Abstracts

In this paper, we describe, analyze and evaluate an algorithm for estimating portfolio loss probabilities using Monte Carlo simulation. This investigation is motivated by the incremental risk charge (IRC) introduced by the Basel Committee on Banking Supervision, in which one needs to calculate $99.9\%$ confidence Value-at Risk (VaR) over a 1-year horizon in the presence of constraints on credit default and migration risks. The IRC assumes that a portfolio is managed dynamically to a target level of risk, with constraints on discrete rebalancing intervals (e.g., monthly or quarterly) as a rough measure of potential illiquidity in underlying assets. By taking these factors into consideration, we propose an efficient Monte Carlo simulation algorithm to estimate such loss probabilities which is essential to calculating VaR, a quantile of the loss distribution. The method employs importance sampling for Markov chains and a spherical Monte Carlo method for further variance reduction.
To have efficient importance sampling for Markov chains, we develop a general account for finding the optimal tilting probability measure. Our approach consists of constructing an exponential tilting family of probability distributions around the original probability and then minimizing the variance of the importance sampling estimator within this family.
By using the device of conjugate probability measure and a solution of a Poisson equation, we obtain a simple and explicit expression of the optimal alternative distribution. Simulation results indicate that an appropriate combination of importance sampling for Markov chains and spherical Monte Carlo can result in large variance reductions when estimating the probability of portfolio losses.