Mean reversion in a price-earnings ratio and under / overvaluation in the Brazilian stock market

Amorim, Daniel Penido de Lima; Camargos, Marcos Antônio de

doi:10.1590/1808-057x202111780

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Brasil

Revista Contabilidade & Finanças

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Original Article • Rev. contab. finanç. 32 (86) • May-Aug 2021 • https://doi.org/10.1590/1808-057x202111780 copy

Mean reversion in a price-earnings ratio and under / overvaluation in the Brazilian stock market^* * Daniel Penido de Lima Amorim thanks the Coordination for the Improvement of Higher Education Personnel (CAPES) for the essential financial support to carry out this research. The authors would like to thank Aureliano Angel Bressan, the anonymous referees, and the editor of RC&F for the comments that contributed to the improvement of this article.

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

The market price-earnings ratios differ from those of each share. Despite allowing for several pertinent analyses, authors have rarely addressed these valuation ratios in the Brazilian context. We can use it to evaluate whether the stock market is overvalued (undervalued). In this article, we analyze the mean reversion in a price-earnings ratio based on Ibovespa and identify periods of overvaluation (undervaluation) in the Brazilian stock market. We considered the period from December 2004 to June 2018. Until then, there are no studies that sought to identify periods of overvaluation (undervaluation) in this market. In the analyses, we used non-linear econometric methods. We analyzed the mean reversion in the price-earnings ratio using a unit root test that incorporates a Fourier function in the deterministic term. We identified the periods of market overvaluation (undervaluation) through the regime probabilities obtained from a Markov Switching model, estimated with the price-earnings ratio. The results evidenced that the price-earnings ratio based on the Ibovespa has a non-linear trend and exhibits mean reversion. Thus, this valuation ratio should provide information on the future stock market returns, mostly when it is very dispersed in relation to historical standards. We identified four periods of market overvaluation interposed with five periods of market undervaluation. Mean reversion in the price-earnings ratio contraposes the Efficient Markets Hypothesis. There are no other applications of unit root tests with a Fourier function in the Brazilian context. Furthermore, adopting a Markov Switching model to identify periods of market overvaluation (undervaluation) consists of a methodological contribution. Investors can take advantage of the identification of these periods to establish investment strategies.

Keywords:
price-earnings ratio; mean reversion; Markov Switching; overvaluation; behavioral finance

Statistics	P/E1
Mean	11.39
Maximum	26.20
Minimum	2.41
Standard deviation	5.24
Number of observations	163

Variable	Coefficient	Standard error	t-statistics
Intercept	1.8760***	0.3449	5.4389
Trend	0.0383***	0.0083	4.6148
$s i n (2 π 1 t / 163)$	-0.3273	0.2090	-1.5658
$c o s (2 π 1 t / 163)$	-0.2018***	0.0725	-2.7824
$s i n (2 π 2 t / 163)$	-0.3484***	0.1259	-2.7668
$c o s (2 π 2 t / 163)$	-0.3205***	0.1034	-3.1003
$s i n (2 π 3 t / 163)$	-0.1923**	0.0920	-2.0914
$c o s (2 π 3 t / 163)$	-0.0270	0.0941	-0.2870
$s i n (2 π 4 t / 163)$	0.0274	0.0954	0.2876
$c o s (2 π 4 t / 163)$	-0.5867***	0.1400	-4.1912
$s i n (2 π 5 t / 163)$	0.3494**	0.1708	2.0455
$c o s (2 π 5 t / 163)$	0.5087***	0.1020	4.9849
${P / E 1}_{t - 1}$	-0.4392***	0.0674	-6.5189
$∆ {P / E 1}_{t - 1}$	0.4688***	0.0942	4.9785
${∆ P / E 1}_{t - 2}$	0.1434	0.0919	1.5596
${∆ P / E 1}_{t - 3}$	0.1672**	0.0815	2.0506
Diagnostic test statistics
R²			0.3802
Adjusted R²			0.3152
Sum of squares of residuals			93.2525
Akaike Information Criterion			2.5055
F-statistic			5.8475
F-statistic - p-value			0.0000
Breusch-Godfrey autocorrelation LM test - $χ^{2}$ (1)			0.2964
Breusch-Godfrey autocorrelation LM test - p-value			0.5861
ARCH LM heteroskedasticity test - $χ^{2}$ (1)			3.1260
ARCH LM heteroskedasticity test - p-value			0.0771
F linearity test - F-statistic (10, 134)			5.1820
F linearity test - p-value			0.0000

Test statistics	Value	Significance	Critical value (T = 100)	Critical value (T = 200)
t-statistics	−6.52	10%	−3.22	−3.24
		5%	−3.56	−3.56
		1%	−4.20	−4.18

Variable	Coefficient	Standard error	z-statistics
Regime-dependent intercepts
$α_{1}$	5.8950***	0.2847	20.7088
$α_{2}$	2.1680***	0.2201	9.8480
Common terms
Trend	0.0982***	0.0028	35.5549
log $(σ)$	0.2669**	0.1052	2.5375
Transition matrix parameters
$s_{1}$	2.4525***	0.4057	6.0452
$s_{2}$	-3.2882***	0.4068	-8.0839
Constant transition probabilities
p_1\|1			92.07%
p_1\|2			7.93%
p_2\|2			96.40%
p_2\|1			3.60%
Constant expected durations
s₁			12.62
s₂			27.80
Diagnostic test statistics
Likelihood log			-298.7134
Shapiro-Wilk test - z-statistics			1.4811
Shapiro-Wilk test - p-value			0.4768

Periods	Duration	Regime	Market State
2004-12 to 2007-08	33	s ₂	Undervalued
2007-09 to 2008-08	12	s ₁	Overvalued
2008-09 to 2009-07	11	s ₂	Undervalued
2009-08 to 2010-11	16	s ₁	Overvalued
2010-12 to 2012-09	22	s ₂	Undervalued
2012-10 to 2013-12	15	s ₁	Overvalued
2014-01 to 2015-10	22	s ₂	Undervalued
2015-11 to 2016-11	13	s ₁	Overvalued
2016-12 to 2018-06	19	s ₂	Undervalued

Universidade de São Paulo, Faculdade de Economia, Administração e Contabilidade, Departamento de Contabilidade e Atuária Av. Prof. Luciano Gualberto, 908 - prédio 3 - sala 118, 05508 - 010 São Paulo - SP - Brasil, Tel.: (55 11) 2648-6320, Tel.: (55 11) 2648-6321, Fax: (55 11) 3813-0120 - São Paulo - SP - Brazil
E-mail: recont@usp.br

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