Explain verbally the following equation:

Self-Test Questions

N

Coupon Par

a

P .

rd)t (1 rd)N

t 1 (1

Explain what happens to the price of a fixed-rate bond if (1) interest rates rise

above the bond™s coupon rate or (2) interest rates fall below the coupon

rate.

What is a “discount bond”? A “premium bond”? A “par bond”?

What is “interest rate risk?” What two characteristics of a bond affect its inter-

est rate risk?

If interest rate risk is defined as the percentage change in the price of a bond

given a 10% change in the going rate of interest (e.g., from 8 percent to 8.8

percent), which of the following bonds would have the most interest rate

risk? Assume that the initial market interest rate for each bond is 8 percent,

and assume that the yield curve is horizontal.

(1 ) A 30-year, 8 percent coupon, annual payment T-bond.

(2) A 10-year, 6 percent coupon, annual payment T-bond.

(3) A 10-year, zero coupon, T-bond.

28-10 Basic Financial Tools: A Review

Chapter 28

Risk and Return

Risk is the possibility that an outcome will be different from what is ex-

pected. For an investment, risk is the possibility that the actual return (dol-

lars or percent) will be less than the expected return. We will consider two

types of risk for assets: stand-alone risk and portfolio risk. Stand-alone risk

is the risk an investor would bear if he or she held only a single asset. Port-

folio risk is the risk that an asset contributes to a well-diversi¬ed portfolio.

Statistical Measures

To quantify risk, we must enumerate the various events that can happen and

the probabilities of those events. We will use discrete probabilities in our cal-

culations, which means we assume a ¬nite number of possible events and

probabilities. The list of possible events, and their probabilities, is called a

probability distribution. Each probability must be between 0 and 1, and the

sum must equal 1.0. For example, suppose the long-run demand for Mercer

Products™ output could be strong, normal, or weak, and the probabilities of

these events are 30 percent, 40 percent, and 30 percent, respectively.

Suppose further that the rate of return on Mercer™s stock depends on

demand as shown in Table 28-1, which also provides data on another com-

pany, U.S. Water. We explain the table in the following sections.

Expected Return The expected rate of return on a stock with possible

returns ri and probabilities Pi is found for Mercer with this equation:

Expected rate of return r

ˆ P1r1 P2 r2 Pnrn

n

(28-8)

a Piri

i 1

0.30(100%) 0.40(15%) 0.30( 70%) 15%.

Note that only if demand is normal will the actual 15 percent return equal

the expected return. If demand is strong, the actual return will exceed the

expected return by 100% 15% 85% 0.85, so the deviation from the

mean is 0.85 or 85 percent. If demand is weak, the actual return will be

less than the expected return by 15% ( 70%) 85% 0.85, so the devi-

ation from the mean is 0.85 or 85 percent. Intuitively, larger deviations

Table 28-1 Probability Distribution for Mercer Products and U.S. Water

Mercer Products™ U.S. Water™s

Demand for Products Probability Stock Return Stock Return

Strong 0.30 100.0% 20.0%

Normal 0.40 15.0 15.0

Weak 0.30 (70.0) 10.0

1.00

Expected return 15.0% 15.0%

Standard deviation 65.8% 3.9%

28-11

Risk and Return

signify higher risk. Notice that the deviations for U.S. Water are considerably

smaller, indicating a much less risky stock.

Variance Variance measures the extent to which the actual return is likely

to deviate from the expected value, and it is de¬ned as the weighted average

of the squared deviations:

n

2

r )2Pi.

a (ri ˆ

Variance (28-9)

i 1

In our example, Mercer Products™ variance, using decimals rather than per-

centages, is 0.30(0.85)2 0.40(0.0)2 0.30( 0.85)2 0.4335. This means

that the weighted average of the squared differences between the actual and

expected returns is 0.4335 43.35 percent.

The variance is not easy to interpret. However, the standard deviation,

or , which is the square root of the variance and which measures how far

the actual future return is likely to deviate from the expected return, can be

20.4335 0.658

interpreted easily. Therefore, is often used as a measure of risk. In gen-

eral, if returns are normally distributed, then we can expect the actual

return to be within one standard deviation of the mean about 68 percent

of the time.

For Mercer Products, the standard deviation is

65.8%. Assuming that Mercer™s returns are normally distributed, there is

about a 68 percent probability that the actual future return will be between

15% 65.8% 50.8% and 15% 65.8% 80.8%. Of course, this

also means that there is a 32 percent probability that the actual return will

be either less than 50.8 percent or greater than 80.8 percent.

The higher the standard deviation of a stock™s return, the more stand-

alone risk it has. U.S. Water™s returns, which were also shown in Table 28-1,

also have an expected value of 15 percent. However, U.S. Water™s variance

is only 0.0015, and its standard deviation is only 0.0387 or 3.87 percent.

