The following data apply to problems 1-4:
Consider a market with three mutual funds. The first is a stock fund, the second is a long-term corporate bond fund, and the third is a T-bill money market fund that yields a rate of 0.5 percent. All investors have identical beliefs and the expected returns and standard deviations of returns are:
|
|
Expected Return
|
Standard Deviation
|
|
Stock fund (S) Bond fund (B)
|
.165
.08
|
.50
.25
|
The correlation between the fund returns is .4.
1. Solve numerically for the proportions of each asset in the optimal risky portfolio P (portfolio that includes only risky assets). Find the expected return and standard deviation of this portfolio.
2. You require that your complete portfolio (portfolio that includes T-bills and risky funds) yield an expected return of 19 percent and be efficient on the best feasible CAL.
a. What is the proportion invested in the T-bill fund and each of the two risky funds?
b. What is the standard deviation of your complete portfolio?
3. If you were to use only the two risky funds (no T-bills) and will require an expected return of 19 percent, what must be the investment proportions of your portfolio? Compare its standard deviation to that of the optimized portfolio in problem 2. What do you conclude?
4. Now assume that Michael has coefficient of risk aversion equal to 1. Find the weight of each security in his complete portfolio (portfolio that includes T-bills and risky portfolio P).
5. An investor with a coefficient of risk aversion of 3 can design a risky portfolio based on two stocks, K and L. Stock K has an expected rate of return of 25% and a standard deviation of return of 30%. Stock L has an expected rate of return of 10% and a standard deviation of return of 12%. The correlation coefficient between the two stocks is 0.4. The risk free lending rate is 4% while the risk free borrowing rate is 7%.
a. What is the weight of stock K in the optimal risky portfolio if an investor lends? What are the expected return and the standard deviation of return on this portfolio?
b. What is the weight of stock K in the optimal risky portfolio if an investor borrows? What are the expected return and the standard deviation of return on this portfolio?
c. Find weights of all assets in the optimal complete portfolio for the investor whose coefficient of risk aversion is 3.
6. Consider portfolio P that is comprised from two stocks, A and B. Stock A has a standard deviation of return (σA) of 15% while stock B has a standard deviation of return (σB ) of 40%. The correlation coefficient between the two stocks (ρ) is 0.6.
a. Graph (sketch) the relation between the weight of stock A and the standard deviation of portfolio P. Plot the weight of stock A along the horizontal axis.
b. Derive the formula for the weights of stocks A and B at which the variance of portfolio P is minimal. Hint: Consider the variance of portfolio P as a function of the weight of stock A, wA, and minimize this function with respect to this weight. Find the values of weights.
c. Go to yahoo finance and download weakly prices for IBM and INTEL (INTC) for year 2014 to excel file. Find weekly returns for these two companies using adjusted closing prices. (Unfortunately, yahoo finance provides excel files without dividends. You can use approximate returns by ignoring dividends.) Assume that these returns can be used to estimate expectation, variance and covariance of future returns. Find estimates of variance of the returns of each security and the covariance between these returns.
Consider portfolio with IBM and INTEL. Use excel solver to find the weights of each security in this portfolio at which the variance of the portfolio is minimal. (On Windows, Solver may be added in by going to File > Options > Add-ins, and under the Manage drop-down choosing Excel Add-ins and pressing the Go button. Check the Solver Add-In box and press OK.) Compare your result with the prediction found in question b. ( In the submitted solution show the last 10 weekly prices and returns for year 2014, the calculated variances and covariance, and the inputs used in excel solver.)
7. Assume the following data on the returns of market, rM , and returns of stock j, rj :
|
State
|
Probability
|
rM
|
rj
|
|
1
|
0.1
|
-0.15
|
-0.30
|
|
2
|
0.3
|
0.05
|
0.00
|
|
3
|
0.35
|
0.15
|
0.20
|
|
4
|
0.25
|
0.2
|
0.50
|
What should be the expected return on stock j according to CAPM if rf = 3%.
Show all your calculations for the full credit.
8. Go to yahoo finance and download monthly prices for S&P/TSX Composite index and Magna International (MG.TO) for 10 years starting from January 2004 to excel file. Assume that S&P/TSX Composite index represents the market portfolio. Find monthly returns for the market portfolio and Magna stock using adjusted closing prices. (Unfortunately, yahoo finance provides excel files without dividends. You can use approximate returns by ignoring dividends.) Assume that these returns can be used to estimate expectation, variance and covariance of future returns. Use excel to answer the following questions (in the submitted solution show the last 12 monthly returns for the two securities along with the inputs and results for the following questions):
a. What are the expected rates of return on Magna stock and the market?
b. What is the beta of Magna stock?
c. If the T-bill rate is 2 percent, draw the SML for this economy.
d. Plot Magna stock on the SML graph. What is the alpha's of this stock?