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we will use a hypothetical portfolio for which monthly returns in the past five years resulted
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%.
Bodie’Kane’Marcus: VI. Active Investment 20. Performance Evaluation © The McGraw’Hill
Essentials of Investments, Management and Active Portfolio Companies, 2003
Fifth Edition Management




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
Bodie’Kane’Marcus: VI. Active Investment 20. Performance Evaluation © The McGraw’Hill
Essentials of Investments, Management and Active Portfolio Companies, 2003
Fifth Edition Management




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.
Bodie’Kane’Marcus: VI. Active Investment 20. Performance Evaluation © The McGraw’Hill
Essentials of Investments, Management and Active Portfolio Companies, 2003
Fifth Edition Management




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
Bodie’Kane’Marcus: VI. Active Investment 20. Performance Evaluation © The McGraw’Hill
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

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