History shows us that long-term bonds have been riskier investments than investments in

Treasury bills and that stock investments have been riskier still. On the other hand, the riskier

investments have offered higher average returns. Investors, of course, do not make all-or-

nothing choices from these investment classes. They can and do construct their portfolios

using securities from all asset classes. Some of the portfolio may be in risk-free Treasury bills

and some in high-risk stocks.

The most straightforward way to control the risk of the portfolio is through the fraction of

the portfolio invested in Treasury bills and other safe money market securities versus risky as-

sets. This is an example of an asset allocation choice”a choice among broad investment

asset allocation

classes, rather than among the specific securities within each asset class. Most investment pro-

Portfolio choice

fessionals consider asset allocation the most important part of portfolio construction. Consider

among broad

this statement by John Bogle, made when he was the chairman of the Vanguard Group of

investment classes.

Investment Companies:

The most fundamental decision of investing is the allocation of your assets: How much should

you own in stock? How much should you own in bonds? How much should you own in cash re-

serves? . . . That decision [has been shown to account] for an astonishing 94% of the differences

Bodie’Kane’Marcus: II. Portfolio Theory 5. Risk and Return: Past © The McGraw’Hill

Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

149

5 Risk and Return: Past and Prologue

in total returns achieved by institutionally managed pension funds. . . . There is no reason to be-

lieve that the same relationship does not also hold true for individual investors.4

Therefore, we start our discussion of the risk-return trade-off available to investors by exam-

ining the most basic asset allocation choice: the choice of how much of the portfolio to place

in risk-free money market securities versus other risky asset classes.

We will denote the investor™s portfolio of risky assets as P, and the risk-free asset as F. We

will assume for the sake of illustration that the risky component of the investor™s overall port-

folio comprises two mutual funds: one invested in stocks and the other invested in long-term

bonds. For now, we take the composition of the risky portfolio as given and focus only on the

allocation between it and risk-free securities. In the next chapter, we turn to asset allocation

and security selection across risky assets.

The Risky Asset

When we shift wealth from the risky portfolio (P) to the risk-free asset, we do not change the

relative proportions of the various risky assets within the risky portfolio. Rather, we reduce the

relative weight of the risky portfolio as a whole in favor of risk-free assets.

A simple example demonstrates the procedure. Assume the total market value of an in-

vestor™s portfolio is $300,000. Of that, $90,000 is invested in the Ready Assets money market

fund, a risk-free asset. The remaining $210,000 is in risky securities, say $113,400 in the Van-

guard S&P 500 index fund (called the Vanguard 500 Index Fund) and $96,600 in Fidelity™s In-

vestment Grade Bond Fund.

The Vanguard fund (V) is a passive equity fund that replicates the S&P 500 portfolio. The

Fidelity Investment Grade Bond Fund (IG) invests primarily in corporate bonds with high

safety ratings and also in Treasury bonds. We choose these two funds for the risky portfolio in

the spirit of a low-cost, well-diversified portfolio. While in the next chapter we discuss port-

folio optimization, here we simply assume the investor considers the given weighting of V and

IG to be optimal.

The holdings in Vanguard and Fidelity make up the risky portfolio, with 54% in V and 46%

in IG.

wV 113,400/210,000 0.54 (Vanguard)

wIG 96,600/210,000 0.46 (Fidelity)

complete

The weight of the risky portfolio, P, in the complete portfolio, including risk-free as well as

portfolio

risky investments, is denoted by y, and so the weight of the money market fund is 1 y.

The entire portfolio

y 210,000/300,000 0.7 (risky assets, portfolio P)

including risky and

1 y 90,000/300,000 0.3 (risk-free assets) risk-free assets.

The weights of the individual assets in the complete portfolio (C) are:

Vanguard 113,400/300,000 0.378

Fidelity 96,600/300,000 0.322

Portfolio P 210,000/300,000 0.700

Ready Assets F 90,000/300,000 0.300

Portfolio C 300,000/300,000 1.000

Suppose the investor decides to decrease risk by reducing the exposure to the risky portfo-

lio from y 0.7 to y 0.56. The risky portfolio would total only 0.56 300,000 $168,000,

requiring the sale of $42,000 of the original $210,000 risky holdings, with the proceeds used to

4

John C. Bogle, Bogle on Mutual Funds (Burr Ridge, IL: Irwin Professional Publishing, 1994), p. 235.

