Problem 27 - Calculus and its applications

Question

An athlete signs a contract that guarantees a \(\$ 9\) -million salary 6 yr from now. Assuming that money can be invested at \(4.7 \%,\) with interest compounded continuously, what is the present value of that year's salary?

Step-by-Step Solution

Answer

The present value is approximately \( 6.7932 \) million dollars.

Step 1: Identify the Formula

To find the present value of a future amount of money with continuously compounded interest, we use the formula: \[ PV = FV imes e^{-rt} \]where \( PV \) is the present value, \( FV \) is the future value, \( r \) is the annual interest rate (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

In this exercise, the future value \( FV \) is \( 9 \) million dollars, the rate \( r \) is 4.7% (or 0.047 as a decimal), and the time \( t \) is 6 years. Substitute these values into the formula:\[ PV = 9 imes e^{-0.047 imes 6} \]

Step 3: Calculate the Exponent

Calculate the value of the exponent:\[ -0.047 imes 6 = -0.282 \]

Step 4: Calculate the Value of \( e \) Raised to the Exponent

Calculate \( e^{-0.282} \). Using a calculator, you find that:\[ e^{-0.282} \approx 0.7548 \]

Step 5: Calculate the Present Value

Substitute the calculated value back into the formula to get the present value:\[ PV = 9 imes 0.7548 \]

Step 6: Complete the Calculation

Proceed by multiplying:\[ PV = 6.7932 \] million dollars. Therefore, the present value of the \( 9 \)-million-dollar salary is approximately \( 6.7932 \) million dollars.

Key Concepts

Continuously Compounded Interest

Continuous compounding is a concept in finance and mathematics where interest is calculated and added to the principal continuously, allowing the amount to grow at every possible instant. This is different from typical compounding methods like annual or monthly compounding, which occur at set intervals.

In continuously compounded interest, the formula used is \[ A = P \cdot e^{rt} \]where:

• \( A \) is the amount of money accumulated after time \( t \), including interest.
• \( P \) is the principal amount (the initial amount of money).
• \( r \) is the annual interest rate expressed as a decimal.
• \( t \) is the time in years.
• \( e \) is Euler's number (approximately 2.71828), which serves as the base for natural logarithms.

This method of compounding maximizes growth as the interest calculation happens at every small interval, theoretically infinite times per year. Therefore, the principal grows faster under continuous compounding compared to periodic compounding methods.

Understanding continuous compounding is crucial for calculations like present value and future value, as it helps in accurately assessing investments and loans where time and rates play a vital role.

Future Value

Future value is the monetary value of an investment or cash flow at a specific point in the future, considering a certain interest rate over the time period.

To determine the future value of an amount of money with continuous compounding, you can use the formula:\[ FV = PV \cdot e^{rt} \]where:

• \( FV \) is the future value.
• \( PV \) is the present value or the initial amount of money.
• \( r \) is the annual interest rate as a decimal.
• \( t \) represents the time in years.
• \( e \) is Euler's number.

The future value signifies how much the current investment will grow over time with a given interest rate.

Future value is fundamental in assessing the profitability of investments and in planning long-term financial goals. It enables predictions about the growth of money when left to compound, which is essential for retirement planning, analyzing savings plans, or understanding debts.

Natural Logarithm

The natural logarithm is a specific type of logarithm with the base \( e \), where \( e \) is approximately equal to 2.71828.

Natural logarithms are denoted as \( \ln(x) \) and are frequently used in various fields, including science, engineering, and finance, because of their properties.

• Natural logarithms are essential in solving problems involving continuous growth or decay, as they help find the time required for an investment to reach a particular value.
• They simplify the equations used in continuous compounding because of the direct relationship with the base \( e \).
• Natural logarithms help solve the problem of finding the exponent when the base \( e \) is involved. This is crucial for reversing the continuous compounding formula, allowing one to solve for time, interest rate, or other variables.

In essence, the natural logarithm is a powerful mathematical tool when working with exponential growth and decay patterns, including continuously compounded interest. Its intimate relationship with continuous growth functions like those involving Euler's number \( e \) makes it indispensable in finance.