Problem 15 - Calculus and its applications

Question

Find the accumulated present value of each continuous income stream at rate \(R(t),\) for the given time \(T\) and interest rate \(k\) compounded continuously. $$R(t)=\$ 800,000, \quad T=20 \mathrm{yr}, \quad k=2.3 \%$$

Step-by-Step Solution

Answer

The accumulated present value is approximately \$12,835,085.

Step 1: Understanding the Problem

We are given a continuous income stream with a constant rate of \( R(t) = \$800,000 \) per year, over a period of \( T = 20 \) years. The interest rate is \( k = 2.3\% \) (0.023 as a decimal) compounded continuously. We need to find the accumulated present value of this income stream.

Step 2: Set Up the Integral for Present Value

The formula for the present value \( PV \) of a continuous income stream is \[ PV = \int_{0}^{T} R(t) e^{-kt} \, dt, \] where \( T \) is the total time period, and \( R(t) \) is the income at time \( t \). In this scenario, \( R(t) \) is constant at \$800,000, so the integral becomes \[ PV = \int_{0}^{20} 800,000 e^{-0.023t} \, dt. \]

Step 3: Evaluate the Integral

First, integrate the function: \[ \int 800,000 e^{-0.023t} \, dt = -\frac{800,000}{0.023} e^{-0.023t} + C. \] Evaluating the definite integral, we substitute the limits of integration, 0 and 20:\[ PV = \left[ -\frac{800,000}{0.023} e^{-0.023t} \right]_{0}^{20}. \]

Step 4: Calculate the Values

Calculate the expression:\[ -\frac{800,000}{0.023} (e^{-0.023(20)} - e^{-0.023(0)}). \]Compute the values:\[ e^{-0.023 \times 20} \approx 0.631, \quad e^{0} = 1. \]So the final expression becomes:\[ PV = -34,782,609.09 (0.631 - 1). \]

Step 5: Final Calculation

Simplify and calculate the final amount:\[ 34,782,609.09 \times (1 - 0.631) \approx 34,782,609.09 \times 0.369 \approx \$12,835,085. \]

Key Concepts

Present Value

The concept of present value is fundamental in finance, and it refers to the current worth of a future sum of money or stream of cash flows given a specified rate of return. This concept relies on the idea that a sum of money today is more valuable than the same sum in the future due to its earning capacity. By calculating the present value, one can determine how much future income is worth today.

The present value is calculated by discounting future cash flows back to their value today using an appropriate interest rate. For a continuous income stream, the present value is found using the integral formula:

• \[PV = \int_{0}^{T} R(t) e^{-kt} \, dt,\]

where:- \( T \) is the total time period,- \( R(t) \) is the income rate as a function of time,- \( k \) is the interest rate.This formula sums up all future incomes, appropriately adjusted for the time value of money.

Continuous Income Stream

A continuous income stream represents cash flows that recur regularly without interruption. In many financial models, such income can be approximated as continuous to simplify calculations and precision. For instance, rental income or ongoing business earnings can be considered continuous if they are consistent over time.

When dealing with a continuous income stream, it is essential to set up the problem correctly. Assign a function \(R(t)\) that models the rate at which income is earned over time. In our exercise, \(R(t) = \$800,000\) per year is constant, which makes the calculations straightforward. By considering this rate constant, we can integrate over the desired period, from 0 to \(T\), the total time, to find the present value.

This setup is important as it helps in determining how much this stream of income is worth today based on the present value formula.

Integration

Integration is a mathematical process of finding the accumulated sum of continuous values. It is one of the core concepts of calculus and involves finding the integral of a function. For problems involving continuous income streams, integration allows us to sum the present value of income over time.

In our problem, we need to integrate the function \(800,000 e^{-0.023t}\) over the interval from 0 to 20 years. This helps us capture the present value of the income:

• Calculate the indefinite integral: \[\int 800,000 e^{-0.023t} \, dt = -\frac{800,000}{0.023} e^{-0.023t} + C,\]
• Select the bounds (0 to 20 years) to evaluate the definite integral by substituting these limits back into the integrated function.

Integration, in this context, not only adds up all future income but also appropriately discounts it back to its present worth.

Interest Rate

Interest rates form the basis for many financial calculations, including the calculation of present value. It represents the cost of borrowing money or the gain from investing. The interest rate for continuous compounding is expressed as a constant rate \(k\), which is applied over every infinitesimal time period.

Using continuous compounding means calculating the present amount of money that would result from a series of continuous investments. This method assumes that interest effectively compounds at every possible instant—ideal for simplifying calculations in mathematics.

• In our problem, the interest rate \(k\) is given as 2.3% or 0.023 in decimal form.
• This rate helps in determining the scale at which future cash flows are reduced to calculate their present value.

Understanding interest rates is crucial, as even small changes in them can significantly impact the computed present value of financial streams.