UDD Question

[RedSoxFan]

I'm a veteran of courses 3 and 4, but (as you'll find out) you forget most of what you learn. What I'm after is how to apply the uniform distribution of deaths.

Let's say I have an annuity-immediate life contingent factor for a 65 year old of 12.3 (pays 1 at the start of each year for life). I also have a factor for a 66 year old of 11.4. How can I use UDD to determine the appropriate factor for someone 65 years and 3 months old?

Thanks in advance..

[Gandalf]

Assuming that the annuity to age 65.25 will have payments commencing one year from now (at age 66.25) and at one year intervals thereafter, it is just .75 * age 65 factor + .25 * age 66 factor, where factors are for immediate annuities.

If the annuity to age 65.25 will have the first payment at exact age 66 and at one year intervals thereafter, then it is a 3/4 year deferred annuity due. For the deferral, multiply by ((1 - q) / (1 - .25q)) * v^.75, where q is for age 65. Multiply that times an annuity-due to 66.

Your wording is confusing since you are referring to immediate annuities and start of year. I didn't want to put in numerical calculations since I'm not certain which factors you have and what the benefits are.

[Pi Man]

having worked with other annuity actuaries, my bet is that immediate is as opposed to deferred; not 'due'... could be wrong.

[Gandalf]

could be; but in the case of immediate as opposed to deferred annuities purchased from life insurance companies they typically are truly immediate, not due: the first payment is not at the start, but one period after the start.

In pension plan situations the annuities might have a payment immediately; also in situations where a death benefit or cash value were being settled with periodic payments.

In any case, the correct answer depends on the payment pattern, which has not been clearly specified.

[Been There Done That]

The OP asked about an annuity that paid at the start of each year of life -- I will assume that this means it pays on the birthday. For a life age 65 we are given the annuity value of 12.3 and for a life age 66 the value is 11.4.

We also have the formula:

a(65) = 1 + vp(65)a(66)

That is: The annuity for the 65-year old is equivalent to the dollar that he gets today plus an annuity for a 66-year old in one year, if he survives.

From these we conclude that:

vp(65) = 11.3/11.4 = 0.9912

Assuming non-negative interest rates we get bounds on v and p(65) of

.9912<=v<=1 and .9912<= p(65) <= 1.00

In other words, interest rates are almost 0 and age 65 mortality is quite small. Very little can be said about mortalities at other ages.

The formula in Gandalf's second paragraph can now be used to bound the annuity value in that case. Of course, if you know the interest rate assumption you can compute q(65) exactly. Interest rates above 81 basis points will imply negative mortality.

By the way, people seldom forget what they learned (but they generally forget what they memorized).

[Gandalf]

Assuming non-negative interest rates we get bounds on v and p(65) of

.9912<=v<=1 and .9912<= p(65) <= 1.00

In other words, interest rates are almost 0 and age 65 mortality is quite small. Very little can be said about mortalities at other ages.


Given that the annuity factor at 66 is 11.4, we can draw some conclusions about mortalities at other ages: they're a lot higher than age 65.

For example, if you assume that they are a constant multiple of the A2000 male basic table, that multiple is on the order of 2.6 (depending on where in the range of possible values you put v). The multiple at specifically age 65 is 1 or less (again, depending on v).

If the annuity factors are made up, from an exam or study material, that's fine. If these are supposed to be real life values, it's possible that the annuity factors are not as simple as we assumed, which could complicate the formulas. E.g., annuity factors based on non-level interest; annuity factors based on select-and-ultimate mortality; annuity factors based on projected mortality improvements (which is actually done in practice in insurance company valuation of group payout annuities).