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Bodie−Kane−Marcus:
Essentials of Investments,
Fifth Edition
VI. Active Investment
Management
20. Performance Evaluation
and Active Portfolio
Management
© The McGraw−Hill
Companies, 2003
where E(r
M
) Ϫ r
f
is the risk premium on M, and ␴
M
is the standard deviation of M. To make a
rational allocation of funds requires an estimate of ␴
M
and E(r
M
), so even a passive investor
needs to do some forecasting.
Forecasting E(r
M
) and ␴
M
is complicated further because security classes are affected
by different environment factors. Long-term bond returns, for example, are driven largely by
changes in the term structure of interest rates, while returns on equity depend also on changes
in the broader economic environment, including macroeconomic factors besides interest rates.

Once you begin considering how economic conditions influence separate sorts of investments,
you might as well use a sophisticated asset allocation program to determine the proper mix for
the portfolio. It is easy to see how investors get lured away from a purely passive strategy.
Even the definition of a “pure” passive strategy is not very clear-cut, as simple strategies
involving only the market index portfolio and risk-free assets now seem to call for market
analysis. Our strict definition of a pure passive strategy is one that invests only in index funds
and weights those funds by fixed proportions that do not change in response to market condi-
tions: a portfolio strategy that always places 60% in a stock market index fund, 30% in a bond
index fund, and 10% in a money market fund, regardless of expectations.
Active management is attractive because the potential profit is enormous, even though
competition among managers is bound to drive market prices to near-efficient levels. For
prices to remain efficient to some degree, decent profits to diligent analysts must be the rule
rather than the exception, although large profits may be difficult to earn. Absence of profits
would drive people out of the investment management industry, resulting in prices moving
away from informationally efficient levels.
Objectives of Active Portfolios
What does an investor expect from a professional portfolio manager, and how do these ex-
pectations affect the manager’s response? If all clients were risk neutral (indifferent to risk),
the answer would be straightforward: The investment manager should construct a portfolio
with the highest possible expected rate of return, and the manager should then be judged by
the realized average rate of return.
When the client is risk averse, the answer is more difficult. Lacking standards to proceed
by, the manager would have to consult with each client before making any portfolio decision
in order to ascertain that the prospective reward (average return) matched the client’s attitude
toward risk. Massive, continuous client input would be needed, and the economic value of
professional management would be questionable.
Fortunately, the theory of mean-variance efficiency allows us to separate the “product deci-
sion,” which is how to construct a mean-variance efficient risky portfolio, from the “con-
sumption decision,” which describes the investor’s allocation of funds between the efficient
risky portfolio and the safe asset. You have learned already that construction of the optimal

risky portfolio is purely a technical problem and that there is a single optimal risky portfolio
appropriate for all investors. Investors differ only in how they apportion investment between
that risky portfolio and the safe asset.
The mean-variance theory also speaks to performance in offering a criterion for judging
managers on their choice of risky portfolios. In Chapter 6, we established that the optimal
risky portfolio is the one that maximizes the reward-to-variability ratio, that is, the expected
excess return divided by the standard deviation. A manager who maximizes this ratio will sat-
isfy all clients regardless of risk aversion.
Clients can evaluate managers using statistical methods to draw inferences from realized
rates of return about prospective, or ex ante, reward-to-variability ratios. The Sharpe measure,
20 Performance Evaluation and Active Portfolio Management 701
Bodie−Kane−Marcus:
Essentials of Investments,
Fifth Edition
VI. Active Investment
Management
20. Performance Evaluation
and Active Portfolio
Management
© The McGraw−Hill
Companies, 2003
or the equivalent M
2
, is now a widely accepted way to track performance of professionally
managed portfolios:
S
P
ϭ
The most able manager will be the one who consistently obtains the highest Sharpe meas-
ure, implying that the manager has real forecasting ability. A client’s judgment of a manager’s

