Historical Simulation Systematically Underestimates the Expected Shortfall

Expected Shortfall (ES) is a risk measure that is acquiring an increasingly relevant role in financial risk management. In contrast to Value-at-Risk (VaR), ES considers the severity of the potential losses and reflects the benefits of diversification. ES is often calculated using Historical Simulation (HS), i.e., using observed data without further processing into the formula for its calculation. This has advantages like being parameter-free and has been favored by some regulators. However, the usage of HS for calculating ES presents a potentially serious drawback: It strongly depends on the size of the sample of historical data, being typically reasonable sizes similar to the number of trading days in one year. Moreover, this relationship leads to systematic underestimation: the lower the sample size, the lower the ES tends to be. In this letter, we present examples of this phenomenon for representative stocks and bonds, illustrating how the values of the ES and their averages are affected by the number of chosen data points. In addition, we present a method to mitigate the errors in the ES due to a low sample size, which is suitable for both liquid and illiquid financial products. Our analysis is expected to provide financial practitioners with useful insights about the errors made using Historical Simulation in the calculation of the Expected Shortfall. This, together with the method that we propose to reduce the errors due to finite sample size, is expected to help avoid miscalculations of the actual risk of portfolios.