I understood the cumulative (aka unconditional) probability of default to be the probability of defaulting in a given period eg: between years 1 and 5. Further $\pi_{cumulative} = 1-e^{-\lambda*t}$ where lambda is a hazard rate.
I understood the marginal (aka conditional) probability of default to be the probability of defaulting at time $T$ given survival up to that point. Further $\pi_{marginal} = \lambda e^{-\lambda*t}$ where lambda is a hazard rate.
Attempting to solve the following problem, I came up with a close but off value.
Problem
1 year hazard rate = 0.1. What is the probability of surviving in the first year followed by defaulting in the second?
My solution was to calculate the marginal probability of default = $0.1\lambda e^{0.1*2}$ = 8.19%
But the given answer was 8.61% arrived at by:
1 year cumulative (also called unconditional) PD = 1 - e^(- hazard*time) = 9.516%
2 year cumulative (also called unconditional) PD = 1 - e^(- hazard*time) = 18.127%
solution - 18.127% - 9.516% = 8.611%
Is my approach incorrect or merely an approximation?