
Acme Annuities recently offered an anuity that pays 6.3 percent counpouned monthly. What equal monthly deposit should be made into this annuity in order to have $167,000 in 12 years?


To solve this problem, we will use the formula for the future value of an ordinary annuity:

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

Where:
- $ FV $ is the future value of the annuity (which is $167,000 in this case),
- $ P $ is the monthly deposit (the unknown we are trying to find),
- $ r $ is the monthly interest rate (annual rate divided by 12),
- $ n $ is the total number of periods (months in this case).

### Step 1: Identify the given values
- Future value $ FV = 167,000 $
- Annual interest rate $ 6.3\% = 0.063 $
- Time period = 12 years
- Compounding frequency = monthly

### Step 2: Calculate the monthly interest rate
The monthly interest rate is:

\[
r = \frac{0.063}{12} = 0.00525
\]

### Step 3: Calculate the total number of periods
Since the payments are made monthly for 12 years, the total number of periods is:

\[
n = 12 \times 12 = 144 \text{ months}
\]

### Step 4: Plug the values into the future value formula and solve for $ P $
The future value equation is:

\[
167,000 = P \times \frac{(1 + 0.00525)^{144} - 1}{0.00525}
\]

Now, let's solve this step-by-step.

#### First, calculate $ (1 + r)^{144} $:
\[
(1 + 0.00525)^{144} = 2.106977
\]

#### Now, compute the rest of the formula:
\[
\frac{2.106977 - 1}{0.00525} = 210.0928
\]

#### Finally, solve for $ P $:
\[
P = \frac{167,000}{210.0928} = 795.03
\]

### Final Answer:
The equal monthly deposit that should be made into the annuity is **$795.03**.

---

Would you like more details on any part of this calculation or another example? Here are 5 related questions you could explore:

1. What happens if the interest rate changes to 5% compounded monthly?
2. How much would you need if the annuity only lasted for 10 years?
3. What would the total interest earned be over 12 years?
4. How does the result change if deposits are made at the beginning of each month instead?
5. What if you wanted to reach $200,000 instead in the same time period?

**Tip:** The longer the investment period, the more benefit you gain from compounding interest!

Acme Annuities recently offered an annuity that pays 6.3 percent compounded monthly. What equal monthly deposit should be made into this annuity in order to have $167,000 in 12 years?

Learn how to calculate the equal monthly deposit needed for an annuity that pays 6.3% compounded monthly to reach $167,000 in 12 years. This finance-related problem covers key concepts like compound interest, annuities, and algebra. Suitable for college or advanced high school students.

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