## Risk Tolerance And Asset Allocation

We have shown how to develop the CAL, the graph of all feasible risk-return combinations available from different asset allocation choices. The investor confronting the CAL now must choose one optimal portfolio, C, from the set of feasible choices. This choice entails a trade-off between risk and return. Individual investor differences in risk aversion imply that, given an identical opportunity set (that is, a risk-free rate and a reward-to-variability

2 Margin purchases require the investor to maintain the securities in a margin account with the broker. If the value of the securities declines below a "maintenance margin," a "margin call" is sent out, requiring a deposit to bring the net worth of the account up to the appropriate level. If the margin call is not met, regulations mandate that some or all of the securities be sold by the broker and the proceeds used to reestablish the required margin. See Chapter 3, Section 3.6, for further discussion.

CHAPTER 7 Capital Allocation between the Risky Asset and the Risk-Free Asset 191

(1) |
(2) |
(3) |
(4) |

y |
E(rc) |
U | |

0 |
7 |
0 |
7.00 |

0.1 |
7.8 |
2.2 |
7.70 |

0.2 |
8.6 |
4.4 |
8.21 |

0.3 |
9.4 |
6.6 |
8.53 |

0.4 |
10.2 |
8.8 |
8.65 |

0.5 |
11.0 |
11.0 |
8.58 |

0.6 |
11.8 |
13.2 |
8.32 |

0.7 |
12.6 |
15.4 |
7.86 |

0.8 |
13.4 |
17.6 |
7.20 |

0.9 |
14.2 |
19.8 |
6.36 |

1.0 |
15.0 |
22.0 |
5.32 |

ratio), different investors will choose different positions in the risky asset. In particular, the more risk-averse investors will choose to hold less of the risky asset and more of the risk-free asset.

In Chapter 6 we showed that the utility that an investor derives from a portfolio with a given expected return and standard deviation can be described by the following utility function:

where A is the coefficient of risk aversion and 0.005 is a scale factor. We interpret this expression to say that the utility from a portfolio increases as the expected rate of return increases, and it decreases when the variance increases. The relative magnitude of these changes is governed by the coefficient of risk aversion, A. For risk-neutral investors, A = 0. Higher levels of risk aversion are reflected in larger values for A.

An investor who faces a risk-free rate, rf, and a risky portfolio with expected return E(rP) and standard deviation oP will find that, for any choice of y, the expected return of the complete portfolio is given by equation 7.1:

From equation 7.2, the variance of the overall portfolio is oC = y2op

The investor attempts to maximize utility, U, by choosing the best allocation to the risky asset, y. To illustrate, we use a spreadsheet program to determine the effect of y on the utility of an investor with A = 4. We input y in column (1) and use the spreadsheet in Table 7.1 to compute E(rc), oC, and U, using equations 7.1-7.4.

Figure 7.4 is a plot of the utility function from Table 7.1. The graph shows that utility is highest at y = .41. When y is less than .41, investors are willing to assume more risk to increase expected return. But at higher levels of y, risk is higher, and additional allocations to the risky asset are undesirable—beyond this point, further increases in risk dominate the increase in expected return and reduce utility.

To solve the utility maximization problem more generally, we write the problem as follows:

Max U = E(rc) - .005AoC = rf + y[E(rP) - rf] - .005Ay2oP

Figure 7.4 Utility as a function of allocation to the risky asset, y.

Figure 7.4 Utility as a function of allocation to the risky asset, y.

Students of calculus will remember that the maximization problem is solved by setting the derivative of this expression to zero. Doing so and solving for y yields the optimal position for risk-averse investors in the risky asset, y*, as follows:3

This solution shows that the optimal position in the risky asset is, as one would expect, inversely proportional to the level of risk aversion and the level of risk (as measured by the variance) and directly proportional to the risk premium offered by the risky asset.

Going back to our numerical example [rf = 7%, - (rP) = 15%, and oP = 22%], the optimal solution for an investor with a coefficient of risk aversion A = 4 is y* = 15 - 7 =41

In other words, this particular investor will invest 41% of the investment budget in the risky asset and 59% in the risk-free asset. As we saw in Figure 7.4, this is the value of y for which utility is maximized.

