Problem Set 1 with solution

FNCE3302
Investments and Security Analysis
Problem Set 1
Question 1
Suppose you have the following expectations about the market condition and the returns on Stocks
X and Y.
Market Condition
Bear Market
Normal Market
Bull Market
Probability
0.3
0.5
0.2
Return on Stock X
-3%
3%
8%
Return on Stock Y
-5%
5%
15%
a) What are the expected returns for Stocks X and Y, E(rX) and E(rY)?
E(rX) = 0.3 × (-3) + 0.5 × 3 + 0.2 × 8 = 2.2%
E(rY) = 0.3 × (-5) + 0.5 × 5 + 0.2 × 15 = 4%
b) What are the standard deviations of the returns for Stocks X and Y, σX and σY?
σ2X = 0.3 × (-0.03 – 0.022)2 + 0.5 × (0.03 – 0.022)2 + 0.2 × (0.08 – 0.022)2 = 0.001516
σX = √0.001516 = 0.0389 = 3.89%
σ2Y = 0.3 × (-0.05 – 0.04)2 + 0.5 × (0.05 – 0.04)2 + 0.2 × (0.15 – 0.04)2 = 0.0049
σY = √0.0049 = 0.07 = 7.0%
c) Suppose you have $1000 to invest, and decide to invest $700 in Stock X and $300 in Stock Y.
What are the expected return and standard deviation of the return on your portfolio, E(r P) and
σP?
E(rP ) = wXE(rX) + wYE(rY ) = 0.7 × 2.2 + 0.3 × 4 = 2.74%
σ2P = w2X σ2X + w2Yσ2Y + 2wXwY Cov(rX, rY)
Cov(rX, rY) = 0.3×(–3 – 2.2)×(–5 – 4) + 0.5×(3 – 2.2)×(5 – 4) +0.2×(8 – 2.2)×(15 – 4) = 27.2
σ2P = 0.72 × 15.16 + 0.32 × 49 + 2 × 0.7 × 0.3 × 27.2 = 23.26
σP = √23.26 = 4.82%
Question 2
The expected returns and standard deviation of returns for two securities are as follows:
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Expected Return
Standard Deviation
Security Z
15%
20%
Security Y
35%
40%
The correlation between the returns is +0.25.
a) Calculate the expected return and standard deviation for the following portfolios:
i) All in Z
ii) 0.75 in Z and 0.25 in Y
iii) 0.5 in Z and 0.5 in Y
iv) 0.25 in Z and 0.75 in Y
v) All in Y
Recall that the general formula of expected return for a two-security portfolio is:
E(Rp) = w1E(R1) + w2E(R2)
Also, the variance of return for a two-security portfolio is:
σp2 = w12σ12 + w22σ22 + 2w1w2σ1σ2ρ
Substituting the numerical values given in the question into these equations produces the following
results:
Expected Return
Standard Deviation
i)
0.15
0.200
ii)
0.20
0.200
iii)
0.25
0.245
iv)
0.30
0.316
v)
0.35
0.400
b) Draw the mean-standard deviation frontier.
Plot x, y plane, y being expected return, and x being standard deviation
c) Which portfolios might not be held by an investor who likes high expected return and low
standard deviation? (In other words, which portfolios are on the efficient frontier?)
Portfolio (i). (i) is dominated by Portfolio (ii) because (ii) produces a higher expected return for
the same level of risk. Therefore, a risk-averse investor does not want to hold Portfolio (i)
Question 3
Suppose that a fund that tracks the S&P has mean E(RM) = 16% and standard deviation σM = 10%,
and suppose that the T-bill rate Rf = 8%. Answer the following questions about efficient portfolios:
a) What is the expected return and standard deviation of a portfolio that has 50% of its wealth in
the risk-free asset and 50% in the S&P?
Expected return is given by:
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E(Rp) = wME(RM) + wfE(Rf) = (0.5)(0.16) + (0.5)(0.08) = .12
Because the standard deviation of the return on the risk free asset is 0, the standard deviation of
the portfolio is:
σp = wMσM = (0.5)(0.10) = 0.05 = 5%
b) What is the expected return and standard deviation of a portfolio that has 125% of its wealth in
the S&P, financed by borrowing 25% of its wealth at the risk-free rate?
The standard deviation of return will be equal to:
σp = wMσM = (1.25)(.10) = 0.125 = 12.5%
Expected return will be equal to:
E(Rp) = wME(RM) + (1 – wM)Rf = 1.25(.16) + (– .25)(.08) = .18
This result can also be obtained using:
𝐸(𝑅𝑃 ) = 𝑅𝑓 +
𝐸(𝑅𝑀 ) − 𝑅𝑓
0.16 − 0.08
𝜎𝑃 = 0.08 +
× 0.125 = 0.18
𝜎𝑀
0.1
c) What are the weights for investing in the risk-free asset and the S&P that produce a standard
deviation for the entire portfolio that is twice the standard deviation of the S&P? What is the
expected return on that portfolio?
From above we have:
σp = wMσM
for the risk of the portfolio. The question asks for wM and wf that produces σp = 2σM. Substituting
σp for 2σM into the equation gives:
2σM = wMσM
This implies
wM = 2
We also know that
wf = 1 – wM = 1 – 2 = –1
This says the following in words: To produce a portfolio that is twice as risky as the market, invest
double your net worth in M (wM = 2), financed by borrowing 100% of your net worth by selling
short the risk-free asset (wf = –1). The standard deviation of the portfolio is
σp = 2 × 0.1 = 0.2 = 20%
The expected return on that portfolio is given by:
E(Rp) = wfRf + wMRM = –1(0.08) + 2(0.16) = 0.24
The expected return of 24% makes sense since it is double the return on the market minus the
financing cost of borrowing at the risk-free rate.
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d) Assume investors’ preferences are characterized by the utility function U = E[r] – 0.5Aσ2. What
would be the optimal allocation, i.e. the investment weights on S&P and T-bill, for an investor
with a risk-aversion coefficient of A=4? What is the expected return and standard deviation of
this optimal portfolio?
Recall that the weight of the risky asset in the optimal portfolio on the Capital Allocation
Line (CAL) is given by
𝐸(𝑅𝑃 ) − 𝑅𝑓
𝑤=
𝐴𝜎𝑃2
Therefore, w = (0.16 – 0.08)/(4×0.12) = 2. This is exactly the portfolio found in the previous part
(c), with expected return of 24% and standard deviation of 20%.
Question 4
Consider the following data:
Russell Fund
Windsor Fund
S&P Fund
Expected Return
16%
14%
12%
Standard Deviation
12%
10%
8%
The correlation between the returns on the Russell Fund and the S&P Fund is 0.7. The rate on Tbills is 6%. Which of the following portfolios would you prefer to hold in combination with Tbills and why?
(a) Russell Fund
(b) Windsor Fund
(c) S&P Fund
(d) A portfolio of 60% Russell Fund and 40% S&P Fund.
The answer is D. The reason is that the 60-40 portfolio combination of the Russell Fund and the
S&P 500 has the highest Sharpe ratio. In other words, this portfolio gives the best investment
opportunity set together with the risk-free Treasury bill. More specifically, when you calculate the
Sharpe ratio, that is, the slope of the capital allocation line, (E(RM) – Rf) / σM, for each of the three
mutual funds as well as the 60-40 combination, you find that the 60-40 combination has the highest
slope. Here are the calculations (where M represents the mutual fund in each case):

