Introduce the hazard rate, then employ it to form probability functions for when default occurs and to derive default probabilities that are conditional as well as unconditional.
The idea
The hazard rate, or default intensity, is written λ(t). It is defined so that λ(t) × dt is the probability of default between t and t + dt, conditional on no default before t. With a constant hazard rate, the default probability by time t is one minus survival. The survival probability is Survival = exp(−λ × t). The unconditional probability of default between times t1 and t2, as seen today, is Unconditional = exp(−λ × t1) − exp(−λ × t2). The conditional probability of default in the next period, given survival to t1, is Conditional = 1 − exp(−λ × Δt). When the hazard rate varies, replace λ × t by the integral of the hazard rate, called the cumulative hazard. The common mistake is mixing the two kinds of probability. Conditional probabilities are larger than unconditional ones for the same year, because they are divided by the survival probability. The hazard rate itself is not a probability.
Formulas
Survival probability with a constant hazard rate
S(t) = e^(−λt)
- λ
- the constant hazard rate per year
- t
- the time in years
One minus this is the cumulative default probability. With a varying hazard, replace λt by the cumulative hazard.
Unconditional default probability between two dates
PD(t₁ to t₂) = e^(−λt₁) − e^(−λt₂)
- λ
- the constant hazard rate
- t₁, t₂
- the start and end of the period, in years
As seen today. It is smaller than the conditional probability for the same period, which is divided by survival to t₁.
Conditional default probability over the next period
conditional PD = 1 − e^(−λ × Δt)
- λ
- the constant hazard rate per year
- Δt
- the length of the period, in years
Given survival to its start; with a constant hazard it is the same every year. The hazard rate itself is not this probability.
A practice question
Which statement defines the hazard rate lambda(t)?
- lambda(t) times a short interval dt is the probability of default between t and t + dt, conditional on no default before t. - correct
- It is the probability of surviving from today until time t.Why not: That is the survival probability Q(t), not the hazard rate.
- It is the cumulative probability of default from today until time t.Why not: That is the cumulative default probability, 1 - Q(t).
- It is the unconditional probability of default between t and t + dt, as seen from today.Why not: Omits the conditioning on survival to t; that is the default density, lambda(t) x Q(t).
The hazard rate is a conditional default intensity; the unconditional probability multiplies it by the survival probability.
A harder one
A firm has a constant hazard rate of 3% per year. As seen today, what is the probability that it defaults during year three (between t = 2 and t = 3)?
- 2.96%Why not: Gives the conditional probability of default in year three given survival to t = 2.
- 8.61%Why not: Gives the cumulative probability of default by year three.
- 2.78% - correct
- 3.00%Why not: Treats the hazard rate as the default probability in the year.
Unconditional PD = exp(-0.06) - exp(-0.09) = 0.94176 - 0.91393 = 2.78%; the conditional figure is 1 - exp(-0.03) = 2.96%.
The hardest tier
A firm's hazard rate is 1.5% in year one, 2.5% in year two and 4.0% in year three. As seen today, what is the probability that it defaults during year three?
- 3.77% - correct
- 3.92%Why not: Gives the conditional year-three probability given survival to the end of year two.
- 7.69%Why not: Gives the cumulative probability of default by the end of year three.
- 2.49%Why not: Applies the three-year average hazard rate to every year instead of the year-specific rates.
Cumulative hazard is 4.0% at t = 2 and 8.0% at t = 3, so unconditional PD = exp(-0.04) - exp(-0.08) = 3.77%.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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