Importance sampling for integrated market and credit portfolio models

Abstract: Standard credit portfolio models do not model market risk factors, such as risk-free interest rates or credit spreads, as stochastic variables. Various studies have documented that a severe underestimation of economic capital can be the consequence. However, integrating market risk factors into credit portfolio models increases the computational burden of computing credit portfolio risk measures. In this paper, the application of various importance sampling techniques to an integrated market and credit portfolio model are presented and the effectiveness of these approaches is tested by numerical experiments. The main result is that importance sampling can reduce the standard error of the percentile estimators, but it is rather difficult to make statements about when the IS approach is especially effective. Besides, the combination of importance sampling techniques originally developed for pure market risk portfolio models with techniques originally developed for pure default mode credit risk portfolio models is less effective than simpler two step-IS approaches

Credit cycles and macro fundamentals

We study the relation between the credit cycle and macro economic fundamentals in an intensity based framework. Using rating transition and default data of U.S. corporates from Standard and Poor’s over the period 1980–2005 we directly estimate the credit cycle from the micro rating data. We relate this cycle to the business cycle, bank lending conditions, and financial market variables. In line with earlier studies, these variables appear to explain part of the credit cycle. As our main contribution, we test for the correct dynamic specification of these models. In all cases, the hypothesis of correct dynamic specification is strongly rejected. Moreover, accounting for dynamic mis-specification, many of the variables thought to explain the credit cycle, turn out to be insignificant. The main exceptions are GDP growth, and to some extent stock returns and stock return volatilities. Their economic significance appears low, however. This raises the puzzle of what macro-economic fundamentals explain default and rating dynamics. JEL Classification: G11, G2

Monte Carlo Simulation in the Integrated Market and Credit Portfolio Model

Credit granting institutions deal with large portfolios of assets. These assets represent credit granted to obligors as well as investments in securities. A common size for such a portfolio lies from anywhere between 400 to 10,000 instruments. The essential goal of the credit institution is to minimize their losses due to default. By default we mean any event causing an asset to stop producing income. This can be the closure of a stock as well as the inability of an obligor to pay their debt, or even an obligor's decision to pay out all his debt. Minimizing the combined losses of a credit portfolio is not a deterministic problem with one clean solution. The large number of factors influencing each obligor, different market sectors, their interactions and trends, etc. are more commonly dealt with in terms of statistical measures. Such include the expectation of return and the volatility of each asset associated with a given time horizon. In this sense, we consider in the following the expected loss and risk associated with the assets in a credit portfolio over a given time horizon of (typically) 10 to 30 years. We use a Monte Carlo approach to simulate the loss of a portfolio in multiple scenarios, which leads to a distribution function for the expected loss of the portfolio over that time horizon. Second, we compare the results of the simulation to a Gaussian approximation obtained via the Lindeberg-Feller Theorem. Consistent with our expectations, the Gaussian approximation compares well with a Monte Carlo simulation in case of a portfolio of very risky assets. Using a model which produces a distribution of expected losses allows credit institutions to estimate their maximum expected loss with a certain confidence interval. This in turn helps in taking important decisions about whether to grant credit to an obligor, to exercise options or otherwise take advantage of sophisticated securities to minimize losses. Ultimately, this leads to the process of credit risk management

The use of portfolio credit risk models in Central Banks.

This report summarises the findings of the task force. It is organised as follows. Section 2 starts with a discussion of the relevance of credit risk for central banks. It is followed by a short introduction to credit risk models, parameters and systems in Section 3, focusing on models used by members of the task force. Section 4 presents the results of the simulation exercise undertaken by the task force. The lessons from these simulations as well as other conclusions are discussed in Section 5.

Credit Risk Monte Carlos Simulation Using Simplified Creditmetrics' Model: the joint use of importance sampling and descriptive sampling

Monte Carlo simulation is implemented in some of the main models for estimating portfolio credit risk, such as CreditMetrics, developed by Gupton, Finger and Bhatia (1997). As in any Monte Carlo application, credit risk simulation according to this model produces imprecise estimates. In order to improve precision, simulation sampling techniques other than traditional Simple Random Sampling become indispensable. Importance Sampling (IS) has already been successfully implemented by Glasserman and Li (2005) on a simplified version of CreditMetrics, in which only default risk is considered. This paper tries to improve even more the precision gains obtained by IS over the same simplified CreditMetrics' model. For this purpose, IS is here combined with Descriptive Sampling (DS), another simulation technique which has proved to be a powerful variance reduction procedure. IS combined with DS was successful in obtaining more precise results for credit risk estimates than its standard form.

