“Social Security—Adequacy,Equity, and Progressiveness: A Review of Criteria Based on Experience In Canada and the United States”, Robert L. Brown,January 2000

Abstract

We consider the valuation of credit default swaps (CDSs) under an extended version of Merton’s structural model for a firm’s corporate liabilities. In particular, the interest rate process of a money market account, the appreciation rate, and the volatility of the firm’s value have switching dynamics governed by a finite-state Markov chain in continuous time. The states of the Markov chain are deemed to represent the states of an economy. The shift from one economic state to another may be attributed to certain factors that affect the profits or earnings of a firm; examples of such factors include changes in business conditions, corporate decisions, company operations, management strategies, macroeconomic conditions, and business cycles. In this article, the Esscher transform, which is a well-known tool in actuarial science, is employed to determine an equivalent martingale measure for the valuation problem in the incomplete market setting. Systems of coupled partial differential equations (PDEs) satisfied by the real-world and risk-neutral default probabilities are derived. The consequences for the swap rate of a CDS brought about by the regimeswitching effect of the firm’s value are investigated via a numerical example for the case of a two-state Markov chain. We perform sensitivity analyses for the real-world default probability and the swap rate when different model parameters vary. We also investigate the accuracy and efficiency of the PDE approach by comparing the numerical results from the PDE approach to those from the Monte Carlo simulation.     

Product market competition and credit risk

This study theoretically and empirically investigates effects of product market competition on credit risk. We first develop a real-options-based structural model in a homogeneous oligopoly and show that credit spreads are positively related to the number of firms in an industry. The disparity of firm size in an industry is relevant to both product market competition and credit risk, and we therefore extend the model to an asymmetric duopoly case. In particular, we demonstrate that credit spreads of relatively small (large) firms within an industry are positively (negatively) related to Herfindahl-Hirschman index, and the relative firm size in an industry is an important determinant of credit risk. The models’ implications are empirically scrutinized by a reduced-form hazard model and generally supported. By performing out-of-sample analyses, the results demonstrate that firm size together with the interaction terms between intra-industry firm size dummies and competition intensity can effectively predict default.     

The analytic pricing of asymmetric defaultable swaps

Swaps where both parties are exposed to credit risk still lack convincing pricing mechanisms. This article presents a reduced-form model where the event of default is related to structural characteristics of each party. The cash flows submitted to credit risk are identified before the swap is priced. Analytical pricing formulas for interest rate and currency swaps are computed using a Gaussian model for risky bonds. Currency swaps exhibit additional correlation risk. The benefits from netting depend on the balance between exposures and market conditions in valuation. We show that sources of credit risk asymmetries are also likely to impact on credit spreads.     

Insider trading in an equilibrium model with default: a passage from reduced-form to structural modelling

We study, in the framework of Back [Rev. Financial Stud. 5(3), 387–409 (1992)], an equilibrium model for the pricing of a defaultable zero coupon bond issued by a firm. The market consists of a risk-neutral informed agent, noise traders, and a market maker who sets the price using the total order. When the insider does not trade, the default time possesses a default intensity in the market’s view as in reduced-form credit risk models. However, we show that, in equilibrium, the modelling becomes structural in the sense that the default time becomes the first time that some continuous observation process falls below a certain barrier. Interestingly, the firm value is still not observable. We also establish the no expected trade theorem that the insider’s trades are inconspicuous.      

A jump to default extended CEV model: an application of Bessel processes

We develop a flexible and analytically tractable framework which unifies the valuation of corporate liabilities, credit derivatives, and equity derivatives. We assume that the stock price follows a diffusion, punctuated by a possible jump to zero (default). To capture the positive link between default and equity volatility, we assume that the hazard rate of default is an increasing affine function of the instantaneous variance of returns on the underlying stock. To capture the negative link between volatility and stock price, we assume a constant elasticity of variance (CEV) specification for the instantaneous stock volatility prior to default. We show that deterministic changes of time and scale reduce our stock price process to a standard Bessel process with killing. This reduction permits the development of completely explicit closed form solutions for risk-neutral survival probabilities, CDS spreads, corporate bond values, and European-style equity options. Furthermore, our valuation model is sufficiently flexible so that it can be calibrated to exactly match arbitrarily given term structures of CDS spreads, interest rates, dividend yields, and at-the-money implied volatilities.     

Does modeling framework matter? A comparative study of structural and reduced-form models

This study provides a rigorous empirical comparison of structural and reduced-form credit risk frameworks. The literature differentiates between structural models that are based on modeling of the evolution of the balance sheet of the issuer, and reduced-form models that specify credit risk exogenously by a hazard rate process. Until now, there has been no common agreement in academia and practice on which model framework better captures credit risk. As major difference we focus on the discriminative modeling of the default time. In contrast to the previous literature, we calibrate both approaches to the same data set, apply comparable estimation techniques, and assess the out-of-sample prediction quality on the same time series of credit default swap prices. As our empirical implementations of both approaches rely on the same market information we are able to judge whether empirically the model structure itself makes an important difference. Interestingly, our study shows that the models’ prediction power are quite close on average indicating that for pricing purposes the modeling type does not greatly matter compared to the input data used. Still, the reduced-form approach outperforms the structural for investment-grade names and longer maturities. In contrast the structural approach performs better for shorter maturities and sub-investment grade names.     