Bodie’Kane’Marcus: II. Portfolio Theory 5. Risk and Return: Past © The McGraw’Hill

Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

150 Part TWO Portfolio Theory

purchase more shares in Ready Assets. Total holdings in the risk-free asset will increase to

300,000 (1 0.56) $132,000 (the original holdings plus the new contribution to the money

market fund: 90,000 42,000 $132,000).

The key point is that we leave the proportion of each asset in the risky portfolio unchanged.

Because the weights of Vanguard and Fidelity in the risky portfolio are 0.54 and 0.46 respec-

tively, we sell 0.54 42,000 $22,680 of Vanguard and 0.46 42,000 $19,320 of

Fidelity. After the sale, the proportions of each fund in the risky portfolio are unchanged.

113,400 22,680

wV 0.54 (Vanguard)

210,000 42,000

96,600 19,320

wIG 0.46 (Fidelity)

210,000 42,000

This procedure shows that rather than thinking of our risky holdings as Vanguard and

Fidelity separately, we may view our holdings as if they are in a single fund holding Vanguard

and Fidelity in fixed proportions. In this sense, we may treat the risky fund as a single risky

asset, that asset being a particular bundle of securities. As we shift in and out of safe assets, we

simply alter our holdings of that bundle of securities commensurately.

Given this simplification, we now can turn to the desirability of reducing risk by changing

the risky/risk-free asset mix, that is, reducing risk by decreasing the proportion y. Because we

do not alter the weights of each asset within the risky portfolio, the probability distribution of

the rate of return on the risky portfolio remains unchanged by the asset reallocation. What will

change is the probability distribution of the rate of return on the complete portfolio that is

made up of the risky and risk-free assets.

>

6. What will be the dollar value of your position in Vanguard and its proportion in your

Concept

complete portfolio if you decide to hold 50% of your investment budget in Ready

CHECK Assets?

The Risk-Free Asset

The power to tax and to control the money supply lets the government, and only government,

issue default-free bonds. The default-free guarantee by itself is not sufficient to make the

bonds risk-free in real terms, since inflation affects the purchasing power of the proceeds from

an investment in T-bills. The only risk-free asset in real terms would be a price-indexed gov-

ernment bond. Even then, a default-free, perfectly indexed bond offers a guaranteed real rate

to an investor only if the maturity of the bond is identical to the investor™s desired holding

period.

These qualifications notwithstanding, it is common to view Treasury bills as the risk-free

asset. Because they are short-term investments, their prices are relatively insensitive to inter-

est rate fluctuations. An investor can lock in a short-term nominal return by buying a bill and

holding it to maturity. Any inflation uncertainty over the course of a few weeks, or even

months, is negligible compared to the uncertainty of stock market returns.

In practice, most investors treat a broader range of money market instruments as effectively

risk-free assets. All the money market instruments are virtually immune to interest rate risk

(unexpected fluctuations in the price of a bond due to changes in market interest rates) because

of their short maturities, and all are fairly safe in terms of default or credit risk.

Money market mutual funds hold, for the most part, three types of securities: Treasury bills,

bank certificates of deposit (CDs), and commercial paper. The instruments differ slightly in

their default risk. The yields to maturity on CDs and commercial paper, for identical maturi-

ties, are always slightly higher than those of T-bills. A history of this yield spread for 90-day

CDs is shown in Figure 2.3 in Chapter 2.

Bodie’Kane’Marcus: II. Portfolio Theory 5. Risk and Return: Past © The McGraw’Hill

Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

151

5 Risk and Return: Past and Prologue

Money market funds have changed their relative holdings of these securities over time, but

by and large, T-bills make up only about 15% of their portfolios. Nevertheless, the risk of such

blue-chip, short-term investments as CDs and commercial paper is minuscule compared to

that of most other assets, such as long-term corporate bonds, common stocks, or real estate.

