Converting Marginal Default Probabilities to Conditional Probabilities

Summary

The document explains how to relate a borrower's marginal default probabilities across periods to the probability of default in a given period conditional on survival to its start. The motivating example comes from logistic-regression estimates of unconditional default probabilities over successive annual periods, and asks how to obtain the survival-conditioned quantity.

The answer defines the target as the probability that default occurs within an interval, given that the borrower has not defaulted by its beginning. Bayes’ rule and the complement of default probability provide the calculation from marginal probabilities. To apply it over multiple periods, the survival condition must be represented by the probability of avoiding default before the interval; the resulting conditional probability depends on the precise definitions and timing of the supplied marginal estimates. The document gives the probability statement and method at a high level, but no numerical worked example or discussion of dependence and competing events.

Key ideas

• A conditional default probability measures default over an interval given survival to its start.
• Bayes’ rule converts marginal event probabilities into a survival-conditioned probability.
• The probability of surviving to the interval’s beginning is needed for the conversion.
• Clear definitions of interval boundaries and marginal probabilities are necessary for consistent calculations.

Tags

# credit risk - marginal default probability


# credit risk - marginal default probability












I have been working on an assignment trying to calculate marginal/conditional probability of default. Using a logistic regression framework, I was able to compute the 12-month unconditional PD for each borrower for a duration of four years as shown in the image. For example 0.055 in the image refers to probability of default of borrower 1 during the period 2005 to 2006.

Could someone provide me an insight on how to convert this unconditional probability of default to conditional probability of default. i.e For example, probability of default between 2005 to 2006 given, the borrower has survived till 2005

## Answer by byouness (score 1)

https://quant.stackexchange.com/a/39577

The conditional default probability between $t_1$ and $t_2$ is the probability of default between those dates assuming survival until $t_1$. Denoting $\tau$ the default time, this is: $$P(\tau \in [ t_1, t_2 ] | \tau \geq t_1)$$

A simple application of Bayes formula (https://en.m.wikipedia.org/wiki/Bayes%27_theorem) and the fact that $P(\bar{A}) = 1 - P(A)$ will give you the answer from your marginal default probabilities.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.