## M

Figure 5.6 Various possible relationships for expected return and standard deviation when the minimum variance portfolio and Colonel Motors are combined.

9If the correlation between U and V equals +1, they will be on the straight line. If it is less than +1, the risk must be less, so combinations must be above the straight line.

Figure 5.7 Various possible relationships between expected return and standard deviation of return when the minimum variance portfolio is combined with portfolio S.

### The Efficient Frontier with No Short Sales

In theory we could plot all conceivable risky assets and combinations of risky assets in a diagram in return standard deviation space. We used the words "in theory," not because there is a problem in calculating the risk and return on a stock or portfolio, but because there are an infinite number of possibilities that must be considered. Not only must all possible groupings of risky assets be considered, but all groupings must be considered in all possible percentage compositions.

If we were to plot all possibilities in risk-return space, we would get a diagram like Figure 5.8. We have taken the liberty of representing combinations as a finite number of points in constructing the diagram. Let us examine the diagram and see if we can eliminate any part of it from consideration by the investor. In Chapter 4 we reasoned that an investor would prefer more return to less and would prefer less risk to more. Thus, if we could find a set of portfolios that

1. offered a bigger return for the same risk, or

2. offered a lower risk for the same return, we would have identified all portfolios an investor could consider holding. All other portfolios could be ignored.

Let us take a look at Figure 5.8. Examine portfolios A and B. Note that portfolio B would be preferred by all investors to portfolio A because it offers a higher return with the same

Risk

Figure 5.8 Risk and return possibilities for various assets and portfolios.

level of risk. We can also see that portfolio C would be preferable to portfolio A because it offers less risk at the same level of return. Notice that at this point in our analysis we can find no portfolio that dominates portfolio C or portfolio B. It should be obvious at this point that an efficient set of portfolios cannot include interior portfolios. We can reduce the possibility set even further. For any point in risk-return space we want to move as far as possible in the direction of increasing return and as far as possible in the direction of decreasing risk. Examine the point D, which is an exterior point. We can eliminate D from further consideration since portfolio E exists, which has more return for the same risk. This is true for every other portfolio as we move up the outer shell from D to point C. Point C cannot be eliminated since there is no portfolio that has less risk for the same return or more return for the same risk. But what is point C? It is the global minimum variance portfolio.10 Now examine point F. Point F is on the outer shell, but point E has less risk for the same return. As we move up the outer shell curve from point F, all portfolios are dominated until we come to portfolio B. Portfolio B cannot be eliminated for there is no portfolio that has the same return and less risk or the same risk and more return than point B. Point B represents that portfolio (usually a single security) that offers the highest expected return of all portfolios. Thus the efficient set consists of the envelope curve of all portfolios that lie between the global minimum variance portfolio and the maximum return portfolio. This set of portfolios is called the efficient frontier.

Figure 5.9 represents a graph of the efficient frontier. Notice that we have drawn the efficient frontier as a concave function. The proof that it must be concave follows logically from the earlier analysis of the combination of two securities or portfolios. The efficient l0The global minimum variance portfolio is that portfolio that has the lowest risk of any feasible portfolio.

frontier cannot contain a convex region such as that shown in Figure 5.10 since as argued jarlier U and V are portfolios and combinations of two portfolios must be concave.11

Figure 5.9 The efficient frontier.

Figure 5.10 An impossible shape for the efficient frontier.

1 'Furthermore, there can be linear segments if the two efficient portfolios are perfectly correlated. Since a linear relationship is both concave and convex, we can still refer to the efficient frontier as concave.

Up to this point we have seen that the efficient frontier is a concave function in expected return standard deviation space that extends from the minimum variance portfolio to the maximum return portfolio. The portfolio problem, then, is to find all portfolios along this frontier. The computational procedures necessary to do so will be examined in Chapter 6.

### The Efficient Frontier with Short Sales Allowed

In the stock market (and many other capital markets), an investor can often sell a security that he or she does not own. This process is called short selling and is described in Chapter 3; however, the mechanics of short sales are worth repeating here. It involves in essence taking a negative position in a security. Short sales exist in sizable amounts on the New York Stock Exchange (as well as other securities markets) and the amount of short sales in New York Stock Exchange stocks is reported in the New York Times every Monday. In a moment we will discuss the incorporation of short sales into our analysis. Before we do so, however, it is worthwhile pointing out that we have not been wasting our time by studying the case where short sales are disallowed. There are two reasons why this is true. The first is that most institutional investors do not short sell. Many institutions are forbidden by law from short selling, whereas still others operate under a self-imposed constraint forbidding short sales. The second is that the incorporation of short sales into our analysis involves only a minor extension of the analysis we have developed up to this point.

In this section we will employ a simplified description of the way short sales work. This has been the general description of short sales in the literature, but in footnotes and in Chapter 6 we present both the deficiencies of this description and an alternative, more realistic description of short sales. Our description of short sales, which treats short sales as the ability to sell a security without owning it, assumes that there are no special transaction costs involved in this process. Let us see how this process might work.

Let us assume an investor believed that the stock of ABC company, which currently sells for \$100 per share, is likely to be selling for \$95 per share (expected value) at the end of the year. In addition, the investor expects ABC company to pay a \$3.00 dividend at the end of the year. If the investor bought one share of ABC stock, the cash flow would be -\$100.00 at time zero when the stock is purchased and +\$3.00 from the dividend, plus +\$95.00 from selling the stock at time 1. The cash flows are

Time