Abstract

We begin with a review of (a) the pricing theory of multiname credit derivatives to hedge the credit risk of a portfolio of corporate bonds and (b) current approaches to modeling correlated default intensities. We then consider pricing of insurance contracts using credibility theory in actuarial science. After a brief discussion of the similarities and differences of both pricing theories, we propose a new unified approach, which uses recent advances in dynamic empirical Bayes modeling, to evolutionary credibility in insurance rate-making and default modeling of credit portfolios.

1. Introduction

Credit markets provide liquidity to the underlying financial system. They include not only banks that give loans to consumers and businesses, but also financial institutions and corporations that develop and trade securities and derivative contracts derived from these loans and credit-sensitive investments such as corporate bonds, and insurance companies and their subsidiaries that issue insurance and reinsurance contracts to protect against adverse credit events. The recent financial crisis in the USA was associated with unpreparedly high default rates of mortgage loans and defaultable bonds in credit markets, beginning with subprime mortgage loans in 2007 and culminating in the collapse of large financial institutions such as Bear Stearns and Lehman Brothers in 2008. In Sections 2 and 3, we give a brief review of the conventional approaches to modeling default risk and of widely used derivative contracts to hedge credit risk, respectively. In particular, we point out in Section 2 the similarities and differences between the intensity models in credit risk and those in survival analysis.

Parallel to the increasing volume of subprime mortgage loans whose value was estimated to be $1.3 trillion by March 2007, an important development in financial markets during the past decade was the rapid growth of credit derivatives, culminating in $32 trillion worth of notional principal for outstanding credit derivatives by December 2009. These derivative contracts are used to hedge against credit loss of either a single one or a portfolio of corporate bonds. A “credit default swap” (CDS) is a contract between a buyer, who obtains the right to sell bonds issued by a company or country (a “single name” that is called the reference entity), and a seller (of the CDS contract) who agrees to buy the bonds for their face values when a credit event occurs. The face value of a coupon-bearing bond is the principal that the issuer repays at maturity if it does not default. The total face value of the bonds is the notional principal of the CDS. The buyer of the CDS makes periodic payments (usually quarterly) to the seller until the expiration date of the CDS or until a credit event occurs. If there is no credit event up to time , the buyer receives no payoff. If there is a credit event at time , the buyer's payoff is the difference between the face value and the mid-market value of the cheapest deliverable bond determined by an auction process several days after the credit event if the contract specifies cash settlement, or between the face value and what the buyer receives by selling the bond if the contract specifies physical settlement. The credit event is defined in the CDS contract and includes failure to pay according to schedule, agency downgrade, bankruptcy, and debt restructuring. The credit risk of a portfolio of corporate bonds can be mitigated by using a multiname credit derivative. Stochastic models of default intensities are used to price these derivatives. For multiname credit derivatives, joint modeling of the default intensities has been a challenging statistical problem. The review in Section 3.3 describes the correlated default models that have been widely used prior to the financial crisis and points out some of their limitations and deficiencies.

In July 2007, Bear Stearns disclosed that two of its subprime hedge funds, the High-Grade Structured Credit Fund and the High-Grade Structured Credit Strategies Enhanced Leverage Master Fund, which were invested in CDOs, had lost nearly all their value following a rapid decline in the subprime mortgage market. Lawsuits were brought by investors against former managers of the hedge funds, and Standard & Poor's (S&P) downgraded the company's credit rating. In March 2008, the Federal Reserve Bank of New York initially agreed to provide a $25 billion collateralized 28-day loan to Bear Stearns, but subsequently changed the deal to make a $30 billion loan to J. P. Morgan Chase to purchase Bear Stearns at $2 a share, or less than 7% of Bear Stearns' market value just two days before. After a class action suit was filed on behalf of the shareholders, J. P. Morgan Chase raised the offer to $10 per share, which was approved by the shareholders in May 2008. Lehman Brothers also suffered unprecedented loss for its large positions in subprime and other lower-rated mortgage-backed securities in 2008. After attempts to sell it to Korea Development Bank and then to Bank of America and to Barclays failed, it filed for Chapter 11 bankruptcy protection on September 15, 2008, making the largest bankruptcy filing, with over $600 billion in assets, in US history.

A day after Lehman's collapse, American International Group (AIG) needed bailout by the Federal Reserve Bank, which gave the insurance company a secured credit facility of up to $85 billion to enable it to meet collateral obligations after its credit ratings were downgraded below AA, in exchange for a stock warrant for 79.9% of its equity. AIG's London unit had sold credit protection in the form of CDS and CDO to insure $44 billion worth of securities originally rated AAA. As Lehman's stock price was plummeting, investors found that AIG had valued its subprime mortgage-backed securities at 1.7 to 2 times the values used by Lehman and lost confidence in AIG. Its share prices had fallen over 95% to just $1.25 by September 16, 2008, from a 52-week high of $70.13. The “contagion” phenomenon, from increased default probabilities of subprime mortgages to those of counterparties in credit derivative contracts whose values vary with credit ratings, was mostly neglected in the models of joint default intensities that were used to price CDOs and mortgage-backed securities. These models also failed to predict well the “frailty” traits of latent macroeconomic variables that underlie the mortgages and mortgage-backed securities. Section 4 reviews recent works in the literature to incorporate frailty and contagion in modeling joint default intensities.

The major business of AIG is insurance, and insurance contracts rather than credit derivatives contracts are what it sells primarily. These insurance contracts provide protection against loss due to automobile accidents and other unexpected events such as fire and floods in home insurance, just like credit derivatives that provide protection against loss due to default or other credit events. On the other hand, as will be explained in Section 5, whereas pricing credit derivatives involves the risk-neutral stochastic dynamics of the credit events that are priced by the market, pricing insurance contracts involves real-world (rather than risk-neutral) statistical models of the events. Section 5 discusses the similarities and differences between both pricing theories and also reviews credibility theory in actuarial science, which aims at deriving the premium of an insurance contract that balances the experience of an individual risk with the class risk experience. In particular, pricing a CDO that provides CDS protection for a portfolio of companies and setting the premium of automobile insurance for drivers in the same risk class both involve predicting the probability of occurrence and the loss incurred for certain future events. Estimation of the probability and the loss can pool information from the individual (company or driver) and the collective (portfolio of companies or risk group) levels.

Such pooling of information is the essence of empirical Bayes (EB) methodolgy in statistics, introduced by Robbins [1] and Stein [2]. The review of credibility theory in Section 5 also reviews EB and points out their connections. Lai et al. [3] have recently shown how the predictive performance of conventional EB methods in estimating the accident-proneness of a driver in a future period, or other insurance applications such as workers' compensation losses due to permanent partial disability, can be improved by using a dynamic EB approach to longitudinal data. Section 6 describes this new approach and relates it to generalized linear mixed models (GLMM) in biostatistics introduced by Breslow and Clayton [4]. It also reviews recent work applying the dynamic EB approach via GLMM to credibility theory, credit portfolios, and default probability forecasts of subprime mortgage loans. Thus the dynamic EB methodology provides a unified statistical approach to credit portfolios and credibility theory, and Section 7 provides further discussion and concluding remarks.

