## Portfolios Of One Risky Asset And One Riskfree Asset

In this section we examine the risk-return combinations available to investors. This is the "technological" part of asset allocation; it deals only with the opportunities available to in-

CHAPTER 7 Capital Allocation between the Risky Asset and the Risk-Free Asset 187

Figure 7.1 Spread between three-month CD and T-bill rates.

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vestors given the features of the broad asset markets in which they can invest. In the next section we address the "personal" part of the problem—the specific individual's choice of the best risk-return combination from the set of feasible combinations.

Suppose the investor has already decided on the composition of the risky portfolio. Now the concern is with the proportion of the investment budget, y, to be allocated to the risky portfolio, P. The remaining proportion, 1 — y, is to be invested in the risk-free asset, F.

Denote the risky rate of return by rP and denote the expected rate of return on P by E(rP) and its standard deviation by op. The rate of return on the risk-free asset is denoted as rf. In the numerical example we assume that E(rP) = 15%, ctp = 22%, and that the risk-free rate is rf = 7%. Thus the risk premium on the risky asset is E(rP) — rf = 8%.

With a proportion, y, in the risky portfolio, and 1 - y in the risk-free asset, the rate of return on the complete portfolio, denoted C, is rC where rc = yrP + (1 — y)rf

Taking the expectation of this portfolio's rate of return,

This result is easily interpreted. The base rate of return for any portfolio is the risk-free rate. In addition, the portfolio is expected to earn a risk premium that depends on the risk premium of the risky portfolio, E(rP) — rf, and the investor's position in the risky asset, y. Investors are assumed to be risk averse and thus unwilling to take on a risky position without a positive risk premium.

As we noted in Chapter 6, when we combine a risky asset and a risk-free asset in a portfolio, the standard deviation of the resulting complete portfolio is the standard deviation of the risky asset multiplied by the weight of the risky asset in that portfolio. Because the standard deviation of the risky portfolio is ctp = 22%,

Figure 7.2 The investment opportunity set with a risky asset and a risk-free asset in the expected return—standard deviation plane.

Figure 7.2 The investment opportunity set with a risky asset and a risk-free asset in the expected return—standard deviation plane.

which makes sense because the standard deviation of the portfolio is proportional to both the standard deviation of the risky asset and the proportion invested in it. In sum, the rate of return of the complete portfolio will have expected value E(rC) = rf + y[E(rP) - rf] = 7 + 8y and standard deviation aC = 22y.

The next step is to plot the portfolio characteristics (given the choice for y) in the expected return-standard deviation plane. This is done in Figure 7.2. The risk-free asset, F, appears on the vertical axis because its standard deviation is zero. The risky asset, P, is plotted with a standard deviation, aP = 22%, and expected return of 15%. If an investor chooses to invest solely in the risky asset, then y = 1.0, and the complete portfolio is P. If the chosen position is y = 0, then 1 - y = 1.0, and the complete portfolio is the risk-free portfolio F.

What about the more interesting midrange portfolios where y lies between zero and 1? These portfolios will graph on the straight line connecting points F and P. The slope of that line is simply [E(rP) - r]/aP (or rise/run), in this case, 8/22.

The conclusion is straightforward. Increasing the fraction of the overall portfolio invested in the risky asset increases expected return according to equation 7.1 at a rate of 8%. It also increases portfolio standard deviation according to equation 7.2 at the rate of 22%. The extra return per extra risk is thus 8/22 = .36.

To derive the exact equation for the straight line between F and P, we rearrange equation 7.2 to find that y = aC/aP, and we substitute for y in equation 7.1 to describe the expected return-standard deviation trade-off:

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

CONCEPT CHECK ^ QUESTION 2

Thus the expected return of the complete portfolio as a function of its standard deviation is a straight line, with intercept rf and slope as follows:

Figure 7.2 graphs the investment opportunity set, which is the set of feasible expected return and standard deviation pairs of all portfolios resulting from different values of y. The graph is a straight line originating at rf and going through the point labeled P.

This straight line is called the capital allocation line (CAL). It depicts all the risk-return combinations available to investors. The slope of the CAL, denoted S, equals the increase in the expected return of the complete portfolio per unit of additional standard deviation—in other words, incremental return per incremental risk. For this reason, the slope also is called the reward-to-variability ratio.

A portfolio equally divided between the risky asset and the risk-free asset, that is, where y = .5, will have an expected rate of return of E(rC) = 7 + .5 X 8 = 11%, implying a risk premium of 4%, and a standard deviation of oC = .5 X 22 = 11%. It will plot on the line FP midway between F and P. The reward-to-variability ratio is S = 4/11 = .36, precisely the same as that of portfolio P,8/22.

Can the reward-to-variability ratio, S = [E(rC) — rf]/oC, of any combination of the risky asset and the risk-free asset be different from the ratio for the risky asset taken alone, [E(rP) — r]/oP, which in this case is .36?

What about points on the CAL to the right of portfolio P? If investors can borrow at the (risk-free) rate of rf = 7%, they can construct portfolios that may be plotted on the CAL to the right of P.

Suppose the investment budget is \$300,000 and our investor borrows an additional \$120,000, investing the total available funds in the risky asset. This is a leveraged position in the risky asset; it is financed in part by borrowing. In that case

and 1 — y = 1 — 1.4 = — .4, reflecting a short position in the risk-free asset, which is a borrowing position. Rather than lending at a 7% interest rate, the investor borrows at 7%. The distribution of the portfolio rate of return still exhibits the same reward-to-variability ratio:

E(rC) = 7% + (1.4 X 8%) = 18.2% oC = 1.4 X 22% = 30.8%

30.8

As one might expect, the leveraged portfolio has a higher standard deviation than does an unleveraged position in the risky asset.

Of course, nongovernment investors cannot borrow at the risk-free rate. The risk of a borrower's default causes lenders to demand higher interest rates on loans. Therefore, the nongovernment investor's borrowing cost will exceed the lending rate of rf = 7%. Suppose the borrowing rate is rf = 9%. Then in the borrowing range, the reward-to-variability ratio, the slope of the CAL, will be [E(rP) — rfB]/oP = 6/22 = .27. The CAL will therefore be "kinked" at point P, as shown in Figure 7.3. To the left of P the investor

Figure 7.3 The opportunity set with differential borrowing and lending rates.

Figure 7.3 The opportunity set with differential borrowing and lending rates.

CONCEPT CHECK ^ QUESTION 3

is lending at 7%, and the slope of the CAL is .36. To the right of P, where y > 1, the investor is borrowing at 9% to finance extra investments in the risky asset, and the slope is .27.

In practice, borrowing to invest in the risky portfolio is easy and straightforward if you have a margin account with a broker. All you have to do is tell your broker that you want to buy "on margin." Margin purchases may not exceed 50% of the purchase value. Therefore, if your net worth in the account is \$300,000, the broker is allowed to lend you up to \$300,000 to purchase additional stock.2 You would then have \$600,000 on the asset side of your account and \$300,000 on the liability side, resulting in y = 2.0.

Suppose that there is a shift upward in the expected rate of return on the risky asset, from 15% to 17%. If all other parameters remain unchanged, what will be the slope of the CAL for y < 1 and y > 1?

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