Two-state and intensity credit models

CM2 · Financial Engineering & Loss Reserving · 12 min read

Three ways to model a default

The Merton model explains default from the inside: the company's assets fall below its debts and it fails. That is a structural model, and its virtue is that it says why. Its weakness is that the assets are not observable and the spreads it produces are too small, especially at short maturities.

The alternatives give up on explaining and concentrate on fitting.

Reduced-form, or intensity-based, models treat default as an event that arrives at random with some intensity, like a claim in a Poisson process. No balance sheet appears. The intensity is whatever the market price says it is.

Credit-migration models sit between the two. The company occupies one of several credit states — the rating grades, with default as an absorbing state — and moves between them according to a transition matrix. They are the multi-state generalisation, and they are what a portfolio credit model usually uses, because they capture a downgrade as well as a default.

Knowing which family a question is in, and being able to say what each buys and gives up, is most of what a comparison question wants.

The two-state model

The simplest reduced-form model has two states, healthy and defaulted, and default is absorbing — once there, always there. The company leaves the healthy state at a constant rate λ, the default intensity or hazard rate:

P(still healthy at time t) = e^−λt
P(default by time t) = 1 − e^−λt

In words: default arrives like the first event of a Poisson process, so the time to default is exponential with mean 1/λ.

That is the same mathematics as the two-state survival model, and deliberately so: a company defaulting and a life dying are the same problem with different words. λ is a force of mortality, the survival function is the same exponential, and anything that generalises one generalises the other — a time-varying λ(t) for a credit curve, a stochastic λ for correlated defaults.

The second ingredient is the recovery rate: on default, the lender receives a fraction of what was promised rather than nothing. Writing the recovery as a proportion δ of the nominal, a zero-coupon corporate bond of nominal N maturing at T is worth

B₀ = e^−rT[ e^−λTN + (1 − e^−λT)δN ]

In words: discount the two possible payoffs — the full nominal if it survives, the recovery if it does not — weighted by their probabilities.

Running it backwards, which is how it is actually used

Nobody knows λ. What is known is the bond's price, so the model is inverted: assume a recovery rate and solve for the λ that reproduces the observed price. That is calibration, and it is the defining feature of the reduced-form approach — the model is fitted to the market rather than tested against it.

Two consequences follow and both are examinable.

The model cannot be wrong about the price, because the price is an input. So it is useless as evidence that the bond is cheap or dear. What it is good for is translating a price into a quantity that can be compared — across bonds, across maturities, against a rating agency's default statistics.

And λ and δ cannot be separated from one price. Only the product of the intensity and the loss given default is identified: a high intensity with a high recovery fits the same price as a low intensity with a low one. The recovery rate therefore has to be assumed, from the seniority of the debt and from historical recoveries on similar bonds, and the assumption should be stated as an assumption. Separating the two needs more instruments than one.

The spread, and the approximation worth knowing

The credit spread is the extra yield over the risk-free rate:

spread = −(1/T)ln(B₀/N) − r

There is also a short approximation that is used constantly in practice:

spread ≈ λ(1 − δ)

In words: the spread is roughly the default intensity multiplied by the fraction lost when default happens.

It is worth knowing both why that works and why it is only approximate. It works because over a short interval the expected loss rate is the chance of defaulting multiplied by the loss, and a spread is a loss rate. It is approximate because over a long horizon the survival probability compounds, so the average loss rate over the period is below the instantaneous one. The approximation therefore overstates the spread, and by more the longer the bond: in the example below it is 16 basis points high on a five-year bond.

There is one case where it is exact, and it makes a perfect check. With zero recovery the bond is worth Ne^−(r+λ)T, so the spread is exactly λ, at every maturity. If your arithmetic does not reproduce that, something is wrong.

What the model gets right and wrong

Right. It fits observed spreads exactly, by construction. It handles short maturities, where Merton fails badly, because default can arrive at any instant rather than requiring the asset value to drift down. It extends naturally: a term structure of intensities fits a whole credit curve, and a stochastic intensity produces correlated defaults across a portfolio. And it needs no balance-sheet data, so it works for a borrower whose assets nobody can value.