2. Intensity and Structural Models of Default Risk

2.1. Survival Analysis, Intensity Processes, and Intensity Models for Default Risk

We begin with a brief review of survival analysis in biostatistics and reliability theory and then show its connections with widely used intensity models for pricing corporate bonds and other credit-sensitive securities. To pinpoint these connections, our review focuses on the martingale approach to survival analysis, as described in [5, 6]. A major theme of the statistical problems in survival analysis is to study the distribution of a failure time , based on possibly censored data from either a homogeneous population or a regression model with covariates. The survival function of is , where and . Clearly, is a nonincreasing left-continuous function of with and . If is absolutely continuous, the probability density function of is , and the hazard (or intensity) function is Integrating (2.1) with respect to yields where is called the cumulative hazard function. Without assuming to be continuous, we can define the cumulative hazard function by the Stieltjes integral Regarding the indicator functions , as a stochastic process, an important property of is that where . Fleming and Harrington [5] and Andersen et al. [6] give overviews of continuous-time martingales and their applications, via (2.4) and its analogs, to survival analysis. Based on independent observations sampled from , the nonparametric maximum likelihood estimate of is the empirical distribution function . However, in applications of survival analysis to biomedical and econometric data, the may not be fully observable because of censoring. Some subjects (or firms) may not fail during the observation period or may have been lost in followup during the period; the data on these individuals are said to be right-censored.

Right-censoring can be formulated by introducing censoring variables that indicate the time of loss to followup. The censoring variable can also indicate the length of the observation period. The observations, therefore, are , where and is the censoring indicator that indicates whether is an actual failure time or is censored. Subject is “at risk” at time if (i.e., has not failed and has not been lost to followup prior to ). Let Note that is the risk set size and is the number of observed deaths at time , and that is the analog of . We use to denote the jump size of at (i.e., ) and also use the convention . The cumulative hazard function can be estimated by for the censored data . This is called the Nelson-Aalen estimator [7, 8]. Analogous to (2.4), for every left-continuous stochastic process ; note that is left-continuous. From this and the martingale central limit theorem, as , where denotes convergence in distribution; see [5, 6].

Altman [9] and others have developed mortality tables for loans and bonds by using methods in actuarial science. These mortality tables (also called life tables) are used by actuaries to set premiums for life insurance policies. Partition time into disjoint intervals , and so forth. A life table summarizes the mortality results of a large cohort of subjects as follows:   number of subjects alive at the beginning of ,   number of deaths during , number lost to followup during .It estimates by so that the actuarial (life-table) estimate of is the product Without discretizing the failure times, Kaplan and Meier [10] introduced the product-limit estimator of : Since has at most jumps, the product in (2.10) is finite. Note that by (2.6) and (2.10). As shown in [6], The product in (2.11) is called the product integral of the nondecreasing function on ; it is defined by the limit of , where is a partition of and the product is in the natural order from left to right, as the mesh size approaches 0. Moreover, by making use of the martingale central limit theorem, it is shown in [5, 6] that

2.1.1. Regression Models for Hazard Functions with Covariates

In applications, one often wants to use a model for to predict future failures from a vector of predictors based on current and past observations; is called a time-varying covariate. When does not depend on , it is called a time-independent (or baseline) covariate. In practice, some predictors may be time-independent, while other components of may be time-varying. Since prediction of future default from is relevant only if (i.e., if default has not occurred at or before ), one is interested in modeling the conditional distribution of given , for example, by relating the hazard function to . Noting that is the probability that , Cox [11] introduced the proportional hazards (or Cox regression) model Putting in (2.13) shows that is also a hazard function; it is called the baseline hazard. Cox regression fits (2.13) to the observed data . Lane et al. [12] apply Cox regression to predict bank default, using time-independent covariates.

The parameters of (2.13) are and the cumulative hazard function , which is an infinite-dimensional parameter. One way to estimate is to specify via a parameter vector so that can be estimated by maximum likelihood. Since an observed failure contributes a factor to the likelihood, where is the density function of the while a censored failure contributes a factor to the likelihood, and since and , the log-likelihood function can be written as see [13]. Commonly used parametric models for failure times are the exponential density that has constant hazard and the Weibull density that has hazard function and cumulative hazard function , with . Lee and Urrutia [14] use a Weibull distribution of default times to predict insolvency in the property-liability insurance industry. Hazard models with time-varying covariates have been used by McDonald and Van de Gucht [15] to model the timing of high-yield bond default and of exercising call options. Related survival analysis to predict bankruptcy has been carried out in [16–19].

