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