The amount of an annuity for 10 years is $8000. What is the size of it's annual payment if the rate is 4% compounded annually
To find the size of the annual payment, or the annuity payment, we need to use the formula for the present value of an annuity. The present value of an annuity (PVA) is given by:
PVA = PMT × (1 - (1 + r)^-n) / r
where
- PMT = annual payment (what we are solving for)
- r = annual interest rate (4% or 0.04 in decimal form)
- n = number of periods (10 years)
We have been given that the present value of the annuity (PVA) is $8,000, the annual interest rate (r) is 4%, and the number of periods (n) is 10 years.
Rearranging the formula to solve for PMT:
PMT = PVA / ((1 - (1 + r)^-n) / r)
Let's plug in the values to compute PMT.
PMT = $8,000 / ((1 - (1 + 0.04)^-10) / 0.04)
Now, let's calculate the right part of the equation (1 - (1 + 0.04)^-10) / 0.04:
(1 + 0.04)^-10 = (1.04)^-10 ≈ 0.67556468
1 - (1.04)^-10 ≈ 1 - 0.67556468 ≈ 0.32443532
Now, divide that by the interest rate:
0.32443532 / 0.04 ≈ 8.110883
Finally, to get the PMT:
PMT ≈ $8,000 / 8.110883 ≈ $986.03
Therefore, the approximate size of the annual payment for the annuity is $986.03. Keep in mind that this is an approximation due to rounding, and to get the precise figure, a financial calculator or software that handles such calculations could be used to provide a more accurate answer.