Thus, all points within this space arefeasible investments. Note that Figures 6.2 and 6.3A depict an opportunity set that is a solid mass of investment opportunities within the scalloped quarter-moon-shaped set. This solid mass of investment opportunities is achieved by not using any negative portfolio weights.
6.3 MARKOWITZ DIVERSIFICATION
The phrase Markowitz diversification refers to its creator, Harry M. Markowitz (1952, 1959, 2010). Markowitz diversification differs from simple (or random) diversification, used widely by security salesmen and some investment books. These sources define diversification as ‘‘not putting all your eggs in one basket’’ or
‘‘spreading your risks.’’ The possible benefits of naive diversification will be discussed later. Naive diversification ignores the covariance between securities and results in superfluous diversification.
isρ12=0. Combining securities 1 and 2 with the weightsw1=2/3 andw2 =1/3 produces a portfolio with expected return of 8.3 percent.
E rp
=(2/3) (0.05)+(1/3) (0.15)=0.083 or 8.3%
σp=
(2/3)2(0.20)2+(1/3)2(0.40)2+2(2/3) (1/3) ρ12
(0.20) (0.40)
=
0.0356+0.0356 ρ12
Although the expected return of this portfolio is fixed at 8.3 percent for the investment weights assumed for securities 1 and 2, the risk of the portfolio varies with the correlation coefficient, ρ12. Thus, if ρ12= +1, then σp=26.7 percent. If ρ12=0, then σp=18.9 percent. And ifρ12= −1, thenσp=0 percent. That is, if the correlation coefficient decreases from ρ12= +1 to ρ12 =0, the risk decreases from 26.7 percent to 18.9 percent, although the assets’ expected returns remain unchanged. The locus of portfolios corresponding to all possible proportions for securities 1 and 2 (w1 and w2) is plotted in Figures 6.4A and 6.4B for three correlation values, ρ12= +1, 0, and −1, and with short sales not allowed and allowed, respectively. The equations for a relationship between E
rp
andσp when the correlation coefficients are+1 and−1, respectively, are given in an appendix at the end of this chapter.
Figure 6.4A graphically depicts how Markowitz diversification affects the risk of the portfolio when borrowing and/or short sales are not allowed.2 It shows that lowering ρ12 reduces risk when combining securities 1 and 2 into a portfolio. The straight line between securities 1 and 2 defines the locus of E
rp
and σp from all possible portfolios of securities 1 and 2 when ρ12 = +1. This straight line is the upper bound for the portfolio’s risk (standard deviation). The lower the value
15%
10%
8.3%
5%
Asset 1
Asset 2
20% 40% sp
r12 = −1
r12 = +1 r12 = 0 E (rp)
FIGURE 6.4A Markowitz Diversification without Short Sales
15%
10%
8.3%
5% Asset 1
Asset 2
20% 40%
sp r12 = −1
r12 = +1 r12 = 0 E (rp)
FIGURE 6.4B Markowitz Diversification with Borrowing or Short Sales
ofρ12, the more the curve connecting securities 1 and 2 curves or bows to the left.
When the lowest possible correlation is encountered,ρ12= −1, the portfolio’s risk is minimized. Typically,−1< ρ12<+1, and, as a result, the line is usually curved to the left, meaning the line is typically concave. This bowing or curvature property of portfolio variance is called the covariance effect or the correlation effect. The dotted line segments in Figure 6.4B indicate a negative weight for one security (that is, short sales or leverage).
The previous illustration of the diversification effect shows that the expected return of the portfolio equals the weighted average of two securities’ expected returns. But the standard deviation of the portfolio is less than or equal to the weighted average of two securities’ standard deviations. That is, whileσp2=w21σ12+ w22σ22+2w1w2ρ12σ1σ2is true, the inequality (6.8) is also true.
σp≤w1σ1+w2σ2 (6.8)
The equality holds only when the correlation coefficient between the assets equals positive one,ρ12= +1.
Figure 6.4A illustrates how Markowitz diversification can reduce the risk (variability of return) on the owner’s equity. The risk and expected return of the individual assets is not affected by their inclusion in a diversified portfolio. In modern portfolio theory the focus is ultimately on (1) the expected rate of return on the owner’s equity, and (2) the variability of return (risk) on the owner’s equity. The individual securities are merely the objects of choice that a portfolio manager uses to form into dominant portfolios. The risk and return statistics on the individual securities being considered for possible inclusion in the portfolio are exogenous constants that are estimated by a statistically oriented security analyst. In contrast, the security weights are thedecision variablesthat are determined endogenously by the portfolio manager (or the portfolio manager’s computer program).
Most investors realize that diversification reduces risk. However, few people have ever thought about risk analytically. If an investor’s concept of risk is not sufficiently well defined to be measured empirically, it follows logically that the investor must be working with only a vague concept of how to diversify. Furthermore, some investors erroneously develop perverted notions about diversification. For instance, some people erroneously believe that low-risk (for example, public utility) stocks and (say, government) bonds, both of which may have low returns, are the best ways to reduce a portfolio’s risk.
Risk can be reduced by means of Markowitz diversification without decreas- ing return at all—in fact, such diversification can be optimal. Simple (random) diversification can usually reduce risk. However, simple diversification cannot be expected to minimize risk because it ignores the correlation (or covariance) between assets. Simple diversification concentrates on owning different assets; that is, not putting all your eggs in one basket. In contrast, Markowitz diversification provides an analytical approach to diversification.
6.4 EFFICIENT FRONTIER WITHOUT THE RISK-FREE ASSET
Figure 6.5 contains an opportunity set derived from many individual securities. The upper left quadrant between pointsEandFin Figure 6.5 is theefficient frontierof the opportunity set. It is comprised of efficient portfolios.