Investment Theory and Risk Management, + Website

Estimating the Hazards of Downside Risks

Monitoring sample movements in the parameters of extreme value distributions are uninformative in that we do not know why a parameter, say, location has shifted further into the tail of the distribution of returns. For this reason, we next devise a hazard function for downside risk conditional on a set of three factors we believe to influence this risk. The hazard rate is the probability of instantaneous failure—the likelihood that a return will exceed the threshold, right now, conditional on the risk state embodied in the values of the risk factors (see Kiefer 1988). We assume the hazard follows an exponential distribution so that the probability of failure is not duration dependent, that is, it is not a function of the length of time elapsed since the last failure; rather it is conditioned only on the level of market liquidity, volatility, and general default risk. There is no reason to restrict the analysis to these three risk factors. We did so for purely illustrative purposes but otherwise encourage more careful study of factors in general. The objective is to attribute the likelihood of extreme returns to changes in factors that we believe drive risks and that we can also observe. Thus, with a forward view on the underlying risk factors, we can form forward views on the probability of failures. A change in this particular risk barometer would, for example, lead portfolio managers to adjust the risk levels targeted in their portfolios ...