At the end of each quarter, a 52-year-old individual puts $1300 in a retirement account that pays 7.3% interest compounded quarterly. When the individual reaches age 60, what is the value of the account? ﻿﻿﻿﻿If no further deposits or withdrawals are made to the account, what is the value of the account when the individual reaches age 65?

This problem involves calculating the future value of an annuity (for the first part) and then the future value of a lump sum (for the second part). Let's break this into two parts:

### Part 1: Value at Age 60
The individual is making regular quarterly contributions of $1300 for 8 years (from age 52 to 60) into an account that compounds interest quarterly at an annual rate of 7.3%. The future value $FV$ of an annuity can be calculated using the formula:

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

Where:
- $P$ is the quarterly contribution ($1300),
- $r$ is the quarterly interest rate (7.3% annual compounded quarterly, so $r = \frac{7.3\%}{4} = \frac{0.073}{4} = 0.01825$),
- $n$ is the total number of quarters (8 years × 4 quarters per year = 32 quarters).

Now, let’s calculate the future value of the account when the individual reaches age 60.

### Part 2: Value at Age 65
After reaching age 60, the individual stops making deposits. The balance from age 60 will grow with the same interest rate (7.3% compounded quarterly) for an additional 5 years (from age 60 to 65), but no further deposits are made. The future value of a lump sum is given by:

\[
FV = A \times (1 + r)^n
\]

Where:
- $A$ is the account balance at age 60,
- $r$ is the same quarterly interest rate (0.01825),
- $n$ is the total number of quarters for 5 years (5 years × 4 quarters per year = 20 quarters).

Let me calculate the values for both parts.### Results:
1. The value of the retirement account when the individual reaches age 60 is **$55,830.94**.
2. If no further deposits or withdrawals are made, the value of the account when the individual reaches age 65 is **$80,161.03**.

Would you like further details or have any questions?

Here are five related questions to expand on this topic:
1. How would the future value change if the contribution amount increased each quarter?
2. What if the interest rate was compounded monthly instead of quarterly?
3. How would delaying the retirement savings by 2 years affect the final amount?
4. What if the individual started saving at age 45 instead of 52?
5. How would early withdrawals impact the account's growth?

**Tip:** Starting early in retirement savings allows compound interest to work in your favor for a longer time, significantly increasing your final balance.

Calculate the future value of a retirement account with quarterly contributions of $1300 at 7.3% interest compounded quarterly. Learn how to determine the account value at age 60 and the future value at age 65 if no further deposits are made. Ideal for high school students learning about compound interest and annuities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation