Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that a company expects its annual profits \(t\) years from now to be \(f(t)\) dollars and that interest is considered to be compounded continuously at an annual rate \(r\). Then the present value of all future profits can be shown to be $$ F P=\int_{0}^{\infty} e^{-r t} f(t) d t $$ Find \(F P\) if \(r=0.08\) and \(f(t)=100,000\).

Step-by-Step Solution

Answer

The present value is $1,250,000.

Step 1: Understand the Problem Setup

We are asked to find the present value of future profits given a specific profit function and interest rate. The formula involves an integral that incorporates continuous compounding of interest.

Step 2: Identify Values and Functions

Identify the rate of interest \( r = 0.08 \) and the profit function \( f(t) = 100,000 \) which is a constant function.

Step 3: Set Up the Integral

Write the integral of the formula for present value. The integral is set up as \[FP = \int_{0}^{\infty} e^{-0.08t} \times 100,000 \, dt\]

Step 4: Evaluate the Integral

Since \( f(t) = 100,000 \) is a constant, pull it out of the integral:\[FP = 100,000 \times \int_{0}^{\infty} e^{-0.08 t} \, dt\]

Step 5: Solve the Integral

Evaluate the integral \( \int_{0}^{\infty} e^{-0.08 t} \, dt \). The integral of \( e^{-rt} \) from 0 to \( \infty \) is \[\left[ -\frac{1}{r} e^{-rt} \right]_{0}^{\infty}\].

Step 6: Apply Limits to the Integral

Calculate the limits:\(-\frac{1}{0.08} e^{-0.08t} |_{0}^{\infty} = \left(0 - (-\frac{1}{0.08})\right) \):The value at infinity is 0 and the value at 0 is \( -\frac{1}{0.08}\).

Step 7: Calculate the Present Value

Substitute the result of the integral back into the equation:\[FP = 100,000 \times \frac{1}{0.08}\]Calculate this value to find:\[ FP = 1,250,000 \]

Key Concepts

Continuous Compounding

Continuous compounding is a concept used in finance to determine how much money grows when interest is added at every possible instant. Unlike simple interests, which are calculated at specific intervals, continuous compounding considers the limit where the compounding periods become infinitely small. In mathematical terms, for an amount of money with a principal \( P \) and a continuous compounding rate \( r \), the future value can be represented using the formula:\[ FV = P \cdot e^{rt} \]where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. The concept is powerful for calculating the present value of future profits, as it allows us to capture the growth of these profits over time.

Future Profits

Future profits refer to the earnings a company expects to receive in the coming years. These earnings are crucial for planning and investment decisions, as they can be brought to present value to assess their worth in today's terms. Understanding future profits helps companies anticipate cash flow and allocate resources efficiently. When estimating future profits, it's essential to consider factors such as economic conditions, industry trends, and company strategies. Companies use various mathematical models and assumptions, like the one in your exercise, to predict future profits, making it easier for investors or stakeholders to understand the company's potential growth.

Integral Evaluation

Integral evaluation is the process of calculating the total accumulation of quantities over a continuous range. In the context of the exercise, the integral \[ \int_{0}^{\infty} e^{-0.08t} \cdot 100,000 \, dt \]is used to calculate the present value of a series of future cash flows that occur continuously over time. This specific integral involves exponential decay, and the evaluated result tells us the total value when considering the continuous compounding effect. The forms like \( e^{-rt} \) appear frequently in continuous growth or decay processes, where the rate of change is proportional to the current state. Calculating such an integral involves using limits, and for the given integral, the steps led to a definite value which when calculated gives \( FP = 1,250,000 \).

Exponential Decay

Exponential decay is a mathematical process where quantities decrease at a rate proportional to their current value. In financial mathematics, it's often used to describe the discounting of cash flows to determine their present value. Exponential decay is illustrated by the function:\[ e^{-rt} \]where \( r \) is the decay rate (interest rate in our context), and \( t \) is time. In present value calculations, we're interested in understanding how much future cash flows or profits are worth in today's terms if they shrink at this rate. The effect is like reversing the growth of money under continuous compounding and tells us how money diminishes in value as time progresses. By using exponential decay in our present value formula, we integrate various components' worth at earlier times as they would appear today.