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Levy and Post, Investments © Pearson Education Limited 2005

Slide 22.1

Investments

Chapter 22: Performance Evaluation


Slide 22.2

General Tools for Performance Evaluation

• Compare performance with risk-adjusted performance indexes.

• Compare performance against benchmark portfolios.

• Use of performance attribution methods.

• Use of comparison universe methods.


Slide 22.3

04/10/23 3

How Should Investors Measure Risk?

• Standard Deviation (absolute risk)

– Investors with limited holdings

• Beta (relative risk)

– Investors with a wide array of holding


Slide 22.4

Words of Caution when using Performance Evaluation

• Performance evaluation is an historical exercise while most investors are interested in the future performance of portfolios.

• Correcting the performance for risk is very difficult.

• It is very difficult to obtain reliable estimates of the risk and return characteristics of individual securities.

• The portfolio compositions change over time.


Slide 22.5

Are Mutual Funds Markowitz Efficient Investments?

• The mutual funds are all inefficient investments

• Funds tend to group into clusters corresponding to their investment goals– Mutual funds are required to

publish written goal statements– In a few cases fund’s stated

objective and performance differed

This income and growth fund

performed in the same league as the growth

funds.


Slide 22.6Scrutinizing Mutual Funds Goal Statements

# of funds claiming each goal Category’s average rate of return

BetaGrowt

h

Growth &

income

Income &

Growth

Income, Growth

& Stability

Growth

Growth &

income

Income &

Growth

Income, Growth

& Stability

0.5 to 0.7

3 5 4 16 6.9% 10.1% 9.7% 9.1%

0.7 to 0.9

15 24 7 7 11.2% 10.0% 10.0% 12.2%

0.9 to 1.1

20 1 None 1 13.8% 9.5% None 13.5%

Risk Class

Range of Betas

# of funds

Average Beta

Average Variance

Average Rate of Return

Low 0.5 to 0.7

28 0.619 0.000877 9.1%

Medium

0.7 to 0.9

53 0.786 0.001543 10.6%

High 0.9 to 1.1

22 0.992 0.002304 13.5%Portfolio’s SDs and Betas were better

indicators of portfolio’s actual performance than their goal statements.

There are some funds which claim they are growth funds, but get lower return than income

funds.


Slide 22.7

Mutual Fund Performance

• The empirical evidence finds consistently that mutual fund managers on average lag behind the market if one corrects for risk and costs.

• Also, there seems to be little consistency in the performance of mutual funds over time.


Slide 22.8

Risk-adjusted Performance Measures

• Sharpe’s Performance Index.

• Treynor’s Performance Index.

• Jensen’s Performance Index.

• APT-based performance measures.


Slide 22.9

Sharpe’s Performance Index

• Based on the Slope of the CML

• Uses Standard Deviation to Measure Risk (i.e. the interest

is to minimize total risk)

• The Higher the Index, the better the performance

• Investors Hold Only the Mutual Fund

• May wish to rank portfolios’ performances

• Need a measure that includes both risk and return

– Sharpe measured the reward to variability index


Slide 22.10

Sharpe’s Performance Index

Based on the CAPM:

Where riskless borrowing and lending is possible at interest rate r, and where

and


Slide 22.11

04/10/23 11

Performance Example (Francis & Ibbotson)

Year Avon Blair Market

1 10.0 10.0 10.0

2 30.0 40.0 30.0

3 -20.0 -20.0 -20.0

4 -10.0 -10.0 -10.0

5 20.0 40.0 20.0

6 10.0 20.0 30.0

7 0.0 -20.0 -10.0

8 30.0 20.0 20.0

9 -10.0 10.0 0.0

10 20.0 40.0 30.0

R 8.0 13.0 10.0

16.6 22.4 17.9

0.8125 1.156 1


Slide 22.12

SHARPE Example

• The Avon Fund earned an average return of 8% annually with a standard deviation of 16.6%, while the Blair Fund earned 13.00% annually with a standard deviation of 22.4%. During the same time period the average risk-free rate was 4%.

