Credit Risk: Modeling the Chance of Default
In Chapter 39 we valued default risk by modeling the firm’s assets, earnings, and cash. That is the “look inside the company” approach. Chapter 40 takes a completely different path. Instead of trying to understand why a company might default, just model default as a random external event. Roll a die. If you get a 1, the company defaults. Simple.
Wilmott is upfront about his skepticism here. He says “I’m not wild about this idea but it is very popular.” And it is popular for good reason: the exogenous approach is much easier to calibrate and use in practice, even if it feels philosophically wrong.
Risky Bonds: The Basic Problem
A company issues a bond. You lend them money today, they promise to pay you back with interest later. But there is a chance they go bankrupt first. Because of this risk, you demand a higher interest rate than if you were lending to the US government.
The real situation is messy. Bonds have different seniority levels determining who gets paid first in bankruptcy. Recovery takes years. You might get some money back, maybe 50%, maybe 17%, depending on how senior your claim is. Moody’s data shows recovery rates ranging from 53.8% for senior secured debt down to 17.1% for junior subordinated, with massive standard deviations in all categories.
But we start simple.
The Poisson Process and Constant Default Risk
The simplest model assumes a constant instantaneous risk of default $p$. In any small time interval $dt$, there is a probability $p , dt$ that the company defaults. Nothing happens for a while, then BOOM. Default. This is a Poisson process.
If $p$ is constant, the probability of surviving (no default) from time $t$ to $T$ is:
$$P(t; T) = e^{-p(T-t)}$$
The value of a risky zero-coupon bond is the expected present value of the payoff:
$$Z^* = e^{-p(T-t)} \cdot Z$$
where $Z$ is the equivalent risk-free bond. The effect on yield? Simply add a spread of $p$ to every maturity. Clean, simple, and constant across all maturities.
The Pricing PDE
To price derivatives on risky bonds, you consider two scenarios at each time step:
- No default (probability $1 - p , dt$): The portfolio changes like normal, with the usual stochastic interest rate terms.
- Default (probability $p , dt$): You lose the risky bond entirely.
Taking expectations across both scenarios, the risky bond satisfies:
$$\frac{\partial V}{\partial t} + \frac{1}{2}w^2 \frac{\partial^2 V}{\partial r^2} + (u - \lambda w)\frac{\partial V}{\partial r} - (r + p)V = 0$$
Notice the discounting term is $r + p$ instead of just $r$. The default risk adds a spread to the discount rate. That is the key result. Default risk and interest rates play similar mathematical roles.
Testing Against Reality: The Tequila Effect
Wilmott tests this constant-$p$ model against real Brady bond data from Argentina, Brazil, Mexico, and Venezuela. Brady bonds are dollar-denominated emerging market bonds created under the 1989 Brady Plan.
He backs out the implied risk of default from market prices for each country. If the constant-$p$ model were correct, this implied $p$ would be stable over time. But the graphs show it is anything but stable.
Venezuela’s risk peaked in July 1994. Then came the Tequila crisis in December 1994 when Mexico devalued its peso by 50%. The implied default risk for ALL Latin American countries spiked, peaking around April 1995 before gradually recovering by late 1996.
Conclusion: the constant $p$ model is too simple. The market clearly believes default risk changes over time, and it is correlated across countries.
From Constant to Stochastic Default Risk
Making $p$ time-dependent lets you fit a term structure of hazard rates. A plausible structure starts low, peaks at some intermediate horizon, then falls off. But to really match market prices, you need $p$ itself to be random:
$$dp = \gamma(p, t) , dt + \delta(p, t) , dX_2$$
Now the risky bond depends on three variables: $V(r, p, t)$. The pricing equation becomes a two-factor PDE, beautifully symmetric in $r$ and $p$:
$$\frac{\partial V}{\partial t} + \frac{1}{2}w^2 \frac{\partial^2 V}{\partial r^2} + \rho w \delta \frac{\partial^2 V}{\partial r \partial p} + \frac{1}{2}\delta^2 \frac{\partial^2 V}{\partial p^2} + (u - \lambda w)\frac{\partial V}{\partial r} + \gamma \frac{\partial V}{\partial p} - (r+p)V = 0$$
The interest rate $r$ and the hazard rate $p$ appear in almost identical ways. They both contribute to discounting (through the $(r+p)V$ term). They both have their own drift and volatility. The only difference is the choice of model for each.
