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Assignment 4: Portfolio Statistics and the CAPM

Due date: December 7, 1995 at the beginning of class.

Note: This problem set can be completed in groups of 1--4 people. If a group contains more than two people, then each group member must turn in sealed envelopes with a group evaluation form. This form should assign proportions (totaling 100%) to the other group members. I encourage you do as many of the assignment questions as possible on your own. If you can do the questions in this assignment, then you are in good shape for the final exam.

Question 1

Your security analyst says that the expected return and standard deviation for Federal Express (ticker symbol FDX) are 20% and 40%, respectively. For Duke Power (ticker symbol DUK), the expected return is 12%, with a standard deviation of 15%. The correlation between DUK's return and FDX's return is 0.30.
  1. What is the covariance of FDX's return with DUK's?
  2. Compute the means and standard deviations of portfolios of FDX and DUK with the following percentages in FDX, and the complement in DUK: (i) 100%, (ii) 60%, (iii) 30%, and (iv) 0%.
  3. Plot the points in (b) and smooth a curve of potential means and standard deviations that could be achieved with portfolios of FDX and DUK.

Question 2

If you have monthly data for a 5--year period on the total returns of Citicorp and of Bank America, how would you compute estimates of their (a) mean annual returns, (b) standard deviations of annual returns, (c) their return covariance and (d) their return correlation? Outline your approach, giving formula where you can.

Question 3

How many different correlations do you have to estimate to fully describe the correlations among 5 risky assets? Explain.

Question 4

Briefly explain the difference between a person who has a low risk tolerance and one with a high risk tolerance. Relate risk tolerance to (a) the slope of the person's indifference curves, (b) the risky asset mix that the individual chooses, (c) the borrowing or lending done by the person, and (d) the mean and standard deviation of the person's optimal portfolio. Also, (e) do relatively risk tolerant people like to take risk--just for the fun of it?

Question 5

An individual invests $20,000 total in 3 assets, with portfolio weights of (.3, .5, .2), respectively. If the price of asset 1 is $30 per share, asset 2 is $150 per share and asset 3 is $50 per share, how many shares of each does he hold?

Question 6

6. Let the prices at time 0, and time 1, and the dividends paid at time 1 be given as in the table below. What were the rates of return on each investment during the period? 

	Asset	P_0	D_1	P_1


	1	$60	$4.00	$71.00
	2	$10	$0.30	$13.00
	3	$30	$2.00	$52.00
	4	$50	$6.00	$50.00

Question 7

  1. If you had a portfolio with weights of 0.2, 0.1, 0.6 and 0.3 in the 4 risky assets of problem 6, what borrowing or lending did you do?
  2. With a riskless rate of 8%, what was your portfolio's return for the period?
  3. If the expected returns on the risky securities were 12%, 20%, 15%, and 9%, respectively, what was the expected return on your portfolio?

Question 8

Consider 2 portfolios of 3 risky assets that have the following covariance matrix and mean vector: 
V=
.04	-.03	.005
-.03	.09	.01
.005	.01	.01

E=
.20  .18  .12
Portfolio A has weights (0.5 0.5 0), and portfolio B has weights (0 0.5 0.5).
  1. What is the standard deviation of each individual asset? What are the variances?
  2. What are the correlations of each asset with each other asset? Covariances?
  3. What are the standard deviations of portfolios A and B? What is their covariance? What is their correlation? What are the portfolios' means?

Question 9

  1. Briefly, what is an efficient portfolio? What is an inefficient portfolio?
  2. Portfolio A, B, and C have means and standard deviations given by {.12, .20}; {.15, .30}; and {.14, .19}; respectively. Can you say for sure that any of these portfolios are inefficient? Explain.

Question 10

The mean absolute deviation (MAD) of a portfolio's return is measure of portfolio volatility. It is calculated by taking the average of the absolute values of the realized returns minus the expected return. Is this the same as standard deviation? Why or why not?

Question 11

True or false? Why? Two assets have a positive correlation of .5, with the same mean (15%) and the same standard deviation of (30%). If one considers all possible portfolios of these two assets, will one find means and standard deviations that are generally above or below those of the individual assets? Explain. Why so?

