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  3. Radboud Universiteit Nijmegen (RU)
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  5. Investment Management (MANBCU3022)

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Summary Investment Management; Tutorial Answers ()

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  • Vak
  • Investment Management (MANBCU3022)
  • Instelling
  • Radboud Universiteit Nijmegen (RU)
  • Boek
  • ISE Investments

Solutions to all the questions covered in the tutorials of the course; Investment Management. Useful to check your own work or to look something up when you don't know the answer, with extensive explanations

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  • Onbekend
  • 16 maart 2021
  • 34
  • 2020/2021
  • Samenvatting

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  • radboud
  • bedrijfskunde
  • economie
  • bedrijfseconomie
  • investment
  • investment management
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  • economics
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Voorbeeld van de inhoud

Answers Tutorials Investment Management
2020-2021
Tutorial 1
CFA 5.3;
What are the expected rates of return for Stocks X and Y?




Expected rate of return; a probability-weighted average of the rates of return in each scenario.
𝐸(𝑟) = ∑! 𝑝(𝑠)𝑟(𝑠)
- 𝑝(𝑠) ; the probability of each scenario
- 𝑟(𝑠) ; the HPR in each scenario

Stock X;
𝐸(𝑟" ) = (0.2 ∙ −0.20) + (0.5 ∙ 0.18) + (0.3 ∙ 0.50) = 0.2 = 20%

Stock Y;
𝐸(𝑟# ) = (0.2 ∙ −0.15) + (0.5 ∙ 0.20) + (0.3 ∙ 0.10) = 0.1 = 10%

CFA 5.4;
What are the standard deviations of returns on Stocks X and Y?

Standard deviation; a measure of the amount of variation or dispersion of a set of values
𝜎 = √𝜎 $
- Variance; 𝜎 $ = ∑ 𝑝(𝑠)[𝑟(𝑠) − 𝐸(𝑟)]$

Stock X;
𝜎"$ = (0.20[−0.20 − 0.20]$ ) + (0.50[0.18 − 0.20]$ ) + (0.30[0.50 − 0.20]$ ) = 0.0592
𝜎" = :𝜎"$ = √0.0592 = 0.2433 = 24.33%

Stock Y;
𝜎#$ = (0.20[−0.15 − 0.10]$ ) + (0.50[0.20 − 0.10]$ ) + (0.30[0.10 − 0.10]$ ) = 0.0175
𝜎# = :𝜎#$ = √0.0175 = 0.1323 = 13.23%

CFA 5.5;
Assume that of your $10,000 portfolio, you invest $9,000 in Stock X and $1,000 in Stock Y. What is the expected return
on your portfolio?

Stock X; $9,000 / $10,000 = 0.90 = 90% (𝑤" )
Stock Y; $1,000 / $10,000 = 0.10 = 10% (𝑤# )

𝐸(𝑟% ) = 𝑤" 𝐸(𝑟" ) + 𝑤# 𝐸(𝑟# )
𝐸(𝑟) = (0.90 ∙ 0.20) + (0.10 ∙ 0.10) = 0.19 = 19%
- Expected return of the portfolio = 19%

$10,000 ∙ 0.19 = $1900 in returns

,Question 6.11;
Calculate the utility levels of each portfolio of Problem 10 for an investor with A = 2. What do you conclude?

&
Utility value; 𝑈 = 𝐸(𝑟) − $ 𝐴𝜎 $
- 𝐴 ; index of investor’s risk aversion
- 𝜎 ; standard deviation

WBills WIndex rPortfolio sPortfolio s2Portfolio U(A = 2)
0.0 1.0 0.130 0.20 0.0400 0.0900
0.2 0.8 0.114 0.16 0.0256 0.0884
0.4 0.6 0.098 0.12 0.0144 0.0836
0.6 0.4 0.082 0.08 0.0064 0.0756
0.8 0.2 0.066 0.04 0.0016 0.0644
1.0 0.0 0.050 0.00 0.0000 0.0500
The column labeled U(A = 2) implies that investor with A = 2 prefer a portfolio that is invested 100% in the market
index to any of the other portfolios in the table
- In the table U(A = 2) is 0.0900 the highest number, thus this gives the highest utility. This is when
WBills = 0 and WIndex = 1.0

Question 6.13;
Use these inputs for Problems 13 through 19:
You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate
is 8%.

Your client chooses to invest 70% of a portfolio in your fund and 30% in an essentially risk-free money market fund.
What is the expected value and standard deviation of the rate of return on his portfolio?

