Amy is now (year 0) 25 years old and planning for her retirement at age 65. She would like to have $2 million for her retirement by then. She will make her first deposit in her retirement savings account next year (year 1) and continue until her last deposit at age 65 (year 40). All deposits are made on the last day of the year. Amy expects her income to increase at an annual rate of 2.64% (nominal) and thus will increase her savings at the same rate. Her savings will earn an annual return of 6.73% (nominal).

To find out how much Amy needs to deposit each year, we must determine the initial deposit amount and the annual increase that matches her goal of $2 million at retirement, considering the annual return and the increase rate of deposits.

Let's define the following variables:
- $ P $ = Initial deposit amount (the amount she deposits at the end of Year 1)
- $ r $ = Rate of return on the retirement savings account (6.73% nominal rate)
- $ g $ = Annual increase rate of Amy's savings (2.64% nominal rate)
- $ n $ = Number of years until retirement (40 years)
- $ FV $ = Future value of the retirement savings (Amy wants \$2,000,000)

### Step-by-Step Calculation

1. **Find the future value of an increasing annuity:**

Amy's savings form an increasing annuity, where each deposit increases by a constant rate $ g $. The formula for the future value of an increasing annuity compounded annually is:

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

where:
- $ P $ = Initial deposit amount
- $ r $ = Annual return rate on investment = 6.73% = 0.0673
- $ g $ = Annual increase rate in savings = 2.64% = 0.0264
- $ n $ = 40 years

We need to solve for $ P $ to ensure that the future value $ FV = 2,000,000 $.

Let's compute $ P $.Amy's initial deposit amount, made at the end of the first year, should be approximately **$7,645.34**. Each subsequent deposit will increase by 2.64% annually to reach her goal of $2 million by retirement.

Would you like further details on this calculation or have any other questions?

### Related Questions to Explore:
1. How would the initial deposit change if the expected return rate was different?
2. What would happen if Amy decided to retire earlier or later than planned?
3. How does the annual increase rate affect the total amount saved over the 40 years?
4. What if Amy wanted to make deposits at the beginning of each year instead?
5. How would inflation impact the real value of the $2 million goal?

**Tip:** Regularly reviewing and adjusting your retirement savings plan can help ensure you stay on track to meet your goals, especially if market conditions change.

Learn how to calculate retirement savings using compound interest and an increasing annuity. Amy plans to save $2 million by making annual deposits that increase over time. Explore the detailed solution with formulas and calculations.

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