The following table shows the average returns for some of the largest mutual funds commonly found in retirement plans. (Assume end-of-month deposits and withdrawals and monthly compounding, and assume that the quoted rate of return continues indefinitely.)
Mutual
Fund	Fidelity
Growth
Company	Vanguard
500 Index	PIMCO
Total Return	Vanguard
Total Bond
Market Index
Rate of
Return	14.83%	13.25%	2.77%	3.67%
Type	Stock fund	Stock fund	Bond fund	Bond fund
How much would be accumulated after 35 years in a retirement account invested entirely in the Vanguard stock fund with payments of $480 per month? HINT [See Quick Example 1.] (Round your answer to the nearest cent.)

To determine how much will be accumulated after 35 years with monthly payments of $480 into the **Vanguard 500 Index stock fund**, we will use the **future value of an annuity formula**. This formula is appropriate when you are making regular payments and earning interest on those payments.

The formula for the future value of an ordinary annuity is:

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

Where:
- $FV$ is the future value of the annuity (the total accumulated value).
- $P$ is the monthly payment ($480 in this case).
- $r$ is the monthly interest rate (annual rate divided by 12).
- $n$ is the total number of payments (number of years multiplied by 12).

### Step-by-step solution:

1. **Identify the variables**:
   - Monthly payment ($P$): 480
   - Annual interest rate ($r_{\text{annual}}$): 13.25% = 0.1325
   - Monthly interest rate ($r$): $\frac{0.1325}{12} = 0.0110417$
   - Time ($t$): 35 years
   - Number of months ($n$): $35 \times 12 = 420$

2. **Apply the values to the formula**:

\[
FV = 480 \times \frac{(1 + 0.0110417)^{420} - 1}{0.0110417}
\]

3. **Calculate the future value**:

   - First, calculate $ (1 + 0.0110417)^{420} $:

     \[
     (1.0110417)^{420} \approx 74.679
     \]

   - Now, subtract 1 from this value:

     \[
     74.679 - 1 = 73.679
     \]

   - Now, divide by the monthly interest rate $0.0110417$:

     \[
     \frac{73.679}{0.0110417} \approx 6673.405
     \]

   - Finally, multiply by the monthly payment of $480:

     \[
     480 \times 6673.405 \approx 3,202,034.40
     \]

### Final answer:
After 35 years, the total amount accumulated in the retirement account would be approximately **$3,202,034.40**.

Would you like further details or explanations? Here are some related questions:

1. How would the result change if the interest rate was slightly lower or higher?
2. What if the monthly payment was increased to $600, what would the accumulated amount be?
3. How does monthly compounding affect the accumulation compared to annual compounding?
4. What happens to the total accumulation if the investment is in a bond fund with a lower return rate?
5. How does the length of the investment period (e.g., 30 or 40 years) affect the total accumulation?

**Tip:** The power of compounding becomes much more pronounced over longer periods of time, as you can see in this example!

How much would be accumulated after 35 years in a retirement account invested entirely in the Vanguard stock fund with payments of $480 per month? Assume monthly compounding and an annual return rate of 13.25%.

Find out how much you would accumulate in a retirement account with monthly $480 deposits in the Vanguard stock fund over 35 years. Learn the calculations using compound interest and the future value of an annuity formula, assuming a 13.25% annual return rate. Ideal for high school students in Grades 11-12.

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