CHAPTER 8

Exercise 10. Here are inflation rates and US stock market and Treasury bill returns between 1929 and 1933:

Year Inflation Stock Market

Return T-bill Return 1929 -2 -14.5 4.8 1930 -6.0 -28.3 2.4 1931 -9.5 -43.9 1.1 1932 -10.3 -9.9 1.0 1933 0.5 57.3 0.3

a) What was the real return on the stock market each year?

Recall from Chapter 4 that: (1 + rnominal) = (1 + rreal) × (1 + inflation rate)

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

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

1929: -12.8% 1930: -23.7% 1931: -38.0% 1932: 0.4% 1933: 56.5%

b) What was the average real return?

From the results for Part (a), the average real return was: -3.51%

c) What was the risk premium in each year? The risk premium for each year was:

Year

Stock Market Return

T-bill Return MRP

1929 -14,50% 4,80% -19,30%1930 -28,30% 2,40% -30,70%1931 -43,90% 1,10% -45,00%1932 -9,90% 1,00% -10,90%

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1933 57,30% 0,30% 57,00%

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

e) What was the standard deviation of the risk premium?

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

[ 2222 0.0978))0.450(0978))0 0.307(0.0978))0.193(15

1σ −−−+−−−+−−−×⎟⎟⎠

⎞⎜⎜⎝

⎛−

= (.((

0.1557390.0978))(0.5700.0978))0.109( 22 =−−+−−−+ ]((

39.46%0.3946370.155739σ ===

Exercise 14. Lonesome Gulch Mines has a standard deviation of 42% per year and a beta of +0.10. Amalgamated Copper has a standard deviation of 31% a year and a beta of +0.66. Explain why Lonesome Gulch is the safer investment for a diversified investor.

In the context of a well-diversified portfolio, the only risk characteristic of a single security that matters is the security’s contribution to the overall portfolio risk. This contribution is measured by beta. Lonesome Gulch is the safer investment for a diversified investor because its beta (+0.10) is lower than the beta of Amalgamated Copper (+0.66). For a diversified investor, the standard deviations are irrelevant.

Exercise 15. Lambeth Walk invests 60% of her funds in stock I, and the balance in stock J. The standard deviation of returns on I is 10% and on J is 20%. Calculate the variance of portfolio returns, assuming: a) The correlation between the returns is 1.0

xI = 0.60 σI = 0.10 xJ = 0.40 σJ = 0.20

1ρIJ =

)]σσρx2(xσxσx[σ JIIJJI2

J2

J2

I2

I2

p ++=

2

0.0196]0)(0.20)40)(1)(0.12(0.60)(0.(0.20)0.40)((0.10)(0.60)[ 2222 =++=

b) The correlation is 0.5 xI = 0.60 σI = 0.10 xJ = 0.40 σJ = 0.20

0.50ρIJ =

)]σσρx2(xσxσx[σ JIIJJI2

J2

J2

I2

I2

p ++=

0.0148])0.10)(0.2040)(0.50)(2(0.60)(0.(0.20)0.40)((0.10)(0.60)[ 2222 =++=

c) The correlation is 0 xI = 0.60 σI = 0.10 xJ = 0.40 σJ = 0.20

Exercise 21. Your eccentric Aunt Gerlinda has left you €50,000 in Alcan shares plus €50,000 cash. Unfortunately her will requires that the Alcan stock not to be sold for one year and the €50,000 cash must be entirely invested in one of the stocks shown in table 8.9. What is the safest attainable portfolio under these restrictions?

0ρij =

)]σσρx2(xσxσx[σ JIIJJI2

J2

J2

I2

I2

p ++=

0.0100]0)(0.20)40)(0)(0.12(0.60)(0.(0.20)0.40)((0.10)(0.60)[ 2222 =++=

Correlation Coefficients

Alcan BP Deutsche

Bank Fiat Heineken LVMH Nestlé Standard Deviation

Alcan 1,00 0,34 0,53 0,30 0,20 0,53 0,08 29,7% BP 1,00 0,44 0,26 0,20 0,27 0,29 18,4%

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Deutsche Bank 1,00 0,52 0,22 0,56 0,24 30,1% Fiat 1,00 0,17 0,42 0,26 35,9% Heineken 1,00 0,33 0,50 17,2% LVMH 1,00 0,31 31,0% Nestlé 1,00 13,8%

Table 8.9 Standard Deviation of Returns and correlation coefficients for a sample of seven stocks. Note: Correlations and standard deviations are calculated using returns in each country’s own currency; in other words, they assume that the investor is protected against exchange risk.

“Safest” means lowest risk; in a portfolio context, this means lowest variance of return. Half of the portfolio is invested in Alcan stock, and half of the portfolio must be invested in one of the other securities listed. Thus, we calculate the portfolio variance for six different portfolios to see which is the lowest. The safest attainable portfolio is comprised of Alcan and Nestlé.

Stocks Portfolio Variance BP 0.039806

Deutsche 0.068393 Fiat 0.070266

Heineken 0.034557 LVMH 0.070476 Nestlé 0.028453

Exercise 22. There are few, if any, real companies with negative betas. But suppose you found one with ß= –0.25. a) How would you expect this stock’s rate of return to change if the overall market rose by an extra 5%? What if the market fell by an extra 5%?

In general, we expect a stock’s price to change by an amount equal to (beta × change in the market). Beta equal to –0.25 implies that, if the market rises by an extra 5%, the expected change in the stock’s rate of return is –1.25%. If the market declines an extra 5%, then the expected change is +1.25%. b) You have $1 million invested in a well-diversified portfolio of stocks. Now you receive an additional $20,000 bequest. Which of the following actions will yield the safest overall portfolio return:

i) Invest $20,000 in Treasury Bills (which have β=0) ii) Invest $20,000 in stocks with β=1 iii) Invest $20,000 in stocks with β=-0.25

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Explain your answer. “Safest” implies lowest risk. Assuming the well-diversified portfolio is invested in typical securities, the portfolio beta is approximately one. The largest reduction in beta is achieved by investing the $20,000 in a stock with a negative beta. Answer (iii) is thus correct. Exercise 23: See BMA for the question text Expected portfolio return = xA E[RA ] + xB E[R B ] = 12% = 0.12

Let xB = (1 – xA )

xA (0.10) + (1 – xA) (0.15) = 0.12 ⇒ xA = 0.60 and xB = 1 – xA = 0.40

Portfolio variance = xA2 σA

2 + xB2 σB

2 +2(xA xB ρAB σA σB )

= (0.60 2 ) (20 2 ) + (0.40 2 ) (40 2 ) + 2(0.60)(0.40)(0.50)(20)(40) = 592

Standard deviation = 24.33%592σ ==

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