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Journal of Portfolio
Management, Winter 1992, 40
pp. 7“19.







Average tracking error (%/month)

frequency distribution of average residuals across 636 mutual funds. The distribution has the
familiar bell shape with a slightly negative mean of .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.

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
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Fifth Edition Management

20 Performance Evaluation and Active Portfolio Management

F I G U R E 20.10
Sharpe ratio
percentile in category Rankings based on
1 Morningstar™s category
RARs and excess return
Sharpe ratios
Source: William F. Sharpe,
0.6 Performance Measures,”

Category RAR
0 percentile in
1 category
0 0.2 0.4 0.6 0.8

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.

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.
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Essentials of Investments, Management and Active Portfolio Companies, 2003
Fifth Edition Management

710 Part SIX Active Investment Management

5. Analysts follow several steps to make up the active portfolio and forecast its
a. Estimate the characteristic line of each analyzed security and obtain its beta and
residual variance. From the beta and the macro forecast, E(rM) rf, determine
the required rate of return of the security.
b. Determine the expected return. Subtracting the required return yields the expected
abnormal return (alpha) of the security.
c. Use the estimates for the values of alpha, beta, and residual risk to determine the
optimal weight of each security in the active portfolio.
d. Estimate the alpha, beta, and residual variance for the active portfolio according to
the weights of the securities in the portfolio.
6. The macroeconomic forecasts for the passive index portfolio and the composite forecast
for the active portfolio are used to determine the optimal risky portfolio, which will be
a combination of the passive and active portfolios.
Although some sophisticated investment managers use the Treynor-Black model, it has
not taken the industry by storm. This is unfortunate for several reasons:
An optimizing 1. Just as even imperfect market-timing ability has enormous value, security analysis of the
model for portfolio sort Treynor and Black propose has similar potential value. Even with far-from-perfect
managers who use
security analysis, active management can add value.
security analysis in a
2. The Treynor-Black model is easy to implement. Moreover, it is useful even relaxing
nearly efficient
market. some of its simplifying assumptions.
3. The model lends itself to use with decentralized decision making, which is essential to
efficiency in complex organizations.

Portfolio Construction
Assuming all securities are fairly priced and using the index model as a guideline for the rate
of return on securities, the rate of return on security i is given by
ri rf i (rM rf ) ei (20.1)

where ei is the zero mean, firm-specific (nonsystematic) component.
Absent security analysis, Treynor and Black take Equation 20.1 to represent the rate of re-
turn on all securities and assume the index portfolio (M) is efficient. For simplicity, they also
assume the nonsystematic components of returns, ei, are independent across securities. Mar-
ket timing is incorporated in the terms rM and M, representing index portfolio forecasts. The
overall investment in the risky portfolio will be affected by the optimism or pessimism re-
flected in these numbers.
Assume a team of security analysts investigates a subset of the universe of available secu-
rities, with the objective of forming an active portfolio. That portfolio will then be mixed with
the index portfolio to improve diversification. For each security, k, that is researched, we write
the rate of return as
rk rf k (rM rf ) ek (20.2)

where k represents the extra (abnormal) expected return attributable to the mispricing of the
security. Thus, for each security analyzed, the research team estimates the parameters
k, k, (ek)
If all the k turn out to be zero, there would be no reason to depart from the passive
strategy, and the index portfolio would remain the manager™s choice. But this is a remote
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Essentials of Investments, Management and Active Portfolio Companies, 2003
Fifth Edition Management

20 Performance Evaluation and Active Portfolio Management

F I G U R E 20.11
E(r) The optimization
process with active
and passive portfolios



possibility. In general, there will be a significant number of nonzero values, some positive
and some negative.
Consider first how you would use the active portfolio once you found it. Suppose the
active portfolio
active portfolio (A) has been constructed and has the parameters
In the context of the
A, A, (eA)
Treynor-Black model,
the portfolio formed
The total variance of the active portfolio is the sum of its systematic variance, 2 2 , plus the
by mixing analyzed
nonsystematic variance, (eA). These three parameters, plus the mean and variance of the
stocks with perceived
index portfolio, are sufficient to identify the opportunity set generated by the active and pas- nonzero alpha values.
sive portfolios. This portfolio is
Figure 20.11 shows the optimization process with active and passive portfolios. The dashed ultimately mixed with
the passive market
efficient frontier line represents the universe of all securities, assuming they are all fairly
index portfolio.
priced, that is, that all alphas are zero. By definition, the market index (M) is on this efficient
frontier and is tangent to the (dashed) capital market line (CML). In practice, our analysts do
not need to (indeed cannot) know this frontier, but they need to forecast the index portfolio
and construct the optimal risky portfolio using the index and active (A) portfolios. The opti-
mal portfolio (P) will lie on the capital allocation line (CAL) that lies above the CML.
From the viewpoint of an investor with superior analysis, the index portfolio will be in-
efficient; that is, the active portfolio (A) constructed from mispriced securities will lie above
the CML.
The optimal combination of the active portfolio with the passive portfolio takes off from
the construction of an optimal risky portfolio from two risky assets that we first encountered
in Chapter 6. As the active portfolio is not perfectly correlated with the index, further diversi-
fication”that is, mixing it with the index”is likely to be beneficial.
We can judge the success of active management, and the contribution of the active portfolio
(A), by the Sharpe measure (ratio of reward to variability) of the resultant risky portfolio (P),
compared with that of the index portfolio (M).
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Essentials of Investments, Management and Active Portfolio Companies, 2003
Fifth Edition Management


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