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This section shows how to model the term structure of historical default rates from historical raw data. The technique allows us to assign probabilities of default at various forward periods. As mentioned above, there are various default rates, arithmetic or value weighted. The first step is to define the number of firms surviving at the beginning of the period. Statistics follow from this number.

Tables from rating agencies provide cumulative default probabilities from inception of a cohort up to any future date. A cohort groups all rated firms at the beginning of the observation period. We use here only one cohort, created at a past date 0. In fact, there are as many cohorts as there are years of observation.

The following presentation ignores some technicalities, well documented in technical notes of rating agencies. These include rating withdrawals and variations of defaults across cohorts due to changing economic conditions. Rating withdrawals are not defaults and require adjustments to use the actual numbers of rated firms between any two dates. In addition, some cohorts were created at the beginning of an economic cycle, some at a peak cycle and others at the end of a cycle. Figures vary from one cohort to another, reflecting the varying economic conditions across cohorts. We assume that all firms having a rating at date 0 still have one at the current date. We focus only on the calculation of marginal and cumulative default rates over a given population of firms. Finally, we ignore value weighting or issues that defaulted, and limit the presentation to arithmetic calculations.

For the cohort created in 0, counting a number N(0) of firms, we observe defaults every year. The arithmetic default rate at any year t, beginning at date t and ending at date t + 1, compares the count of defaults between t and t + 1 to the original number of firms within the cohort formed at date 0 and the surviving firms of this cohort at t .We use the following notation: capital letters designate numbers and lowercase letters are rates or percentage ratios. S(0, t) is the number of firms surviving at date t. S(0, 0) = N(0) is the original number of firms of a cohort created at date 0. From these definitions, we derive the cumulative count of defaults from 0 up to t,or D(0,t), and the corresponding default rate d(0, t), as well as the number of surviving firms at t,or S(0, t) and the survival rate s(0, t).

The cumulative number of surviving firms at t is S(0,t) = N(0) — D(0,t). The number of firms S(0, t) surviving at t is lower than the original number of firms N(0). The total number of defaults is the summation of all defaults from 0 to the end of period t,or D(0,t) = N(0) — S(0,t). The ratio of the surviving firms at t, S(0, t), to the original number in the cohort, N(0), is the cumulative survival rate from 0 to t, a percentage: s(0,t) = S(0,t)/N(0) = [N(0) — D(0,t)]/N(0) = 1 — D(0,t)/N(0). Dividing D(0, t) by N(0) provides the cumulative default rate at t: d(0,t) = [N(0) — S(0,t)]/N(0) =

1 — s(0,t).

D(t, t + k) is the number of defaults between dates t and t + k. It is a forward count of defaults. For k = 1,D(t,t + 1) is the number of defaults within the year starting at date t and ending at t + 1, or the marginal rate of default from t to t + 1. The marginal default rate between t and t + 1, or d(t, t + 1), is the percentage of the number of firms defaulting between t and t + 1, or D(t, t + 1), to the number of surviving firms at t, S(0, t). The number of surviving firms at t is S(0,t) = N(0) — D(0,t). The count of defaults between dates t and t + 1is D(t, t + 1) and the forward, or marginal default rate at t up to t + 1is d(t, t + 1) = D(t, t + 1)/S(0,t) = D(t, t + 1)/[N(0) — D(0,t)].

Historical survival rates, marginal or cumulated, or default rates, marginal or cumulated, can serve as survival or default probabilities when looking forward. When looking at historical data, we obtain historical estimates of probabilities. These are estimates of natural or real probabilities.