4.9Finding the Minimum Variance Portfolio

This section illustrates how to compute the weights of the minimum variance portfo-

lio, a portfolio that is of interest for a variety of reasons. Investors who are extremely

risk averse will select this portfolio if no risk-free investment is available. In addition,

this portfolio is useful for understanding many risk management problems.¹³This sec-

tion discusses insights about covariance as a marginal variance presented in the last

section to develop a set of equations that, when solved, identify the weights of this

portfolio.

Properties of a Minimum Variance Portfolio

The previous section discussed how to adjust the portfolio weights of stocks in a port-

folio to lower the portfolio’s variance by following these steps:

1.Take two stock returns that have different covariances with the portfolio’s

return.

2.Take on a small additional positive investment (that is, a slightly larger

portfolio weight) in the low covariance stock financed and an additional

negative offsetting position (i.e., a slightly lower portfolio weight) in the high

covariance stock.

With this process, the portfolio’s variance can be lowered until all stocks in the

portfolio have identical covariances with the portfolio’s return. When all stocks have

the same covariance with the portfolio’s return, more tinkering with the portfolio

weights at the margin will not reduce variance, implying that a minimum variance port-

folio has been obtained.

Result 4.10

The portfolio of a group of stocks that minimizes return variance is the portfolio with areturn that has an equal covariance with every stock return.

Identifying the Minimum Variance Portfolio of Two Stocks

Example 4.18’s two-stock problem illustrates the procedure for finding this type of port-

folio.

Example 4.18:Forming a Minimum Variance Portfolio forAsset Allocation

Historically, the return of the S&P 500 Index (S&P) has had a correlation of .8 with the

return of the Dimensional Fund Advisors small cap fund, which is a portfolio of small

stocks that trade mostly on Nasdaq.S&P has a standard deviation of 20 percent per year;

that is, .2.The DFA small cap stock return has a standard deviation of 39 per-

S&P

cent per year;that is, .39.What portfolio allocation between these two investments

DFA

minimizes variance?

Answer:Treat the two stock indexes as if they were two individual stocks.If xis the

weight on the S&P, the covariance of the portfolio with the S&P index (using Result 4.5) is

cov(x˜(1x)˜, r˜) x cov(˜, r˜)(1x)cov(˜, r˜)

rrrr

S&PDFAS&PS&PS&PDFAS&P

2

.2x(.2)(.39)(.8)(1x)

.022x.062

¹³See

Chapter 22 for a discussion of risk management.

Grinblatt267Titman: Financial	II. Valuing Financial Assets	4. Portfolio Tools	© The McGraw267Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

124Part IIValuing Financial Assets

The covariance of the portfolio with the return of the DFA fund is

(xr˜(1x)˜, r˜) x cov(˜, r˜)(1x)cov(˜, r˜)

covrrr

S&PDFADFAS&PDFADFADFA

2

(.2)(.39)(.8)x(1x)(.39)

.090x.152

Setting the two covariances equal to each other and solving for xgives

.022x.062 .090x.152, or

x 1.32 (approximately)

Thus, placing weights of approximately 132 percent on the S&P index and 32 percent on

the DFA fund minimizes the variance of the portfolio of these two investments.

Example 4.18 implies that a short position in the DFAsmall cap fund reduces vari-

ance relative to a portfolio with a 100 percent position in the S&P. Indeed, until we reach

the 132 percent investment position in the S&Pindex, additional shorting of the DFAfund

to finance the more than 100 percent position in the S&Pindex reduces variance.

For example, consider what happens to the variance of a portfolio that is 100 per-

cent invested in the S&P500 when its weights are changed slightly. Increase the posi-

tion to 101 percent invested in the S&P, the increase financed by selling short 1 per-

cent in the DFAsmall cap fund. The covariance of the DFAsmall cap fund with the

initial position of 100 percent invested in S&Pis .06 (.8)(.39)(.2), while the covari-

ance of the S&Pwith itself is a lower number .04 (.2)(.2). Moreover, variances do

not matter for such small changes, only covariances. Hence, increasing the S&Pposi-

tion from 100 percent and reducing the DFAposition from 0 percent reduces variance.