REVISITING THE SIZE EFFECT IN THE BOVESPA

The size effect has been analyzed in numerous stock markets using different approaches. However, there are few studies focused on its practical applicability. In this context, the aim of this study is two-fold. First, we examine price and volatility linkages among large, medium, and small firms employing a multivariate VAR-BEKK model. Second, we provide the out-of-sample performance of optimal portfolios constructed on the basis of time-varying return and volatility forecasts from this specification approach. Our overall results show that optimal portfolios are primarily composed of medium and small firms. Moreover, our findings reveal that using this technique, it is possible to reduce risk and outperform the naïve rule, which is usually employed by foreign investors interested in the Brazilian stock market. These findings are relevant not only for academics but also for practitioners because it is important an in-depth knowledge of stock market patterns in order to develop correct trading strategies.

KEYWORDS
Multivariate GARCH; optimal strategies; size effect; statistical and economic significance; Bovespa

	Large	Medium	Small
Average	0.0002	0.0007	0.0024
Standard deviation	0.0327	0.0212	0.0256
Skewness	-1.6935	-1.3052	1.4956
Kurtosis	11.9434	19.0896	16.3579
Jarque-Bera	3974.517	11546.39	8143.228
Probability	(0.0000)	(0.0000)	(0.0000)

	Large		Medium		Small
	Coeff.	t-stat	Coeff.	t-stat	Coeff.	t-stat
ε²_1,it-1	0.000	(0.03)	0.000	(0.17)	0.042	(1.50)
ε²_2,it-1	0.069	(0.65)	0.751	(4.75)	0.109	(0.79)
ε²_3,it-1	0.002	(0.28)	0.038	(1.50)	0.014	(0.42)
ε_1,t-1ε_2,t-1	-0.002	(0.06)	-0.035	(0.33)	-0.135	(1.20)
ε_{1, t -1}ε_{3, t -1}	0.000	(0.06)	0.008	(0.33)	0.049	(0.74)
ε_{2, t -1}ε_{3, t -1}	-0.023	(0.43)	-0.339	(2.52)	-0.079	(0.60)
h _11,t-1	0.907	(17.58)	0.000	(0.10)	0.006	(0.88)
h _22,t-1	0.000	(0.06)	0.113	(1.33)	0.007	(0.20)
h _33,t-1	0.000	(0.39)	0.007	(0.96)	0.937	(16.38)
h _12,t-1	0.039	(0.12)	0.007	(0.20)	0.013	(0.46)
h _13,t-1	0.035	(0.77)	0.002	(0.19)	-0.152	(1.78)
h _23,t-1	0.001	(0.14)	0.058	(2.10)	-0.157	(0.40)
η ² _1,t-1	0.000	(0.02)	0.010	(0.31)	0.006	(0.11)
η ² _2,t-1	0.003	(0.07)	0.007	(0.05)	0.042	(0.26)
η ² _3,t-1	0.006	(0.47)	0.010	(0.24)	0.114	(1.50)
η _1,t-1 η _2,t-1	0.001	(0.04)	-0.017	(0.10)	-0.033	(0.19)
η _1,t-1 η _3,t-1	-0.001	(0.04)	0.020	(0.41)	0.054	(0.23)
η _2,t-1 η _3,t-1	-0.008	(0.13)	-0.016	(0.09)	-0.139	(0.46)

Fundação Getulio Vargas, Escola de Administração de Empresas de S.Paulo Av 9 de Julho, 2029, 01313-902 S. Paulo - SP Brasil, Tel.: (55 11) 3799-7999, Fax: (55 11) 3799-7871 - São Paulo - SP - Brazil
E-mail: rae@fgv.br

