Time horizon trading and the idiosyncratic risk puzzle

Abstract

We analyse whether the idiosyncratic risk puzzle reported by Ang et al. can be explained by the existence of market participants with different investment horizons. We adopt a wavelet multiresolution analysis to decompose the returns distribution for different time scales. Our approach divides the nonlinear link between expected returns and idiosyncratic risk into two linear relationships, a positive one for long-run investors and a negative one for short-run investors, indicating that the puzzle disappears as the wavelet scale increases (long-term horizons). Our results are robust to several types of wavelets, to different definitions of short-term investors and to various measures of idiosyncratic risk.

Keywords:

• Asset pricing
• Systematic risk
• Wavelets
• Time-scaling risk

JEL Classification:

• G00
• G12

Acknowledgements

We are grateful for helpful comments and suggestions on Wavelets Multiresolution Analysis from Agnieska Jack. David Moreno ([email protected]) acknowledges financial support from Ministerio de Ciencia y Tecnología grant ECO2010-17158. Rosa Rodríguez ([email protected]) acknowledges financial support from Ministerio de Ciencia y Tecnología grant ECO2009-10796. Corresponding author Juliana Malagón ([email protected]) acknowledges financial support from Comunidad de Madrid grant CCG10-UC3M/HUM-5237.

Notes

1 Although the recent debate has renewed the discussion on the predictability of returns using idiosyncratic risk, the issue has been extensively discussed in the past. Both Lintner (1965) and Douglas (1969) find significant explanatory power of the variance of the residuals from a market model in the cross-section of average stock returns. Miller and Scholes (1972) and Fama and Macbeth (1973) argue for statistical problems. Finally, Lehmann (1990) reaffirms Douglas’ results after a careful econometric revision.

2 Wavelet analysis is relatively new in economics and finance, although the literature on wavelets is growing rapidly. Applications in these fields include the study of systematic risk in the capital asset pricing model (Gençay et al. 2003, Rhaiem et al. 2007), the multi-scale relationship between stock returns and inflation (Kim and In 2005), the relation between returns and systematic co-kurtosis and co-skewness (Galagedera and Maharaja 2008), a multi-scale hedge ratio (In and Kim 2006), studies in portfolio management (In et al. 2008, Bowden and Zhu 2010) and portfolio allocation (Kim and In 2010), the analysis of co-movements in stock markets (Rua and Nunes 2009), Value at Risk measures (Fernandez 2005) and credit portfolio losses (Masdemont and Ortiz-Gracia 2011).

3 In particular, the beta of thinly traded securities increases as the return interval rises, whereas the beta of frequently traded securities falls. Furthermore, the estimated beta of highly capitalized firms decreases as the return interval increases, whereas the beta of low-cap firms increases.

4 Heterogeneous agent models (Müller et al. 1993, 1997, LeBaron 2000) are theoretical explanations for empirical stylized facts based on the existence of differences in investors. Müller argues that differences can be observed in perceptions, institutional constraints, risk profiles, prior beliefs and geographical location. We focus on the idea of differences in time horizons because it offers the possibility of the mathematical treatment we show.

5 O’Hara (2003) discusses the impact of diverging information within the classical asset pricing model assumptions. Her main conclusion is that asymmetric information derives from a group of uninformed (noise) traders who, even if they systematically lose to better-informed ones, make portfolio choices so that their risk exposure to wins of informed investors is lower.

6 They have been obtained from Kenneth French’s website http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

7 We also verify the puzzle using 6 months for the regression in equation (1) to address the critique of error-in-variance exposed by Malkiel and Xu (2002). The results do not change. They are available upon request.

8 Although not reported here, the significance of the [5–1] difference is sensitive to the time period considered. Only for three months in the sample from November 1990 to December 2009 is the difference not significant at either the 5% or 10% levels. These results are available upon request.

9 The same conclusion was found by Ang et al. (2006) using another methodology. They included a double-sorting control for momentum, after which the puzzle still holds.

10 Notice that, as stated previously, the time scale increases by multiples of 2^j, and we are working with daily data. Therefore, the time scale increases by 2^j days each time: 2–4 days, 4–8 days, 8–16 days, and so on.

11 It has been shown that technical analysis is mostly used for short-term forecasting (Frankel and Froot 1990). However, we prefer to let the exact nature of investors be an open issue and limit our classification to the relative frequency of trading of each group of investors considered in the MRA. This is because we assume that investors use all tools available for decision making no matter how frequently they trade.

12 Adopting O’Hara’s idea would require separating short-term investors from uninformed ones. However, we do not have sufficient evidence to do this. The Fractal Market Hypothesis proposes that information is more closely related to market sentiment and technical factors in the short term than in the long term and that short-term price movements are likely to be the result of crowd behavior (Blackledge 2010).

13 The authors argue that retail investors are especially interested in these stocks because of their speculative character: high skewness and high volatility.

14 Although many other possibilities exist, in this paper we consider only the Haar and the Daubechies wavelet families. We consider the Haar family our benchmark because many of the previous studies available on risk loadings in asset pricing models use it. Keeping the same family facilitates comparisons. The Daubechies family is a natural extension in that the Haar wavelet is the Daubechies wavelet of minimum length. It is also a common wavelet family for studies in economics and finance. See, for example, Huang and Wu (2008) or Fan and Gençay (2010).

15 Results are available upon request.