• Which fund was the better performer?

Avon

0.08 - 0.04

0.166 0.241

SHARPE

Blair

0.13 - 0.04

0.224 0.4018

SHARPE

Since SHARPEBlair > SHARPEAvon, Blair was

the better performer on a risk-adjusted basis.


Slide 22.13

04/10/23 13

SHARPE Example

Avon

RFR

13%

8%

22.4%16.6%

Standard Deviation of Returns

Exp

ecte

d R

etur

n, E

(r)

Blair Slope is 0.4018 for REVARBlair

Slope is 0.241 for REVARAvon


Slide 22.14

Treynor’s Performance Index

• Based on SML• Uses Beta to measure Risk (i.e. the interest is to minimize

the market risk)• The Higher the Index, the better the performance• Investors Hold Many Assets• For Investors Only Interested in Whether They Beat the

Market• Treynor devised measure to evaluate performance that

uses systematic risk (beta) rather than total risk (standard deviation)


Slide 22.15

Treynor’s Performance Index

Based on the APT:

Where riskless borrowing and lending is possible at interest rate r, and where

and

T

tmrtm

T

ttmitmttti

i

rRrRT

rRrRrRrRT

1

22,

1,

)()(1

)()()()(1


Slide 22.16

TREYNOR Example

• The Avon Fund earned an average return of 8% annually (Characteristic LineAVON: Intercept: -0.00125; Beta: 0.8125), while the Blair Fund earned 13.00% annually (Characteristic LineBLAIR: Intercept: 0.014; Beta: 1.156). During the same time period the average risk-free rate was 4%.

• Which fund was the better performer?

Avon

0.08 - 0.04

0.8125 0.0492

TREYNOR

Blair

0.13 - 0.04

1.156 0.0779

TREYNOR

Since TREYNORBlair > TREYNORAvon, Blair was

the better performer on a risk-adjusted basis.


Slide 22.17

TREYNOR Example

• TREYNOR measures the desirability of fund in a SML context

Avon

RFR

13%

8%

1.156.8125

Beta

Exp

ecte

d R

etur

n, E

(r)

Blair TREYNORBlair = 0.0778

TREYNORAvon = 0.049

SML


Slide 22.18

TREYNOR Example

• Notice that the SML gives slightly different return for both funds! None of them is on the SML!

• The Avon Fund earned an average return of 8% annually because the Characteristic LineAVON: -0.00125 + 0.8125*(10% ) + , where 10% = Rm .

• According to the SML, the return to Avon Fund is:• 4% + 0.8125*(10% - 4%) = 8.875%• Similar differences are for the Blair Fund


Slide 22.19 Jensen’s Alpha Performance Index

• Based on CAPM• Uses Beta to Measure Risk (equal to the vertical distance

to SML)• Determines How Much One Fund Outperforms or

Underperforms Another Fund (neither Sharpe nor Treynor indicate by how much a fund has outperformed or underperformed the index)

• Determines the Significance of Results• Investors Hold Many Assets


Slide 22.20

An Investment’s Alpha

• Jensen modified the characteristic line equation – Rather than using periodic rates of return, he uses

periodic risk-premiums

)1(u)RR(aRR t,pft,mpft,p p

With expected values (1) gets:

)RR(Ea)RR(E ft,mpft,p p


Slide 22.21

Jensen’s alpha represents excess returns from asset Can be +, 0 or – If asset is correctly priced, Jensen’s alpha = 0 If alpha > 0, asset has earned return greater

than appropriate for its level of undiversifiable risk (beta)

• Asset is underpriced If alpha < 0, asset’s returns are lower than

appropriate for its level of risk

• Asset is overpriced

Explanation of an Investment’s Alpha


Slide 22.22

Jensen’s Alpha Example

)04.010.0(8125.0a04.008.0 Avon

Using data (risk premiums, not returns) from Table 16-3 (previous slide) for the Avon and Blair Funds:

Characteristic LineAvon

•Jensen’s alpha: -0.00875

•Beta: 0.8125

Characteristic LineBlair

•Jensen’s alpha: +0.02062

•Beta: 1.1562Blair earned

positive excess returns.