A bridge from ruin theory to credit risk

The structural model uses the firm-value process and the default threshold to obtain the implied credit spread. Merton’s (J Finance 29:449–470, 1974) credit spread is reported too small compared to the observed market spread. Zhou (J Bank Finance 25:2015–2040, 2001) proposes a jump-diffusion firm-value process and obtains a credit spread that is closer to the observed market spread. Going in a different direction, the reduced-form model uses the observed market credit spread to obtain the probability of default and the mean recovery rate. We use a jump-diffusion firm-value process and the observed credit spread to obtain the implied jump distribution. Therefore, the discrepancy in credit spreads between the structural model and the reduced-form model can be removed. From the market credit spread, we obtain the implied probability of default and the mean recovery rate. When the solvency-ratio process in credit risk and the surplus process in ruin theory both follow jump-diffusion processes, we show a bridge between ruin theory and credit risk so that results developed in ruin theory can be used to develop analogous results in credit risk. Specifically, when the jump is Logexponentially distributed, it results in a Beta distributed recovery rate that is close to market experience. For bonds of multiple seniorities, we obtain closed-form solutions of the mean and variance of the recovery rate. We prove that the defective renewal equation still holds, even if the jumps are possibly negative. Therefore, we can use ruin theory as a methodology for assessing credit ratings.      

Valuation of default-sensitive claims under imperfect information

We propose a valuation method for financial assets subject to default risk, where investors cannot observe the state variable triggering the default but observe a correlated price process. The model is sufficiently general to encompass a large class of structural models and can be seen as a generalization of the model of Duffie and Lando (Econometrica 69:633–664, [2001]). In this setting we prove that the default time is totally inaccessible in the market’s filtration and derive the conditional default probabilities and the intensity process. Finally, we provide pricing formulas for default-sensitive claims and illustrate in particular examples the shapes of the credit spreads.      

Do Credit Spreads Reflect Stationary Leverage Ratios?

Most structural models of default preclude the firm from altering its capital structure. In practice, firms adjust outstanding debt levels in response to changes in firm value, thus generating mean-reverting leverage ratios. We propose a structural model of default with stochastic interest rates that captures this mean reversion. Our model generates credit spreads that are larger for low-leverage firms, and less sensitive to changes in firm value, both of which are more consistent with empirical findings than predictions of extant models. Further, the term structure of credit spreads can be upward sloping for speculative-grade debt, consistent with recent empirical findings.     

Macroeconomic Conditions, Firm Characteristics, and Credit Spreads

We study a structural model that allows us to examine how credit spreads are affected by the interaction of macroeconomic conditions and firm characteristics. Unlike most other structural models, our model explicitly incorporates equilibrium macroeconomic dynamics and models a firm's cash flow as primitive processes. Corporate securities are priced as contingent claims written on cash flows. Default occurs when the firm's cash flow cannot cover the interest payments and the recovery rate is dependent on the economic condition at default. Our model produces the following predictions: (i) credit spread is mostly negatively correlated with interest rate; (ii) credit spread yield curves are upward sloping for low-grade bonds; (iii) firm characteristics have significant effects on credit spreads and these effects also vary with economic conditions. These predictions are consistent with the available empirical evidence and generate implications for further empirical investigation.     

A jump-diffusion model for pricing corporate debt securities in a complex capital structure

This paper proposes a jump-diffusion model, in closed form, to price corporate debt securities, senior and junior, with the same maturity and violation of the absolute priority rule. We take the structural approach that the firm's asset value follows a jump-diffusion process in a stochastic interest rate economy. Default occurs only if the firm value at the maturity of the corporate debts is less than the sum of the prespecified face values. Unlike previous models in the structural approach, our model is consistent with the current term structures of credit spreads for both senior and junior debts. In particular, it captures realistic short maturity credit spreads observed in the market. The key idea is to allow the jump intensity to be a time-dependent function. As an application, valuation of credit spread options is presented.     

Pricing credit derivatives under incomplete information: a nonlinear-filtering approach

This paper considers a general reduced-form pricing model for credit derivatives where default intensities are driven by some factor process X. The process X is not directly observable for investors in secondary markets; rather, their information set consists of the default history and of noisy price observations for traded credit products. In this context the pricing of credit derivatives leads to a challenging nonlinear-filtering problem. We provide recursive updating rules for the filter, derive a finite-dimensional filter for the case where X follows a finite-state Markov chain, and propose a novel particle-filtering algorithm. A numerical case study illustrates the properties of the proposed algorithms.     

A Simple Approach to Valuing Risky Fixed and Floating Rate Debt

We develop a simple approach to valuing risky corporate debt that incorporates both default and interest rate risk. We use this approach to derive simple closed-form valuation expressions for fixed and floating rate debt. The model provides a number of interesting new insights about pricing and hedging corporate debt securities. For example, we find that the correlation between default risk and the interest rate has a significant effect on the properties of the credit spread. Using Moody's corporate bond yield data, we find that credit spreads are negatively related to interest rates and that durations of risky bonds depend on the correlation with interest rates. This empirical evidence is consistent with the implications of the valuation model.     

