Chapter 8 Risk and Rates of Return 243Stocks W and M can be combined to form a riskless portfolio because their re-
turns move countercyclically to each other—when W’s fall, M’s rise, and vice
versa. The tendency of two variables to move together is called correlation, and
the correlation coef! cient, % (pronounced “rho”), measures this tendency.^16 In sta-
tistical terms, we say that the returns on Stocks W and M are perfectly negatively
correlated, with % $ !1.0. The opposite of perfect negative correlation is perfect posi-
tive correlation, with % $ "1.0. If returns are not related to one another at all, they
are said to be independent and % $ 0.
The returns on two perfectly positively correlated stocks with the same ex-
pected return would move up and down together, and a portfolio consisting of
these stocks would be exactly as risky as the individual stocks. If we drew a graph
like Figure 8-4, we would see just one line because the two stocks and the portfolio
would have the same return at each point in time. Thus, diversi! cation is completely
useless for reducing risk if the stocks in the portfolio are perfectly positively correlated.
We see then that when stocks are perfectly negatively correlated (% $ !1.0), all
risk can be diversi" ed away; but when stocks are perfectly positively correlated
(% $ "1.0), diversi" cation does no good. In reality, most stocks are positively cor-
related but not perfectly so. Past studies have estimated that on average, the corre-
lation coef" cient between the returns of two randomly selected stocks is about
0.30.^17 Under this condition, combining stocks into portfolios reduces risk but does not
completely eliminate it.^18 Figure 8-5 illustrates this point using two stocks whose cor-
relation coef" cient is % $ "0.35. The portfolio’s average return is 15%, which is the
same as the average return for the two stocks; but its standard deviation is 18.62%,
which is below the stocks’ standard deviations and their average #. Again, a ratio-
nal, risk-averse investor would be better off holding the portfolio rather than just
one of the individual stocks.
In our examples, we considered portfolios with only two stocks. What would
happen if we increased the number of stocks in the portfolio?
As a rule, portfolio risk declines as the number of stocks in a portfolio increases.
If we added enough partially correlated stocks, could we completely eliminate
risk? In general, the answer is no. For an illustration, see Figure 8-6 on page 246,
which shows that a portfolio’s risk declines as stocks are added. Here are some
points to keep in mind about the " gure:
- The portfolio’s risk declines as stocks are added, but at a decreasing rate; and
once 40 to 50 stocks are in the portfolio, additional stocks do little to reduce risk. - The portfolio’s total risk can be divided into two parts, diversi! able risk and
market risk.^19 Diversi" able risk is the risk that is eliminated by adding stocks.
Correlation
The tendency of two
variables to move
together.
Correlation
Coefficient, %
A measure of the degree of
relationship between two
variables.Correlation
The tendency of two
variables to move
together.
Correlation
Coefficient, %
A measure of the degree of
relationship between two
variables.Diversifiable Risk
That part of a security’s
risk associated with
random events; it can be
eliminated by proper
diversification. This risk is
also known as company-
specific, or unsystematic,
risk.
Market Risk
The risk that remains
in a portfolio after
diversification has
eliminated all company-
specific risk. This risk is
also known as non-
diversifiable or systematic
or beta risk.Diversifiable Risk
That part of a security’s
risk associated with
random events; it can be
eliminated by proper
diversification. This risk is
also known as company-
specific, or unsystematic,
risk.
Market Risk
The risk that remains
in a portfolio after
diversification has
eliminated all company-
specific risk. This risk is
also known as non-
diversifiable or systematic
or beta risk.(^16) The correlation coe% cient, ρ, can range from +1.0, denoting that the two variables move up and down in per-
fect synchronization, to !1.0, denoting that the variables move in exactly opposite directions. A correlation coef-
! cient of zero indicates that the two variables are not related to each other—that is, changes in one variable are
independent of changes in the other. It is easy to calculate correlation coe% cients with a! nancial calculator. Sim-
ply enter the returns on the two stocks and press a key labeled “r.” For W and M, ρ $ !1.0. See our tutorial on the
text’s web site or your calculator manual for the exact steps. Also note that the correlation coe% cient is often de-
noted by the term r. We use ρ here to avoid confusion with r used to denote the rate of return.
(^17) A study by Chan, Karceski, and Lakonishok (1999) estimated that the average correlation coe% cient between
two randomly selected stocks was 0.28, while the average correlation coe% cient between two large-company
stocks was 0.33. The time period of their sample was 1968 to 1998. See Louis K. C. Chan, Jason Karceski, and Josef
Lakonishok, “On Portfolio Optimization: Forecasting Covariance and Choosing the Risk Model,” The Review of
Financial Studies, Vol. 12, no. 5 (Winter 1999), pp. 937–974.
(^18) If we combined a large number of stocks with ρ = 0, we could form a riskless portfolio. However, there are not
many stocks with ρ = 0—stocks’ returns tend to move together, not to be independent of one another.
(^19) Diversi! able risk is also known as company-speci" c, or unsystematic, risk. Market risk is also known as non-
diversi" able or systematic or beta risk; it is the risk that remains in the portfolio after diversi! cation has eliminated
all company-speci! c risk.