)04.010.0(1562.1a04.013.0 Blair


Slide 22.23

Jensen’s Alpha Example

Econometric estimates of (1)

)1(u)RR(aRR t,pt,ft,mpt,ft,p p

u)04.010.0(825.00125.004.008.0

Assume actual observations on all funds, market portfolio and risk free returns give the following alpha and beta values:

In that case u = 0.03. This is the unik = specific risk of Avon fund


Slide 22.24 Caveats About Alphas

• Jensen’s alpha cannot be used to rank performance of different assets unless it’s adjusted for the assets’ risks (same alphas does not imply same performance, because the vertical distance to the SML might be the same, but one fund might have much higher risk)

• The appraisal ratio divides Jensen’s alpha by the standard error of the estimate (SE(u)) which then allows for rankings

SE

alpha sJensen' Ratio Appraisal

u

Notice that the alpha = intercept of the original characteristic line (used to estimate our beta) is not the same as Jensen’s alpha and should not be used for investment performance evaluation


Slide 22.25Caveats About Ranking

• Sharpe, Jensen & Treynor rank funds performances differently!

• If there are two funds (A, B) and the market index (M), and Treynor ranks for instance, A > M > B, so does Jensen.

• If there are many funds and the market index (M), Treynor may rank A > B first, while Jensen may rank K > L first.


Slide 22.26 Performance Indexes With APT

• One or More Factors Determine Risk

• Jensen’s Performance Measure

• Examine the Difference Between

– Actual and expected average rate of return

• Determines the Significance of Results

• For Investors Who Want to Compare Their Performance With Other Fund Managers


Slide 22.27

Empirical Evidence For MFs

• MFs performance Fall Behind the Market• MFs can not Outperform the simple strategy:

– Buy-the-market and-hold policy

• International MFs Tend to do Better and:– Outperform the S&P 500

– Choice of market portfolio critical

• Bond Funds Underperform the Indexes– Relationship

• underperformance and the expense ratio


Slide 22.28 Caution About Performance Indexes

• Problems– Historical performance is used to infer future performance– Difficult to measure the risk of activity traded accounts– Beta is not stable

• Depends on the choice of market index– Overall performance indexes cannot identify

• What activities of the portfolio manager resulted in the performance


Slide 22.29

Style Analysis: I

• An umbrella term for a set of tools for determining the investment style of portfolios and for measuring the performance of portfolios given their investment style.

• Money managers are evaluated, in part, based on how well they have offered what they promised or were told to do.


Slide 22.30

Style Analysis: II

• Holdings-based Style AnalysisDetermines the investment style of a portfolio by examining the characteristics of the individual securities in the portfolio.

• Returns-based Style AnalysisDetermines the investment style of a portfolio by analyzing its co-movements with indexes that proxy for different styles.


Slide 22.31

Selection of Money Manager

• These institutions/individuals must select a money manager

• This part presents tools for measuring and ranking money managers’ performances

– Aids in the selection process

– Money managers also use these tools to appraise and improve their skills


Slide 22.32

Analyzing a Portfolio Manager’s Style

• In 1992 Sharpe introduced a model to analyze a portfolio manager’s style (i.e., growth vs value investing, etc.)– Uses modest amount of public information about funds

• Uses price indexes for 12 asset classes as explanatory variables for a mutual fund’s return

– Sample explanatory factors such as» Soloman Brothers 90-day Treasury bill index» Lehman Brothers Intermediate-Term Government Bond Index» Sharpe/BARRA Value Stock Index


Slide 22.33

04/10/23 33

Analyzing a Portfolio Manager’s Style

– Uses factor analysis• The factor loadings are estimates of the weights that

a fund invests in the twelve asset categories• R2 of 0.70 are common

• Sharpe also suggests that same type of analysis could be done using a ‘rolling’ regression– Repeating regression when new data is released—dropping

oldest data and adding newest data


Slide 22.34

Performance AttributionThe assessment of the performance of portfolio management decisions.

