PRINCIPLES BOX: DEFINITION OF AN EFFICIENT INVESTMENT
3.4 DIVERSIFICATION
Markowitz diversification is the particular form of diversification activity implied by portfolio analysis. This type of diversification differs fromnaive diversification, used widely by security salespeople and even in some investment publications.
These sources define diversification as ‘‘not putting all your eggs in one basket’’
or ‘‘spreading your risks.’’ Naive diversification ignores the covariance between securities and results in superfluous diversification.
Markowitz diversificationinvolves combining assets with less-than-perfect pos- itive correlations in order to reduce risk in the portfolio without sacrificing any of the portfolio’s return. In general, the lower the correlation between the assets in a portfolio, the less risky the portfolio will be. This is true regardless of how risky the assets of the portfolio are when analyzed individually. Markowitz explains his approach to diversification as follows:8
Not only does [portfolio analysis] imply diversification, it implies the
‘‘right kind’’ of diversification for the ‘‘right reason.’’ The adequacy of diversification is not thought by investors to depend on the number of different securities held. A portfolio with sixty different railway securities, for example, would not be as well diversified as the same size portfolio with some railroad, some public utility, mining, various sorts of manufacturing, etc. The reason is that it is generally more likely for firms within the same industry to do poorly at the same time than for firms in dissimilar industries.
Similarly, in trying to make variance [of returns] small it is not enough to invest in many securities. It is necessary to avoid investing in securities with high covariances [or correlations] among themselves.
3.4.1 Diversification Illustrated
Consider the characteristics of the two securities in Table 3.2.
Reconsider equation (2.12) from Chapter 2. If Securities A and B are combined in the proportions wA=2/3 and wB=1/3, the expected return of the resulting portfolio is 8.3 percent.
E rp
=wAE rA
+wAE rA
(2.12)
=(2/3)0.05+(1/3)0.15=0.083=8.3 percent
TABLE 3.2 Two Investment Assets
Investments: E(r) σ
A 5% 20%
B 15% 40%
s E(r)
rAB = −1.0
rAB = 0 20%
15%
10%
8.3%
5%
20% 40%
B
A
0 60%
rAB = +1.0 p
FIGURE 3.10 The Effects of Markowitz Diversification
The risk of the portfolio is given by equation (2.14) from Chapter 2.
σp2=w2AσA2+w2BσB2+2wAwB[σAB] (2.14)
=(2/3)2(0.2)2+(1/3)2(0.4)2+2(2/3) (1/3)[ρAB(0.2) (0.4)]
=0.0356+0.0356×ρAB
σp=
0.0356+0.0356×ρAB
Although the expected return of this portfolio is fixed at 8.3 percent with these proportions of A and B regardless of the correlation coefficient,ρAB, the risk of the portfolio varies with ρAB. Thus, if ρAB= +1, then σp=√
0.0712= 26.7 percent.
IfρAB=0, thenσp=√
0.0356=18.7 percent. And, if the correlation is−1, then σp=0. The locus of all possible proportions (wAandwB) for investments A and B are plotted in Figure 3.10 for three different values ofρAB:+1, 0, and−1.
3.4.2 Risky A + Risky B = Riskless Portfolio
Consider two risky securities named A and B. Table 3.3 lays out annual rates of return from the two assets over four consecutive years, as well as the returns from portfoliop, which is composed half and half of A and B. The single-period returns from portfolio p for each year are calculated with 50–50 weights on A and B, according to the following formula:
rpt=0.5rAt+0.5rBt for t=1, 2, 3, 4 years
TABLE 3.3 Riskless Portfoliopis Constructed from Risky Assets A and B
t=1 t=2 t=3 t=4 Variances Asset A (rAt) 10.6% 8.3% 5.1% 14.2% σ2A>0 Asset B (rBt) 6.6% 8.3% 11.5% 2.4% σ2B>0 Portfoliop(rpt) 8.3% 8.3% 8.3% 8.3% σ2p=0
wheretis a time period counter in Table 3.3. A glance at Table 3.3 reveals that assets A and B experience considerable variability of return from year to year—thus, they are risky assets. In contrast, the diversified portfoliopexperiences zero variance of returns, which conforms with our definition of ariskless asset. The question is: How can the portfoliopbe riskless when it is constructed from the two risky assets? The key to understanding this paradoxical question lies in understanding the correlation between the two assets’ returns.
Assets A and B have perfectly negatively correlated returns, ρAB= −1. Their prices always move inversely. Any gains on A are always exactly offset by equal losses from B, and vice versa. As a result, portfoliopexperiences zero variability of returns. In short, a riskless portfolio can be constructed from two risky assets whose prices are perfectly inversely correlated. A portfolio likepis illustrated at pointpin Figure 3.10.
3.4.3 Graphical Analysis
Figure 3.10 illustrates how Markowitz diversification determines the risk of a 2-asset portfolio.9Figure 3.10 shows that the lower the correlation coefficient ρAB
, the more risk is reduced by combining A and B into a portfolio. The straight line between A and B defines the locus of E(r) and σ pairs for all possible portfolios of A and B when ρAB= +1. Considering the effects of diversification depicted in Figure 3.10, the reader is invited to reexamine Figure 3.9. The curves convex toward theE(r) axis in Figure 3.10 suggest the opportunity set in Figure 3.9 results from risk-reducing diversification.
Figure 3.10 also illustrates graphically that Markowitz diversification reduces the risk (variability of return) only on the owner’s equity. The risk and expected return of the individual assets, A and B, are not affected by the formation of a diversified portfolio in which A and B happen to be a part. Throughout this portfolio analysis monograph the focus is ultimately on (1) the expected rate of return on the owner’s equity, and (2) the variability of return (the risk) of the owner’s equity. The individual assets are mere objects of choice that portfolio analysis endeavors to form into investment portfolios that dominate the individual assets. The risk and return statistics on the individual assets being considered for possible inclusion in the portfolio areexogenous constantsthat arestatistical inputs for portfolio analysis.