Spectral risk measure of holding stocks in the long run

Abstract

We investigate how the spectral risk measure associated with holding stocks rather than a risk-free 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.

New Classes of Distortion Risk Measures and Their Estimation

“…The function ψ represents the user's risk attitude; see Acerbi (2002) and Dowd et al (2008). For the applications of distortion risk measures, see Sereda et al (2010) and Bihary et al (2020). Wang (1995Wang ( , 2000 proposed two classes of distortion operators or transformation for pricing financial and insurance risk: power distortion or the proportional hazards transform defined by g(w) = w a , 0 ≤ a ≤ 1, and the Wang transform defined by g(w) = Φ Φ −1 (w) + λ , where Φ(•) is the standard normal cdf and λ is a scalar.…”