We show that pro-cyclicality is inherent in risk measure estimates based on historical data. Taking the example of VaR, we show that the empirical VaR measure is mean-reverting over a 1-year horizon when the portfolio is held fixed. It means that a capital requirement rule based on historical measurements of VaR tends in calm times to understate future required capital and tends in volatile times to overstate it. To quantify this pro-cyclicality, we develop a simple and efficient methodology, which we apply to major equity market indices. We make the interesting point that the pro-cyclicality property holds true even in a world with constant volatility, though the empirical magnitude of the mean-reversion is greater than what would be observed in that special case.
BRÄUTIGAM, M., DACOROGNA, M. et KRATZ, M. (2023). Pro-cyclicality beyond business cycle. Mathematical Finance, 33(2), pp. 308-341.