Instead of assuming a parametric model to estimate the baseline hazard function , Cox [11, 20] introduced a semiparametric method to estimate the finite-dimensional parameter in the presence of an infinite-dimensional nuisance parameter ; it is semiparametric in the sense of being nonparametric in but parametric in (in terms of ). Cox's partial likelihood method decomposes the likelihood function into two factors, with one involving only and the other involving both and . It estimates by maximizing the following partial likelihood, which is a factor that only involves . Order the observed censored failure times as , with . Let denote the set of censored 's in the interval , and let denote the individual failing at , noting that with probability 1 there is only one failure at because the failure time distributions have density functions. Let denote the risk set at . Cox's regression estimator is the maximizer of the partial log-likelihood or equivalently, is the solution of . Martingale theory can be used to show that where is the Hessian matrix of second partial derivatives . One, therefore, can perform usual likelihood inference, treating the partial likelihood as a likelihood function. Moreover, even though is based on partial likelihood, it has been shown to be asymptotically efficient; see [6]. Modifying in (2.5) to leads to the Breslow estimator [21] of , which is again given by (2.6) but with defined by (2.17).

2.1.2. Intensity-Based Models for Pricing Default Risk

Besides the important roles they have played in the statistical analysis of failure-time data as explained in the preceding section, intensity modeling has also become a standard approach to pricing corporate bonds and other credit-sensitive securities since the 1990s. Intensity models provide an elegant way to combine term-structure modeling for default-free bonds with the default risk and recovery features of corporate bonds. Chapter 10 of [22] gives an overview of interest rate markets and the term structure of interest rates, statistical methods to estimate the zero-coupon yield curve from the current prices of default-free Treasury bonds, arbitrage-free pricing theory of interest rate derivatives, short rate and other models under the risk-neutral (instead of the real-world) measure entailed by arbitrage-free pricing, and calibration of these models to market data involving actively traded interest rate derivatives. For corporate bonds, besides the interest rate processes, we also include an additional stochastic process for the default intensity of the bond. All the stochastic processes are defined on the same probability space in which the probability measure is a risk-neutral (or martingale) measure associated with arbitrage-free pricing. Expectations taken under this measure will be denoted by or , while is used to denote expectation under the real-world (or physical) measure .

A Cox process (also called stochastic Poisson process) is often used to model the default intensity of the bond's issuer. The default intensity is assumed to be governed by an exogenous stochastic process , so that and the stochastic dynamics of is specified through . Let be the default time and . Then is distributed as an exponential random variable with mean 1 that is independent of since         . Hence we can use to generate by . Consider a zero-coupon bond, with maturity date and face value 1, issued by a firm at time 0. Assume that there is a short-rate process under the risk-neutral measure such that the default-free bond price is given by see [22, page 257]. Assuming to be the intensity process for the default time of the firm and assuming zero recovery at default, the price of the defaultable bond at time 0 is since , noting that and that is independent of . Thus (2.19) replaces the short rate in the default-free bond pricing formula by the default-adjusted short rate . Moreover, closed-form expressions are available for if one uses affine models for both and ; see [23, 24]. The short rate process and the intensity process have unknown parameters which can be encoded in a vector and can be estimated from the bond prices by using least squares in the same way as that for Treasury bonds; see [22]. Note that unlike survival analysis in biostatistics that uses censored failure-time data to estimate the intensity in (2.13) or its integral in (2.3), estimation of the parameters of the intensity process is based not on observed defaults but rather on the bond prices that involve the risk-neutral (rather than the real-world) measure.

