Notice that the reward-to-variability ratio is the same for risky portfolio P and the complete

available by varying

portfolio allocation portfolio that was formed by mixing P and the risk-free asset in equal proportions.

between a risk-free

asset and a risky

portfolio.

Expected Risk Standard Reward-to-

Return Premium Deviation Variability Ratio

reward-to-

8

variability ratio Portfolio P: 15% 8% 22% 0.36

22

Ratio of risk premium

4

Portfolio C: 11% 4% 11% 0.36

to standard deviation.

11

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Fifth Edition

153

5 Risk and Return: Past and Prologue

In fact, the reward-to-variability ratio is the same for all complete portfolios that plot on the

capital allocation line. While the risk-return combinations differ, the ratio of reward to risk is

constant.

What about points on the line to the right of portfolio P in the investment opportunity

set? If investors can borrow at the (risk-free) rate of rf 7%, they can construct complete

portfolios that plot on the CAL to the right of P. They simply choose values of y greater

than 1.0.

Suppose the investment budget is $300,000, and our investor borrows an additional

$120,000, investing the $420,000 in the risky asset. This is a levered position in the risky asset,

EXAMPLE 5.6

which is financed in part by borrowing. In that case

Levered

420,000

y 1.4

Complete

300,000

Portfolios

and 1 y 1 1.4 0.4, reflecting a short position in the risk-free asset, or a borrowing

position. Rather than lending at a 7% interest rate, the investor borrows at 7%. The portfolio

rate of return is

E(rC) 7 (1.4 8) 18.2

Another way to find this portfolio rate of return is as follows. Your income statement will show

that you expect to earn $63,000 (15% of $420,000) and pay $8,400 (7% of $120,000) in

interest on the loan. Simple subtraction yields an expected profit of $54,000, which is 18.2%

of your investment budget of $300,000.

Your portfolio still exhibits the same reward-to-variability ratio:

1.4 22 30.8

C

E(rC) rf 11.2

S 0.36

30.8

C

As you might have expected, the levered portfolio has both a higher expected return and a

higher standard deviation than an unlevered position in the risky asset.

Of course, nongovernment investors cannot borrow at the risk-free rate. The risk of a bor-

rower™s default leads lenders to demand higher interest rates on loans. Therefore, the non-

government investor™s borrowing cost will exceed the lending rate of rf 7%.

Suppose the borrowing rate is rB 9%. Then, for y greater than 1.0 (the borrowing range),

the reward-to-variability ratio (the slope of the CAL) will be: [E(rP) rB]/ P 6/22 0.27.

Here, the borrowing rate (rB) replaces the lending rate (rf), reducing the “reward” (numerator)

in the reward-to-variability ratio. The CAL will be “kinked” at point P as in Figure 5.6. To the

left of P, where y 1, the investor is lending at 7% and the slope of the CAL is 0.36. To the

right of P, where y 1, the investor is borrowing (at a higher than risk-free rate) to finance

extra investments in the risky asset, and the slope is 0.27.

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 you want to buy

“on margin.” Margin purchases may not exceed 50% of the purchase value. For example, 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. 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.

<

8. Suppose there is a shift upward in the expected rate of return on the risky asset, Concept

from 15% to 17%. If all other parameters remain unchanged, what will be the

CHECK

slope of the CAL for y 1 and y 1?

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

Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

154 Part TWO Portfolio Theory

E(r)

CAL

P S(y > 1) = .27

E(rP) = 15%

rB = 9%

S(y ¤ 1) = .36

r’ = 7%

σ

σP = 22%

F I G U R E 5.6

The opportunity set with differential borrowing and lending rates

Risk Tolerance and Asset Allocation

We have developed the CAL, the graph of all feasible risk-return combinations available from

allocating the complete portfolio between a risky portfolio and a risk-free asset. The investor

confronting the CAL now must choose one optimal combination from the set of feasible choices.

This choice entails a trade-off between risk and return. Individual investors with different levels

of risk aversion, given an identical capital allocation line, will choose different positions in the

risky asset. Specifically, the more risk-averse investors will choose to hold less of the risky asset

and more of the risk-free asset.

Graphically, more risk-averse investors will choose portfolios near point F on the capital

allocation line plotted in Figure 5.5. More risk-tolerant investors will choose points closer to

P, with higher expected return and higher risk. The most risk-tolerant investors will choose

portfolios to the right of point P. These levered portfolios provide even higher expected re-

turns, but even greater risk.

The nearby box contains a further discussion of this risk-return trade-off, which sometimes

is characterized as a decision to “eat well,” versus “sleep well.” You will eat well if you earn

a high expected rate of return on your portfolio. However, this requires that you accept a large

risk premium and, therefore, a large amount of risk. Unfortunately, this risk may make it dif-

ficult to sleep well.

The investor™s asset allocation choice also will depend on the trade-off between risk and re-

turn. If the reward-to-variability ratio increases, then investors might well decide to take on

riskier positions. For example, suppose an investor reevaluates the probability distribution of

the risky portfolio and now perceives a greater expected return without an accompanying in-

crease in the standard deviation. This amounts to an increase in the reward-to-variability ratio

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

Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

The Right Mix: Make Money vs. Sleep Soundly

Plunged into doubt? and 40% bonds, remains a firm favorite with many in-

Amid the recent market turmoil, maybe you are vestment experts.

wondering whether you really have the right mix of in- A balanced portfolio isn™t a bad bet. But if you want

vestments. Here are a few thoughts to keep in mind: to calm your stock portfolio, I would skip bonds and in-

stead add cash investments such as Treasury bills and

money market funds. Ibbotson calculates that, over the

TAKING STOCK

past 25 years, a mix of 75% stocks and 25% Treasury

If you are a bond investor who is petrified of stocks, bills would have performed about as well as a mix of

the wild price swings of the past few weeks have prob- 60% stocks and 40% longer-term government bonds,

ably confirmed all of your worst suspicions. But the and with a similar level of portfolio price gyrations.

truth is, adding stocks to your bond portfolio could Moreover, the stock“cash mix offers more certainty,

bolster your returns, without boosting your portfolio™s because you know that even if your stocks fall in value,

overall gyrations. your cash never will. By contrast, both the stocks and

How can that be? While stocks and bonds often bonds in a balanced portfolio can get hammered at the

move up and down in tandem, this isn™t always the case, same time.

and sometimes stocks rise when bonds are tumbling.

Indeed, Chicago researchers Ibbotson Associates PATIENCE HAS ITS REWARDS, SOMETIMES

figure a portfolio that™s 100% in longer-term govern-

ment bonds has the same risk profile as a mix that in- Stocks are capable of generating miserable short-run re-

cludes 83% in longer-term government bonds and 17% sults. During the past 50 years, the worst five-calendar-

in the blue-chip stocks that constitute Standard & year stretch for stocks left investors with an annualized

Poor™s 500 stock index. loss of 2.4%.

The bottom line? Everybody should own some But while any investment can disappoint in the short

stocks. Even cowards. run, stocks do at least sparkle over the long haul. As a

long-term investor, your goal is to fend off the dual

threats of inflation and taxes and make your money

PADDING THE MATTRESS

grow. And on that score, stocks have been supreme.

On the other hand, maybe you™re a committed stock

market investor, but you would like to add a calming in-

SOURCE: Abridged from Jonathan Clements, “The Right Mix: Fine-

fluence to your portfolio. What™s your best bet?

Tuning a Portfolio to Make Money and Still Sleep Soundly,” The Wall

When investors look to mellow their stock portfolios, Street Journal, July 23, 1996. Reprinted by permission of Dow Jones

they usually turn to bonds. Indeed, the traditional bal- & Company, Inc. via Copyright Clearance Center, Inc. © 1996 Dow

anced portfolio, which typically includes 60% stocks Jones & Company, Inc. All rights reserved Worldwide.

or, equivalently, an increase in the slope of the CAL. As a result, this investor will choose a

higher y, that is, a greater position in the risky portfolio.

One role of a professional financial adviser is to present investment opportunity alterna-

tives to clients, obtain an assessment of the client™s risk tolerance, and help determine the ap-

propriate complete portfolio.5

5.6 PASSIVE STRATEGIES AND THE CAPITAL

MARKET LINE

The capital allocation line shows the risk-return trade-offs available by mixing risk-free as-

sets with the investor™s risky portfolio. Investors can choose the assets included in the risky

5

“Risk tolerance” is simply the flip side of “risk aversion.” Either term is a reasonable way to describe attitudes

toward risk. We generally find it easier to talk about risk aversion, but practitioners often use the term risk tolerance.

155

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

Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

156 Part TWO Portfolio Theory

Risk Premium (%)

TA B L E 5.5

Standard Reward-to-

Average rates of

Mean Deviation Variability Ratio

return, standard

deviations, and the

1926“1944 8.03 28.69 0.2798

reward-to-variability

1945“1963 14.41 18.83 0.7655

ratio of the risk

1964“1982 2.22 17.56 0.1265

premiums of large

1983“2001 9.91 14.77 0.6709

common stocks

1926“2001 8.64 20.70 0.4176

over one-month

bills over

1926“2001

and various

subperiods.

Source: Prepared from data in Table 5.3.

passive strategy portfolio using either passive or active strategies. A passive strategy is based on the premise

that securities are fairly priced and it avoids the costs involved in undertaking security analy-

Investment policy that

sis. Such a strategy might at first blush appear to be naive. However, we will see in Chapter 8

avoids security

that intense competition among professional money managers might indeed force security

analysis.

prices to levels at which further security analysis is unlikely to turn up significant profit op-

portunities. Passive investment strategies may make sense for many investors.

To avoid the costs of acquiring information on any individual stock or group of stocks, we

may follow a “neutral” diversification approach. A natural strategy is to select a diversified

portfolio of common stocks that mirrors the corporate sector of the broad economy. This re-