Problem 40 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Since he was 22 years old, Ben has been depositing $$\$ 200$$ at the end of each month into a taxfree retirement account earning interest at the rate of \(6.5 \%\) /year compounded monthly. Larry, who is the same age as Ben, decided to open a tax-free retirement account 5 yr after Ben opened his. If Larry's account earns interest at the same rate as Ben's, determine how much Larry should deposit each month into his account so that both men will have the same amount of money in their accounts at age 65 .

Step-by-Step Solution

Answer

Larry should deposit approximately $$\$252.39$$ at the end of each month into his account to have the same amount of money as Ben at the age of 65.

Step 1: Determine the number of payments for Ben and Larry

At age 65, both Ben and Larry will have made payments over different periods. For Ben, he starts saving at age 22. So, he will be making payments for 65 - 22 = 43 years. For Larry, he starts saving at age 27 (5 years after Ben). So, he will be making payments for 65 - 27 = 38 years. As payments are made monthly, we need to convert these years into months: - For Ben, number of payments (n) = 43 years * 12 months = 516 payments - For Larry, number of payments (n) = 38 years * 12 months = 456 payments

Step 2: Calculate the monthly interest rate

The annual interest rate is given as \(6.5\%\). To calculate the monthly interest rate, we'll divide the annual interest rate by 12: \(r = \frac{6.5\%}{12} = \frac{6.5}{100*12} = 0.00542\) (approximately)

Step 3: Calculate the Future Value of Ben's account

Now, we'll calculate the Future Value (FV) of Ben's account at age 65 using the Ordinary Annuity formula: \(FV = P \cdot \frac{(1+r)^n - 1}{r} \) Plugging in the values for Ben: \(FV_B = 200 \cdot \frac{(1+0.00542)^{516} - 1}{0.00542} \) On calculating, we get: \(FV_B \approx \$337410.15\) The Future Value of Ben's account at age 65 is approximately $$\$337410.15$$.

Step 4: Calculate Larry's monthly deposit

We know the Future Value of Ben's account at age 65 and want Larry's account to have the same Future Value. So, we'll use the Ordinary Annuity formula to solve for Larry's monthly deposits (P). Using the same formula and plugging in the values for the future value of Larry's account, number of payments, and the interest rate: \(337410.15 = P \cdot \frac{(1+0.00542)^{456} - 1}{0.00542} \) Now, solve for P: \(P = \frac{337410.15 \cdot 0.00542}{(1+0.00542)^{456} - 1} \) On calculating, we get: \(P \approx \$252.39\) So, Larry should deposit approximately $$\$252.39$$ at the end of each month into his account to have the same amount of money as Ben at the age of 65.

Key Concepts

Compound Interest

Compound interest is a powerful financial concept that allows money to grow over time by earning interest on both the initial principal and the accumulated interest. This compounding effect can substantially increase the value of an investment over a long period.

In the context of Ben and Larry's retirement accounts, compound interest is applied monthly, meaning the interest is calculated each month on the initial principal as well as any interest previously earned.

• Annual interest rate: 6.5%.
• Monthly interest rate: 6.5% divided by 12.
• Interest compounding: monthly.

Because interest is compounded more frequently, even a small monthly interest rate can result in significant growth over several years. The formula used for calculating this growth is embedded in the Ordinary Annuity formula, where compounding plays a key role.

Retirement Savings

Saving for retirement is crucial for ensuring financial security during your non-working years. Starting early is one of the most effective ways to build a substantial retirement fund, as demonstrated by Ben's and Larry's savings journey.

Actions that can impact retirement savings significantly include:

• Starting early: Ben began at 22, which gave him a longer period of compounding interest.
• Consistency: Regular monthly deposits help in accumulating a sizable amount over time.
• Maximizing the interest rate: A higher interest rate accelerates the growth of savings through compound interest.

For both Ben and Larry, despite the difference in start times, their goal is to have equivalent savings by retirement age. Larry compensates for his later start by making higher monthly deposits.

Ordinary Annuity Formula

The Ordinary Annuity formula is used to calculate the future value of a series of equal payments made at regular intervals, such as monthly deposits into a retirement account. This formula helps determine how much regular saving contributes to the accumulated amount. The formula is expressed as: \[FV = P \cdot \frac{(1 + r)^n - 1}{r}\]where:

• **FV** - Future Value of the annuity
• **P** - payment amount per period
• **r** - interest rate per period
• **n** - number of periods

For Ben, his monthly deposits of $$200 were used in this formula to calculate his future retirement savings. For Larry, this formula was used to ascertain how much he needs to save each month to achieve the same future value as Ben.Understanding the Ordinary Annuity formula is fundamental when planning regular savings for long-term goals like retirement. It takes into account factors such as duration, compound interest, and payment amounts to provide a clear picture of future financial standing.