Problem 3 - Calculus Concepts

Question

a. Write the flow rate of the income stream. b. Calculate the 5 -year future value c. Calculate the 5 -year present value. Company C showed a profic of \(\$ 1.8\) million last year. The CEO of the company expects the profit to decrease by \(7 \%\) each year over the next five years and the profits will be continuously invested in an account bearing a \(4.75 \%\) APR compounded continuously.

Step-by-Step Solution

Answer

Flow rate decreases by 7%; future value and present value require integration of profits over 5 years with compounding.

Step 1: Determine the Flow Rate of the Income Stream

The flow rate of the income stream is the continuous amount decreasing over time. Since the profits decrease by 7% each year, the flow rate at time \( t \) can be given by the function: \[ P(t) = 1.8 imes e^{-0.07t} \] million dollars per year.

Step 2: Calculate the Future Value of the Income Stream After 5 Years

To calculate the future value of a continuously decreasing income stream compounded at 4.75%, use the formula for continuous compounding and integration: \[ FV = \int_0^5 P(t) \times e^{0.0475(5-t)} \, dt \]Substitute \( P(t) = 1.8 \times e^{-0.07t} \):\[ FV = \int_0^5 1.8 \times e^{-0.07t} \times e^{0.0475(5-t)} \, dt \]Simplify and compute the integral to find the future value.

Step 3: Calculate the Present Value of the Income Stream Over 5 Years

The present value is the amount you would need to invest now at 4.75% to achieve the same value after 5 years as the income stream. Compute this using:\[ PV = \int_0^5 P(t) \times e^{-0.0475t} \, dt \]Substitute \( P(t) = 1.8 \times e^{-0.07t} \):\[ PV = \int_0^5 1.8 \times e^{-0.07t} \times e^{-0.0475t} \, dt \]Solve the integral for the present value over the duration.

Key Concepts

Future Value Calculation

The concept of future value is all about predicting how much money will grow over time if it is continuously reinvested in an account. This account earns interest at a certain rate. Let's break it down with a simple example related to income streams. Imagine you have profits that are being invested and compounding continuously at a 4.75% annual rate.

To find out the future value after a specific period, say 5 years, you have to consider both the change in the principal amount over time and the interest rate. Hence, an integral calculation is required. The formula used is:

• \[FV = \int_0^5 P(t) \times e^{0.0475(5-t)} \, dt\]

Here, \( P(t) \) refers to the income stream function that decreases exponentially over time, \( e^{-0.07t} \), and \( e^{0.0475(5-t)} \) represents the compounding effect over time period \( t \) to 5 years. You integrate this function from 0 to 5 to compute how large the investment becomes in 5 years.

This process of using integration is essential when dealing with continuous compounding of declining income streams.

Present Value Calculation

Present value helps us determine the current worth of a future income stream, assuming a specific interest rate over time. It can tell you how much you need to invest right now to achieve a desired financial goal in the future.

In the case of continuously compounding interest, the present value calculation involves a similar integration technique as the future value but with a slight twist in the formula. You need to discount back the income stream over the 5 years at the given continuous compounding rate, which in this case is 4.75%.

• \[PV = \int_0^5 P(t) \times e^{-0.0475t} \, dt\]

Substituting our profit equation \( P(t) = 1.8 \times e^{-0.07t} \), the formula adjusts to reflect how much current cash is required today to equate to the future declining income. Solving this integral simplifies our understanding of the real value needed currently to match the expected profit flow over the full period of 5 years.

This calculation is pivotal in finance as it assists in planning investments where income streams may fluctuate over time.

Income Stream Analysis

Income stream analysis is used to evaluate the performance and potential of a revenue-generating entity. It is especially useful in financial decision-making processes, where continuous changes in income streams are prevalent.

In our scenario, the flow rate of the income stream initially requires an accurate depiction of how profits are expected to change over time. Here, profits are predicted to decrease by 7% each year, represented by the function \( P(t) = 1.8 \times e^{-0.07t} \). The continual decrease in profits emphasizes the importance of analyzing the stream of income accurately.

Key takeaways for effective analysis of an income stream include:

• Understanding the rate at which revenue is changing: In this case, a 7% decrease yearly.
• Using appropriate mathematical models: Employing exponential functions to model changes in revenue over time.
• Evaluating future and present values: Ensuring that financial planning aligns with such changes.

Understanding these aspects allows businesses to make informed decisions about current investments and future growth. This analysis provides clarity on how an ongoing income stream will affect financial forecasts.