Russell Fund:

Windsor Fund:

S&P Fund:
𝐸(𝑅𝑀 ) − 𝑅𝑓 0.16 − 0.06
=
= .8333
𝜎𝑀
0.12
𝐸(𝑅𝑀 ) − 𝑅𝑓 0.14 − 0.06
=
= .8
𝜎𝑀
0.10
𝐸(𝑅𝑀 ) − 𝑅𝑓 0.12 − 0.06
=
= .75
𝜎𝑀
0.08
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
Portfolio of .6 in Russell + .4 in S&P 500.
We have to calculate E(RM) and σM of this 60:40 portfolio. To do this, we use the formulas for the
mean and standard deviation for a two-asset portfolio. Say that asset 1 is the Russell fund and asset
2 is the S&P.
First, we calculate E(RM):
E(RM) = .6E(R1) + .4E(R2) = .6(.16) + .4(.12) = .144
And now we calculate σM:
σM2 = w12σ12 + w22σ22 + 2w1w2σ1σ2ρ
= (.6)2(.12)2 + (.4)2(.08)2 + 2(.6)(.4)(.7)(.12)(.08) = .0094336
Thus,
σM = .097127
Therefore, for the 60-40 combination we have:
𝐸(𝑅𝑀 ) − 𝑅𝑓 0.144 − 0.06
=
= .8648
𝜎𝑀
0.097127
Since .8648 > .8333 > .8 > .75, the slope for the capital allocation line with the 60-40 mutual fund
combination is largest.