Credit Risk and the Term Structure of Lease Rates: A Reduced Form Approach

Previous research either assumes default free leases or leases subject to default risk using a structural approach. However, structural credit risk models suffer from a common criticism that the firm’s asset value process is unobservable. We develop a reduced form credit risk model for leases that avoids making assumptions regarding unobservable asset valuation processes. Furthermore, we assume a correlated market and credit risk that provides us with a simple analytic formula for valuing defaultable lease contracts. Numerical analysis reveals that tenant credit risk can have a substantial impact on the term structure of leases. Finally, we use the model to demonstrate the implied lease term structure for a set of retail and financial firms in the Fall of 2000.

Impact of multiple curve dynamics in credit valuation adjustments under collateralization

We present a detailed analysis of interest rate derivatives valuation under credit risk and collateral modeling. We show how the credit and collateral extended valuation framework presented in Pallavicini et al. [Funding valuation adjustment: FVA consistent with CVA, DVA, WWR, collateral, netting and re-hyphotecation, 2011], and the related collateralized valuation measure, can be helpful in defining the key market rates underlying the multiple interest rate curves that characterize current interest rate markets. A key point is that spot Libor rates are to be treated as market primitives rather than being defined by no-arbitrage relationships. We formulate a consistent realistic dynamics for the different rates emerging from our analysis and compare the resulting model performances to simpler models used in the industry. We include the often neglected margin period of risk, showing how this feature may increase the impact of different rates dynamics on valuation. We point out limitations of multiple curve models with deterministic basis considering valuation of particularly sensitive products such as basis swaps. We stress that a proper wrong way risk analysis for such products requires a model with a stochastic basis and we show numerical results confirming this fact.     

Tempered stable structural model in pricing credit spread and credit default swap

In this paper, we explore the features of a structural credit risk model wherein the firm value is driven by normal tempered stable (NTS) process belonging to the larger class of Lévy processes. For the purpose of comparability, the calibration to the term structure of a corporate bond credit spread is conducted under both NTS structural model and Merton structural model. We find that NTS structural model provides better fit for all credit ratings than Merton structural model. However, it is noticed that probabilities of default derived from the calibration of the term structure of a bond credit spread might be overestimated since the bond credit spread could contain non-default components such as illiquidity risk or asymmetric tax treatment. Hence, considering CDS spread as a reflection of the pure credit risk for the reference entity, we calibrate it in order to obtain more reasonable probability of default and obtain valid results in calibration of the market CDS spread with NTS structural model.     

Capital Structure and Debt Priority

We study a defaultable firm's debt priority structure in a simple structural model where the firm issues senior and junior bonds and is subject to both liquidity and solvency risks. Assuming that the absolute priority rule prevails and that liquidation is immediate upon default, we determine the firm's interior optimal priority structure along with its optimal capital structure. We also obtain closed‐form solutions for the market values of the firm's debt and equity. We find that the magnitude of the spread differential between junior and senior bond yields is positively, but not linearly related to the total debt level and the riskiness of assets. Finally, we provide an in‐depth analysis of probabilities of default and the term structure of credit spreads.     

Currency total return swaps: valuation and risk factor analysis

Abstract

Currency total return swaps (CTRS) are hybrid derivative instruments that allow us to simultaneously hedge against credit and currency risks. We develop a structural credit risk model to evaluate CTRS premia. An empirical test on a sample of 23,005 price observations from 59 underlying issuers yields an average percentage error of around 10%. This indicates that, beyond interest rate risk, firm-specific factors are major drivers of the variations in the valuation of these instruments. Regression analysis of residuals shows that exchange rate determinants account for up to 40% of model pricing errors, indicating that a currency risk premium affects the CTRS price significantly but only marginally, which confirms the prevalence of credit risk in the pricing of CTRS.     

Modeling CDS spreads: A comparison of some hybrid approaches

According to the credit risk model proposed by Cathcart and El-Jahel (2006), default can occur either expectedly, when a certain signaling variable breaches a lower barrier, or unexpectedly, as the first jump of a Poisson process, whose intensity depends on the signaling variable itself and on the interest rate. In the present paper we test the performances of such a model and of other three models generalized by it in fitting the term structure of credit default swap (CDS) spreads. In order to do so, we derive a semi-analytical formula for pricing CDSs and we use it to fit the observed term structures of 65 different CDSs. The analysis reveals that all the model parameters yield a relevant contribution to credit spreads. Moreover, if the dependence of the default intensity on both the signaling variable and the interest rate is removed, the pricing of CDSs becomes very simple, from both the analytical and the computational standpoint, while the goodness-of-fit is reduced by only a few percentage points. Therefore, when using the credit risk model proposed by Cathcart and El-Jahel (2006), assuming a constant default intensity provides an interesting and efficient compromise between parsimony and goodness-of-fit. Furthermore, by fitting the term structure of CDS spreads on a period of about twelve years, we find that the parameters of the model with constant default are rather stable over time, and the goodness-of-fit is maintained high.