Problem 18 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Robin, who is self-employed, contributes $$\$5000$$/year into a Keogh account. How much will he have in the account after \(25 \mathrm{yr}\) if the account earns interest at the rate of \(8.5 \% /\) year compounded yearly?

Step-by-Step Solution

Answer

Robin will have approximately $$\$420,480$$ in his Keogh account after \(25\) years.

Step 1: Identify the variables in the problem

In this problem, we have the following variables: 1. \(P = 5000\) (Annual contribution) 2. \(r = 8.5 \% = 0.085\) (Interest rate per year, in decimal form) 3. \(n = 25\) (Number of years)

Step 2: Plug the variables into the future value formula for an ordinary annuity

We will now substitute these values into the annuity formula: \( FV = 5000 \times \frac{(1 + 0.085)^{25}-1}{0.085} \)

Step 3: Calculate the future value

Now, we'll perform the calculations to find the future value of Robin's Keogh account after \(25\) years: \( FV = 5000 \times \frac{(1.085)^{25}-1}{0.085} \) \( FV = 5000 \times \frac{7.1487}{0.085} \) \( FV \approx 5000 \times 84.096 \) \( FV \approx \$420,480 \) So, Robin will have approximately $$\$420,480$$ in his Keogh account after \(25\) years.

Key Concepts

Ordinary Annuity

An ordinary annuity is a series of equal payments made at regular intervals at the end of each period. This is a common financial product used in retirement accounts, like the one Robin contributes to. In the exercise, Robin's Keogh account receives an annual contribution of \(\$5000\). Since the payments are made at the end of each year, this type of annuity is classified as 'ordinary'. It's important to differentiate between an ordinary annuity and an annuity due, where payments are made at the beginning of each period. The future value formula for an ordinary annuity takes into consideration that each contribution has less time to earn interest when compared to the first payment.

For example, the last payment made into Robin's account will only have one year to accrue interest, while the first payment will accumulate interest across the full 25 years.

Present Value

Present value is the current value of a future sum of money or stream of cash flows given a specified rate of return. It's based on the principle that money available now is worth more than the same amount in the future due to its potential earning capacity. If you were to reverse Robin's scenario and wanted to understand what a future amount is worth in today's dollars, you'd calculate the present value. However, for gauging the sum that will be in the account in the future based on current contributions, we use the future value formula rather than present value calculations.

It's a key principle that affects investment and financing decisions, as well as a fundamental idea behind discounted cash flow valuation methods and pension fund management, just like Robin's Keogh account.

Compound Interest

Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. In the context of Robin's account, compound interest means that each year's interest earnings will themselves earn interest in subsequent years. This concept is sometimes referred to as 'interest on interest' and can significantly affect the amount of money saved or owed over time.

Compounded annually, as in the example given, means that the interest is calculated and added to the account balance once a year. The formula incorporates the principle of compound interest which is why we raise the base (1 + interest rate) to the power of the number of compounding periods.

Annual Contribution

The annual contribution refers to the amount of money someone like Robin deposits into an investment account each year. In our scenario, the contribution is fixed at \(\$5000\) per year. When calculating the future value of an annuity, it's critical to consider the effect of each annual contribution. The consistency of these contributions, combined with the power of compound interest, leads to a substantial sum over the course of 25 years.

When planning for retirement or any long-term financial goal, calculating the impact of regular contributions can give you a realistic projection of your financial progress. Understanding this impact is vital for personal financial planning and making informed decisions about investments and savings.

Time Value of Money

The time value of money (TVM) is the concept that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. It highlights the benefits of receiving money sooner rather than later. In the case of Robin, every dollar contributed to the Keogh account now is presumed to be worth more than a dollar in the future because of its potential to earn interest over time. This principle underpins the formula used to calculate the future value of the annuity.

The TVM is pivotal in financial decision-making because it provides the rationale for investing now rather than waiting. The formula used in the calculation accounts for both the timing and the amount of each cash flow, driving home the importance of early and consistent investing.