In practice, recovery rates also need to be considered and the vector may include predictors that make and much more complex than the affine model that gives an explicit formula for the integral in (2.19). Monte Carlo methods are used instead to evaluate (2.19) in these more complex situations. Since corporate bonds typically promise cash flows, which mimic those of Treasury bonds, it is natural to consider yields when one compares the effects of different recovery assumptions at default. The yield at date of a zero-coupon bond with price , face value , and maturity is defined by and the difference between the yield of a defaultable bond and a corresponding Treasury bond is referred to as the credit spread or yield spread. The first type of recovery assumptions is recovery of market value. This measures the change in market prices before and after the default, representing the loss in the bond's value associated with the default. More generally, consider the price process of a defaultable contingent claim promising a payoff of at . The claim is said to have a fractional recovery of market value of at default time if the amount recovered in the event of default is equal to where is the value of the claim just prior to default and . With this recovery assumption, the price at of the defaultable contingent claim (if there has not been a default up to date ) is which is an extension of (2.19). A variant of recovery of market value is recovery of face value, which is the closest to legal practice in the sense that debt with the same priority is assigned a fractional recovery depending on the outstanding notional amount but not on maturity or coupon. It is also the measure typically used in rating-agency studies. The third type of recovery assumptions is recovery of treasury, under which the corporate bond in default is replaced with a treasury bond with the same maturity but a reduced payment. Unlike recovery of face value, it tries to correct for the fact that amounts of principal with long maturity should be discounted more than principal payments with short maturity.

2.2. Econometric Modeling of Default Risk

The intensity-based models in the preceding section are sometimes called reduced-form models, using a terminology in econometrics that refers to models of the relationship of an economic variable to exogenous variables. In contrast, structural models attempt to derive the relationship from some underlying economic theory. Structural models for the default of a firm assume that default is triggered when the firm's asset and debt values enter some region whose boundary is called a “default barrier.”

2.2.1. Merton's Model of Debt and Equity Values and the KMV Credit Monitor Model

As in the Black-Scholes model described in Section 8.1 of [22], Merton [25] assumes a market with continuous trading, no transaction costs, unlimited short selling and perfectly divisible assets, and with a risk-free asset that has constant interest rate . The asset value of the firm is assumed to follow a geometric Brownian motion (GBM) . Suppose at time 0 the firm has issued two kinds of contingent claims: equity and debt, in which debt is assumed to be a zero-coupon bond with face value and maturity date . We can regard the firm as being run by equity owners. At maturity date of the bond, equity owners pay to retain ownership if , and would not pay if since they have limited liability. In the latter case, bond holders receive a recovery of instead of the promised payment . Therefore, the equity value at time is given by the Black-Scholes formula where is the risk-neutral measure under which the drift of the GBM is instead of , and see [22, page 186]. Similarly, the debt value at time is since .

In the 1990s, the KMV Corporation of San Francisco (now merged into Moody's KMV) introduced its Credit Monitor Model for default prediction using Merton's framework. To begin with, the GBM for the asset value can be expressed in terms of the Brownian motion by ; see [22, page 66]. Therefore, the probability of default on the debt at maturity is This leads to KMV's distance to default defined by so that the predicted probability of default under Merton's model is . Since is the standard deviation of , DD can be regarded as the number of standard deviations of the log asset value from the log liability. Instead of using the normal distribution for to predict the default probability as in (2.26), the KMV Credit Monitor Model uses a large proprietary database of firms and firm defaults to construct an EDF (expected default frequency) score for a firm with standard deviations away from the default (i.e., DD ) at the beginning of a future period. The idea is to stratify the firms based on their DD's and estimate empirically the default probability of the firm from its associated stratum. The firm's DD is evaluated from its balance sheet and option models for its debt.

2.2.2. Black-Cox Framework of Default Barriers and Other Extensions of Merton's Model

Black and Cox [26] extend the Merton model by allowing defaults to occur prior to the maturity of the bond; specifically, when the asset value hits a nonrandom, continuous boundary . They also allow the equity owners to be paid dividends, at rate , so that under the risk-neutral measure the GBM for has drift ; see [22, page 186]. Let . The payoff to bond holders is in which the first summand reflects that the bond holders receive at the bond's maturity if default has not occurred up to that time, while the second summand reflects that the bond holders receive the recovery of upon default. Similarly, the payoff to equity holders is .

