AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
Important notes:
-
-
Please keep in mind that there are often different ways to solve a problem. In the answers, I usually
merely propose one way. If you are unsure whether your way of solving the problem is correct, please
see me or one of my TAs.
In the solutions, you’ll see that, sometimes, I was being lazy and provided answers only to the second
decimal. In the exam, unless it’s a dollar amount, please provide four (4) decimals in your answers (e.g.,
10.21% or 0.1021). It helps my TAs and I, grade the exams. For dollar amounts, two decimals are fine
(e.g., $15.32).
Unless further specified, when I provide a required rate of return of X%, it’s the return that investors
demand on an annual basis. For instance, “investors require a return of 10%” ≈ investors want their
investment to grow by 10% every year.
Slide Deck 6
1.
Bavarian Sausage just issued a 10-year, 12% coupon bond. The face value of the bond is $1,000 and the bond
makes ANNUAL coupon payments. If the required return on the bond is 10%, what is the bond’s price?
a.
b.
c.
d.
ANS:
2.
$815.16
$1,000
$1,122.89
$1067.24
C
FV: 1,000, PMT: 120, I/Y: 10, N: 10  PV: 1,122.89
You are offered a zero-coupon bond with a $1,000 face value and 5 years left to maturity. If the required
return on the bond is 8%, what is the most you should pay for this bond?
a.
b.
c.
d.
ANS:
$752.69
$680.58
$1,000
$1,126.94
B
FV: 1,000, PMT: 0, I/Y: 8, N: 5  PV: 680.58
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
3.
A one-year bond offers an annualized yield of 6% and a three-year bond offers an annualized yield of 7.5%.
Under the expectations theory, what should be the annualized yield on a two-year bond in one year?
a.
b.
c.
d.
ANS:
4.
5.95%
8.26%
3.06%
12.49%
B
(1.075)3 = (1.06) (1+y)2  y = .0826
Collegetown Pizza has a $1,000 face value bond with a 11% coupon rate; coupon payments are made once a
year. The bond matures in nine years. Bonds similar to that of Collegetown Pizza have a required rate of return
of 10%. At what price should the Collegetown Pizza bond be trading at?
a.
b.
c.
d.
ANS:
5.
$890.00
$1,053.35
$1,000.00
$1,057.59
D
FV: 1,000, N: 9, I/YR: 10, PMT: 110  PV: 1,057.59
A bond’s coupon rate
a.
b.
c.
d.
ANS:
6.
is equal to its annual coupon payment divided by the bonds’ current market price.
can be changed during the life of the bond.
is equal to its annual coupon payment divided by its face value.
both a and b are correct.
C
A bond will mature in 10 years. The bond has a face value of $1,000. Further, the bond has a coupon rate of
9%; coupon payments are made ANNUALLY. What should be the trading price for the bond if investors require
a 12% return/year on their investment?
a.
b.
c.
d.
ANS:
$1,192.53
$830.49
$827.95
$508.52
B
N: 10, I/Y: 12, PMT: 90, FV: 1,000  PV: 830.49
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
7. Bond ratings:
a.
b.
c.
d.
e.
ANS:
8.
have no impact on bond prices.
are based upon the bond rating agencies' assessment of the borrower's default risk.
result in yield spreads between different quality bonds.
Both (b) and (c)
All of the above
D
Hai Hong Bonds have 14 years to maturity and a coupon rate of 8% with coupons being paid ANNUALLY. If the
discount rate is 12%, what is the current yield of Hai Hong Bonds? The bond has a face value of $1,000.
a.
b.
c.
d.
ANS:
9.
8.33%
9.00%
10.89%
12.00%
C
N: 14, I/Y: 12, PMT: 80, FV: 1,000  PV: 734.87  Current Yield = 80 / 734.87 = 10.89%
Aladdin’s Bonds will mature in 8 years. The coupon rate of the bond is 6% with coupons being paid SEMIANNUALLY. If the discount rate is 2% every six months, what is the value of the bond? The bond has a face
value of $1,000.
a.
b.
c.
d.
ANS:
$1,135.78
$1,293.02
$1,073.25
$1,543.11
A
N: 16, PMT: 30, FV: 1,000, I/Y: 2  PV: 1,135.78
10. Which one statement is true?
a.
b.
c.
d.
ANS:
Callable bonds and puttable bonds always have a difference in yields by at least 2%.
