Consider a portfolio of size (N) of fully discrete whole life policies. For simplicity, we assume that they are all issued to a life aged (x=40) with a face equal to one and that mortality follows the Illustrative Life Table. Initially, the insurer faces an (i=5%) interest environment and so collects premiums
begin{eqnarray*}
P_{40} = frac{A_{40}}{ddot{a}_{40}}= frac{0.2079}{16.6331} = 0.012502
end{eqnarray*}
from each of the (N) policyholders. Let (N_1) be the random number who die during the first year, a binomial random variable with a probability of (q_{40}=0.0027812) of failure. Thus, at the end of the first policy year, the fund associated with this portfolio is
begin{eqnarray*}
FUND_1 = N P_{40}(1.05) – N_1 + sum_{j=1}^{N-N_1} ~_1 L_j,
end{eqnarray*}
where (~_1 L_j = v^{K_{x,j}} – P_{40} ddot{a}_{overline{K_{x,i}|}}) is the random future loss for the (j)th contract at time 1 for those that have survived. Note that if an (i=5%) interest environment prevails, then the expected fund is
begin{eqnarray*}
textrm{E}~FUND_1 &=& N P_{40}(1.05) – N q_{40} – N (1-q_{40}) textrm{E}~ L_{1,j} \
&=& N left( P_{40}(1.05) – q_{40} – p_{40} ~_1 V_{40} right) = 0,
end{eqnarray*}
using the recursive reserve formulation.