Charta is on Google Play: practise on the go. Get the app
Charta

Pick out which default probabilities are used when a CDS is valued.

FRM® Part II · Credit Risk Measurement and Management · Credit Derivatives · FRM2.CR.13.c

The idea

Two families of default probabilities exist. Risk-neutral (market-implied) probabilities are backed out of bond yields or CDS spreads and include the premium investors demand for bearing default risk. They are higher than real-world probabilities, which are estimated from historical default experience of rated firms. A CDS is valued with risk-neutral probabilities so that the model reproduces market prices. Real-world probabilities belong in risk management work such as credit VaR, economic capital and scenario analysis, where you want the expected frequency of actual losses. A useful approximation links the spread to the hazard rate λ and the recovery rate R. Spread = λ × (1 − R). So the hazard rate is about the spread divided by one minus the recovery rate, in annual terms. Over t years, the cumulative probability of default that follows from this hazard rate is PD = 1 − exp(−λt). Never price with historical probabilities, or measure capital with market-implied ones.

Formulas

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

Gives a risk-neutral intensity, right for pricing a CDS and wrong for capital. Use real-world intensities from default history for loss estimates.

Cumulative default probability from a constant hazard rate

PD = 1 − e^(−λT)

λ
the constant annual hazard rate
T
the horizon in years

Risk-neutral if λ came from spreads. Reporting e^(−λT), the survival probability, is the other slip.

A practice question

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

Which default probabilities should be used to value a CDS?

  1. Real-world probabilities estimated from the historical default rates of rating agencies.Why not: Real-world probabilities suit risk management; a price consistent with the market needs market-implied (risk-neutral) probabilities.
  2. The default probability of the protection buyer, since the buyer is the party making the spread payments.Why not: Confuses whose default matters: the payoff is triggered by the reference entity's default, not the buyer's.
  3. The higher of the real-world and the risk-neutral estimates, so that the valuation is conservative.Why not: Valuation relies on one consistent measure that reproduces market prices; picking the higher figure is not a pricing method.
  4. Risk-neutral probabilities, implied from market prices such as bond yields and CDS spreads. - correct

A CDS price must be consistent with other traded credit instruments, so hazard rates are backed out of market prices. These are risk-neutral probabilities and include a premium for bearing default risk.

A harder one

A five-year CDS on a company is quoted at 180 bp and the expected recovery rate is 40 percent. Using the approximation that spread = hazard rate x (1 - R), the average risk-neutral annual hazard rate is closest to:

  1. 4.50%Why not: Divides by the recovery rate instead of the loss given default (1 - R).
  2. 1.08%Why not: Multiplies the spread by (1 - R) instead of dividing; the spread is already loss-weighted.
  3. 3.00% - correct
  4. 1.80%Why not: Equates the hazard rate with the spread, ignoring that only (1 - R) of exposure is lost on default.

Hazard rate = spread / (1 - R) = 1.80% / 0.60 = 3.00% per year.

The hardest tier

A bond yields 130 bp over the risk-free rate and the expected recovery rate is 40 percent. Historical data for this rating show a real-world default intensity of 0.60 percent per year. Treating the whole spread as compensation for default risk, the expected annual excess return over the risk-free rate from holding the bond is closest to:

  1. 94 bp - correct
  2. 70 bpWhy not: Subtracts the default intensity without applying the loss given default; the expected loss rate is hazard x (1 - R).
  3. 106 bpWhy not: Uses the recovery rate in place of loss given default when computing the expected loss.
  4. 166 bpWhy not: Adds the expected loss to the spread; the expected loss is deducted from the promised spread.

Expected loss rate = 0.60% x 0.60 = 36 bp. Excess return = 130 - 36 = 94 bp, the reward for bearing default risk.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Credit Derivatives