Non-Parametric Inference on Risk Measures for Integrated Returns

When evaluating the market risk of long-horizon equity returns, it is always difficult to provide a statistically sound solution due to the limitation of the sample size. To solve the problem for the value-at-risk (VaR) and the conditional tail expectation (CTE), Ho et al. (2016, 2018) introduce a general multivariate stochastic volatility return model from which asymptotic formulas for the VaR and the CTE are derived for integrated returns with the length of integration increasing to infinity. Based on the formulas, simple non-parametric estimators for the two popular risk measures of the long-horizon returns are constructed. The estimates are easy to implement and shown to be consistent and asymptotically normal. In this chapter, we further address the issue of testing the equality of the CTEs of integrated returns. Extensive finite-sample analysis and real data analysis are conducted to demonstrate the efficiency of the test statistics we propose.