Distressed debt prices and recovery rate estimation

This paper has two purposes. First, it uses distressed debt prices to estimate recovery rates at default. In this regard, estimates are obtained for three recovery rate models: recovery of face value, recovery of Treasury, and recovery of market value. We show that identifying the “economic” default date, as distinct from the recorded default date, is crucial to obtaining unbiased estimates. The economic default date is defined to be the first date when the market prices the firm’s debt as if it has defaulted. An implication is that the standard industry practice of using 30-day post default prices to compute recovery rate yields biased estimates. Second, we construct and estimate a distressed debt pricing model. We use this model to implicitly estimate the parameters of the embedded recovery rate process and to price distressed debt. Our distressed debt pricing model fits market prices well, with an average pricing error of less than one basis point.      

Pricing default swaps: Empirical evidence

In this paper we compare market prices of credit default swaps with model prices. We show that a simple reduced form model outperforms directly comparing bonds' credit spreads to default swap premiums. We find that the model yields unbiased premium estimates for default swaps on investment grade issuers, but only if we use swap or repo rates as proxy for default-free interest rates. This indicates that the government curve is no longer seen as the reference default-free curve. We also show that the model is relatively insensitive to the value of the assumed recovery rate.     

Credit risk measurement: Developments over the last 20 years

This paper traces developments in the credit risk measurement literature over the last 20 years. The paper is essentially divided into two parts. In the first part the evolution of the literature on the credit-risk measurement of individual loans and portfolios of loans is traced by way of reference to articles appearing in relevant issues of the Journal of Banking and Finance and other publications. In the second part, a new approach built around a mortality risk framework to measuring the risk and returns on loans and bonds is presented. This model is shown to offer some promise in analyzing the risk-return structures of portfolios of credit-risk exposed debt instruments.     

Corporate bond default risk: A 150-year perspective

We study corporate bond default rates using an extensive new data set spanning the 1866-2008 period. We find that the corporate bond market has repeatedly suffered clustered default events much worse than those experienced during the Great Depression. For example, during the railroad crisis of 1873-1875, total defaults amounted to 36% of the par value of the entire corporate bond market. Using a regime-switching model, we examine the extent to which default rates can be forecast by financial and macroeconomic variables. We find that stock returns, stock return volatility, and changes in GDP are strong predictors of default rates. Surprisingly, however, credit spreads are not. Over the long term, credit spreads are roughly twice as large as default losses, resulting in an average credit risk premium of about 80 basis points. We also find that credit spreads do not adjust in response to realized default rates.     

Selecting credit rating models: a cross-validation-based comparison of discriminatory power

In commercial banking, various statistical models for corporate credit rating have been theoretically promoted and applied to bank-specific credit portfolios. In this paper, we empirically compare and test the performance of a wide range of parametric and nonparametric credit rating model approaches in a statistically coherent way, based on a ‘real-world’ data set. We repetitively (k times) split a large sample of industrial firms’ default data into disjoint training and validation subsamples. For all model types, we estimate k out-of-sample discriminatory power measures, allowing us to compare the models coherently. We observe that more complex and nonparametric approaches, such as random forest, neural networks, and generalized additive models, perform best in-sample. However, comparing k out-of-sample cross-validation results, these models overfit and lose some of their predictive power. Rather than improving discriminatory power, we perceive their major contribution to be their usefulness as diagnostic tools for the selection of rating factors and the development of simpler, parametric models.

Modeling the business risk of financially weakened firms: A new approach for corporate bond pricing

Most structural models of default risk assume that the firm's asset return is normally distributed, with a constant volatility. By contrast, this article details the properties that the process of assets should have in the case of financially weakened firms. It points out that jump-diffusion processes with time-varying volatility provide a refined and accurate perspective on the business risk dimension of default risk. Representative Arrow-Debreu state price densities (SPD) and term structures of credit spreads are then explored. The credit curves show that the business uncertainties play a major in the pricing of corporate liabilities.     

