Credit risk is dependent upon many variables, making its pricing a complex issue. Consider a plain vanilla gas swap with counterparty XYZ: Our primary exposure is to the risk of XYZ going bankrupt, resulting in loss of the current mark-to-market (MTM) value, less any recovery value. This exposure is dynamic, subject to underlying price curve movements and the amount we recover upon bankruptcy . Any increases in the MTM value will increase our credit exposure, whereas increases in our expected recovery rate will reduce our exposure. This analysis describes the steps we would take to quantify these risks, in order to accurately model the credit risk in the deal. The three steps are 1) to calculate approximate probabilities of bankruptcy for counterparty XYZ; 2) estimate the future MTM value lost upon bankruptcy; and 3) assign a recovery value.

Many varied methods facilitate the estimation of bankruptcy probabilities. The bankruptcy swap is the only traded instrument which enables direct computation of implied rates, however this market is still in its infancy. More traditional methods include using bond and credit derivative prices; the option-theoretic models of Merton and KMV; statistical estimation using financial ratios; and historical analysis.

The bankruptcy swap is a digital credit derivative, whereby the payoff occurs upon the bankruptcy of the reference entity, and the payoff equals the full notional value of the contract. The simplicity of the structure enables the direct computation of a bankruptcy probability curve, using a bootstrapping algorithm based upon observed market prices at different tenors.

Credit default swaps are currently more liquid instruments than bankruptcy swaps, so the breadth of credit information available in this market is greater. However, the credit event underlying these contracts is typically default on an underlying bond, and the payoff is par less recovery following default. This requires two further assumptions to extract bankruptcy probabilities. First the recovery rate on the underlying bond must be estimated. Secondly the discount to transform bond default probabilities into bankruptcy probabilities must be estimated. This measure, called the “Event Discount”, can be estimated from the ratio between historical default and bankruptcy rates.

Using bonds we can derive bond default probabilities. The spread between the yields on risky corporate and risk-free bonds reflects the risk that the corporate will fail to repay the coupons and principal. However pricing information on corporate bonds is often scarce, so generic “Fair Market” yield curves for the credit rating and industry of the counterparty can be used as a proxy.

¹The real risk on the contract is failure-to-pay rather than bankruptcy. However failure-to-pay is a contract-specific event, whereas bankruptcy occurs in the public domain. Also failure-to-pay often results in bankruptcy so the two events are closely tied. For these reasons we focus on bankruptcy as our credit event.

Issuer-specific yield curves can be constructed using exponential splines (Vasicek and Fong), or affine models such as Hull-White or Cox-Ingersoll-Ross. These work well when a wide spectrum of liquid bonds from consistent pricing sources is available. When the number of bonds for a particular issuer is small however, more ad hoc techniques must be employed. One useful technique is to define a universe of curves “parallel” to the Fair Market curves, then find the curve which best prices the input bonds.

Given the risky yield curve, we must now define where the risk-free curve lies in order to generate credit spreads. Traditionally the government curve has been used. However, in recent months, this has become problematic, especially in the UK and US. Budget surpluses have resulted in the buy-back of government debt, inverting the yield curve and creating a scarcity premium at long maturities. Many analysts and traders now consider the LIBOR or swap curve to be a closer proxy to the true risk-free rate. This is, however, still not ideal as many high-rated credits trade below the swap curve, creating negative probabilities. Agency rates (Fannie Mae, Freddie Mac) and AAA credits (GE) are also vying for the status of the global benchmark risk-free curve, but neither has gained widespread acceptance.

Once we have the credit spreads, we can derive bond-default probabilities using the following formula; where r is the risky forward interest rate, rf the risk-free forward interest rate, R the risky bond’s recovery rate upon default, and p the probability of default.

This formula says that the default-risk adjusted return on the risky bond equals the return on the risk-free bond, which assumes risk-neutrality of investors. The probability of default is backed out as:

This method is described further in Buy, Kaminski, Krishnarao, and Shanbhogue. Other methods for estimating bankruptcy probabilities from credit spreads include adjusting the actual cashflows from the bonds for default risk (Fons), or by using no-arbitrage arguments about the price of the bond rather than the interest rates embedded within it (Jarrow and Turnbull).

Again we have the issues described earlier regarding the estimation of the recovery rate and event discount parameters, in order to convert the estimated default probability into a bankruptcy probability.

This approach, pioneered by Black, Scholes and Merton, views the equity of a firm as a call option on the assets of the firm, where the exercise price and maturity correspond to the face value and maturity of the outstanding debt. Equivalently, using put-call parity, the debt-holders of the firm can be considered to have sold a put option based on the firm’s assets to the equity holders. The put option will be exercised whenever the asset value of the firm falls below the debt level. The probability of default of the firm then equals the probability that the put option is exercised.

Using the book value of debt and the market value of equity, along with an estimate of equity volatility, it is then possible to reverse engineer the implied asset value and volatility. An assumption of normality for asset returns then allows the direct computation of the probability of default as the probability of assets dropping below the debt level at some point before the debt’s maturity.