Stochastic models for the default-free interest rate dynamics (see [22, Chapter 10]) have been used to replace the constant in Merton's model. Another extension is to allow coupon payments to the bond holders. More general jump diffusion models have been introduced to replace GBM in Merton's model. Lando [24] provides an overview of these extensions, which also have been considered in conjunction with default barriers.

2.3. Credit Ratings and Credit Value at Risk

Credit scoring has been used in the banking industry since the 1940s to assess the creditworthiness of loan applicants. The basic idea is to assign scores to loan applications based on their attributes such as gender, marital status, education level, income, profession, and credit history. Such scoring systems, called scorecards, are usually developed by in-house credit departments and/or consulting firms. One of the most widely used credit scoring systems was developed by Fair Isaac & Company (FICO), which built credit scoring systems for US banks and retail stores in the 1960s and 1970s and then expanded those systems to meet the needs of other industries and to evaluate the credit of borrowers. Credit scoring systems have now been developed for virtually all types of credit analysis, from consumer credit to commercial loans. The idea is to preidentify certain key factors that determine the probability of default and combine or weight them into a quantitative score.

Credit ratings of the creditworthiness of corporate bonds are provided by rating agencies such as Moody's, S&P, and Fitch. Using Moody's system, the ratings are Aaa, Aa, A, Baa, Ba, B, Caa, Ca, and C and “investment grade” ratings are Baa or above. The corresponding S&P and Fitch ratings are AAA, AA, A, BBB, BB, B, CCC, CC, and C. For finer ratings, Moody's divides the Aa rating category into Aa1, Aa2, Aa3, the A category into A1, A2, A3, and so forth, and S&P and Fitch divide AA into AA+, AA, and AA−, A into A+, A, and A−, and so forth. Although methodologies and standards differ from one rating agency to another, regulators generally do not make distinctions among the agencies. Furthermore, although there is a strong similarity between the rating systems of Moody's and S&P, different agencies might assign slightly different ratings for the same bond. The ratings given by rating agencies are used by several credit risk software models such as CreditMetrics of J. P. Morgan, a system that evaluates risks individually or across an entire portfolio. Most corporations approach rating agencies to request a rating prior to sale or registration of a debt issue. S&P assigns and publishes ratings for all public corporate debt issues over $50 million, with or without a request from the issuer; but in all instances, S&P analytical staff first contact the issuer to call for meetings.

2.3.1. CreditMetrics, Credit Ratings Migration, and Product-Limit Estimators

CreditMetrics was introduced in 1997 by J. P. Morgan and its cosponsors (including KMV, UBS, Bank of America, and others). The system measures the risk and potential losses of loans and portfolios of bonds and loans over a period (e.g., a year) by simulation and analytic methods. In particular, it uses Monte Carlo simulations to estimate credit rating changes of all counterparties. Historical data provided by rating agencies on credit rating changes can be used to estimate the transition probability matrix for credit rating changes. In the Basel Accord for bank regulation, banks are required to construct these transition probability matrices based on their own data to stress test their portfolios. Some risk management systems such as CreditMetrics use these credit rating transition matrices for Monte Carlo simulations of credit losses due to credit ratings migration in a future period. A Markov chain model of ratings migration assumes a finite state space consisting of different rating classes, in which state 1 denotes the best credit rating class and state represents the default case. The probability transition matrix of the Markov chain is in which for all and represents the actual probability of moving to state from initial rating state in one time step. Note that the default state is an absorbing state. The estimates of credit transition matrices published by rating agencies use this Markov chain framework.

Suppose there are firms in a given rating category at the beginning of the period and that firms migrate to the category at the end of the period. An obvious estimate of is for . However, there are serious difficulties with this estimate in practice. First, if no transition from rating class to is observed in the period, then the estimated transition probability becomes 0. Second, the assumption of homogeneity over time is too strong for credit transition matrices. The product-limit estimator (2.10) for the survival function has been generalized by Aalen [27], Aalen and Johansen [28], and Fleming [29] to estimate the transition probability matrix of a nonhomogeneous continuous-time Markov process on a finite state-space. Analogous to (2.11), the transition probability matrix from time to can be expressed as the product integral , and the components of the cumulative transition intensity matrix can be estimated by the Nelson-Aalen estimator of the form (2.6); see Section  IV.4 of [6].