Creditor recovery: The macroeconomic dependence of industry equilibrium

This paper reconciles the state of the economy with industry conditions in driving asset liquidation values and, therefore, recovery rates on defaulted debt securities. Evidence to date downplays the economywide effect in favor of industry and debt characteristic explanations. This paper shows that macroeconomic effects are important but operate differentially at the industry level. Industries whose sales growth is more correlated with GDP growth recover less during recessions. And industries that are more dependent on external finance recover less when the stock market falls. These findings expose how economywide shocks are transmitted to industry downturns, providing a framework for the role of aggregate risk in recovery risk and for macroeconomic stress testing.     

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.     

Determinants of yield spread dynamics: Euro versus US dollar corporate bonds

This paper presents a systematic comparison between the determinants of euro and US dollar yield spread dynamics. The results show that US dollar yield spreads are significantly more affected by changes in the level and the slope of the default-free term structure and the stock market return and volatility. Surprisingly, euro yield spreads are strongly affected by the US (and not the euro) level and slope. This confirms the dominance of US interest rates in the corporate bond markets. Interestingly, I find that liquidity risk is higher for US dollar corporate bonds than euro corporate bonds. For both regions, the effect of changes in the bid-ask spread is mainly significant during periods of high liquidity risk. Finally, the results indicate that the credit cycle as measured by the region-specific default probability significantly increases US yield spreads. This is not the case for euro yield spreads.     

On the calibration of structural credit spread models

Empirical findings are mixed about the performance of structural models for term structure of credit spreads. It is commonly believed that all structural models have equally poor performance after calibration. However, proper calibration is not a trivial issue, especially for highly structural models. This paper proposes a more accurate procedure for calibrating two models: Leland–Toft (J Finance 51:987–1019, 1996) and Collin-Dufresne and Goldstein (J Finance 56:2177–2208, 2001). Using rating-based bond data, we find that the Leland–Toft model has significantly greater explanatory power for credit spreads across rating categories than previously reported. We provide theoretical explanations for these findings, and further extend our empirical analysis to include 286 individual senior bonds. Our findings help clarify the controversies over the performance of structural models in general and that of the Leland–Toft model in particular. In addition, we offer a rigorous procedure that can be used for calibrating other structural models more effectively.      

Assessing the credit risk of money market funds during the eurozone crisis

This paper measures credit risk in prime money market funds (MMFs) and studies how such credit risk evolved during the eurozone crisis of 2011–2012. To accomplish this, we estimate the annualized expected loss on each fund's portfolio. We also calculate by Monte Carlo the cost of insuring a fund against losses amounting to over 50 basis points. We find that credit risk of prime MMFs, though small, doubled from 12 basis points in June 2011 to 23 basis points in December 2011 before receding in 2012. Contrary to common perceptions, this did not primarily reflect funds’ credit exposure to eurozone banks because funds took measures to reduce this exposure. Instead, credit risk in prime MMFs rose because of the deteriorating credit outlook of banks in the Asia/Pacific region. We conclude that the increase in the credit risk of prime MMFs in the second half of 2011 reflected contagion in the worldwide banking system coupled with slowing global economic growth, not actions taken by MMFs.     

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.     

Evaluating the sovereign and household credit risk in Singapore: A contingent claims approach

This study evaluates the credit risk of the household and government (sovereign) sectors in Singapore using the contingent claims approach (CCA). The CCA model estimates the default probability of both sectors based on the market value of the assets and liabilities of the sectors. Compared to the traditional credit rating system, this model is able to provide numerical estimates of the exposures and default probabilities. We find that from the year 2000 to 2013, variations in the credit risk measures correspond to the economic growth of Singapore. In addition, we suggest that the main factor affecting the credit risks in the government and household sectors in Singapore is the volatility of the assets held by both sectors, given that the asset-to-distress barrier ratios are relatively stable over the past 14 years for both sectors.     

