The Size Effect

Brett BatesGreg Chedwick

Chris FerreMatt Karam

The Size Effect 2 Articles by Marc Reinganum:

o “Abnormal Returns in Small Firm Portfolios” (1981)o “Portfolio Strategies Based on Market Capitalization’

(1983) The Capital Asset Pricing Model (CAPM) asserts

that two assets with the same beta will have the same expected return

The model implies that small firms will only command higher risk premiums if they have higher betas

Do abnormal returns exist, that are not explained by Beta?

Capital Asset Pricing Model

Security Market LineE(Ri) = Rf + βi(E(Rm) −

Rf)

E(Ri) = the expected return on the capital asset

Rf = the risk-free rate of interest βi =(beta coefficient) = the

sensitivity of the asset returns to market returns

E(Rm) = the expected return of the market

E(Rm) - Rf = the market premium or risk premium

The Test CAPM implies that any two assets with the same

Beta will possess identical expected returns Since the Beta of the Market Portfolio by

definition = 1.0, the difference between the return of another portfolio with a Beta near 1.0 and that of the market portfolio measures Abnormal Return

If CAPM is correct, over long time periods the difference in returns should be zero.

Simple test of the CAPM is to form portfolios with Betas near 1.0 and determine whether the mean abnormal returns are statistically different from zero.

The Data1. Collected NYSE & AMEX stock prices from 1962 - 19752. Ranked all stocks by December 31 stock market values, and

divided into 10 equally equally-weighted portfolios.3. Combined daily returns of securities to obtain portfolio returns.4. Equal weights were applied to all securities and portfolios were

adjusted for beta risk5. Re-balanced portfolio by repeating step 2 at the end of each

year.6. Calculated abnormal returns (Daily returns of portfolios minus

daily return of the equally-weighted NYSE/AMEX index)7. Analyzed portfolios in 2 ways:

1. Computed Average Rates of Return for year subsequent to formation2. Computed Average Rates of Return for second year after formation

Mean Abnormal Daily Returns of 10

Market Value Portfolios  During First Year During Second Year

PortfolioMean Abnormal

Return %'s BetaAvg Median Market

Value ($M)Mean Abnormal

Return %'s BetaAvg Median Market

Value ($M)

1             6.42  1.00               8.30  6.38 1.02                  8.30 

2             3.47  1.02             20.00  2.66 1.01                20.00 

3           (0.71) 1.00             34.10  -0.33 0.99                34.10 

4           (1.11) 1.00             54.50  -0.97 0.99                54.50 

5           (2.60) 0.94             86.10  -1.95 0.93                86.10 

6           (4.18) 0.88          138.30  -4.88 0.87             138.30 

7           (3.99) 0.90          233.50  -4.86 0.91             233.50 

8           (4.00) 0.83          413.00  -4.79 0.82             413.00 

9           (5.24) 0.83          705.30  -5.83 0.83             705.30 

10           (4.79) 0.82       1,759.90  -5.63 0.83          1,759.90 

Results: Persistence of small firm abnormal returns reduces the

chance that the results are due to market inefficiencies. Portfolio with smallest firms on average experienced returns

>20% a year higher than portfolio with largest firms. Investors can form portfolios that systematically earn

abnormal returns based on firm size. CAPM does not adequately describe stock return behavior. The persistence of positive abnormal returns for small firm

portfolios seriously violates the null hypothesis that the mean abnormal returns associated with the simple one-period CAPM are zero.

Portfolio Strategies Based on Market Capitalization – Background

Inspired by size effect theory posed by R. W. Banz, Mark Reinganum corroborated size effect in 1981.o Banz divided the stocks on the NYSE into quintiles based

on market capitalization. The returns from 1926 to 1980 for the smallest quintile outperformed the other quintiles

Reinganum (1983) takes it a step furthero CAPM is deficient in accounting for the differences in

rates of returns with equivalent beta risk

Does market capitalization have an effect on the rate of return of a portfolio over time?

Do actively managed portfolios outperform passively managed portfolios?

