Pricing defaultable bonds: a middle-way approach between structural and reduced-form models

In this paper we present a valuation model that combines features of both the structural and reduced-form approaches for modelling default risk. We maintain the cause and effect or ‘structural’ definition of default and assume that default is triggered when a state variable reaches a default boundary. However, in our model, the state variable is not interpreted as the assets of the firm, but as a latent variable signalling the credit quality of the firm. Default in our model can also occur according to a doubly stochastic hazard rate. The hazard rate is a linear function of the state variable and the interest rate. We use the Cox et al. (A theory of the term structure of interest rates. Econometrica, 1985, 53(2), 385–407) term structure model to preclude the possibility of negative probabilities of default. We also horse race the proposed valuation model against structural and reduced-form default risky bond pricing models and find that term structures of credit spreads generated using the middle-way approach are more in line with empirical observations.     

A simple continuous measure of credit risk

This paper introduces a simple parameterization for the risk-neutral default probability distributions for risky firms that are easily computed from quoted bond prices. The corresponding expected times to default have a particularly simple form and are proposed as a measure for credit risk. Being continuous in nature, times to default provide a much finer measure of risk than those provided by ratings agencies. Comparison with the ratings provided by Moody's and the distance to default measures calculated using the Merton [Merton, R. (1974). On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 2(2), 449-470] model shows that the highest rank correlation is found between the proposed time to default measure and Moody's ratings.     

Pricing Derivatives on Financial Securities Subject to Credit Risk

This article provides a new methodology for pricing and hedging derivative securities involving credit risk. Two types of credit risks are considered. The first is where the asset underlying the derivative security may default. The second is where the writer of the derivative security may default. We apply the foreign currency analogy of Jarrow and Turnbull (1991) to decompose the dollar payoff from a risky security into a certain payoff and a “spot exchange rate.” Arbitrage-free valuation techniques are then employed. This methodology can be applied to corporate debt and over the counter derivatives, such as swaps and caps.     

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.     

Credit derivatives with multiple debt issues

We evaluate the most actively traded types of credit derivatives within a unified pricing framework that allows for multiple debt issues. Since firms default on all of their obligations, total debt is instrumental in the likelihood of default and therefore in credit derivatives valuation. We use a single factor interest rate model where the exponential default frontier is based on total debt and is made coherent with observed bond prices. Analytical formulae are derived for credit default swaps, total return swaps (both fixed-for-fixed and fixed-for-floating), and credit risk options (CROs). Price behaviors and hedging properties of all these credit derivatives are investigated. Simulations document that credit derivatives prices may be significantly affected by terms of debt other than those of the reference obligation. The analysis of CROs indicates their superior ability to fine-tune the hedging of magnitude and arrival risks of default.     

Currency-Induced Credit Risk in a Dollarized Economy

We analyze the effects of exchange rate fluctuations on corporate credit default in a dollarized economy. The application, before an exogenous exchange rate shock, of a new regulation concerning currency-induced credit risk (CICR) in the Peruvian banking system created natural conditions for a comparison between exposed and unexposed corporate borrowers. We use firm-level data to find that CICR and debt dollarization have opposite effects on credit risk. While CICR increases default, debt dollarization reduces it. Our results suggest that banks transfer exchange risk as a hedging mechanism by lending to such borrowers in dollars only.     

Covered interest parity with default risk

In this paper we derive a simple model of covered interest parity (CIP) with the assumption that interbank money market rates are risky. The model assumes that the default risk of uncollateralised loans can be hedged perfectly by credit default swap contracts. We show that the no-arbitrage condition is satisfied by a band. The location of this no-arbitrage band depends on the relative riskiness of the two counterparties in the CIP trade. We present evidence on the performance of the model for developed currency pairs in 2008–2011. We find that FX swap spreads (CIP deviations calculated from interbank interest rates of two countries) either fluctuated within the no-arbitrage bands or were close to the edges of the no-arbitrage bands.     

