FIN-469 Investments Analysis
Professor Michel A. Robe
Practice Set #4 and Solutions.
What to do with this practice set?
To help students prepare for the assignment and the exams, practice sets with solutions will be
handed out. These sets contain worked-out end-of-chapter problems from BKM3 and BKM4.
These sets will not be graded, but students are strongly encouraged to try hard to solve them and
to use office hours to discuss any problems they may have doing so. One of the best self-tests
for a student of his or her command of the material before a case or the exam is whether he or
she can handle the questions of the relevant practice sets. The questions on the exam will cover
the reading material, and will be very similar to those in the practice sets.
Consider the following statement to answer Questions 1 to 5:
You manage a risky portfolio with an expected rate of return of 17% and a standard deviation of
27%. The Treasury-bill rate is 7%.
Question 1:
One of your clients chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money
market fund. What is the expected value and standard deviation of the rate of return on your
client’s portfolio?
Question 2:
Suppose that your client decides to invest in your portfolio a proportion y of the total investment
budget so that the overall portfolio will have an expected rate of return of 15%.
(a) What is the proportion y?
(b) Further suppose that your risky portfolio includes the following investments in the given
proportions: Stock A (27%), Stock B (33%), and Stock C (40%). What are your client’s
investment proportions in your three stocks and the T-bill fund.
Question 3:
Now suppose that your client prefers to invest in your fund a proportion y that maximizes the
expected return on the overall portfolio subject to the constraint that the overall portfolio’s
standard deviation will not exceed 20%.
(a) What is the investment proportion, y?
(b) What is the expected rate of return on the overall portfolio?
Question 4:
Suppose that your client’s degree of risk aversion is A = 3.5. What proportion, y, of the total
investment should you suggest that he invest in your fund?
Question 5:
You estimate that a passive portfolio (that is, one entirely invested in a risky portfolio that
mimics the S&P 500 stock index) yields an expected rate of return of 13% with a standard
deviation of 25%.
(a) What is the slope of the CML?
(b) Characterize in one short paragraph the advantage of your fund over the passive fund.
(c) Your client is considering whether to switch to the passive portfolio the 70% of his wealth
currently invested in your fund. Show your client that he is better off staying with you.
(Hint 1: show your client the maximum fee you could charge -- as a percentage of the
investment in your fund deducted at the end of the year -- that would still leave him
at least as well off investing in your fund as in the passive one)
(Hint 2: the fee will lower the slope of your client’s CAL by reducing the expected return net
of the fee)
Question 6:
Suppose that there are many stocks in the market and that the characteristics of Stocks A and B
are given as follows:
Stock
Expected Return
Standard Deviation
------------------------------------------------------------------------------------------A
10%
5%
B
15%
10%
------------------------------------------------------------------------------------------Correlation = -1
------------------------------------------------------------------------------------------Suppose that it is possible to borrow at the risk-free rate, Rf. What must be the value of the riskfree rate?
(Hint: think about constructing a risk-free portfolio from Stocks A and B).
Question 7:
The market price of a security is $40. Its expected rate of return is 13%. The risk-free rate is 7%
and the market risk premium is 8%. What will be the market price of the security if its
covariance with the market portfolio doubles (and all other variables remain unchanged)?
Assume that the stock is expected to pay a constant dividend in perpetuity.
Question 8:
Consider the following table, which gives a security analyst’s expected return on two stocks for
two particular market returns:
Market Return
Aggressive Stock
Defensive Stock
------------------------------------------------------------------------------------------5%
2%
3.5%
20%
32%
14%
------------------------------------------------------------------------------------------(a) What are the betas of the two stocks?
(b) What is the expected rate of return on each stock if the market return is equally likely to be
5% or 20%?
(c) If the T-bill rate is 8% and the market return is equally likely to be 5% or 20%, draw the
SML for this economy.
(d) Plot the two securities on the SML graph. What are the alphas of each?
(e) What hurdle rate should be used by the management of the aggressive firm for a project with
the risk characteristics of the defensive firm’s stock?
Question 9:
Two investment advisors are comparing performance. One averaged a 19% rate of return and the
other a 16% rate of return. However, the beta of the first investor was 1.5, whereas that of the
second was 1.
(a) Can you tell which investor was a better predictor of individual stocks (aside from the issue
of general movements in the market)?
(b) If the T-bill rate were 6% and the market return during the period were 14%, which investor
would be the superior stock selector?
(c) What if the T-bill rate were 3% and the market return were 15%?
FIN-469 Investments Analysis
A. Robe
Professor
Michel
Practice Set #4: Solutions.
Question 1:
Mean = (0.30 x 7%) + (0.7 x 17%) = 14% per year.
Standard deviation = 0.70 x 27% = 18.9% per year.
Question 2:
(a) Mean return on portfolio = Rf + (Rp - Rf)y
= 7% + (17% - 7%)y = 7% + 10%y
If the mean of the portfolio is equal to 15%, then solving for y we will get:
15% = 7% +10%y
=>
y = (15% - 7%)/10% =>
y = 0.8
Thus, in order to obtain a mean return of 15%, the client must invest 80% of total funds in the
risky portfolio and 20% in Treasure bills.
(b) Investment proportions of the client’s funds:
• 20% in T-bills
• 0.8 x 27% = 21.6% in Stock A
• 0.8 x 33% = 26.4% in Stock B
• 0.8 x 40% = 32.0% in Stock C
Question 3:
(a) Portfolio standard deviation = y x 27%. If your client wants a standard deviation of 20%,
then
y = (20%/27%) = 0.7407 = 74.07% in the risky portfolio.
