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?

• Do you have an idea ? or a formula?. Please share it.
– user16651
Jul 16 '16 at 8:11
• Is this the only information you have? Are you allowed to use default curves of other companies too? Ie sector curves?
– will
Jul 16 '16 at 9:34

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 $15.36\%$.