Measuring Long-Term Tail Risk: Evaluating the Performance of the Square-Root-of-Time Rule

Abstract

This paper focuses on risk over long time horizons and within extreme percentiles, which have attracted considerable recent interest in numerous subfields of finance. Value at risk (VaR) aggregates several components of asset risk into a single quantitative measurement and is commonly used in tail risk management. Due to realistic data limits, many practitioners might use the square-root-of-time rule (SRTR) to compute long-term VaR. However, serial dependence and heavy-tailedness can bias the SRTR. This paper addresses two deficiencies of the study by Wang et al. (2011), who propose the modified-SRTR (MSRTR) to partially correct the serial dependence and use subsampling estimation as the benchmark to verify the performance of MSRTR. First, we investigate the validity of the subsampling approach through numerical simulations. Second, to reduce the heavy-tailedness bias, we propose a new MSRTR approach (MSRTR∗) in light of the Central Limit Theorem (CLT). In the empirical study, 28 country-level exchange-traded funds (ETFs) from 2010 to 2015 are considered to estimate the 30-day VaR. After modifying both serial dependence and heavy-tailedness, our approach reduces the bias from 26.46% to 5.97%, on average, compared to the SRTR. We also provide a backtesting analysis to verify the robustness of the MSRTR∗. This new approach should be considered when estimating long-term VaR using short-term VaR.