Risky portfolio;
𝐸(𝑟% ) = 0.18 Client chooses;
𝜎% = 0.28 𝑦 = 0.70
𝑟' = 0.08 1 − 𝑦 = 0.30

Expected return;
𝐸(𝑟( ) = 𝑦𝐸(𝑟% ) + (1 − 𝑦)𝑟' = 𝑟' + 𝑦N𝐸(𝑟% ) − 𝑟' O
𝐸(𝑟( ) = 0.08 + 0.70[0.18 − 0.08] = 0.15 = 15%

Standard Deviation;
𝜎( = 𝑦𝜎%
𝜎( = 0.70 ∙ 0.28 = 0.196 = 19.6%

Question 6.14;
Suppose that your risky portfolio includes the following investments in the given proportions:
- Stock A 25%
- Stock B 32%
- Stock C 43%
What are the investment proportions of your client’s overall portfolio, including the position in T-bills?

Risky portfolio;
𝐸(𝑟% ) = 0.18 Client chooses;
𝜎% = 0.28 𝑦 = 0.70
𝑟' = 0.08 1 − 𝑦 = 0.30

0.70 ∙ 0.25 = 0.175 = 17.5% in Stock A
0.70 ∙ 0.32 = 0.224 = 22.4% in Stock B
0.70 ∙ 0.43 = 0.301 = 30.1% in Stock C
30.0% in T-bills

,Question 6.15;
What is the reward-to-volatility (Sharpe) ratio (S) of your risky portfolio? Your client’s?
Steps to determine the Sharpe Ratio complete portfolio;
1) Expected return; 𝐸(𝑟( ) = 𝑦𝐸(𝑟% ) + (1 − 𝑦)𝑟' = 𝑟' + 𝑦N𝐸(𝑟% ) − 𝑟' O
2) Standard Deviation; 𝜎( = 𝑦𝜎%
)(+!)-+"
3) Sharpe Ratio; 𝑆% =
.!


Your risky portfolio;
Expected return; 𝐸(𝑟% ) = 0.18
Standard Deviation; 𝜎% = 0.28
)(+! )-+" /.&1-/./1
Sharpe Ratio; 𝑆% = .!
= /.$1
= 0.3571

Client’s portfolio;
Expected return; 𝐸(𝑟( ) = 0.15
Standard Deviation; 𝜎( = 0.196
)(+#)-+" /.&2-/./1
Sharpe Ratio; 𝑆% = = = 0.3571
.# /.&34

Thus the Sharpe Ratio of your risky portfolio and your clients are the same!

Question 6.16;
Draw the CAL of your portfolio on an expected return–standard deviation diagram. What is the slope of the CAL? Show
the position of your client on your fund’s CAL.
)(+! )-+"
Slope CAL; 𝑆% = = 0.3571
.!
Your Sharpe ratio can be seen as a reward-to-risk ratio. Since the client invested only in the portfolio and risk-free
bonds, he has a lower risk and a lower reward in such a way that the ratio stays the same

Question 6.17;
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 16%.
a) What is the proportion y?
b) What is your client’s investment proportions in your three stocks and the T-bill fund?
c) What is the standard deviation of the rate of return on your client’s portfolio?
a)
Expected return;
𝐸(𝑟( ) = 𝑦𝐸(𝑟% ) + (1 − 𝑦)𝑟' = 𝑟' + 𝑦N𝐸(𝑟% ) − 𝑟' O
𝐸(𝑟( ) = 0.16
/./1
0.16 = 0.08 + 𝑦[0.18 − 0.08] ----> 0.10𝑦 = 0.08 ----> 𝑦 = = 0.8 = 80%
/.&/

Therefore, in order to have a portfolio with expected rate of return equal to 16% the client must invest 80% of total
funds in the risky portfolio and 20% in T-bills.

b)
Stock A 25%
Stock B 32%
Stock C 43%

0.8 ∙ 0.25 = 0.200 = 20.0% in Stock A
0.8 ∙ 0.32 = 0.256 = 25.6% in Stock B
0.8 ∙ 0.43 = 0.344 = 34.4% in Stock C
100% - 20.0% - 25.6% - 34.4% = 20% (Invest 20% in T-bills)

c)
Standard Deviation;
𝜎( = 𝑦𝜎%
𝜎( = 0.8 ∙ 0.28 = 0.224 = 22.4%

,Question 6.18;
Suppose that your client prefers to invest in your fund a proportion 𝑦 that maximizes the expected return on the
complete portfolio subject to the constraint that the complete portfolio’s standard deviation will not exceed 18%.
a) What is the investment proportion, y?
b) What is the expected rate of return on the complete portfolio?

Risky portfolio;
𝐸(𝑟% ) = 0.18
𝜎% = 0.28
𝑟' = 0.08
𝜎( = 0.18

a)
Standard Deviation;
𝜎( = 𝑦𝜎%
0.18 = 𝑦 ∙ 0.28
𝑦 = 0.6429 = 64.29%

b)
Expected return;
𝐸(𝑟( ) = 𝑟' + 𝑦N𝐸(𝑟% ) − 𝑟' O
𝐸(𝑟( ) = 0.08 + 0.6429[0.18 − 0.08] = 0.14429 = 14.429%

Question 6.19;
Your client’s degree of risk aversion is A = 3.5.
a) What proportion, y, of the total investment should be invested in your fund?
b) What is the expected value and standard deviation of the rate of return on your client’s optimized portfolio?

a)
Utility maximization;
)6+$ 7-+"
𝑦∗ = 8.!%
/.&1-/./1
𝑦∗ = = 0.3644 = 36.44%
9.2∙(/.$1)%


b)
Expected return;
𝐸(𝑟( ) = 𝑟' + 𝑦 ∗ N𝐸(𝑟% ) − 𝑟' O
𝐸(𝑟( ) = 0.08 + 0.3644[0.18 − 0.08] = 0.1164 = 11.64%

Standard Deviation;
𝜎( = 𝑦 ∗ 𝜎%
𝜎( = 0.3644 ∙ 0.28 = 0.1020 = 10.20%

, Question 6.27;
For Problems 27 through 28:
You estimate that a passive portfolio, for example, one 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%. You manage an active portfolio with
expected return 18% and standard deviation 28%. The risk-free rate is 8%.

Draw the CML and your funds’ CAL on an expected return–standard deviation diagram.
a) What is the slope of the CML?
b) Characterize in one short paragraph the advantage of your fund over the passive fund.

a)
Passive portfolio; Active portfolio;
𝐸(𝑟% ) = 0.13 𝐸(𝑟% ) = 0.18
𝜎% = 0.25 𝜎% = 0.28
)(+! )-+" /.&9-/./1 )(+! )-+" /.&1-/./1
Sharpe Ratio; 𝑆% = .!
= /.$2
= 0.20 Sharpe Ratio; 𝑆% = .!
= /.$1
= 0.357

My fund allows an investor to achieve a higher mean for any given standard deviation than would a passive strategy,
i.e., a higher expected return for any given level of risk

Question 6.28;
Your client ponders whether to switch the 70% that is invested in your fund to the passive portfolio.
a) Explain to your client the disadvantage of the switch.
b) Show him the maximum fee you could charge (as a percentage of the investment in your fund, deducted at the end
of the year) that would leave him at least as well-off investing in your fund as in the passive one. (Hint: The fee will
lower the slope of his CAL by reducing the expected return net of the fee.)

Active portfolio; Passive portfolio;
𝐸(𝑟% ) = 0.18 𝐸(𝑟% ) = 0.13
𝜎% = 0.28 𝜎% = 0.25
)(+! )-+" /.&1-/./1 )(+! )-+" /.&9-/./1
Sharpe Ratio; 𝑆% = .!
= /.$1
= 0.357 Sharpe Ratio; 𝑆% = .!
= /.$2
= 0.20

70% of my fund; 70% of passive portfolio;
𝑦 = 0.70 𝑦 = 0.70
𝐸(𝑟; ) = 𝑦𝐸(𝑟% ) + (1 − 𝑦)𝑟' 𝐸(𝑟; ) = 𝑦𝐸(𝑟% ) + (1 − 𝑦)𝑟'
𝐸(𝑟; ) = 0.70 ∙ 0.18 + (1 − 0.70) ∙ 0.08 = 15% 𝐸(𝑟; ) = 0.70 ∙ 0.13 + (1 − 0.70) ∙ 0.08 = 11.5%

𝜎( = 𝑦 ∙ 𝜎% 𝜎( = 𝑦 ∙ 𝜎%
𝜎( = 0.70 ∙ 0.28 = 19.60% 𝜎( = 0.70 ∙ 0.25 = 17.50%

With 70% of his money invested in my fund’s portfolio, the client’s expected return is 15% per year and standard
deviation is 19.6% per year. If he shifts that money to the passive portfolio his overall expected return becomes 11.5%
and standard deviation is 17.50%

Therefore, the shift entails a decrease in mean from 15% to 11.5% and a decrease in standard deviation from 19.6%
to 17.5%. Since both mean return and standard deviation decreases, it is not yet clear whether the move is beneficial.

The disadvantage of the shift is that, if the client is willing to accept a mean return on his total portfolio of 11.5%, he
can achieve it with a lower standard deviation using my fund rather than the passive portfolio.

To achieve a target mean of 11.5%;
0.115 = 𝑦𝐸(𝑟% ) + (1 − 𝑦)𝑟'
0.115 = 𝑦 ∙ 0.18 + (1 − 𝑦) ∙ 0.08
𝑦 = 0.35

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