Hazard Rates And Survival Probabilities

Ask "what's the probability this company defaults sometime in the next 10 years" and you get one number for a long stretch of time. Ask instead "given the company is still solvent right this instant, what's the probability it defaults in the next split second" and you get a different kind of number entirely — a rate, not a probability, because it's measured per unit time and can be compared moment to moment. That's the hazard rate: exactly like a smoke detector's alarm probability isn't "the chance the building ever burns down," it's "the chance of an alarm going off right now, given no alarm has gone off yet." Chain that instantaneous, conditional rate together continuously over time and you can reconstruct the full probability of surviving to any horizon.

From hazard rate to survival probability

If $h(t)$ is the hazard rate at time $t$ (the instantaneous default intensity, conditional on survival to $t$), the probability of surviving past time $T$ is

$$Q(T) = \exp\left(-\int_0^T h(t)\, dt\right).$$

In plain English: survival probability decays exponentially with the accumulated hazard, not the hazard at any single point — a company with a low, steady hazard rate over a long horizon can end up with the same survival probability as one with a brief but intense spike, because it's the total area under the hazard curve that matters, not its shape. When $h(t) = h$ is constant, this simplifies to $Q(T) = e^{-hT}$, the familiar constant-hazard case.

Worked example 1 — constant hazard rate

A company has a constant hazard rate of $h = 2\%$ per year (a rough proxy for a BBB-rated credit). Probability of surviving 5 years: $Q(5) = e^{-0.02 \times 5} = e^{-0.10} = 0.9048$, about 90.5%. Probability of surviving 10 years: $Q(10) = e^{-0.02 \times 10} = e^{-0.20} = 0.8187$, about 81.9%. Note this isn't $2\% \times 10 = 20\%$ cumulative default probability exactly — it's $1 - 0.8187 = 18.13\%$, slightly less than the naive linear estimate, because the hazard only applies to the shrinking pool of survivors at each instant, not the original full population.

Worked example 2 — a hazard rate that jumps

Suppose the market suddenly reprices the same company's credit risk: hazard rate stays at 2% for the first 3 years, then jumps to 6% for years 3 through 5 (reflecting new information — say, a ratings downgrade). Survival to year 3: $e^{-0.02 \times 3} = e^{-0.06} = 0.9418$. Survival to year 5, building on that: $0.9418 \times e^{-0.06 \times 2} = 0.9418 \times e^{-0.12} = 0.9418 \times 0.8869 = 0.8352$. Compare to the constant-2%-forever case at year 5, $e^{-0.10} = 0.9048$ — the jump in hazard rate after year 3 drags 5-year survival probability down from 90.5% to 83.5%, a nearly 7-percentage-point swing driven entirely by 2 years of elevated hazard.

Default timing under a hazard-rate model behaves like the arrivals of a Poisson-type process with a possibly time-varying rate — the plot above shows how a rate parameter shapes an arrival-count distribution, the same intensity idea that drives when (not if) a hazard model expects a default to land.

What this means in practice

Every CDS and corporate bond desk works backward from market prices to a hazard-rate curve (the reverse of the calculation above), because it's the standard, curve-like way to represent an entire term structure of default risk in one object, comparable across issuers and directly usable for pricing any instrument whose payoff depends on default timing.