Problem 4 - Applied Calculus

Question

If you need $$\$ 20,000$$ in your bank account in 6 years, how much must be deposited now? The interest rate is \(10 \%\), compounded continuously.

Step-by-Step Solution

Answer

Approximately $10,981.45 must be deposited now.

Step 1: Understand the Problem

We need to determine the present value of a deposit that will grow to $20,000 in 6 years at an interest rate of 10% compounded continuously.

Step 2: Identify the Formula

For continuous compounding, the future value can be calculated using the formula \( FV = PV \times e^{rt} \), where \( FV \) is the future value, \( PV \) is the present value, \( r \) is the interest rate, and \( t \) is the time in years. We need to find \( PV \).

Step 3: Rearrange the Formula

Rearrange the formula to solve for \( PV \): \( PV = \frac{FV}{e^{rt}} \).

Step 4: Substitute Known Values

Substitute the known values into the formula: \( FV = 20000 \), \( r = 0.10 \), \( t = 6 \).

Step 5: Calculate the Exponential Term

Calculate the value of the exponential part \( e^{(0.10 \times 6)} \). This gives approximately 1.8221.

Step 6: Calculate Present Value

Plug the exponential value into the rearranged formula: \( PV = \frac{20000}{1.8221} \), which simplifies to approximately $10,981.45.

Key Concepts

Present Value Calculation

Calculating the present value of a future sum is essential when making financial decisions. It helps determine how much money you need to set aside today to reach a certain goal in the future. In our continuous compound interest scenario, we aim to find how much we must deposit now to have \(20,000 in six years.

The main idea is to use the present value formula for continuous compounding, which is: \[ PV = \frac{FV}{e^{rt}} \]Where:

• \( PV \) is the present value we want to calculate.
• \( FV \) is the future value or the amount we want to receive, which is \)20,000.
• \( r \) is the annual interest rate; in this case, 0.10 for 10%.
• \( t \) is the number of years until the money matures, which is 6 years.

First, we calculate the exponential term \( e^{rt} \). Then, we divide the future value \( FV \) by this exponential part to find the present value. This method lets us work backward, ensuring the deposited amount grows to meet our financial target.

Exponential Growth Formula

Exponential growth plays a pivotal role in compound interest calculations, especially when interest is compounded continuously. Unlike simple interest, exponential growth considers the constant accumulation of interest, which compounds at every possible moment.

For our purpose, the formula \( FV = PV \times e^{rt} \) helps us understand how a present sum of money grows over time under continuous compounding. This formula illustrates:

• How financial scenarios involve growth proportional to the current amount.
• That interest not only grows but also compounds on itself continuously.
• The use of \( e \), the base of the natural logarithms, which represents the quantity of continuous growth.

By understanding this formula, you can visualize how an initial deposit will expand over time, resulting in a higher final balance when continuously compounding.

Interest Rate Applications

Interest rates impact how quickly investments grow. For continuous compounding, the interest rate directly influences the rate at which your deposited amount increases over time.

In our scenario, the given annual interest rate is 10%, or 0.10 expressed as a decimal. The unique aspect of continuous compounding means the interest is not just calculated annually, quarterly, or monthly, but at every conceivable moment.

• This results in a slightly higher amount of interest earned compared to periodic compounding at the same nominal rate.
• Continuous compounding is useful in theoretical finance models to illustrate maximum potential growth.
• It demonstrates how sensitive the growth of money is to changes in the interest rate as even a small change can compound significantly over time.

Understanding interest rates in this context aids in leveraging them effectively for financial planning. It's crucial for estimating how much the current funds must be to reach particular investment goals in the future.