SoA MLC #236<\/p>\n
For a fully discrete insurance of 1000 on \\((x)\\), you are given:<\/p>\n
* \\(~_4 AS = 396.63\\) is the asset share at the end of year 4.
\n * \\(~_5 AS = 694.50\\) is the asset share at the end of year 5.
\n * \\(G = 281.77\\) is the gross premium.
\n * \\(~_5 CV = 572.12\\) is the cash value at the end of year 5.
\n * \\(c_4 = 0.05\\) is the fraction of the gross premium paid at time 4 for expenses.
\n * \\(e_4 = 7.0\\) is the amount of per policy expenses paid at time 4.
\n * \\(q_{x+4}^{(1)} = 0.09\\) is the probability of decrement by death.
\n * \\(q_{x+4}^{(2)} = 0.26\\) is the probability of decrement by withdrawal.<\/p>\n
Calculate \\(i\\).<\/p>\n
Solution.<\/p>\n
At the beginning of the year, the asset share plus net income available is
\n\\begin{eqnarray*}
\n~_4 AS + G(1-c_4) – e_4 = 396.63 +281.77(1-0.05) – 7 = 657.3115 .
\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+4}^{(\\tau)} ~_5 AS + 1000 q_{x+4}^{(1)} + ~_5 CV q_{x+4}^{(2)} \\\\
\n&~~~~~~~~=& (1 – 0.09 – 0.26) (694.50) + 1000 (0.09) + (572.12) (0.26) \\\\
\n&~~~~~~~~=& 690.1762 .
\n\\end{eqnarray*}
\nUsing the relation \\( 657.3115 (1+i) =690.1762\\), we get \\(i = 4.9999% approx 5% \\).<\/p>\n