in the following statistics. We also present comparable data for the market portfolio for the

same period.

Portfolio Market

Average return 16% 14%

Standard deviation 20% 24%

Beta 0.8 1.0

Finally, suppose the average return on risk-free assets during the five-year period was 6%.

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686 Part SIX Active Investment Management

The Sharpe measure, the Treynor measure, and the Jensen measure are three risk-adjusted

performance statistics. The Sharpe measure is calculated as follows:

rp

¯ rf

¯

p

The Sharpe measure divides average portfolio excess return over the sample period by the

Sharpe measure

standard deviation of returns over that period. The numerator is the incremental return the

Reward-to-volatility

portfolio earned in comparison with an alternative investment in the risk-free asset, and the de-

ratio; ratio of

nominator is the increment in portfolio volatility compared with the risk-free alternative.

portfolio excess return

to standard deviation. Therefore, the ratio measures the reward to (total) volatility trade-off. (The bars over rp as well

as rf denote the fact that, because the risk-free rate may not be constant over the measurement

period, we are taking a sample average of both.) Using our numbers, the Sharpe measure for

the portfolio is (16 6)/20 0.5, while for the market it is (14 6)/24 0.33.

In contrast, the Treynor measure is given as follows:

rp

¯ rf

¯

p

Like Sharpe™s, the Treynor measure gives average excess return per unit of risk incurred, but

Treynor measure

it uses systematic risk instead of total risk. The Treynor measure for the portfolio over this

Ratio of portfolio

period is (16 6)/0.8 12.5, while for the market portfolio it is (14 6)/1.0 8.

excess return to beta.

In contrast to these two methods, the Jensen measure is as follows:

rp

¯ [¯f

r p(¯M

r rf )]

¯

p

The Jensen measure is the average return on the portfolio over and above that predicted

Jensen measure

by the CAPM, given the portfolio™s beta and the average market return. The Jensen

The alpha of

measure is the portfolio™s alpha value. Using our numbers, the Jensen measure is

an investment.

16 [6 0.8(14 6)] 3.6%.

Each measure has its own appeal. In this instance, all three measures are consistent in re-

vealing that the portfolio outperformed the market benchmark on a risk-adjusted basis. How-

ever, this need not be the case. As the following Concept Check illustrates, the three measures

do not necessarily provide consistent assessments of relative performance, as the approach

used to adjust returns for risk differ substantially.

>

1. Consider the following data for a particular sample period:

Concept

CHECK

Portfolio P Market M

Average return 35% 28%

Beta 1.2 1.0

Standard deviation 42% 30%

Calculate the following performance measures for portfolio P and the market:

Sharpe, Jensen (alpha), and Treynor. The T-bill rate during the period was 6%. By

which measures did portfolio P outperform the market?

The M2 Measure of Performance

While the Sharpe ratio can be used to rank portfolio performance, its numerical value is not

easy to interpret. Comparing the ratios for portfolios M and P in Concept Check 1, you should

have found that SP .69 and SM .73. This suggests that portfolio P underperformed the

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687

20 Performance Evaluation and Active Portfolio Management

F I G U R E 20.2

E(r)

The M2 of portfolio P

CML

CAL(P)

rP 35% P

M

rM 28% 2

M = rP* rM 1.3%

P*

rP* 26.7%

rf 6%

σ

σM σP

30% 42%

market index. But is a difference of .04 in the Sharpe ratio economically meaningful? We are

used to comparing rates of return, but these ratios are difficult to interpret.

A variant of Sharpe™s measure was introduced by Graham and Harvey and by Leah

Modigliani of Morgan Stanley and her grandfather Franco Modigliani, past winner of the No-

bel Prize for economics.1 Their approach has been dubbed the M 2 measure (for Modigliani-

squared). Like the Sharpe ratio, the M 2 measure focuses on total volatility as a measure of

risk, but its risk-adjusted measure of performance has the easy interpretation of a differential

return relative to the benchmark index.

To compute the M 2 measure, we imagine that a managed portfolio, P, is mixed with a po-

sition in T-bills so that the complete, or “adjusted,” portfolio matches the volatility of a mar-

ket index such as the S&P 500. For example, if the managed portfolio has 1.5 times the

standard deviation of the index, the adjusted portfolio would be two-thirds invested in the

managed portfolio and one-third invested in bills. The adjusted portfolio, which we call P*,

would then have the same standard deviation as the index. (If the managed portfolio had lower

standard deviation than the index, it would be leveraged by borrowing money and investing

the proceeds in the portfolio.) Because the market index and portfolio P* have the same stan-

dard deviation, we may compare their performance simply by comparing returns. This is the

M 2 measure:

M2 rP* rM

In the example of Concept Check 1, P has a standard deviation of 42% versus a market

standard deviation of 30%. Therefore, the adjusted portfolio P* would be formed by mixing

bills and portfolio P with weights 30/42 .714 in P and 1 .714 .286 in bills. The ex-

pected return on this portfolio would be (.286 6%) (.714 35%) 26.7%, which is

1.3% less than the market return. Thus portfolio P has an M 2 measure of 1.3%.

A graphical representation of the M 2 measure appears in Figure 20.2. We move down the

capital allocation line corresponding to portfolio P (by mixing P with T-bills) until we reduce

1

John R. Graham and Campbell R. Harvey, “Grading the performance of market timing newsletter,” Financial Ana-

lysts Journal 53 (November/December 1997), pp. 54“66; and Franco Modigliani and Leah Modigliani, “Risk-

Adjusted Performance,” Journal of Portfolio Management, Winter 1997, pp. 45“54.

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Risky Business: Is Your Favorite

Mutual Fund Walking a High Wire?

The Modigliani approach allows investors to analyze

The top-performing stock funds of the past decade in-

a bunch of disparate funds as if they had all shown the

clude many funds that took big risks to produce their

same volatility. It does that by hypothetically blending

outsized gains.

shares of a volatile fund with cash or by using borrowed

But investors worried that the next decade won™t be

money to leverage up the jumpiness of a more sedate

as sweet as the last may want to focus on a different

fund.

approach to ranking the “best” funds”one identifying

Ms. Modigliani™s effort may suggest some interest-

the funds that delivered the highest returns relative to

ing funds for investors to investigate. At the same time,

the amount of risk they took.

though, the ranking points out some limitations of the

Such a list has just been published by Morgan Stan-

M-squared measure in particular and mechanical,

ley Dean Witter. Leah Modigliani, one of the stock

mathematical screenings in general.

strategists at the firm, a unit of Morgan Stanley, Dean

For instance, Ms. Modigliani and others say in-

Witter, Discover, ranked all stock funds with a 10-year

vestors shouldn™t take the top ranking of Fidelity Select

history using a measure of “risk adjusted” performance

Food as a clear signal to rush out and buy that fund.

she developed with her grandfather, Nobel laureate

The top ranking confirms that stocks of food com-

Franco Modigliani.

panies and other household-name consumer-products

While the Modigliani-Modigliani or “M-squared”

businesses have been strong and steady performers for

measure is only one of several ways to gauge risk-

the past decade. But the performance of those stocks

adjusted performance, the topic is an important one

could obviously be far different in coming years.

given today™s roaring stock prices. Many investment

“Screens by their very nature are backward-looking,

advisers and fund executives fret that investors aren™t

not forward-looking,” notes Rick Spillane, a senior vice

paying enough attention to portfolio risks that will loom

president in Fidelity™s equity division. While Fidelity Se-

large if and when the booming stock market turns ugly.

lect Food hasn™t been extraordinarily risky”in the sense

Neither a high-risk fund nor a low-risk fund is inher-

of being volatile, that is”this and other “sector” funds

ently better, Ms. Modigliani says, but “we want to be

clearly take a lot of risk by concentrating in only one

sure that we are being rewarded for the risks that we

slice of the economy.

take.” The Modigliani approach defines risk as the vari-

That criticism gets to the limitations of M-squared

ability or unpredictability of quarterly fund returns. A

and other backward-looking measures, including plain

fund™s M-squared return is the hypothetical return an

old total return. They don™t tell which investment styles

investor would have earned in a particular period if the

will be in and out of favor in the future. They can

fund™s risk had been adjusted to match that of a bench-

be misleading when a mutual fund has changed its

mark such as Standard & Poor™s 500-stock index.

the standard deviation of the adjusted portfolio to match that of the market index. The

M 2 measure is then the vertical distance (i.e., the difference in expected returns) between port-

folios P* and M. You can see from Figure 20.2 that P will have an M 2 measure below that of

the market when its capital allocation line is less steep than the capital market line, that is,

when its Sharpe ratio is less than that of the market index.

The nearby box reports on the growing popularity of the M 2 measure in the investment

community.

Choosing the Right Measure of Risk

Because different risk adjustment procedures can yield different implications for performance

evaluation, it is essential that you choose the appropriate measure for the task. For example,

suppose you are a pension fund manager who is selecting potential portfolio managers to over-

see investment of the fund™s assets. If you envision hiring one investment manager to manage

all the fund™s assets, then you must be concerned with the total variability of investment per-

formance. Both the systematic and firm-specific risk remaining in the portfolio will affect total

risk because the pension fund is not diversified across managers. The manager™s portfolio will

688

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Essentials of Investments, Management and Active Portfolio Companies, 2003

Fifth Edition Management

investment style or its portfolio manager or grown a lot number for a particular fund, Ms. Modigliani says, “you

in size. While you can look at an attractive historical are not sure to get that return going forward.”

Stock Funds: Risk-Adjusted Winners

Among stock funds with a 10-year history, these scored highest on the Modigliani-Modigliani (M-squared)

measure of risk-adjusted performance.

Returns

1988“1997 % Cash