Find the periodic payments PMT necessary to accumulate the given amount in an annuity account. HINT [See Quick Example 2.] (Assume end-of-period deposits and compounding at the same intervals as deposits. Round your answer to the nearest cent.)
$10,000 in a fund paying 4% per year, with monthly payments for 5 years

To find the periodic payments (PMT) needed to accumulate a given amount in an annuity, we use the **future value of an ordinary annuity formula** since the deposits are made at the end of each period. The formula is:

\[
FV = PMT \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $ FV $ = future value of the annuity (\$10,000)
- $ PMT $ = the regular payment amount (what we need to find)
- $ r $ = interest rate per period (monthly interest rate)
- $ n $ = total number of payments (number of periods)

### Given values:
- $ FV = 10,000 $
- Annual interest rate = 4%, so monthly interest rate $ r = \frac{0.04}{12} = 0.003333$
- Duration = 5 years, so $ n = 5 \times 12 = 60 $ periods

### Solve for $ PMT $:
Reorganize the formula to solve for $ PMT $:

\[
PMT = \frac{FV \cdot r}{(1 + r)^n - 1}
\]

Now, plug in the values:

\[
PMT = \frac{10,000 \cdot 0.003333}{(1 + 0.003333)^{60} - 1}
\]

Let's compute this:

1. Calculate $ (1 + 0.003333)^{60} $.
2. Subtract 1 from the result.
3. Multiply the future value $ 10,000 $ by $ 0.003333 $.
4. Divide the product by the result from step 2.

Let's do the calculations:

1. $ (1 + 0.003333)^{60} \approx 1.22139 $
2. $ 1.22139 - 1 = 0.22139 $
3. $ 10,000 \times 0.003333 = 33.33 $
4. $ \frac{33.33}{0.22139} \approx 150.63 $

### Final Answer:
The periodic payment (PMT) is approximately **\$150.63**.

Would you like a more detailed explanation or have any questions?

### Related Questions:
1. How would the payment change if the interest rate were 5% instead?
2. What if the payments were made at the beginning of each period? How would that affect the formula?
3. How is the formula derived for future value of an ordinary annuity?
4. What would be the total amount of money deposited over the 5 years?
5. How much interest is earned in the account after 5 years?

### Tip:
Always ensure you convert annual interest rates to the correct periodic rates when dealing with annuities, based on the frequency of payments (e.g., monthly, quarterly).

Find the periodic payments PMT necessary to accumulate the given amount in an annuity account. $10,000 in a fund paying 4% per year, with monthly payments for 5 years.

Learn how to calculate the monthly payments required to accumulate $10,000 in an annuity account with a 4% annual interest rate over 5 years. This problem uses the future value of ordinary annuities formula and covers the key concepts of interest compounding and annuities.

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