<<

. 37
( 193 .)



>>

RISK-FREE PORTFOLIOS
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.

<<

. 37
( 193 .)



>>