Dynamical analysis of corporate bonds based on the yield spread term-quality surface

Our aim of this research is to propose a model which estimates implied relative credit reliability from the yield spread of defaultable bonds and evaluates their spread risk. We introduce “yield spread term-quality surface” (YSTQS) which is defined on the space of duration and credit reliability of the issuers, and express their yield spread. First, we review the general pricing theorem of defaultable bonds with unpredictable recovery in the no-arbitrage context based on the external hazard rates. Second, we show that the dynamics of state variables determine the shape of the YSTQS, and they drive the YSTQS if the loss-adjusted hazard rates are described by a function of them. Finally, we show an empirical analysis of our model with daily yield spread, duration, and the credit ratings of corporate bonds.

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.

Banking systemic vulnerabilities: A tail-risk dynamic CIMDO approach

This study proposes a novel framework which combines marginal probabilities of default estimated from a structural credit risk model with the consistent information multivariate density optimization (CIMDO) methodology and the generalized dynamic factor model (GDFM) supplemented by a dynamic t-copula. The framework models banks’ default dependence explicitly and captures the time-varying non-linearities and feedback effects typical of financial markets. It measures banking systemic credit risk in the three forms categorized by the European Central Bank: (1) credit risk common to all banks; (2) credit risk in the banking system conditional on distress on a specific bank or combinations of banks; and (3) the buildup of banking system vulnerabilities over time which may unravel disorderly. In addition, the estimates of the common components of the banking sector short-term and conditional forward default measures contain early warning features, and the identification of their drivers is useful for macroprudential policy. Finally, the framework produces robust out-of-sample forecasts of the banking systemic credit risk measures. This paper advances the agenda of making macroprudential policy operational.     

Bayesian estimation of truncated data with applications to operational risk measurement

Data insufficiency and reporting threshold are two main issues in operational risk modelling. When these conditions are present, maximum likelihood estimation (MLE) may produce very poor parameter estimates. In this study, we first investigate four methods to estimate the parameters of truncated distributions for small samples—MLE, expectation-maximization algorithm, penalized likelihood estimators, and Bayesian methods. Without any proper prior information, Jeffreys’ prior for truncated distributions is used. Based on a simulation study for the log-normal distribution, we find that the Bayesian method gives much more credible and reliable estimates than the MLE method. Finally, an application to the operational loss severity estimation using real data is conducted using the truncated log-normal and log-gamma distributions. With the Bayesian method, the loss distribution parameters and value-at-risk measure for every cell with loss data can be estimated separately for internal and external data. Moreover, confidence intervals for the Bayesian estimates are obtained via a bootstrap method.     

On the convergence rate of fixed-width sequential confidence intervals

Abstract

Let X _f1, X _f2, ... be a sequence of i.i.d. random variables with mean µ and variance σ²∈ (0, ∞). Define the stopping times N(d)=min {n:n ^?1 Σ ⁿ _i=1} (X _i&#x2212;X _n)²+n ^?1?nd ²/a ²}, d>0, where X n =n ^?1 Σ ⁿ _i=1} Xi and (2π)^?½ ∫ ^a _?a exp (?u ²/2) du=α ∈(0,1). Chow and Robbins (1965) showed that the sequence I_n,d =[X_n ?d, X _n + d], n=1,2, ... is an asymptotic level -α fixed-width confidence sequence for the mean, i.e. lim_d→0 P(µ∈I_N(d),d )=α. In this note we establish the convergence rate P(µ∈I_N(d),d )=α + O(d½?δ) under the condition E|X₁|^3+?+5/(28) < ∞ for some δ ∈ (0, ½) and ??0. The main tool in the proof is a result of Landers and Rogge (1976) on the convergence rate of randomly selected partial sums.     

A Time Series Analysis of Financial Fragility in the UK Banking System

This paper extends the model proposed by Goodhart, Sunirand, and Tsomocos (2004, 2005, 2006) to an infinite horizon setting. Thus, we are able to assess how the model conforms with the time series data of the UK banking system. We conclude that, since the model performs satisfactorily, it can be readily used to assess financial fragility given its flexibility, computability, and the presence of multiple contagion channels and heterogeneous banks and investors. JEL Classification Numbers C68, E4, E5, G11, G21 We are grateful to seminar participants at the Bank of England, European Central Bank, University of Oxford, University of Pireaus, 59th International Atlantic Economic Conference, London and especially an anonymous referee for helpful comments and suggestions