You are 

2222 

years old and decide to start saving for your retirement. You plan to save 

$ 4 comma 500$4,500 

at the end of each year​ (so the first deposit will be one year from​ now), and will make the last deposit when you retire at age 

6868. 

Suppose you earn 

4 %4% 

per year on your retirement savings.

**a.** How much will you have saved for​ retirement?

**b.** How much will you have saved if you wait until age 

3232 

to start saving​ (again, with your first deposit at the end of the​ year)?

To answer your retirement savings questions, we need to break down the two parts of the problem and calculate the future value of an annuity for each case.

### **Part a: Saving from Age 22 to Age 68**

You plan to save $4,500 per year from age 22 to age 68, which means you'll make deposits for $ 68 - 22 = 46 $ years. The interest rate on your savings is 4% per year. 

We will use the **future value of an annuity formula** to calculate how much you will have saved by retirement at age 68:

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

Where:
- $ FV $ is the future value of the annuity (the total amount saved),
- $ P $ is the annual payment ($4,500),
- $ r $ is the annual interest rate (4% = 0.04),
- $ n $ is the number of payments (46 years).

### **Part b: Saving from Age 32 to Age 68**

Now, if you start saving at age 32 instead of 22, you will make deposits for $ 68 - 32 = 36 $ years. Again, the interest rate is 4%, and the annual savings remain $4,500.

We will use the same future value of annuity formula but with $ n = 36 $.

Let me calculate the future values for both scenarios.### Results:

- **a.** If you start saving at age 22, by the time you retire at age 68, you will have saved approximately **$570,917.55**.
- **b.** If you start saving at age 32, by the time you retire at age 68, you will have saved approximately **$349,192.41**.

### Key Takeaways:
Starting to save 10 years earlier results in a significantly larger retirement fund due to the power of compound interest.

Would you like more details, or do you have any questions? Here are 5 related questions to deepen your understanding:

1. How would your savings change if the interest rate were 5% instead of 4%?
2. What if you increased your annual savings from $4,500 to $6,000 starting at age 22?
3. How does the time period affect savings when considering different retirement ages?
4. What happens if you make monthly deposits instead of annual ones?
5. Can you calculate how much of the total savings is interest earned?

**Tip:** Starting early allows compound interest to work in your favor, exponentially growing your savings over time.

You are 22 years old and decide to start saving for your retirement. You plan to save $4,500 at the end of each year (so the first deposit will be one year from now), and will make the last deposit when you retire at age 68. Suppose you earn 4% per year on your retirement savings. a. How much will you have saved for retirement? b. How much will you have saved if you wait until age 32 to start saving (again, with your first deposit at the end of the year)?

Learn how to calculate the future value of annuities for retirement savings if you start at age 22 or 32, with annual deposits of $4,500 at a 4% interest rate. Understand the impact of compound interest on long-term savings. Suitable for college-level finance students or advanced high school math learners.

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