Problem 2 - 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 B showed a profit of \(\$ 1.8\) million last year. The CEO of the company expects the profit to increase by 0.02 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: \( R(t) = 1.8 + 0.02t \). Use integration to find the future and present values.

Step 1: Determine the Flow Rate of the Income Stream

The flow rate of the income stream is the change in profit per year. The profit last year was \(1.8\) million, and it is expected to increase by \(0.02\) million each year. Therefore, the flow rate can be described by the function: \( R(t) = 1.8 + 0.02t \), where \( t \) represents the number of years.

Step 2: Calculate the 5-Year Future Value

To calculate the future value of a continuous income stream compounded continuously, use the formula: \( FV = \int_{0}^{T} R(t) e^{r(T-t)} \ dt \), where \( T = 5 \), and the annual interest rate \( r = 0.0475 \). Plugging in the flow function, we get: \[ FV = \int_{0}^{5} (1.8 + 0.02t) e^{0.0475(5-t)} \ dt \].First, evaluate the integral part of the formula to find the future value in 5 years.

Step 3: Calculate the 5-Year Present Value

To find the present value of the stream, use the formula: \( PV = \int_{0}^{T} R(t) e^{-rt} \ dt \), with \( r = 0.0475 \) and \( T = 5 \). This changes the integral to:\[ PV = \int_{0}^{5} (1.8 + 0.02t) e^{-0.0475t} \ dt \].Evaluate this integral to find the present value of the income stream for 5 years.

Key Concepts

Future Value Calculation

Future value in calculus involves determining the worth of a current sum at a future date, taking into account a specific rate of return. This is particularly useful when dealing with continuous income streams. The future value (FV) of such a stream, compounded continuously, can be calculated using:\[FV = \int_{0}^{T} R(t) e^{r(T-t)} \ dt\]Where:

• \(R(t)\) is the income stream rate at time \(t\),
• \(e\) is the exponential function,
• \(r\) is the continuous compounding interest rate,
• \(T\) is the total investment period.

In our scenario, we're working with an annual increase in profits, continuously deposited into an account at a 4.75% annual interest rate. By evaluating the integral, we compute the future value in 5 years. This calculation shows how investments grow over time when subject to consistent additions and compounding interest.
Such analysis assists in financial planning and decision making for long-term investments.

Present Value Calculation

Present value is the current worth of a stream of cash flows expected in the future, discounted back to the present using a specific interest rate. The formula for calculating the present value (PV) of a continuous income stream is:\[PV = \int_{0}^{T} R(t) e^{-rt} \ dt\]Where:

• \(R(t)\) is the rate of income at time \(t\),
• \(e\) is the base of natural logarithms,
• \(r\) is the continuous compounding rate,
• \(T\) is the time horizon of the income stream.

By solving this integral with our given parameters—such as a flow rate function and a period of 5 years with a 4.75% interest rate—we determine the present worth of the future income stream. This calculation is vital for understanding how much a stream of future earnings is worth now, aiding businesses in making informed investment choices.

Continuous Compounding

Continuous compounding refers to the scenario where interest is calculated and added to the principal balance of an investment perpetually at every possible moment. This maximizes the potential growth of the investment, as interest is constantly being calculated and included into the new principal balance.In mathematical terms, the effect of continuous compounding is expressed by the formula:\[A = P \, e^{rt}\]Where:

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

Continuous compounding is a critical concept within calculus and financial mathematics because it represents the limit as the compounding frequency becomes infinitely high.
It provides accurate models for growth processes such as investments or loans, offering a precise calculation of future value by accounting for constantly occurring interest additions.