Hazard rate: Definition & Examples

The hazard rate is the instantaneous rate of default conditional on having survived so far. It converts a default-time model into the survival probabilities used in credit-risk measurement and pricing. A hazard of 0.02 per year does not mean an exact 2% probability of default over the next year.

Also called a failure rate, it describes how quickly probability leaves the surviving population. In a simple deterministic credit model it is often called default intensity. With richer information, default intensity can be a stochastic process rather than a fixed function of elapsed time.

The denominator is whoever has survived

Let $\tau$ be the default time, measured in years, with no default at time zero. Define the survival function $S(t)=\mathbb{P}(\tau>t)$ . At a time where the following limit exists and $S(t)>0$ ,

\lambda(t)=\lim_{h\downarrow0} \frac{\mathbb{P}(t<\tau\le t+h\mid\tau>t)}{h}.

For a small positive interval

h

, the conditional default probability is approximately

\lambda(t)h

. The rate has units per year; the probability is dimensionless. A rate can exceed one per year without violating a probability bound. For an absolutely continuous default-time law with density

f

\lambda(t)=\frac{f(t)}{S(t)}.

The density measures default mass per unit time relative to the original population; dividing by survival changes the reference population to those still at risk. In this continuous model, default at any one specified instant has probability zero, despite a positive hazard there. NIST's hazard definition makes the conditional rate and its time units explicit.

Accumulate the rate, then convert it

For this absolutely continuous model, the cumulative hazard is

H(t)=\int_0^t\lambda(u)\,du=-\log S(t).

It is dimensionless, nonnegative and nondecreasing. It is not a cumulative default probability: that quantity is

F(t)=1-S(t)

. The conversion is

S(t)=e^{-H(t)},\qquad F(t)=1-e^{-H(t)}.

The area under the hazard curve drives survival. Replacing the exact default probability by $H(t)$ is a small-cumulative-hazard approximation; it overstates it whenever $H(t)>0$ .

Suppose a synthetic model stipulates a constant hazard

\lambda=0.02

per year. Its default time has an exponential distribution. No issuer data or market calibration is assumed.

Horizon from today	Cumulative hazard $H(t)$	Survival $S(t)$	Default by horizon $F(t)$
0 years	0	100%	0%
1 year	0.02	98.0199%	1.9801%
5 years	0.10	90.4837%	9.5163%

At five years, $S(5)=e^{-0.10}\approx0.904837$ and $F(5)=1-e^{-0.10}\approx0.095163$ . Multiplying the annual hazard by five gives the cumulative hazard 0.10, which corresponds to a 10% linear approximation, rather than the exact five-year PD.

Synthetic hazard 0.02 per year stays flat; cumulative hazard reaches 0.1000 and default probability 0.095163 at five years. Survival is 0.904837.

Original deterministic model illustration. The rate has units per year; its integral and the default probability are dimensionless and differ even when their curves look close.

Conversely, if a one-year PD is specified as exactly 2%, the constant hazard that reproduces it is

-\log(0.98)\approx0.0202027

per year. A quoted annual PD and an annual hazard therefore require a conversion even when their rounded values look similar.

Default during a later year

For

0\le s<t

with

S(s)>0

, conditional probability gives

\mathbb{P}(s<\tau\le t\mid\tau>s) =1-\frac{S(t)}{S(s)} =1-\exp\!\left(-\int_s^t\lambda(u)\,du\right).

In the flat-hazard example, default during year five conditional on surviving four years has probability

1-e^{-0.02}\approx1.9801\%

. From today's starting point, the unconditional probability of default during that same year is

\mathbb{P}(4<\tau\le5)=S(4)-S(5)\approx1.8279\%.

The smaller number accounts for the possibility of default before year five begins. Constant hazard makes equal-length conditional intervals equally risky; it does not make the unconditional default mass in every year equal. With a varying hazard, the conditional interval probability depends on the entire rate curve over that interval.

State the probability measure

A physical hazard under

\mathbb{P}

describes default likelihood under a real-world statistical model. A risk-neutral hazard under a pricing measure

\mathbb{Q}

is used to value discounted contingent cash flows. The same rate-to-survival formulas apply under each measure, using that measure's hazard and probabilities consistently.

A hazard inferred from bond or CDS prices depends on a pricing model, recovery assumptions and contract cash flows. It is not automatically a forecast of realised defaults. The synthetic 0.02 rate above demonstrates the conversion under whichever measure is stipulated; it establishes neither a market calibration nor a physical estimate.

For the next step, Reduced-Form Models: Hazard Rates and Risk-Neutral Default Probabilities develops conditional survival and simplified bond/CDS pricing. The public page is a preview; the full lesson requires membership.

Sources

NIST/SEMATECH, Failure (or hazard) rate, §8.1.2.3: conditional rate, cumulative hazard and survival identities.
NIST/SEMATECH, Exponential, §8.1.6.1: constant hazard and exponential survival.
McNeil, Frey and Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, revised edition, Princeton University Press, 2015, §10.4.1, pp.391–393; §10.4.2, pp.394–395: default-time hazards, conditional survival and risk-neutral pricing.

The denominator is whoever has survived

Accumulate the rate, then convert it

Default during a later year

State the probability measure

Sources

Related terms

Conditional probability

Credit default swap

Default probability

Exponential distribution

Probability density function

Risk-neutral probability