Is size a priced factor in Germany?

Presented by :

Wen Huang

Savang Kittikhoun

Kim-Tuan Nguyen

Yong Yao

Tianxue Zhang

Introduction

• CAPM :– Excess return = ß * market premium

• Empirical studies show that stock price not fully explained by CAPM– Returns of small firms turn out higher– Returns of big firms turn out lower

Introduction

Does the size of a firm affect its return ?

Methodology

• Excess return = ß * market premium + ??

• Period of analysis : Jan 1991 to Dec 2000

• Randomly pick 200 out of 800 firms

• Sort by size (market capitalization)– Group in 10 portfolios of 20 firms each– Re-balance portfolio every year

Attribute portfolio Quintiles

Portfolio 1(smallest)

2 3 4 5 6 7 8 9 10(largest)

Mean ExcessReturn

1.315% 0.180% 0.466% 0.175% -0.051% 0.297% 0.570% 0.527% 0.029% 0.334%

Std Deviation 0.058 0.043 0.045 0.046 0.038 0.044 0.090 0.041 0.035 0.036Market Beta 0.290 0.251 0.344 0.281 0.254 0.276 0.140 0.370 0.376 0.411

Attribute Portfolio - Graph

Portfolio Mean Excess Returns

0.000%

0.200%

0.400%

0.600%

0.800%

1.000%

1.200%

1.400%

0 1 2 3 4 5 6 7 8 9 10

Portfolio n

Mea

n Ex

cess

Ret

urn

Regression

• Construction of a factor-mimicking spread portfolio

• Small minus Big (SMB)– Returns of smallest 30% - returns of biggest

30%– Spread portfolio has no market risk

Regression - cont’d

• rnt = n+ nm rmt + nA SMBt+ unt

• Where:– rnt : average excess return

n : intercept;

– rmt : excess return of German market index

nm : market beta of portfolio n

– SMBt : return of spread portfolio

nA : sensitivity beta of the return of portfolio n to this size attribute

– unt : the error term

Regression results

n

nm

nA P-value(nA)Portfolio 1 0.0083 0.3855 1.0585 5.2E-14Portfolio 2 -0.0008 0.2977 0.5126 3.6E-06Portfolio 3 0.0014 0.3993 0.6097 1.6E-08Portfolio 4 0.0010 0.2806 -0.0054 0.9646Portfolio 5 -0.0009 0.2484 -0.0627 0.5342Portfolio 6 0.0023 0.2722 -0.0404 0.7252Portfolio 7 0.0048 0.1525 0.1374 0.5855Portfolio 8 0.0056 0.3380 -0.3513 0.0004Portfolio 9 -8.5E-05 0.3615 -0.1631 0.0417Portfolio 10 0.0034 0.3830 -0.3045 9.5E-05

Regression results - GraphRegression Result

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10

Portfolio n

Bet

as

Market Beta

Size Attribute Beta

Linear (Market Beta)

Linear (Size AttributeBeta)

Another test

• rnt = 0t+ mt nm + 1t nA + 2t An + vnt

nm , nA taken from previous regression

– An is the log (base 10) of the size of the firm

• If 1t significant, but 2t not significant :

– > nA fully captures the attribute induced risk

We cannot conclude that nA fully captures the risk induced attribute

Variables Coefficients Standard Error P-value

Intercept 0t -0.001594 0.005635 0.786776

nm -0.000217 0.016449 0.989888

nA 0.008905 0.004775 0.111458An 0.002090 0.002344 0.406798

Regression results...

Sensitivity analysis - SMB

• Variation of returns of SMB could change nA substantially

• Independance International Associates (IIA)– Set of 5 portfolios for each country– Covers 75% of market capitalisation– Big cap portfolio : 70% of total market cap– Small cap portfolio : 30% of total market cap

• Repeat time series analysis

Results

n

nm

nA P-value(nA)

Portfolio 1 0.00856 0.27185 0.15127 0.53174Portfolio 2 0.00420 0.41371 0.58813 0.01341Portfolio 3 0.00085 0.44374 0.12574 0.48858Portfolio 4 0.00145 0.29281 0.56750 0.02544Portfolio 5 -0.00225 0.39060 0.39130 0.07189Portfolio 6 0.00371 0.35050 0.25180 0.28332Portfolio 7 0.00255 0.28145 0.42717 0.04515Portfolio 8 0.00557 0.41249 0.12683 0.55018Portfolio 9 0.00068 0.41557 0.11850 0.41257Portfolio 10 0.00112 0.53128 0.19309 0.14497

Sensitivity analysis - SMBSensitivity Analysis

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 10

Portfolio n

Be

ta

IIA Beta

Linear (IIA Beta)

Sensitivity Analysis - Time

• Divide our samples into two sub-periods :– January 1991 to December 1995– January 1996 to December 2000

• Purpose of check :– Does the correlation between portfolio returns

and the two risk factors time dependent ?

Regression Results

Intercept n nm

nA

1st 2nd 1st 2nd 1st 2nd

n

n nm

nm nA

nA nA nA nA nA nA nA nA

1 0.0055 0.0114 0.3418 0.3992 0.9057 1.1254

2 -0.0021 0.0018 0.293 0.2667 0.1761 0.6642

3 0.0026 0.0015 0.4747 0.3253 0.4744 0.6756

4 0.0006 0.0006 0.2239 0.3358 0.064 -0.0406

5 -0.0023 0.0017 0.2573 0.2096 -0.3491 0.0673

6 0.0025 0.0028 0.3073 0.2379 -0.0809 -0.0198

7 0.0032 0.0079 0.2187 0.071 -0.0376 0.2217

8 0.0055 0.0065 0.3148 0.3339 -0.6309 -0.2273

9 -0.0007 0.0014 0.4022 0.3137 -0.2636 -0.1148

10 0.0012 0.0068 0.3924 0.3435 -0.5494 -0.1927

Conclusion

• Our work provides evidence that :– Larger firms have lower returns– Smaller firms have higher returns

• We encountered a few inconclusive results– May require larger sample– Return could depend on other attributes

• Our model :– concurs with empirical observations– Still incomplete