I know it is not quite as simple as this but…
If you get a $100,000 20 year term policy at age 35, the obvious way to calculate the premium is use a mortality table and calculate the probability that you will die before age 55, multiply this by 100000 and divide by 240 to get the monthly premium.
But it seems to me that if you die early, you don’t make the premium payments until age 55 so this way of calculating premiums. My thought was assuming dying in month 6 of a year and letting the premium = n, then a fair premium is the solution n to the equation n x 6 x p(dying your 35th year) + n x 18 x p(dying your 36th year) + n x 30 x p(dying your 37th year) + … + n x 234 x p(dying your 55th year) = 100000 x p(dying before age 55).
Is this more accurate or just more complicated?
It’s nowhere near complicated enough.
Money now is more valuable than money 20 years from now, so every premium payment and the possible death benefit must be discounted for the time value of money, using a discount rate appopriate to the time until payment.
For simplicity assume that both premium payments and death benefits are paid at the end of each year. The first year’s premium payment is worth Pp1d1 where P is the premium, p1 is the probability of dying in the first year, and d1 is the discount rate for a payment one year from now. The possible death benefit in the first year is worth DB*(1-p1)*d1, where DB is the death benefit.
Extrapolating out 20 years, you would have a stream of 20 premium values and 20 death benefit values. The premium payments, of course, occur unless you’ve died in the year at issue or any preceding year. The death benefit occurs if you die in the year at issue.
You set a “fair” premium so that the sums of the two streams are equal. Of course, in real life you also have to load in expenses and profit.