Holding all else equal, puttable bonds always trade at lower prices than non-puttable bonds.
One of the key benefits to investors from holding puttable bonds is the ability to retire the bond should
investors no longer wish to hold the bond.
Holding all else equal, callable bonds always trade at higher prices than non-callable bonds.
C
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
11. Which of the following is true about bonds with redemption features, given all else is held equal?
a.
b.
c.
d.
ANS:
The price of a callable bond is lower than the price of a non-callable bond.
The price of a callable bond is greater than the price of a non-callable bond.
The price of a puttable bond is lower than the price of a non-puttable bond.
The price of a puttable bond is lower than the price of a callable bond.
A
12. The spot rate for every 6 months is 6% (the yield curve is flat) and the bond’s coupon rate is 5%. The bond
pays coupons semi-annually, matures in 10 years, and has a face value of $1,000. What is the bond’s price
today?
ANS:
598.55
N=20, Y=0.06, “Annuity factor” = 11.4699
 Price of bond: 25* (11.4699) + 1,000/1.0620 = 598.55
13. If a zero-coupon bond matures 3 years from today and is currently trading at $921.50, what is the 3-year spot
rate for that bond?
ANS:
921.50 =
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏
(𝟏𝟏+𝒚𝒚𝒚𝒚)𝟑𝟑
 𝒚𝒚𝟑𝟑 = 𝟐𝟐. 𝟕𝟕𝟕𝟕%
14. If a bond with a par value of $1000 and a 10% coupon rate is trading today for $990, which of the following is
true?
a.
b.
c.
d.
ANS:
The bond is trading at a premium and the yield-to-maturity (YTM) is greater than the coupon rate
The bond is trading at par and the YTM is equal to the coupon rate
The bond is trading at a discount and the YTM is greater than the coupon rate
The bond is trading at a premium and the yield-to-maturity (YTM) is lower than the coupon rate
C
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
15. Which one of the following statements is/are true?
I. Default risk refers to the risk that a debt issuer fails to make the promised payments (coupon payments or
face value)
II. Current yield is the annual coupon amount divided by the current bond price
III. Current yield is the annual coupon amount divided by the face value
a.
b.
c.
d.
ANS:
I
I, II
II, III
I, II, III
B
16. You invest in a bond with a ½ -year spot rate of 6% and the coupon rate is 10%. The bond makes semi-annual
coupon payments and will mature in 5 years. What is the bond’s price?
a.
b.
c.
d.
ANS:
$926.4
$957.9
$1,294.4
$1,168.5
A
FV: 1,000, PMT: 50, I/Y: 6, N: 10
 PV: 926.4
Now assume that the ½ -year spot rate becomes 4%, what will the new price of the bond be?
a.
b.
c.
d.
ANS:
$922.8
$1,081.1
$1,044.5
$1,486.6
B
FV: 1,000, PMT: 50, I/Y: 4, N: 10
 PV: 1,081.1
17. A bond issued at a premium has a coupon rate that is ______ than its yield to maturity and a market price that
will gradually ______ until maturity
a.
b.
c.
d.
ANS:
greater; increase
lower; decrease
lower; increase
greater; decrease
D
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
18. One year ago, Dos Amigos issued five-year bonds with an 8% coupon rate (with coupons payments being
made semi-annually). The bond’s face value is $1000. Investors’ required semi-annual rate of return was 4%;
the yield curve is flat.
Back to present day: after the arrival of Old Mexico Express, Dos Amigos downgraded its earnings
expectations, prompting concerned investors to now require a 5% semi-annual return on Dos Amigos bonds.
What is the dollar amount increase (decrease) in the price of the bonds over the past one year?
ANS:
$(64.63)
a)
Investors get $40 in coupon payments every six months; their semi-annual required rate of return
is 4%  Bond trades at par value of $1000
b) Now: FV: 1,000, N: 8, I/Y: 5, PMT: 40  PV: 935.37
Difference: Price after shift – Price before shift = 935.37 – 1000 = (64.63)
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
Slide Deck 7
19. Toyota stock has the following probability distribution of prices one year from now:
Probability
25%
40%
35%
a.
If you buy Toyota today for $55 and Toyota is guaranteed to pay a dividend of $4 per share, what is your
expected return on Toyota?
ANS:
b.
