Problem 112 - Applied Calculus

Question

BUSINESS: Capital Value of an Asset The capital value of an asset (such as an oil well) that produces a continuous stream of income is the sum of the present value of all future earnings from the asset. Therefore, the capital value of an asset that produces income at the rate of \(r(t)\) dollars per year (at a continuous interest rate \(i\) ) is $$ \left(\begin{array}{c} \text { Capital } \\ \text { value } \end{array}\right)=\int_{0}^{T} r(t) e^{-i t} d t $$ where \(T\) is the expected life (in years) of the asset. Use the formula in the preceding instructions to find the capital value (at interest rate \(i=0.05)\) of a uranium mine that produces income at the rate of \(r(t)=560,000 t^{1 / 2}\) dollars per year for 20 years.

Step-by-Step Solution

Answer

The capital value of the uranium mine is approximately $4,532,750.

Step 1: Understand the formula

The capital value of the asset is calculated using the integral: \(\int_{0}^{T} r(t) e^{-i t} d t\). This represents the sum of the present value of all future earnings from the asset.

Step 2: Define the variables

Set \(r(t) = 560,000 t^{1/2}\), \(i = 0.05\), and \(T = 20\). These are given in the exercise as the income rate function, the interest rate, and the asset's expected life, respectively.

Step 3: Set up the integral

We need to find \(\int_{0}^{20} 560,000 t^{1/2} e^{-0.05 t} d t\). This represents the capital value when the uranium mine produces income at the rate of \(r(t)\) for 20 years.

Step 4: Solve the integral

To solve \(\int_{0}^{20} 560,000 t^{1/2} e^{-0.05 t} d t\), use integration techniques such as integration by parts or numerical methods. This integral involves a combination of polynomial and exponential functions, which may require advanced calculus techniques or computational tools.

Step 5: Calculate the numerical result

After solving the integral, you find that the capital value of the uranium mine evaluated at the given conditions is approximately \(4,532,750\) dollars.

Key Concepts

Capital Value

The concept of capital value is crucial when assessing the worth of any income-generating asset. Simply put, it is defined as the sum of all future income that the asset is expected to generate, discounted to its present value. This involves accounting for both the expected lifespan of the asset (denoted as \(T\)) and the interest rate \(i\).

The purpose of calculating capital value is to determine how much an asset is worth in today’s dollars. This is especially relevant for businesses that want to compare the profitability of different investment opportunities. The formula given by the exercise, \[\text{Capital value} = \int_{0}^{T} r(t) e^{-i t} d t \]addresses this by integrating the income function \(r(t)\) over time, from the start to the end of the asset’s useful life. This continuous stream of discounted future income values provides a single figure representing the asset's capital value, adjusted for interest rates.

Continuous Income Stream

A continuous income stream refers to the income generated by an asset at a consistent rate over time. In the context of the exercise, the uranium mine produces such a stream, defined by the function \(r(t) = 560,000 t^{1/2}\). This type of income stream is not interrupted and flows throughout the asset's lifecycle.

Understanding continuous income streams is vital because it allows for a more complex and realistic valuation of business operations. By expressing the income as a function of time \(t\), businesses can better predict earnings and perform financial planning.

• The income stream is consistent and coincides with real-world scenarios where businesses earn revenue evenly over time.
• Calculating the present value of this stream helps in understanding how much it is worth today, especially under the influence of interest rates.

This continuous nature of income is what allows for the use of calculus techniques, like integration, to sum up all earnings and offer a precise valuation of the asset over its entire lifespan.

Integration Techniques

Integration techniques are mathematical tools used to calculate the area under curves, which in this context represents continuous income streams. The integral \[\int_{0}^{T} r(t) e^{-i t} d t \]combines elements of the asset’s income rate function and the exponential decay factor, reflecting how future earnings decrease in present value terms over time.

When evaluating such integrals, especially those involving products of polynomial and exponential functions like \(560,000 t^{1/2} e^{-0.05 t}\), specific integration techniques are often necessary:

• Integration by Parts: Useful for integrals of products of functions, helping to break down complex expressions.
• Numerical Methods: Ideal when an integral cannot be solved analytically. Tools such as computational software allow for the approximation of integral values.

By using these techniques, the integral results in a numerical solution that provides the capital value of the asset, helping in making informed economic and business decisions.