As stated earlier, an annuity is a series of payments and
the present value of the annuity is the total of the present values of each of
the individual payments. To find the present value of a series of payments where
each payment is made only if the designated payer or recipient is alive to pay
or receive it, we can now use the method for calculating the present value of a
single payment.
For Example:
The present value at age 35 of a life annuity of $100 per year for three years
(with the first payment due at age 36) can be represented by the following line
diagram:
The present value of each of
the three payments can be calculated individually, as follows:
The Payment Due at Age 36
Basic equation:
The Payment Due at Age 37
Basic equation:
The Payment Due at Age 36
Basic equation:
The present value at
age 35 of this annuity is the total of these three expressions. The common
multiplier ($100) can be factored out. The fractions to be added together have
a common denominator (l35). Hence the present
value of the annuity can be expressed as:
The numerator of this
expression represents the total to be paid out to the survivors at each age,
with each such amount being discounted at interest to the evaluation date. The
denominator represents the number of persons alive on the evaluation date,
among whom this total present value to be paid in must be allocated.
If, for example, given
table and 3% interest were used, the present value of the annuity would be
calculated as follows:
Age (x)

lx

35

9373807

36

9350279

37

9325594

38

9299482

From above:
To Illustrate: Using the given table
and 6% interest, calculate the present value at age 45 of a life annuity of $30
per year for four years, first payment due at age 46.
Age (x)

lx

45

9048999

46

9000587

47

8948114

48

8891204

49

8829410

Solution:
The line diagram for
this life annuity appears as follows:
$30 $30 $30 $30
Age 45 *
46 47 48 49
The expression
for the present value will have a numerator representing the total to be paid
out to the survivors at each age, with each such amount being discounted at
interest to the evaluation date:
Thank you for posting these equations! These will help calculate the value of specific payments. I have been looking for something like this to truly understand my annuity. Thanks!
ReplyDeleteJeff
So I just heard from a friend that you can sell your annuity, how does that work? I was going to look into it.
ReplyDelete