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What if our risky assets are still confined to the bond and stock funds, but now we can also invest in risk-free T-bills yielding 5%? We start with a graphical solution. Figure 8.6 shows the opportunity set based on the properties of the bond and stock funds, using the data from Table 8.1.
Two possible capital allocation lines (CALs) are drawn from the risk-free rate (rf = 5%) to two feasible portfolios. The first possible CAL is drawn through the minimum-variance portfolio A, which is invested 82% in bonds and 18% in stocks (Table 8.3, bottom panel). Portfolio A's expected return is 8.90%, and its standard deviation is 11.45%. With a T-bill rate of 5%, the reward-to-variability ratio, which is the slope of the CAL combining T-bills and the minimum-variance portfolio, is
11.45
RECIPE FOR SUCCESSFUL INVESTING: FIRST, MIX ASSETS WELL
If you want dazzling investment results, don't start your day foraging for hot stocks and stellar mutual funds. Instead, say investment advisers, the really critical decision is how to divvy up your money among stocks, bonds, and supersafe investments such as Treasury bills.
In Wall Street lingo, this mix of investments is called your asset allocation. "The asset-allocation choice is the first and most important decision," says William Droms, a finance professor at Georgetown University. "How much you have in [the stock market] really drives your results."
"You cannot get [stock market] returns from a bond portfolio, no matter how good your security selection is or how good the bond managers you use," says William John Mikus, a managing director of Financial Design, a Los Angeles investment adviser.
For proof, Mr. Mikus cites studies such as the 1991 analysis done by Gary Brinson, Brian Singer and Gilbert Beebower. That study, which looked at the 10-year results for 82 large pension plans, found that a plan's asset-allocation policy explained 91.5% of the return earned.
Designing a Portfolio
Because your asset mix is so important, some mutual fund companies now offer free services to help investors design their portfolios.
Gerald Perritt, editor of the Mutual Fund Letter, a Chicago newsletter, says you should vary your mix of as sets depending on how long you plan to invest. The further away your investment horizon, the more you should have in stocks. The closer you get, the more you should lean toward bonds and money-market instruments, such as Treasury bills. Bonds and money-market instruments may generate lower returns than stocks. But for those who need money in the near future, conservative investments make more sense, because there's less chance of suffering a devastating short-term loss.
"One of the most important things people can do is summarize all their assets on one piece of paper and figure out their asset allocation," says Mr. Pond.
Once you've settled on a mix of stocks and bonds, you should seek to maintain the target percentages, says Mr. Pond. To do that, he advises figuring out your asset allocation once every six months. Because of a stock-market plunge, you could find that stocks are now a far smaller part of your portfolio than you envisaged. At such a time, you should put more into stocks and lighten up on bonds.
When devising portfolios, some investment advisers consider gold and real estate in addition to the usual trio of stocks, bonds and money-market instruments. Gold and real estate give "you a hedge against hyperinflation," says Mr. Droms. "But real estate is better than gold, because you'll get better long-run returns."
Source: Jonathan Clements, "Recipe for Successful Investing: First, Mix Assets Well," The Wall Street Journal, October 6, 1993. Reprinted by permission of The Wall Street Journal, © 1993 Dow Jones & Company, Inc. All Rights Reserved Worldwide.
Now consider the CAL that uses portfolio B instead of A. Portfolio B invests 70% in bonds and 30% in stocks. Its expected return is 9.5% (a risk premium of 4.5%), and its standard deviation is 11.70%. Thus the reward-to-variability ratio on the CAL that is supported by Portfolio B is
which is higher than the reward-to-variability ratio of the CAL that we obtained using the minimum-variance portfolio and T-bills. Thus Portfolio B dominates A.
But why stop at Portfolio B? We can continue to ratchet the CAL upward until it ultimately reaches the point of tangency with the investment opportunity set. This must yield the CAL with the highest feasible reward-to-variability ratio. Therefore, the tangency portfolio, labeled P in Figure 8.7, is the optimal risky portfolio to mix with T-bills. We can read the expected return and standard deviation of Portfolio P from the graph in Figure 8.7.
In practice, when we try to construct optimal risky portfolios from more than two risky assets we need to rely on a spreadsheet or another computer program. The spreadsheet we present later in the chapter can be used to construct efficient portfolios of many assets. To start,
220 PART II Portfolio Theory
Figure 8.7 The opportunity set of the debt and equity funds with the optimal CAL and the optimal risky portfolio.
Figure 8.7 The opportunity set of the debt and equity funds with the optimal CAL and the optimal risky portfolio.
however, we will demonstrate the solution of the portfolio construction problem with only two risky assets (in our example, long-term debt and equity) and a risk-free asset. In this case, we can derive an explicit formula for the weights of each asset in the optimal portfolio. This will make it easy to illustrate some of the general issues pertaining to portfolio optimization.
The objective is to find the weights wD and wE that result in the highest slope of the CAL (i.e., the weights that result in the risky portfolio with the highest reward-to-variability ratio). Therefore, the objective is to maximize the slope of the CAL for any possible portfolio, p. Thus our objective function is the slope that we have called Sp:
For the portfolio with two risky assets, the expected return and standard deviation of Portfolio p are
= 8wd + 13 wE ap = [wDvD + wWe + 2wDWECov(rD, rE)]1/2 = [144wD + 400wE + (2 X 72wdwe)]1/2
When we maximize the objective function, Sp, we have to satisfy the constraint that the portfolio weights sum to 1.0 (100%), that is, wD + wE = 1. Therefore, we solve a mathematical problem formally written as
E(rP) - rf Max Sp = —p-f wi p Op subject to %wi = 1. This is a standard problem in optimization.
