Expected shortfall and tail conditional expectation with the Pearson type IV distribution

Abstract

The Basel Committee of Banking Supervision has used value-at-risk as a measure of market risk in the trading book for two decades in several accords. After the global financial crisis of 2008-2009 value-at-risk was criticised on the grounds that as a measure of risk is not sub-additive, in the sense that it behaves very erratically when banks or regulators try to aggregate compartmentalised risk across all branches of a large diverse bank. Expected shortfall emerged as a natural alternative of value-at-risk fulfilling all four axioms of a coherent risk measure and belongs to the category of spectral risk measures which are not elicitable unless they reduce to minus the expected value. Another critical issue is that the Basel committee indicates that risk managers should conduct the risk measurement tests based on a dataset which is very small for high confidence levels calculations. In this work we calculate and apply the theoretical tail conditional expectation of the standardised Pearson type IV distribution. This is a coherent measure of risk and assuming it describes appropriately the data generation process it can provide the risk manager with reliable results.

Value-at-risk and related measures for the Bitcoin

“…In cases of few data points, the VaR and ES methodology for high confidence levels might not be applicable, and one solution is to use the theoretical ES or theoretical tail conditional expectation (TTCE) emerging from the cumulative distribution function of the probability density of equation (3). This was shown to be (Stavroyiannis, 2016): …”