KMV (Vasicek, Crosbie) differ from the traditional Merton approach in the final distributional assumption. Rather than assume normality, they define the “distance-to-default” as the distance between asset and debt levels, scaled for asset volatility, then use a database of historic distances-to-default and subsequent defaults to estimate the probability. This overcomes one of the major flaws in the Merton approach – that short-term probabilities are too small – but leaves the estimated probability as a historic measure, which is unsuitable for mark-to-market trading. However the resulting expected default frequencies (EDF) do provide excellent leading indicators of financial distress as relayed through the equity markets.

Statistical techniques based on financial ratios and accounting data are either simplistic, or the most general framework available for credit pricing, depending on your point of view. Often criticised for lacking any overriding theory, they compensate for this by being the most flexible and the most appropriate for estimating default risk on small firms without any traded debt. By replicating the credit analyst or traders’ thinking when trying to rate or price a company, the methodology provides this closest fit to reality.

The Altman Z-score model is the most famous. Using multivariate discriminant analysis based upon variables such as in Table 1, the model classifies firms into one of two groups – bankrupt or non-bankrupt. Other models based on logistic and linear regression techniques can also be used for classifying companies into ratings and for direct estimation of their bankruptcy risk.

Working Capital / Total Assets

EBIT / Total Assets

Long-term Debt / Total Capital

Return on Assets

Operating Income / Sales

EBIT / Interest Expense

Total Assets / Total Common Equity

Table 1. Example Financial Ratios

A new approach gaining widespread popularity for both rating and pricing companies is based on neural networks. Using the same set of financial ratios, alongside equity based indicators such as distance-to-default and equity volatility, neural networks allow the identification of non-linear relationships between the input and output variables. Success in this area can be seen in the proliferation of software products currently hitting the market (IQ Financial, Moody’s Public Firm Model) and in recent academic interest (Albanis, Coats and Fant).

Using data collected by ratings agencies (Moody’s, Standard and Poors), direct estimation of historical default and bankruptcy rates is possible. Scarcity of data does not permit detailed conclusions to be made, beyond rates estimated per rating per year. The best application for the data is to determine the risk premium observed in the credit markets, by comparing rates implied from bonds or credit derivatives and historical rates. Using historical rates for direct credit risk analysis in a mark-to-market environment is not suitable as credit costs calculated internally will never match the current market cost-of-credit, and thus hedging of credit exposure will be uneconomic.

We can now assume that bankruptcy rates calculated from any of the above methods are either static or stochastic. A widely used model for stochastic bankruptcy rates is taken from the literature on birth-death processes and survival times. We model the time of bankruptcy of XYZ as the arrival rate of a Poisson process with intensity h, which is called a hazard rate. By positing a stochastic process for h(t), such as Cox-Ingersoll-Ross or Geometric Brownian Motion (GBM), we introduce volatility into our bankruptcy rates. From each month’s bankruptcy rate realisation, simulating a bankruptcy time simply involves drawing a random number from [0,1] and comparing it with the rate. For example, if the bankruptcy rate realisations for years 1 and 2 were 0.1 and 0.05, and the random number draw turned out as 0.12, then bankruptcy would be simulated in year 2 since 0.10 < 0.12 < 0.10+0.05.

Using either static or stochastic bankruptcy rates we build a distribution of bankruptcy times from a Monte Carlo simulation. In most simulations counterparty XYZ remains solvent, but whenever it does go bankrupt, the next step is to arrive at a MTM value for the deal at the time of bankruptcy. This firstly relies upon traditional price curve simulation for example using GBM, Heath-Jarrow-Morton (HJM) or trend-reversion processes; then revaluation of the deal, which can use an approximation or the full valuation model. The key aspect to remember is that credit risk is asymmetric – we will not gain from default on any of our out-of-the-money contracts. So the credit exposure equals the maximum of the MTM value and zero, as shown in the following graph:

Graph 1: Credit Exposure against Underlying Swap Price

The option-like payoff structure displays two important credit phenomena. At-the-money swap contracts will have positive credit risk; and credit risk will be positively correlated to underlying price volatility. These often-overlooked facts are critical for thorough credit risk analysis.

Now two steps of the modelling are complete – bankruptcy and market risk. The final and most subjective pieces of the puzzle are the recovery rate embedded within the bonds underpinning the implied methods above, and the specific contract recovery rate following bankruptcy. Historical analysis can allow the estimation of average recovery rates for different debt seniorities and industries, but again the data is scarce and often unreliable due to regime shifts. However until a liquid market develops which allows the implication of recovery rates from market prices, historical estimates are the best we have, alongside our own judgement.

Applying the recovery rate to the credit exposure gives the loss following bankruptcy. A full Monte Carlo simulation of bankruptcy and price curves allows us to build a distribution of credit losses. The resultant distribution will be heavily skewed with a concentration at zero (representing no bankruptcy or loss) and a long thin tail. The mean of the distribution then represents the “Expected Loss” which is taken as the credit reserve on the deal. The 95th or 99th percentile loss is the Credit VaR. The latter measure is more important from a portfolio viewpoint, and can be used as a guideline for insurance purposes against multiple bankruptcies during periods of extreme market stress.

These tools enable the risk manager to model the credit risk arising from certain deals and combinations of deals. However they must always be aware of the saying “garbage in, garbage out” when assessing the results. The numerical inputs to these models are often fairly approximate and subjective, and rarely model exactly the risk on the deal. The risk manager must be aware of these problems, before they can gain any benefit from the results.

Measuring Credit Risk on Derivative Exposures

Ben Parson - Enron Europe Research Group