## Finding Funds That Zig When The Blue Chips

Investors hungry for lower risk are hearing some surprising recommendations from financial advisers:

• mutual funds investing in less-developed nations that many Americans can't immediately locate on a globe.

• funds specializing in small European companies with unfamiliar names.

• funds investing in commodities.

All of these investments are risky by themselves, advisers readily admit. But they also tend to zig when big U.S. stocks zag. And that means that such fare, when added to a portfolio heavy in U.S. blue-chip stocks, actually may damp the portfolio's ups and downs.

Combining types of investments that don't move in lock step "is one of the very few instances in which there is a free lunch—you get something for nothing," says Gary Greenbaum, president of investment counselors Greenbaum & Associates in Oradell, N.J. The right combination of assets can trim the volatility of an investment portfolio, he explains, without reducing the expected return over time.

Getting more variety in one's holdings can be surprisingly tricky. For instance, investors who have shifted dollars into a diversified international-stock fund may not have ventured as far afield as they think, says an article in the most recent issue of Morningstar Mutual Funds. Those funds typically load up on European blue-chip stocks that often behave similarly and respond to the same world-wide economic conditions as do U.S. corporate giants. . . .

Many investment professionals use a statistical measure known as a "correlation coefficient" to identify categories of securities that tend to zig when others zag. A figure approaching the maximum 1.0 indicates that two assets have consistently moved in the same direction. A correlation coefficient approaching the minimum, negative 1.0, indicates that the assets have consistently moved in the opposite direction. Assets with a zero correlation have moved independently.

Funds invested in Japan, developing nations, small European companies, and gold stocks have been among those moving opposite to the Vanguard Index 500 over the past several years.

Source: Karen Damato, "Finding Funds That Zig When Blue Chips Zag," The Wall Street Journal, June 17, 1997. Excerpted by permission of The Wall Street Journal, © 1997 Dow Jones & Company, Inc. All Rights Reserved Worldwide.

8.1 shows that expected return is unaffected by correlation between returns. Therefore, other things equal, we will always prefer to add to our portfolios assets with low or, even better, negative correlation with our existing position. The nearby box from The Wall Street Journal makes this point when it advises you to find "funds that zig when blue chip [stocks] zag."

Because the portfolio's expected return is the weighted average of its component expected returns, whereas its standard deviation is less than the weighted average of the component standard deviations, portfolios of less than perfectly correlated assets always offer better risk-return opportunities than the individual component securities on their own. The lower the correlation between the assets, the greater the gain in efficiency.

How low can portfolio standard deviation be? The lowest possible value of the correlation coefficient is -1, representing perfect negative correlation. In this case, equation 8.5 simplifies to vp = (Wd - WE and the portfolio standard deviation is vp = Absolute value (wD vD - wEvE)

When p = -1, a perfectly hedged position can be obtained by choosing the portfolio proportions to solve

The solution to this equation is

CHAPTER 8 Optimal Risky Portfolios

Table 8.3 Expected Return and Standard Deviation with Various Correlation Coefficients

CHAPTER 8 Optimal Risky Portfolios

Table 8.3 Expected Return and Standard Deviation with Various Correlation Coefficients

