Finding the Minimum Variance Portfolio
This section illustrates how to compute the weights of the minimum variance portfolio, a portfolio that is of interest for a variety of reasons. Investors who are extremely risk averse will select this portfolio if no riskfree investment is available. In addition, this portfolio is useful for understanding many risk management problems.13 This section 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 portfolio 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 portfolio has been obtained.
Result 4.10 The portfolio of a group of stocks that minimizes return variance is the portfolio with a return that has an equal covariance with every stock return.
Identifying the Minimum Variance Portfolio of Two Stocks
Example 4.18's twostock problem illustrates the procedure for finding this type of portfolio.
Example 4.18: Forming a Minimum Variance Portfolio for Asset 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, ctg&p = .2. The DFA small cap stock return has a standard deviation of 39 percent per year; that is, adfa = .39. What portfolio allocation between these two investments minimizes variance?
Answer: Treat the two stock indexes as if they were two individual stocks. If x is the weight on the S&P, the covariance of the portfolio with the S&P index (using Result 4.5) is cov(xrS&p + (1  x)Fdfa, fs&p) = x cov(rs&p, rS&p) + (1  x)cov(^fa, rS&p)
13See Chapter 22 for a discussion of risk management.
124 Part II Valuing Financial Assets
The covariance of the portfolio with the return of the DFA fund is cov(xrS&p + (1  x)aDfa, ¿dfa) = x cov(rS&p, ¿ofa) + (1  x)cov(/Dfa, ¿ofa)
Setting the two covariances equal to each other and solving for x gives
 .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 DFA small cap fund reduces variance 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&P index, additional shorting of the DFA fund to finance the more than 100 percent position in the S&P index reduces variance.
For example, consider what happens to the variance of a portfolio that is 100 percent invested in the S&P 500 when its weights are changed slightly. Increase the position to 101 percent invested in the S&P, the increase financed by selling short 1 percent in the DFA small cap fund. The covariance of the DFA small cap fund with the initial position of 100 percent invested in S&P is .06 = (.8)(.39)(.2), while the covariance of the S&P with itself is a lower number .04 = (.2)(.2). Moreover, variances do not matter for such small changes, only covariances. Hence, increasing the S&P position from 100 percent and reducing the DFA position from 0 percent reduces variance.
Identifying the Minimum Variance Portfolio of Many Stocks
With N stocks, we recommend a twostep process, based on Result 4.10, to find the portfolio that has the same covariance with every stock. First, solve N equations with N unknowns. Then, rescale the portfolio weights. Example 4.19, using a portfolio of three stocks, illustrates the technique:
Example 4.19: Finding the Minimum Variance Portfolio
Amalgamate Bottlers, a U.S. softdrink bottler, wants to branch out internationally. Recognizing that the markets in Japan and Europe are difficult to break into, it is contemplating capital investments to open franchises in India (investment 1), Russia (investment 2), and China (investment 3). Given that Amalgamate has a fixed amount of capital to invest in foreign franchising, and recognizing that such investment is risky, Amalgamate wants to find the minimum variance investment proportions for these three countries. Solve Amalgamate's portfolio problem. Assume that the returns of the franchise investments in the three countries have covariances as follows:
Covariance with  
India 
Russia 
China  
India 
.002 
.001 
0 
Russia 
.001 
.002 
.001 
China 
0 
.001 
.002 
Chapter 4 Portfolio Tools
Answer: Treat franchise investment in each country as a portfolio investment problem.
Step 1. Solve for the "weights" that make the covariance of each of the three country returns a constant. These "weights" are not true weights because they do not necessarily sum to 1. (Some constant will result in weights that sum to 1. However, it is easier to first pick any constant and later rescale the weights.) Let us use "1" as that constant. The first step is to simultaneously solve three equations
.002x1 + .001 x2 + 0x3 = 1 .001 x, + .002x2 + .001 x3 = 1 0x1 + .001 x2 + .002x3 = 1
The lefthand side of the first equation is the covariance of the return of a portfolio with weights x1, x2, x3 with the return of stock 1 (using Result 4.5). The first equation shows that this covariance must equal 1. The other two equations make identical statements about covariances with stocks 2 and 3.
Now, solve these equations with the substitution method. Rewrite the first and third equations as x1 = 500  — 1 2
X2 2
Substituting these values of x and x2 into the second equation yields or
Substitution of this value into the remaining two equations implies that x1 = 500 x3 = 500
Step 2. Rescale the portfolio weights so they add to 1. After rescaling, the computed solution is x1 = .5 x2 = 0 x3 = .5
Solving for the minimum variance portfolio of a large number of stocks usually requires a computer.14
14Many software packages, including spreadsheets, can be used to obtain a numerical solution to this type of problem. The solution usually requires setting up a matrix (or array) of covariances, where the row i and column j of the matrix is the covariance between the returns of stocks i and j, denoted Oj. Then instruct the software to first invert the matrix, which is an important step that the computer uses to solve systems of linear equations, then to sum the columns of the inverted covariance matrix, which is the same as summing the elements in each row. The sum of the columns is then rescaled so that the entries sum to 1. The matrix inversion method is partly analogous to the substitution method in Example 4.19. In Microsoft Excel, matrix inversion uses the function MINVERSE.
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