Wrong, or at least silent. It explains nothing. There is no link to leverage, to profitability or to anything a credit analyst would look at, so it cannot say what would happen if the company issued more debt. The intensity is a parameter rather than a consequence. And because it is calibrated to prices, it inherits whatever the market has wrong.

The honest summary is the division of labour: structural models to understand, reduced-form models to price, migration models to manage a portfolio. A good comparison answer says that rather than preferring one.

Risk-neutral and real-world again

An intensity implied from a bond price is a risk-neutral intensity, and it is materially larger than the rate at which such companies actually default. Investors require compensation for bearing default risk, not only for the expected loss, so the price is lower than the expected-loss calculation alone would justify, and the implied intensity is correspondingly higher.

The ratio between the two is substantial — implied intensities are commonly several times historical default rates for investment-grade debt. So an implied intensity is the right input for pricing another instrument on the same name, and the wrong input for a capital model that is meant to describe what will happen. This is the same distinction as on the simulation page, and it is the one most often got wrong in practice.

Where marks are lost

Forgetting the recovery. With a recovery rate, the loss on default is N(1 − δ), not N. Treating default as a total loss overstates the implied intensity substantially.

Using 1 − λT for the survival probability. It is e^−λT. The linear version is a decent approximation for small λT and is noticeably wrong over five or ten years.

Quoting the approximation as the spread. λ(1 − δ) is an approximation. If the question asks for the spread, compute it from the price; quote the approximation alongside as a check, and say which is which.

Discounting the recovery at the corporate yield. Both payoffs are discounted at the risk-free rate, because the probabilities have already done the credit adjustment. Discounting at the risky yield adjusts twice.

Calling the implied intensity a default probability. λ is a rate per unit time; the probability of default by T is 1 − e^−λT. They are close for small λT and diverge: in the example below λ is 5.14% a year and the five-year default probability is 22.7%, not 25.7%.

Questions students ask

Why does the spread fall with maturity in the example, when there is more time to default?

Because a spread is a rate per year, not a total. There is indeed more cumulative default probability over ten years than over one, but it is spread over ten years and the survival probability compounds downwards, so the average annual loss rate falls. A constant intensity therefore produces a gently downward-sloping spread curve. Real investment-grade curves slope upwards, which tells you the constant-intensity assumption is too simple and that λ(t) should rise with term — which is exactly how a real credit curve is fitted.

How is this different from the chain of states in a migration model?

Only in the number of states. The two-state model has healthy and defaulted; a migration model has AAA down to C plus default, with a transition matrix giving the one-year probabilities of moving between them. The two-state model is the migration model with every non-default grade collapsed into one. The cost of the extra states is a matrix of parameters to estimate; the gain is that a downgrade, which changes the bond's value without any default, becomes visible.

Can the intensity be made stochastic?

Yes, and that is how correlated defaults are produced. If every company's intensity depends on a common factor, their defaults become correlated without any direct link between the companies — which is both realistic and the mechanism that makes portfolio credit risk so much worse than the sum of its parts. A constant intensity per name implies independent defaults, and that is the assumption that failed most expensively in 2007 and 2008.

Which model would an actuary use for a bond portfolio?

Usually a migration model for the portfolio and intensity models for individual pricing, with a structural model in the background as a sanity check on whether a spread makes sense given the borrower's balance sheet. The useful answer to this question is that they are complementary and that the choice follows the purpose: explaining, pricing, or capital.

A worked example

A five-year zero-coupon corporate bond with a nominal value of ₹100 trades at ₹64.00. The continuously compounded risk-free rate is 6%. On default, holders are assumed to recover 40% of the nominal, paid at maturity.

The bond and every figure are invented for this example. The recovery rate in particular is an assumption, not an observation.

Step 1: what it would be worth without credit risk

e^{−0.06 × 5} = e^−0.30 = 0.7408182
risk-free value = 100 × 0.7408182 = ₹74.08

It trades at ₹64.00, so the market is charging ₹10.08 for the credit risk — 13.6% of the risk-free value. Note that 64.00 must be below 74.08 and above the recovery's present value of 40 × 0.7408182 = ₹29.63, and it is. A quoted price outside that range would mean the recovery assumption or the risk-free rate is wrong before any model is fitted.

