Problem 23 - Calculus and its applications

Question

At age \(35,\) Rochelle earns her MBA and accepts a position as vice president of an asphalt company. Assume that she will retire at the age of \(65,\) having received an annual salary of \(\$ 95,000,\) and that the interest rate is \(5 \%,\) compounded continuously. a) What is the accumulated present value of her position? b) What is the accumulated future value of her position?

Step-by-Step Solution

Answer

Present value: \(\$1,476,129\); Future value: \(\$6,615,230\).

Step 1: Understanding the Scenario

Rochelle earns a constant annual salary of \( \$95,000 \) for 30 years, from age 35 to age 65. We want to determine the accumulated present value at age 35 and the accumulated future value at age 65, using a continuous interest rate of \( 5\% \).

Step 2: Formula for Present Value

The present value of a continuous annual salary \( C \) over \( t \) years, at an interest rate \( r \), is given by the formula: \[ PV = C \times \frac{1 - e^{-rt}}{r} \] Here, \( C = 95000 \), \( r = 0.05 \), and \( t = 30 \) years.

Step 3: Calculating Present Value

Substituting values in the present value formula: \[ PV = 95000 \times \frac{1 - e^{-0.05 \times 30}}{0.05} \] Evaluating \( e^{-0.05 \times 30} \) gives \( e^{-1.5} \approx 0.2231 \).Thus, the present value is: \[ PV = 95000 \times \frac{1 - 0.2231}{0.05} \approx 95000 \times 15.5382 \approx 1,476,129 \]

Step 4: Formula for Future Value

The future value of a continuous annual salary \( C \), compounded continuously, is given by: \[ FV = C \times \frac{e^{rt} - 1}{r} \] Using \( C = 95000 \), \( r = 0.05 \), and \( t = 30 \) years.

Step 5: Calculating Future Value

Substituting the values in the future value formula: \[ FV = 95000 \times \frac{e^{0.05 \times 30} - 1}{0.05} \] Calculating \( e^{1.5} \approx 4.4817 \), thus:\[ FV = 95000 \times \frac{4.4817 - 1}{0.05} \approx 95000 \times 69.634 \approx 6,615,230 \]

Step 6: Conclusion

The accumulated present value of Rochelle's position is approximately \( \\(1,476,129 \) and the accumulated future value is approximately \( \\)6,615,230 \).

Key Concepts

Present Value

Present value (PV) is an important concept in finance that helps us understand the current worth of a future income or cash flow, given a specific interest rate. In Rochelle's scenario, her future earnings across 30 years are taken into consideration to estimate how much they are worth as of age 35.

- The formula used to calculate the present value for continuous compounding is: \[PV = C \times \frac{1 - e^{-rt}}{r}\]where: - \(C\) is the annual salary, here \(95,000\) - \(r\) is the annual interest rate, here \(0.05\) - \(t\) is the total number of years, here \(30\)

By plugging the values into the formula, we find that the present value of Rochelle's earnings as of age 35 is approximately \(1,476,129\). This number signifies the worth today of the total earnings she expects to make over her career, assuming they are influenced by the continuous 5% interest rate.

Continuous Compounding

Interest can be compounded in various ways, but continuous compounding is a special case that assumes interest compounds at every possible moment.

- With continuous compounding, growth is calculated using the exponential function, ensuring more frequent interest application than any other form of compounding.

The formula for continuous compounding is expressed as:\[A = PV \cdot e^{rt}\]- where \(A\) is the future value,- \(PV\) is the present value,- \(r\) is the interest rate,- \(t\) is the time in years.

Notice that in the problems of accumulated value discussed, we work with exponential functions, like \(e^{rt}\) and \(e^{-rt}\). This demonstrates how continuous compounding effectively uses the power of calculus to maximize the compounding frequency infinitely. In practical terms, it provides a more detailed approximation in financial computations.

Interest Rate Calculation

Calculating interest rates can greatly affect how we understand both the present and future values of earnings or investments. In Rochelle’s case, the interest rate is a key variable in determining how her future salary streams convert into present and future values.

- The interest rate is always depicted as a percentage and influences how money grows over time.- It allows us to observe the potential growth by relating to how quickly money can be expected to accrue if reinvested continuously.

In Rochelle's example, a continuous compounding at a 5% interest rate helps estimate both the present and future values over 30 years. For present value, it uses \(-rt\), indicating discounting back in time, while for future value, it applies \(+rt\), illustrating growth forward in time.

Understanding these dynamics helps in making sound financial decisions, whether you're evaluating savings or assessing investments, by showing how small changes in interest rates can significantly impact long-term outcomes.