Rport) = (0.7)(0.15) + (0.3)(0.2) = 0.165
7.26 Portfolios with More Than One Asset: You know that the covariance in the return on
two assets is -0.0025. Without knowing the expected return of the two assets, explain
what that covariance means.
Solution:
The covariance measure is dependent upon the expected return of the two assets in
questions so without the expected return of the two assets, it is difficult to characterize the
scale of the covariance. However, since the covariance is negative, we can say that
generally the two assets move in opposite directions, with respect to their own means,
from each other in given states of nature.
7.27 Portfolios with More Than One Asset: In order to fund your retirement, you require a
portfolio with an expected return of 12 percent per year over the next 30 years. You have
decided to invest in Stocks 1, 2, and 3, with 25 percent in Stock 1, 50 percent in Stock 2,
and 25 percent in Stock 3. If Stocks 1 and 2 have expected returns of 9 percent and 10
percent per year, respectively, then what is the minimum expected annual return for Stock
3 that will enable you to achieve your investment requirement?
Solution:
The formula for the expected return of a 3-stock portfolio is:
Therefore, we can solve as below:
0.12 = 0.25(0.09) + 0.5(0.1) + 0.25E(R3)
0.19 = E(R3)
7.28 Portfolios with More Than One Asset: If you are putting together a portfolio of 10
stocks in equal proportions, what is the relative importance of the variance for each stock
versus the covariance for the pairs of stocks? For this exercise, ignore the actual values
of the variance and covariance terms and explain their importance conceptually.
Solution:
The variance of the portfolio will be composed of 10 (n = 10) individual stock variance
terms and 45 ((n2 –n)/2) covariance terms (really 90). Therefore, the vast majority of the
portfolio variance calculation will be determined by the covariance terms of the portfolio
in most cases.
7.29 Why Systematic Risk is all that Matters: Explain why investors who have diversified
their portfolios will determine the price and consequently the expected return on an asset.
Solution:
If we assume that all investors will seek to be compensated (generate returns) for the level
of risk that they are bearing, then we can see that undiversified investors will require a
greater return for a given investment than diversified investors will. Given that, we can
see that diversified investors will be willing to pay a greater price for an asset than
undiversified investors. Therefore, the diversified investor is the marginal investor
whose purchase will determine the equilibrium price, and therefore the equilibrium return
for an asset.
