Problem 59 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

DEBT REPAYMENT You have a debt of \(\$ 10,000\), which is scheduled to be repaid at the end of 10 years. If you want to repay your debt now, how much should your creditor demand if the prevailing annual interest rate is: a. \(7 \%\) compounded monthly b. \(6 \%\) compounded continuously

Step-by-Step Solution

Answer

a. \$ 4980.52\, b. \$ 5488.14\

Step 1: Understanding Present Value Concept

To determine how much should be paid now to repay the debt, we need to calculate the present value (PV) of the debt. Present value is the current worth of a future sum of money given a specific interest rate.

Step 2: Formula for Monthly Compounding

For compounding interest monthly, the present value formula is: \[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \ \text{where} \ FV = 10000 \ r = 0.07 \ n = 12 \ t = 10 \]

Step 3: Calculate Monthly Compounded Present Value

Plugging the values into the formula: \[ PV = \frac{10000}{(1 + \frac{0.07}{12})^{12 \times 10}} = \frac{10000}{(1 + 0.005833)^{120}} = \frac{10000}{2.006768} \approx 4980.52 \]

Step 4: Formula for Continuous Compounding

For compounding interest continuously, the present value formula is: \[ PV = FV \times e^{-rt} \ \text{where} \ FV = 10000 \ r = 0.06 \ t = 10 \]

Step 5: Calculate Continuously Compounded Present Value

Plugging the values into the formula: \[ PV = 10000 \times e^{-0.06 \times 10} = 10000 \times e^{-0.6} \approx 10000 \times 0.5488 = 5488.14 \]

Key Concepts

Compounded Interest

Interest that is calculated on the initial principal, which also includes all the previously accumulated interest from prior periods, is known as compounded interest. This is a versatile and widely used concept in finance.
When interest is compounded, you calculate the interest for each period, add it to the principal and then calculate interest for the next period on this new total. Here's why it's important:

• Increases the total amount of interest paid or earned
• Makes savings grow faster
• Impacts loan repayment schedules significantly

To calculate compounded interest, you need to know:

• The principal amount: The initial amount you borrow or invest
• The interest rate: Usually represented as an annual percentage rate
• The number of compounding periods: How often the interest is added to the principal

Monthly Compounding

Monthly compounding is when the interest is calculated and added to the principal every month. This means 12 compounding periods in a year. Using the monthly compounding formula, you can calculate how much you owe or how much your investment will grow.
The formula used is:

Continuous Compounding

This is the theoretical limit of compounded interest as the compounding periods grow infinitely smaller. The interest is compounded an infinite number of times per period. Thus, instead of applying interest monthly, daily, or hourly, it's applied continuously.
The formula to calculate the present value when interest is compounded continuously is:

Debt Repayment Calculation

Debt repayment calculations are crucial for understanding how much you owe at different times and how your payments will be structured. If you intend to repay a debt before its due date, you'll need to calculate its present value. It shows you the current worth of all future payments.
For different compounding methods, the calculations will slightly differ: