Problem 12 - Precalculus

Question

A finance company offers a promotion on \(\$ 5000\) loans. The borrower does not have to make any payments for the first three years, however interest will continue to be charged to the loan at \(29.9 \%\) compounded continuously. What amount will be due at the end of the three year period, assuming no payments are made? If the promotion is extended an additional three years, and no payments are made, what amount would be due?

Step-by-Step Solution

Answer

After three years, the amount due is \( \$ 12,261.26 \). After six years, the amount is \( \$ 30,097.96 \).

Step 1: Identify the formula

The problem involves continuous compounding interest. The formula for the future value using continuous compounding is given by \( A = Pe^{rt} \), where \( A \) is the amount due at the end of the period, \( P \) is the principal amount (initial loan), \( r \) is the annual interest rate, and \( t \) is the time in years.

Step 2: Calculate amount due after three years

Substitute the values into the formula with \( P = 5000 \), \( r = 0.299 \), and \( t = 3 \). Thus, we have:\[A = 5000 \times e^{(0.299 \times 3)}\]Calculate \( e^{(0.299 \times 3)} \), which is approximately \( e^{0.897} \approx 2.452252 \). Therefore,\[A \approx 5000 \times 2.452252 = 12261.26\]So, the amount due after three years is approximately \( \$ 12,261.26 \).

Step 3: Calculate amount due after six years

For an additional three years, the total time period becomes 6 years. Substitute these values into the formula with \( t = 6 \):\[A = 5000 \times e^{(0.299 \times 6)}\]Calculate \( e^{(0.299 \times 6)} \), which is approximately \( e^{1.794} \approx 6.019592 \). Therefore,\[A \approx 5000 \times 6.019592 = 30097.96\]So, the amount due after six years is approximately \( \$ 30,097.96 \).

Key Concepts

Future Value Calculation

When dealing with finance, especially loans, we often want to know the future value of a sum of money. This future value calculation helps us understand how much a current investment or loan will be worth at a specific time in the future.
In the context of continuous compounding interest, the future value is calculated using the formula \( A = Pe^{rt} \). Here, \( A \) is the future amount you will owe or be paid, \( P \) is the principal or the initial amount, \( e \) is the base of the natural logarithm approximately equal to 2.718, \( r \) is the annual interest rate, and \( t \) is the time period in years.

• Let’s take the example of a \( \\(5000 \) loan. After three years of continuous compounding at a 29.9% interest rate, the future value, or the amount due, would be approximately \( \\)12,261.26 \), calculated as \( 5000 \times e^{0.897} \).
• If this period is extended to six years without any payments, the amount due becomes \( \$30,097.96 \), calculated using \( 5000 \times e^{1.794} \).

Future value calculation is crucial for both borrowers and lenders to understand the financial impact of the terms of a loan or investment.

Exponential Growth

Exponential growth occurs when the increase of a quantity is proportional to its current value. This is a hallmark of interest in continuous compounding scenarios.
The formula \( A = Pe^{rt} \) encapsulates this idea, as the factor \( e^{rt} \) represents the exponential growth of your initial amount over time at an ongoing rate.

• This type of growth is powerful, as it means earnings (or debt) will repeatedly increase at the same rate per time period.
• For example, using a 29.9% continuous interest rate, the money borrowed grows much faster than simple interest.
• Each additional year adds a larger absolute amount as the balance itself becomes a larger base for future interest accumulation.

It's important to understand that exponential growth can quickly escalate balances, making it crucial to recognize that even small changes in rates can have significant impacts over time.

Finance Mathematics

Finance mathematics is the backbone of many economic activities, from evaluating loans to forming investment strategies. It focuses on the calculations and theories guiding money growth, value, and timing.
A significant component revolves around understanding different interest models, such as continuous compounding. These models help predict financial outcomes, optimizing financial decision-making.

• Mathematical expressions like \( A = Pe^{rt} \) allow us to bridge the gap between theoretical predictions and real-world scenarios, helping us understand how money compounds over time.
• Key financial metrics assessed through these calculations include future value, present value, interest rates, and growth factors.
• This knowledge equips individuals and businesses to make informed choices, whether involving taking out loans, investing savings, or managing financial risks.

Mastering finance mathematics empowers anyone to navigate the complexities of financial systems effectively.