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Equity premium over different investment horizons

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Abstract

This paper studies the adequate size of equity premium over different investment horizons based on spatial dominance. We find that the puzzle with respect to the size of equity premium disappears as investment horizons get longer in terms of the spatial dominance; therefore, the adequate size of equity premium should be dependent upon the investment horizon.

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Notes

  1. For instance, (Lewis 1991; Handa et al. 1993) show that asset pricing models cannot be rejected at longer holding intervals, and (Daniel and Marshall 1997; Parker and Julliard 2005) find that the risk premium can be explained better by long horizon consumption-based models.

  2. For example, in order to analyse the effect of time diversification, the mean–variance framework was used in Lloyd and Haney (1980); Lloyd and Modani (1983); Lee (1990); Thorley (1995) and Hodges et al. (1997), while (Hodges and Yoder 1996) applied the stochastic dominance analysis.

  3. We would like to thank an anonymous referee for addressing this issue.

  4. For more details about multi-period stochastic dominance, readers are referred to Levy (1973); Levy and Paroush (1974a) and Levy and Paroush (1974b).

  5. Hodges and Yoder (1996) provided the empirical study based on the stochastic dominance analysis with the assumption of \(i.i.d.\) returns, but they cannot provide any evidence for the relationship between dominance and the different investment horizons.

  6. For instance, the sample analogue estimators of the integrated local time and the integrated integrated local time are expressed as

    $$\begin{aligned} \hat{{L}}(T,x)=\Delta \sum _{i=1}^n {e^{-ri\Delta }} 1\{X_{i\Delta } \le x\},I\hat{{L}}(T,x)=\Delta \sum _{i=1}^n {e^{-ri\Delta }(x-} X_{i\Delta } )1\{X_{i\Delta } \le x\}, \end{aligned}$$

    where \(\Delta \) and \(\hbox {n}(=T/\Delta )\) indicate an observation interval and the number of observations for a given time period from 0 up to T, respectively.

  7. For instance, the sample observation inside the k-th holding period is denoted by (\(X^{k}_{i\varDelta })=\{ X^{k}_{1\varDelta , }X^{k}_{2\varDelta ,\ldots .,}X^{k}_{n\varDelta ,}\}\) for i = 1,...n, k = 1,...N.

  8. We also performed the test using daily T-bill rates simply obtained by dividing by 252, and the results of the second spatial dominance analysis with this data were very similar to the ones reported in the paper. We appreciate the referee who suggested using continuously compounding annual returns for the riskless asset in our analysis.

  9. The method in Capozza and Cornell (1979) was used to obtain the continuously compounding returns.

  10. We thank an anonymous referee for suggesting this point. In general, the investors should consider the conditional distribution of returns on information at the time of the investment decision. However, it might only apply to the 3-month maturity of the risk-free bond since there is no risk-free rate to compare for longer horizons.

  11. We use the overlapping returns for all the sampling intervals from 3 months to 10 years not just for 10 years for consistency of tests. In addition, the results with short horizons are very similar when we do not use overlapping data.

  12. When there exists serial dependence in the data, the form of the sampling distribution is generally unknown and depends on the unknown underlying distributions. To deal with this problem, the sampling distribution of the test statistics is approximated using a resampling scheme based on subsampling methods. See Politis and Romano (1999) and Linton et al. (2005).

  13. We also analysed the first spatial dominance test over various time periods with S&P500, DJIA and NASDAQ as risky assets, but cannot find any dominance in the first order. We do not report the specific results to save space.

  14. This finding seems in contrast to the traditional equity premium puzzle. Our findings are similar to the evidence of a reverse puzzle in Lim et al. (2006) who found that T-bill stochastically dominates equities. The approach and data sets used in our paper are different from the ones in the literature since we used high-frequency data whereas most of the empirical evidence for equity premium is based on the annual or quarterly data since the consumption data are only available for low frequency. Moreover, our spatial dominance tests are based on asset returns and not consumption data.

  15. We do not present the figures of the integrated spatial distribution functions of the cumulative returns for the NASDAQ as a risky asset since the patterns are very similar to those in Figs. 1 and 2.

  16. We would like to thank to an anonymous referee who suggested a sub-period analysis.

  17. In this case, the spatial dominance tests were employed for 3-month to 3- year investment horizons since we have small number of observations and we tried the subsample size in the range of [\(N^{0.3 } N ^{0.65}\)] so that the number of different subsample sizes is 30 and reported the median value of corresponding \(p\) values.

  18. Merton (1973) suggested that variation in expected returns over time could potentially introduce horizon effect. Barberis (2000) and Gil-Bazo (2006) addressed this issue based on the returns predictability.

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Acknowledgments

We would like to thank Joon Y. Park for many helpful discussions and suggestions. We also thank the editor and two anonymous referees for helping us to improve the quality of this paper substantially. This paper was supported by Samsung Research Fund, Sungkyunkwan University, 2013.

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  1. Department of Economics, Sungkyunkwan University, Jongro-gu, Seoul, 110-745, Korea

    Eunhee Lee, Chang Sik Kim & In-Moo Kim

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  1. Eunhee Lee
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  2. Chang Sik Kim
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Correspondence to Chang Sik Kim.

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Lee, E., Kim, C.S. & Kim, IM. Equity premium over different investment horizons. Empir Econ 48, 1169–1187 (2015). https://doi.org/10.1007/s00181-014-0812-z

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