Hence, we treat money market funds as representing the most easily accessible risk-free asset

for most investors.

Portfolio Expected Return and Risk

Now that we have specified the risky portfolio and the risk-free asset, we can examine the

risk-return combinations that result from various investment allocations between these two

assets. Finding the available combinations of risk and return is the “technical” part of asset

allocation; it deals only with the opportunities available to investors given the features of the

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, given his or her level of risk aversion.

Since we assume the composition of the optimal risky portfolio (P) already has been deter-

mined, the concern here is with the proportion of the investment budget (y) to be allocated to

the risky portfolio. The remaining proportion (1 y) is to be invested in the risk-free asset (F).

We denote the actual risky rate of return by rP, the expected rate of return on P by E(rP),

and its standard deviation by P. The rate of return on the risk-free asset is denoted as rf. In the

numerical example, we assume E(rP) 15%, P 22%, and rf 7%. Thus, the risk premium

on the risky asset is E(rP) rf 8%.

Let™s start with two extreme cases. If you invest all of your funds in the risky asset, that is,

if you choose y 1.0, the expected return on your complete portfolio will be 15% and the

standard deviation will be 22%. This combination of risk and return is plotted as point P in

Figure 5.5. At the other extreme, you might put all of your funds into the risk-free asset, that

E(r)

CAL = Capital

allocation

line

P y = 1.25

E(rP) = 15%

y = .50 E(rP) “ r’ = 8%

S = 8/22

r’ = 7% F

σ

σP = 22%

F I G U R E 5.5

The investment opportunity set with a risky asset and a risk-free asset

Bodie’Kane’Marcus: II. Portfolio Theory 5. Risk and Return: Past © The McGraw’Hill

Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

152 Part TWO Portfolio Theory

is, you choose y 0. In this case, your portfolio would behave just as the risk-free asset, and

you would earn a riskless return of 7%. (This choice is plotted as point F in Figure 5.5.)

Now consider more moderate choices. For example, if you allocate equal amounts of your

overall or complete portfolio, C, to the risky and risk-free assets, that is, if you choose y 0.5,

the expected return on the complete portfolio will be an average of the expected return on

portfolios F and P. Therefore, E(rC) 0.5 7% 0.5 15% 11%. The risk premium of

the complete portfolio is therefore 11% 7% 4%, which is half of the risk premium of P.

The standard deviation of the portfolio also is one-half of P™s, that is, 11%. When you reduce

the fraction of the complete portfolio allocated to the risky asset by half, you reduce both the

risk and risk premium by half.

To generalize, the risk premium of the complete portfolio, C, will equal the risk premium

of the risky asset times the fraction of the portfolio invested in the risky asset.

E(rC) rf y[E(rP) rf] (5.9)

The standard deviation of the complete portfolio will equal the standard deviation of the risky

asset times the fraction of the portfolio invested in the risky asset.

y (5.10)

C P

In sum, both the risk premium and the standard deviation of the complete portfolio increase in

proportion to the investment in the risky portfolio. Therefore, the points that describe the risk

and return of the complete portfolio for various asset allocations, that is, for various choices

of y, all plot on the straight line connecting F and P, as shown in Figure 5.5, with an intercept

of rf and slope (rise/run) of

E(rP) rf 15 7

S 0.36 (5.11)

22

P

>

7. What are the expected return, risk premium, standard deviation, and ratio of risk

Concept

premium to standard deviation for a complete portfolio with y 0.75?

CHECK

The Capital Allocation Line

The line plotted in Figure 5.5 depicts the risk-return combinations available by varying asset

capital allocation, that is, by choosing different values of y. For this reason, it is called the capital

allocation line allocation line, or CAL. The slope, S, of the CAL equals the increase in expected return that

an investor can obtain per unit of additional standard deviation. In other words, it shows extra

Plot of risk-return

return per extra risk. For this reason, the slope also is called the reward-to-variability ratio.