Stockholm School of Economics in Riga Financial Economics, Spring 2010 Jevgenijs Babaicevs

Problem Set III: The Capital Asset Pricing Model Problem 1: The ABC's of the Capital Asset Pricing Model Assume there are only three stocks traded on the stock market. All the necessary information regarding these stocks is summarized in the table below. Stock

Expected Return

Beta-Coefficient

Stock 1

12.0%

1.8

Stock 2

7.0%

0.8

Stock 3



1.2

a) Assume that a risk-free asset is also available. Explain what is meant by the Security Market Line (SML) in the context of CAPM? b) Construct a SML from the information above and interpret the values of its coefficients. Calculate the expected rate of return on stock 3 according to the CAPM model? How big is the rate of return on the risk-free asset? What is the expected rate of return on the market portfolio? c) Two more stocks have been just introduced, stock 4 with  4  2.0 and stock 5 with  5  1.05 . Empirical evidence reveals that the average rate of return of stock 4 is 16.0%, and the average rate of return of stock 5 is 7%. What inference can you draw from this information? Explain!

Problem 2: SML or CML? Consider the following two equations of the Security Market Line (SML) and the Capital Market Line (CML):

1

SML :

ri  rF   i   rM  rF 

 2

CML :

r r  ri  rF   i   M F   M 

You also have the following information: rM  15%, rF  6%,  M  15%. Answer the following questions, assuming that the CAPM holds. a) Which equation would you use to determine the expected return on an individual security with a standard deviation of returns of 50% and a beta of 2; that is,   50%,   2 . What is the expected return for such security?

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b) Which equation would you use to determine the expected return on a portfolio knowing that it is an efficient portfolio? If you were told that the standard deviation of returns on such portfolio is equal to  M , what is the expected return on such portfolio? c) What is a beta of the portfolio in b)? d) Given your answers above, explain what type of risky assets equation (1) can be used for, and what type of risky assets equation (2) can be used for?

Problem 3: Capital Budgeting Lembergs Ltd., a new conglomerate in Riga, has three operating divisions outlined in the following table. Division Food Electronics Chemicals

Percentage of Firm Value 50 30 20

You want to estimate the cost of capital for each division. As part of your project in Market Research, you identified three principal competitors shown in the table below.

Competitor

Estimated Equity Beta,

Rimi Foods Sony Electronics Dow Chemicals

0.8 1.6 1.2

E

D DE 0.3 0.2 0.4

Answer the following questions, assuming that betas are estimated accurately and the CAPM holds. a) Assuming that the debt of these firms is risk-free, estimate the asset beta for each of Lembergs' divisions. D -ratio is 0.4. If your estimates of divisional betas are correct, what is DE Lembergs' equity beta?

b) Lembergs'

c) Assume that the risk-free interest rate is 7% and the expected return on the market portfolio is 15%. Estimate the cost of capital for each of Lembergs' divisions. d) How much would your estimates of each of Lembergs division's cost of capital change if you assumed that debt has a beta of 0.2?

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Problem 4: The CAPM in Practice - Part I Note: This is a slightly modified problem from Re-Exam 2008! Don Arnisimo, a young specialist in the field of pension economics, offers you three funds to invest for your future pension: (i) a Money-Market Fund which invests into 3-month Treasury bills with a return of 2% per annum; (ii) an S&P 500 Index Fund, which is a good proxy for the market portfolio, delivers a premium of 8% and a standard deviation of 20% per annum; (iii) and Investment Unlimited Equity Fund, managed by LittleDima&Co, the returns of which can be described by the following equation:

 rt  rF        rMt  rF    t

1

where rt and rMt are returns of the fund and market portfolios respectively; rF is a return on the riskfree asset,  is a constant, and  t is the share of the fund’s returns not explained by the market. Don Arnisimo has observed that the performance of the fund over the past years yields:

  0.0

  1.0

R 2  0.50

a) Calculate the expected rate of return of the Investment Unlimited Equity Fund using the CAPM model? b) Calculate the Sharpe Ratios for the S&P 500 Index Fund and Investment Unlimited Equity Fund respectively. c) Assume that CAPM holds! What is a composition of the optimal portfolio, i.e. how much you should invest in each of the assets, to achieve an expected return of 8% per annum? d) Don Arnisimo found a mistake in his calculations! He has re-estimated equation 1 and discovered that an estimate of  is 0.02 (2% per annum) with a standard error of 0.002. Does this mistake matter for the composition of the optimal portfolio calculated in c)? If no, then explain why not; if yes - explain why so and calculate the composition of a new optimal portfolio (with the same expected return of 8%).

Problem 5: The CAPM in Practice - Part II Note: This is a slightly modified problem from Re-Exam 2009! Suppose that the local stock market in Finlandia is made up of only two kinds of stocks, small stocks and large stocks. If you regress the excess return on a small stock, indexed by i, on the excess market return, you find the following relation:

 Rit  RFt   0.03  1.3   RMt  RFt    it Every small stock behaves in this way, but different small stocks have different  it which are uncorrelated with each other. Similarly, if you regress the excess return on a large stock, indexed by j, on the excess market return, you find the following relation:

R

jt

 RFt   0.01  0.9   RMt  RFt    jt

Again, every large stock behaves in this way, but different large stocks have different  jt which are uncorrelated with each other.

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a) What fraction of the stock market value in Finlandia is accounted for by large stocks and what fraction by small stocks? b) Show that if you have an infinite number of stocks available, and you can freely trade in small and large stocks, then there is an arbitrage opportunity. c) Does the Capital Asset Pricing Model hold in Finlandia? (Hint: Your answer should be a qualitative, verbal explanation in one sentence!) d) Now suppose that  it of different small stocks have a correlation  with each other, and  jt of different large stocks have also correlation  with each other. How does this affect your answer in part b)?

Additional Problem for Your own Study Problem 6: The Arbitrage Pricing Theory Imagine that there are only two pervasive macroeconomic factors. Securities X, Y and Z have sensitivities to these factors summarizes in the following table. Securities

1

2

X

1.75

0.25

Y

-1.00

2.00

Z

2.00

1.00

Assume that the expected risk premium is 4% on factor 1 and 8% on factor 2 (Treasury Bills obviously offer zero risk premium). a) What is the risk premium on each of the three securities according to the Arbitrage Pricing Theory (APT)? b) Suppose you buy $80 of security X and $60 of security Y and sell $40 of security Z. What is the sensitivity of your portfolio to each of the two factors? What is the expected risk premium? c) Suppose you buy $160 of security X and $20 of security Y and sell $80 of security Z. What is the sensitivity of your portfolio to each of the two factors? What is the expected risk premium? d) Suppose that the APT did not hold and that security X offered a risk premium of 8%, security Y offered a premium of 14%, and security Z offered a premium of 16%. Suggest an investment strategy that has zero sensitivity to each factor and that has a negative risk premium. Answer: Let ri be the risk premium on security i, and  j - the portfolio's sensitivity to the factor j. a) rX  9.0% rY  12.0% rZ  16.0%

b) 1  0 2  1.0 rp  8%

c) 1  1.0 2  0 rp  4%

d) wX  2.0 wY  0.5 wZ  1.5

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rp  1%  0