Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Karen has been depositing $$\$ 150$$ at the end of each month in a tax-free retirement account since she was \(25 .\) Matt, who is the same age as Karen, started depositing $$\$ 250$$ at the end of each month in a taxfree retirement account when he was 35 . Assuming that both accounts have been and will be earning interest at the rate of \(5 \% /\) year compounded monthly, who will end up with the larger retirement account at the age of 65 ?

Step-by-Step Solution

Answer

Karen will end up with a larger retirement account, as her account's future value at the age of 65 is \(\$334,986.37\), compared to Matt's account future value, which is \(\$304,052.46\).

Step 1: Calculate the number of payments for each person

We need to find out how many payments Karen and Matt will make until they reach the age of 65. For Karen: She started depositing at the age of 25, so the number of payments would be: Payments = (65 - 25) * 12 = 40 * 12 = 480 For Matt: He started depositing at the age of 35, so the number of payments would be: Payments = (65 - 35) * 12 = 30 * 12 = 360

Step 2: Calculate the monthly interest rate

The annual interest rate is 5% compounded monthly, so we need to calculate the equivalent monthly interest rate. Monthly interest rate, \(r\), can be calculated using: \(r = (1 + annual\_interest\_rate)^{\frac{1}{12}} - 1\) Plugging in the annual interest rate of 0.05: \(r = (1 + 0.05)^{\frac{1}{12}} - 1 \approx 0.004074\)

Step 3: Calculate the future value of annuity for Karen and Matt

We will use the future value of annuity formula, which is given by: \(FV = P \frac{(1 + r)^{n} - 1}{r}\) For Karen: Dollar Payment Amount, \(P = 150\) Payments, \(n = 480\) Monthly Interest Rate, \(r = 0.004074\) Plugging in the values: \(FV_K = 150 \frac{(1 + 0.004074)^{480} - 1}{0.004074} \approx \$334,986.37\) For Matt: Dollar Payment Amount, \(P = 250\) Payments, \(n = 360\) Monthly Interest Rate, \(r = 0.004074\) Plugging in the values: \(FV_M = 250 \frac{(1 + 0.004074)^{360} - 1}{0.004074} \approx \$304,052.46\)

Step 4: Compare the account values

Karen's account future value: \(\$334,986.37\) Matt's account future value: \(\$304,052.46\) Since Karen's account has a higher future value at the age of 65, Karen will end up with a larger retirement account.

Key Concepts

Compound Interest

Understanding compound interest is crucial when it comes to growing your savings over time. It's the process where the interest you earn on an investment is reinvested to earn more interest, leading to exponential growth. Think of it as 'interest on interest' which can significantly increase the value of your initial investment.

In the monthly savings example, an interest rate of 5% per year compounded monthly means that each month's interest is calculated not only on the initial principal amount but also on the accumulated interest from the previous months. As a result, the account balance grows at an increasing rate, especially over long periods. This is why it's beneficial for Karen and Matt to start saving early—the longer their contributions compound, the larger the future value of their annuities.

The math behind it employs the compound interest formula where the future value (\(FV\)) depends on the regular payment (\(P\)), the interest rate per period (\(r\)), and the total number of payments (\(n\)): \[FV = P \dfrac{(1 + r)^n - 1}{r}\] This formula helps to calculate exactly how much savings like Karen's and Matt's can grow over time.

Retirement Savings

Retirement savings are critical for a secure financial future. By setting aside a portion of your income regularly, you create a financial cushion that can support you when you are no longer earning a regular income from employment. To maximize the benefits of retirement savings, it is generally advisable to start as early as possible. The power of compounding interest works best when it has more time to operate, meaning that the earlier you start saving, the more you benefit from the interest accumulating on your savings over time.

In our example, although Matt saves a larger amount per month than Karen, he starts 10 years later. This delay results in a considerable difference in the end balance of their retirement accounts, emphasizing the importance of early and consistent savings contributions. Such strategies can be part of individual planning or employer-sponsored retirement plans, like 401(k) plans in the United States.

Time Value of Money

The time value of money (TVM) is a financial concept that tells us money available today is worth more than the same amount in the future due to its earning capacity. This principle underscores the idea of earning interest and how money can grow over time. For instance, if you can earn a 5% return per year on your investment, a dollar invested today will be worth more in the future because of the interest it will earn.

Applying the time value of money to the example of Karen and Matt, we see Karen's smaller but earlier deposits accumulate to a larger sum than Matt's larger, later contributions. This illustrates TVM's impact—Karen's money has more time to work for her, compounding and growing due to being invested longer. When planning for things like retirement, the time value of money is a motivating factor to begin investing as soon as possible.

Monthly Contributions

Monthly contributions are the amounts of money you add to your savings or investment accounts regularly, typically every month. These contributions are key to building significant savings, especially when paired with compound interest. It's a strategy that can be particularly effective for long-term goals, such as retirement planning, as it takes advantage of time and the compounding effect.

For example, consistent monthly contributions enable you to spread your investment over time. This not only makes it more manageable compared to lump-sum investments but also allows you to benefit from dollar-cost averaging, reducing the impact of market volatility. Furthermore, making regular contributions can become a disciplined savings habit, ensuring you stay on track with your financial goals, like Karen, who diligently made her monthly contributions starting at a younger age than Matt, ultimately ending up with a larger retirement savings account.