Problem 1 - Applied Calculus

Question

Find the present and future values of an income stream of \(\$ 3000\) per year over a 15 -year period, assuming a \(6 \%\) annual interest rate compounded continuously.

Step-by-Step Solution

Answer

The present value is approximately $32,684.18, and the future value is approximately $60,973.88.

Step 1: Understanding Present Value of an Income Stream

The present value (PV) of an income stream involves finding the current worth of a series of payments received in the future when considering a specific interest rate. Here, we use the formula for continuous compounding.

Step 2: Continuously Compounded Present Value Formula

The present value of the income stream can be calculated using the formula: \( PV = P \times \int_{0}^{T} e^{-rt} \ dt \), where \(P\) is the annual payment, \(r\) is the interest rate, and \(T\) is the time period. Given: \( P = 3000, r = 0.06, T = 15 \).

Step 3: Solving the Integral for Present Value

Evaluate the integral \( \int_{0}^{15} e^{-0.06t} \ dt \), which is equivalent to \( \left[-\frac{1}{0.06} e^{-0.06t}\right]_{0}^{15} \). Calculate this to find the present value.

Step 4: Calculating the Present Value

The integral evaluates to \( \left[-\frac{1}{0.06} e^{-0.06 \times 15} + \frac{1}{0.06} e^{-0}\right] = \left[-\frac{1}{0.06} e^{-0.9} + \frac{1}{0.06}\right]\). Calculate each part using a calculator to find the approximate value.

Step 5: Determining the Present Value (PV)

Substitute back into the formula with \( P = 3000 \) to get \( PV = 3000 \times \left[\frac{1}{0.06}(1 - e^{-0.9})\right] \). Using a calculator, determine the result.

Step 6: Understanding Future Value of an Income Stream

The future value (FV) of an income stream refers to the amount of money an income stream will grow to by the end of a specified period, at a given interest rate and compounding condition.

Step 7: Future Value Formula with Continuous Compounding

The future value of the income stream is calculated by: \( FV = P \times e^{rT} \). For each payment, time until the end of the term diminishes from 15 years to 1 year.

Step 8: Applying the Formula for Each Year

To calculate the future value, sum the value for each future payment term. Calculate for each year: \( 3000 \times e^{0.06 \times (15-n)} \) for \( n = 0 \) to \( n = 14 \).

Step 9: Calculate and Sum Future Values

Utilize the formula for each year and add them together using a calculator to obtain the total future value at the end of 15 years.

Key Concepts

Continuous Compounding

Continuous compounding refers to the process where interest is added to the principal balance of an investment instantly, at every moment. It results in the highest possible amount of interest being accumulated on a given principal, given the same nominal rate and time period, compared to other compounding frequencies like annual, semi-annual, or monthly.
Here's why continuous compounding is powerful:

• Interest is calculated and added in tiny increments, theoretically an infinite number of times within any time period.
• It maximizes returns compared to other compounding methods because it works on the principle of earning "interest on interest" continuously.

A fundamental mathematical constant that plays a role in continuous compounding is Euler's number, represented as \( e \). This is used in exponential calculations, enabling continuous compounding formulas.

Present Value Formula

The present value (PV) formula for continuous compounding helps in understanding how much a future income stream is worth today. Understanding this concept is key for investors and analysts who want to determine the current value of future cash flows, like payments or receipts.
For money compounded continuously, the present value is given by the integral: \[PV = P \times \int_{0}^{T} e^{-rt} \, dt\] where:

• \(P\) is the annual income or cash flow.
• \(r\) is the annual interest rate (expressed as a decimal).
• \(T\) is the duration of the income stream in years.

This formula involves integrating an exponential decay function, which reflects the process of discounting future cash flows back to present value terms.

Integral Calculus

Integral calculus plays a crucial role in computing present values when interest is compounded continuously. Integral calculus allows us to evaluate integrations, which are fundamental in calculating accumulated income streams over a period.
In the context of the present value formula using continuous compounding, integration helps determine how incremental future payments accumulate or discount over time:

• We compute the integral \(\int_{0}^{T} e^{-rt} \, dt\) to evaluate the total discounting effect over the timeline \(T\).
• Solving such an integral gives profound insight into how returns grow and helps finesse exact values for financial forecasting.

This involves mathematical rigor but becomes simpler when using integration rules, especially for exponential functions.

Future Value Formula

The future value (FV) formula in the context of continuous compounding helps investors estimate how much an income stream will amount to at a specific future point. It provides a measure of how initial investments grow over time.
For an income stream compounded continuously, the future value is calculated using: \[FV = P \times e^{rT}\] This formula implies that:

• Each payment grows at a rate \( r \) for the period of \( T \) years.
• The exponential function \( e^{rT} \) accounts for the compounding's cumulative effect over time.

When calculating the future value of a stream of periodic payments, each amount is compounded for a decreasing number of years until the term's end. Adding these gives the total future value at a given time horizon.