Spectral risk measure of holding stocks in the long run

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 risk. However, using realistic models, we find numerically that the risk increases for a few decades and reaches zero at around 100 years. Therefore, we conclude that holding stocks is risky for all practically relevant periods.