Suppose you start saving for retirement when you are 35 years old. You invest ​$6 comma 000 the first year and increase this amount by 4​% each year to match inflation. How much money will you have saved after 20 years if the interest rate earned on the retirement account is 8​% per​ year?
b) On the other​ hand, you will have saved ​$
  
enter your response here if the interest rate was 4​% instead of the original 8​%.

To solve this problem, we need to account for two key factors:

1. **Growing annual contributions**: You contribute $6,000 the first year, but this contribution increases by 4% each year to match inflation. This forms a growing annuity.

2. **Compound interest**: The retirement account earns interest over time, either at 8% or 4%, depending on the scenario.

### Part (a) — Interest rate of 8% per year:
We’ll first calculate the future value of the contributions with an 8% interest rate. Since the annual contributions grow by 4% each year, we’ll use the formula for the **future value of a growing annuity**:

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

Where:
- $ P = 6,000 $ (the initial contribution)
- $ r = 0.08 $ (the interest rate per year)
- $ g = 0.04 $ (the growth rate of contributions)
- $ n = 20 $ (the number of years)

### Part (b) — Interest rate of 4% per year:
Now we repeat the calculation with a lower interest rate of 4%. We’ll apply the same formula, but now $ r = 0.04 $.

Let me calculate both parts.It seems that my formula encountered a division error because the interest rate $ r $ and the growth rate $ g $ are equal in the second scenario ($ r = g = 0.04 $).

To resolve this, we use a modified formula for when $ r = g $, which is:

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

I'll apply this correction and calculate the values for both scenarios again.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, if you'd like, I can guide you through the process of calculating this manually or clarify any details. Would you like that?

Here are 5 related questions you might want to explore:
1. How do you calculate the future value of an annuity without growing contributions?
2. What happens if contributions stop growing but still accumulate interest?
3. How would different interest rates impact retirement savings over 30 or 40 years?
4. How does inflation affect retirement savings in real terms?
5. Can you calculate retirement savings with irregular contributions?

**Tip**: Understanding the formula for the future value of a growing annuity helps predict long-term savings accurately.

Suppose you start saving for retirement when you are 35 years old. You invest $6,000 the first year and increase this amount by 4% each year to match inflation. How much money will you have saved after 20 years if the interest rate earned on the retirement account is 8% per year? On the other hand, how much will you have saved if the interest rate was 4% instead of 8%?

Discover how to calculate your retirement savings when contributions grow by 4% annually, and the interest rate is either 8% or 4%. This detailed guide walks you through using formulas for the future value of a growing annuity. Suitable for students and professionals in finance or math.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation