Simulation and Optimization in Finance by Dessislava Pachamanova & Frank J. Fabozzi

Author:Dessislava Pachamanova & Frank J. Fabozzi
Language: eng
Format: epub
ISBN: 9780470882122
Publisher: Wiley
Published: 2012-10-30T16:00:00+00:00

These complications actually make simulation a good way to approach the modeling of credit-risky positions. The procedure for evaluating the VaR or CVaR of a credit risky bond is similar to the procedure for a nondefaultable bond described earlier in this section. However, to evaluate future cash flows, we simulate the realizations of binary variables that tell us whether the cash flows are actually received. (Once a bond has defaulted, we make sure that no future cash flows are received.)

Estimating the VaR or the CVaR of a portfolio of credit risky bonds (as opposed to a position in a single bond) is more complicated because we need to simulate the dependencies between the defaults of the different bonds. Such dependencies have been often modeled through different types of copula functions.7 Through transformations of the original variables, copula functions allow for modeling dependence between the random variables in a tractable way. The Gaussian copula is one such example.8 Gaussian copula dependency models in particular, however, have received bad press after their failure to account for the high tail losses observed during the financial crisis of the fall of 2008. In addition, calibrating (i.e., finding the input parameters for) such models is very difficult in practice, because historically there have been few observations of multiple defaults happening simultaneously.

Let us now consider an example that takes advantage of both simulation and optimization to manage the credit risk of a bond portfolio. Anderson, Mausser, Rosen, and Uryasev (2001) describe an approach to optimizing bond portfolios so as to minimize the losses stemming from credit risk exposure. They consider a portfolio of 197 bonds from 29 different countries with a market value of $8.8 billion and duration of approximately 5. The goal is to rebalance the portfolio in order to minimize credit risk over a one-year horizon, that is, to minimize losses resulting from default and from a decline in market value because of downgrades in credit ratings.

To address the problem, they first generate scenarios for the future values of the losses due to credit migration (i.e., changes in the credit rating of obligors). This is done by simulation. They simulate 20,000 scenarios of joint credit states for bond issuers and the related losses. Based on these scenarios, they can evaluate the total losses on all bonds in the portfolio, equal to the sum of the differences between the value of each bond without credit migration and the value of that bond with credit migration. Then, they solve a CVaR minimization problem using the optimization formulation that we presented in section 8.3.3 of Chapter 8.

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