PROFESSOR: Mr. Konstantinos Kanellopoulos, MSc (L.S.E.), M.B.A. COURSE: MBA-680-50-SUII12 Corporate Financial TheorySEMESTER: Summer Session II

Exercises 7 - 8 – 9(with solutions)

Konstantinos Kanellopoulos8th August 2012

PART I EXERCISES FROM CHAPTERS 7 , 8 and 9 – with solutions

Exercise 1

Here are inflation rates and U.S. stock market and Treasury bill returns between 1929 and 1933:

Year Inflation Stock Market

T-Bill Return

1929 -2 -14,5 4,81930 -6 -28,3 2,41931 -9,5 -43,9 1,11932 -10,3 -9,9 11933 0,5 57,3 0,3

a) What was the real return on the stock market in each year?b) What was the average real return?c) What was the risk premium in each year?d) What was the average risk premium?e) What was the standard deviation of the risk premium?

Suppose that five-year government bonds are selling on a yield of 4%. Value a five-year bond with a 6% coupon. Start by assuming that the bond is issued by a continental European government and makes annual coupon payments. Then rework your answer assuming that the bond is issued by the U.S. Treasury, that the bond pays semiannual coupons, and the yield refers to a semiannually compounded rate.

Solution

Recall that:

(1 + rnominal) = (1 + rreal) (1 + inflation rate)

Therefore:

rreal = [(1 + rnominal)/(1 + inflation rate)] – 1

a. The real return on the stock market in each year was:

1929: -14.7%1930: -23.7%1931: -38.0%1932: 0.5%1933: 56.5%

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b. From the results for Part (a), the average real return was: -3.89%

c. The risk premium for each year was:

1929: -19.3%1930: -30.7%1931: -45.0%1932: -10.9%1933: 57.0%

d. From the results for Part (c), the average risk premium was: –9.78%

e. The standard deviation () of the risk premium is calculated as follows:

Exercise 2

Each of the following statements is dangerous or misleading. Explain why.

a) A long-term United States government bond is always absolutely safeb) All investors should prefer stocks to bonds because stocks offer higher long-run rates of

returnc) The best practical forecast of future rates of return on the stock market is a 5- or 10-year

average of historical returns

Solution

a. A long-term United States government bond is always absolutely safe in terms of the dollars received. However, the price of the bond fluctuates as interest rates change and the rate at which coupon payments received can be invested also changes as interest rates change. And, of course, the payments are all in nominal dollars, so inflation risk must also be considered.

b. It is true that stocks offer higher long-run rates of return than do bonds, but it is also true that stocks have a higher standard deviation of return. So, which investment is preferable depends on the amount of risk one is willing to tolerate. This is a complicated issue and depends on numerous factors, one of which is the investment time horizon. If the investor has a short time horizon, then stocks are generally not preferred.

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c. Unfortunately, 10 years is not generally considered a sufficient amount of time for estimating average rates of return. Thus, using a 10-year average is likely to be misleading.

Exercise 3

Mark Harrywitz proposes to invest in two shares, X and Y. He expects a return of 12%from X and 8% from Y. The standard deviation of returns is 8% for X and 5% for Y.The correlation coefficient between the returns is .2.

a. Compute the expected return and standard deviation of the following portfolios:Portfolio Percentage in X Percentage in Y1 50 502 25 753 75 25

b. Sketch the set of portfolios composed of X and Y.c. Suppose that Mr. Harrywitz can also borrow or lend at an interest rate of 5%. Show on your

sketch how this alters his opportunities. Given that he can borrow or lend, what proportions of the common stock portfolio should be invested in X and Y?

Solution

a.

Portfolio R

1 10.0% 5.1%2 9.0 4.63 11.0 6.4

b. See the figure below. The set of portfolios is represented by the curved line. The five points are the three portfolios from Part (a) plus the following two portfolios: one consists of 100% invested in X and the other consists of 100% invested in Y.

c. See the figure below. The best opportunities lie along the straight line. From the diagram, the optimal portfolio of risky assets is portfolio 1, and so Mr. Harrywitz should invest 50 percent in X and 50 percent in Y.

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Exercise 4

The Treasury bill rate is 4%, and the expected return on the market portfolio is 12%. Using the capital asset pricing model:

a. Draw a graph showing how the expected return varies with beta.b. What is the risk premium on the market?c. What is the required return on an investment with a beta of 1.5?d. If an investment with a beta of 0.8 offers an expected return of 9.8%, does it have to be a

positive NPV?e. If the market expects a return of 11.2% from Stock X, what is its beta?

Solution

a.

