The Two-Parameter Long-Horizon Value-at-Risk

Value-at-Risk (VaR) has become a standard measure for risk management and regulation. In the case of a two-parameter distribution, a common method among practitioners is first to calculate the daily VaR and then to apply it to a longer investment horizon by using the Square Root Rule (SRR). We show that the SRR is theoretically incorrect and propose a correct measure. The error from employing the SRR is positive for short horizons, inducing an overestimation of the true VaR, and negative for longer horizons, inducing underestimation of the true VaR. This error is relatively small for conservative portfolios and for short horizons. However, for risky portfolios and for long horizons – where accurate VaR is most important – the underestimation error is both substantial and systematic