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Average M-Squared Needed to
Fund Name Annual Risk-Adjusted* Adjust**

Fidelity Select Food & Agriculture 21.5% 20.5% 6%
Putnam Fund for Gr. & Inc. (A shares) 16.7 20.1 30
Mutual Beacon (Z shares) 17.2 20.0 24
Gateway Index Plus 11.8 20.0 129
Fidelity Contrafund 23.0 19.4 21
Hancock Regional Bank (B shares) 26.1 19.3 33
Mutual Qualified (Z shares) 17.4 19.1 15
Benchmark: S&P 500 17.9 17.9 0

*The return an investor would have earned by combining fund shares with cash”or alternatively, borrowing
money”to create a hypothetical portfolio with the same volatility as the benchmark S&P 500.
**Percent of the hypothetical portfolio needed to be invested in cash equivalents to match the benchmark™s
volatility”or, if a negative figure, amount of borrowing.
Source: Morgan Stanley Dean Witter, using quarterly returns from Morningstar.

SOURCE: Karen Damato,“Risky Business: Is Your Favorite Mutual Clearance Center, Inc. © 1998 Dow Jones & Company, Inc. All Rights
Fund Walking a Highwire?” The Wall Street Journal, March 20, 1998. Reserved Worldwide.
Reprinted by permission of Dow Jones & Company, Inc. via Copyright




be the entire portfolio, with no further opportunities for diversification. In this case, the Sharpe
measure (and M 2) is the appropriate basis on which to evaluate the portfolio manager. Because
that manager is in charge of the entire portfolio, she must be attentive to diversification of
firm-specific risk and should be judged on her achievement of excess return to total portfolio
volatility.
In contrast, suppose your pension fund is large, and you envision hiring many managers,
giving each a fraction of the total assets of the plan. You hope that hiring a set of managers,
each with an investment specialty, will enhance returns. This means the pension plan effec-
tively ends up with a portfolio of portfolio managers. Each manager can pursue his or her spe-
cialty without paying much attention to issues of diversification because the plan as a whole
will have diversified returns across the several managed portfolios. With assets spread across
many portfolio managers, the residual firm-specific risks of each portfolio become irrelevant
because of diversification across portfolios. In this case, only nondiversifiable risk should
matter. Such circumstances call for the use of a beta-based risk adjustment, and the Treynor
measure would be appropriate.
We can use a variant of the Treynor measure to express the difference in portfolio per-
formance in terms of rates of return. Consider a portfolio P, with the characteristics sum-
marized in Table 20.2. To convert the difference between the Treynor measure of P and that
of the market to an easier-to-interpret percentage return basis, we will consider a measure
689
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Essentials of Investments, Management and Active Portfolio Companies, 2003
Fifth Edition Management




690 Part SIX Active Investment Management


Portfolio P Portfolio Q Market
TA B L E 20.2
Excess return, r rf 13% 20% 10%
Portfolio
Beta 0.80 1.80 1.0
performance
Alpha* 5% 2% 0
Treynor measure 16.25 11.11 10

*Alpha Excess return (Beta 3 Market excess return)
(r rf) (rM rf)
r [rf (rM rf)]


analogous to M 2 that we call Treynor-Square (T 2). To start, we will create a portfolio P* by
mixing P and T-bills to match the beta of the market, 1.0. (In contrast, to obtain the M 2 meas-
ure we mixed bills and P to match the standard deviation of the market.)
Recall that both the beta and the excess return of T-bills are zero. Therefore, if we construct
portfolio P* by investing w in portfolio P and 1 w in T-bills, the excess return and beta of
P* will be:
RP* wRP , w
P* P

We can construct P* from portfolio P and T-bills with any desired beta. If you desire a beta
equal to P*, simply set: w P* / P; hence, RP* wRP RP P* / P. Because the market
beta is 1.0, P* can be constructed to match the market beta by setting
w M/ P 1/ 1/0.8
P

Therefore, RP* wRP (1/ P) RP 13%/0.8 16.25%. Notice that w 1 implies that
P* is a leveraged version of P. Leverage is necessary to match the market beta since the beta
of P is less than that of the market. Notice also that RP* is in fact the Treynor measure of port-
folio P”it equals the excess return of P divided by beta.
Portfolio P* is constructed to have the same beta as the market, so the difference between
the return of P* and the market is a valid measure of relative performance when systematic
risk is of concern to investors. Therefore, as an analogy to the M 2 measure of performance, we
define the Treynor-square measure as
2
TP RP* RM RP / RM 13/0.8 10 6.25%.
P

Figure 20.3A shows a graphical representation of the T 2 measure for portfolio P. Portfolio
P* is a leveraged version of portfolio P with the same beta as the market, 1.0. The T 2 measure
is simply the difference in expected returns at this common beta.
To compare the T 2 measure for the two portfolios, P and Q, described in Table 20.2, we
compute for portfolio Q
2
TQ 20/1.8 10 1.11%
which is smaller than the T 2 measure of P by 5.14%. Figure 20.3B shows the positions of
P and Q. Portfolios on a given line from the origin (notice that the vertical axis measures ex-
cess returns), all have the same Treynor measure. P is located on a steeper line because it has
a higher Treynor measure. As a result, P* is farther above the SML than Q*, and the difference
between the T 2 measures, the line segment P*Q*, measures the difference in the systematic
risk-adjusted percent return.
It is clear that we must adjust portfolio returns for risk before evaluating performance. The
nearby box shows how important such adjustments can be. It reports on the results of a series
of investment “contests” between investment professionals and randomly chosen stocks (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




