Lambda Value at Risk and Regulatory Capital: A Dynamic Approach to Tail Risk

Abstract

This paper presents the first methodological proposal of estimation of the ΛVaR. Our approach is dynamic and calibrated to market extreme scenarios, incorporating the need of regulators and financial institutions in more sensitive risk measures. We also propose a simple backtesting methodology by extending the VaR hypothesis-testing framework. Hence, we test our ΛVaR proposals under extreme downward scenarios of the financial crisis and different assumptions on the profit and loss distribution. The findings show that our ΛVaR estimations are able to capture the tail risk and react to market fluctuations significantly faster than the VaR and expected shortfall. The backtesting exercise displays a higher level of accuracy for our ΛVaR estimations.

A Discussion on Recent Risk Measures with Application to Credit Risk: Calculating Risk Contributions and Identifying Risk Concentrations

“…However, a careful risk manager would prefer to choose a probability/loss function which accepts extreme losses only with very low probabilities. Hitaj and Peri (2015) and Hitaj et al (2017) deliver the first empirical studies for the LVaR in the context of market risk. It is straightforward to prove that LVaR in general is a monotone and law invariant risk measure.…”

Backtesting lambda value at risk

“…In this way, we are able to assess the correctness of ΛV aR more precisely than Hitaj, Mateus, and Peri (2015). Notice that a rejection of H K 0 in (7) implies a rejection of H 0 in (9).…”

“…During the analysis of the results, Test 1 and Test 2 have pointed out an issue of estimation in the ΛV aR models proposed by Hitaj, Mateus, and Peri (2015). In particular, the authors do not discuss in details the choice of the Λ minimum, min x Λ(x), that seems to be set equal to 0.1% after empirical experimentations.…”

“…When we have run for the first time Test 1 and 2 using the choice of Hitaj, Mateus, and Peri (2015), min x Λ(x) = 0.1%, we have noticed that the increasing ΛV aR models ΛV aR 1% (decr) (VaR 5%) 100 % 75 % 8 % 100 % 92 % 67 % 100 % 83 % 17 % 100 % 100 % 75 % (VaR 1%) 92 % 83 % 25 % 100 % 100 % 67 % 100 % 83 % 42 % 100 % 100 % 83 % 96 % 79 % 17 % 100 % 96 % 67 % 100 % 83 % 29 % 100 % 100 % 79 % Thus, we have studied how the probability of infraction λ t evolves in the different ΛV aR models and we have observed that in most of the cases it obtains the minimal value. This happens especially during crisis periods, when the cumulative distribution function of the assets shifts on the left and intersects the Λ function at the minimum level.…”

“…We propose a third test that is inspired to the approach used by Acerbi and Szekely (2014) Finally, we conduct an empirical analysis where we experiment and compare the results of our backtesting proposals for ΛV aR, computed using the same dynamic benchmark approach proposed by Hitaj, Mateus, and Peri (2015). The backtesting exercise has been performed along six different time windows throughout all the global financial crisis (2006)(2007)(2008)(2009)(2010)(2011).…”