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An investor holds a portfolio of stocks and is considering investing in the XYZ Company. The firm's prospects look neutral and you estimate the following probability distribution of possible returns:

Returns on ABC P Returns on XYZ
-5% 0.10 -40%
3% 0.20 -10%
8% 0.40 20%
10% 0.20 32%
12% 0.10 45%

How much is the expected return for XYZ? How much is the coefficient of variation for XYZ? 99.73% of the time in what range (what specific values) would you expect the returns for XYZ, using Empirical Rule. You add ABC to your portfolio; you sell 30% of XYZ to ABC. How much is your expected return for this portfolio? How much is the coefficient of variation for the new portfolio? Do you consider this portfolio more or less risky than the individual stocks? Why?

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Reid Wolff
Reid WolffLv2
29 Sep 2019

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Question 1.

Consider a portfolio that has equal amounts of $10 invested in two assets. Suppose returns on the two assets are jointly normally distributed. The annual expected returns and variance or return on the first asset are given by

m1 = 0.10 s1^2 = 0.04

And those on the second asset are given by

m2 = 0.05 s2^2 = 0.03

Consider three cases:

The correlation between the return is r = 0

The correlation between the return is r = +0.50

The correlation between the return is r = -0.50

For each case, identify the 99% Value-at-Risk of the portfolio. Explain the pattern of dependence of VaR on the correlation.

Question 2.

You are given a portfolio of three assets whose returns are jointly normally distributed with the following mean vector and covariance matrix:

0.20 0.08 0.02 0.02

0.10 0.02 0.06 0.03

0.15 0.02 0.03 0.07

a. Compute the 95% VaR for the portfolio if we invest $1 in the first asset. $2 in the second asset, and $3 in the third asset.

b. How much does each asset’s holding contribute to the overall VaR risk?

Question 3.

You are given a portfolio of two assets whose returns are jointly distributed with the following vector and covariance matrix.

0.20 0.08 0.04

0.10 0.04 0.06

a. Compute the 95% VaR of the portfolio if $1 is invested in the first asset and $1 is invested in the second.

b. Compute the risk-contribution of each asset to the VaR.

c. Is the current portfolio weighting optimal? If not, suggest a better one.

Stocks X and Y have the following probability distributions of expected future returns:

Probability X Y
0.1 -15% -25%
0.2 6 0
0.3 10 24
0.3 20 27
0.1 33 40

Calculate the expected rate of return, rY, for Stock Y (rX = 12.00%.) Round your answer to two decimal places.
%

Calculate the standard deviation of expected returns, σX, for Stock X (σY = 18.21%.) Round your answer to two decimal places.
%

Now calculate the coefficient of variation for Stock Y. Round your answer to two decimal places.

Is it possible that most investors might regard Stock Y as being less risky than Stock X?

A. If Stock Y is less highly correlated with the market than X, then it might have a lower beta than Stock X, and hence be less risky in a portfolio sense.

B. If Stock Y is less highly correlated with the market than X, then it might have a higher beta than Stock X, and hence be more risky in a portfolio sense.

C. If Stock Y is more highly correlated with the market than X, then it might have a higher beta than Stock X, and hence be less risky in a portfolio sense.

D. If Stock Y is more highly correlated with the market than X, then it might have a lower beta than Stock X, and hence be less risky in a portfolio sense.

E. If Stock Y is more highly correlated with the market than X, then it might have the same beta as Stock X, and hence be just as risky in a portfolio sense.

1. If we are able to eliminate all of the unsystematic or diversifiable risk in a portfolio then, what is the result?

a.

a risk-free portfolio
b.
a portfolio that has an expected return of zero
c.

a portfolio that contains only systematic risk

2. A portfolio consists 20% of a risk-free asset and 80% of a stock. The risk-free return is 4%. The stock has an expected return of 15% and a standard deviation of 30%. What’s the expected return of the portfolio?

a.
9.5%
b.
15.0%
c.
12.8%

d.
2%


3. 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 is its expected return?


a.
12%
b.
19.5%
c.

17%

4. For what value of the correlation coefficient for the returns of a pair of stocks will the benefits of diversification the largest?


a.

1.0
b.

-1.0
c.

0.0
d.

0.5


5. Which of the following is an example of systematic risk?
a.
IBM posts lower than expected earnings.
b.
The overall stock market becomes more risky due to super fasting trading algorithms.
c.

Intel announces record earnings.

6. According to the CAPM (capital asset pricing model), the security market line is a straight line. The intercept of this line should be equal to

a.
zero
b.
the expected risk premium on the market portfolio
c.
the expected return on the market portfolio
d.
the risk-free rate


7. 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.
0.80
b.
1.06
c.
1.27


8. Suppose you can borrow and lend at the risk free-rate of 3%. Which of the following four risky portfolios has the best return relative to risk. (Hint: Calculate Sharpe Ratios the ratio of risk premium to standard deviation for each portfolio)

a.
portfolio with a standard deviation of 15% and an expected return of 12%
b.
portfolio with a standard deviation of 12% and an expected return of 9%
c.
portfolio with a standard deviation of 25% and an expected return of 18%
d.
portfolio with a standard deviation of 19% and an expected return of 15%


9. Which statements are TRUE regarding risk and return?

Statement I:
Diversification is the process of removing systematic risk from a portfolio.
Statement II:
In general, the greater the risk, the greater the return required by an investor.
Statement III:
Investors should focus on real returns if they are concerned about the purchasing power of their wealth.


a.
Statements II and III only
b.
Statement I only
c.
Statements I and III only
d.
Statements I and II only

10. The beta of the risk-free asset is:
a.

1.0
b.

0.5
c.

0.0
d.

-1.0

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