Price
$50
$60
$70
E(r) = .25 (50-55+4)/55 + .40 (60-55+4)/55 + .35 (70-55+4)/55 = 18.18%.
What is the standard deviation for Toyota?
ANS:
StdDev =[.25 (((50-55+4)/55) - 0.1818)2 + .40 (((60-55+4)/55) - 0.1818)2
+ .35 (((70-55+4)/55) - 0.1818)2]1/2 = 14%
20. Assume you are a fund manager. 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% (σTreasury bill rate=0%).
a.
Your client chooses to invest 70% of his/her wealth in your risky portfolio and 30% in Treasury bills. What is
the expected return and standard deviation of your client’s portfolio?
ANS:
b.
E(rP) = (0.3 * 0.07) + (0.7 * 0.17) = 0.14
StDev(rP) = (0.7 * 0.27) = 0.189
Suppose your risky portfolio includes the following investments in the given proportions:
Stock
Stock A
Stock B
Stock C
Proportions
27%
33%
40%
What are the investment proportions of your client’s overall portfolio in Stock A, Stock B, Stock C and Treasury
bills?
ANS:
Security
T-Bills
Stock A
Stock B
Stock C
0.7 × 27% =
0.7 × 33% =
0.7 × 40% =
Investment
30.0%
18.9%
23.1%
28.0%
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
21. Consider a treasury bill with a rate of return of 2% and the following risky securities:
Security A: E(r) = .10; StdDev = .10
Security B: E(r) = .12; StdDev = .13
Security C: E(r) = .14; StdDev = .16
Security D: E(r) = .14; StdDev = .20
The investor must develop a combined portfolio by combining the risk-free asset with one of the securities
above. The security the investor should choose as part of his/her combined portfolio to achieve the best
Capital Allocation Line would be _________.
a. Security A
b. Security B
c. Security C
d. Security D
ANS:
A
The Capital Allocation Line through A has the highest slope (Sharpe Ratio).
22. You are looking to form a combined portfolio of a Treasury Bill (2.5% rate of return) and one risky security.
Which of the following securities would you choose as the risky asset?
a.
b.
c.
d.
E(r) = .35; variance = .12
E(r) = .20; variance = .075
E(r) = .12; variance = .05
E(r) = .40; variance = .195
ANS:
A
a.
b.
c.
d.
Sharpe Ratio = (.35 - .025)/.34641 = .94
Sharpe Ratio = (.20 - .025)/.27386 = .64
Sharpe Ratio = (.12 - .025)/.22361 = .42
Sharpe Ratio = (.40 - .025)/.44159 = .85
23. Consider an investment opportunity set formed with two securities that are perfectly negatively correlated. If
two securities are perfectly negatively correlated, it is always possible to construct a global minimum variance
portfolio with a standard deviation that is _________.
a. equal to the sum of the securities standard deviations
b. equal to -1
c. equal to 0
d. greater than 0
ANS:
C
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
24. Stock A’s expected return and standard deviation are E[rA] = 8% and σA= 15%, while stock B’s expected return
and standard deviation are E[rB] = 12% and σB= 21%.
Determine the expected return and standard deviation of the return on a portfolio with weights wA=.35 and
wB=.65 for the following two alternative values of correlation between A and B: pAB=0.6 and pAB= -0.4.
ANS:
E[rP]
= wA E[rA] +wB E[rB]
= 0.35 (0.08) + 0.65 (0.12) = 0.106
regardless of the correlation.
Using σ p2 = ωA2σ A2 + ωB2σ B2 + 2ωAωB ρ ABσ Aσ B ,
for correlationAB = 0.6:
σp2
= (.35)2(.15)2+(.65)2(.21)2+2(.35)(.65)(0.6)(.15)(.21)
= 0.029988
σp
= 17.32%
For correlationAB = -0.4:
σp2
= (.35)2(.15)2+(.65)2(.21)2+2(.35)(.65)(-0.4)(.15)(.21)
= 0.015656
σp
= 12.51%
You can see low correlations allow you to lower the volatility of your portfolio without
sacrificing any of our expected returns!
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
25. Stock A’s expected return and standard deviation are E[rA] = 8% and σA= 15%, while stock B’s expected return
and standard deviation are E[rB] = 12% and σB= 21%.
Assume that pAB=-1.0. Find the portfolio p of stocks A and B that has no risk (i.e. such that σp=0).
