Suppose payments were made at the end of each month into an ordinary annuity earning interest at the rate of 3.5%/year compounded monthly. If the future value of the annuity after 12 years is $50,000, what was the size of each payment? (Round your answer to the nearest cent.)
To find the size of each payment, we can use the future value of an ordinary annuity formula:
\[ FV = P \times \left( \frac{{(1 + r)^n - 1}}{r} \right) \]
Where:
- FV is the future value of the annuity
- P is the size of each payment
- r is the interest rate per compounding period
- n is the number of compounding periods
In this case, we are given:
- FV = $50,000
- r = 3.5%/year = 0.035/12 per month
- n = 12 years = 12 \times 12 = 144 months
Substituting the values into the formula, we have:
\[ \$50,000 = P \times \left( \frac{{(1 + \frac{{0.035}}{12})^{144} - 1}}{\frac{{0.035}}{12}} \right) \]
To find P, we can rearrange the equation and solve for P:
\[ P = \frac{{\$50,000}}{{\frac{{(1 + \frac{{0.035}}{12})^{144} - 1}}{\frac{{0.035}}{12}}}} \]
Let's calculate this using a calculator:
\[ P ≈ \frac{{\$50,000}}{{\frac{{(1 + \frac{{0.035}}{12})^{144} - 1}}{\frac{{0.035}}{12}}}} \approx \$255.41 \]
Therefore, the size of each payment, rounded to the nearest cent, is approximately $255.41.
To find the size of each payment, we can use the formula for the future value of an ordinary annuity:
FV = P * (1 + r)^n - 1 / r
Where:
FV = Future Value
P = Payment
r = Interest rate per compounding period
n = Number of compounding periods
In this case, the future value (FV) is given as $50,000, the interest rate (r) is 3.5% per year compounded monthly, and the annuity duration (n) is 12 years.
First, we need to convert the interest rate to the rate per compounding period:
r = 3.5% / 12
r = 0.035 / 12
r = 0.00291667
Now, we can substitute the values into the formula:
50,000 = P * (1 + 0.00291667)^12 - 1 / 0.00291667
Next, we need to solve this equation for P.
Rounding to the nearest cent, the size of each payment is approximately $379.82.