Jump persistence and temporal aggregation of tail risk

Major events can have a lasting impact on the financial markets and affect the temporal aggregation of tail risk. We capture the dynamics of jump intensity using a generalized autoregressive conditional heteroskedasticity with autoregressive jump intensity (GARCH-ARJI) model, derive analytical formulas for the first four moments of cumulative returns, and utilize them to calculate VaR based on the Johnson distribution method. Our numerical experiments reveal that skewness decreases sharply while excess kurtosis rises in the short term, particularly when initial jump intensity is high. In the long term, the time diversification effect causes skewness and excess kurtosis to converge slowly to zero. Our out-of-sample backtesting analysis on S&P 500, FTSE 100, and DAX 30 total return indexes shows that it is important to incorporate the time-varying jump intensity when forecasting tail risk.