Problem 22 - Calculus and its applications

Question

In \(16 \mathrm{yr},\) Claire Beasley is to receive \(\$ 180,000\) under the terms of a trust established by her aunt. Assuming an interest rate of \(4.2 \%,\) compounded continuously, what is the present value of Claire's trust?

Step-by-Step Solution

Answer

The present value of Claire's trust is approximately $91,955.

Step 1: Understand the Formula

For continuous compounding, we use the present value formula \( PV = FV \times e^{-rt} \), where \( PV \) is the present value, \( FV \) is the future value, \( e \) is the base of the natural logarithm (approximately 2.71828), \( r \) is the annual interest rate in decimal form, and \( t \) is the time in years.

Step 2: Identify the Known Values

In this problem, the future value \( FV = 180,000 \), interest rate \( r = 4.2\% = 0.042 \), and time \( t = 16 \) years. We need to substitute these values into the formula.

Step 3: Substitute Values Into the Formula

Insert the known values into the present value formula: \[ PV = 180,000 \times e^{-0.042 \times 16} \].

Step 4: Evaluate the Exponent

Calculate the exponent \(-0.042 \times 16 = -0.672\) and thus the expression becomes \( e^{-0.672} \).

Step 5: Calculate \( e^{-0.672} \)

Using a calculator, compute \( e^{-0.672} \approx 0.51086 \).

Step 6: Solve for Present Value

Multiply the future value by the result from Step 5: \( PV = 180,000 \times 0.51086 = 91,954.80 \approx 91,955 \) dollars. Thus, the present value is approximately \( \$91,955 \).

Key Concepts

Continuous Compounding

Have you ever wondered how your money can grow faster over time? Using a concept called continuous compounding, you can see how an investment can accumulate more effectively. Continuous compounding calculates interest on an investment or loan by applying the interest rate at every moment in time, rather than at set intervals, such as annually or monthly. This idea is akin to the snowball effect, where interest adds on the principal, and then the next interest calculation includes both the principal and the previously generated interest.
The magic behind continuous compounding lies in its formula:

• For present value, we use: \[ PV = FV \times e^{-rt} \] where:
• \(PV\) is the present value,
• \(FV\) is the future value,
• \(e\) stands for the base of the natural logarithm, roughly 2.71828,
• \(r\) is the interest rate (in decimal form),
• and \(t\) is the time in years.

Continuous compounding allows money to grow at an accelerating rate, making it a powerful tool in finance.

Interest Rate

The interest rate is a critical element in the world of finance. It determines how quickly your investment grows. In the case of Claire's trust, the interest rate is given as 4.2%.
To use it in financial formulas, it's essential to convert the percentage to a decimal, which is done by simply dividing by 100. Thus, 4.2% becomes 0.042.
An interest rate can significantly impact how much money you earn or owe over time, showcasing its importance whether you're saving money, taking a loan, or evaluating an investment. It can also vary greatly depending on economic conditions and lender policies. That’s why understanding your rate in the context of your financial goals is essential—both for investments like Claire's and for everyday financial decision-making.

Future Value

The future value represents what an investment or amount of money is expected to be worth after a specific period. In our example, Claire is expected to receive $180,000 in 16 years, which is her future value.
Knowing the future value is crucial for investors, as it helps in planning and understanding how much they can expect to earn or need to pay.
Future value calculations consider aspects like the amount invested, the interest rate applied, and the duration of the investment. A higher interest rate or longer period can increase the future value significantly. This makes future value a handy concept, especially when deciding on long-term investments or savings plans.
Having insight into future value empowers you to visualize how decisions today can affect your financial standing tomorrow.

Exponential Functions

In finance, understanding exponential functions is crucial when dealing with compounding interest, especially continuous compounding. An exponential function describes situations where growth or decay is not constant but instead accelerates or decelerates over time.
The formula for continuous compounding \[ PV = FV \times e^{-rt} \] is a perfect example of an exponential function in action.

• The base \(e\), approximately equal to 2.71828, is fundamental in exponential growth models.
• These functions can represent numerous natural processes, not just financial ones, but any situation where growth accelerates.

Expanding your understanding of exponential functions can drastically improve financial literacy, as they are foundational in evaluating investments, comparing loans, and forecasting financial futures. Their distinct nature highlights how small changes in variables like interest rates or time can lead to significant differences in outcomes.