With 41% invested in the risky portfolio, the rate of return of the complete portfolio will have an expected return and standard deviation as follows:

The risk premium of the complete portfolio is — (rC) — rf = 3.28%, which is obtained by taking on a portfolio with a standard deviation of 9.02%. Notice that 3.28/9.02 = .36, which is the reward-to-variability ratio assumed for this problem.

Another graphical way of presenting this decision problem is to use indifference curve analysis. Recall from Chapter 6 that the indifference curve is a graph in the expected return-standard deviation plane of all points that result in a given level of utility. The curve displays the investor's required trade-off between expected return and standard deviation.

3 The derivative with respect to y equals — (rP) — rf — .01yA op. Setting this expression equal to zero and solving for y yields equation 7.5.

Table 7.2 | |||||

Spreadsheet |
A = 2 |
A |
= 4 | ||

Calculations of |
CT |
U = 5 |
U = 9 |
U= 5 |
U= 9 |

Indifference | |||||

Curves. (Entries in |
0 |
5.000 |
9.000 |
5.000 |
9.000 |

columns 2-4 are |
5 |
5.250 |
9.250 |
5.500 |
9.500 |

expected returns |
10 |
6.000 |
10.000 |
7.000 |
11.000 |

necessary to |
15 |
7.250 |
11.250 |
9.500 |
13.500 |

provide specified |
20 |
9.000 |
13.000 |
13.000 |
17.000 |

utility value.) |
25 |
11.250 |
15.250 |
17.500 |
21.500 |

30 |
14.000 |
18.000 |
23.000 |
27.000 | |

35 |
17.250 |
21.250 |
29.500 |
33.500 | |

40 |
21.000 |
25.000 |
37.000 |
41.000 | |

45 |
25.250 |
29.250 |
45.500 |
49.500 | |

50 |
30.000 |
34.000 |
55.000 |
59.000 |

To illustrate how to build an indifference curve, consider an investor with risk aversion A = 4 who currently holds all her wealth in a risk-free portfolio yielding rf = 5%. Because the variance of such a portfolio is zero, equation 7.4 tells us that its utility value is U = 5. Now we find the expected return the investor would require to maintain the same level of utility when holding a risky portfolio, say with ct = 1%. We use equation 7.4 to find how much E(r) must increase to compensate for the higher value of ct:

This implies that the necessary expected return increases to required E(r) = 5 + .005 X A X ct2 (7.6)

We can repeat this calculation for many other levels of ct, each time finding the value of E(r) necessary to maintain U = 5. This process will yield all combinations of expected return and volatility with utility level of 5; plotting these combinations gives us the indifference curve.

We can readily generate an investor's indifference curves using a spreadsheet. Table 7.2 contains risk-return combinations with utility values of 5% and 9% for two investors, one with A = 2 and the other with A = 4. For example, column (2) uses equation 7.6 to calculate the expected return that must be paired with the standard deviation in column (1) for an investor with A = 2 to derive a utility value of U = 5. Column 3 repeats the calculations for a higher utility value, U = 9. The plot of these expected return-standard deviation combinations appears in Figure 7.5 as the two curves labeled A = 2.

Because the utility value of a risk-free portfolio is simply the expected rate of return of that portfolio, the intercept of each indifference curve in Figure 7.5 (at which ct = 0) is called the certainty equivalent of the portfolios on that curve and in fact is the utility value of that curve. In this context, "utility" and "certainty equivalent" are interchangeable terms. Notice that the intercepts of the indifference curves are at 5% and 9%, exactly the level of utility corresponding to the two curves.

Given the choice, any investor would prefer a portfolio on the higher indifference curve, the one with a higher certainty equivalent (utility). Portfolios on higher indifference curves offer higher expected return for any given level of risk. For example, both indifference

Figure 7.5 Indifference curves for U = 5 and U = 9 with A = 2 and A = 4.

