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Chapter 8 Portfolio Theory and the Capital Asset Pricing Model 197By holding different proportions of the 10 securities, you can obtain an even wider selec-
tion of risk and return, but which combination is best? Well, what is your goal? Which
direction do you want to go? The answer should be obvious: you want to go up (to increase
expected return) and to the left (to reduce risk). Go as far as you can, and you will end up with
one of the portfolios that lies along the red line. Markowitz called them efficient portfolios.
They offer the highest expected return for any level of risk.
We will not calculate the entire set of efficient portfolios here, but you may be interested
in how to do it. Think back to the capital rationing problem in Section 5-4. There we wanted
to deploy a limited amount of capital investment in a mixture of projects to give the highest
NPV. Here we want to deploy an investor’s funds to give the highest expected return for a
given standard deviation. In principle, both problems can be solved by hunting and pecking—
but only in principle. To solve the capital rationing problem, we can employ linear program-
ming; to solve the portfolio problem, we would turn to a variant of linear programming known
as quadratic programming. Given the expected return and standard deviation for each stock,
as well as the correlation between each pair of stocks, we could use a standard quadratic com-
puter program to calculate the set of efficient portfolios.
Three of these efficient portfolios are marked in Figure 8.4. Their compositions are sum-
marized in Table 8.1. Portfolio C offers the highest expected return: It is invested entirely in
one stock, Ford. Portfolio A offers the minimum risk; you can see from Table 8.1 that it has
large holdings in Consolidated Edison, Walmart, and Campbell Soup, which have the low-
est standard deviations. However, the portfolio also has a sizeable holding in Newmont even
though it is individually risky. The reason? On past evidence the fortunes of gold-mining
shares, such as Newmont, are almost uncorrelated with those of other stocks and so provide
additional diversification.
Table 8.1 also shows the compositions of a third efficient portfolio with intermediate levels
of risk and expected return.
Of course, large investment funds can choose from thousands of stocks and thereby achieve
a wider choice of risk and return. This choice is represented in Figure 8.5 by the shaded,
broken-egg-shaped area. The set of efficient portfolios is again marked by the red curved line.We Introduce Borrowing and Lending
Now we introduce yet another possibility. Suppose that you can also lend or borrow money at
some risk-free rate of interest rf. If you invest some of your money in Treasury bills (i.e., lend
money) and place the remainder in common stock portfolio S, you can obtain any combina-
tion of expected return and risk along the straight line joining rf and S in Figure 8.5. SinceBEYOND THE PAGEmhhe.com/brealey12ePortfolio selection
in practice◗ FIGURE 8.5
Lending and borrowing extend the
range of investment possibilities. If
you invest in portfolio S and lend or
borrow at the risk-free interest rate,
rf, you can achieve any point along
the straight line from rf through S.
This gives you the highest expected
return for any level of risk. There is
no point in investing in a portfolio
like T.Expected return (rfr^ ), %S BorrowingLendingTStandard deviation (σ)