Acompanhe os números deste periódico no seu leitor de RSS

Found evidence of the occurrence of the anomaly
Author	Analysis period	Methods and tests used
Eid and Romaro (2002)Eid, W., & Romaro, P. (2002). Efeito tamanho na Bovespa. Anais do XXIII CLADEA, Porto Alegre, 1, 1-12.	1995 to 1998	Linear regression, average, variance, skewness and kurtosis, method of least squares, Durbin-Watson, Breusch-Godfrey, Q Lyung-Box, Goldfeld and ARCH-LM tests
Matsumoto and Lima (2004)Matsumoto A., & Lima, F. (2004). Efeito tamanho: Evidências no mercado acionário brasileiro. In IV Congresso USP de Controladoria e Contabilidade, 2004, São Paulo (SP). Anais do IV Congresso USP de Controladoria e Contabilidade.	1996 to 2003	Linear regression
Lima et al. (2005)Lima, F., Costa, M., & Bruni, A. (2005). Anomalias em mercados de capitais: Uma análise do efeito tamanho na bolsa de valores de São Paulo com o uso do CAPM e do modelo de mercado. Revista Gestão e Planejamento, 11, 23-28.	1995 to 2003	Linear regression, CAPM, market model
Found no evidence of the occurrence of the anomaly
Author	Analysis period	Methods and tests used
Lucena and Figueiredo (2004)Lucena, P., & Figueiredo, A. (2004). Pressupostos de eficiência de mercado: Um estudo empirico na Bovespa. Revista Eletrônica de Gestão Organizacional, 2(3). Retrieved from https://periodicos.ufpe.br/revistas/gestaoorg/index). https://periodicos.ufpe.br/revistas/gest...	1994 to 2003	Linear regression, model of three factors, autoregressive models of Box and Jenkins, and Grinblatt and Moskowitz model
Antunes et al. (2006)Antunes, G. A., Lamounier, W. M., & Bressan, A. A. (2006). Análise do efeito tamanho nos retornos das ações de empresas listadas na BOVESPA. Revista Contabilidade & Finanças, 17(40), 87-101. doi: 10.1590/S1519-70772006000100007 https://doi.org/10.1590/S1519-7077200600...	1998 to 2004	Simple arithmetic average, CAPM and Jensen’s alpha indicator
Flister et al. (2011)Flister F., Bressan, A., & Amaral, H. (2011). CAPM condicional no mercado brasileiro: Um estudo dos efeitos momento, tamanho e book-to-market entre 1995 e 2008. Revista Brasileira de Finanças, 9(1), 105-129.	1995 to 2008	Linear regression and CAPM
Saturnino et al. (2012)Saturnino, O., Saturnino, V., Lucena, P., Carmona, C., & Araújo, L. F. (2012). Investimento em valor contrário no Brasil: Overreaction ou efeito tamanho? Revista de Finanças Aplicadas, 1, 1-20.	1995 to 2010	Linear regression and CAPM
Machado and Machado (2014)Machado, M., & Machado, M. (2014). Liquidez e precificação de ativos: Evidências do mercado brasileiro. Brasilian Business Review, 11, 73-95.	1995 to 2008	Linear regression, model of two factors, model of three factors and CAPM

