Multiple lives question

[dsmith3]

peewee,

This comes from ASM 36.20. For an insurance on (55:55) that pays a death benefit of 1000 at the moment of the first death, you are given:
(i) Future lifetimes are independent.
(ii) Mortality on both lives follow de Moivre's law with w=100.
(iii) force of interest is .04

Calculate the net single premium for the insurance.

I know this is not a very difficult problem, especially in the single life. But with multiple lives the integral gets very ugly. Is there another way to do this problem other than integrals? Thanks. Have a great Easter weekend.

[Steve Paris]

Hey D-Rat,

Sorry for not checking this for a few days, but I spent this weekend at the beach.

Here's how I would do this problem (no evaluation of integrals):

The integral that would give this NSP is 1000 * Int(from 0 to 45) [v^t * t_p_xy * mu_xy(t)] dt. Notice that because of independence, t_p_xy = t_p_x * t_p_y, and mu_xy(t) = mu_x(t) + mu_y(t). So when you multiply t_p_xy * mu_xy(t), you'll get the sum of two terms: t_p_x * t_p_y * mu_x(t) + t_p_x * t_p_y * mu_y(t).

Now recognize that under DML, t_p_x * mu_x(t) = 1/(w-x) = 1/45 in this case. Likewise t_p_y * mu_y(t) = 1/45. So the sum of the two terms in the last paragraph is 1/45 * t_p_y + 1/45 * t_p_x = 2/45 * t_p_55 since both x and y are 55.

Factoring out the constant 2/45, we get the NSP = 1000 * 2/45 * Int(from 0 to 45) [v^t * t_p_55] dt. Now recognize the integral as being a(bar)_55 = (1 - A(bar)_55) / delta. Using DML, A(bar)_55 = 1/45 * a_angle45 = 1/45 * (1-e^(-45*.04))/.04 = .463723.

So I get NSP = 1000 * 2/45 * (1 - .463723)/.04 = 596.

P.S. Is this the manual's answer? Please let me know what their answers to the questions are from now on, so that I'll know if I've miscalculated something.