Why the Horizon Matters
A statement such as “the default probability is 4%” is incomplete unless it specifies over what period and, where relevant, conditional on what information. A 4% probability over one year is not the same quantity as a 4% probability over five years. Credit risk measures must therefore be aligned to the horizon used for the decision or comparison.
This tutorial focuses on the probability that a defined credit event occurs by a specified time, and the complementary probability that it has not occurred by then. It also shows how to build a multi-period probability from conditional probabilities for successive periods. The calculations concern the event definition and observation horizon; they do not require assumptions about the size of any loss if default occurs.
The phrase “survives through \(t\)” means that default has not occurred at or before \(t\). Thus default in the interval up to and including \(t\) is counted in \(F(t)\), while survival is represented by \(T>t\). Consistent treatment of the horizon boundary matters when probabilities come from data recorded at specific dates.
A cumulative probability is unconditional from the original measurement date. By contrast, an interval probability can be conditional on surviving to the start of that interval. These quantities are related, but they are not interchangeable.
Building a Horizon Probability From Interval Probabilities
Suppose a sequence of periods has conditional default probabilities \(q_1,q_2,\ldots,q_n\). Here \(q_k\) means the probability of default during period \(k\), conditional on having survived to its beginning. Survival through all \(n\) periods requires surviving each period in turn. The chain rule for conditional probability therefore gives:
This product does not require independent default events across years. The \(q_k\) values are conditional probabilities, each defined for borrowers who have survived to the relevant period. The product follows from conditioning successively on survival, not from assuming that yearly default outcomes are independent.
Write down the time interval being measured and confirm that all probabilities use the same default definition and starting population.
Determine whether it is cumulative from the original date, unconditional for an interval, or conditional on survival to the interval's start.
For a cumulative default probability \(F(t)\), use \(S(t)=1-F(t)\). For sequential conditional probabilities, multiply the conditional survival probabilities.
Report the cumulative default probability and survival probability with the same clearly named measurement period.
Why the Product Formula Holds
Let \(A_k\) be the event that the obligor survives period \(k\), given survival to that period's start. The probability of surviving all periods can be expanded by the chain rule:
By definition, each conditional default probability \(q_k\) applies to an obligor that has survived through period \(k-1\). Consequently, the conditional probability of surviving period \(k\) for that group is \(1-q_k\). Substituting these values into the chain-rule expression yields \(S_n=\prod_{k=1}^{n}(1-q_k)\). Since default by the end of period \(n\) and survival through period \(n\) are complementary outcomes, \(F_n=1-S_n\).
If instead a problem provides cumulative probabilities \(F(t)\) at multiple dates, the conditional probability for a later interval must account for the probability of having survived to the interval's beginning. Directly subtracting the cumulative probabilities gives an unconditional interval probability, not the conditional probability among survivors.
Worked Example: Constant Annual Conditional Default Probability
A hypothetical lender models a borrower with a 2.0% conditional probability of default in each year, provided the borrower has survived to that year's start. Under this constant annual conditional probability assumption, calculate the cumulative default probability and survival probability over three years. Give probabilities to four decimal places.
Method and assumption. The annual conditional default probability is \(q=0.020\) in each of the three successive years. Thus the conditional survival probability in each year is \(1-q=0.980\). The constant conditional probability is a model assumption; it is not implied merely by knowing a one-year default probability.
Year 1 survival: \(S_1=0.980\).
Year 2 survival: \(S_2=0.980\times0.980=0.9604\).
Year 3 survival: \(S_3=0.9604\times0.980=0.941192\). Equivalently, \(S_3=0.980^3\).
Therefore the three-year cumulative default probability is \(F_3=1-S_3=1-0.941192=0.058808\).
Result: Rounded only at the final step, the three-year survival probability is 0.9412 and the three-year cumulative default probability is 0.0588.
Check: \(0.9412+0.0588=1.0000\), subject to rounding. A common mistake is to add \(2.0\%+2.0\%+2.0\%=6.0\%\). That treats each conditional annual probability as though it applied to the original population. The correct calculation allows the population at risk to decline as defaults occur; it gives 5.8808% before rounding, not 6.0%.
Worked Example: Different Conditional Probabilities by Year
For a hypothetical portfolio segment, the conditional default probability is 4% in year 1, 5% in year 2 given survival through year 1, and 8% in year 3 given survival through year 2. Calculate the three-year survival probability and cumulative default probability. Give probabilities to four decimal places.
Method. These are successive conditional probabilities, so multiply the corresponding conditional survival probabilities. The assumptions are that the supplied values apply to the stated years and to obligors that survive to each year's start.
The conditional survival probabilities are \(1-0.04=0.96\), \(1-0.05=0.95\), and \(1-0.08=0.92\).
