Suppose payments were made at the end of each month into an ordinary annuity earning interest at the rate of 8%/year compounded monthly. If the future value of the annuity after 14 yr is $70,000, what was the size of each payment? (Round your answer to the nearest cent.)
This time your formula is
Amount = paym( (1+i)^n - 1]/i
give it a try, following the steps I used in your previous post.
(If you are studying this topic, you should be able to do these type of questions, they are routine questions.)
To find the size of each payment, we can use the future value of an ordinary annuity formula. The formula is:
FV = P * [(1 + r)^n - 1] / r
where:
FV = future value of the annuity
P = size of each payment
r = interest rate per compounding period
n = number of compounding periods
In this case, we are given the future value (FV = $70,000), the interest rate (r = 8%/year = 0.08/12 = 0.00667/month), and the number of compounding periods (n = 14 years * 12 months/year = 168 months). We need to solve for the size of each payment (P).
Plugging in the values into the formula:
$70,000 = P * [(1 + 0.00667)^168 - 1] / 0.00667
To solve for P, we can multiply both sides of the equation by 0.00667:
$70,000 * 0.00667 = P * [(1 + 0.00667)^168 - 1]
466.9 = P * [1.00667^168 - 1]
Now, we need to calculate 1.00667^168 using a calculator or a spreadsheet.
1.00667^168 ≈ 2.5196
So, the equation becomes:
466.9 = P * (2.5196 - 1)
Simplifying further:
466.9 = P * 1.5196
Now, to solve for P, we divide both sides of the equation by 1.5196:
P = 466.9 / 1.5196
P ≈ $307.32
Therefore, the size of each payment, rounded to the nearest cent, is approximately $307.32.