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Table 5.1 presents the return on a portfolio for selected values of Xc and Figure 5.1 presents a graph of this relationship. Note that the relationship is a straight line. The equation of the straight line could easily be derived as follows. Utilizing the equation presented above for <jp to solve for Xc yields

Substituting this expression for Xc into the equation for Rp and rearranging yields4

In the case of perfectly correlated assets, the return and risk on the portfolio of the two assets is a weighted average of the return and risk on the individual assets. There is no reduction in risk from purchasing both assets. This can be seen by examining Figure 5.1 and noting that combinations of the two assets lie along a straight line connecting the two assets. Nothing has been gained by diversifying rather than purchasing the individual assets.

Figure 5.1 Relationship between expected return and standard deviation when p = +1.

4An alternative way to derive this equation is to substitute the appropriate values for the two firms into the equation derived in footnote 3. This yields

Case 2—Perfect Negative Correlation (p = — 1.0)

We now examine the other extreme: two assets that move perfectly together but in exactly opposite directions. In this case the standard deviation of the portfolio is [from Equation (5.4) with p = -1.0]

<7, = [x2c02c + (l- xcf(72s -2xc(1 - xc)ocosf (5.6)

Once again the equation for standard deviation can be simplified. The term in the brackets is equivalent to either of the following two expressions:

Thus up is either or ap = -xcac + (l - Xc)crs (5.8)

Since we took the square root to obtain an expression for <jp and since the square root of a negative number is imaginary, either of the above equations holds only when its right-hand side is positive. A further examination shows the right-hand side of one equation is simply -1 times the other. Thus, each equation is valid only when the right-hand side is positive. Since one is always positive when the other is negative (except when both equations equal zero), there is a unique solution for the return and risk of any combination of securities c and s. These equations are very similar to the ones we obtained when we had a correlation of +1. Each also plots as a straight line when up is plotted against xc. Thus, one would suspect that an examination of the return on the portfolio of two assets as a function of the standard deviation would yield two straight lines, one for each expression for Up. As we observe in a moment, this is, in fact, the case.5

The value of up for Equation (5.7) or (5.8) is always smaller than the value of <jp for the case where p = +1 [Equation (5.5)] for all values of xc between 0 and 1. Thus the risk on a portfolio of assets is always smaller when the correlation coefficient is -1 than when it is +1. We can go one step further. If two securities are perfectly negatively correlated (i.e., they move in exactly opposite directions), it should always be possible to find some combination of these two securities that has zero risk. By setting either Equation (5.7) or (5.8) equal to 0, we find that a portfolio with xc = vsk<js + ctc) will have zero risk. Since crs > 0 and <js + (tc > (rs, this implies that 0 < xc < 1 or that the zero risk portfolio will always involve positive investment in both securities.

Now let us return to our example. Minimum risk occurs when xc = 3/(3 + 6) = Furthermore, for the case of perfect negative conrelation,

5This occurs for the same reason that the analysis for p = +1 led to one straight line and the mathematical proof is analogous to that presented for the case of p = +1.

there are two equations relating <jp to Xc Only one is appropriate for any value of Xc. The appropriate equation to define ap for any value of Xc is that equation for which <sp > 0. Slote that if up > 0 from one equation, then <yp < 0 for the other. Table 5.2 presents the return on the portfolio for selected values of Xc and Figure 5.2 presents a graph of this relationship.6

Notice that a combination of the two securities exists that provides a portfolio with zero risk. Employing the formula developed before for the composition of the zero-risk portfolio, Xc should equal 3/(3 + 6) or We can see this is correct from Figure 5.2 or by substituting j for Xc in the equation for portfolio risk given previously. We have once again demonstrated the most powerful result of diversification: the ability of combinations of securities to reduce risk. In fact, it is not uncommon for combinations of two securities to lave less risk than either of the assets in the combination.

We have now examined combinations of risky assets for perfect positive and perfect negative correlation. In Figure 5.3 we have plotted both of these relationships on the same graph. From this graph we should be able to see intuitively where portfolios of these two stocks should lie if correlation coefficients took on intermediate values. From the expression for the standard deviation [Equation (5.4)] we see that for any value for Xc between 0 and 1 the lower the correlation the lower is the standard deviation of the portfolio. The standard deviation reaches its lowest value for p = — 1 (curve SBC) and its highest value for p = +1 (curve SAC). Therefore, these two curves should represent the limits within which all portfolios of these two securities must lie for intermediate values of the correlation coefficient. We would speculate that an intermediate correlation might produce a curve such as SOC in Figure 5.3. We demonstrate this by returning to our example and constructing the relationship between risk and return for portfolios of our two securities when the correlation coefficient is assumed to be 0 and +0.5.

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