Background Material

A life annuity is a contract where the annuitant receives a payment of $1 at the end of each year that he or she survives in return for a lump sum payment.
The formula for the value of the contract at age 35 is expressed as: a₃₅ = N₃₆ ÷ D₃₅ (÷ = divide)
where D₃₅ = v³⁵ X l₃₅(multiplication sign, X, is usually omitted) and
where v³⁵ = 1 ÷ (1 + i)³⁵ and where i = the interest rate and l₃₅ = the number of people living
at age 35 according to the mortality table being used and where N₃₆ is equal to
D₃₆ + D₃₇ + D₃₈ + .........to end of table.
Usually annuity payments are made once a month rather than once a year at the end of the year.
Obviously the value is different. Here is an explanation to help you understand how much the
difference might be. If the payments were made at the beginning of the year rather than the end
we could quickly recognize the the value would be exactly $1 more than if the payments were made
at the end of the year because then there would be a payment of $1 right away followed by payments
at the end of each year that the annuitant survived. We call a life annuity where the first payment
is made at the beginning of the year a life annuity due. It is represented by ä_x (Note two dots over the a)
and we have have the relationship a_x + 1 = ä_x. Monthly payments would be
worth somewhere between a_x and ä_x so we estimate it at a_x + ½
A temporary 20 year life annuity at age 35, by the way, is represented by
a_35: ₂₀ _| = (N₃₆ - N₅₆) ÷ D₃₅
We usually create commutation columns for each function as follows:
x=Age
v^x = 1 ÷ (1+i)^x
l_x = number living at age x
D_x = v^xl_x
N_x = D_x+1 + D_x+2 + D_x+3 + ........to end of table.

So what shall we do with this?

I have all these functions available for the CM90 Mortality Table which is used by the IRS for it's
actuarial evaluations and I have used 5% as the interest rate. The following functions are
germane to this discussion.
Suppose we have an apartment to rent for $995 a month. Suppose further that we offer the tenant
a choice of renting the apartment or buying life use of it. If he chooses to buy life use we will
charge him $400 a month maintenance fee. By life use I mean charge him a one time
fee and then let him live there for the rest of his life. How much shall we charge him?
Is it not obvious that we are going to pay him a life annuity of ($995 - $400) a month? Isn't he going
to pay us $595 a month more for life if he rents? (assuming he is not going to move out until he dies).
One way to find out how much he should pay is to look up the cost of a $595 life annuity at his age in
the rate book of some insurance company. If he is married we'll look up a joint and survivor life annuity.
We can also do it the hard way with that CM90 mortality table that the IRS uses.
If he is 61 years of age we find that N₆₂ ÷ D₆₁ = a₆₁
All we have to do is add 50^¢ to it and multiply by (595 X 12).
I have done the math...we should charge him $85,037.40.
If he is 81 instead of 61 we should charge him $43,768.20
If he is married we should use a joint life function and charge him something different.
If there are additional benefits, eg. refunds upon death, the values must be calculated.

Are you beginning to feel uneasy?

I am told that a Connecticut lady landlord sold life use of rooms in her home to several tenants who
lived longer than she anticipated. They lived so long that she ran out of funds because she had not
charged them enough money for life use. To extricate herself from her predicament she murdered
some of the tenants. Following this the legislatures of several states enacted legislation
requiring sellers of life use to consult with a Fellow of The Society of Actuaries and charge according to
his/her estimates of the costs. The Connecticut Legislature did not act.
There is another matter, the federal law on discrimination against older
Americans. To charge the same price for life use to an age 81 as an age 61
seems to me to be a problem.
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