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

We need to find the present value of future profits, given that the interest rate is compounded continuously at an 8% rate and profits are modeled by the constant function \(f(t) = 100,000\). We will use the formula given for the present value which is an integral.

Step 2: Set up the integral

Plug the values \(r = 0.08\) and \(f(t)=100,000\) into the present value formula. The integral becomes:\[FP = \int_{0}^{\infty} e^{-0.08t} \times 100,000 \, dt\]This represents the present value of continuous future income discounted by an 8% interest rate.

Step 3: Simplify the integration

The integral simplifies to:\[FP = 100,000 \int_{0}^{\infty} e^{-0.08t} \, dt\]This is because the \(100,000\) is a constant and can be factored outside the integral.

Step 4: Solve the integral

We need to solve the integral \(\int_{0}^{\infty} e^{-0.08t} \, dt\). The antiderivative of \(e^{-0.08t}\) is \(-\frac{1}{0.08}e^{-0.08t}\). Evaluate this from 0 to infinity:\[-\frac{1}{0.08}\left[ e^{-0.08t}\right]_0^{\infty}\]

Step 5: Evaluate the antiderivative at the bounds

First, evaluate the limit as \(t\) approaches infinity for \(e^{-0.08t}\), which is 0, then evaluate at 0:\(-\frac{1}{0.08}(0 - 1) = \frac{1}{0.08}\).This results from:- \(\lim_{t \to \infty} e^{-0.08t} = 0\)- \(e^{-0.08 \times 0} = 1\)

Step 6: Calculate the present value

Substituting our solution from the integral back, we get:\[FP = 100,000 \times \frac{1}{0.08} = 100,000 \times 12.5 = 1,250,000\]The present value is thus \(1,250,000\) dollars.

Key Concepts

Integral Calculus

In the exercise above, integral calculus plays a crucial role in determining the present value of future profits. With integral calculus, we can sum up an infinite number of small quantities, which in this case represents the discounted future profits over time.
When we see an integral symbol like this \(\int_{0}^{\infty} e^{-0.08t} \times 100,000 \, dt\), it signifies that we are calculating the total present value by accumulating everything from time \(t = 0\) to infinity.
Integrals are used to calculate areas under curves on a graph. For financial problems like this, it's about finding the value of cash flows that will happen continuously over time.
By solving the integral, we find out how much all those future cash flows are worth right now.

Continuous Compounding

Continuous compounding is a fundamental concept in this exercise, as it refers to the formula used to calculate the future value of an asset. Unlike simple or annual compounding, continuous compounding assumes that interest is constantly being added.
In our integral, \(e^{-0.08t}\), the term \(-r\) in the exponent, represents the continuous compounding rate, which is 8% or 0.08 annually in this problem.
Continuous compounding means that even the smallest increment of time has its interest, giving us a more exact calculation of the present value of the future cash flows. It results in a smoother, continuous increase in your investment value over time.
Thus, the exponential function \(e^{-rt}\) in the integral steps is the heart of continuous compounding mathematics.

Antiderivative

An antiderivative essentially reverses the process of differentiation. In this context, finding an antiderivative helps us solve the integral integral calculation required in this problem.
The integral \(\int_{0}^{\infty} e^{-0.08t} \, dt\) depends on finding the antiderivative of \(e^{-0.08t}\), which is \(-\frac{1}{0.08}e^{-0.08t}\).
When solving integrals, you're effectively asking for a function whose derivative results in the given function. By applying antiderivation, we can evaluate the integral from 0 to infinity.
With the antiderivative calculated, we substitute back into our integral limits to compute the final present value.

Discounted Cash Flow

The concept of discounted cash flow (DCF) is central to determining the present value of future cash flows like those of profits expected by a company. DCF allows us to estimate what a certain amount of money in the future is worth in today's terms.
We apply a discount rate—in this case, the continuous compounding rate of 8%—which accounts for the time value of money. The time value of money suggests that money presently available is worth more than the same amount in the future due to its potential earning capacity.
The formula \(FP = \int_{0}^{\infty} e^{-0.08t} f(t) dt\) expresses this, taking \(f(t) = 100,000\) for every year as the future constant profit, which is discounted back to the present value.
In essence, DCF combines the other mathematical tools discussed to provide a comprehensive financial evaluation.