Suppose assets x and y have the following state dependent returns:

Suppose assets x and y have the following state dependent returns:


State Probability Asset x Asset y
A 0.15 18% 8%
B 0.15 -5% -3%
C 0.1 8% 10%
D 0.2 10% 6%
E 0.25 15% -5%
F 0.15 -20% 2%
  • Calculate the mean and variance of each of these variables, and the covariance between
  • Calculate the mean and variance for the following portfolios

% in x     125    100    75    50    25      0       -25

% in y      -25       0      25    50    75    100    125

Plot the portfolio profiles in the space of mean and standard deviation.

  • What is the covariance between the minimum variance portfolio and any other portfolio in part (b)?
  • If the risk-free rate is 0.5%, what are the optimal risky portfolio weights in x

and y?


  1. Suppose you are trying to evaluate a portfolio manager with the following infor- mation


Probability Portfolio Market
0.1 -0.15 -0.30
0.3 0.05 0.00
0.4 0.15 0.20
0.2 0.20 0.25


The risk-free rate is 6%

  • Write the equation of the security market


  • Does the manager under-perform or over-perform the market expectation?
  • Do you want to long or short the portfolio? Why?
  • If you were to take the same amount of systematic risk as the manager, how much you would invest in the market portfolio and the risk free asset (a passive strategy)?
  1. As a manager with a risk aversion of 15, you select securities for investment based on the single-index model:

R˜  rf  = α + β(R˜m   rf ) +  .

Suppose you have obtained the following information:


Stock α β  
A 0.0087 0.1618 0.1050
B 0.0173 1.0524 0.2667
C 0.0017 0.6189 0.1637
D 0.0055 1.0446 0.0929
E 0.0016 1.1048 0.1645
F 0.0072 0.7845 0.0840
M 0.0000 1.0000 0.0000


The expected return on the index portfolio is 9% with a standard deviation of 7.5%. The risk free rate is 3%.

  • Calculate the expected returns of the
  • Calculate the covariance matrix of the
  • Calculate the optimal risky portfolio weights in the selected securities. What is the optimal Sharpe ratio?
  • Breakdown your complete investment portfolio in the selected
  1. A stock is currently traded at $60 per share with a volatility of 30%. The risk-free interest rate is 3% with continuous
    • Use the Black-Scholes model to evaluate a European call option with strike price $50 and 6-month
    • Based on the information, use a two-period binomial model to evaluate an America put option with strike price $70 and 6-month


  • Will the put option in part (b) be exercised before its maturity?
  1. A firm has a market value of $10 million and outstanding debt of $6 million that matures in 5 The firm asset grows at a rate of 15% with a standard deviation of 30%, and the risk-free rate is 5% with continuous compounding.
    • What is the fair value of the equity?
    • Express the market value of the debt as a function of a put option on the firm’s asset.
    • If the risk-free rate is 10% and everything else is unchanged, will the market value of equity and the market value of debt be changed? How?
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