Figure 7.5 Indifference curves for U = 5 and U = 9 with A = 2 and A = 4.

curves for the A = 2 investor have the same shape, but for any level of volatility, a portfolio on the curve with utility of 9% offers an expected return 4% greater than the corresponding portfolio on the lower curve, for which U = 5%.

Columns (4) and (5) of Table 7.2 repeat this analysis for a more risk-averse investor, one with A = 4. The resulting pair of indifference curves in Figure 7.5 demonstrates that the more risk-averse investor has steeper indifference curves than the less risk-averse investor. Steeper curves mean that the investor requires a greater increase in expected return to compensate for an increase in portfolio risk.

Higher indifference curves correspond to higher levels of utility. The investor thus attempts to find the complete portfolio on the highest possible indifference curve. When we superimpose plots of indifference curves on the investment opportunity set represented by the capital allocation line as in Figure 7.6, we can identify the highest possible indifference curve that touches the CAL. That indifference curve is tangent to the CAL, and the tan-gency point corresponds to the standard deviation and expected return of the optimal complete portfolio.

To illustrate, Table 7.3 provides calculations for four indifference curves (with utility levels of 7, 7.8, 8.653, and 9.4) for an investor with A = 4. Columns (2)-(5) use equation 7.6 to calculate the expected return that must be paired with the standard deviation in column (1) to provide the utility value corresponding to each curve. Column (6) uses equation 7.3 to calculate E(rC) on the CAL for the standard deviation aC in column (1):

Figure 7.6 graphs the four indifference curves and the CAL. The graph reveals that the indifference curve with U = 8.653 is tangent to the CAL; the tangency point corresponds to the complete portfolio that maximizes utility. The tangency point occurs at aC = 9.02% and E(rC) = 10.28%, the risk/return parameters of the optimal complete portfolio with y* = 0.41. These values match our algebraic solution using equation 7.5.

CHAPTER 7 Capital Allocation between the Risky Asset and the Risk-Free Asset

CT |
U = 7 |
U = 7.8 |
U = 8.653 |
U = 9.4 |
CAL |

0 |
7.00 |
7.80 |
8.65 |
9.40 |
7.00 |

2 |
7.08 |
7.88 |
8.73 |
9.48 |
7.73 |

4 |
7.32 |
8.12 |
8.97 |
9.72 |
8.45 |

6 |
7.72 |
8.52 |
9.37 |
10.12 |
9.18 |

8 |
8.28 |
9.08 |
9.93 |
10.68 |
9.91 |

9.02 |
8.63 |
9.43 |
10.28 |
11.03 |
10.28 |

10 |
9.00 |
9.80 |
10.65 |
11.40 |
10.64 |

12 |
9.88 |
10.68 |
11.53 |
12.28 |
11.36 |

14 |
10.92 |
11.72 |
12.57 |
13.32 |
12.09 |

18 |
13.48 |
14.28 |
15.13 |
15.88 |
13.55 |

22 |
16.68 |
17.48 |
18.33 |
19.08 |
15.00 |

26 |
20.52 |
21.32 |
22.17 |
22.92 |
16.45 |

30 |
25.00 |
25.80 |
26.65 |
27.40 |
17.91 |

Figure 7.6 Finding the optimal complete portfolio using indifference curves.

Figure 7.6 Finding the optimal complete portfolio using indifference curves.

The choice for y*, the fraction of overall investment funds to place in the risky portfolio versus the safer but lower-expected-return risk-free asset, is in large part a matter of risk aversion. The box on the next page provides additional perspective on the problem, characterizing it neatly as a trade-off between making money, but still sleeping soundly.

a. If an investor's coefficient of risk aversion is A = 3, how does the optimal asset mix change? What are the new E(rC) and ctc?

b. Suppose that the borrowing rate, rf = 9%, is greater than the lending rate, rf = 7%. Show graphically how the optimal portfolio choice of some investors will be affected by the higher borrowing rate. Which investors will not be affected by the borrowing rate?

CONCEPT CHECK ^ QUESTION 4

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