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State the meaning of recovery rate and default rate, and tell a dollar default rate from an issue default rate.

FRM® Part I · Financial Markets and Products · Corporate Bonds · FRM1.FM.17.h

The idea

The default rate measures how often issuers fail. The recovery rate measures how much of the claim is recovered afterwards. Recovery rate is the value recovered as a percentage of par, typically measured by the market price shortly after default. Loss = 1 − recovery rate. The issue default rate is the number of issues that defaulted divided by the number of issues outstanding at the start of the period. The dollar default rate is the par value of defaulted debt divided by the par value outstanding, so it weights large issues more heavily. The two differ whenever defaulting issues are larger or smaller than the average issue. Expected loss = default probability × (1 − recovery rate) × exposure. When averaging recovery across several defaults, weight by par amount, not by count.

Formulas

Issue default rate

issue default rate = Nd ÷ N

Nd
the number of issues that defaulted in the period
N
the number of issues outstanding at the start of the period

Count-based, so every issue weighs the same whatever its size.

Dollar default rate

dollar default rate = Vd ÷ V

Vd
the par value of debt that defaulted in the period
V
the par value outstanding at the start of the period

Par-weighted, so it exceeds the issue rate when the defaulters are larger than average.

Loss given default from the recovery rate

loss given default = 1 − RR

RR
the recovery rate, value recovered as a share of par

Recovery is usually measured by the market price shortly after default. Average recoveries by par amount, not by count.

A practice question

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

The recovery rate on a defaulted bond is best defined as:

  1. The annual probability that the issuer will default.Why not: Defines the default rate, not recovery.
  2. One minus the percentage of par recovered.Why not: Defines the loss given default, the complement of the recovery rate.
  3. The coupon rate on the bond after default.Why not: Confuses recovery with the post-default coupon.
  4. The value recovered after default, as a percentage of par. - correct

Recovery rate = recovery value / par. Loss given default = 1 - recovery rate.

A harder one

In a year, 400 bond issues with total par value USD 60 billion were outstanding at the start. 12 issues with total par USD 2.4 billion defaulted. What is the dollar default rate?

  1. 4.0% - correct
  2. 3.0%Why not: Computes the issue default rate (12/400), not the dollar default rate.
  3. 0.6%Why not: Divides the number of defaults by the par amount in billions, mixing counts and dollars.
  4. 20.0%Why not: Divides the defaulted par by the average par of 12 defaults, not by the par outstanding.

Dollar default rate = 2.4 / 60 = 4.0%. The issue default rate would be 12 / 400 = 3.0%.

The hardest tier

A rating cohort has 250 issues with USD 40 billion of par at the start of the year. Five issues with total par USD 1.8 billion default. Which statement is correct?

  1. The issue default rate is 4.5% and the dollar default rate is 2.0%, which shows the defaulters were smaller than average.Why not: Swaps the two rates, reading the issue rate as the dollar rate.
  2. The issue rate is 2.0% and the dollar rate is 4.5%, so defaulters were larger than average. - correct
  3. Both rates are 2.0%, because five defaults out of 250 issues is 2.0% on either basis.Why not: Uses counts for both and ignores par values.
  4. The issue default rate is 2.0% and the dollar default rate is 0.7%, so defaulters were smaller than average.Why not: Divides defaulted par by par per issue incorrectly; the dollar rate is 1.8 / 40.

Issue rate = 5 / 250 = 2.0%. Dollar rate = 1.8 / 40 = 4.5%. The dollar rate exceeds the issue rate, so the defaulted issues averaged USD 360 million against USD 160 million for all issues.

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