Hint:
Note:
ANS:
(a2 + b2 - 2ab) = (a – b)2
I consider this question to be challenging.
From σ p = ω Aσ A + ω Bσ B + 2ω Aω Bσ Aσ B ρ AB , we want to find ωA such that σp2 = 0.
2
2
2
2
2
Since ωB = 1-ωA and pAB=-1.0, we need to solve
ω 2Aσ 2A + (1− ω A ) 2 σ B2 − 2ω A (1− ω A )σ A σ B = 0
(ω σ ) + ((1 − ω )σ ) − 2ω σ (1 − ω )σ = 0
2
A
2
A
A
B
A
A
A
B
or with a = wAσA and b = (1-wA)σB [see hint above]
0
(ω σ − (1 − ω )σ ) =
0
(ω σ − (1 − ω )σ ) =
A
A
This has the solution
ωA =
A
A
A
A
B
2
B
σB
0.21
=
= 0.58333 .
σ A + σ B 0.15 + 0.21
The remaining weight is ωB=1-ωA=0.41666.
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
26. You invest a total of $100 between a risky asset with an expected rate of return of 0.11 and a standard deviation
of 0.21 and a risk-free asset with a rate of return of 0.045.
a.
How would you form a portfolio that has an expected outcome of $114?
ANS:
b.
You want an expected return of 14% ($100 $114)
Solving for 0.14 = w1(0.11) + (1 - w1)(0.045)
0.14 = .11w1 + .045 - .045w1
0.095 = 0.065w1
=> w1 = 1.46  $146 and wf = (1- w1) = -0.46  -$46.
So the portfolio is formed by borrowing $46 at the risk-free rate and investing the total amount of $146
in the risky asset.
What is the slope of the Capital Allocation Line formed by combining the risky asset with the risk-free asset?
ANS:
The slope of the Capital Allocation Line formed by combining the risky asset with the risk-free asset equals
(0.11 - 0.045)/0.21 = 0.3095.
27. You are presented with three distinct investment opportunities involving Treasury Bills, Corporate Bonds with
AAA credit rating and Small Cap Stocks. You are told that each of these investments is expected to produce
$100 one year from now. Which asset should be the least expensive today?
Hint: Which one is the riskiest and, thus, the one with the highest required rate of return?
a.
b.
c.
ANS:
Treasury Bills
Corporate Bonds
Small Cap Stocks (i.e., stocks of small companies)
C
Small Cap Stocks: Greatest risk  greatest discount rate  lowest price
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
28. An investor plans to invest $42,500 in a combined portfolio. The portfolio has a risky asset (Security A) with an
expected return of 18% and a standard deviation of 25%. There is also a risk-free asset offering 5%.
How much money should be invested into the risky asset to ensure a return of 10%?
ANS:
E(rc)
=
(1-wR) * 0.05 + wR * 0.18 = 0.10
= 0.10
0.05 - 0.05wR + 0.18wR
= 0.10
0.05 + 0.13wR
0.05
0.05/0.13
= 0.13wR
= wR
38.46% of $42,500 should be invested, so $16,346.15
You’re just about to invest in Security A (along with the risk-free asset), when a friend makes you aware of
Security B, which has an expected return of 20% and a standard deviation of 31%. If you could only invest in
Security A along with the risk-free asset, or Security B along with the risk-free asset, which combination is better?
ANS:
Security A
Sharpe Ratio = (0.18 - 0.05)/ 0.25 = 0.5200
Security B
Sharpe Ratio = (0.20-0.05)/ 0.31 = 0.4839
Security A is the better choice due to the higher Sharpe Ratio
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
Take a look at this portfolio for questions 29-30:
Security
1
2
3
Weight
35%
40%
Expected Return
7%
9%
29. What is the weight of Security 1?
a.
b.
c.
d.
ANS:
25%
35%
45%
55%
A
100% - 35% - 40% = 25%
30. If the expected return on the portfolio is 9.7%, what is the expected return for Security 3?
a.
b.
c.
d.
ANS:
10%
11%
12%
13%
C
0.25*0.07 + 0.35*0.09 + 0.4*X = 0.097  X = .12
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
Take a look at this portfolio for questions 31-34:
Security
1
2
3
$ Invested
$5,000
$7,000
$9,000
Expected Return
7%
9%
12%
31. What is the weight of Security 1?
a.
b.
c.
d.
