The original rider was priced in 1971, using mortality tables containing data from 1953. An actuary is asked to use more recent mortality data to estimate the cost of the rider and determine whether a change in price is warranted. Also, the company wants to know if it can now include coverage for infants age 0–14 days without a change in price.

The Original Cost Estimate
The actuary assumes that the company sells 1,000 rider policies. Then the actual number of children covered will be about 2.3 times 1,000 because the average number of children per family in the United States is 2.3 in 1953. In a population of 1,000 newborns, approximately 46 will die before reaching age 21. Twenty-seven of these deaths occur in the first two weeks of life and are not covered. If the company sells 1,000 rider policies offering a death benefit of $1,000 per child, the cost (just the death benefit) per family incurred over the 21 years is roughly

.

Divide by 21, the number of years in which the 19 deaths occur, to get a cost of roughly $2.08 per family per year.

In the last half-century, improvements in medicine have significantly reduced child mortality. This change is reflected in child mortality data from 1999.

Table: Improvement in Child Mortality

1953 Mortality Data					1999 Mortality Data
Age		Age			Age		Age
1	0.02870	11	0.0005		1	0.00706	11	0.00013
2	0.00230	12	0.0005		2	0.00053	12	0.00013
3	0.00140	13	0.0005		3	0.00036	13	0.00017
4	0.00100	14	0.0006		4	0.00027	14	0.00026
5	0.00080	15	0.0007		5	0.00022	15	0.00038
6	0.00080	16	0.0009		6	0.00020	16	0.00051
7	0.00070	17	0.0010		7	0.00019	17	0.00063
8	0.00060	18	0.0011		8	0.00018	18	0.00073
9	0.00050	19	0.0012		9	0.00016	19	0.00079
10	0.00050	20	0.0013		10	0.00014	20	0.00084
		21	0.0013				21	0.00088

For example, whereas 28.7 out of 1,000 children in the United States died before reaching age one in 1953, only about seven out of 1,000 died at the same age in 1999.

The actuary sees cost reduction in the child mortality data from 1999: Only 34 children out of 1,000 will die before age 21. In addition, the number of children per family is smaller in 1999 (about 1.7) than it was in 1953, reducing costs further. These cost reductions allow the insurance company to extend coverage to include the first 14 days of life — heretofore excluded — for a total cost of about $2.75 per year per family. Because the increased coverage more than justifies the increased cost, the actuary recommends that the company drop the clause limiting coverage on children age 14 days and under.

This is a significant change, especially for the agents who will sell the policy. The actuary must explain the change to the agents in a way the agents can understand and explain to their customers.

Extensions
There are many simplifying assumptions made in the above calculations. Here are a few:

• All of the children in the population are the same age.
• All of the children in all of the families were born on the first day of the year.
• The premium is paid at the start of the year, and deaths all occur at the end of the year.
• There is no inflation, and the insurance company earns no interest on reserves.
• No other benefits are included in the policy.
• The average number of children per family for the insured population matches the U.S. population.

All of these can be replaced by more realistic assumptions. For example, the actuary would compute the present value of a stream of future premium payments as the sum of a finite geometric series rather than a back-of-the-envelope calculation. It is perhaps just as important to be able to deduce which assumptions tend to underestimate costs and which tend to overestimate costs. For example, the census data giving 2.3 children per family is probably not a good estimate for the population insured by the company (in 1953 or today). The expected number of children in a family, given that the family has purchased life insurance, is probably larger than the expected number for the full U.S. population. In real applications, the trade-off is often between getting a "pretty good" solution today and a "very good" solution next month.