From Log-Optimal Portfolio Theory to Risk Measures: Logarithmic Expected Shortfall

Historically, risk measures have been used for single-period investments, and this has prevented their application by investors willing to reinvest their wealth for more than one period. Due to its intuitive definition and coherent properties, one of the most widely adopted risk measures is expected shortfall (ES). This is defined as a tail expectation; it therefore gives the same weight to all of the losses involved in an average calculation. Here, we propose a modification of ES that does not treat all losses equally. We do this in order to represent the worries surrounding big drops that are typical of multiperiod investors. Our version of ES exploits Kelly’s logarithmic utility function to penalize heavier losses. This enables us to avoid investments that may expose the investor to bad circumstances from which it would be too hard to recover. An analysis of our results using synthetic and real data is offered in order to certify the viability of the proposed method.