Chapter 11 Risk and Return in Capital Markets 133

Chapter 11
Risk and Return in Capital Markets

Note: All problems in this chapter are available in MyFinanceLab. An asterisk (*) indicates problems with a higher level of difficulty.

1. Plan: Compute the realized return on this equity investment.

Execute:

Evaluate: The realized return on the equity investment is 12%.

2. Plan: Split the realized return into the dividend and capital gain yields.

Execute:

Evaluate: The dividend yield is 2% and the capital gain yield is 10%, thus the bulk of the return came from price appreciation.

3. Plan: Compute the capital gain and dividend yield under the assumption that the stock price has fallen to $45.

Execute:

a. Yes, the capital gain is different, because the difference between the current price and the purchase price is different than in Problem 1.

b. The dividend yield does not change, because the dividend is the same as in Problem 1.

Evaluate: The capital gain changes with the new lower price; the dividend yield does not change.

4. Plan: Compute the future sale price that is necessary to produce a 10% return.

Execute:

Evaluate: The selling price immediately after the dividend would need to be 21.50 for you to earn a 10% return on the investment.

5. Plan: Compute each period’s return as the price change + dividend divided by the initial price (see Eq. 11.1). Then compute the annual realized return as the product of 1 + each period’s return and then subtract off the 1 (see Eq. 11.2):

Execute:

R = (1 + 0.12)(1 - 0.027)(1 + 0.076)(1 + 0.009) -1 = 1.183 = 0.183

Evaluate:

In this case, the annual realized return is the compound return of the quarterly returns, taking into account both the dividends and price changes.

6. Plan: Download the Excel spreadsheet data and analyze it.

Execute: a/b. Using Excel:

S&P 500 / Small Stocks / Corp Bonds / World Portfolio / Treasury Bills / CPI
Average / 2.553% / 16.550% / 5.351% / 2.940% / 0.859% / -1.491%
Variance: / 0.1018 / 0.6115 / 0.0013 / 0.0697 / 0.0002 / 0.0022
Standard
Deviation: /
31.904% /
78.195% /
3.589% /
26.398% /
1.310% /
4.644%

Evaluate:

c. The riskiest assets were the small stocks. Intuition tells us that this asset class would be the riskiest.

7. Plan: For part (a), to compute the arithmetic average, use Eq.11.3. For part (b), to compute the geometric average, take the product of 1 + each return and then take the 10th root of that product (see the box on page 327). For part (c), realize that the total return computed in part (b) before taking the average can be applied directly to the $100.

Execute:

a. Using Eq. 11.3:

(-0.1993 + 0.166 + 0.18 - 0.5 + 0.433 + 0.012 - 0.165 + 0.456 + 0.452 -
0.03)/10 = 0.0805

*b. (0.801)(1.166)(1.180)(0.500)(1.433)(1.012)(0.835)(1.456)(1.452)(0.970) = 1.3683

. Subtracting the 1, we get the geometric average of 0.0319.

c. In part (b) we computed the total realized return as the product of 1 + each year’s return.
We would have earned that return on the $100, so the answer is $100(1.3683) = $136.83.

Evaluate:

The geometric average return is a better representation of what actually happened. However,
the arithmetic average is a better estimate of what you can expect to happen in any given year
(if you were trying to forecast the return for next year, for example).

Historical Stock and Dividend Data
Date / Price / Dividend
1/2/03 / 33.88
2/5/03 / 30.67 / 0.17
5/14/03 / 29.49 / 0.17
8/13/03 / 32.38 / 0.17
11/12/03 / 39.07 / 0.17
1/2/04 / 41.99

Plan: Calculate the realized return for each period and then compound those returns.

Execute: Return from 1/2/03 ® 2/5/03

Return from 2/5 ® 5/14

Return from 5/14 ® 8/13

Return from 8/13 ® 11/12

Return from 11/12 ® 1/2

Return for the year is:

Evaluate:

Taking into account both dividends and price changes, the return on this stock from January 2, 2003 to January 2, 2004 was 26.55%.

9. Plan: Calculate the dividend yield and capital gain for the stock over the same time period as in the previous problem.

