  We introduce a new numerical approximation method for functionals of factor
credit portfolio models based on the theory of mod-$\phi$ convergence and
mod-$\phi$ approximation schemes. The method can be understood as providing
correction terms to the classic Poisson approximation, where higher order
corrections lead to asymptotically better approximations as the number of
obligors increases. We test the model empirically on two tasks: the estimation
of risk measures ($\mathrm{VaR}$ and $\mathrm{ES}$) and the computation of CDO
tranche prices. We compare it to other commonly used methods -- such as the
recursive method, the large deviations approximation, the Chen--Stein method
and the Monte Carlo simulation technique (with and without importance sampling)
-- and we show that it leads to more accurate estimates while requiring less
computational time.
