Instead, suppose that beginning at time 1, the insurer faces one of three interest rate scenarios given by
\n\\begin{eqnarray*}
\n\\textbf{i} = \\left\\{
\n\\begin{array}{cr}
\ni_1 = 4 & \\textrm{with probability 0.25} \\\\
\ni_2 = 5 & \\textrm{with probability 0.50} \\\\
\ni_3 = 6 & \\textrm{with probability 0.25} \\\\
\n\\end{array} \\right.
\n\\end{eqnarray*}
\nFor a fixed interest scenario \\(i_s\\), we have that the random future loss variable at policy duration \\(k\\) is
\n\\begin{eqnarray*}
\n_k L(i_s) = v_s^{K+1-k} – P_{40} \\ddot{a}_{\\overline{K+1-k|}} = \\left(1+\\frac{P_{40}}{1-v_s}\\right) v_s^{K+1-k} – \\frac{P_{40}}{1-v_s} ,
\n\\end{eqnarray*}
\nwhere \\(v_s=1\/(1+i_s)\\). This has conditional (on the interest scenario) mean
\n\\begin{eqnarray*}
\n\\textrm{E}(_k L(i_s)|i_s) = A_{40+k}@i_s – P_{40} \\ddot{a}_{40+k}@i_s,
\n\\end{eqnarray*} and variance
\n\\begin{eqnarray*}
\n\\textrm{Var}(_k L(i_s)|i_s) = \\left(1+\\frac{P_{40}}{1-v_s}\\right)^2 \\left( ^2A_{40+k}@i_s – A_{40+k}^2 @i_s\\right).
\n\\end{eqnarray*}
\nThe following table summarizes the calculations at duration \\(k=1\\) based on the Illustrative Life Table (recall \\(P_{40} = 0.125021\\) was set at contract initiation and so is constant across scenarios).<\/p>\n
| <\/th>\n | <\/th>\n | <\/th>\n | <\/th>\n | Conditional<\/th>\n | <\/th>\n | Conditional<\/th>\n<\/tr>\n |
|---|---|---|---|---|---|---|
| \\(i\\)<\/td>\n | prob<\/td>\n | \\(A_{41}\\)<\/td>\n | \\(\\ddot{a}_{41}\\)<\/td>\n | Exp Loss<\/td>\n | \\(^2 A_{41}\\)<\/td>\n | Var Loss<\/td>\n<\/tr>\n |
| 0.04<\/td>\n | 0.25<\/td>\n | 0.282<\/td>\n | 18.658<\/td>\n | 0.049<\/td>\n | 0.105<\/td>\n | 0.042<\/td>\n<\/tr>\n |
| 0.05<\/td>\n | 0.50<\/td>\n | 0.216<\/td>\n | 16.461<\/td>\n | 0.010<\/td>\n | 0.072<\/td>\n | 0.034<\/td>\n<\/tr>\n |
| 0.06<\/td>\n | 0.25<\/td>\n | 0.169<\/td>\n | 14.686<\/td>\n | -0.015<\/td>\n | 0.052<\/td>\n | <\/td>\n<\/tr>\n<\/table>\n For example, at \\(i=0.04\\), the conditional mean loss is (0.282 – 0.012502 (18.658) = 0.049). The conditional variance is |