691
20 Performance Evaluation and Active Portfolio Management




F I G U R E 20.3
A. T 2 of portfolio P
Treynor-Square
R r rf
measure


P* SML
RP/βP 16.25
RP*
2
T
P
13
RP
±P 13 8 5%
10
RM M
8




β
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
βP βM

B. Comparison of P and Q
R r rf
Q
20.00
18.00
16.25 P*
SML
P
13.00 Q*
11.11
10.00 M
8.00




0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
βP βM βQ β




dartboard portfolio) sponsored by The Wall Street Journal. While the professionals have
tended to win the contest, the box shows that risk-adjustment nearly wipes out the differential.

Risk Adjustments with Changing Portfolio Composition
One potential problem with risk-adjustment techniques is that they all assume that portfolio
risk, whether it is measured by standard deviation or beta, is constant over the relevant time
period. This isn™t necessarily so. If a manager attempts to increase portfolio beta when she
thinks the market is about to go up and to decrease beta when she is pessimistic, both the stan-
dard deviation and the beta of the portfolio will change over time. This can wreak havoc with
our performance measures.
Bodie’Kane’Marcus: VI. Active Investment 20. Performance Evaluation © The McGraw’Hill
Essentials of Investments, Management and Active Portfolio Companies, 2003
Fifth Edition Management




Putting Away the Darts After 14 Years:
The Wall Street Journal™s Dartboard Ends Its Run
average 10.2% investment gain. The darts managed
After 14 years and hundreds of attempts to pick win-
just a 3.5% six-month gain, on average, over the same
ning stocks, Wall Street Journal staffers are putting
period, while the Dow industrials posted an average
away their darts.
rise of 5.6%.
The darts had a good run, even if they didn™t often
What do 14 years of throwing darts prove? Not
win. More than 200 investment professionals good-
much, says the man who inspired the competition.
naturedly pitted their stock-picking skills against the re-
Dr. Malkiel, who gamely threw the first dart when the
sults of a portfolio assembled by throwing darts at the
competition began in October 1988, said from the out-
stock tables, as well as the performance of the Dow
set that this contest wasn™t a fair test of the efficient
Jones Industrial Average. But the contest is now going
market theory. The reasons, he maintains, are the
to be retired.
small number of stocks involved in the contest and the
It was all a light-hearted test of the efficient market
fact that the “publicity effect” of an article in The Wall
theory. That theory, popularized by
Street Journal might make the stocks selected perform
Princeton University economics profes- INVESTMENT
better than others, at least temporarily.
sor Burton Malkiel in his 1973 book, DARTBOARD
“The darts were a nice metaphor, but four darts
“A Random Walk Down Wall Street,”
were not what I recommended,” Dr. Malkiel explains.
holds that since all available information is quickly fac-
The idea of the efficient market theory, he adds, is to
tored into stock prices, all stocks present equal chances
buy a broad cross-section of stocks.
for gains.
He also says that the pros tend to pick stocks that
“Taken to its logical extreme,” Dr. Malkiel wrote, the
are riskier, and thus rise faster when the market is going
theory “means that a blindfolded monkey throwing
up, as it was for most of the 14 years of the competi-
darts at a newspaper™s financial pages could select a
tion. Still, he says, the contest “was fun.” [In a paper
portfolio that would do just as well as one carefully se-
co-authored with Gilbert Metcalf, Malkiel found that
lected by the experts.”
pros™ selection had a high beta of 1.4, implying that on
The Wall Street Journal never used monkeys, and
a risk-adjusted basis, the pros™ margin of superiority
the dart throwers weren™t blindfolded”too risky, per-
shrunk to a statistically insignificant 0.4%.]
haps”as they threw the darts at stock tables posted on
our office walls. But in the end, after 142 six-month
SOURCE: Georgette Jasen, The Wall Street Journal, April 18, 2002.
contests, the pros came out ahead, racking up an




Suppose the Sharpe measure of the passive strategy (investing in a market index fund) is
0.4. A portfolio manager is in search of a better, active strategy. Over an initial period of, say,
20.1 EXAMPLE four quarters, he executes a low-risk or defensive strategy with an annualized mean excess
return of 1% and a standard deviation of 2%. This makes for a Sharpe measure of 0.5, which
Risk Measurement
beats the passive strategy.
with Changing
Over the next period of another four quarters, this manager finds that a high-risk strategy
Portfolio
is optimal, with an annual mean excess return of 9% and standard deviation of 18%. Here
Composition
again the Sharpe measure is 0.5. Over the two years, our manager maintains a better-than-
passive Sharpe measure.
Figure 20.4 shows a pattern of (annualized) quarterly returns that is consistent with our
description of the manager™s strategy over two years. In the first four quarters, the excess re-
turns are 1%, 3%, 1%, and 3%, making for an average of 1% and standard deviation of
2%. In the next four quarters, the returns are 9%, 27%, 9%, and 27%, making for an av-
erage of 9% and standard deviation of 18%. Thus, each year undoubtedly exhibits a Sharpe

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