Rating-Based Default Probability Term Structures

Why Default Probability Term Structures Need Care

A one-year default probability answers a specific question: what is the chance that an obligor in a given rating defaults during the next year? A term structure answers related questions at several future horizons. Those answers are not interchangeable. The probability of having defaulted by the end of year 3 is cumulative, while the probability of default during year 3 among those that survived the first two years is conditional. Confusing these measures can materially distort a credit-risk assessment.

A rating-based term structure follows an initial rating cohort as obligors migrate among ratings and may eventually default. Its calculation uses the annual transition matrix and, when default is an absorbing state, the probability of occupying that state at each horizon. The key consistency check is that cumulative default probability and survival probability sum to one.

From an Annual Matrix to a Term Structure

Assume a time-homogeneous Markov model: the same annual transition matrix \(P\) applies in every year, and the next year's rating depends on the current rating rather than on the full migration history. List the nondefault ratings and then the default state in a fixed order. If the initial rating is \(i\), let \(e_i\) be the row vector with 1 in position \(i\) and 0 elsewhere. The distribution over all states at the end of year \(t\) is \(e_iP^t\).

With default absorbing, the entry of \(e_iP^t\) corresponding to \(D\) is the cumulative default probability through year \(t\). Write this as \(C_i(t)\), and define \(S_i(t)\) as the probability of surviving without default through that year. Then:

The marginal probability of default during year \(t\) is the increase in cumulative probability from the previous horizon. The conditional probability for that year divides this increase by the probability of surviving to the start of the year:

Here \(C_i(0)=0\) and \(S_i(0)=1\) when the initial obligor is not in default. The quantity \(h_i(t)\) is sometimes called a cohort hazard or conditional default rate. It is conditional on the initial cohort surviving to year \(t-1\), not on being in the same rating as at the start.

The cohort hazard can also be interpreted using the ratings occupied by survivors. Let \(q_{t-1}(j)\) be the probability that an obligor starting in \(i\) is in nondefault rating \(j\) at the end of year \(t-1\), without having defaulted. The total probability of survival is \(S_i(t-1)=\sum_j q_{t-1}(j)\). Conditional on survival, the share in rating \(j\) is \(q_{t-1}(j)/S_i(t-1)\). Therefore:

This is a weighted average of the one-year default probabilities for the ratings held by surviving members of the original cohort. It is generally not equal to the initial rating's one-year default probability after the first year, because the survivors may have migrated.

Why the Matrix Calculation Works

The multiplication \(P^2\) sums the probabilities of all possible intermediate ratings. In particular, the probability of moving from \(i\) to \(j\) in two years is \(\sum_k P_{ik}P_{kj}\): each product is the probability of the path through intermediate state \(k\), and the paths are mutually exclusive. Repeating this argument gives \(P^t\) for \(t\) annual transitions.

For the default column, the absorbing-state condition makes the entry at year \(t\) the probability of having defaulted on or before year \(t\), not merely the probability of a new default in that year. Every path that entered \(D\) earlier stays there. Thus the difference between the default-column entries at two consecutive horizons isolates the probability of first entering default during the intervening year. Dividing that difference by prior survival conditions on the cohort still being at risk at the start of that year.

Worked Term-Structure Calculations

Interpretation and Common Exam Traps

A term structure should be read with its conditioning event stated explicitly. The cumulative probability \(C_i(t)\) is unconditional on survival to earlier years: it describes default by the horizon for the original cohort. The conditional probability \(h_i(t)\) is conditional on survival to the beginning of that year. The marginal probability \(m_i(t)\) is an unconditional probability of first default during that particular year. All three can be useful, but they answer different questions.

A frequent error is to subtract cumulative probabilities and report the result as a conditional probability. For example, in Worked Example 1 the year-2 increment is \(0.120-0.050=0.070\). Reporting 0.070 as the conditional year-2 default probability is wrong because it is still measured against the original cohort. The correct conditioning divides by year-1 survival:

Another trap is to use the initial rating's one-year default probability as the cohort's conditional probability in later years. By then, surviving obligors can occupy several nondefault ratings. Their conditional default probability is the weighted average over their current rating mix, as shown above. A third error is to multiply annual transition matrices in reverse chronological order when the matrices change by year. Finally, a matrix with rows that do not sum to 1, or a default state that is not absorbing, requires investigation before interpreting its default column as cumulative default. If default is not absorbing, cumulative default must instead be calculated as the probability of first entering default by the horizon.

These distinctions are central to rating-based default probability term structures and are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus. For any reported horizon, show the starting rating, the time convention, whether the probability is cumulative, marginal, or conditional, and the survival check. That makes the result interpretable and exposes common calculation errors.