Reduced-Form Default Intensity Models

A light bulb doesn't have a scheduled death date. It has a failure rate — some chance per hour of burning out, given that it hasn't already. Bond default is priced the same way. Instead of asking why a company would default (its assets falling below its debts, the approach behind structural models like Merton's), a reduced-form model just asks: at any moment, what is the rate at which default could strike out of nowhere?

That rate is the default intensity, usually written $\lambda$. It's the credit-market cousin of a Poisson arrival rate — the same maths used for radioactive decay or customers walking into a shop.

The survival probability

If $\lambda$ is constant, the probability a firm has not defaulted by time $t$ is:

$$S(t) = e^{-\lambda t}$$

In words: survival probability decays exponentially, and it decays faster the higher the intensity. This is the same formula that describes how many atoms of a radioactive sample remain after time $t$ — default risk behaves like decay, not like a scheduled event.

The probability of defaulting sometime before $t$ is just $1 - S(t)$, and note this is not simply $\lambda \times t$ except for very small $t$ — that's the classic mix-up.

Worked example

A firm's CDS spread implies a constant default intensity of $\lambda = 2\%$ per year. What is the probability it survives 5 years, and the probability it defaults at some point in those 5 years?

1. Survival. $S(5) = e^{-0.02 \times 5} = e^{-0.10} \approx 0.905$, so about 90.5%.
2. Cumulative default probability. $1 - 0.905 = 0.095$, about 9.5% — noticeably less than the naive $2\% \times 5 = 10\%$, because each year's 2% only applies to firms still alive.

Drag the intensity ($\lambda$) in the explorer above and watch how the probability mass of "number of default-like jumps by a given time" shifts — a higher intensity front-loads probability toward earlier, more frequent jumps, exactly the mechanism priced into a CDS spread.

Where this is actually used

Reduced-form models are the workhorse for pricing single-name CDS and corporate bonds because they don't need a view on a firm's balance sheet — a trader can back $\lambda$ straight out of observed market spreads (bootstrapping a credit curve, one tenor at a time) and use it immediately to price other instruments on the same name. Structural models need equity volatility and leverage as inputs and are better suited for a fundamentals-driven view of why spreads should move; reduced-form models are better suited for taking spreads as given and pricing everything consistently off them, which is why trading desks lean on the reduced-form approach for day-to-day marking and hedging.

Term structures matter too: $\lambda$ doesn't have to be constant. A distressed firm often has a high near-term intensity that falls once it survives an immediate crunch, producing the same kind of curve shapes seen in the credit-curves discussion.