ability will affect the fraction of investment funds allocated to this manager; the client can in-
vest the remainder with competing managers and in a safe fund.
If managers’ Sharpe measures were reasonably constant over time, and clients could reli-
ably estimate them, allocating funds to managers would be an easy decision.
Actually, the use of the Sharpe measure as the prime measure of a manager’s ability re-
quires some qualification. We know from the discussion of performance evaluation earlier in
this chapter that the Sharpe ratio is the appropriate measure of performance only when the
client’s entire wealth is managed by the professional investor. Moreover, clients may impose
additional restrictions on portfolio choice that further complicate the performance evaluation
problem.
20.4 MARKET TIMING
Consider the results of three different investment strategies, as gleaned from Table 5.3:
1. Investor X, who put $1 in 30 day T-bills (or their predecessors) on January 1, 1926, and
always rolled over all proceeds into 30-day T-bills, would have ended on December 31,
2001, 76 years later, with $16.98.
2. Investor Y, who put $1 in large stocks (the S&P 500 portfolio) on January 1, 1926,
and reinvested all dividends in that portfolio, would have ended on December 31, 2001,
with $1,987.01.
3. Suppose we define perfect market timing as the ability to tell with certainty at the
beginning of each year whether stocks will outperform bills. Investor Z, the perfect timer,
shifts all funds at the beginning of each year into either bills or stocks, whichever is
going to do better. Beginning at the same date, how much would Investor Z have ended
up with 76 years later? Answer: $115,233.89!
3. What are the annually compounded rates of return for the X, Y, and perfect-timing
strategies over the period 1926–2001?
These results have some lessons for us. The first has to do with the power of compounding.
Its effect is particularly important as more and more of the funds under management represent
pension savings. The horizons of pension investments may not be as long as 76 years, but they
are measured in decades, making compounding a significant factor.
The second is a huge difference between the end value of the all-safe asset strategy

($16.98) and of the all-equity strategy ($1,987.01). Why would anyone invest in safe assets
given this historical record? If you have absorbed all the lessons of this book, you know the
reason: risk. The averages of the annual rates of return and the standard deviations on the all-
bills and all-equity strategies were
Arithmetic Mean Standard Deviation
Bills 3.85% 3.25%
Equities 12.49 20.30
E(r
P
) Ϫ r
f
␴
P
702 Part SIX Active Investment Management
market timing
Asset allocation in
which the investment
in the market is
increased if one
forecasts that
the market will
outperform bills.
Concept
CHECK
>
Bodie−Kane−Marcus:
Essentials of Investments,
Fifth Edition
VI. Active Investment
Management

20. Performance Evaluation
and Active Portfolio
Management
© The McGraw−Hill
Companies, 2003
The significantly higher standard deviation of the rate of return on the equity portfolio is
commensurate with its significantly higher average return. The higher average return reflects
the risk premium.
Is the return premium on the perfect-timing strategy a risk premium? Because the perfect
timer never does worse than either bills or the market, the extra return cannot be compensa-
tion for the possibility of poor returns; instead it is attributable to superior analysis. The value
of superior information is reflected in the tremendous ending value of the portfolio. This value
does not reflect compensation for risk.
To see why, consider how you might choose between two hypothetical strategies. Strat-
egy 1 offers a sure rate of return of 5%; strategy 2 offers an uncertain return that is given by
5% plus a random number that is zero with a probability of 0.5 and 5% with a probability of
0.5. The results for each strategy are
Strategy 1 (%) Strategy 2 (%)
Expected return 5 7.5
Standard deviation 0 2.5
Highest return 5 10
Lowest return 5 5
Clearly, strategy 2 dominates strategy 1, as its rate of return is at least equal to that of strat-
egy 1 and sometimes greater. No matter how risk averse you are, you will always prefer strat-
egy 2 to strategy 1, even though strategy 2 has a significant standard deviation. Compared to
strategy 1, strategy 2 provides only good surprises, so the standard deviation in this case can-
not be a measure of risk.
You can look at these strategies as analogous to the case of the perfect timer compared with
either an all-equity or all-bills strategy. In every period, the perfect timer obtains at least as
good a return, in some cases better. Therefore, the timer’s standard deviation is a misleading