Therefore, assuming U.S. Water™s returns are normally distributed, then

there is a 68 percent probability that its actual return will be in the range of

11.13 percent to 18.87 percent. The returns data in Table 28-2 clearly indi-

cate that Mercer Products is much riskier than U.S. Water.

Coefficient of Variation The coef¬cient of variation, calculated using

Equation 28-10, facilitates comparisons between returns that have different

expected values:

Coefficient of variation CV . (28-10)

ˆ

r

Dividing the standard deviation by the expected return gives the standard

deviation as a percentage of the expected return. Therefore, the CV measures

Return Ranges for Mercer Products and U.S. Water

Table 28-2 if Returns Are Normal

Mercer U.S. Water Probability of

Return Range Returns Returns Return in Range

ˆ

r 1 50.8% to 80.8% 11.1% to 18.9% 68%

28-12 Basic Financial Tools: A Review

Chapter 28

the amount of risk per unit of expected return. For Mercer, the standard

deviation is over four times its expected return, and its CV is 0.658/0.15

4.39. U.S. Water™s standard deviation is much smaller than its expected

return, and its CV is only 0.0387/0.15 0.26. By the coef¬cient of variation

criterion, Mercer is 17 times riskier than U.S. Water.

Portfolio Risk and Return

Most investors do not keep all of their money invested in just one asset;

instead, they hold collections of assets called portfolios. The fraction of the

total portfolio invested in an individual asset is called the asset™s portfolio

ˆ

weight, wi. The expected return on a portfolio, rp, is the weighted average of

the expected returns on the individual assets:

Expected return on a portfolio rp

ˆ w1r1

ˆ w2 r2

ˆ wnrn

ˆ

n

(28-11)

a wi ri

ˆ

i 1

ˆ

Here the ri values are the expected returns on the individual assets.

The variance and standard deviation of a portfolio depend not only on the

variances and weights of the individual assets in the portfolio, but also on

the correlation between the individual assets. The correlation coef¬cient

between two assets i and j, ij, can range from 1.0 to 1.0. If the correlation

coef¬cient is greater than 0, the assets are said to be positively correlated,

while if the correlation coef¬cient is negative, they are negatively corre-

lated.10 Returns on positively correlated assets tend to move up and down

together, while returns on negatively correlated assets tend to move in oppo-

2w2

site directions. For a two-asset portfolio with assets 1 and 2, the portfolio

standard deviation, p, is calculated as follows:

2

w2 2 (28-12)

2w1w2 1 2 1,2.

p 1 1 2 2

Here w2 (1 w1), and 1,2 is the correlation coef¬cient between assets

1 and 2. Notice that if w1 and w2 are both positive, as they must be unless

one asset is sold short, then the lower the value of 1,2, the lower the value

of p. This is an important concept: Combining assets that have low corre-

lations results in a portfolio with a low risk. For example, suppose the cor-

relation between two assets is negative, so when the return on one asset

falls, then that on the other asset will generally rise. The positive and nega-

tive returns will tend to cancel each other out, leaving the portfolio with

very little risk. Even if the assets are not negatively correlated, but have a

correlation coef¬cient less than 1.0, say 0.5, combining them will still be

bene¬cial, because when the return on one asset falls dramatically, that on

the other asset will probably not fall as much, and it might even rise. Thus,

the returns will tend to balance each other out, lowering the total risk of the

portfolio.

10

See Chapter 3 for details on the calculation of correlations between individual assets.

28-13

Risk and Return

To illustrate, suppose that in August 2003, an analyst estimates the fol-

lowing identical expected returns and standard deviations for Microsoft and

General Electric:

Expected Return, ˆ

r Standard Deviation,

Microsoft 13% 30%

General Electric 13 30

Suppose further that the correlation coef¬cient between Microsoft and GE

is M,GE 0.4. Now if you have $100,000 invested in Microsoft, you will

have a one-asset portfolio with an expected return of 13 percent and a stan-

dard deviation of 30 percent. Next, suppose you sell half of your Microsoft

and buy GE, forming a two-asset portfolio with $50,000 in Microsoft and

$50,000 in GE. (Ignore brokerage costs and taxes.) The expected return on

this new portfolio will be the same 13.0 percent:

ˆ ˆ ˆ

rp wM r M wGE rGE

0.50(13%) 0.50(13%) 13.0%.

Since the new portfolio™s expected return is the same as before, what™s the

2w2 2

point of the change? The answer, of course, is that diversi¬cation reduces

2(0.5)2 (0.3)2 (0.5)2 (0.3)2 2(0.5)(0.5)(0.3)(0.3)(0.4)

risk. As noted above, the correlation between the two companies is 0.4, so

20.0630

the portfolio™s standard deviation is found to be 25.1 percent:

w2 2 2wMwGE M GE M,GE

p MM GE GE