Question 12

What is the "investment opportunity set?" What does it look like if there is a riskless asset along with risky assets? If there is no riskless asset, what is its shape? What is the efficient frontier?

Question 13

The riskless rate is 7%, and there are two risky assets --A and B. You may borrow or lend risklessly and invest in either A or B, but not both. If A and B have means and standard deviations given as follows, which risky asset would you choose to go along with the riskless? Show your calculations and make your arguments carefully. The mean and standard deviation of A are 15% and 15%, respectively. The mean and standard deviation of B are 20% and 30%, respectively. Show graphically what is going on.

Question 14

Show in words and graphs (step-by-step) how you as an investment advisor could help your client to find the best portfolio to hold. At each step, explain what information you would need. How is your client's portfolio individually tailored? What determines whether your client is a borrower or a lender?

Question 15 ( For the finance specialists, i.e. optional)

You are considering investing in 3 risky assets and a riskless asset with expected returns, E covariance matrix V and covariance matrix inverse V^-1 as given below: 
E=
IBM        .12
Polaroid   .15
Gold       .08
Riskless   .06

V=
.0625  .05  .0037 
.05    .16  .006 
.0037  .006 .0225

V^-1=
21.3   -6.6   -1.7 
-6.6    8.34  -1.14 
-1.7   -1.14  44.9
  1. What are the proportions in the optimal portfolio of risky assets (for IBM, Polaroid, and Gold)? Show some of your calculations.
  2. If you decided that you wish to have a portfolio standard deviation of 30%, what is the maximum expected return that you can achieve? (Hint: compute the capital market line first.)
  3. What portfolio would you hold to achieve your goal in (b)?
  4. If someone else looked at these same assets and agreed with you on the covariance, but thought that the mean returns could be (.20 .15 .06 .06), respectively, what optimal mix of risky assets should she hold? Compare this portfolio to that in (a). Is it different in sensible ways?

Question 16

True or false? Why? In a realistic economy such as the U.S. economy today, a large number of stocks in a portfolio is likely to diversify away almost all risks.

Question 17

Explain with examples the concepts of (a) market risk, (b) extra- market covariance, and (c) specific risk. Explain the effects of diversification on each of these in a large portfolio.

Question 18

  1. Briefly, what is the CAPM?
  2. What does the CAPM say about risk measurement for assets? Explain what the relevant risk of an asset's return is for investors who behave optimally.
  3. Why is it that an asset's covariance shows up in risk measurement?

Question 19

  1. What is "alpha" for a security. Why is it used in performance evaluation?
  2. What is the "characteristic line" for a security?
  3. If security A has a return of 6% when the market has a return of -5% and the riskless rate is 7%, what is the security A's abnormal return?

Question 20

  1. Using the CAPM and Value Line (which is available in the library), explain how you a novice investment advisor, can make rational estimates for the expected returns on 1700 stocks. Use 7% for the risk free return and 13% as the expected market return.
  2. After scanning Value Line, what range will cover most stocks' expected returns?

Question 21

  1. The riskless rate is 7% and the expected return on the market is 13%. Given the stocks and their betas listed below, find the equilibrium expected returns on all of these assets. 
    
				 Beta
	American Express		1.45
	AT\&T				0.80
	CBS				1.05
	Chase Manhattan			1.10
	Cal Fed (S\&L)			1.80
	Duke Power			0.65
	Delta Airlines			1.02
	Eastern Airlines(!!!)		1.50
	Exxon				0.80
	Financial Corp. of America	1.95
	General Motors			1.15
	IBM				1.00
	Merrill Lynch			1.90
	Marriott Corp.			1.11
	First Boston			1.40
	Sony				1.15
	Nationsbank			0.95
	Fuqua Industries		1.10
	First Wachovia			0.80
  2. What is the beta for a portfolio with the following weights: American Express 20%, Merrill Lynch 30%, First Boston 30%, and Eastern Airlines 20%?

Question 22

What is the security market line? How well should the line fit?

Question 23

What is the capital market line? What does it tell investors?