After year 1, survival is \(S_1=0.96\). After year 2, it is \(S_2=0.96\times0.95=0.912\). After year 3:
The cumulative default probability is \(F_3=1-0.83904=0.16096\). Rounded only at the final step, \(S_3=\mathbf{0.8390}\) and \(F_3=\mathbf{0.1610}\).
Check and trap. The results sum to 1.0000 after rounding. Adding 4%+5%+8%=17% is incorrect: the 5% and 8% probabilities apply only to borrowers who survive to their respective years. Multiplication of conditional survival probabilities gives the correct cumulative result.
Recovering a Conditional Probability From Cumulative Probabilities
Cumulative probabilities are often reported at several horizons. Their difference identifies the probability of default in the interval on the original population scale, but a conditional probability among survivors requires division by the probability of reaching that interval.
Worked Example: Converting Cumulative Probabilities to a Conditional Interval Probability
A hypothetical credit model reports cumulative default probabilities of 7% by the end of year 1 and 12% by the end of year 2. Calculate (a) the unconditional probability of default during year 2, measured from the original date, and (b) the conditional probability of default during year 2 given survival through year 1. Give probabilities to four decimal places.
Method and assumptions. The cumulative probabilities refer to the same starting date, the same defined default event, and horizons at the ends of years 1 and 2. Since \(F(1)=0.07\) and \(F(2)=0.12\), the unconditional year-2 interval probability is:
Thus 5% of the original population defaults during year 2. To find the conditional probability among those who survive year 1, first calculate \(S(1)=1-F(1)=1-0.07=0.93\). Then:
Rounded only at the final step, the unconditional year-2 interval probability is 0.0500, and the conditional year-2 default probability given survival through year 1 is 0.0538.
Check. The implied year-2 survival probability among year-1 survivors is \(1-0.053763\ldots=0.946237\ldots\). Multiplying by year-1 survival gives \(0.93\times0.946237\ldots=0.88\), so the cumulative default probability by year 2 is \(1-0.88=0.12\), as reported. A common error is to call 5% the conditional year-2 probability; it is the unconditional interval probability on the original population.
Aligning Probabilities to the Measurement Period
A probability is usable for a calculation only when its horizon matches the question, or when a stated model allows conversion between horizons. For example, an annual conditional probability cannot simply be used as a five-year cumulative probability. Likewise, a probability for a six-month period cannot be compared directly with a two-year cumulative probability without accounting for the difference in horizon.
If the model explicitly assumes a constant conditional probability over equal-length intervals, one can apply the product formula over the number of intervals. If that assumption is not supplied, one-year information alone does not determine the probability over a longer horizon: conditional default probabilities may change over time. Avoid imposing a constant rate merely to make a calculation possible.
| Quantity provided | Meaning | Use |
|---|---|---|
| Cumulative \(F(t)\) | Default by time \(t\), from the original measurement date | Convert to survival using \(S(t)=1-F(t)\) |
| Conditional interval probability \(q_k\) | Default in interval \(k\), given survival to its start | Multiply interval survival probabilities \(1-q_k\) |
| Difference \(F(t)-F(s)\) | Default in \((s,t]\), measured from the original date | Divide by \(S(s)\) to obtain the conditional interval probability |
In practice, horizon alignment also requires a consistent starting date and event definition. A one-year probability estimated for one population or under one definition of default is not automatically comparable with a multi-year probability for a different population or definition. State what is measured before interpreting or combining the numbers.
Check Your Understanding
For each problem, identify the horizon and whether the supplied probabilities are cumulative or conditional before calculating.
- A hypothetical lender assumes a constant 4% conditional default probability in each year. Calculate the four-year survival probability and cumulative default probability. Give probabilities to four decimal places. (Answer: survival \(0.8493\); cumulative default probability \(0.1507\).)
- A model gives a 9% cumulative default probability by the end of year 2 and a 6% conditional default probability in year 3 given survival through year 2. Find the cumulative default probability by the end of year 3. (Answer: \(0.1446\), or 14.46%.)
- Conditional default probabilities are 1%, 3%, and 5% in years 1, 2, and 3, respectively, each conditional on survival to that year's start. Calculate three-year survival and cumulative default probabilities. (Answer: survival \(0.9123\); cumulative default probability \(0.0877\).)
- A hypothetical model assumes the conditional default probability is 2% in every six-month interval. Calculate the cumulative default probability over 18 months. Give the probability to four decimal places. (Answer: \(0.0588\), or 5.88%.)
- Cumulative default probabilities are 7% by year 1 and 12% by year 2. Calculate the conditional probability of default during year 2 given survival through year 1. (Answer: \(0.0538\), or 5.38%.)