rˆpw 1 rˆ 1 w 2 rˆ 2 w 3 ˆr 3 w 4 rˆ 4
0.25(12%) 0.25(11.5%) 0.25(10%) 0.25(9.5%)
10.75%.
Of course, after the fact and a year later, the actual realized rates of return, ,on the
individual stocks—the i, or “r-bar,” values—will almost certainly be different from
their expected values, so pwill be different fromˆrp10.75%. For example, Coca-
Cola might double and provide a return of 100%, whereas Microsoft might have a
terrible year, fall sharply, and have a return of 75%. Note, though, that those two
events would be somewhat offsetting, so the portfolio’s return might still be close to its
expected return, even though the individual stocks’ actual returns were far from their
expected returns.Portfolio Risk
As we just saw, the expected return on a portfolio is simply the weighted average of the
expected returns on the individual assets in the portfolio. However, unlike returns, the
risk of a portfolio, p, is generally notthe weighted average of the standard deviations
of the individual assets in the portfolio; the portfolio’s risk will almost always be smaller
than the weighted average of the assets’ ’s. In fact, it is theoretically possible to com-
bine stocks that are individually quite risky as measured by their standard deviations to
form a portfolio that is completely riskless, with p0.
To illustrate the effect of combining assets, consider the situation in Figure 3-5.
The bottom section gives data on rates of return for Stocks W and M individually, and
also for a portfolio invested 50 percent in each stock. The three top graphs show plots
of the data in a time series format, and the lower graphs show the probability distri-
butions of returns, assuming that the future is expected to be like the past. The two
stocks would be quite risky if they were held in isolation, but when they are combined
to form Portfolio WM, they are not risky at all. (Note: These stocks are called W and
M because the graphs of their returns in Figure 3-5 resemble a W and an M.)
The reason Stocks W and M can be combined to form a riskless portfolio is that
their returns move countercyclically to each other—when W’s returns fall, those
of M rise, and vice versa. The tendency of two variables to move together is called
correlation,and the correlation coefficient measures this tendency.^6 The symbol
for the correlation coefficient is the Greek letter rho, (pronounced roe). In statistical
terms, we say that the returns on Stocks W and M are perfectly negatively correlated,
with 1.0.
The opposite of perfect negative correlation, with 1.0, is perfect positive corre-
lation,with 1.0. Returns on two perfectly positively correlated stocks (M andrrrRisk in a Portfolio Context 115(^6) The correlation coefficient, , can range from 1.0, denoting that the two variables move up and down in
perfect synchronization, to 1.0, denoting that the variables always move in exactly opposite directions. A
correlation coefficient of zero indicates that the two variables are not related to each other—that is, changes
in one variable are independentof changes in the other.
The correlation is called R when it is estimated using historical data. Here is the formula to estimate the
correlation between stocks i and j ( is the actual return for stock i in period t and is the average re-
turn during the period; similar notation is used for stock j):
Fortunately, it is easy to calculate correlation coefficients with a financial calculator. Simply enter the re-
turns on the two stocks and then press a key labeled “r.” In Excel, use the CORRELfunction.
R
a
n
t 1
( ri,trAvgi) (rj,trAvgj)
Ba
n
t 1
( ri,trAvgi)^2 a
n
t 1
(rj,trAvgj)^2
.
ri,t rAvgi