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b. Market risk premium = rm – rf = 0.12 – 0.04 = 0.08 = 8.0%

c. Use the security market line:

r = rf + (rm – rf)

r = 0.04 + [1.5 (0.12 – 0.04)] = 0.16 = 16.0%

d. For any investment, we can find the opportunity cost of capital using the security market line. With = 0.8, the opportunity cost of capital is:

r = rf + (rm – rf)

r = 0.04 + [0.8 (0.12 – 0.04)] = 0.104 = 10.4%

The opportunity cost of capital is 10.4% and the investment is expected to earn 9.8%. Therefore, the investment has a negative NPV.

e. Again, we use the security market line:

r = rf + (rm – rf)

0.112 = 0.04 + (0.12 – 0.04) = 0.9

Exercise 5

The following table shows estimates of the risk of two well-known Canadian stocks:

Cash Flows, $ ThousandsSt. Deviat. % R2 Beta Standard

Error of Beta

Alcan 29 .37 1.58 .27Canadian Pacific 22 .15 0.75 .23

a) What proportion of each stock’s risk was market risk, and what proportion was unique risk?

b) What is the variance of Alcan? What is the unique variance?c) What is the confidence level on Canadian Pacific’s beta?d) If the CAPM is correct, what is the expected return on Alcan? Assume a risk-free interest

rate of 5% and an expected market return of 12%.e) Suppose that next year the market provides a zero return. Knowing this, what return would

you expect from Alcan?

Solution

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a. The R2 value for Alcan was 0.37, which means that 37% of total risk comes from movements in the market (i.e., market risk). Therefore, 63% of total risk is unique risk.

The R2 value for Canadian Pacific was 0.15, which means that 15% of total risk comes from movements in the market (i.e., market risk). Therefore, 85% of total risk is unique risk.

b. The variance of Alcan is: (29)2 = 841Market risk for Alcan: 0.37 × 841 = 311.17Unique risk for Alcan: 0.63 × 841 = 529.83

c. The t-statistic for CP is: 0.75/0.23 = 3.26

This is significant at the 1% level, so that the confidence level is 99%.

d. rAL = rf + AL (rm – rf) = 0.05 + [1.58 (0.12 – 0.05)] = 0.1606 = 16.06%

e. rAL = rf + AL (rm – rf) = 0.05 + [1.58 (0 – 0.05)] = −0.0290 = −2.90%

Exercise 6

A project has the following forecasted cash flows:

Cash Flows, $ ThousandsCo C1 C2 C3

-100 +40 +60 +50

The estimated project beta is 1.5. The market return rm is 16%, and the risk-free rate rf is 7%.a) Estimate the opportunity cost of capital and the project’s PV (using the same rate to

discount each cash flow).b) What are the certainty-equivalent cash flows in each year?c) What is the ratio of the certainty-equivalent cash flows in each year?d) Explain why this ratio declines.

Solution

a. Using the Security Market Line, we find the cost of capital:

r = 0.07 + [1.5 (0.16 – 0.07)] = 0.205 = 20.5%

Therefore:

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b. CEQ1 = 40(1.07/1.205) = 35.52CEQ2 = 60(1.07/1.205)2 = 47.31CEQ3 = 50(1.07/1.205)3 = 35.01

c. a1 = 35.52/40 = 0.8880a2 = 47.31/60 = 0.7885a3 = 35.01/50 = 0.7002

d. Using a constant risk-adjusted discount rate is equivalent to assuming that at

decreases at a constant compounded rate.

Exercise 7

Nero Violins has the following capital structure:

Security Beta Total Market Value ($ millions)

Debt 0 $100Preferred stock 0.20 40Common stock 1.20 299

a) What is the firm’s asset beta? (Hint: What is the beta of a portfolio of all the firm’s securities?)

b) Assume that the CAPM is correct. What discount rate should Nero set for investments that expand the scale of its operations without changing its asset beta? Assume a risk-free interest rate of 5% and a market risk premium of 6%.

Solution

a.

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b. r = rf + (rm – rf) = 0.05 + (0.836 0.06) = 0.10016 = 10.016%

Exercise 8

Look at the following table: What would the nine countries’ betas be if the correlation coefficient for each was 0.5? Do the calculations and explain.

Solution

Ratio of ’s Correlation Beta

Argentina 2.36 0.5 1.18

Brazil 2.10 0.5 1.05

China 1.96 0.5 0.98

Egypt 1.49 0.5 0.75

India 1.80 0.5 0.90

Indonesia 1.71 0.5 0.86

Mexico 1.36 0.5 0.68

Sri Lanka 2.07 0.5 1.04

Turkey 2.96 0.5 1.48

The betas generally increase compared to those reported in Table 10.2 because the returns for these markets are now, in most cases, more highly correlated with the U.S. market. Thus, the contribution to overall market risk generally becomes greater.

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