SoA MLC # 242<\/p>\n
For a fully discrete whole life insurance of 10,000 on \\((x)\\), you are given:<\/p>\n
* \\(~_{10} AS =1600\\) is the asset share at the end of year 10.
\n * \\(G = 200\\) is the gross premium.
\n * \\(~_{11} CV =1700\\) is the cash value at the end of year 11.
\n * \\( c_{10} = 0.04\\) is the fraction of gross premium paid at time 10 for expenses.
\n * \\(e_{10} = 70\\) is the amount of per policy expense paid at time 10.
\n * Death and withdrawal are the only decrements.
\n * \\(q_{x+10}^{(d)} = 0.02 \\)
\n * \\(q_{x+10}^{(w)} = 0.18 \\)
\n * \\(i = 0.05\\)<\/p>\n
Calculate \\(~_{11} AS \\), the asset share at the end of year 11.<\/p>\n
Solution.<\/p>\n
At the beginning of the year, the asset share plus net income available is
\n\\begin{eqnarray*}
\n~_{10} AS + G(1-c_{10}) – e_{10} = 1600 + 200(1-0.04) – 70 = 1722.
\n\\end{eqnarray*}
\nAt the end of the year, funds must be sufficient to pay those who survive, die and withdraw:
\n\\begin{eqnarray*}
\n& & p_{x+10}^{(\\tau)} ~_{11} AS + 10000 q_{x+10}^{(d)} + ~_{11} CV q_{x+10}^{(w)} \\\\
\n&~~~~~~~~=& (1 – 0.02 – 0.18) ~_{11} AS + 10000 (0.02) + (1700) (0.18) \\\\
\n&~~~~~~~~=& 0.8 ~_{11} AS +506.
\n\\end{eqnarray*}
\nThus, with \\(i = 0.05\\), we have \\((1.05) 1722 = 0.8 ~_{11} AS +506\\), or
\n\\(~_{11} AS = 1627.625\\).<\/p>\n