Problem 40 - Applied Calculus

Question

A business associate who owes you 3000 offers to pay you 2800 now, or else pay you three yearly installments of 1000 each, with the first installment paid now. If you use only financial reasons to make your decision, which option should you choose? Justify your answer, assuming a 6 \%$ interest rate per year, compounded continuously.

Step-by-Step Solution

Answer

Choose Option 2, as it has a higher present value.

Step 1: Understand the Options

You have two payment options: (1) receive $2800 now, or (2) receive $1000 every year for three years (starting today). The decision hinges on the present value of future payments.

Step 2: Calculate Present Value of Option 1

With Option 1, you are offered $2800 immediately. The present value of receiving $2800 now is already $2800, as there is no waiting period for future payments.

Step 3: Introduce Present Value Formula

To compare payments occurring at different times, use the present value formula for continuous compounding: \( PV = FV \times e^{-rt} \), where \( PV \) is present value, \( FV \) is future value, \( r \) is the interest rate, and \( t \) is time in years.

Step 4: Calculate Present Value of Option 2

For the installment option, calculate the present value of each \(1000 payment:- Today (year 0): Present value is \)1000.- Year 1: \( PV = 1000 \times e^{-0.06\times 1} = 1000\times e^{-0.06} \approx 1000\times 0.941764 = 941.76 \).- Year 2: \( PV = 1000 \times e^{-0.06\times 2} = 1000\times e^{-0.12} \approx 1000\times 0.886920 = 886.92 \).The total present value from Option 2 is: \(1000 + 941.76 + 886.92 = 2828.68 \).

Step 5: Compare the Present Values

With a present value of $2800 from Option 1 and $2828.68 from Option 2, Option 2 is financially more advantageous, as it offers a higher present value.

Key Concepts

Financial Decision Making

When making financial decisions, it is essential to weigh different options by considering their present values. Present value helps you assess the value of future payments today. In financial decision making, you aren't just deciding based on face values; you're assessing the worth of money received at different times by converting future cash flows into today's value. This benchmark allows you to make informed choices that align with your financial goals.

Here's why understanding present value is crucial:

• It helps compare financial options that are spread out over time.
• It considers the time value of money, recognizing that funds available today can be invested to earn interest.
• It supports strategic decision-making, ensuring you maximize potential returns.

In the given example, the decision is between taking $2800 now or receiving three payments of $1000 one each year for three years. To decide clearly, each option's monetary value at present should be calculated and analyzed.

Continuous Compounding

Continuous compounding is a method used to calculate interest where the interest is constantly accumulated, leading to the exponential growth of the investment or debt. In other words, the more frequent the compounding, the more interest you'll earn on your investment over time. With continuous compounding, the interest is added to the principal at every possible instant, theoretically. This results in the formula: \[ PV = FV \times e^{-rt} \]
Where:

• \( PV \) is the present value, meaning the worth of future cash flows at the present moment.
• \( FV \) represents the future value, or the amount the cash flow will grow into.
• \( r \) denotes the interest rate, and it dictates how fast the future value grows.
• \( t \) is the time period in years.

Continuous compounding offers a distinctive advantage in the scenario outlined in the original exercise, where future payments are compared to an immediate payment.

Interest Rate Calculation

Interest rates play a significant role in financial decision-making and are crucial when calculating the present value of future payments. In the context of continuous compounding, the interest rate represents the speed at which money grows. The interest rate is a critical factor influencing the exponential growth of investments and loans.Interest rate calculation helps determine:

• The potential growth of an investment.
• The cost of borrowing money.
• Comparative profitability of financial options.

In the practice problem, you used a 6% interest rate compounded continuously to determine the present value of \(1000 received at different times. Each payment underwent the calculation:\[ PV = 1000 \times e^{-0.06 \times n} \] Where \( n \) denotes the respective year (0, 1, or 2). This approach allows you to ascertain the individual present values, sum them up, and make an informed decision comparing it against other financial options like taking \)2800 now.