## Three Examples

Let us return to the two examples discussed in Chapter 4. Consider first the allocation between equity and debt. The minimum variance portfolio is given by Equation (5.9). The estimated inputs for bonds and stocks are

4^=12-5% GS&P = U.9% ps&p B = .45 Rk = 6% (7„ = 4.8%

CHAPTER s delineating efficient portfolios 93

Plugging the values for standard deviation and correlation into Equation (5.9) gives

skP (4.8)2 + (14.9)2 - (2)(.45)(4.8)(14.9) xs&p = -051

Thus the minimum variance portfolio involves short selling stock. The associated standard deviation is 4.75%, which is slightly less than the standard deviation associated with investing 100% in bonds. The dots in Figure 5.17 are plots of all combinations of the S&P index and Lehman Brothers aggregate bond index, ranging from the global minimum variance portfolio to the portfolio representing 150% in common stock and —50% in bonds. [ he dot next to the global minimum variance portfolio represents the expected return and standard deviation of the portfolio with 0% in common stocks. As we move to the right each dot represents the expected return and standard deviation of a portfolio with 10% more in common stock. This is the efficient frontier with short sales allowed (although it would continue to the right). The efficient frontier with no short sales is Figure 4.4. In this case the global minimum variance portfolio is 100% in bonds.

At the time of this revision the interest rate on Treasury bills was about 5%. Using this as a riskless lending and borrowing rate, the tangency portfolio is portfolio T shown in figure 5.17. We will see how this is calculated in the next chapter. The expected return and risk for portfolio T as read from the graph are 13.54% and 16.95%, respectively. Thus the slope of the line connecting the tangency portfolio and the efficient frontier is

and the equation of the efficient frontier with riskless lending and borrowing is

Once we know the expected return of portfolio T we can easily determine its composition. Simply recall that

Therefore

The second example we examined in Chapter 4 was a combination of a domestic portfolio represented by the S&P index and an international portfolio represented by an average international fund. All combinations of these two funds without short sales were represented by Figure 4.5. Note that part of these combinations is inefficient. The estimated inputs were

^„=12.5% as&P=U.9% ps&p ¡M = 0.33 R* =10.5% crinl = 14.0%

Solving for the global minimum variance portfolio we have

(14)2 - 0.33(14)(14.9) SSlF (14.9)2 + (14)2 + (2)(.33)(14)(14.9) XS&P = 0.45

Thus the global minimum variance portfolio is obtained by investing 0.45 in the S&P index and 0.55 in the foreign portfolio. The resulting standard deviation is 11.76, which is less than the standard deviation of both portfolios. This is an example of how diversification can reduce risk. Note that it is inefficient to hold the foreign portfolio by itself. An investor wishing to accept the risk of 14% on the foreign portfolio could obtain an expected return of 12.31% by putting 90.7% in the S&P index and 9.3% in the foreign portfolio. Thus at a 14% standard deviation the increase in expected return from using the optimum combination is 1.81% with no increase in risk. The efficient frontier with no short sales is the scatter of dots in Figure 5.18 from the global minimum variance portfolio to 100% in the S&P index. The dot to the right of the global minimum variance portfolio is the expected return and standard deviation of return when there is 50% in the S&P index. Each dot as we move to the right represents the expected return and standard deviation of return as we increase the amount in the S&P index by 10%. The efficient frontier with short sales allowed is the complete scatter of dots shown in Figure 5.18 (although it would continue to the right).

If the riskless lending and borrowing rate is 5%, then the tangency portfolio is 61% in the S&P index and 39% in the international portfolio. The associated mean return is 11.72% and standard deviation of return is 12.04%. Thus the slope of the efficient frontier with riskless lending and borrowing is