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 determine the amount Amy should deposit in her retirement savings account each year to reach her goal of $2 million by age 65, we need to calculate the required initial deposit and how it increases annually due to her expected income growth. Here are the key variables and steps we'll use:

### Given Information:
- **Current age of Amy (Year 0):** 25 years
- **Retirement age:** 65 years
- **Time until retirement:** $ 65 - 25 = 40 $ years
- **Future Value (FV) required at retirement:** $2,000,000
- **Annual return on savings (nominal):** 6.73%
- **Annual increase in savings (nominal):** 2.64%
- **Number of deposits:** 40 (from year 1 to year 40)

### Understanding the Problem:
Amy will make 40 annual deposits into her retirement account. Each deposit will increase by 2.64% each year due to her increasing income. The account earns an annual nominal return of 6.73%. We need to find the initial deposit (let's call it $ D_1 $) that will grow over time to $2 million by the end of year 40.

### Steps to Calculate the Initial Deposit:

1. **Convert nominal rates to effective annual rates:**

   The effective annual rate (EAR) adjusts for compounding, but since Amy's deposits are made annually, we can directly use the nominal rates for our calculation.

2. **Formulate the future value of a growing annuity:**

   The future value (FV) of a growing annuity, where the payment grows at a constant rate $ g $ and the account earns an annual return $ r $, is given by:

   \[
   FV = D_1 \times \frac{(1 + r)^{n} - (1 + g)^{n}}{r - g}
   \]

   Where:
   - $ D_1 $ = Initial deposit
   - $ r $ = Annual return on savings (6.73% or 0.0673)
   - $ g $ = Annual increase in savings (2.64% or 0.0264)
   - $ n $ = Number of deposits (40)

3. **Plug in the values to solve for $ D_1 $:**

   Given that $ FV = 2,000,000 $, we solve for $ D_1 $:

   \[
   2,000,000 = D_1 \times \frac{(1 + 0.0673)^{40} - (1 + 0.0264)^{40}}{0.0673 - 0.0264}
   \]

   Let's calculate this step by step.

### Calculation:

1. **Calculate the terms:**
   - $ (1 + 0.0673)^{40} $
   - $ (1 + 0.0264)^{40} $
   - $ r - g = 0.0673 - 0.0264 = 0.0409 $

2. **Solve for $ D_1 $:**

Let's perform the calculation to find $ D_1 $.The initial deposit $ D_1 $ that Amy should make in her retirement savings account next year (Year 1) is approximately **$7,645.34**.

### Summary:
Amy needs to deposit $7,645.34 in her first year and increase this amount by 2.64% annually to achieve her goal of $2 million by the time she retires at age 65.

### Follow-up Questions:
1. How much will Amy's deposit be in her 10th year?
2. What is the total amount Amy will have deposited by the time she retires?
3. How does the final amount compare if the annual return on savings is 5% instead of 6.73%?
4. What would the initial deposit be if Amy planned to retire at age 60 instead of 65?
5. How would the required deposit change if her expected income increase was 3% instead of 2.64%?

### Tip:
To effectively plan for retirement, it's crucial to periodically review and adjust your contributions based on changes in income, expected return rates, and financial goals.

Learn how to calculate retirement savings using the Future Value of a Growing Annuity formula. Amy, aged 25, plans to save $2 million by age 65 with annual deposits increasing by 2.64% and earning a 6.73% annual return. Discover the initial deposit and annual increase needed to achieve this goal.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation