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Do not put all your eggs in one (time) basket

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Abstract

We show that it is easy to enhance an investment’s Sharpe ratio at no additional cost by purchasing the risky asset in a few installments instead of all at once. A similar argument holds for selling the risky asset. In the simple case of a geometrical Brownian motion (GBM), we prove analytically that such a strategy decreases the variance of returns without changing the expected returns, relative to the one-shot strategy. We demonstrate the benefits of this strategy by bootstrapping daily S&P-500 prices for the 1985–2013 period and using Monte Carlo simulations of GBM and jump-diffusion processes. The results are statistically significant. We show that the strategy is more effective for short investment horizons and that performance improves with the number of installments used.

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Notes

  1. Among the supporters are Malkiel and Ellis (2010), Rubin and Spaht (2011) and Trainor (2005). Among the opponents are Shtekhman et al. (2012), Leggio and Lien (2001, 2003a, b), Runkle (1997), Constantinides (1979), Knight and Mandell (1993), Rozeff (1994), Thorley (1994), Milevsky and Posner (2003) and Hayley (2012).

  2. In a market where positive surprises are common, a lump-sum investment might be beneficial. In a market where negative jumps are likely (this is the common view regarding large indices; hence their implied volatility smirk), it might seem somewhat more prudent to spread the investment so as to gain from the negative jump (avoiding negative returns and capturing potential low price purchases, on average).

  3. Dodson (1989) briefly describes various diversification modes. In her discussion of “diversification across time,” she marginally describes a variant of DCA that is an example of this paper’s TD strategy (using one-fifth of the investment amount over five consecutive months to purchase a designated asset).

  4. Although we are aware that there are other methods of proving TD’s superiority over OS, for example the approaches taken by Merton (1972), Samuelson (1967), and others, we prefer developing explicit expectations and variance for the two strategies. The explicit quantitative expressions of this paper allow calculating specific improvement ratios that can be compared to empirical results of real market data and Monte Carlo simulations.

  5. For simplicity and clarity, we ignore the case of a stochastic return on the “risk-free” asset.

  6. See, for example, Glasserman (2004).

  7. \(x^{\prime }\) denotes the transpose of vector x.

  8. Again ignoring the common factor of the risk-free asset returns, which is identical for all installments and for the one-shot strategy (under our assumption of a fixed rate r for the risk-free asset).

  9. Since \(\frac{2}{N^{2}}\sum \nolimits _{i=1}^{N-1} {\left( {N-i} \right) } =\frac{N-1}{N}\) and \(e^{\sigma ^{2}\left( {T-i\cdot \Delta t} \right) }<e^{\sigma ^{2}\textit{T}}\) for \(\Delta {t} > 0\) and \(i\ge 1\) (which occurs when N \(\ge \) 2). N \(=\) 1 is the trivial one-shot case.

  10. This Sharpe ratio improvement is valid on average only, since it is possible that certain realizations would exhibit a reverse relation.

  11. Data source is Yahoo! Finance.

  12. We also calculate the downside measures—semi-deviations and the Sortino ratio—which yield results similar to the standard deviation and Sharpe ratio and thus are omitted from this paper to save space.

  13. The parameters are the estimation results of Chacko and Viceira (2003), found in their Table 2. For the JD process simulations, we use code from the Jondeau and Rockinger website (https://doi.org/www.hec.unil.ch/matlabcodes/) modified to meet our needs (SimExpJump.m and SimJumpDiff.m).

  14. A more rigorous test (Jung-Box Q-test for residual autocorrelation) rejects the null hypothesis of a white noise process; nevertheless, this characteristic of JD (and GBM) does not affect the tests on real market data.

  15. For details on the Sortino ratio and semi-deviation, see, for example, Estrada (2006). We use a threshold of 3 % for the Sortino ratio.

  16. We ignore here cases of day trading or investment foresight (information) that may favor an OS strategy.

  17. The term volatility here refers to the random variability of the asset price process, and not to its more common and more narrow meaning of the \(\sigma \) parameter in the GBM.

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Acknowledgments

I thank Simon Benninga, Koresh Galil, Gady Jacoby, and an anonymous reviewer for their helpful comments. The idea for this paper was triggered by a recommendation of Rafi Gamish (an investment fund manager and private investor) to execute an investment decision in a few installments aiming to “test the market” instead of a one-shot purchase of the entire amount. The responsibility for the paper is, of course, mine.

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  1. Department of Business Administration, Guilford Glazer Faculty of Business and Management, Ben-Gurion University of the Negev, Beer Sheva, Israel

    Zvika Afik

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  1. Zvika Afik
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Afik, Z. Do not put all your eggs in one (time) basket. Financ Mark Portf Manag 29, 251–269 (2015). https://doi.org/10.1007/s11408-015-0252-6

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