Abstract: When evaluating the tail risk of stock portfolio returns, providing statistically sound solutions for long return horizons is important, but difficult. Furthermore, there are drawbacks to using traditional parametric methods that rely on strong model assumptions or simulations. This study investigates the problem by focusing on an important risk measure, the conditional tail expectation (CTE), under a general multivariate stochastic volatility model. To overcome the estimation difficulties caused by the long period, we derive an asymptotic formula to approximate the CTE. Based on this formula, we propose a simple nonparametric estimate of the unconditional CTE, and show that it is both consistent and asymptotically normal. Next, we forecast the CTE using a modified form of the nonparametric estimator. With the help of the asymptotic formula, we evaluate the accuracy of the CTE predictor by treating it as an interval forecast for furure returns. Simulation studies demonstrate the applicability of our approach. Lastly, we apply the proposed estimation and predictor to daily S&P 500 index returns.

Key words and phrases: Asymptotic normality, conditional tail expectation, integrated process, interval forecast, long-horizon returns, stochastic volatility model.