Portfolios of Two Perfectly Positively Correlated or Perfectly Negatively Correlated Assets
Perfect Positive Correlation. Exhibit 4.4 shows the plotted means and standard deviations obtainable from portfolios of two perfectly positively correlated stocks. Points A and B on the line, designated, respectively, as "100% in stock 1" and "100% in stock 2," correspond to the mean and standard deviation pairings achieved when 100 percent of an investor's wealth is held in one of the two investments. Bold segment AB graphs the means and standard deviations achieved from portfolios with positive weights on the two perfectly correlated stocks. Moving up the line from point A toward point B
Part II Valuing Financial Assets
Exhibit 4.4 MeanStandard Deviation Diagram: Portfolios of Two Perfectly Positively Correlated Stocks
Mean Short in
Mean Short in
places more weight on the investment with the higher expected return (stock 2). Above point B, the portfolio is selling short stock 1 and going "extra long" (weight exceeds 1) in stock 2. Below point A, the portfolio is selling short stock 2.
Exhibit 4.4 demonstrates that it is possible to eliminate risk—that is, to achieve zero variance—with a portfolio of two perfectly positively correlated stocks.9 To do this, it is necessary to be long in one investment and short in the other in proportions that place the portfolio at point C.
The graph of portfolios of two perfectly positively correlated, but risky, investments has the same shape (a pair of straight lines) as the graph for a portfolio of a riskless and a risky investment. This should not be surprising because a portfolio of the two perfectly correlated investments is itself a riskless asset. The feasible means and standard deviations generated from portfolios of, say, the riskless combination of stocks 1 and 2 on the one hand, and stock 2 on the other, should be identical to those generated by portfolios of stocks 1 and 2 themselves. Also, when the portfolio weights on both investments are positive, the mean and the standard deviation of a portfolio of two perfectly correlated investments are portfolioweighted averages of the means and standard deviations of the individual assets. Consistent with Result 4.6, such portfolios are on a line connecting the two investments whenever the weighted average of the standard deviations is positive.
Perfect Negative Correlation. For similar reasons, a pair of perfectly negatively correlated risky assets graphs as a pair of straight lines. In contrast to a pair of perfectly positively correlated investments, when two investments are perfectly negatively correlated, the investor eliminates variance by being long in both investments.
9This also was shown in Example 4.13.
Chapter 4 Portfolio Tools 119
Example 4.16: Forming a Riskless Portfolio from Two Perfectly Negatively Correlated Securities
Over extremely short time intervals, the return of IBM is perfectly negatively correlated with the return of a put option on the company. The standard deviation of the return on IBM stock is 18 percent per year, while the standard deviation of the return on the option is 54 percent per year. What portfolio weights on IBM stock and its put option create a riskless investment over short time intervals?
Answer: The variance of the portfolio, using equation (4.9b), is
.182x2 + .542(1  x)2  2x(1  x)(.18)(.54) = [.18x  .54(1  x)]2
This expression is 0 when x = .75. Thus, the weight on IBM stock is 3/4 and the weight on the put option is 1/4.
The Feasible Means and Standard Deviations from Portfolios of Other Pairs of Assets
We noted in the last subsection that with perfect positive correlation, the standard deviation of a portfolio with positive weights on both stocks equals the portfolioweighted average of the two standard deviations. Now, consider the case where risky investments have less than perfect correlation (p < 1). Result 4.2 states that the lower the correlation, the lower the portfolio variance. Therefore, the standard deviation of a portfolio with positive weights on both stocks is less than the portfolioweighted average of the two standard deviations, which gives the curvature to the left shown in Exhibit 4.5. The
Exhibit 4.5 Mean Standard Deviation Diagram: Portfolios of Two Risky Securities with Arbitrary Correlation, p
Mean Return
100% in stock 2
100% in stock 2
Standard Deviation of Return ctp
Part II Valuing Financial Assets degree to which this curvature occurs depends on the correlation between the returns. Consistent with Result 4.2, the smaller the correlation, p, the more distended the curvature. The ultimate in curvature is the pair of lines generated with perfect negative correlation, p = 1, which is the smallest correlation possible.10
The combination of the two stocks that has minimum variance is a portfolio with positive weights on both stocks only if the correlation between their returns is not too large. For sufficiently large correlations, the minimum variance portfolio, the portfolio of risky investments with the lowest variance, requires a long position in one investment and a short position in the other.11
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