In the case of two risky assets, the solution for the weights of the optimal risky portfolio, P, can be shown to be as follows:6
[E(D - foE + [E(^) - rjoD - [E(rD) - rf + E(rE) - rf]Cov(rD, rE)
Substituting our data, the solution is w =_(8 - 5)400 - (13 - 5)72_=
The expected return and standard deviation of this optimal risky portfolio are
oP = [(.42 X 144) + (.62 X 400) + (2 X .4 X .6 X 72)]1/2 = 14.2%
The CAL of this optimal portfolio has a slope of
which is the reward-to-variability ratio of Portfolio P. Notice that this slope exceeds the slope of any of the other feasible portfolios that we have considered, as it must if it is to be the slope of the best feasible CAL.
In Chapter 7 we found the optimal complete portfolio given an optimal risky portfolio and the CAL generated by a combination of this portfolio and T-bills. Now that we have constructed the optimal risky portfolio, P, we can use the individual investor's degree of risk aversion, A, to calculate the optimal proportion of the complete portfolio to invest in the risky component.
An investor with a coefficient of risk aversion A = 4 would take a position in Portfolio P of7
E(rP) - rf 11 - 5 y = rnXXli = .01 X 4 X 14.22 = .7439 (8.8)
Thus the investor will invest 74.39% of his or her wealth in Portfolio P and 25.61% in T-bills. Portfolio P consists of 40% in bonds, so the percentage of wealth in bonds will be yWD = .4 X .7439 = .2976, or 29.76%. Similarly, the investment in stocks will be ywE = .6 X .7439 = .4463, or 44.63%. The graphical solution of this asset allocation problem is presented in Figures 8.8 and 8.9.
6 The solution procedure for two risky assets is as follows. Substitute for E(rp) from equation 8.1 and for ^p from equation 8.5. Substitute 1 — wD for wE. Differentiate the resulting expression for Sp with respect to wD, set the derivative equal to zero, and solve for wD.
7 As noted earlier, the .01 that appears in the denominator is a scale factor that arises because we measure returns as percentages rather than decimals. If we were to measure returns as decimals (e.g., .07 rather than 7%), we would not use the .01 in the denominator. Notice that switching to decimals would reduce the scale of the numerator by a multiple of .01 and the denominator by .012.
222 PART II Portfolio Theory
Figure 8.8 Determination of the optimal overall portfolio.
Expected return (%)
Expected return (%)
Once we have reached this point, generalizing to the case of many risky assets is straightforward. Before we move on, let us briefly summarize the steps we followed to arrive at the complete portfolio.
1. Specify the return characteristics of all securities (expected returns, variances, covariances).
2. Establish the risky portfolio:
a. Calculate the optimal risky portfolio, P (equation 8.7).
b. Calculate the properties of Portfolio P using the weights determined in step (a) and equations 8.1 and 8.2.
3. Allocate funds between the risky portfolio and the risk-free asset:
a. Calculate the fraction of the complete portfolio allocated to Portfolio P (the risky portfolio) and to T-bills (the risk-free asset) (equation 8.8).
b. Calculate the share of the complete portfolio invested in each asset and in T-bills.
Before moving on, recall that our two risky assets, the bond and stock mutual funds, are already diversified portfolios. The diversification within each of these portfolios must be credited for a good deal of the risk reduction compared to undiversified single securities. For example, the standard deviation of the rate of return on an average stock is about 50% (see Figure 8.2). In contrast, the standard deviation of our stock-index fund is only 20%,
CHAPTER 8 Optimal Risky Portfolios
Figure 8.9 The proportions of the optimal overall portfolio.
CHAPTER 8 Optimal Risky Portfolios
Figure 8.9 The proportions of the optimal overall portfolio.
about equal to the historical standard deviation of the S&P 500 portfolio. This is evidence of the importance of diversification within the asset class. Optimizing the asset allocation between bonds and stocks contributed incrementally to the improvement in the reward-to-variability ratio of the complete portfolio. The CAL with stocks, bonds, and bills (Figure 8.7) shows that the standard deviation of the complete portfolio can be further reduced to 18% while maintaining the same expected return of 13% as the stock portfolio.
CONCEPT CHECK ^ QUESTION 3
Expected Return |
Standard Deviation | |
A |
10% |
20% |
B |
30 |
60 |
T-bills |
5 |
0 |
The correlation coefficient between funds A and B is -.2.
a. Draw the opportunity set of Funds A and B.
b. Find the optimal risky portfolio, P, and its expected return and standard deviation.
c. Find the slope of the CAL supported by T-bills and Portfolio P.
d. How much will an investor with A = 5 invest in Funds A and B and in T-bills?
The correlation coefficient between funds A and B is -.2.
a. Draw the opportunity set of Funds A and B.
b. Find the optimal risky portfolio, P, and its expected return and standard deviation.
c. Find the slope of the CAL supported by T-bills and Portfolio P.
d. How much will an investor with A = 5 invest in Funds A and B and in T-bills?
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