Pricing and Reserving a 10-Year Level Term Product

Background

Sundial Life is launching a new 10-year level term product for healthy applicants issue age 45, face amount $250,000. The pricing actuary has been given a select mortality extract (issue age 45, per-1,000 rates during the select period) and must set the net premium, a gross premium with a modest expense loading, and project the net premium reserve at two future durations for the appointed actuary's opinion.

Sundial's valuation basis for this cohort is an annual effective interest rate of $i = 4\%$, with mortality following the extract below (independent of eventual improvement studies, this table is treated as the valuation and pricing basis). Premiums and benefits are annual: premiums are paid at the start of each policy year (an annuity-due) contingent on survival, and the death benefit is assumed payable at the end of the year of death (a discrete/curtate model, consistent with introductory FAM treatment).

Mortality extract (per 1,000, select duration 0 = issue)

Age $x$	$1000\,q_x$
45	1.87
46	2.02
47	2.19
48	2.38
49	2.60
50	2.85
51	3.14
52	3.47
53	3.84
54	4.26

Starting from a radix $l_{45} = 100{,}000$, successive survivorship is $l_{x+1} = l_x(1-q_x)$. Carrying this through the ten mortality rates above gives the survivorship column used throughout this case:

$x$	45	46	47	48	49	50	51	52	53	54	55
$l_x$	100,000.00	99,813.00	99,611.38	99,393.23	99,156.67	98,898.87	98,617.00	98,307.35	97,966.22	97,590.03	97,174.30

Pricing

Discounting at $v = 1/1.04$, the actuarial present value (APV) of the death benefit per $1,000 of face over the 10-year term is

$\text{APV}_{\text{ben}}/1000 = \sum_{k=0}^{9} v^{k+1}\, \frac{l_{45+k}-l_{46+k}}{l_{45}} = 22.2675$

and the APV of a 10-year temporary life annuity-due of 1 per year (premiums) is

$\ddot a_{45:\overline{10}|} = \sum_{k=0}^{9} v^{k}\, \frac{l_{45+k}}{l_{45}} = 8.3527.$

The net annual premium on $250,000 of face is$P = 250{,}000 \times 22.2675/1000 \div 8.3527 = $666.48$ per year, payable for 10 years (or until earlier death).

For pricing purposes Sundial loads for a **flat $50 per-policy expense** each year the policy is in force plus a **6$G$ solves

$G\cdot \ddot a_{45:\overline{10}|}\,(1-0.06) = \text{APV}_{\text{ben}} + 50\cdot \ddot a_{45:\overline{10}|},$

which gives G = \762.21$ per year.

Reserves

The prospective net premium reserve at the end of policy year $k$ (attained age $45+k$), using the net premium of $666.48$, is

$\,{}_kV = 250{,}000 \cdot \sum_{j=k}^{9} v^{\,j-k+1}\frac{l_{45+j}-l_{46+j}}{l_{45+k}} \;-\; 666.48\cdot \sum_{j=k}^{9} v^{\,j-k}\frac{l_{45+j}}{l_{45+k}}.$

Evaluating this recursion gives \,{}_3V = \587.98$(end of year 3, attained age 48) and$,{}_5V = $784.96$ (end of year 5, attained age 50). Note that the reserve is increasing with duration even though the product is level term — this is the normal pattern for a net premium reserve on a term plan whose mortality cost rises faster than the level premium, i.e., early premiums pre-fund later, more expensive years.

Your task

You are the assistant actuary reviewing this pricing and reserving work before it goes to the appointed actuary. Use the figures above (treat them as exact) to answer the questions that follow. Where a question asks you to extend the calculation (e.g., a different duration or a shock to assumptions), show your work using the same recursive/first-principles approach as above.