measure of risk when you compare perfect timing to an all-equity or all-bills strategy.
Valuing Market Timing as an Option
Merton (1981) shows that the key to analyzing the pattern of returns of a perfect market timer
is to compare the returns of a perfect foresight investor with those of another investor who
holds a call option on the equity portfolio. Investing 100% in bills plus holding a call option
on the equity portfolio will yield returns identical to those of the portfolio of the perfect timer
who invests 100% in either the safe asset or the equity portfolio, whichever will yield the
higher return. The perfect timer’s return is shown in Figure 20.5. The rate of return is bounded
from below by the risk-free rate, r
f
.
To see how the value of information can be treated as an option, suppose the market index
currently is at S
0
and a call option on the index has exercise price of X ϭ S
0
(1 ϩ r
f
). If the
market outperforms bills over the coming period, S
T
will exceed X; it will be less than X other-
wise. Now look at the payoff to a portfolio consisting of this option and S
0
dollars invested
in bills.
Payoff to Portfolio
Outcome: S
T
Յ XS

T
Ͼ X
Bills S
0
(1 ϩ r
f
) S
0
(1 ϩ r
f
)
Option 0 S
T
Ϫ X
Total S
0
(1 ϩ r
f
) S
T
20 Performance Evaluation and Active Portfolio Management 703
Bodie−Kane−Marcus:
Essentials of Investments,
Fifth Edition
VI. Active Investment
Management
20. Performance Evaluation
and Active Portfolio
Management
© The McGraw−Hill

Companies, 2003
The portfolio returns the risk-free rate when the market is bearish (that is, when the market
return is less than the risk-free rate) and pays the market return when the market is bullish and
beats bills. This represents perfect market timing. Consequently, the value of perfect timing
ability is equivalent to the value of the call option, for a call enables the investor to earn the
market return only when it exceeds r
f
.
Valuation of the call option embedded in market timing is relatively straightforward using
the Black-Scholes formula. Set S ϭ $1 (to find the value of the call per dollar invested in the
market), use an exercise price of X ϭ (1 ϩ r
f
) (the current risk-free rate is about 3%), and
a volatility of ␴ϭ.203 (the historical volatility of the S&P 500). For a once-a-year timer,
T ϭ 1 year. According to the Black-Scholes formula, the call option conveyed by market tim-
ing ability is worth about 8.1% of assets, and this is the annual fee one could presumably
charge for such services. More frequent timing would be worth more. If one could time
the market on a monthly basis, then T ϭ
1
⁄12 and the value of perfect timing would be 2.3%
per month.
The Value of Imperfect Forecasting
But managers are not perfect forecasters. While managers who are right most of the time
presumably do very well, “right most of the time” does not mean merely the percentage of the
time a manager is right. For example, a Tucson, Arizona, weather forecaster who always
predicts “no rain” may be right 90% of the time, but this “stopped clock” strategy does not
require any forecasting ability.
Neither is the overall proportion of correct forecasts an appropriate measure of market
forecasting ability. If the market is up two days out of three, and a forecaster always predicts
a market advance, the two-thirds success rate is not a measure of forecasting ability. We need

to examine the proportion of bull markets (r
M
Ͼ r
f
) correctly forecast and the proportion of
bear markets (r
M
Ͻ r
f
) correctly forecast.
If we call P
1
the proportion of the correct forecasts of bull markets and P
2
the proportion
for bear markets, then P
1
ϩ P
2
Ϫ 1 is the correct measure of timing ability. For example, a
forecaster who always guesses correctly will have P
1
ϭ P
2
ϭ 1 and will show ability of 1
(100%). An analyst who always bets on a bear market will mispredict all bull markets
(P
1
ϭ 0), will correctly “predict” all bear markets (P
2

ϭ 1), and will end up with timing
ability of P
1
ϩ P
2
Ϫ 1 ϭ 0. If C denotes the (call option) value of a perfect market timer, then
(P
1
ϩ P
2
Ϫ 1)C measures the value of imperfect forecasting ability.
704
Part SIX Active Investment Management
FIGURE 20.5
Rate of return of a
perfect market time
r
f
r
f
r
M
Rate of return
Bodie−Kane−Marcus:
Essentials of Investments,
Fifth Edition
VI. Active Investment
Management
20. Performance Evaluation
and Active Portfolio

Management
© The McGraw−Hill
Companies, 2003
The incredible potential payoff to accurate timing versus the relative scarcity of billionaires
should suggest to you that market timing is far from a trivial exercise and that very imperfect
timing is the most that we can hope for.
4. What is the market timing score of someone who flips a fair coin to predict the
market?
Measurement of Market Timing Performance
In its pure form, market timing involves shifting funds between a market index portfolio and
a safe asset, such as T-bills or a money market fund, depending on whether the market as a
whole is expected to outperform the safe asset. In practice, most managers do not shift fully
between bills and the market. How might we measure partial shifts into the market when it is
expected to perform well?
To simplify, suppose the investor holds only the market index portfolio and T-bills. If the
weight on the market were constant, say 0.6, then the portfolio beta would also be constant,
and the portfolio characteristic line would plot as a straight line with a slope 0.6, as in Figure
20.6A. If, however, the investor could correctly time the market and shift funds into it in peri-
ods when the market does well, the characteristic line would plot as in Figure 20.6B. The idea
is that if the timer can predict bull and bear markets, more will be shifted into the market when
the market is about to go up. The portfolio beta and the slope of the characteristic line will be
higher when r
M
is higher, resulting in the curved line that appears in 20.6B.
Treynor and Mazuy (1966) tested to see whether portfolio betas did in fact increase prior
to market advances, but they found little evidence of timing ability. A similar test was imple-
mented by Henriksson (1984). His examination of market timing ability for 116 funds in
20 Performance Evaluation and Active Portfolio Management 705
Concept
CHECK

<
FIGURE 20.6
Characteristic lines
A: No market timing,
beta is constant
B: Market timing,
beta increases with
expected market
excess return
Steadily
increasing
slope
r
P
– r
f
r
M
– r
f
Slope = 0.6
r
P
– r
f
r
M
– r
f
A. No Market Timing, Beta Is Constant

B. Market Timing, Beta Increases with Expected Market
Excess Return
Bodie−Kane−Marcus:
Essentials of Investments,
Fifth Edition
VI. Active Investment
Management
20. Performance Evaluation
and Active Portfolio
Management
© The McGraw−Hill
Companies, 2003
1968–1980 found that, on average, portfolio betas actually fell slightly during the market ad-
vances, although in most cases the response of portfolio betas to the market was not statisti-
cally significant. Eleven funds had statistically positive values of market timing, while eight
had significantly negative values. Overall, 62% of the funds had negative point estimates of
timing ability.
In sum, empirical tests to date show little evidence of market timing ability. Perhaps this
should be expected; given the tremendous values to be reaped by a successful market timer, it
would be surprising to uncover clear-cut evidence of such skills in nearly efficient markets.
20.5 STYLE ANALYSIS
Style analysis was introduced by Nobel laureate William Sharpe.
3
The popularity of the con-
cept was aided by a well-known study
4
concluding that 91.5% of the variation in returns of
82 mutual funds could be explained by the funds’ asset allocation to bills, bonds, and stocks.
Later studies that considered asset allocation across a broader range of asset classes found that
as much as 97% of fund returns can be explained by asset allocation alone.

