Problem 5 - 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 E showed a profit of \(\$ 1.8\) million last year. The CEO of the company expects the profit to decrease by 0.04 million dollars each year over the next 5 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: \(1.8 - 0.04t\). Calculate integrals for future and present values.

Step 1: Define the Income Stream

The income stream can be represented by a function, \( R(t) \), where \( t \) is the time in years. Since the profit decreases by 0.04 million each year from an initial stream of 1.8 million, the flow rate of the income stream can be written as \( R(t) = 1.8 - 0.04t \) million dollars per year.

Step 2: Calculate the 5-Year Future Value

To calculate the future value, integrate the income flow over the interval from 0 to 5 years. The future value \( FV \) is given by: \[ FV = \int_0^5 R(t) e^{r(5-t)} \, dt \] Where \( r \) is the continuous compound rate, 4.75%, or 0.0475 when expressed as a decimal. Compute this integral.

Step 3: Simplify the Integral for Future Value

Substitute \( R(t) = 1.8 - 0.04t \) into the integral: \[ FV = \int_0^5 (1.8 - 0.04t) e^{0.0475(5-t)} \, dt \] This can be split into two integrals: \[ FV = \int_0^5 1.8 e^{0.0475(5-t)} \, dt - \int_0^5 0.04t e^{0.0475(5-t)} \, dt \] Evaluate these integrals separately.

Step 4: Calculate the 5-Year Present Value

The present value is found by integrating the income stream without compounding over the 5-year period: \[ PV = \int_0^5 R(t) e^{-0.0475t} \, dt \] Where the negative sign in the exponent accounts for discounting back to present value. Substitute \( R(t) = 1.8 - 0.04t \) and evaluate the integral.

Step 5: Evaluate Future and Present Integrals

Using a software tool or integration techniques, compute the value of both integrals for \( FV \) and \( PV \). Express the results in millions. To find the future value, evaluate both parts of the \( FV \) integral and sum them. Do the same for the present value, integrating \( PV \) as a whole.

Key Concepts

Future Value Calculation

The concept of future value (FV) is vital when examining how money can grow over time through investments. In our exercise, future value determines how much the gradually diminishing profits from Company E will amount to after being continuously invested for five years.

To calculate the future value, we need to account for the income stream, represented by the function \( R(t) = 1.8 - 0.04t \). Here, \( t \) is the time in years, meaning profits decrease over time. The future value is found by integrating this income stream from 0 to 5 years, applying continuous compounding using an annual percentage rate (APR) of 4.75%.

The formula for the future value is:
\[ FV = \int_0^5 R(t) e^{r(5-t)} \, dt \]
Where: - \( R(t) \) is our income function, \( 1.8 - 0.04t \) - \( r \) is the continuous compound rate as a decimal, \( 0.0475 \)

This approach takes each year's income, adjusts for continuous interest from the time it was earned until the end of five years, and accumulates it. By solving this integral, you determine the total future wealth from these investments.

Present Value Calculation

The present value (PV) brings future cash flows into today's monetary value framework. It assesses what those decreasing profits are worth right now, considering that money today typically holds more purchasing power than the same amount in the future.

To find the present value of Company E's income stream over five years, integrate the function \( R(t) = 1.8 - 0.04t \) while discounting each amount back to the present using a continuous discount rate of 4.75%.

The formula for present value is:
\[ PV = \int_0^5 R(t) e^{-0.0475t} \, dt \]
Here: - \( R(t) \) is our profit function, \( 1.8 - 0.04t \) - The \(-0.0475t \) factor in the exponential reflects the compounding effect over time

This step involves calculating how much each year's adjusted value contributes to the total present worth of the company's future earnings, providing a sum that reflects today's value of future income.

Continuous Compounding

Continuous compounding is an essential concept in finance, influencing how investments grow at the highest potential rate. It involves calculating interest in a manner that recognizes continually earned interest on prior interest—it is the natural growth model of financial investments.

In our problem scenario, Company E’s decreasing profits are continuously compounded as they are reinvested at an APR of 4.75%. Continuous compounding uses the mathematical constant \( e \), approximately 2.71828, to reflect this ongoing accumulation.

The general formula for continuous compounding is given by:
\[ A = Pe^{rt} \]
Where:

• \( A \) is the amount of money accumulated after n years
• \( P \) is the principal amount (initial investment)
• \( r \) is the annual interest rate (as a decimal)
• \( t \) is the time the money is invested for, in years

By compounding continuously, each slice of time multiplies the investment a bit more, simulating constant growth and leading to higher returns compared to simple or even regular interval compounding methods. This characteristic is used to project both the future and present values in our exercise, enhancing the understanding of how continuously reinvested profits evolve.