AMLCR Example 8.9. The following table is an excerpt from a multiple decrement table for an insurance policy offering benefits on death or diagnosis of critical illness. The insurance expires on the earliest event of death (j=1), surrender (j=2), and critical illness diagnosis (j=3).
| (x) | (l_x) | (d_x^{(1)}) | (d_x^{(2)}) | (d_x^{(3)}) |
| 40 | 100,000 | 51 | 4,784 | 44 |
| 41 | 95,121 | 52 | 4,526 | 47 |
| 42 | 90,496 | 53 | 4,268 | 50 |
| 43 | 86,125 | 54 | 4,010 | 53 |
| 44 | 82,008 | 55 | 3,753 | 56 |
| 45 | 78,144 | 56 | 3,496 | 59 |
| 46 | 74,533 | 57 | 3,239 | 62 |
| 47 | 71,175 | 57 | 2,983 | 65 |
| 48 | 68,070 | 58 | 2,729 | 67 |
| 49 | 65,216 | 58 | 2,476 | 69 |
| 50 | 62,613 | 58 | 2,226 | 70 |
(a) Calculate (i) (_3 p_{45}^{00}), (ii) (p_{40}^{01}) and (iii)(_5 p_{41}^{03}).
Solution.
begin{eqnarray*}
_3 p_{45}^{00} = frac{l_{48}}{l_{45}}=frac{68,070}{78,144}= 0.871084.
end{eqnarray*}
begin{eqnarray*}
p_{40}^{01} &=& frac{d_{40}^{(1)}}{l_{40}}=frac{51}{100,000}= 0.00051.
end{eqnarray*}
begin{eqnarray*}
_5 p_{41}^{03}&=& frac{d_{41}^{(3)}+d_{42}^{(3)}+d_{43}^{(3)}+d_{44}^{(3)}+d_{45}^{(3)}}{l_{41}} \
&=& frac{47+50+53+56+59}{95,121} = frac{265}{95,121} = 0.002786.
end{eqnarray*}
(b) Calculate the probability that a policy issued to a life aged 45 generates a claim for death or critical illness before age 47.
Solution.
begin{eqnarray*}_2 p_{45}^{01}+_2 p_{45}^{03}&=& frac{d_{45}^{(1)}+d_{46}^{(1)}+d_{45}^{(3)}+d_{46}^{(3)}}{l_{45}} \
&=& frac{56+57+59+62}{78,144} = frac{234}{78,144} = 0.002994.
end{eqnarray*}
(c) Calculate the probability that a policy issued to a life age 40 is surrendered between ages 45 and 47.
Solution.
begin{eqnarray*}
_5 p_{40}^{00} times _2 p_{45}^{02}&=& frac{l_{45}}{l_{40}} times frac{d_{45}^{(2)}+d_{46}^{(2)}}{l_{45}}\
&=& frac{d_{45}^{(2)}+d_{46}^{(2)}}{l_{40}} = frac{3,496+3,239}{100,000} = 0.06735.
end{eqnarray*}