Exhibit 22.7 Flow chart of the top-down money-management processSource: From Introduction to Investments, 2nd edn, by Levy. © 1999. Reprinted with permission of South-Western, a division of Thomson Learning: www.thomsonrights.com. Fax 800 730-2215.


Slide 22.35

35

Rolling Style Analysis

• Ibbotson Associates uses a rolling regression period of 60 months– Deleting oldest month and adding new month as data becomes available

Some fixed-income securities entered this

growth stock fund in mid 1990s—this is interesting

because Magellan’s published investment

objective is a growth stock fund.


Slide 22.36

Benefits From Using Quantitative Management Style Analysis

• Quantitative style analysis is important due to:– Investment holdings are usually not reported publicly until

months after they are made—too late for investors to react in a timely manner

– Mutual funds can report misleading investment goals

• Can also provide better forecasts of mutual fund’s risk/return than subjective comments in newspapers, etc.


Slide 22.37

Analyzing Performance Statistics

• Mutual funds with the highest average rate of return might not have the highest rank because– A highly aggressive fund may earn higher

returns than a less aggressive fund but the higher returns may not be sufficient to compensate for the extra risk taken


Slide 22.38 Analyzing Performance Statistics

PossibleInvestmen

ts

ExpectedReturn

StandardDeviatio

n

Yak Fund 30% 20%

Zebra Fund

15% 5%

RFR 4% 0%

While the Yak Fund earned twice as much as the Zebra

Fund it is four times as risky.


Slide 22.39

Analyzing Performance Statistics• By multiplying Zebra’s low SD by 4, we could create a new

portfolio on Zebra’s Asset Allocation Line (AAL) with the same high SD as Yak Fund

• Borrow 4 times as much as the initial equity, invest in Zebra, and achieve the following E(RZebra):

• (4*0.15 - 4*0.04)= 0.48

3w48.015).w1(w04.

R)w1(wrR zebrafAALzebra


Slide 22.40 Analyzing Performance Statistics

• Check that the SD is the same as for Yak Fund

(see notes in SML, the risk free interest rate has zero variance)

20.005.0*4)w1( zebraAALzebra

20.005.0

04.015.004.048.0

rR,or

AALzebra

AALzebra

AALzebra

rR

fAALzebra zebra

fzebra


Slide 22.41

Analyzing Performance Statistics

Yak

RFR

48%

30%

20%5%

Standard Deviation

Exp

ecte

d R

etur

n, E

(r)

Zebra

Zebra’s SHARPE = 2.2

Yak’s SHARPE =

1.3

15%Yak’s

AALZe

bra’

s AAL

The leveraged Zebra portfolio

dominates the Yak Fund; thus Zebra

is a better fund even though Yak

has a higher average return.

Perhaps both Treynor and

Jensen would give the same ranking

in this case


Slide 22.42

General Discussion of Performance Measurement Tools

• When investors analyze merits of alternative investments, usually concerned with– Asset selection

• Portfolio manager’s ability to select good investments and to not select poor investments

– Sharpe, Treynor & Jensen’s Alpha are good tools to evaluate this issue

– Market timing• Portfolio manager’s ability to buy low/sell high and manager’s ability

to react to changes in market’s direction– Sharpe, Treynor & Jensen’s Alpha are not good tools for evaluating market timing unless

theoretical framework is extended


Slide 22.43

Evaluating Timing Decisions

• Treynor & Mazuy included a second-order term in the characteristic line to evaluate market-timing

t,invest2

t,Minvest,2t,Minvest,1investt.,invest RRerceptintR


Slide 22.44

04/10/23 44

Evaluating Timing Decisions

• A successful market timer will– Shift into high beta securities when bull market begins

– Shift into low beta securities when bear market begins• If portfolio manager does this, beta2,investment > 0

• If portfolio manager cannot outguess market turns, beta2,investment = 0 (statistically)

• If portfolio manager incorrectly predicts market turns, beta2,investment <

0


Slide 22.45

Do Winners Repeat?

