Spectral risk measure of holding stocks in the long run
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Spectral risk measure of holding stocks in the long run Zsolt Bihary1 · Péter Csóka1,2
· Dávid Zoltán Szabó3
© The Author(s) 2020
Abstract We investigate how the spectral risk measure associated with holding stocks rather than a riskfree deposit, depends on the holding period. Previous papers have shown that within a limited class of spectral risk measures, and when the stock price follows specific processes, spectral risk becomes negative at long periods. We generalize this result for arbitrary exponential Lévy processes. We also prove the same behavior for all spectral risk measures (including the important special case of Expected Shortfall) when the stock price grows realistically fast and when it follows a geometric Brownian motion or a finite moment log stable process. This result would suggest that holding stocks for long periods has a vanishing downside risk. However, using realistic models, we find numerically that spectral risk initially increases for a significant amount of time and reaches zero level only after several decades. Therefore, we conclude that holding stocks has spectral risk for all practically relevant periods. Keywords Coherent risk measures · Downside risk · Lévy processes · Finite moment log stable model · Time diversification JEL Classification G11
We would like to thank Peter Farkas, Gabor Kondor, Adam Zawadowski, and participants of the 8th Annual Financial Market Liquidity Conference for helpful comments. Péter Csóka was supported by the ÚNKP-17-4-III New National Excellence Program of the Ministry of Human Capacities and also thanks funding from National Research, Development and Innovation Office— NKFIH, K-120035.
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Péter Csóka [email protected] Zsolt Bihary [email protected] Dávid Zoltán Szabó [email protected]
1
Department of Finance, Corvinus University of Budapest, Budapest, Hungary
2
“Momentum” Game Theory Research Group, Centre for Economic and Regional Studies, Budapest, Hungary
3
Independent Researcher, Budapest, Hungary
123
Annals of Operations Research
1 Introduction There has been tremendous interest both from practitioners and researchers on the question of how risky it is to hold stocks rather than a risk-free deposit in the long run. The existence of time diversification, meaning that the risk of holding stocks decreases with the time horizon, has profound effects on all long-term investments such as pension savings and target date funds. Industry practitioners hold the common wisdom that the longer the intended holding period for a portfolio, the more it should contain risky, rather than risk-free investments. The academic literature on the topic is more controversial (Bennyhoff 2009). Siegel (1998) showed that for investment horizons of at least 15 years, the risk of holding stocks (measured by realized variance) is lower than the risk of holding bonds or Treasury bills, and the risk of holding stocks is decreasing over time. However, Pástor and Stambaugh (2012) argue that stocks are more volatile in the long run from the per
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