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Management, Winter 1992, 40

pp. 7â€“19.

30

20

10

0

1.00

0.50

0.00

0.50

1.00

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 Sharpeâ€™s methodology. Many portfolio managers utilize

websites that help investors identify their style and stock selection performance.

20.6 MORNINGSTARâ€™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 fundâ€™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

0â€“10 1

10â€“32.5 2

32.5â€“67.5 3

67.5â€“90 4

90â€“100 5

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Fifth Edition Management

709

20 Performance Evaluation and Active Portfolio Management

F I G U R E 20.10

Sharpe ratio

percentile in category Rankings based on

1 Morningstarâ€™s category

RARs and excess return

Sharpe ratios

0.8

Source: William F. Sharpe,

â€œMorningstar

0.6 Performance Measures,â€

www.wsharpe.com.

0.4

0.2

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 1994â€“1996. Sharpe notes that this period is characterized by high returns that contribute

to a good fit.

20.7 SECURITY SELECTION:

THE TREYNOR-BL ACK 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.

Bodieâˆ’Kaneâˆ’Marcus: VI. Active Investment 20. Performance Evaluation Â© The McGrawâˆ’Hill

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Fifth Edition Management

710 Part SIX Active Investment Management

5. Analysts follow several steps to make up the active portfolio and forecast its

performance:

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

Treynor-Black

not taken the industry by storm. This is unfortunate for several reasons:

model

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)

k

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

2

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 managerâ€™s choice. But this is a remote

Bodieâˆ’Kaneâˆ’Marcus: VI. Active Investment 20. Performance Evaluation Â© The McGrawâˆ’Hill

Essentials of Investments, Management and Active Portfolio Companies, 2003

Fifth Edition Management

711

20 Performance Evaluation and Active Portfolio Management

F I G U R E 20.11

CAL

E(r) The optimization

CML

process with active

and passive portfolios

E(rA)

A

P

M

Ïƒ

ÏƒA

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

2

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

AM

by mixing analyzed

2

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-

ficationâ€”that is, mixing it with the indexâ€”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).

Bodieâˆ’Kaneâˆ’Marcus: VI. Active Investment 20. Performance Evaluation Â© The McGrawâˆ’Hill

Essentials of Investments, Management and Active Portfolio Companies, 2003

Fifth Edition Management

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