Problem 62 - College Algebra

Question

Rachael deposits \(\$ 1,500\) into a retirement fund each year. The fund earns \(8.2 \%\) annual interest, compounded monthly. If she opened her account when she was 19 years old, how much will she have by the time she is 55 ? How much of that amount will be interest earned?

Step-by-Step Solution

Answer

Rachael will have $221,097.22 by age 55, with $167,097.22 being interest.

Step 1: Understand the Problem Requirements

Rachael makes annual deposits into a retirement account with an 8.2% annual interest rate compounded monthly. She wants to know the amount in the account when she is 55, starting at age 19.

Step 2: Calculate the Number of Years

First, calculate the total number of years Rachael will invest in the retirement fund. Since she starts at age 19 and continues until she is 55, the period is 55 - 19 = 36 years.

Step 3: Determine the Compounding Periods

Since the interest is compounded monthly, convert the annual interest rate into a monthly one. The monthly interest rate is 8.2% divided by 12, which is approximately 0.6833% per month.

Step 4: Calculate the Future Value of Annuity

Use the future value of annuity formula: \[ FV = P \left( \frac{(1 + r)^n - 1}{r} \right) \text{, where} \] - \( P \) is the annual deposit (\$1,500)- \( r \) is the monthly interest rate in decimal (0.082/12)- \( n \) is the total number of compounding periods (36 years x 12 months = 432)- Substitute the values into the formula to compute the future value.

Step 5: Compute Total Interest Earned

The total money deposited is \( P \times \text{number of years} = 1500 \times 36 = 54,000 \, \text{dollars} \). The interest earned is the final amount in the account minus the total deposits.

Key Concepts

future value of an annuity

Calculating the future value of an annuity is crucial when you want to plan for long-term financial goals, such as retirement. An annuity involves making regular payments over time, in this case, depositing $1,500 annually into a retirement fund. The future value of an annuity provides an estimate of the total amount you'll accumulate, considering both these regular payments and the interest earned over time.

The formula used is:\[FV = P \left( \frac{(1 + r)^n - 1}{r} \right)\]where:

• \(P\) is the payment amount made in each period.
• \(r\) is the interest rate per period. Since the interest is compounded monthly, we must convert the annual interest rate into a monthly rate. For a rate of 8.2%, this becomes \(\frac{0.082}{12}\).
• \(n\) is the total number of payment periods. Given 36 years until retirement with monthly compounding, this equates to \(36 \times 12 = 432\) payment periods.

Substituting your known values into this formula will give you the total future value of the annuity.

retirement fund

A retirement fund is an essential tool for ensuring financial security in your later years. Establishing one early has the benefit of allowing compound interest to work in your favor. An effective retirement fund strategy involves consistent, timely deposits just as Rachael does, starting at the age of 19 and planning up to the age of 55.

To calculate how much you'll have by retirement, you need to know:

• Your annual deposit amount.
• The interest rate your fund accrues.
• Your investment period.

Rachael's consistent deposits, paired with a steady 8.2% return, showcase how even small, regular contributions can grow substantially over decades. Understanding your projected retirement fund helps outline your financial trajectory and ascertain adequate savings for future needs.

monthly compounding

The concept of monthly compounding is central to understanding how a retirement fund grows over time. Compounding refers to the process where the interest earned on your investment itself earns interest.
Monthly compounding means that this process happens twelve times a year, which can significantly increase the amount of interest earned compared to annual compounding.

Here's why it's impactful:

• The interest is calculated each month on not just your deposits but also any interest that has accumulated from previous months.
• With each passing month, the balance that earns interest grows larger, due to the previous month's interest being added.

In Rachael's case, the annual interest rate of 8.2% is effectively compounded each month, making the effective yield calculation: \(0.082/12\) per month. This compounding effect can exponentially increase your future retirement fund amount.