Sharpe considered 12 asset class (style) portfolios. His idea was to regress fund returns on
indexes representing a range of asset classes. The regression coefficient on each index would
then measure the implicit allocation to that “style.” Because funds are barred from short posi-
tions, the regression coefficients are constrained to be either zero or positive and to sum to
100%, so as to represent a complete asset allocation. The R-square of the regression would
then measure the percentage of return variability attributed to the effects of security selection.
To illustrate the approach, consider Sharpe’s study of the monthly returns on Fidelity’s
Magellan Fund over the period January 1985 through December 1989, shown in Table 20.7.
706 Part SIX Active Investment Management
TABLE 20.7
Sharpe’s style
portfolios for the
Magellan fund
Regression
Coefficient*
Bills 0
Intermediate bonds 0
Long-term bonds 0
Corporate bonds 0
Mortgages 0
Value stocks 0
Growth stocks 47
Medium-cap stocks 31
Small stocks 18
Foreign stocks 0
European stocks 4
Japanese stocks 0
Total 100.00
R-squared 97.3
*Regressions are constrained to have nonnegative coefficients and to have coefficients

that sum to 100%.
Source: William F. Sharpe, “Asset Allocation: Management Style and Performance
Evaluation,” Journal of Portfolio Management, Winter 1992, pp. 7–19.
3
William F. Sharpe, “Asset Allocation: Management Style and Performance Evaluation,” Journal of Portfolio Man-
agement, Winter 1992, pp. 7–19.
4
Gary Brinson, Brian Singer, and Gilbert Beebower, “Determinants of Portfolio Performance,” Financial Analysts
Journal, May/June 1991.
Bodie−Kane−Marcus:
Essentials of Investments,
Fifth Edition
VI. Active Investment
Management
20. Performance Evaluation
and Active Portfolio
Management
© The McGraw−Hill
Companies, 2003
While there are 12 asset classes, each one represented by a stock index, the regression
coefficients are positive for only 4 of them. We can conclude that the fund returns are well
explained by only four style portfolios. Moreover, these three style portfolios alone explain
97.3% of returns.
The proportion of return variability not explained by asset allocation can be attributed to
security selection within asset classes. For Magellan, this was 100 Ϫ 97.3 ϭ 2.7%. To evalu-
ate the average contribution of stock selection to fund performance we track the residuals from
the regression, displayed in Figure 20.7. The figure plots the cumulative effect of these resid-
uals; the steady upward trend confirms Magellan’s success at stock selection in this period.
Notice that the plot in Figure 20.7 is far smoother than the plot in Figure 20.8, which shows
Magellan’s performance compared to a standard benchmark, the S&P 500. This reflects the

fact that the regression-weighted index portfolio tracks Magellan’s overall style much better
than the S&P 500. The performance spread is much noisier using the S&P as the benchmark.
Of course, Magellan’s consistently positive residual returns (reflected in the steadily
increasing plot of cumulative return difference) is hardly common. Figure 20.9 shows the
20 Performance Evaluation and Active Portfolio Management 707
FIGURE 20.7
Fidelity Magellan
Fund cumulative
return difference:
fund versus style
benchmark
Source: William F. Sharpe,
“Asset Allocation:
Management Style and
Performance Evaluation,”
Journal of Portfolio
Management, Winter 1992,
pp. 7–19.
1986 1987 1988 1989 1990
30
25
20
15
10
5
0
FIGURE 20.8
Fidelity Magellan
Fund cumulative
return difference:

fund versus S&P 500
Source: William F. Sharpe,
“Asset Allocation:
Management Style and
Performance Evaluation,”
Journal of Portfolio
Management, Winter 1992,
pp. 7–19.
1986 1987 1988 1989 1990
12
10
8
6
4
2
0
Ϫ2
Bodie−Kane−Marcus:
Essentials of Investments,
Fifth Edition
VI. Active Investment
Management
20. Performance Evaluation
and Active Portfolio
Management
© The McGraw−Hill
Companies, 2003
frequency distribution of average residuals across 636 mutual funds. The distribution has the
familiar bell shape with a slightly negative mean of Ϫ.074% per month.
Style analysis has become very popular in the investment management industry and has

