Forecasting the distribution of long-horizon returns with time-varying volatility

The study of long-horizon returns has received a great deal of attention in recent years (see, for example, Boudoukh, Richardson, and Whitelaw (2008), Neuberger (2012) and Lee (2013), Fama and French (2018)). While most of the discussions are concerned with some practical issues in investment, few have touched the important aspect on risk management. The approach adopted in this article is to predict the future distribution of the returns of a fixed long-horizon by which the risk measures of interest that come in the form of a distributional functional such as the value at risk (VaR) and the conditional tail expectation (CTE) can be easily derived. The characteristic feature of our approach which requires no specification of the volatility dynamics nor parametric assumptions of the shock distribution extends the work by Ho et al. (2016) and Ho ( 2017) to a more general volatility dynamics that includes both the widely-used SV model and the GARCH model (Bollerslev, 1986) as special cases.