FIN 320
Fall 2023
PROBLEM SET # 2
Important 1: When you submit your solutions on Blackboard include the number of your group on the file name as follows:
“Problem_Set_2_GROUP_3.xlsx”
Important 2: Submit Excel files
Important 3: Please solve each problem on a separate Excel tab
Question 1:
Consider the following scenarios for the one year return on a Stock Fund and on a Bond Fund.
|
Scenario |
Probability |
Stock Fund Return |
Bond Fund Return |
|
Recession |
0.20 |
-21% |
12% |
|
Normal |
0.60 |
12% |
7% |
|
Boom |
0.20 |
30% |
-8% |
a) Calculate the expected return and the standard deviation of returns of the Stock Fund and of the Bond Fund.
b) Calculate the correlation coefficient between the Stock and the Bond Fund.
c) Consider a portfolio that has 50% in the Stock Fund and 50% in the Bond Fund. What is the return of this portfolio in each of the three scenarios?
d) Calculate the expected return and the standard deviation of returns for a portfolio with the 50% in the Stock Fund and 50% in the Bond Fund.
d.1) Using the answers to items a) and b)
d.2) Using the answer to item c)
Question 2:
Consider two risky assets, S and B, with the following characteristics:
E(rS)= 9% , σS=20%
E(rB)= 5%, σB= 5% and ρBS= - 1
a) Is it possible to combine the two assets in a portfolio such that the portfolio has zero risk (i.e. zero standard deviation)? If so, what is the composition of the zero risk portfolio?
b) Suppose that in addition to trading in the risky assets S and B, investors can also freely buy, sell or short-sell a risk-free asset with risk-free rate rf. What must be the risk-free rate rf ?
What would happen otherwise?
Question 3:
Go to Yahoo!Finance. Download monthly prices for GE, IBM, and Exxon-Mobil stocks over the last 5 years.
a) Use the adjusted closing prices to calculate monthly returns. Use the sample to estimate expected returns, variances, standard deviations, covariances, and correlation of returns.
b) Annualize expected returns and variances by multiplying with 12.
c) Annualize volatilities by multiplying with 12^0.5 .
d) Assume that stock returns over the next year are drawn from a normal distribution with the expected returns, standard deviations and correlations calculated in a. What is the probability that an equally-weighted portfolio of GE, IBM and XOM will produce a negative return over the next year?
Question 4:
Consider the following economic scenarios for the next year, with the associated returns of a Stock and a Bond portfolio.
|
Scenario |
Economic Explanation |
Probability |
Return of Stock Portfolio rS |
Return of Bond Portfolio rB |
|
Moderate growth Moderate inflation |
Most likely Scenario |
50% |
18.1% |
8.4% |
|
Strong Growth High Inflation |
High Corporate Profits
Pure Yield Curve Shifts Up |
15% |
24.5% |
– 7.8% |
|
Low Growth Low Inflation |
Low Corporate Profits Pure Yield Curve Shifts Down |
15% |
– 17.5% |
24.5% |
|
Strong Growth Low Inflation |
High Corporate Profits Pure Yield Curve Shifts Down |
10% |
45.5% |
11.4% |
|
Weak Growth High Inflation |
Low Corporate Profits Pure Yield Curve Shifts Up |
10% |
– 46.5% |
– 18.4% |
a) Calculate the expected returns of the Stock and Bond Portfolios.
b) Calculate the standard deviations of the Stock and Bond Portfolios.
c) Calculate the correlation between the returns on the Stock and Bond Portfolios.
d) Consider a combined portfolio with 50% on the Stock Portfolio and 50% on the Bond Portfolio. What is the return of this portfolio across all scenarios? From this returns, calculate the expected return and standard deviation of the combined portfolio.
e) Using the formulas for the expected return and standard deviation of a portfolio, calculate the expected return and standard deviation of the combined portfolio with 50% in Stocks and 50% in Bonds.
f) Suppose that the correlation coefficient between the two assets is 1, but the standard deviations are the same as above. Is it possible to combine the two assets in a portfolio such that the portfolio has zero risk (i.e. zero standard deviation)? If so, what is the composition of the zero risk portfolio?
g) Suppose that, in addition to trading in the risky assets S and B, investors can also freely buy, sell or short-sell a risk-free asset with risk-free rate rf. What must be the risk-free rate rf ? What would happen otherwise?