BT FRM

Assume the following performance for a bond cohort:

Year T	Value of bonds outstanding at the beginning of Year T	Dollar value of bonds defaulted on during Year T
1	$1,000	$45
2		$55
3		$80

Questions:

• Calculate the marginal mortality rate (MMR) in year 3 for the following class of issuers
• Calculate the cumulative mortality rate (CMR) over the three year period

Answers:

According to Caouette, Altman et al:

The individual mortality rate [MMR(t)] of bonds in a specific rating class for each year (marginal mortality rate or MMR) is given by:
MMR(t)= Total value of defaulting debt in the year(t) / Total value of the population of bonds at the start of the year(t)

The cumulative mortality rate (CMR) is measured over a specic time period (1, 2, ......, T years) by subtracting the product of the surviving population of each of the previous years from one (1.0); i.e.,
CMR (t) = 1 - [Survival rate, year t] * [Survival rate, year t-1] * ... * [Survival rate, year 1]

Here are the calculations in a simple EditGrid spreadsheet.
Note the CMR is calculated per above, but also equals: cumulative defaults ($180) divided by $1,000 = 18%

D. Harper note:
MMR/CMR is analogous to Saunder’s marginal default probability (1 - p) and cumulative default probability (Cp):
Saunders’ (1-p) is an ex ante probability of default; MMR is an (ex post) realized default (mortality)
Saunders’ Cp is an ex ante cumulative probability of default; CMR is the realized default (mortality)