Problem 8 - Calculus and its applications

Question

Find the present value \(P_{0}\) of each amount \(P\) due \(t\) years in the future and invested at interest rate \(k\), compounded continuously. $$ P=\$ 2,000,000, \quad t=20 \mathrm{yr}, \quad k=3.5 \% $$

Step-by-Step Solution

Answer

The present value is approximately $993,200.

Step 1: Understanding the Formula

To find the present value, we use the formula for continuous compounding: \[ P_0 = P imes e^{-kt} \] where \( P \) is the future amount, \( k \) is the interest rate (expressed as a decimal), \( t \) is the time in years, and \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Step 2: Substitute Given Values

We substitute the values \( P = 2,000,000 \), \( k = 0.035 \), and \( t = 20 \) into the formula: \[ P_0 = 2,000,000 imes e^{-0.035 imes 20} \]

Step 3: Calculate the Exponent

Calculate the exponent \(-0.035 imes 20\) which equals \(-0.7\). This gives us: \[ P_0 = 2,000,000 imes e^{-0.7} \]

Step 4: Compute Exponential Term

Evaluate the exponential term \( e^{-0.7} \). Using a calculator, \( e^{-0.7} \approx 0.4966 \). Thus, the equation becomes: \[ P_0 = 2,000,000 imes 0.4966 \]

Step 5: Find the Present Value

Multiply \( 2,000,000 \) by \( 0.4966 \) to find the present value: \[ P_0 = 2,000,000 imes 0.4966 = 993,200 \]

Key Concepts

Continuous Compounding

Continuous compounding brings a unique twist to the world of finance and investing. Unlike simple or periodic compounding, which occur at set intervals, continuous compounding assumes that the interest is constantly calculated and added back to the principal every moment.
In simpler terms, imagine your money growing every second, minute, and day. This method is represented by the mathematical constant \( e \), approximately equal to 2.71828. This constant is used to calculate the exponential growth of investments in scenarios involving constant compounding.
Having the formula \( P_0 = P \times e^{-kt} \) gives us a powerful tool for determining the present value of future cash flows. Here, \( P \) is the future value you wish to have, \( k \) is the interest rate, and \( t \) is the number of years until receipt. The presence of the negative sign in the exponent indicates the process of discounting a future value back to the present.

Exponential Growth

Exponential growth is a concept that can initially seem daunting but is all around us! In finance, it helps us understand how investments can grow over time, especially with compounding.
The principle is straightforward: growth rate remains constant, and the amount grows by that constant percentage.

• This makes the growth curve steepen over time.
• In the context of continuous compounding, exponential growth is driven by the constant \( e \).

When calculating present value using continuous compounding, we're essentially reversing exponential growth. Instead of "compounding forwards" to see how much future value grows, we "discount backwards," shrinking the number to see its worth at a specified earlier date. This duality of growth and discounting is at the heart of understanding exponential functions in finance.

Interest Rate

The interest rate is a pivotal element in any investment or loan decision. It's essentially the cost of borrowing money or the reward for saving money, expressed as a percentage.
Interest rates can be annual, semi-annual, or any other frequency. In continuous compounding, the rate is treated as seamlessly constant over time.

• For present value calculations, this rate \( k \) is converted into a decimal.
• It plays a vital role in determining both compound interest and present value outcomes.

In our formula \( P_0 = P \times e^{-kt} \), the interest rate \( k \) directly affects how much our future value shrinks to become the present value. The bigger the rate, the greater the discount factor becomes, reducing the present value more significantly.