Dynamic asset (and liability) management under market and credit risk

We introduce a modelling paradigm which integrates credit risk and market risk in describing the random dynamical behaviour of the underlying fixed income assets. We then consider an asset and liability management (ALM) problem and develop a mul- tistage stochastic programming model which focuses on optimum risk decisions. These models exploit the dynamical multiperiod structure of credit risk and provide insight into the corrective recourse decisions whereby issues such as the timing risk of default is appropriately taken into consideration. We also present a index tracking model in which risk is measured (and optimised) by the CVaR of the tracking portfolio in relation to the index. Both in- and out-of-sample (backtesting) experiments are undertaken to validate our approach. In this way we are able to demonstrate the feasibility and flexibility of the chosen framework

Credit Cycles and Macro Fundamentals

We study the relation between the credit cycle and macro economic fundamentals in an intensity based framework. Using rating transition and default data of U.S. corporates from Standard and Poor’s over the period 1980–2005 we directly estimate the credit cycle from the micro rating data. We relate this cycle to the business cycle, bank lending conditions, and financial market variables. In line with earlier studies, these variables appear to explain part of the credit cycle. As our main contribution, we test for the correct dynamic specification of these models. In all cases, the hypothesis of correct dynamic specification is strongly rejected. Moreover, accounting for dynamic mis-specification, many of the variables thought to explain the credit cycle, turn out to be insignificant. The main exceptions are GDP growth, and to some extent stock returns and stock return volatilities. Their economic significance appears low, however. This raises the puzzle of what macro-economic fundamentals explain default and rating dynamics.Credit Cycles, Business Cycles, Bank Lending Conditions, Unobserved Component Models, Intensity Models, Monte Carlo Likelihood

Analytic results and weighted Monte Carlo simulations for CDO pricing

We explore the possibilities of importance sampling in the Monte Carlo pricing of a structured credit derivative referred to as Collateralized Debt Obligation (CDO). Modeling a CDO contract is challenging, since it depends on a pool of (typically about 100) assets, Monte Carlo simulations are often the only feasible approach to pricing. Variance reduction techniques are therefore of great importance. This paper presents an exact analytic solution using Laplace-transform and MC importance sampling results for an easily tractable intensity-based model of the CDO, namely the compound Poissonian. Furthermore analytic formulae are derived for the reweighting efficiency. The computational gain is appealing, nevertheless, even in this basic scheme, a phase transition can be found, rendering some parameter regimes out of reach. A model-independent transform approach is also presented for CDO pricing.Comment: 12 pages, 9 figure

Reducing Asset Weights' Volatility by Importance Sampling in Stochastic Credit Portfolio Optimization

The objective of this paper is to study the effect of importance sampling (IS) techniques on stochastic credit portfolio optimization methods. I introduce a framework that leads to a reduction of volatility of resulting optimal portfolio asset weights. Performance of the method is documented in terms of implementation simplicity and accuracy. It is shown that the incorporated methods make solutions more precise given a limited computer performance by means of a reduced size of the initially necessary optimization model. For a presented example variance reduction of risk measures and asset weights by a factor of at least 350 was achieved. I finally outline how results can be mapped into business practice by utilizing readily available software such as RiskMetrics� CreditManager as basis for constructing a portfolio optimization model that is enhanced by means of IS. Dieser Beitrag soll die Auswirkung der Anwendung von Importance Sampling (IS) Techniken in der stochastischen Kreditportfoliooptimierung aufzeigen. Es wird ein Modellaufbau vorgestellt, der zu einer deutlichen Reduktion der Volatilität der Wertpapieranteilsgewichte führt. Durch eine Darstellung der verhältnismäßig einfachen Berücksichtigung der Importance Sampling Technik im Optimierungsverfahren sowie durch ein empirisches Beispiel wird die Leistungsfähigkeit der Methode dargelegt. In diesem Anwendungsbeispiel kann die Varianz der Schätzer sowohl für die Risikomaße als auch für die optimalen Anteilsgewichte um einen Faktor von mindestens 350 reduziert werden. Es wird somit gezeigt, dass die hier vorgestellte Methode durch eine Reduktion der Größe des ursprünglich notwendigen Optimierungs-problems die Genauigkeit von optimalen Lösungen erhöht, wenn nur eine begrenzte Rechnerleistung zur Verfügung steht. Abschließend wird dargelegt, wie die Lösungsansätze in der Praxis durch eine Ankopplung an existierende Softwarelösungen im Bankbetrieb umgesetzt werden können. Hierzu wird ein Vorgehen skizziert, das auf den Ergebnissen des Programms CreditManager von RiskMetrics ein Portfoliooptimierungsmodell aufbaut. Dieses wird um eine Importance Sampling Technik erweitert.Kreditrisiko ; Stochastische Optimierung; Varianzreduktion ; CVaR; CVaR ; credit risk ; stochastic portfolio optimization ; importance sampling ; CreditMetrics ; CreditManager

Realized volatility

Realized volatility is a nonparametric ex-post estimate of the return variation. The most obvious realized volatility measure is the sum of finely-sampled squared return realizations over a fixed time interval. In a frictionless market the estimate achieves consistency for the underlying quadratic return variation when returns are sampled at increasingly higher frequency. We begin with an account of how and why the procedure works in a simplified setting and then extend the discussion to a more general framework. Along the way we clarify how the realized volatility and quadratic return variation relate to the more commonly applied concept of conditional return variance. We then review a set of related and useful notions of return variation along with practical measurement issues (e.g., discretization error and microstructure noise) before briefly touching on the existing empirical applications.Stochastic analysis