Explain verbally the following equation:
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 bondÔÇ™s coupon rate or (2) interest rates fall below the coupon
What is a ÔÇťdiscount bondÔÇŁ? A ÔÇťpremium bondÔÇŁ? A ÔÇťpar bondÔÇŁ?
What is ÔÇťinterest rate risk?ÔÇŁ 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
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´¬üed portfolio.
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 ´¬ünite 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
ProductsÔÇ™ 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 MercerÔÇ™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
╦ć P1r1 P2 r2 Pnrn
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 ProductsÔÇ™ U.S. WaterÔÇ™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
Expected return 15.0% 15.0%
Standard deviation 65.8% 3.9%
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´¬üned as the weighted average
of the squared deviations:
a (ri ╦ć
In our example, Mercer ProductsÔÇ™ 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
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 MercerÔÇ™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 stockÔÇ™s return, the more stand-
alone risk it has. U.S. WaterÔÇ™s returns, which were also shown in Table 28-1,
also have an expected value of 15 percent. However, U.S. WaterÔÇ™s variance
is only 0.0015, and its standard deviation is only 0.0387 or 3.87 percent.
Therefore, assuming U.S. WaterÔÇ™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´¬ücient of variation, calculated using
Equation 28-10, facilitates comparisons between returns that have different
Coefficient of variation CV . (28-10)
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
r 1 50.8% to 80.8% 11.1% to 18.9% 68%
28-12 Basic Financial Tools: A Review
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. WaterÔÇ™s standard deviation is much smaller than its expected
return, and its CV is only 0.0387/0.15 0.26. By the coef´¬ücient 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 assetÔÇ™s portfolio
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
╦ć w2 r2
a wi ri
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´¬ücient
between two assets i and j, ij, can range from 1.0 to 1.0. If the correlation
coef´¬ücient is greater than 0, the assets are said to be positively correlated,
while if the correlation coef´¬ücient 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-
site directions. For a two-asset portfolio with assets 1 and 2, the portfolio
standard deviation, p, is calculated as follows:
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´¬ücient 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´¬ücient less than 1.0, say 0.5, combining them will still be
bene´¬ücial, 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
See Chapter 3 for details on the calculation of correlations between individual assets.
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
Expected Return, ╦ć
r Standard Deviation,
Microsoft 13% 30%
General Electric 13 30
Suppose further that the correlation coef´¬ücient 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:
╦ć ╦ć ╦ć
rp wM r M wGE rGE
0.50(13%) 0.50(13%) 13.0%.
Since the new portfolioÔÇ™s expected return is the same as before, whatÔÇ™s the
point of the change? The answer, of course, is that diversi´¬ücation 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
the portfolioÔÇ™s standard deviation is found to be 25.1 percent:
w2 2 2wMwGE M GE M,GE
p MM GE GE