Portfolio Standard Deviation for Given Correlation | ||||||

wD |
wE |
E(rp) |
P= -1 |
P = 0 |
p = .30 |
P = 1 |

0.00 |
1.00 |
13.00 |
20.00 |
20.00 |
20.00 |
20.00 |

0.10 |
0.90 |
12.50 |
16.80 |
18.04 |
18.40 |
19.20 |

0.20 |
0.80 |
12.00 |
13.60 |
16.18 |
16.88 |
18.40 |

0.30 |
0.70 |
11.50 |
10.40 |
14.46 |
15.47 |
17.60 |

0.40 |
0.60 |
11.00 |
7.20 |
12.92 |
14.20 |
16.80 |

0.50 |
0.50 |
10.50 |
4.00 |
11.66 |
13.11 |
16.00 |

0.60 |
0.40 |
10.00 |
0.80 |
10.76 |
12.26 |
15.20 |

0.70 |
0.30 |
9.50 |
2.40 |
10.32 |
11.70 |
14.40 |

0.80 |
0.20 |
9.00 |
5.60 |
10.40 |
11.45 |
13.60 |

0.90 |
0.10 |
8.50 |
8.80 |
10.98 |
11.56 |
12.80 |

1.00 |
0.00 |
8.00 |
12.00 |
12.00 |
12.00 |
12.00 |

Minimum Variance Portfolio | ||||||

Wd |
0.6250 |
0.7353 |
0.8200 |
— | ||

wE |
0.3750 |
0.2647 |
0.1800 |
— | ||

E(p) |
9.8750 |
9.3235 |
8.9000 |
— | ||

ap |
0.0000 |
10.2899 |
11.4473 |
— |

These weights drive the standard deviation of the portfolio to zero.3

Let us apply this analysis to the data of the bond and stock funds as presented in Table 8.1. Using these data, the formulas for the expected return, variance, and standard deviation of the portfolio are

E(rp) = 8wd + 13 we u2p = 122wD + 202wE + 2 X 12 X 20 X .3 X wDwE = 144wD + 400wE + 144wdwe

We can experiment with different portfolio proportions to observe the effect on portfolio expected return and variance. Suppose we change the proportion invested in bonds. The effect on expected return is tabulated in Table 8.3 and plotted in Figure 8.3. When the proportion invested in debt varies from zero to 1 (so that the proportion in equity varies from 1 to zero), the portfolio expected return goes from 13% (the stock fund's expected return) to 8% (the expected return on bonds).

What happens when wD > 1 and wE < 0? In this case portfolio strategy would be to sell the equity fund short and invest the proceeds of the short sale in the debt fund. This will decrease the expected return of the portfolio. For example, when wD = 2 and wE = — 1, expected portfolio return falls to 2 X 8 + (—1) X 13 = 3%. At this point the value of the bond fund in the portfolio is twice the net worth of the account. This extreme position is financed in part by short selling stocks equal in value to the portfolio's net worth.

The reverse happens when wD < 0 and wE > 1. This strategy calls for selling the bond fund short and using the proceeds to finance additional purchases of the equity fund.

3 It is possible to drive portfolio variance to zero with perfectly positively correlated assets as well, but this would require short

214 PART II Portfolio Theory

Figure 8.3 Portfolio expected return as a function of investment proportions.

Figure 8.3 Portfolio expected return as a function of investment proportions.

Of course, varying investment proportions also has an effect on portfolio standard deviation. Table 8.3 presents portfolio standard deviations for different portfolio weights calculated from equation 8.5 using the assumed value of the correlation coefficient, .30, as well as other values of p. Figure 8.4 shows the relationship between standard deviation and portfolio weights. Look first at the solid curve for Pde — .30. The graph shows that as the portfolio weight in the equity fund increases from zero to 1, portfolio standard deviation first falls with the initial diversification from bonds into stocks, but then rises again as the portfolio becomes heavily concentrated in stocks, and again is undiversified. This pattern will generally hold as long as the correlation coefficient between the funds is not too high. For a pair of assets with a large positive correlation of returns, the portfolio standard deviation will increase monotonically from the low-risk asset to the high-risk asset. Even in this case, however, there is a positive (if small) value of diversification.

What is the minimum level to which portfolio standard deviation can be held? For the parameter values stipulated in Table 8.1, the portfolio weights that solve this minimization problem turn out to be:4

4 This solution uses the minimization techniques of calculus. Write out the expression for portfolio variance from equation 8.2, substitute 1 — Wd for w£, differentiate the result with respect to Wd, set the derivative equal to zero, and solve for Wd to obtain

Alternatively, with a computer spreadsheet, you can obtain an accurate solution by generating a fine grid for Table 8.3 and observing the portfolio weights resulting in the lowest standard deviation.