ANS:
42.9%
33.3%
23.8%
Cannot be determined with the data given
C
$5,000/$21,000 = .238
32. What is the weight of Security 2?
a.
b.
c.
d.
ANS:
42.9%
33.3%
23.8%
Cannot be determined with the data given
B
$7,000/$21,000 = .333
33. What is the weight of Security 3?
a.
b.
c.
d.
ANS:
42.9%
33.3%
23.8%
Cannot be determined with the data given
A
$9,000/$21,000 = .429
34. What is the expected return on the portfolio?
a.
b.
c.
d.
ANS:
9.81%
9.00%
17.31%
Cannot be determined with the data given
A
5000/21000*0.07 + 7000/21000*0.09 + 9000/21000*0.12 = 0.0981
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
Slide Deck 8
Recap on some definitions/terminology:
Expected return of stock i: E(ri)
Expected return of stock i in excess of the risk-free rate OR Expected excess return of stock i: E(ri)-rf
Market risk premium: E(rm)-rf
In general, you can differentiate between the following two scenarios:
(1) The CAPM holds all the time: E(ri)-rf = βi [E(rm)-rf].
(2) The CAPM holds generally. However, there can be temporary deviations: E(ri)-rf = αi + βi [E(rm)-rf].
In other words, based on the CAPM, you should get an expected excess return of βi [E(rm)-rf].
However, sometimes, you may expect to get an excess return that is higher than that predicted by
the CAPM. In that case, E(ri)-rf > βi [E(rm)-rf] and αi > 0. Other times, you may expect to get an excess
return that is lower than that predicted by the CAPM. In that case, E(ri)-rf < βi [E(rm)-rf] and αi < 0.
When you can assume that you are in the first scenario, I will say “imagine/consider a world where the simple CAPM
holds in its “purest” form (i.e., alpha=0% at any given point in time)”.
When you can assume that you are in the second scenario, I will say “imagine/consider a world where the simple
CAPM holds generally (i.e., there may be positive or negative alphas)”.
For questions 35-45, imagine/consider a world where the simple CAPM holds in its “purest” form (i.e., alpha=0% at
any given point in time).
35. The risk-free rate is 3%. Which of the following four risky portfolios is likely to be the “market portfolio”?
a.
b.
c.
d.
portfolio with a standard deviation of 15% and an expected return of 12%
portfolio with a standard deviation of 19% and an expected return of 15%
portfolio with a standard deviation of 25% and an expected return of 25%
portfolio with a standard deviation of 12% and an expected return of 9%
ANS:
C
The portfolio with the highest Sharpe Ratio.
36. The risk-free rate is 5% and the expected return on the market portfolio is 13%. A stock has a beta of 1.5. What
should be the expected return?
a.
b.
c.
d.
17%
12%
19.5%
24.5%
ANS:
A
E(R) = 5% + 1.5 (13%-5%) = 17%
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
37. A particular stock has an expected return of 11%. If the market risk premium is 8% and the risk-free rate is 5%,
what’s the stock’s CAPM beta?
a.
b.
c.
d.
1.375
0.750
0.846
0.462
ANS:
B
11% = 5% + beta (8%)

beta = 0.75
38. An investor put 40% of his/her money in Stock A and 60% in Stock B. Stock A has a beta of 1.2 and Stock B has a
beta of 1.6. If the risk-free rate is 5% and the expected return on the market is 12%, what’s the investor’s
expected return?
a.
b.
c.
d.
22.28%
14.80%
15.08%
21.80%
ANS:
C
Expected return (stock A) = 5% + 1.2(12%-5%) = 13.4%
Expected return (stock B) = 5% + 1.6(12%-5%) = 16.2%
Portfolio expected return = .4 (13.4%) + .6 (16.2%) = 15.08%
39. The risk-free rate is 6%. The expected return on a stock with a beta of 1.3 is 21.60% What is the expected return
on the market?
a.
b.
c.
d.
6%
15.6%
18%
21.6%
ANS:
C
E[r] = 6% + (1.3) [X - 6%] = 21.60%  X = 18%
40. The risk-free rate is 5% and the expected return on the market is 15%. What is the beta on a stock with an
expected return of 17%?
a.
b.
c.
d.
0.5
0.7
1.0
1.2
ANS:
D
17% = 5% + β[15% - 5%]; β = 1.20
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
41. The expected return on the market is 18%. The expected return on a stock with a beta of 1.2 is 20%. What is the
risk-free rate?
a.
b.
c.
d.
