Suppose that inflation is 2.64% per year and there is no change in Amy's income growth and the return on her savings. What would Amy's first deposit in year 0 have to be in order to achieve the $2,000,000 real value in her retirement savings account balance when she retires?

To find the first deposit Amy needs to make in year 0 to achieve a $2 million *real* value in her retirement savings account by the time she retires at age 65, we need to adjust the future value for inflation. This means that Amy's goal of $2 million should represent the purchasing power equivalent to $2 million today, adjusted for the inflation rate over the 40-year period.

### Step 1: Adjust the future value for inflation
We first calculate the inflation-adjusted future value (real value) of the $2 million:

\[
\text{Real Future Value} = \frac{FV}{(1 + \text{inflation rate})^n}
\]

Where:
- $ FV $ is the nominal future value ($2,000,000),
- $ \text{inflation rate} = 2.64\% $,
- $ n = 40 $ years.

### Step 2: Calculate the required initial deposit
Once we have the real future value, we'll use the same growing annuity formula, but this time with the adjusted target value.

Let's first adjust the future value for inflation, then solve for the initial deposit $ P $.The inflation-adjusted future value, or the real value equivalent to $2 million today, is approximately **$705,284.37**. This means Amy needs to accumulate this amount in today's purchasing power by the time she retires.

### Step 3: Calculate the required initial deposit $ P $
Now, we'll use the real future value in the growing annuity formula to calculate the initial deposit Amy needs to make in year 0 to achieve this real value.

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

Let’s calculate this.The initial deposit Amy needs to make at the end of year 1, adjusted for inflation, is approximately **$2,696.07**. By increasing her savings each year by 2.64% and earning an annual return of 6.73%, this amount will grow to a real value of $2 million in today’s purchasing power by the time she retires.

Would you like to explore this further or ask any other questions? Here are some related questions you might consider:

1. How would changes in the inflation rate affect the required initial deposit?
2. What is the nominal value of her savings at retirement, considering inflation?
3. How does the real value compare if inflation is higher or lower than expected?
4. How can Amy adjust her savings plan if she starts saving later than year 1?
5. What strategies could Amy use to mitigate the impact of inflation on her retirement savings?

**Tip:** It's important to plan for the real value of money when saving for long-term goals, as inflation can significantly erode purchasing power over time.

Explore how to calculate the initial deposit Amy needs to make today to achieve a $2 million real value in her retirement savings account. This problem involves adjusting for inflation over 40 years, using annuity formulas, and considering income growth and savings returns. Learn the step-by-step solution and implications of inflation on long-term financial planning.

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