Size Effect
Matthew Boyce
Huibin Hu
Rajesh Raghunathan
Lina Yang
Introduction
• In this presentation we will review two articles on
how the size of market capitalization affects
portfolio performance.
• The data sets given in both articles support the
basic argument that smaller size results in a
superior performance.
Size Effect
• A theory that holds that smaller firms, or those
companies with a small market capitalization,
outperform larger companies.
• The theory holds that smaller companies have a greater
amount of growth opportunities than larger companies.
• An effect size is typically calculated by taking the
difference in means between two groups and dividing
that number by their combined (pooled) standard
deviation.
Abnormal Returns in Small Firm Portfolio
Marc R. Reinganum
Capital Asset Pricing Model (CAPM)
• E(R)=Rf+Beta*MRP
• Beta
o The most important term in the equation
o A measure of the asset’s covariance with the market as a
whole
o COV (Ri,, Rm)/VAR(Rm), the risk of asset I relative to the
market portfolio
• Implication: any two assets with the same beta will have the
same expected return. In particular, the model implies that
firms will commend higher risk premiums only if they have
Misspecifications of CAPM
• Data on firm size can be used to create a portfolio that
systematically earns abnormal returns.
• Small firms systematically experienced abnormal rates of
return.
• The persistence of these abnormal returns reduces the
likelihood the results are being generated by market
inefficiency.
• CAPM may not adequately describe the behavior of the
stock prices.
Test of Abnormal Return
• Collected stock prices, daily returns and common shares from University of
Chicago’s center for research in Security Prices daily master and return
tapes and the Compustat Merged Annual Industrial tape
• For each year from 1962 through 1975, ranked all firms in the sample on
the basis of their Dec. 31 aggregate stock market values
• Broke down the ranked sample into deciles making all portfolios have betas
near one and combined the daily returns of securities in each decile to form
the daily returns of each portfolio 1 through 10, with 1 corresponding to the
lowest decile and 10 to the highest
• Equal weights were applied to all portfolio and equal weighted NYSEAMEX market index serves as the control portfolio
• Table 1 shows that the portfolio composed of
small firms stand out. On average the smallest
firms experience returns more than 20 percent
per year higher than the returns for the largest
firms. It is not only because of their positive
abnormal returns, but also because each is
heavily traded on the American Stock
Exchange.
On the basis of firm size data, an investor can form portfolios that systematically
earn abnormal returns that persist for at least two years.
Application
The fact that small firms have systematically experienced average rates of
return significantly greater than those of larger firms with equivalent beta risk,
and that these abnormal returns have persisted for at least two years from the
portfolio formation dates, indicates that the simple one-period CAPM is an
inadequate empirical representation of capital market equilibrium. Alternative
models of capital market equilibrium should be seriously considered and
tested.
Portfolio Strategies based on Market Capitalization*
No matter how you slice it, small caps win out
Marc R. Reinganum
This paper explores some simple portfolio strategies suggested by the
empirical relationship between stock returns and market capitalization.
Overview of Method
•
•
•
•
•
Included all stocks traded on NYSE & AMEX.
10 portfolios with 10% of stocks in each.
Portfolios ranked from 1 to 10 (largest).
Funds remaining after delisting were placed in an S&P 500
index fund.
Two holding strategies: yearly balancing & buy and hold
(1962-1980).
Overview of Results: Yearly Balancing
•
•
•
$1 invested in 1962 would equal $46 in smallest
portfolio, $13 in intermediate, and $4 in large.
MV1 averaged 32.77% annual return; mid-sized
averaged about 18%; and, MV10 averaged 9.47%.
Exception to rule found for the period of 1969 to
1973 (rule reversed).
Investment Characteristics of the Ten Market Value
Portfolios
Portfolio
Average Annual
Return
Average Percent on
AMEX
Average Median
Value
Median
Share Price
Estimated Portfolio
Beta
1
32.77
92.19
4.6
5.24
1.58
2
23.51
77.33
10.8
9.52
1.57
3
22.98
52.09
19.3
12.89
1.50
4
20.24
34.05
30.7
16.19
1.46
5
19.08
21.33
47.2
19.22
1.43
6
18.30
12.73
74.2
22.59
1.36
7
15.64
8.37
119.1
26.44
1.28
8
14.24
4.73
209.7
30.83
1.22
9
13.00
3.39
434.6
34.43
1.11
10
9.47
2.25
1102.6
44.94
0.96
Beta
 Measure of volatility of a portfolio in comparison to the
market as a whole.
 β = 1  security’s price will move with the market.
 β < 1  security will be less volatile than the market.
 β > 1  security will be more volatile than the market.
 E.g. If β = 1.2, theoretically that security is 20% more
volatile than the market.
Dimson Beta
•
•
•
•
Why? – need to use the aggregated coefficients technique
because standard market model beta may seriously understate
the true beta of the small firm portfolio because of nontrading.
Suggest that small firms are riskier than the large firms.
