Now we can expand the asset allocation problem to include a risk-free asset. Let us continue to use the input data from the bottom of Spreadsheet 6.5 , but now assume a realistic correlation coefficient between stocks and bonds of 0.20. Suppose then that we are still confined to the risky bond and stock funds, but now can also invest in risk-free T-bills yielding 5%. Figure 6.5 shows the opportunity set generated from the bond and stock funds. This is the same opportunity set as graphed in Figure 6.4 with BS 0.20.
Two possible capital allocation lines (CALs) are drawn from the risk-free rate ( r f 5%) to two feasible portfolios. The first possible CAL is drawn through the minimum-variance portfolio ( A ), which invests 87.06% in bonds and 12.94% in stocks. Portfolio A ’s expected return is 6.52% and its standard deviation is 11.54%. With a T-bill rate ( r f ) of 5%, the rewardto-volatility ratio of portfolio A (which is also the slope of the CAL that combines T-bills with portfolio A ) is
Now consider the CAL that uses portfolio B instead of A. Portfolio B invests 80% in bonds and 20% in stocks, providing an expected return of 6.80% with a standard deviation of 11.68%. Thus, the reward-to-volatility ratio of any portfolio on the CAL of B is
This is higher than the reward-to-volatility ratio of the CAL of the minimum-variance portfolio A.
The difference in the reward-to-volatility ratios is S B S A 0.02. This implies that portfolio B provides 2 extra basis points (0.02%) of expected return for every percentage point increase in standard deviation.
The higher reward-to-volatility ratio of portfolio B means that its capital allocation line is steeper than that of A. Therefore, CAL B plots above CAL A in Figure 6.5 . In other words, combinations of portfolio B and the risk-free asset provide a higher expected return for any level of risk (standard deviation) than combinations of portfolio A and the risk-free asset. Therefore, all risk-averse investors would prefer to form their complete portfolio using the risk-free asset with portfolio B rather than with portfolio A. In this sense, portfolio B dominates A.
But why stop at portfolio B? We can continue to ratchet the CAL upward until it reaches the ultimate point of tangency with the investment opportunity set. This must yield the CAL with the highest feasible reward-to-volatility ratio. Therefore, the tangency portfolio ( O ) in Figure 6.6 is the optimal risky portfolio to mix with T-bills, which may be defined as the risky portfolio resulting in the highest possible CAL. We can read the expected return and standard deviation of portfolio O (for “optimal”) off the graph in Figure 6.6 as
which can be identified as the portfolio that invests 32.99% in bonds and 67.01% in stocks.
These weights may be obtained algebraically from the following formula, which is the solution to the maximization of the reward-to-volatility ratio.
The CAL with our optimal portfolio has a slope of
which is the reward-to-variability ratio of portfolio O. This slope exceeds the slope of any other feasible portfolio, as it must if it is to be the slope of the best feasible CAL.
In the last chapter we saw that the preferred complete portfolio formed from a risky portfolio and a risk-free asset depends on the investor’s risk aversion. More risk-averse investors will prefer low-risk portfolios despite the lower expected return, while more risk-tolerant investors will choose higher-risk, higher-return portfolios. Both investors, however, will choose portfolio O as their risky portfolio since that portfolio results in the highest return per unit of risk, that is, the steepest capital allocation line. Investors will differ only in their allocation of investment funds between portfolio O and the risk-free asset.
Figure 6.7 shows one possible choice for the preferred complete portfolio, C. The investor places 55% of wealth in portfolio O and 45% in Treasury bills. The rate of return and volatility of the portfolio are
In turn, we found above that portfolio O is formed by mixing the bond fund and stock fund with weights of 32.99% and 67.01%. Therefore, the overall asset allocation of the complete portfolio is as follows:
Figure 6.8 depicts the overall asset allocation. The allocation reflects considerations of both efficient diversification (the construction of the optimal risky portfolio, O ) and risk aversion (the allocation of funds between the risk-free asset and the risky portfolio O to form the complete portfolio, C ).
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* Bodie Z., Kane A., Marcus A. (2008), Essentials of Investments, 7th Edition, McGraw-Hill/Irwin, New York, pp.164-167.
Two possible capital allocation lines (CALs) are drawn from the risk-free rate ( r f 5%) to two feasible portfolios. The first possible CAL is drawn through the minimum-variance portfolio ( A ), which invests 87.06% in bonds and 12.94% in stocks. Portfolio A ’s expected return is 6.52% and its standard deviation is 11.54%. With a T-bill rate ( r f ) of 5%, the rewardto-volatility ratio of portfolio A (which is also the slope of the CAL that combines T-bills with portfolio A ) is
This is higher than the reward-to-volatility ratio of the CAL of the minimum-variance portfolio A.
The difference in the reward-to-volatility ratios is S B S A 0.02. This implies that portfolio B provides 2 extra basis points (0.02%) of expected return for every percentage point increase in standard deviation.
The higher reward-to-volatility ratio of portfolio B means that its capital allocation line is steeper than that of A. Therefore, CAL B plots above CAL A in Figure 6.5 . In other words, combinations of portfolio B and the risk-free asset provide a higher expected return for any level of risk (standard deviation) than combinations of portfolio A and the risk-free asset. Therefore, all risk-averse investors would prefer to form their complete portfolio using the risk-free asset with portfolio B rather than with portfolio A. In this sense, portfolio B dominates A.
But why stop at portfolio B? We can continue to ratchet the CAL upward until it reaches the ultimate point of tangency with the investment opportunity set. This must yield the CAL with the highest feasible reward-to-volatility ratio. Therefore, the tangency portfolio ( O ) in Figure 6.6 is the optimal risky portfolio to mix with T-bills, which may be defined as the risky portfolio resulting in the highest possible CAL. We can read the expected return and standard deviation of portfolio O (for “optimal”) off the graph in Figure 6.6 as
which can be identified as the portfolio that invests 32.99% in bonds and 67.01% in stocks.
These weights may be obtained algebraically from the following formula, which is the solution to the maximization of the reward-to-volatility ratio.
In the last chapter we saw that the preferred complete portfolio formed from a risky portfolio and a risk-free asset depends on the investor’s risk aversion. More risk-averse investors will prefer low-risk portfolios despite the lower expected return, while more risk-tolerant investors will choose higher-risk, higher-return portfolios. Both investors, however, will choose portfolio O as their risky portfolio since that portfolio results in the highest return per unit of risk, that is, the steepest capital allocation line. Investors will differ only in their allocation of investment funds between portfolio O and the risk-free asset.
Figure 6.7 shows one possible choice for the preferred complete portfolio, C. The investor places 55% of wealth in portfolio O and 45% in Treasury bills. The rate of return and volatility of the portfolio are
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* Bodie Z., Kane A., Marcus A. (2008), Essentials of Investments, 7th Edition, McGraw-Hill/Irwin, New York, pp.164-167.
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