In the last section we considered naive diversification using equally weighted portfolios of several securities. It is time now to study efficient diversification, whereby we construct risky portfolios to provide the lowest possible risk for any given level of expected return.

Portfolios of two risky assets are relatively easy to analyze, and they illustrate the principles and considerations that apply to portfolios of many assets. We will consider a portfolio comprised of two mutual funds, a bond portfolio specializing in long-term debt securities, denoted D, and a stock fund that specializes in equity securities, E. Table 8.1 lists the parameters describing the rate-of-return distribution of these funds. These parameters are representative of those that can be estimated from actual funds.

PART II Portfolio Theory

Table 8.1 Descriptive Statistics for Two Mutual Funds

PART II Portfolio Theory

Table 8.1 Descriptive Statistics for Two Mutual Funds

Debt |
Equity | ||

Expected return, E(r) |
8% |
13% | |

Standard deviation, a |
12% |
20% | |

Covariance, Cov(rD, rE) |
72 | ||

Correlation coefficient, pDE |
.30 |

A proportion denoted by wD is invested in the bond fund, and the remainder, 1 - wD, denoted wE, is invested in the stock fund. The rate of return on this portfolio, rp, will be rp = WDrD + WErE

where rD is the rate of return on the debt fund and rE is the rate of return on the equity fund.

As shown in Chapter 6, the expected return on the portfolio is a weighted average of expected returns on the component securities with portfolio proportions as weights:

The variance of the two-asset portfolio (rule 5 of Chapter 6) is a p = w D^D + w Ea E + 2wDWECov(rD, rE) (8.2)

Our first observation is that the variance of the portfolio, unlike the expected return, is not a weighted average of the individual asset variances. To understand the formula for the portfolio variance more clearly, recall that the covariance of a variable with itself is the variance of that variable; that is

scenarios

Therefore, another way to write the variance of the portfolio is as follows:

ap = WdWDCOV(^, rD) + WEWECOV(^, rE) + 2WDWECOV(^, rE) (8.4)

In words, the variance of the portfolio is a weighted sum of covariances, and each weight is the product of the portfolio proportions of the pair of assets in the covariance term.

Table 8.2 shows how portfolio variance can be calculated from a speadsheet. Panel A of the table shows the bordered covariance matrix of the returns of the two mutual funds. The bordered matrix is the covariance matrix with the portfolio weights for each fund placed on the borders, that is along the first row and column. To find portfolio variance, multiply each element in the covariance matrix by the pair of portfolio weights in its row and column borders. Add up the resultant terms, and you have the formula for portfolio variance given in equation 8.4.

We perform these calculations in Panel B, which is the border-multiplied covariance matrix: Each covariance has been multiplied by the weights from the row and the column in the borders. The bottom line of Panel B confirms that the sum of all the terms in this matrix (which we obtain by adding up the column sums) is indeed the portfolio variance in equation 8.4.

This procedure works because the covariance matrix is symmetric around the diagonal, that is, Cov(rD, rE) = Cov(rE, rD). Thus each covariance term appears twice.

Table 8.2 Computation of Portfolio Variance from the

Covariance Matrix

CHAPTER 8 Optimal Risky Portfolios 211

CHAPTER 8 Optimal Risky Portfolios 211

Table 8.2 Computation of Portfolio Variance from the

Covariance Matrix

A. Bordered Covariance Matrix | ||

Portfolio Weights |
wD |
wE |

Wd |
Cov(D, D) |
Cov(rD, rE) |

wE |
Cov(E, D) |
Cov(rE, rE) |

B. Border-multiplied Covariance Matrix | ||

Portfolio Weights |
wD |
wE |

wD |
WdWDCOV(D, D) |
wDwECov(rD, rE) |

wE |
wEwDCov(rE, rD) |
wDwDCov(rE, rE) |

wD + wE= 1 |
wDwDCov(rD, rD) + wEwDCov(rE, rD) |
wDwECov(rD, rE) + wEwECov(rE, rE) |

Portfolio variance |
wDwDCov(rD, rD) + wEwDCov(rE, rD) |
wDwECov(rD, rE) + wEwECov(rE, rE) |

This technique for computing the variance from the border-multiplied covariance matrix is general; it applies to any number of assets and is easily implemented on a spreadsheet. Concept Check 1 asks you to try the rule for a three-asset portfolio. Use this problem to verify that you are comfortable with this concept.

a. First confirm for yourself that this simple rule for computing the variance of a two-asset portfolio from the bordered covariance matrix is consistent with equation 8.2.

b. Now consider a portfolio of three funds, X, Y Z, with weights wX, wY, and wZ. Show that the portfolio variance is wX&X + WyOy + wZpZ + 2wXwYCov(rX, rY) + 2wXwZ Cov(rX, rZ) + 2wYwZ Cov(rY, rZ)

Equation 8.2 reveals that variance is reduced if the covariance term is negative. It is important to recognize that even if the covariance term is positive, the portfolio standard deviation still is less than the weighted average of the individual security standard deviations, unless the two securities are perfectly positively correlated.

To see this, recall from Chapter 6, equation 6.5, that the covariance can be computed from the correlation coefficient, pDE, as

Because the covariance is higher, portfolio variance is higher when pDE is higher. In the case of perfect positive correlation, pDE = 1, the right-hand side of equation 8.5 is a perfect square and simplifies to op = (Wd Od + WE Oe)2

Therefore, the standard deviation of the portfolio with perfect positive correlation is just the weighted average of the component standard deviations. In all other cases, the correlation coefficient is less than 1, making the portfolio standard deviation less than the weighted average of the component standard deviations.

A hedge asset has negative correlation with the other assets in the portfolio. Equation 8.5 shows that such assets will be particularly effective in reducing total risk. Moreover, equation

CONCEPT CHECK ^ QUESTION 1

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