Question

1. you are given $l_{75} = 1200$, $d_{75} = 100$, $l_{76} = 1000$, $d_{76} = 130$, and $e_{75}^{\circ} = 14.5$, where the select period is a one-year period.

calculate $e_{76}^{\circ}$ assuming udd in each year of age.

Explanation:

Step1: Recall curtate expectation formula

$e_{[x]}^\circ = e_{[x]} + \frac{1}{2}$

Step2: Find $e_{[75]}$ from given $e_{[75]}^\circ$

$14.5 = e_{[75]} + \frac{1}{2} \implies e_{[75]} = 14$

Step3: Express $e_{[75]}$ via survival counts

$e_{[75]} = \frac{l_{[76]} + l_{77} + l_{78} + ...}{l_{[75]}}$
Substitute known values:
$14 = \frac{1000 + l_{77} + l_{78} + ...}{1200}$
$14 \times 1200 = 1000 + \sum_{k=77}^\infty l_k$
$16800 = 1000 + \sum_{k=77}^\infty l_k \implies \sum_{k=77}^\infty l_k = 15800$

Step4: Calculate $e_{[76]}$

$e_{[76]} = \frac{l_{77} + l_{78} + ...}{l_{[76]}} = \frac{15800}{1000} = 15.8$

Step5: Apply UDD to get $e_{[76]}^\circ$

$e_{[76]}^\circ = e_{[76]} + \frac{1}{2}$

Answer:

$16.3$