Inferring Two-Year Default Risk from a One-Year Probability

Summary

The document asks how to infer the probability of default by year two when only an annual default probability of 8% is given. One answer models survival probabilities as log-linear over time, which corresponds to a piecewise constant hazard assumption. Starting from 92% survival at year one, it derives 84.64% survival at year two and therefore a 15.36% cumulative default probability by that horizon.

A second answer instead assumes the one-year default probability stays constant conditional on survival, producing a different cumulative estimate. The disagreement highlights that a single annual probability does not uniquely determine a multi-year default curve. Both calculations require assumptions about how risk evolves; the log-linear survival approach is a modeling convention, while the constant conditional probability approach is explicitly described as simplistic. The document provides no company-specific data to select between them.

Key ideas

• A one-year default probability alone does not uniquely determine cumulative default risk at year two.
• Log-linear survival interpolation is consistent with a piecewise constant hazard assumption.
• Under that assumption, the stated one-year survival implies 84.64% survival by year two.
• A constant conditional annual default probability is an alternative simplifying assumption.
• Choosing a multi-year estimate requires specifying how default risk changes over time.

Tags

# What is the probability of defaulting in year 2?


# What is the probability of defaulting in year 2?












I was asked this question the other day, but it's been years since I've done this work.

If the probability of a company to default in a year is $8\%$, what is the probability that it will default in year 2?

## Answer by Gordon (score 1)

https://quant.stackexchange.com/a/28112

Generally, survival probability can be interpolated or extrapolated log-linearly, which is, at the least, consistent with the piece-wise constant hazard rate assumption. Specifically, let $x$ be the survival probability at year 2. The year 1 survival probability, 92%, can be log-linearly interpolated from the survival probability 1 at year 0 and the survival probability $x$ at year 2, that is, \begin{align*} \ln (0.92) = \ln (1) + \frac{\ln x - \ln (1)}{2-0}\times(1-0). \end{align*} Then \begin{align*} x = 0.92^2 = 84.64\%. \end{align*} Consequently, the default probability at year 2 is $15.36\%$.

## Answer by user3748386 (score 0)

https://quant.stackexchange.com/a/28109

I guess with that very sparse information all you could do is take the probability that the company still exists in a year (92%) times 8 % (assuming that the probability itself does not change). I believe I had something similar in an exam once. However, it is a really simplistic answer.

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