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
\n\\begin{eqnarray*}
\nP_{40} = \\frac{A_{40}}{\\ddot{a}_{40}}= \\frac{0.2079}{16.6331} = 0.012502
\n\\end{eqnarray*}
\nfrom 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
\n\\begin{eqnarray*}
\nFUND_1 = N P_{40}(1.05) – N_1 + \\sum_{j=1}^{N-N_1} ~_1 L_j,
\n\\end{eqnarray*}
\nwhere \\(~_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
\n\\begin{eqnarray*}
\n\\textrm{E}~FUND_1 &=& N P_{40}(1.05) – N q_{40} – N (1-q_{40}) \\textrm{E}~ L_{1,j} \\\\
\n&=& N \\left( P_{40}(1.05) – q_{40} – p_{40} ~_1 V_{40} \\right) = 0,
\n\\end{eqnarray*}
\nusing the recursive reserve formulation. <\/p>\n