more Annuities

Life Annuities a continuation of PART I

We're following Milevsky & Robinson and, so far, we have:

Probability that an n-year-old will die before t years have elapsed is
F(n,t) = 1 -
where m and c are constants

If F(n,t) gives the probability that an n-year-old will die before t years has elapsed (when s/he is age n+t), then 1 - F(n,t) is the probability that s/he will live beyond age n+t. If we sold Life Annuities to these people (when they were n-year-olds), we'd still be paying these people.
>We stop paying only when they drop dead, right?
Right. So how much should we have charged them for a $1000 per year annuity, way back when they were n years old?
>Well, they're now age n+t and they're still alive and ... uh ...
Discounting the $1000 at a certain annual rate, say r (where 5% means r = 0.05), we should have charged them 1000/(1+r)^t. That's like the Present Value of $1000, over t years, at the annual rate r.

However, in order to make the math more tractable, we do the compounding more frequently:

The discounting for a single period of 1 year is (1+r)^-t.
A rate of r per year means a rate r/12 per month and, over 12t months, that means the discounting would be (1+r/12)^-12t.
A rate of r per year means a rate r/365 per day and, over 365t days, that means the discounting would be (1+r/365)^-365t.
A rate of r per year means a rate r/8760 per hour and ...

>Yeah, yeah. I get it. So what?
So an expression like (1+r/M)^-Mt is indistinguishable from e^{-r t}, when M is large.
In fact, for r = 0.05 and t = 10 years we get (1+0.05/365)^-365(10) = 0.606551 whereas e^-(0.05)(10) = e^-0.5 = 0.606531
so that means, for continuous compounding, we can put that discounting factor at e^{-r t}.

But, as years go by, fewer of these n-year-olds will survive so there are fewer $1000 payments so we must consider the sum of all of these payments, for all future years ...
>For the survivors?
Yes, and that means we add together the present value for all these payments for all future years by integrating like so:

P = e^{-r t} G(n,t) dt where G(n,t) = 1 - F(n,t) =

Here, G(n,t) gives the n-year-old population which survives for t years.
So, what's the answer?
>Huh?
Suppose we rewrite the expression for G(n,t) so it has the form: G(n,t) = K e^{- B e^kt} (K and B are some constants) then
P = K e^{-r t} e^{- B e^kt} dt and we can now substitute x = B e^kt so P now takes the form

P = L x^a-1 e^-x dx (L and a are some constants)

and we now recognize somebody related to the Gamma function, eh?
>Huh?
Remember?
Γ(n+1) = is called the Gamma function and has the value Γ(n+1) = n!
when n is a positive integer, but the integral allows us to define this function for non-integer values of n.

Finally, Γ(a,B) = x^a-1 e^-x dx is called the Incomplete Gamma Function ...

>So Γ(a,0)=a!, right?
Close, but no cigar. Actually Γ(a,0)=(a-1)! and although it would seem reasonable to define Γ(x) so it was x! the definition is such that Γ(x) = (x-1)! instead and that gives math-types the opportunity to say:
"Γ(-n) is infinite for n = 0, 1, 2, ..." because then Γ(0) = (-1)! and Γ(-1) = (-2)! etc. and everybody knows that the factorial of negative integers is infinite so that ...
>zzzZZZ

Figure 1

Figure 2 Maybe a picture will make things more interesting.
If we can use the above stuff to determine the initial Cost of a Life Annuity that pays $1.00 per year (for life) then 1.00/Cost is how much we'd get, per year, for an investment of $1.00 and we can compare with other investment returns.
The chart at the left is such an example, where we've used
r = 0.04 (or 4%) in the earlier formulas ... and get the coloured dots. Well?
>zzzZZZ
See how it compares with the Excel PMT function (the thin lines)?
For example, if you're male, then PMT(0.016,(91-age)/2,-1) gives the annual payments for a $1.00 purchase price (using a 1.6% rate) assuming you live another (91-age)/2 years.

Good, eh? Of course, those who sell annuities may want a larger purchase price and/or a smaller annual payout so the percentages in Figure 2 will probably be smaller and ...

>zzzZZZ