Before revision:  We demonstrate how the uncertainty can be quantified using the full posterior distribution of model parameters obtained from Bayesian inference via Markov chain Monte Carlo (MCMC). Results show that the parameter uncertainty and its impact on capital can be very significant. We have also quantified the effect of diversification for a finite number of borrowers in comparison with the infinitely granular portfolio .
After revision:  We demonstrate how the uncertainty can be quantified using the full posterior distribution of model parameters obtained from Bayesian inference via Markov chain Monte Carlo method. This approach is very convenient for joint estimation of all model parameters and latent systematic factors. Moreover, all relevant uncertainties are easily quantified. Typically available data are annual averages of defaults and recoveries and thus the datasets are small and parameter uncertainty is significant. In this case Bayesian approach is superior to the maximum likelihood method that relies on a large sample limit Gaussian approximation for the parameter uncertainty. As an example, we consider a homogeneous portfolio with one latent factor. However, the approach can be easily extended to deal with non-homogenous portfolios and several latent factors .