When $p$ is constant, this reduces to the simple constant-spread model. When $p$ is independent of $r$ and they are uncorrelated, the solution factors into a product of a risk-free bond value and a pure default component.
Positive Recovery
In practice, default does not mean total loss. Recovery rates range from 17% to 54% depending on seniority. If recovery is a known amount $Q$, the PDE changes: default loses the bond but gives you $Q$, modifying the term from $-pV$ to $-p(V - Q)$.
The Argentine Par Bond Case Study
Wilmott applies the stochastic hazard rate model to the Argentine Par Bond (December 1993 to September 1996). The implied $p$ is extracted daily by matching model and market prices. The time series clearly shows the Tequila effect: $p$ shoots up in early 1995 and gradually recovers.
Credit Ratings
Rating agencies like Standard & Poor’s and Moody’s assign letter grades to bonds as estimates of creditworthiness. S&P uses AAA, AA, A, BBB, BB, B, CCC, and Default. Moody’s uses Aaa through C. Historical data shows a clear relationship between rating and actual default frequency, confirming that the ratings carry real information.
A change in rating is called a migration, and it has major price impact. An upgrade decreases yield, a downgrade increases it.
The Transition Matrix
To model rating changes, you use a transition matrix. This is a square matrix where entry $(i,j)$ gives the probability of moving from rating $i$ to rating $j$ over some time horizon (usually one year).
For example, an A-rated company has a 91.3% chance of still being A after one year, a 2.42% chance of moving to AA, a 0.09% chance of reaching AAA, and so on. Default is an absorbing state, once you are there, you stay there.
Continuous-Time Transitions: Markov Chains
For pricing, you need transitions over arbitrary time periods, not just one year. This requires a continuous-time model using Markov chains.
Over a small time step $dt$, the transition matrix is:
$$\mathbf{P}_{dt} = \mathbf{I} + \mathbf{Q} , dt$$
where $\mathbf{Q}$ is the generator matrix (rows sum to zero, bottom row all zeros). Over a finite period $T$, the solution is:
$$\mathbf{P}(0, T) = e^{T\mathbf{Q}}$$
The matrix exponential is computed via eigenvalue decomposition. If you know the one-year transition matrix, you can recover $\mathbf{Q}$ by diagonalization and taking the matrix logarithm.
Pricing Risky Bonds with Ratings
With the Markov chain model, the bond value becomes a vector $\mathbf{V}$, one entry per rating. The pricing equation is:
$$\frac{d\mathbf{V}}{dt} + \mathbf{Q}^T \mathbf{V} - r\mathbf{V} = 0$$
This is the backward equation with a discounting term. The final condition is a vector of ones (full repayment) with a zero in the default row (no repayment on default). Add stochastic interest rates and you get extra terms, but the structure stays the same.
Credit Risk in Convertible Bonds
Wilmott applies these ideas to convertible bonds (CBs). A CB pays coupons like a bond but can be converted into stock, making it sensitive to both rates and equity prices.
Credit risk hits CBs hard. When the stock is very low, conversion is worthless so the CB acts like a plain bond. But low stock price signals a sick company, so default risk is high. Two approaches work: default when stock hits a barrier level (Merton-style), or use an exogenous Poisson process where $p$ depends on the stock price. Both give lower CB values than a no-default model, matching market behavior.
The Hedging Problem
One uncomfortable truth runs through this chapter: you cannot fully hedge credit risk. The portfolio hedges interest rate moves but not the default event itself. This means the expected return framework may be more appropriate than risk-neutral pricing. The distinction between your assessment of default risk and the market’s becomes critical for trading decisions.
This is part of a series covering “Paul Wilmott on Quantitative Finance”. Next up: Credit Derivatives: CDS, CDOs, and the Products That Blew Up.
Previous: Merton Model: Your Company’s Equity Is Just an Option.