(b) Mean return = 7% + (17% - 7%)y = 7% + 10% (0.7407) = 7% + 7.407% = 14.407%.
Question 4:
y* = (Rp - Rf)/0.01Aσ2 =>
y* = (17 - 7)/(0.01 x 3.5 x 272) = 10/25.515 = 0.3919
Thus, the client’s optimal investment proportions are 39.19% in the risky portfolio and 60.81%
in T-bills.
Question 5:
(a) The slope of the CML = (13% - 7%)/25% = 0.24. You can draw a graph as we did in class.
(b) My fund allows an investor to achieve a higher mean for any given standard deviation than
would a passive strategy, that is, a higher expected return for any given level of risk. Indeed,
my fund’s reward-to-variability ratio is (17% - 7%)/27% = 0.37, which is much better than
0.24 (the RTRR of the S&P 500).
(c) The fee would reduce the reward-to-variability ratio, that is, the slope of the CAL. Clients
will be indifferent between your fund and the passive portfolio if the slope of the after-fee
CAL and the CML are equal. Let f denote the maximum fee. In this case:
Slope of CAL with fee = (17% - 7% - f)/27 = (10 - f)/27%.
Slope of CML (which requires no fee) = (13% - 7%)/25% = 0.24
Setting these slopes equal, then we get:
(10% - f)/27% = 0.24 => 10% - f = 27% x 0.24 => f = 10% - 6.48% = 3.52% per year.
Question 6:
Since Stocks A and B are perfectly negatively correlated, a risk-free portfolio can be constructed
and its rate of return in equilibrium will be the risk-free rate.
To find the proportions of this portfolio (wA invested in Stock A and wB = 1 - wA in Stock B), set
the standard deviation equal to zero. With perfect negative correlation, the portfolio standard
deviation reduces to:
σp = | wAσA - wBσB|
=>
0 = | 5wA - 10 (1-wA) |
=>
The expected rate of return on this risk-free portfolio is:
E(R) = 0.6667 x 10% + (0.3333 x 15%) ≈ 11.67%.
Thus, to avoid arbitrage, the risk-free rate must also be 11.67%.
wA =0.6667
Question 7:
If the covariance of the security doubles, then so will its beta and its risk premium. The current
risk premium is 6% (i.e., 13% - 7%), so the new risk premium would be 12%, and the new
discount rate for the security would be 19% (i.e., 12% + 7%).
If the stock pays a level perpetual dividend, then we know from the original data that the
dividend, D, must satisfy the equation for a perpetuity:
Price = Dividend/Discount rate
=>
$40 = D/0.13 =>
D = $5.20.
At the new discount rate of 19%, the stock would be worth only $5.20/0.19 = $27.37. As a
consequence, the increase in stock risk has lowered the stock value by 31.58%, i.e., ($27.37 $40)/$40.
Question 8:
(a) The beta is the sensitivity of the stock return to the market return movements. Let A be the
aggressive stock and D be the defensive one. Then beta is the change in the stock return per
change in the market return. Therefore, we compute each stock’s beta by calculating the
difference in its return across the two scenarios divided by the difference in the market
return.
βA = (2 - 32)/(5 - 20) = 2.00
βD = (3.5 - 14)/(5 - 20) = 0.70
(b) With equal likelihood of either scenarios, the expected return is an average of the two
possible outcomes.
E(RA) = 0.5 (2 + 32) = 17%
E(RD) = 0.5 (3.5 + 14) = 8.75%
(c) The SML is determined by the market expected return of 0.5 x (20 + 5) = 12.5%, with a beta
of 1, and the T-bill return of 8% with a beta of zero. The equation for the security market line
is:
E(R) = 8% + β(12.5% - 8%)
(graph to be sketched).
(d) The aggressive stock has a fair expected return of:
E(RA) = 8% + 2(12.5% - 8%) = 17%,
and the expected return by the analyst also is 17%. Thus, its alpha is zero. Similarly, the required
return on the defensive stock is:
E(RD) = 8% + 0.7(12.5% - 8%) = 11.5%,
but the analyst’s expected return on Stock D is only 8.75%, and hence,
αD = actual expected return - required return given risk)
= 8.75% - 11.15% = -2.4%.
(e) The hurdle rate is determined by the project beta, 0.70, not by the firm’s beta. The correct
discount rate is 11.15%, the fair rate of return on Stock D.
Question 9:
(a) We know that: R1 = 19%, R2 = 16%, β1 = 1.5, and β1 = 1. To tell which investor was a better
predictor of individual stocks, we should look at their abnormal return, which is the ex-post
alpha, that is, the abnormal return is the difference between the actual return and that
predicted by the SML. Without information about the parameters of this equation (risk-free
rate and the market rate of return) we cannot tell which investor is more accurate.
(b) If Rf = 6% and Rm = 14%, then (using the notation of alpha for the abnormal return):
α1 = 19% - [6% + 1.5(14% - 6%)] = 19% - 18% = 1%
α2 = 16% - [6% + 1(14% - 6%)] = 16% - 14% = 2%.
Here, the second investor has the larger abnormal return, and thus he appears to be a more
accurate predictor. By making better predictions, the second investor appears to have tilted his
portfolio toward underpriced stocks.
(c) If Rf = 3% and Rm = 15%, then
α1 = 19% - [3% + 1.5(15% - 3%)] = 19% - 21% = -2%
α2 = 16% - [3% + 1(15% - 3%)] = 16% - 15% = 1%.
Here, not only does the second investor appear to be a better predictor, but also the first
investor’s prediction appears valueless (or worse).