2%
6%
8%
12%
ANS:
C
20% = rf + (1.2)*(18% - rf); rf = 8.00%
42. If enough investors decide to purchase stocks (and nothing else changes) they are likely to drive up stock prices
thereby causing _____________.
a. expected returns on stocks to fall
b. expected returns on stocks to rise
ANS:
A
43. If all investors become more risk averse, stock prices will _______________ and the Security Market Line will
_______________.
a. fall; have a steeper slope
b. fall; have a flatter slope;
ANS:
A
44. In a world where the CAPM holds in its purest form, which one of the following is NOT a true statement regarding
the capital market line?
a.
c.
c.
d.
The capital market line always has a positive slope
The capital market line is also called the security market line
The capital market line is the best attainable capital allocation line
The capital market line is the line from the risk-free rate through the market portfolio
ANS:
B
45. Consider the simple CAPM in its purest form. According to the CAPM, _________.
a.
b.
c.
d.
all securities returns must lie on the capital market line
all securities returns must lie on the security market line
the slope of the security market line must be less than the market risk premium
any security with a beta of 1 must have an excess return of zero
ANS:
B
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
For questions 46-49, imagine/consider a world where the simple CAPM holds generally (i.e., there may be positive
or negative alphas).
46. A particular asset has a beta of 1.2. The expected return on the market portfolio is 13% and the risk-free is 5%.
For some reason, you expect the asset to yield a return of 10%. Which of the following statements is correct?
a.
b.
c.
d.
This asset lies on the security market line.
This asset lies above the security market line.
This asset lies below the security market line.
Cannot tell from the given information.
ANS:
C
Based on the CAPM, this asset should give you an expected return of 5% + 1.2 (13%-5%) = 14.6%
47. A particular asset has a beta of 1.2. The expected return on the market portfolio is 13% and the risk-free is 5%.
For some reason, you expect the asset to yield a return of 10%. The stock is _______
a.
b.
c.
d.
overpriced
underpriced
appropriately priced
Cannot tell from the given information
ANS:
A
Based on the CAPM, you should be getting 5% + 1.2 (13%-5%) = 14.6%.
You are only expected to get 10%. The stock is overpriced.
48. An asset has a beta of 2.0. The market risk premium is 5% and the risk-free is 7%. For some reason, you expect
the asset to yield a return of 20%. The stock is
a.
b.
c.
d.
overpriced
underpriced
appropriately priced
Cannot tell from the given information
ANS:
B
Based on the CAPM, you should be getting 7% + 2.0 (5%) = 17%.
For some reason, you expect to get more than what you should be getting (given the stock’s level
of risk as captured by the beta). The stock is underpriced.
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
49. You have the following data on the securities of three firms:
Firm A
Firm B
Firm C
Return last year
10%
11%
12%
Beta
0.8
1.0
1.2
If the risk-free rate last year was 3% and the return on the market was expected to be 11%, which firm had the
best performance on a beta-adjusted basis (in other words, which firm had the highest alpha)?
a.
b.
c.
d.
Firm A
Firm B
Firm C
There is no difference in performance on a risk-adjusted basis
ANS:
Return you should be getting based on the CAPM (firm A) = 3% + 0.8(11% - 3%) = 9.4%
Return you should be getting based on the CAPM (firm B) = 3% + 1.0(11% - 3%) = 11%
Return you should be getting based on the CAPM (firm C) = 3% + 1.2(11% - 3%) = 12.6%
 Firm A had an alpha of +0.6%; Firm B had an alpha of 0%; Firm C had an alpha of -0.6%.
So A outperformed the most.
For questions 50-51, it doesn’t matter whether the simple CAPM holds in its “purest” form or whether the CAPM
holds generally.
50. A portfolio has 40% invested in Asset 1 and 60% invested in Asset 2. If Asset 1 has a beta of 1.2 and Asset 2 has
a beta of 1.8, what’s the beta of the portfolio?
a.
b.
c.
d.
e.
1.50
1.56
1.20
1.80
cannot tell from the given information
ANS:
B
0.4 * 1.2 + 0.6 * 1.8
51. A portfolio has 40% invested in Asset 1, 50% invested in Asset 2 and 10% invested in Asset 3. Asset 1 has a beta
of 1.2, Asset 2 has a beta of 0.8 and Asset 3 has a beta of 1.8, what’s the beta of the portfolio?
a.
b.
c.
d.
e.
