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Define the hazard rate, then apply it to find the unconditional probability that a credit asset defaults.

FRM® Part I · Valuation and Risk Models · External and Internal Credit Ratings · FRM1.VR.4.c

The idea

The hazard rate, λ, is the instantaneous default intensity. The quantity λ × dt is the probability of default in the next short interval dt, conditional on survival to that moment. With a constant hazard rate, the survival probability to time t is S = e^(−λt). The cumulative default probability is then PD = 1 − e^(−λt). Between times t1 and t2, the unconditional probability of default is PD = e^(−λt1) − e^(−λt2). If the hazard rate changes by period, add λ × length across the periods in the exponent. A common market approximation is λ = spread ÷ (1 − recovery rate). The usual errors are using λt as the default probability, which overstates it, and confusing the cumulative probability to a date with the probability in one particular year.

Formulas

Survival probability with a constant hazard rate

S(t) = e^(−λt)

λ
the constant hazard rate per year
t
the time in years

Cumulative PD to t is 1 − e^(−λt). Using λt as the default probability overstates it; with changing hazard rates, add λ × length across periods in the exponent.

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

The fall in survival over the period. Do not confuse it with the cumulative probability of default to t₂.

Hazard rate implied by a credit spread

λ = spread ÷ (1 − R)

spread
the CDS spread or bond yield spread over the risk-free rate
R
the expected recovery rate

An approximation that treats the whole spread as compensation for expected default loss. The result is a risk-neutral rate, higher than real-world default frequencies.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement defines the hazard rate, lambda, of a credit asset?

  1. Lambda is the probability today that the asset defaults at any time before maturity.Why not: Describes the cumulative default probability, which is 1 - exp(-lambda x t) under a constant hazard.
  2. Lambda x dt is the chance of default in the next short interval dt, given survival so far. - correct
  3. Lambda is the fraction of the exposure lost when default occurs.Why not: Describes loss given default, which is 1 minus the recovery rate.
  4. Lambda is the annual probability that the rating will fall by at least one notch.Why not: Describes a downgrade probability from a transition matrix, not an instantaneous default intensity.

The hazard rate is the conditional default intensity: given survival to t, the probability of default in the next dt is about lambda x dt.

A harder one

A firm has a constant hazard rate of 2% per year. The probability that it defaults at some point within five years is closest to:

  1. 10.00%Why not: Multiplies lambda by time (2% x 5), which ignores that default probabilities cannot be added over time when survival matters.
  2. 90.48%Why not: Reports the five-year survival probability, exp(-0.10), instead of the default probability.
  3. 9.52% - correct
  4. 1.76%Why not: Reports only the unconditional probability of default in year 5, exp(-0.08) - exp(-0.10).

Cumulative default = 1 - exp(-0.02 x 5) = 1 - exp(-0.10) = 1 - 0.9048 = 9.52%.

The hardest tier

An issuer has a constant hazard rate. Its probability of default within five years is 20%. The unconditional probability that it defaults during year 4 (between t = 3 and t = 4) is closest to:

  1. 4.5%Why not: Stops after backing out lambda = -ln(0.8)/5 = 4.46% and reports it as the year-4 probability.
  2. 16.3%Why not: Reports the cumulative default probability through year 4, 1 - 0.8^(4/5), not the single year.
  3. 3.8% - correct
  4. 4.0%Why not: Divides the 20% cumulative probability by five, assuming evenly spread unconditional probabilities.

Survival to year 5 is 0.80, so lambda = -ln(0.80)/5 = 0.04463. Year-4 unconditional PD = exp(-3 lambda) - exp(-4 lambda) = 0.8746 - 0.8365 = 3.8%.

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