Step 2: imply the default probability

Undo the discounting to get the expected payoff, then solve for the probability.

expected payoff = 64.00 / 0.7408182 = 86.390964
86.390964 = 100(1 − p) + 40p = 100 − 60p
p = (100 − 86.390964) / 60 = 13.609036 / 60 = 0.2268173

So a 22.68% risk-neutral probability of default over five years. The 60 in that calculation is the loss given default, 100 × (1 − 0.40).

Step 3: turn the probability into an intensity

1 − e^−5λ = 0.2268173, so e^−5λ = 0.7731827
5λ = −ln(0.7731827) = 0.2572402
λ = 0.0514480 a year

An intensity of 5.14% a year, with an expected time to default of 1/λ = 19.44 years.

Compare 5.14% a year with the 22.68% five-year probability. Five times 5.14% is 25.72%, which is not 22.68% — the difference is the compounding of survival, and reporting the intensity as though it were an annual probability would overstate the risk by three percentage points over five years.

The check: reprice the bond from λ

Nothing below reuses the price.

B₀ = 0.7408182 × [0.7731827 × 100 + 0.2268173 × 40]
= 0.7408182 × [77.31827 + 9.072692]
= 0.7408182 × 86.390962 = ₹64.000000

Exactly the observed price. That is what calibration means and it is also the warning that goes with it: the fit is guaranteed, so it is no evidence that the model is right.

Step 4: the credit spread, exactly and approximately

yield = −(1/5)ln(64.00/100) = −0.2 × (−0.4462871) = 0.0892574
spread = 0.0892574 − 0.06 = 0.0292574, that is 292.57 basis points

And the approximation:

λ(1 − δ) = 0.0514480 × 0.60 = 0.0308688 = 308.69 basis points

The approximation is 16.11 basis points high, or 5.5% too large. It overstates because it applies the instantaneous loss rate to the whole five years, ignoring that the bond has to survive to be at risk later. Over one year the two agree far more closely; over ten they diverge further. Quote the exact figure and use the approximation to check the order of magnitude.

The spread curve a constant intensity implies

Maturity	Survival probability	Price	Spread
1 year	0.949853	91.3429	305.50 bp
3 years	0.856977	76.3593	299.07 bp
5 years	0.773183	64.0000	292.57 bp
10 years	0.597812	41.6376	276.17 bp

The spread falls with maturity, from 306 to 276 basis points, even though the cumulative chance of default rises from 5.0% to 40.2%. That is not an error: a spread is a rate per year and the annual loss rate declines as survival compounds.

It is also a finding about the model rather than about the bond. Real investment-grade credit curves slope upwards, so a constant intensity is the wrong shape, and fitting a real curve needs λ(t) rising with term. Noticing that the model's implication contradicts the market is exactly the kind of comment a question on model limitations is asking for.

The exact check: zero recovery

Set the recovery to nothing and keep the same intensity. The bond becomes 100e^−(r+λ)T, so its spread must be λ exactly, at every maturity.

price = 100 × e^−0.30 × e^−0.2572402 = ₹57.278785
spread = −(1/5)ln(0.57278785) − 0.06 = 0.0514480
λ = 0.0514480

Identical to seven decimal places. The approximation that was 16 basis points out at a 40% recovery is exact at a zero recovery, which locates precisely where the error comes from: it is the recovery payment, which arrives at maturity and so is discounted differently from the loss it replaces.

Reading the answer against a rating

A 5.14% annual risk-neutral intensity is high — the sort of figure implied by sub-investment-grade debt. Before concluding that the company is in trouble, remember what the figure is: a risk-neutral intensity, which includes the compensation investors demand for bearing default risk, not just the expected loss. The historical default rate for a borrower of this kind would typically be a good deal lower. So the right reading is that the market is charging 293 basis points, and that this corresponds to a 5.14% risk-neutral intensity on a 40% recovery assumption — three statements that are all the same fact, and none of which is a forecast of how often such companies fail.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Two-state and intensity credit models: backing a default rate out of a bond price