Panel A: Mean equation
	Large		Medium		Small
	Coeff.	t-stat	Coeff.	t-stat	Coeff.	t-stat
c_i	-0.0004	(-0.26)	0.0003	(0.35)	0.0023	(1.93)
R_1,t-1	0.0376	(0.73)	0.0701	(2.48)	0.0445	(1.14)
R_1,t-2	0.1207	(2.35)	0.0180	(0.64)	-0.0177	(-0.45)
R_2,t-2	0.0360	(0.37)	0.0998	(1.90)	0.1337	(1.85)
R_2,t-2	0.0997	(1.06)	0.1863	(3.60)	0.3618	(5.08)
R_3,t-1	-0.0134	(-0.22)	0.0168	(0.51)	-0.0818	(-1.80)
R_3,t-2	-0.1226	(-2.02)	-0.0070	(-0.21)	-0.0031	(-0.07)
Panel B: Variance equation
$C = [\begin{matrix} 0.005 \\ (0.00) \\ 0.002 0.011 \\ (0.639) (0.000) \\ (0.133) (0.11) (1.00) \end{matrix}]$			$A = [\begin{matrix} 0.003 & - 0.020 & - 0.205 \\ (0.93) & (0.63) & (0.00) \\ 0263 & 0.866 & 0.330 \\ (0.07) & (0.00) & (0.02) \\ - 0.043 & - 0.196 & - 0.119 \\ (0.43) & (0.00) & (0.24) \end{matrix}]$
$B = [\begin{matrix} 0.953 & - 0.010 & - 0.079 \\ (0.00) & (0.78) & (0.01) \\ 0.020 & 0.334 & - 0.081 \\ (0.86) & (0.00) & (0.57) \\ - 0.019 & - 0.086 & 0.968 \\ (0.27) & (0.01) & (0.00) \end{matrix}]$			$G = [\begin{matrix} 0.006 & - 0102 & - 0.080 \\ (0.96) & (0.38) & (0.75) \\ 0.053 & 0.083 & 0.206 \\ (0.83) & (0.89) & (0.46) \\ - 0.078 & - 0.098 & - 0.337 \\ (0.19) & (0.49) & (0.00) \end{matrix}]$
Panel C: Restriction test on the effects of variance
		Chi-squared			*( p-value* )**
		155.738			(0.00)
		45.437			(0.00)
		29.011			(0.00)
		42.227			(0.00)
		74.325			(0.00)
Panel D: Residual diagnostics
	Large		Medium		Small
	Coeff.	*p-value*	Coeff.	*p-value*	Coeff.	*p-value*
Q(20)	15.169	(0.767)	18.708	(0.541)	20.043	(0.455)
Q²(20)	18.322	(0.566)	9.444	(0.977)	7.322	(0.995)
Sign	-0.298	(0.144)	-0.048	(0.808)	0.198	(0.338)
	Large-medium		Large-small		Medium-small
	Lead	Lag	Lead	Lag	Lead	Lag
Cross (20)	-0.0391	-0.0177	-0.0041	-0.0012	-0.0311	-0.0450

Panel A - without short positions
	Large	Medium	Small
Large/Medium	0.1146	0.6575	--
Large/Small	0.2861	--	0.5185
Medium/Small	--	0.4271	0.4350
Large/Medium/Small	0.0777	0.4346	0.2980
Panel B – with short positions
	Large	Medium	Small
Large/Medium	-0.6382	1.6382	--
Large/Small	0.8020	--	0.1980
Medium/Small	--	3.1950	-2.1950
Large/Medium/Small	0.2611	0.2952	0.4437

Panel A: Profitability
	Naïve strategy	Optimal strategy	Differences optimal - naive
Large/Medium	0.0385	0.1050	0.0665
Large/Small	0.0785	0.1122	0.0337
Medium/Small	0.0732	0.0995	0.0264
Large/Medium/Small	0.0634	0.1208	0.0574
Panel B: Risk
	Naïve strategy	Optimal strategy	Differences optimal - naive
Large/Medium	0.1670	0.1139	-0.0531
Large/Small	0.1549	0.1163	-0.0386
Medium/Small	0.1402	0.1229	-0.0172
Large/Medium/Small	0.1490	0.1108	-0.0381
Panel C: Sharpe ratio
	Naïve strategy	Optimal strategy	Differences optimal - naive
Large/Medium	0.2303	0.9220	0.6917
Large/Small	0.5071	0.9649	0.4578
Medium/Small	0.5219	0.8096	0.2877
Large/Medium/Small	0.4255	1.0898	0.6644

Panel A: Profitability
	Naive strategy	Optimal strategy	Differences optimal - naive
Large/Medium	0.0385	0.2680	0.2296
Large/Small	0.0785	0.3217	0.2432
Medium/Small	0.0732	1.3895	1.3164
Large/Medium/Small	0.0634	0.1823	0.1189
Panel B: Risk
	Naive strategy	Optimal strategy	Differences optimal - naive
Large/Medium	0.1670	1.0766	0.9096
Large/Small	0.1549	0.4219	0.2670
Medium/Small	0.1402	3.5404	3.4002
Large/Medium/Small	0.1490	0.2015	0.0525
Panel C: Sharpe ratio
	Naive strategy	Optimal strategy	Differences optimal - naive
Large/Medium	0.2303	0.2489	0.0187
Large/Small	0.5071	0.7625	0.2554
Medium/Small	0.5219	0.3925	-0.1294
Large/Medium/Small	0.4255	0.9050	0.4795