END-OF-CHAPTER

SOLUTIONS

FUNDAMENTALS OF INVESTMENTS B-1

Chapter 1

A Brief History of Risk and Return

Answers to Questions and Problems

Core Questions

1. No, whether you choose to sell the stock or not does not affect the gain or loss for the

year; your stock is worth what it would bring if you sold it. Whether you choose to do so

or not is irrelevant (ignoring taxes).

2. Capital gains yield = ($31 “ $42)/$42 = “26.19%

Dividend yield = $2.40/$42 = +5.71%

Total rate of return = “26.19% + 5.71% = “20.48%

3. Dollar return = 750($60 “ $42) + 750($2.40) = $15,300

Capital gains yield = ($60 “ $42)/$42 = 42.86%

Dividend yield = $2.40/$42 = 5.71%

Total rate of return = 42.86% + 5.71% = 48.57%

4. a. average return = 5.41%, average risk premium = 1.31%

b. average return = 4.10%, average risk premium = 0%

c. average return = 12.83%, average risk premium = 8.73%

d. average return = 17.21%, average risk premium = 13.11%

5. Jurassic average return = 11.4%; Stonehenge average return = 9.4%

6. A: average return = 6.20%, variance = 0.00627, standard deviation = 7.92%

B: average return = 9.40%, variance = 0.03413, standard deviation = 18.47%

7. For both risk and return, increasing order is b, c, a, d. On average, the higher the risk of

an investment, the higher is its expected return.

8. That™ plus or minus one standard deviation, so about two-thirds of time or two years out

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of three.

9. You lose money if you have a negative return. With a 6 percent expected return and a 3

percent standard deviation, a zero return is two standard deviations below the average.

The odds of being outside (above or below) two standard deviations are 5 percent; the

odds of being below are half that, or 2.5 percent. You should expect to lose money only

2.5 years out of every 100. It™ a pretty safe investment.

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B-2 CORRADO AND JORDAN

Prob( Return < “2.94 or Return > 13.76 ) ≈1/3, but we are only interested in one tail;

10.

Prob( Return < “2.94) ≈1/6.

95%: 5.41 ± 2σ = 5.41 ± 2(8.35) = “11.29% to 22.11%

99%: 5.41 ± 3σ = 5.41 ± 3(8.35) = “19.64% to 30.46%

Intermediate Questions

Expected return = 17.21% ; σ = 34.34%. Doubling your money is a 100% return, so if

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the return distribution is normal, “z” = (100“17.21)/34.31 = 2.41 standard deviations; this

is in between two and three standard deviations, so the probability is small, somewhere

between .5% and 2.5% (why?). (Referring to the nearest “z” table, the actual probability is

≈1%, or once every 100 years.) Tripling your money would be “z” =(200 “ 17.21)/ 34.31

= 5.32 standard deviations; this corresponds to a probability of (much) less than 0.5%, or

once every 200 years. (The actual answer is less than once every 1 million years; don™t

hold your breath.)

12. It is impossible to lose more than “100 percent of your investment. Therefore, return

distributions are cut off on the lower tail at “100 percent; if returns were truly normally

distributed, you could lose much more.

13. Year Common stocks T-bill return Risk premium

1980 32.6% 12.0% 20.6%

1981 “5.0 15.2 “20.2

1982 21.7 11.3 10.4

1983 22.6 8.9 13.7

1984 6.2 10.0 “ 3.8

1985 31.9 7.7 24.2

1986 18.7 6.2 12.5

128.7 71.3 57.4

a. Annual risk premium = Common stock return “ T-bill return (see table above).

b. Average returns: Common stocks = 128.7 / 7 = 18.4% ; T-bills = 71.3 / 7 =

10.2%, Risk premium = 57.4 / 7 = 8.2%

Var = 1/6[ (.326“.184)2 + (“.05“.184)2 + (.217“.184)2 +

c. Common stocks:

(.226“.184)2 + (.062“.184)2 + (.319“.184)2 + (.187“.184)2 ]

= 0.01848

Standard deviation = (0.01848)1/2 = 0.1359 = 13.59%

Var = 1/6[ (.120“.102)2 + (.152“.102)2 + (.113“.102)2 +

T-bills:

(.089“.102)2 + (.100“.102)2 + (.077“.102)2 + (.062“.102)2 ]

= 0.00089

Standard deviation = (0.00089)1/2 = 0.02984 = 2.98%

FUNDAMENTALS OF INVESTMENTS B-3

Var = 1/6[ (.206“.082)2 + (“.202“.082)2 + (.104“.082)2

Risk premium:

+ (.137“.082)2 + (“.038“.082)2 + (.242“.082)2 + (.125“

.082)2 ] = 0.02356

Standard deviation = (0.02356)1/2 = 0.1535 = 15.35%

d. Before the fact, the risk premium will be positive; investors demand compensation

over and above the risk-free return to invest their money in the risky asset. After

the fact, the observed risk premium can be negative if the asset™ nominal return is

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unexpectedly low, the risk-free return is unexpectedly high, or any combination of

these two events.

14. T-bill rates were highest in the early eighties; inflation at the time was relatively high. As

we discuss in our chapter on interest rates, rates on T-bills will almost always be slightly

higher than the rate of inflation.

15. Risk premiums are about the same whether or not we account for inflation. The reason is

that risk premiums are the difference between two returns, so inflation essentially nets out.

16. Returns, risk premiums, and volatility would all be lower than we estimated because

aftertax returns are smaller than pretax returns.

17. We™ seen that T-bills barely kept up with inflation before taxes. After taxes, investors in

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T-bills actually lost ground (assuming anything other than a very low tax rate). Thus, an all

T-bill strategy will probably lose money in real dollars for a taxable investor.

18. It™ important not to lose sight of the fact that the results we have discussed cover well

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over 70 years, well beyond the investing lifetime for most of us. There have been extended

periods during which small stocks have done terribly. Thus, one reason most investors will

chose not to pursue a 100 percent stock strategy is that many investors have relatively

short horizons, and high volatility investments may be very inappropriate in such cases.

There are other reasons, but we will defer discussion of these to later chapters.

B-4 CORRADO AND JORDAN

Chapter 2

Buying and Selling Securities

Answers to Questions and Problems

Core questions