CHAPTER 8 Optimal Risky Portfolios 215

Figure 8.4 Portfolio standard deviation as a function of investment proportions.

Portfolio standard deviation (%)

Portfolio standard deviation (%)

This minimum-variance portfolio has a standard deviation of aMin = [(.822 X 122) + (.182 X 202) + (2 X .82 X .18 X 72)]1/2 = 11.45%

as indicated in the last line of Table 8.3 for the column p = .30.

The solid blue line in Figure 8.4 plots the portfolio standard deviation when p = .30 as a function of the investment proportions. It passes through the two undiversified portfolios of wD = 1 and wE = 1. Note that the minimum-variance portfolio has a standard deviation smaller than that of either of the individual component assets. This illustrates the effect of diversification.

The other three lines in Figure 8.4 show how portfolio risk varies for other values of the correlation coefficient, holding the variances of each asset constant. These lines plot the values in the other three columns of Table 8.3.

The solid black line connecting the undiversified portfolios of all bonds or all stocks, wD = 1 or wE = 1, shows portfolio standard deviation with perfect positive correlation, p = 1. In this case there is no advantage from diversification, and the portfolio standard deviation is the simple weighted average of the component asset standard deviations.

The dashed blue curve depicts portfolio risk for the case of uncorrelated assets, p= 0. With lower correlation between the two assets, diversification is more effective and portfolio risk is lower (at least when both assets are held in positive amounts). The minimum

216 PART II Portfolio Theory

Figure 8.5 Portfolio expected return as a function of standard deviation.

Expected return (%)

Standard deviation (%)

Standard deviation (%)

portfolio standard deviation when p = 0 is 10.29% (see Table 8.3), again lower than the standard deviation of either asset.

Finally, the upside-down triangular broken line illustrates the perfect hedge potential when the two assets are perfectly negatively correlated (p = -1). In this case the solution for the minimum-variance portfolio is

.625

and the portfolio variance (and standard deviation) is zero.

We can combine Figures 8.3 and 8.4 to demonstrate the relationship between portfolio risk (standard deviation) and expected return—given the parameters of the available assets. This is done in Figure 8.5. For any pair of investment proportions, wD, wE, we read the expected

CHAPTER 8 Optimal Risky Portfolios

CONCEPT CHECK ^ QUESTION 2

return from Figure 8.3 and the standard deviation from Figure 8.4. The resulting pairs of expected return and standard deviation are tabulated in Table 8.3 and plotted in Figure 8.5.

The solid blue curve in Figure 8.5 shows the portfolio opportunity set for p = .30. We call it the portfolio opportunity set because it shows all combinations of portfolio expected return and standard deviation that can be constructed from the two available assets. The other lines show the portfolio opportunity set for other values of the correlation coefficient. The solid black line connecting the two funds shows that there is no benefit from diversification when the correlation between the two is positive (p = 1). The opportunity set is not "pushed" to the northwest. The dashed blue line demonstrates the greater benefit from diversification when the correlation coefficient is lower than .30.

Finally, for p = -1, the portfolio opportunity set is linear, but now it offers a perfect hedging opportunity and the maximum advantage from diversification.

To summarize, although the expected return of any portfolio is simply the weighted average of the asset expected returns, this is not true of the standard deviation. Potential benefits from diversification arise when correlation is less than perfectly positive. The lower the correlation, the greater the potential benefit from diversification. In the extreme case of perfect negative correlation, we have a perfect hedging opportunity and can construct a zero-variance portfolio.

Suppose now an investor wishes to select the optimal portfolio from the opportunity set. The best portfolio will depend on risk aversion. Portfolios to the northeast in Figure 8.5 provide higher rates of return but impose greater risk. The best trade-off among these choices is a matter of personal preference. Investors with greater risk aversion will prefer portfolios to the southwest, with lower expected return but lower risk.5

Compute and draw the portfolio opportunity set for the debt and equity funds when the correlation coefficient between them is p = .25.

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