1.27
0.80
1.06
1.20
Cannot tell from given information
ANS:
C
0.4 * 1.2 + 0.5 * 0.8 + 0.1 * 1.8
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
Consider the following information for questions 52-55: The expected return on the Market Portfolio M is E(rM)=15%,
the standard deviation of the Market Portfolio is σM=25% and the risk-free rate is rf=5%. Imagine/consider a world
where the simple CAPM holds in its “purest” form (i.e., alpha=0% at any given point in time).
52. Describe the Capital Market Line (CLM) derived from the above data. Make sure to clarify the intercept and the
slope.
ANS:
The CML is a straight line on the expected return-standard deviation graph (expected return on
the Y-axis and standard deviation on the X-axis). The intercept is the risk-free rate of 5% and the
slope is the Sharpe ratio of the market portfolio:
𝐸𝐸(𝑟𝑟𝑐𝑐 ) = 𝑟𝑟𝑓𝑓 +
𝐸𝐸(𝑟𝑟𝑚𝑚 )−𝑟𝑟𝑓𝑓
𝜎𝜎𝑚𝑚
𝜎𝜎𝑐𝑐 = 0.05 + 0.4 ∗ 𝜎𝜎𝑐𝑐
53. Compute the expected return of two portfolios on the CML, one with StdDev of 18%, and the other with StdDev
of 30%.
E [rM ] − rf
σ C =0.05 + 0.4 ⋅ σ C
ANS:
The CML is E [rC ] =rf +
σM
So a well-diversified portfolio with a standard deviation of 18% has an expected return of
0.05+0.4(.18)=12.2% and a portfolio with standard deviation of 30% has an expected return of
0.05+0.4(.30)=17%.
54. Suppose that a portfolio with a standard deviation 10% has an expected return of 12%. Is this compatible with
the CAPM? Explain carefully.
ANS:
In the simple CAPM in its purest form, no portfolio may be above the CML (recall the CML is
supposed to be the best attainable Capital Allocation Line). An efficient portfolio with a standard
deviation of 10% should have an expected return of 9% to be on the CML (0.05+0.4(.10)=9%).
Therefore, this portfolio lies above the CML. This is not possible.
55. Suppose that a portfolio has a beta of 0.5, an expected return of 10%, and a standard deviation of 15%. Is this
compatible with the CAPM? Explain.
ANS:
An efficient portfolio with a standard deviation of 15% should have an expected return of 11%
to be on the CML (0.05+0.4(.15)=11%). Therefore, the portfolio should have an expected return
of 11% or less (recall, it’s ok for a portfolio to be below the CML).
Therefore, this portfolio, which has an expected return of 10%, is compatible with the CAPM.
The portfolio is possible (though not efficient).
Also, the beta of this portfolio is 0.5.
Plugged into the CAPM equation: 0.05 + 0.5*(0.15-0 .05) = 10%. So we’re all good.
AEM 2240 – Spring 2020 – Take-Home Exam 1 (Practice Problems WITH SOLUTIONS)
56. If the simple CAPM in its purest form is valid (i.e., alpha=0% at any given point in time), which of the situations
below are possible? Please consider the situations individually. It is not necessarily true that only one of them
is possible.
Option 1
A
Market
Option 2
Risk-free
Market
A
Option 3
Risk-free
Market
A
Option 4
Risk-free
Market
A
ANS:
1.
E(r)
20%
25%
Beta
1.4
1.2
E(r)
10%
40%
30%
StdDev(r)
0%
25%
35%
E(r)
10%
18%
16%
StdDev(r)
0%
24%
12%
E(r)
10%
18%
20%
StdDev(r)
0%
24%
12%
Not possible. Asset A has a higher beta than the market portfolio, but the expected return for asset A
is lower.
2.
Possible. The Sharpe Ratio of the market portfolio dominates that of asset A.
3.
Not possible. The Sharpe ratio for asset A is better than that of the market, which is not possible according
to the CAPM, since the CAPM predicts that the market portfolio is the best portfolio. Using the numbers
supplied:
16 − 10
= 0.5
12
18 − 10
= 0.33
SM =
24
SA =
4.
Not possible. Asset A clearly dominates the market portfolio. It has a lower standard deviation and a
higher expected return.