Mean-Variance Mixture Model for Calibrating Loss Given Default (LGD) of a Credit Portfolio

Motivated by the work of Kupiec (2008), we study the sensitivity of Value-at-Risk (VaR) and Tail-Value-at-Risk (TVaR) of credit portfolio of defaultable obligors, to the tail fatness of the loss given default latent variable distribution. We consider a static structural model where the ith obligor defaulting and loss given default (LGD) is governed by latent variables Ai and Bi respectively. We propose the use of the Normal-Variance mixture model (Jamieson (2019)) to model the Bi distribution. These choices make the Bi model able to capture shocks not captured by the Bi model by Kupiec (2008) (we denote Kupiec (2008) Bi model by RM). All the Bi models we considered have the expectation of 0 and are symmetric at 0. We determine the parameters for our proposed Bi models by assuming unity variance for all Bi models. We also assume that all the Bi models have the same kurtosis and are greater than 3. These two assumptions make it possible for us to compare our results to the reference model RM. In this presentation, we show how to determine the threshold values for a step function used in finding LGD values for obligors in the credit portfolio and find the portfolio loss. We also show how the VaR and TVaR change with higher kurtosis for the Bi distribution. Using the law of large numbers we present the limiting distribution of the conditional loss rate.