A practical approach to semideviation and its time scaling in a jump-diffusion process

One of the most popular risk-adjusted fund return measures in the asset management industry is the Sortino ratio. It is an alternative to the Sharpe ratio that differentiates harmful volatility from general volatility by taking into account the standard deviation of negative asset returns, a quantity called semideviation. Indeed, the semideviation is generally preferred to the standard deviation when the distribution of the returns is skewed. A common method to annualize it is to use the square-root-of-time rule, where an estimated quantile of a return distribution is scaled to a lower frequency by the square root of the time horizon. However, this relation does not generally hold for this risk measure and often gives a terrible estimation of it. The aim of this paper is to provide a practical approach to semideviation by explaining how it should be computed. We propose and justify the use of a new model, which delivers a more accurate estimation of the downside risk. It is a generalization of the Ball-Torous approximation of a jump-diffusion process, which can be applied when the volatility is constant or stochastic. In the latter case, we use Markov Chain Monte Carlo (MCMC) methods to fit our stochastic volatility model. We also derive an exact formula for the semideviation when the volatility is kept constant, explaining how it should be scaled when considering a lower frequency. For the tests, we apply our methodology to a highly skewed set of returns based on the Barclays US High Yield Index, where we compare different time scalings for the semideviation. Our work shows that the square-root-of-time rule provides a poor approximation of the semideviation, and that the simplification brought by Ball and Torous should be replaced by our new methodology, as it gives much better results.