sabelle wants to save for retirement. She earns $4400 in income each month, and wishes to deposit 10% of her income into a savings acount each month. If the savings account has a nominal interest rate of 8%, compounded monthly, how much will be in the account if she retires in 38 years?

To calculate how much Isabelle will have in her retirement account in 38 years, we can use the **Future Value of an Annuity formula**. Since she is making regular monthly deposits into an account that compounds interest monthly, this formula is appropriate.

The formula for the future value of an annuity compounded monthly is:

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

Where:
- $ P $ is the monthly deposit.
- $ r $ is the monthly interest rate (annual nominal rate divided by 12).
- $ n $ is the total number of deposits (number of years multiplied by 12).
- $ FV $ is the future value of the annuity.

### Step 1: Calculate the monthly deposit
Isabelle deposits 10% of her monthly income:
\[
P = 0.10 \times 4400 = 440 \text{ dollars per month}
\]

### Step 2: Determine the monthly interest rate
The nominal annual interest rate is 8%, so the monthly interest rate $ r $ is:
\[
r = \frac{8\%}{12} = \frac{0.08}{12} = 0.0066667
\]

### Step 3: Determine the total number of deposits
Isabelle will save for 38 years, and since she deposits every month:
\[
n = 38 \times 12 = 456 \text{ months}
\]

### Step 4: Plug values into the formula
\[
FV = 440 \times \frac{(1 + 0.0066667)^{456} - 1}{0.0066667}
\]

Let’s calculate this step by step.

#### 1. Calculate $ (1 + r)^{n} $:
\[
(1 + 0.0066667)^{456}
\]

#### 2. Subtract 1 from the result:
\[
(1 + 0.0066667)^{456} - 1
\]

#### 3. Divide by $ r $:
\[
\frac{(1 + 0.0066667)^{456} - 1}{0.0066667}
\]

#### 4. Multiply by $ P $:
\[
FV = 440 \times \frac{(1 + 0.0066667)^{456} - 1}{0.0066667}
\]

Let me calculate this for you.Isabelle will have approximately **$1,299,896.44** in her retirement account after 38 years.

Would you like more details or have any questions?

### Related Questions:
1. How would the future value change if Isabelle saved for 40 years instead?
2. What if the interest rate was 6% instead of 8%? How would that affect the future value?
3. What if Isabelle only saved 5% of her income each month? How much would she have in 38 years?
4. How does compounding frequency affect the future value of the account?
5. How much would Isabelle have if she made an initial lump sum deposit in addition to her monthly deposits?
6. What if Isabelle decided to increase her monthly contributions by 2% every year?
7. How would inflation impact the purchasing power of her future savings?
8. What is the effect of taxes on the interest earned in the savings account?

**Tip:** Regularly reviewing your savings strategy and adjusting contributions based on changes in income or financial goals can help maximize retirement savings.

Discover how to calculate the future value of retirement savings with regular monthly deposits and compound interest. Use the Future Value of an Annuity formula to determine Isabelle's retirement account balance after 38 years, with a monthly deposit of $440 and an 8% annual interest rate compounded monthly.

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