1.58 for the smallest - 0.96 for the largest
According to Reinganum, the spread in Dimson betas is not
large enough to account for the observed difference in the
average portfolio returns.
Important Observations
•
•
•
Size effect vs. listing effect (AMEX).
Dimson Beta unable to capture real beta of small
firms.
CAPM is an inadequate model for this study.
ΔE(R) = Δβ[E(RM) – Rf]
=23.3%/.62 = ΔE(R)/Δβ
(Only if risk-free rate = 37.5%)
Take away lesson
Size effects exists even after beta adjustments.
Cumulative Returns with Annual Updating
(Expressed in Percentages)
MVP
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1
17
40
120
110
511
1366
768
627
829
920
531
451
897
1465
1788
2421
3528
4528
2
15
42
104
93
315
599
384
336
433
447
231
156
322
541
696
921
1402
1850
3
22
51
128
104
289
472
353
307
392
414
236
177
390
650
839
1039
1515
2016
4
24
46
95
89
239
355
241
191
248
267
141
84
212
377
490
639
994
1395
5
17
38
83
71
200
303
212
197
249
278
147
94
213
375
463
548
867
1179
6
23
41
94
79
180
268
193
173
248
267
157
90
205
355
442
541
831
1168
7
15
35
74
63
141
200
146
134
179
197
115
59
146
260
308
369
563
782
8
17
39
74
63
122
166
124
117
167
201
126
61
162
271
295
336
479
650
9
19
40
74
65
112
152
124
125
164
187
118
64
155
246
250
280
402
570
10
20
37
54
42
68
89
71
73
99
139
108
53
107
157
141
155
219
312
Cumulative Percent Return for Market Value
Portfolios (1963-1980) with Annual Updating
1
2
3
4
5
6
7
8
9
10
4528 1850 2016 1395 1179 1168 782 650 570 312
Cumulative Returns: By Sub-Period with Annual
Updating
Portfolio
1963-1968
1969-1974
1975-1980
1
1166
-56
739
2
599
-63
661
3
472
-51
663
4
355
-59
712
5
303
-51
559
6
268
-48
567
7
200
-47
454
8
166
-39
365
9
152
-34
308
10
89
-19
169
Summary Results for Annually Updated Market Value Portfolios
•
•
•
Portfolio 9 showed 570% increase, vs. 312% for
portfolio 10.
Smaller portfolios more sensitive to market
volatility.
Smallest portfolio still beats large firms (265% since
1969).
Cumulative Returns without Annual Updating (Expressed in Percentages)
Portfolio
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1
17
48
120
115
346
628
433
355
426
478
300
196
339
499
528
603
807
1026
2
15
35
91
73
303
485
327
277
317
371
260
171
323
475
490
548
734
958
3
22
55
114
104
249
377
278
266
345
392
285
182
308
440
470
505
717
923
4
24
46
104
97
219
314
221
204
265
298
206
134
252
386
412
460
640
835
5
17
35
83
75
186
263
187
181
219
256
172
101
193
293
297
339
438
562
6
23
42
88
72
159
232
165
154
203
243
166
109
217
317
314
341
450
603
7
15
40
85
71
145
206
155
159
196
224
172
106
215
325
334
361
495
667
8
17
38
70
57
113
153
120
123
159
192
132
74
160
245
239
255
333
414
9
19
37
64
52
89
133
99
113
146
185
144
101
198
287
276
300
388
501
10
20
38
55
41
66
93
74
85
105
131
103
60
128
200
182
194
250
328
Summary Results for Market Value Portfolios Without
Annually Updating
•
•
•
Portfolio 1 returns exceeded those of portfolio 10 by almost
700%.
Active is better than passive. ($1 invested smallest in 1963
leads to $46 or $11 return.)
Only the largest portfolio did not benefit from updating.
Distribution of One-Year Holding Period Returns for Securities
within Each Market Value Group
Portfolio
Mean (%)
Median (%)
Tenth
Percentile
Ninetieth
Percentile
Skewness
Kurtosis
1
31.77
10.94
-43.90
120.59
9.97
216.65
2
23.66
9.99
-42.95
98.64
4.25
44.92
3
23.52
11.85
-40.00
95.59
2.66
16.02
4
21.24
11.67
-38.45
89.40
1.92
8.43
5
19.77
11.53
-37.08
82.25
2.41
18.04
6
19.12
11.93
-35.48
76.48
3.21
35.50
7
16.45
9.99
-33.10
72.21
1.35
4.20
8
14.86
10.28
-29.23
62.53
1.43
7.06
9
13.42
8.94
-27.06
58.73
1.16
4.21
10
9.57
7.17
-25.44
44.46
0.88
2.82
Overview of Kurtosis and Skewness
Critique of Article
•
•
•
•
Lacking in data regarding the statistical significance of
findings.
Period is limited (18 years).
Discussion of U.S. economic trends for the period is not
discussed.
Common size classifications used by investors are not
included.
Conclusion
Invest in small market cap portfolio over a longer period of
time!