To efficiently assess the performance of investing in stocks rather than in a bank account for the long run, stochastic interest rate modelling is advocated. We introduce a correlated stochastic interest rate model that addresses this problem. We derive analytic formulas for general spectral risk measures in our setting, and apply our results to Value at Risk, Expected Shortfall and GlueVaR. We characterize the short- and long-term behaviour of these risk measures. We fit our model to financial markets, perform an empirical study and evaluate risk numbers for realistic scenarios in the future. Our results reveal sizeable sensitivities on parameter estimation, but we may conclude that holding stocks for less than a few decades bears significant risk.
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.
We demonstrate that margin requirements of central counterparties show a significantly different behavior when calculated with a portfoliowise treatment instead of taking the weighted sum of the margin requirements of the components without accounting for their correlation structures. This is shown via simulating trajectories of a joint stochastic volatility–stochastic correlation model. Results indicate that an unnecessarily large overmargin requirement is set by regulators when the applied risk measure is not calculated via a portfoliowise treatment. Finally, accounting for the correlation structure of the assets during the margining process would not lead to an overly prudent method, nor would it cause greater procyclicality.
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