spawned quite a few variations on Sharpe’s methodology. Many portfolio managers utilize
websites that help investors identify their style and stock selection performance.
20.6 MORNINGSTAR’S RISK-ADJUSTED RATING
The commercial success of Morningstar, Inc., the premier source of information on mutual
funds, has made its Risk Adjusted Rating (RAR) among the most widely used performance
measures. The Morningstar five-star rating is coveted by the managers of the thousands of
funds covered by the service.
Morningstar calculates a number of RAR performance measures that are similar, although
not identical, to the standard mean-variance measures. The most distinct measure, the Morn-
ingstar Star Rating, is based on comparison of each fund to a peer group. The peer group for
each fund is selected on the basis of the fund’s investment universe (e.g., international, growth
versus value, fixed-income, and so on) as well as portfolio characteristics such as average
price-to-book value, price-earnings ratio, and market capitalization.
Morningstar computes fund returns (adjusted for loads) as well as a risk measure based on
fund performance in its worst years. The risk-adjusted performance is ranked across funds in
a style group and stars are awarded based on the following table:
Percentile Stars
0–10 1
10–32.5 2
32.5–67.5 3
67.5–90 4
90–100 5
708 Part SIX Active Investment Management
FIGURE 20.9
Average tracking error,
636 mutual funds,
1985–1989
Source: William F. Sharpe,
“Asset Allocation:
Management Style and

Performance Evaluation,”
Journal of Portfolio
Management, Winter 1992,
pp. 7–19.
Ϫ1.00
Ϫ0.50
0.00
0.50
1.00
90
80
70
60
50
40
30
20
10
0
Average tracking error (%/month)
Bodie−Kane−Marcus:
Essentials of Investments,
Fifth Edition
VI. Active Investment
Management
20. Performance Evaluation
and Active Portfolio
Management
© The McGraw−Hill
Companies, 2003

The Morningstar RAR method produces results that are similar but not identical to that of
the mean/variance-based Sharpe ratios. Figure 20.10 demonstrates the fit between ranking by
RAR and by Sharpe ratios from the performance of 1,286 diversified equity funds over the pe-
riod 1994–1996. Sharpe notes that this period is characterized by high returns that contribute
to a good fit.
20.7 SECURITY SELECTION:
THE TREYNOR-BLACK MODEL
Overview of the Treynor-Black Model
Security analysis is the other dimension of active investment besides timing the overall mar-
ket and asset allocation. Suppose you are an analyst studying individual securities. Quite
likely, you will turn up several securities that appear to be mispriced and offer positive alphas.
But how do you exploit your analysis? Concentrating a portfolio on these securities entails a
cost, namely, the firm-specific risk you could shed by more fully diversifying. As an active
manager, you must strike a balance between aggressive exploitation of security mispricing and
diversification considerations that dictate against concentrating a portfolio in a few stocks.
Jack Treynor and Fischer Black (1973) developed a portfolio construction model for man-
agers who use security analysis. It assumes security markets are nearly efficient. The essence
of the model is this:
1. Security analysts in an active investment management organization can analyze in depth
only a relatively small number of stocks out of the entire universe of securities. The
securities not analyzed are assumed to be fairly priced.
2. For the purpose of efficient diversification, the market index portfolio is the baseline
portfolio, which is treated as the passive portfolio.
3. The macro forecasting unit of the investment management firm provides forecasts of
the expected rate of return and variance of the passive (market index) portfolio.
4. The objective of security analysis is to form an active portfolio of a necessarily limited
number of securities. Perceived mispricing of the analyzed securities is what determines
the composition of this active portfolio.
20 Performance Evaluation and Active Portfolio Management 709
FIGURE 20.10

Rankings based on
Morningstar’s category
RARs and excess return
Sharpe ratios
Source: William F. Sharpe,
“Morningstar
Performance Measures,”
www.wsharpe.com.
0 0.2 0.4 0.6 0.8 1
1
0.8
0.6
0.4
0.2
0
Sharpe ratio
percentile in category
Category RAR
percentile in
category
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