Evaluating corporate credit risks in emerging markets

Corporate leverage among emerging market firms went up considerably after the 2007–09 Global Financial Crisis (GFC). We investigate how the increased emerging market corporate leverage in the post-GFC period (2010–15) impacted the underlying credit risk, compared to the pre-GFC (2002–2006) and GFC (2007–09) periods. Using firm-level credit risk, financial, and balance sheet data for 350 firms in 23 emerging markets, we find that leverage growth leads to a significant increase in corporate credit default swap (CDS) spreads only in the post-GFC period, and the incremental effect is mainly evident among risky firms (firms with high leverage and idiosyncratic volatility). In contrast, emerging market CDS spreads during the GFC period are mainly driven by global market risk factors. The post-GFC corporate debt vulnerability is mitigated for high growth prospect firms and firms domiciled in countries with high net capital inflows and superior governance. While corporate leverage growth impacts aggregate corporate credit risk, there is no evidence that it increases sovereign credit risk. Our paper contributes to the recent literature on potential sources of default risk in emerging markets.     

From equity to default correlation with taxes

For fixed income investment, the preponderant risk is the clustering of defaults in the portfolio. Accurate prediction of such clustering depends on the knowledge of default correlation. We develop models with exogenous debt and endogenous debt to predict default correlations from equity correlations based on a self-consistent structural framework. We also examine how taxes affect the prediction of default correlations based on the two models. The empirical analysis shows that the corporate taxes tend to decrease default correlations, while personal taxes could increase or decrease default correlations. Our default correlation model with exogenous debt does a better job of predicting default correlations for high quality bonds, while the one with endogenous debt predicts more accurately for lower rated bonds. Our studies not only theoretically improve the modeling of default correlation in the structural setting but also shed new light on various aspects of default correlations and thereby help financial practitioners price credit derivatives more accurately and formulate more effective strategies to manage default risk of credit portfolios.     

Liquidity and Credit Risk

We develop a structural bond valuation model to simultaneously capture liquidity and credit risk. Our model implies that renegotiation in financial distress is influenced by the illiquidity of the market for distressed debt. As default becomes more likely, the components of bond yield spreads attributable to illiquidity increase. When we consider finite maturity debt, we find decreasing and convex term structures of liquidity spreads. Using bond price data spanning 15 years, we find evidence of a positive correlation between the illiquidity and default components of yield spreads as well as support for downward‐sloping term structures of liquidity spreads.     

Interest Tax Shields: A Barrier Options Approach

There is a link between barrier options and tax shields of interest expense. We combine this link with a traditional valuation approach, to present practical valuation formulas for interest tax shields in three debt scenarios with risk of default: (1) constant debt, (2) delayed debt, and (3) debt refinancing. In all cases, default and refinancing are contingent on the random evolution of the income of the firm. For each scenario, we work out sensitivity analysis of the value of tax shields with respect to income, growth, systematic and business risk, risk-free interest rate, interest coverage ratio covenant, and the firm??s refinancing strategy.     

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.     

Leverage Choice and Credit Spreads when Managers Risk Shift

We model the debt and asset risk choice of a manager with performance‐insensitive pay (cash) and performance‐sensitive pay (stock) to theoretically link compensation structure, leverage, and credit spreads. The model predicts that optimal leverage trades off the tax benefit of debt against the utility cost of ex‐post asset substitution and that credit spreads are increasing in the ratio of cash‐to‐stock. Using a large cross‐section of U.S.‐based corporate credit default swaps (CDS) covering 2001 to 2006, we find a positive association between cash‐to‐stock and CDS rates, and between cash‐to‐stock and leverage ratios.     

Structural models of credit risk are useful: Evidence from hedge ratios on corporate bonds

Structural models of credit risk provide poor predictions of bond prices. We show that, despite this, they provide quite accurate predictions of the sensitivity of corporate bond returns to changes in the value of equity (hedge ratios). This is important since it suggests that the poor performance of structural models may have more to do with the influence of non-credit factors rather than their failure to capture the credit exposure of corporate debt. The main result of this paper is that even the simplest of the structural models [Merton, R., 1974. On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance 29, 449–470] produces hedge ratios that are not rejected in time-series tests. However, we find that the Merton model (with or without stochastic interest rates) does not capture the interest rate sensitivity of corporate debt, which is substantially lower than would be expected from conventional duration measures. The paper also shows that corporate bond prices are related to a number of market-wide factors such as the Fama-French SMB (small minus big) factor in a way that is not predicted by structural models.     

