  We consider a large, homogeneous portfolio of life or disability annuity
policies. The policies are assumed to be independent conditional on an external
stochastic process representing the economic-demographic environment. Using a
conditional law of large numbers, we establish the connection between claims
reserving and risk aggregation for large portfolios. Further, we derive a
partial differential equation for moments of present values. Moreover, we show
how statistical multi-factor intensity models can be approximated by one-factor
models, which allows for solving the PDEs very efficiently. Finally, we give a
numerical example where moments of present values of disability annuities are
computed using finite difference methods and Monte Carlo simulations.