Portfolio Strategies Based on Market Capitalization – Issue

Portfolio Strategies Based on Market Capitalization - Sourced

Data Market capitalizations and stock returns came

from the University of Chicago’s CRSP daily tape fileo Data from 1963 to 1980 comprised from stocks listed on

the New York and American Stock Exchanges Data cleansing due to delisting

o Acquisitionso Bankruptcyo Failure to satisfy listing requirements of the exchange

Portfolio Strategies Based on Market Capitalization - Test Design Multipurpose Design

o Firm Size• Ten equally weighted portfolios grouped by market

capitalizationo Active vs. Passive

• Actively managed portfolio were rebalanced every based on year end market capitalization

• Passively managed portfolio compositions were not altered for the duration of the 18 year test

• In both cases, proceeds from delisted securities were reinvested into S&P 500 Index fund

Portfolio Strategies Based on Market Capitalization – The Market

Portfolios

PortfolioAverage

Annual ReturnAverage

Percent on AMEXAverage

Median ValueMedian

Share PriceEstimated

Portfolio BetaMV1 32.77 92.19 4.6 5.24 1.58MV2 23.51 77.33 10.8 9.52 1.57MV3 22.98 52.09 19.3 12.89 1.50MV4 20.24 34.05 30.7 16.19 1.46MV5 19.08 21.33 47.2 19.22 1.43MV6 18.30 12.73 74.2 22.59 1.36MV7 15.64 8.37 119.1 26.44 1.28MV8 14.24 4.73 209.7 30.83 1.22MV9 13.00 3.39 434.6 34.43 1.11MV10 9.47 2.25 1,102.6 44.94 0.96

Investment Charteristics of the Ten Market Value Portfolios

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Market Value Portfolios

Average Annual Returns vs. Average Median Value

Average Annual Return Average Median Value

Active Strategy Results Market Capitalization size returns evident across

the portfolio spectrumo MV1 returned cumulative returns of 4528% o MV2 returned cumulative returns of 1850% o MV3 returned cumulative returns of 2016% o MV5 returned cumulative returns of 1179% o MV10 returned cumulative returns of 312 %

Active Strategy Results

Cumulative ReturnsPortfolio 1963-1968 1969-1974 1975-1980MV1 1166 -56 739MV2 599 -63 661MV3 472 -51 663MV4 355 -59 712MV5 303 -51 559MV6 268 -48 567MV7 200 -47 454MV8 166 -39 365MV9 152 -34 308MV10 89 -19 169

Passive Strategy Results Small Firms did better even without rebalancing

o MV1 returned cumulative returns of 1026% o MV5 returned cumulative returns of 562% o MV10 returned cumulative returns of 328% 

Passive Strategy Results

Considerations Mean returns for small firms is substantially

greater than the mean holding period return for large firms (as much as 22.2% per year)

The odds for small versus large firm doubling in value were 10:1

The downside: a small firm was almost twice as likely to experience a one-year return of 25% or less

Over time, the returns of big winners more than offset the losses within the small portfolio

ConclusionAverage Returns were systematically related to market capitalization.

• Smaller firms outperformed larger ones on average even after adjusting for risk as measured by beta

• Returns Astounding• $1 invested in 1962 in small capitalization became $46 by

1980• Firms earn approximately twice as much as firms with twice

the market capitalization• Firms investing using size strategies should be actively

managed rather than passive • In short market capitalization was an excellent indicator for

long run rates of return

Modern Applications Hedge Funds Index Funds – Mid 1970s Exchange Traded Funds – Early 1990s

o Spiders (SPDR)o Active management ETFs - 2008

Index and Sector Funds

Offer target-specific index tracking Cannot outperform the constituents of their index Standardized survival biases

Index and Sector Funds

9/1/1995 5/28/1998 2/21/2001 11/18/2003 8/14/2006 5/10/2009 2/4/2012

-0.21

-0.16

-0.11

-0.06

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0.14sp500

smallcap

ETFs and Active SPDRs Exchange trading allows intra-day volatility Actively managed and rebalanced Less-Standardized biases (includes alpha)

ETFs and Active SPDRs

2/1/1993 10/29/1995 7/25/1998 4/20/2001 1/15/2004 10/11/2006 7/7/2009

-0.17

-0.12

-0.07

-0.02

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sp500

SPY

ETFs and Active SPDRs

1/1/2006 5/16/2007 9/27/2008 2/9/2010 6/24/2011

-0.24

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SPY

SLY

ETFs and Active SPDRs

2/1/2004 6/15/2005 10/28/2006 3/11/2008 7/24/2009 12/6/2010

-0.23

-0.18

-0.13

-0.08

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SPY

VBK

ETFs and Active SPDRs

12/1/2007 6/18/2008 1/4/2009 7/23/2009 2/8/2010 8/27/2010 3/15/2011 10/1/2011

-0.23

-0.18

-0.13

-0.08

-0.03

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0.07

0.12

0.17

MGK

VBK

Hedge Funds No Standardization (high alpha dependence) No trading, no intra-day volatility Unique investment goals High survival bias Strategy is AUM dependent

Hedge Funds

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Hedge Funds

1,000,000.00 201,000,000.00 401,000,000.00 601,000,000.00 801,000,000.000%

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30% avg returnMoving average (avg return)avg sigmaMoving average (avg sigma)