Wanmo Kang

Fast Simulation of Multifactor Portfolio Credit Risk

This paper develops rare-event simulation methods for the estimation of portfolio credit risk---the risk of losses to a portfolio resulting from defaults of assets in the portfolio. Portfolio credit risk is measured through probabilities of large losses, which are typically due to defaults of many obligors (sources of credit risk) to which a portfolio is exposed. An essential element of a portfolio view of credit risk is a model of dependence between these sources of credit risk: large losses occur rarely and are most likely to result from systematic risk factors that affect multiple obligors. As a consequence, estimating portfolio credit risk poses a challenge both because of the rare-event property of large losses and the dependence between defaults. To address this problem, we develop an importance sampling technique within the widely used Gaussian copula model of dependence. We focus on difficulties arising in multifactor models---that is, models in which multiple factors may be common to multiple obligors, resulting in complex dependence between defaults. Our importance sampling procedure shifts the mean of the common factor to increase the frequency of large losses. In multifactor models, different combinations of factor outcomes and defaults can produce large losses, so our method combines multiple importance sampling distributions, each associated with a shift in the mean of common factors. We characterize “optimal” mean shifts. Finding these points is both a combinatorial problem and a convex optimization problem, so we address computational aspects of this step as well. We establish asymptotic optimality results for our method, showing that---unlike standard simulation---it remains efficient as the event of interest becomes rarer.

Large Deviations in Multifactor Portfolio Credit Risk

The measurement of portfolio credit risk focuses on rare but significant large-loss events. This paper investigates rare event asymptotics for the loss distribution in the widely used Gaussian copula model of portfolio credit risk. We establish logarithmic limits for the tail of the loss distribution in two limiting regimes. The first limit examines the tail of the loss distribution at increasingly high loss thresholds; the second limiting regime is based on letting the individual loss probabilities decrease toward zero. Both limits are also based on letting the size of the portfolio increase. Our analysis reveals a qualitative distinction between the two cases: in the rare-default regime, the tail of the loss distribution decreases exponentially, but in the large-threshold regime the decay is consistent with a power law. This indicates that the dependence between defaults imposed by the Gaussian copula is qualitatively different for portfolios of high-quality and lower-quality credits.

Stress scenario selection by empirical likelihood

This paper develops a method for selecting and analyzing stress scenarios for financial risk assessment, with particular emphasis on identifying sensible combinations of stresses to multiple factors. We begin by focusing on reverse stress testing--finding the most likely scenarios leading to losses exceeding a given threshold. We approach this problem using a nonparametric empirical likelihood estimator (in the sense of Owen (2001)) of the conditional mean of the underlying market factors given large losses. We then scale confidence regions for the conditional mean by a coefficient that depends on the tails of the market factors to estimate the most likely loss scenarios. We provide rigorous justification for the confidence regions and the scaling procedure in three models of the joint distribution of the market factors and portfolio loss with qualitatively different tail behavior: multivariate normal (light-tailed), multivariate Laplace (exponentially tailed), and multivariate-t (regularly varying). The key to this analysis (and the differences across the three cases) lies in the asymptotics of the conditional variances and covariances in extremes. These results also lead to asymptotics for marginal expected shortfall and the corresponding variance, conditional on extreme losses; we combine these results with empirical likelihood significance tests of systemic risk rankings based on marginal expected shortfall. For the problem of selecting macro stress scenarios, we apply our results to estimate the most likely outcome for other variables given a stress in one variable, and thus to gauge the plausibility of particular combinations of stresses to financial and economic factors. Finally, we develop a scenario sampling method, suggested by the empirical likelihood contours, for exploring regions of large losses in generating stress scenarios.

Fast simulation for multifactor portfolio credit risk in the t -copula model

We present an importance sampling procedure for the estimation of multifactor portfolio credit risk for the t -copula model, i.e, the case where the risk factors have the multivariate t distribution. We use a version of the multivariate t that can be expressed as a ratio of a multivariate normal and a scaled chi-square random variable. The procedure consists of two steps. First, using the large deviations result for the Gaussian model in Glasserman, Kang, and Shahabuddin (2005a), we devise and apply a change of measure to the chi-square random variable. Then, conditional on the chi-square random variable, we apply the importance sampling procedure developed for the Gaussian copula model in Glasserman, Kang, Shahabuddin (2005b). We support our importance sampling procedure by numerical examples.

Design of Risk Weights

Banking regulations set minimum levels of capital for banks. These requirements are generally formulated through a ratio of capital to risk-weighted assets. A risk-weighting scheme assigns a weight to each asset or category of assets and effectively functions as a linear constraint on a banks portfolio choice; it also changes the incentives for banks to hold various kinds of assets. In this paper, we investigate the design of risk weights to align regulatory and private objectives in a simple mean-variance framework for portfolio selection. By setting risk weights proportional to profitability rather than risk, the regulator can induce a bank to reduce its overall level of risk without distorting its asset mix. Because the regulator is unlikely to know the true profitability of assets, we introduce an adaptive formulation in which the regulator sets weights by observing a banks portfolio. The adaptive scheme converges to the same combination of weights and portfolio choice that would hold if the regulator knew the asset profitability. We also investigate other objectives, including steering banks to a target mix of assets, adding robustness, mitigating procyclicality, and reducing system-wide risk in a setting with multiple heterogeneous banks.

OR Forum-Design of Risk Weights

Banking regulations set minimum levels of capital for banks. These requirements are generally formulated through a ratio of capital to risk-weighted assets. A risk-weighting scheme assigns a weight to each asset or category of assets and effectively functions as a linear constraint on a banks portfolio choice; it also changes the incentives for banks to hold various kinds of assets. In this paper, we investigate the design of risk weights to align regulatory and private objectives in a simple mean-variance framework for portfolio selection. By setting risk weights proportional to profitability rather than risk, the regulator can induce a bank to reduce its overall level of risk without distorting its asset mix. Because the regulator is unlikely to know the true profitability of assets, we introduce an adaptive formulation in which the regulator sets weights by observing a banks portfolio. The adaptive scheme converges to the same combination of weights and portfolio choice that would hold if the regulator knew the asset profitability. We also investigate other objectives, including steering banks to a target mix of assets, adding robustness, mitigating procyclicality, and reducing system-wide risk in a setting with multiple heterogeneous banks.

Exact Simulation of the Wishart Multidimensional Stochastic Volatility Model

In this article, we propose an exact simulation method of the Wishart multidimensional stochastic volatility (WMSV) model, which was recently introduced by Da Fonseca et al. \cite{DGT08}. Our method is based onanalysis of the conditional characteristic function of the log-price given volatility level. In particular, we found an explicit expression for the conditional characteristic function for the Heston model. We perform numerical experiments to demonstrate the performance and accuracy of our method. As a result of numerical experiments, it is shown that our new method is much faster and reliable than Euler discretization method.