To simulate the distribution of credit losses due to credit rating changes of two counterparties, say one with initial Aaa rating (state 1) and the other with initial B rating, over a 1-year period, we can use the estimate of the transition matrix (2.29) to sample the new ratings for both counterparties in the following way. Without assuming the rating changes of the two counterparties to be independent, a Gaussian copula model, which will be described in Section 3.3, is typically used to simulate the joint probability distribution of the rating changes. Therefore, we first generate two standard normal random variables and with correlation . Noting that is uniform, the new rating of the Aaa company remains Aaa if , moves to states 2 (i.e., Aa) if , and so forth. This is tantamount to generating the state transitions of the Aaa company by the first row of the probability transition matrix . While the state transitions of the B company are likewise generated by the corresponding row of , the state transitions of the two companies are not independent and their dependence structure is inherited from the correlation of and . From the simulated credit rating changes of the two counterparties we can revalue their outstanding contracts to determine the total credit losses during the year. From a large number of such simulations, a probability distribution for credit losses is obtained, which can be used to evaluate credit VaR (Values at Risk) for risk management.

2.3.2. Credit VaR and a One-Factor Default Model

A credit VaR over a period of years is the th quantile of the credit loss within the period. J. P. Morgan's CreditMetrics defines credit losses as those arising not only from defaults but also from credit ratings downgrade. This involves estimating the probability distribution of credit losses by Monte Carlo simulation of defaults and credit rating changes of all counterparties. Instead of Monte Carlo, a closed-form approximation, which does not involve downgrade in credit ratings, to credit VaR for a large portfolio of similar loans is used in the minimum capital requirement formulas of the Basel Committee on Banking Supervision. Suppose the portfolio consists of loans in the same risk class. Letting denote the time to default of the th loan, we can express the proportion of loans that default during the period as . Since the loans belong to the same risk class, their are identically distributed, with common distribution function that is assumed to be continuous. Then is uniform and is standard normal, where is the standard normal distribution function. Vasicek [30, 31] and Schonbucher [32] assume the one-factor Gaussian model in which and are i.i.d. standard normal. Since if and only if , it follows that Conditional on , are i.i.d. and therefore by the strong law of large numbers, as , with probability 1.

Let be the size of the loan portfolio and the recovery rate. The proportional loss given default is and therefore the loss due to loan default in the period is . Using the approximation (2.32), the credit VaR over the period is since the th quantile of is .

3. Credit Derivatives: An Overview

3.1. Pricing and Hedging with Credit Default Swaps

Credit default swaps (CDSs) have been described in the second paragraph of Section 1. The total amount paid per year by the buyer of the CDS, as a percentage of the notional principal per annum, is called the CDS spread. Suppose the reference bond has coupon dates and the protection buyer pays a constant premium at these dates prior to , the time of the credit event. Assume that the face value of the bond is 1 and that the protection seller pays in the case of a credit event. Then the value of the premium leg of the swap is where is the short rate process, is the default intensity process, and is the price of the defaultable bond with zero recovery and maturity , using the same argument as that in (2.19). Similarly the value of the default leg of the swap, to be paid by the protection seller, is where is the density function of under the risk-neutral measure , assuming that the short rate process and the default intensity process are independent under . Note that the survival function of under is Letting be the hazard function of under , it then follows from (3.2) that where is the price of the default-free bond; see (2.18). Note that under this independence assumption on the short rate and default intensity processes, we can further simplify in (3.1) to In the absence of arbitrage, the premium of the default swap is the premium for which . Therefore, combining (3.1) with (3.4) yields