Execute: Dividend Return from 1/2/03 ® 2/5/03

Dividend Return from 2/5 ® 5/14

Dividend Return from 5/14 ® 8/13

Dividend Return from 8/13 ® 11/12

Dividend Return from 11/12 ® 1/2

Dividend Return for the year is:

Capital Gain Return from 1/2/03 ® 2/5/03

Capital Gain Return from 2/5 ® 5/14

Capital Gain Return from 5/14 ® 8/13

Capital Gain Return from 8/13 ® 11/12

Capital Gain Return from 11/12 ® 1/2

Capital Gain Return for the year is:

Evaluate: The total return for the year should be equal to the compound of the capital gain and dividend returns for the year.

The difference is due to rounding.

The dividend yield on the stock from January 2, 2003 to January 2, 2004 was 2.1754%. The capital gain return on the stock from January 2, 2003 to January 2, 2004 was 23.938%.

10. Plan: Compute the arithmetic average return using Eq. 11.3. For part (b), to compute the geometric average, take the product of 1 + each return and then take the 10th root of that product (see the box on page 327). For parts (c) and (d), use Eq. 11.4 and Eq. 11.5 to calculate the variance and standard deviation of returns.

a. (0.05 - 0.02+0.04 + 0.08 - 0.01)/5 = 0.028

b.* (1.05)(0.98)(1.04)(1.08)(0.99) = 1.1442

d. Standard Dev (R) =

Evaluate:

The arithmetic and geometric averages are different (see the box on page 327), but not by much. The standard deviation reveals that the returns are volatile around the average.

11. The answers are different because the arithmetic average return basically assumes that you reset your investment every year. It is the best measure to use to predict the most likely annual return next year. However, it is not the best representation of how your investment actually performed. The geometric average return does that.

12. Plan: Given the data presented, make the calculations requested in the question.

Execute:

Evaluate: The average annual return is 10%. The variance of return is 0.01867. The standard deviation of returns is 13.66%.

13. Plan: Given the data in Figures 11.3 and 11.4, calculate the 95% confidence intervals for
the four securities mentioned. Use Eq. 11.6.

Execute:

Evaluate: The 95% confidence interval is two standard deviations from the right and the left of the mean. For example, for small stocks it ranges from -0.6319 to 1.0729.

14. For this, choose the investments above that have the lower limit of the 95% Confidence Interval that is above -8%. These investments are Corporate Bonds and Treasury Bills.

15. Plan: The range would be a 95% prediction interval that runs from two standard deviations below the average return to two standard deviations above the average return.

Execute: 0.1174 - 2(0.2052) to 0.1174 + 2(.2052) = - 0.2930 to + 0.5278

Evaluate:

In order to be 95% confident about your prediction, you have to have a very wide range. This is because of the substantial volatility in the returns.

16. Plan: Using a 95% prediction interval, the bottom of the prediction range is Two standard deviations below the average return. Compare the bottom of the prediction interval to the minimum return (-30%) that you are willing to tolerate.

Execute:

0.12 - 2(0.20) = 0.12 - 0.40 = - 0.28. Yes, the low end of the 95% prediction interval is -28%, which is greater than -30%.

Evaluate:

Even though the average return is 12%, the returns themselves are volatile enough that you can only just barely be 95% confident that you will not suffer a 30% loss.

17. Plan: Calculate the expected payoff of each bank’s loans. Recognize that Bank A has a portfolio of independent loans, so we would expect diversification to reduce the volatility of its loan portfolio.

Execute:

a. Expected payoff is the same for both banks:

b. Bank A:

Bank B:

One loan:

Now Bank A has 100 loans that are all independent of each other so the standard deviation of the average loan is:

But the bank has 100 such loans so the standard deviation of the portfolio is

which is much lower than Bank B.

Evaluate:

Even though the two banks have the same expected return on their loans, Bank A’s position is much safer because it has diversified its $100 million across 100 independent loans.

18. A risk-averse investor would choose the economy in which stock returns are independent because this risk can be diversified away in a large portfolio.