Finance 261 Workshop 4
Portfolio theory
Questions
1. Suppose you have a project that has a 0.7 chance of doubling your investment in a
year and 0.3 chance of halving your investment in a year. What is the standard
deviation of the rate of return on this investment?
2. The standard deviation of the portfolio is always equal to the weighted average of
the standard deviations of the assets in the portfolio. (True or false?)
3. Stocks A, B, and C have the same expected returns and standard deviations. The
following table shows the correlations between the returns on these stocks.
A
+1
+0.9
+0.1
A
B
C
B
C
+1
-0.4
+1
Given these correlations, the portfolio constructed from these stocks having the
lowest risk is a portfolio:
a.
b.
c.
d.
Equally invested in stocks A and B.
Equally invested in stocks A and C.
Equally invested in stocks B and C.
Totally invested in stock C.
4. You are given the following return probability distribution for Stock X and Y:
Probability
Stock X
Stock Y
Bear market
0.2
-20%
-15%
Normal market
0.5
18%
20%
Bull market
0.3
50%
10%
a. What are the expected rates of return for stocks X and Y?
b. What are the standard deviations of stocks X and Y?
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c. 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?
5. Jane Smith has a $900,000 fully diversified portfolio. She subsequently inherits ABC
company common stock worth $100,000. Her financial adviser provided her with the
following financial information:
Original portfolio
ABC Company
Risk and Return Characteristics
Expected Monthly Returns (%)
Standard Deviation of
Monthly returns (%)
0.67
2.37
1.25
2.95
The correlation coefficient of ABC stock returns with the original portfolio return is
0.40.
The inheritance changes Jane’s overall portfolio and she is deciding whether to keep
the ABC stock. Assuming Jane keeps the ABC stock, calculate the:
A. (i) Expected return of her new portfolio which includes the ABC stock.
(ii) Covariance of ABC stock returns with the original portfolio returns.
(iii) Standard deviation of her new portfolio which includes the ABC stock.
B. If Jane sells the ABC stock, she will invest the proceeds in risk-free government
securities yielding 0.42% monthly. Assuming Jane sells the ABC stock and replaces
it with the government securities, calculate the:
(i) Expected return of her new portfolio which includes the government securities.
(ii) Covariance of the government security returns with the original portfolio
returns.
(iii) Standard deviation of her new portfolio which includes the government
securities.
C. Based on conversations with her husband, Jane is considering selling the $100,000
of ABC stock and acquiring $100,000 of XYZ Company common stock instead. XYZ
stock has the same expected return and standard deviation as stock ABC. Her
husband comments, “It doesn’t matter whether you keep all of the ABC stock or
replace it with $100,000 XYZ stock”. State whether her husband’s comment is
correct or incorrect.
6. Which statement about portfolio diversification is correct?
a. Proper diversification can reduce or eliminate systematic risk.
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b. Diversification reduces the portfolio’s expected return because it reduces a
portfolio’s total risk.
c. As more securities are added to a portfolio, total risk typically would be expected
to fall at a decreasing rate.
d. The risk-reducing benefits of diversification do not occur meaningfully until at least
30 individual securities are included in the portfolio.
7. (2005SC Exam) A portfolio consists of 50 equally weighted independent securities
each with a standard deviation of 20%. Calculate the portfolio standard deviation.
Show your workings.
8. (2013FC Exam) Consider a market where each industry consists of three stocks. All
within-industry variances and covariances are the same and given by the following
variance-covariance matrix:
Variance-covariance Matrix of Stocks within Each Industry
Stock 1
Stock 2
Stock 3
Stock 1
0.1
0.1
0.1
Stock 2
0.1
0.2
0.2
Stock 3
0.1
0.2
0.3
Stocks in different industries are not correlated – i.e., correlation = 0 for stocks in
different industries.
a. An investor wants to form an equally-weighted portfolio of ten three-stock
industries – that is, a portfolio consisting of 30 stocks, 3 stocks in each of 10
industries. Calculate the portfolio standard deviation.
b. Derive a general expression for the portfolio variance of an equally-weighted
portfolio of N industries, an equally-weighted portfolio consisting of 3N stocks, 3
stocks in each of N industries.
c. Does the portfolio variance in part b) increase or decrease when N gets large? What
is the limit of the portfolio variance as N approaches infinity? Explain.
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