• Are the best portfolio managers able to repeat their high performance?– If security markets are perfectly efficient, there should be no

consistency in high performance– When evaluating whether winners repeat, must be careful to

not flaw study in terms of survivorship bias• Market indexes only contain securities that have ‘survived’—not experienced

bankruptcy, merger, etc.• Goetzmann & Ibbotson studied mutual funds

– Mitigated survivorship bias by comparing funds within-sample performances through time


Slide 22.46

Goetzmann & Ibbotson Study

• Database– Monthly total returns of several hundred mutual funds over a 13-year

period– Management fees deducted, but load, exit fees and taxes were not

considered– All cash flows reinvested monthly– Returns measured over 2-year within-sample period, beginning in

1976 to predict out-of-sample performance for subsequent 2-year period

– Only funds in existence for entire 2-year interval included– Every mutual fund categorized as ‘winner’ or ‘loser’ based on whether

it ranked above or below that 2-year sample’s median return


Slide 22.47


1978-1979Winners

1978-1979Losers

1980-1981Winners

1980-1981Losers

1976-1977Winners

84 541978-1979

Winners110 41

1976-1977Losers

50 881978-1979

Losers38 113

1982-1983Winners

1982-1983 Losers

1984-1985Winners

1984-1985Losers

1980-1981Winners

63 961982-1983

Winners104 62

1980-1981Losers

96 631982-1983

Losers71 95

Combined ResultsSuccessive Period

1986-1987Winners

1986-1987Losers

Winners Losers

1984-1985Winners

125 72Initial

Winners486

59.9%325

40.1%

1984-1985Losers

72 125InitialLosers

32740.3%

48459.7%

The combined

results show that there is about a 60%

chance a winner will be a winner

the following period.

But, the repeat-winners pattern

didn’t persist during this subsample.


Slide 22.48


• However, these high-return mutual funds could continue to have high-ranking returns due to high risk, not because they were winners

• G&I replicate study using risk-adjusted returns– Computed Jensen’s Alpha for each fund– Classified fund as a winner or loser if fund’s alpha > or <

period’s median alpha• Results show that winners tend to repeat in all 5 subsamples

• Also, divided sample into growth funds and found similar results• Also, used 1-year subsamples rather than 2-year

– Similar, but weaker, support for the repeat winners hypothesis


Slide 22.49

Other Studies

• Malkiel argues that while repeat winners phenomenon existed in 1970s, it was not present during 1980s

• Carhart finds that winning funds tend to have a winning performance the following year, but not afterwards– Losers have a strong tendency to persist

with the worst performers persisting for years


Slide 22.50The Bottom Line

• About Portfolio Performance Measures– To adequately evaluate a portfolio, must analyze both risk and return– SHARPE measures risk-premium per unit of total risk– TREYNOR measures risk-premium per unit of systematic risk– Jensen’s alpha measures risk-adjusted returns for both portfolios and

individual assets• All three measures tend to rank mutual funds similarly, but not

always exactly– Additional tools are available for measuring a manager’s market

timing skills


Slide 22.51 The Bottom Line

• About mutual fund investments– Average American buying round lots can afford only about 7

different stocks• Not enough to minimize diversifiable risk

– Mutual funds are usually able to reduce their diversifiable risk

– Investors can maintain their desired risk-class by mutual fund investing

– Most investors should focus on a mutual fund’s fees and favor funds charging smallest fees

l pch22

Economy & Finance

performance investors

performance of mutual

mutual fund performance

treynors performance

jensens performance

avon fund

market risk

average return