Tax-Adjusted Discount Rates with Investor Taxes and Risky Debt

This paper derives a tax-adjusted discount rate formula with a constant proportion leverage policy, investor taxes, and risky debt. The result depends on an assumption about the treatment of tax losses in default. We identify the assumption that justifies the textbook approach of discounting interest tax shields at the cost of debt. We contrast this with an alternative assumption that leads to the Sick (1990) result that these should be discounted at the riskless rate. These two approaches represent polar cases. Each generates its results by using a different simplifying assumption, and we explain what determines the correct treatment in practice. We also discuss implementation of the valuation procedure using the capital asset pricing model.     

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.     

Financial statement comparability and credit risk

Prior research shows that firms’ financial statement comparability improves the accuracy of market participants’ valuation judgments and thus may reduce firms’ costs of capital. Distinct from prior research focusing on the equity market, we develop measures of comparability relevant to debt market participants based on the within-industry variability of Moody’s adjustments to reported accounting numbers for the purposes of credit rating. We examine two sets of adjustments: (1) to the interest coverage ratio and (2) to non-recurring income items. We validate these comparability measures by providing evidence that greater comparability is associated with lower frequency and magnitude of split ratings by credit rating agencies. We predict and find that greater comparability is associated with (1) lower estimated bid-ask spreads for traded bonds, (2) lower credit spreads for both bonds and five-year credit default swaps, and (3) a steeper one- to five-year credit default swap term structure. Our results are consistent with financial statement comparability reducing debt market participants’ uncertainty about and pricing of firms’ credit risk.     

Inflation Risk in Corporate Bonds

We argue that corporate bond yields reflect fears of debt deflation. When debt is nominal, unexpectedly low inflation increases real liabilities and default risk. In a real business cycle model with optimal but infrequent capital structure choice, more uncertain or procyclical inflation leads to quantitatively important increases in corporate log yields in excess of default‐free log yields. A panel of credit spread indexes from six developed countries shows that credit spreads rise by 14 basis points if inflation volatility or the inflation‐stock correlation increases by one standard deviation.     

The Impact of the Federal Reserve's Large‐Scale Asset Purchase Programs on Corporate Credit Risk

Estimating the effect of Federal Reserve's announcements of Large‐Scale Asset Purchase (LSAP) programs on corporate credit risk is complicated by the simultaneity of policy decisions and movements in prices of risky financial assets, as well as by the fact that both interest rates of assets targeted by the programs and indicators of credit risk reacted to other common shocks during the recent financial crisis. This paper employs a heteroskedasticity‐based approach to estimate the structural coefficient measuring the sensitivity of market‐based indicators of corporate credit risk to declines in the benchmark market interest rates prompted by the LSAP announcements. The results indicate that the LSAP announcements led to a significant reduction in the cost of insuring against default risk—as measured by the CDX indexes—for both investment‐ and speculative‐grade corporate credits. While the unconventional policy measures employed by the Federal Reserve to stimulate the economy have substantially lowered the overall level of credit risk in the economy, the LSAP announcements appear to have had no measurable effect on credit risk in the financial intermediary sector.     

Valuation of collateralized debt obligations with hierarchical Archimedean copulae

Modelling portfolio credit risk is one of the crucial challenges faced by financial services industry in the last few years. We propose the valuation model of collateralized debt obligations (CDO) based on hierarchical Archimedean copulae (HAC) with up to three parameters, with default intensities calibrated to market data and with random loss given defaults that are correlated with default times. The methods presented are used to reproduce the spreads of the iTraxx Europe tranches. Our approach describes the market prices better than the standard pricing procedure based on the Gaussian distribution. We also obtain a flat correlation smile across tranches thereby solving the implied correlation puzzle.