Premium calculation for whole life policy? d/delta

[jonathanfreund]

We sell a certain whole life policy at work, and the document is hard to read. It says the Net Annual Premium is A(x) divided by a(x). Or rather the present value of a whole life divided by the present value of an annuity. But it is had to read whether it is a continuous whole life or something else. And is it an annuity due, constant, I don't know.

Either way to calculate the Net Investment Premium they multiply this by d/delta, d over delta, and then add some margins to arrive at the gross annual premium.

I though they were maybe multiplying it by d over delta to make it continuous, but that would be i over delta. So I guess maybe I am reading the A(x)/a(x) part wrong. But no matter whether I make it an annuity due or anything else, I can not for the life of me figure out why we are multiplying by d/delta!

Any thoughts to d/delta???

[Gandalf]

I don't recall the specific assumptions behind d/delta but will make some comments that may get you on the right track.

If you start with P, the curtate premium (A(x)/anndue(x)), you can multiply it by i/delta to make it semi-continuous: claims paid at moment of death, premiums paid at the start of the year, no portion of the premium for the year returned if the policyholder dies during the year.

I don't think there is any simple factor to go from the curtate premium or the semicontinous premium to the fully continuous premium. It is not hard to go from one to the other, but there is not a single multiplicative factor applying to P at all ages, just depending on interest.

Yet another possible premium - I forget the term - would be:
(a) Death benefits paid immediately at death
(b) Premiums paid at the beginning of the year
(c) If the policyholder dies after x% of the year, 1-x% of the year's premium is refunded, possibly with some interest adjustment, though interest on that premium would have extremely little impact.

I'm thinking maybe the d/delta would apply to the fully continuous premium. I believe I've seen that formula, maybe only as an approximation, used in calculating mean reserves for valuation purposes. [i.e., saying mean reserve is based on a prior terminal, next terminal, and a discounted beginning of year net premium of d/delta times the valuation net, where both terminals and the valuation net are fully continuous.] No guarantees.

[NoName]

i/delta converts from an EOY payment to a continuous payment.

d/delta converts from a BOY payment to a continuous payment, considering interest only. In the case of life insurance, where the continuous premium would also involve a haircut for mortality, it is only an approximation, but a reasonable one except at high qxs (of course, the whole idea of a continuous payment is not an exact reflection of reality in any case).

[Steve Grondin]

Originally Posted by NoName

i/delta converts from an EOY payment to a continuous payment.

d/delta converts from a BOY payment to a continuous payment, considering interest only. In the case of life insurance, where the continuous premium would also involve a haircut for mortality, it is only an approximation, but a reasonable one except at high qxs (of course, the whole idea of a continuous payment is not an exact reflection of reality in any case).

I am away from textbooks right now, so I am going from memory. I'll probably have to edit for stupidity later.

I think d = i/1+i, and del = ln(1+i). This means that d/del is less than 1. I also think d/del is applied to a fully continuous premium (Abarx/abarx) to get a discounted continuous premium (annual up front equivalent to a fully continuous premium).