Problem 46 - Applied Calculus

Question

\(46-48 .\) BUSINESS: Capital Value of an Asset The capital value of an asset is defined as the present value of all future earnings. For an asset that may last indefinitely (such as real estate or a corporation), the capital value is $$ \left(\begin{array}{c} \text { Capital } \\ \text { value } \end{array}\right)=\int_{0}^{\infty} C(t) e^{-r t} d t $$ where \(C(t)\) is the income per year and \(r\) is the continuous interest rate. Find the capital value of a piece of property that will generate an annual income of \(C(t),\) for the function \(C(t)\) given below, at a continuous interest rate of \(5 \%\). $$ C(t)=8000 \text { dollars } $$

Step-by-Step Solution

Answer

The capital value is $160,000.

Step 1: Understand the Problem

We need to calculate the capital value of an asset that generates a constant annual income of $8000 indefinitely, with a continuous interest rate of 5%. The formula given is:\[\text{Capital value} = \int_{0}^{\infty} C(t) e^{-rt} \ dt\]where \(C(t)\) is the income per year and \(r\) is the interest rate.

Step 2: Define Given Values and Parameters

Identify the given values, which are:- \(C(t) = 8000\) dollars/year (constant annual income)- \(r = 0.05\) (5% continuous interest rate) Substitute these values into the capital value integral formula.

Step 3: Set Up the Integral

Substitute the given values into the formula:\[\text{Capital value} = \int_{0}^{\infty} 8000 \cdot e^{-0.05t} \ dt\]This integral represents the present value of a perpetual income stream of $8000 at a 5% interest rate.

Step 4: Evaluate the Integral

Integrate the function:\[\int_{0}^{\infty} 8000 \cdot e^{-0.05t} \ dt = 8000 \int_{0}^{\infty} e^{-0.05t} \ dt\]Integrate \(e^{-0.05t}\) with respect to \(t\):\[\int e^{-0.05t} \ dt = -\frac{1}{0.05} e^{-0.05t} = -20 e^{-0.05t}\]Evaluate this indefinite integral from 0 to \(\infty\):\[\left[-20 e^{-0.05t}\right]_{0}^{\infty}\]Substitute the limits of integration:

Step 5: Substitute Limits and Simplify

Evaluate the definite integral:\[\left[-20 e^{-0.05t}\right]_{0}^{\infty} = -20 \left(e^{-0.05 \times \infty} - e^{-0.05 \times 0}\right)\]Since \(e^{-0.05 \times \infty}\) approaches 0 as \(t\) goes to infinity, the expression simplifies to:\[-20(0 - 1) = 20\]Thus, the integral evaluates to \(20\). Multiply by \(8000\) to find the capital value:\[8000 \times 20 = 160000\]

Step 6: Conclude the Solution

Thus, the capital value of the property is $160,000. This represents the present value of perpetually receiving $8000 annually at a continuous 5% interest rate.

Key Concepts

Present Value

The concept of present value is crucial in finance and investment. It determines how much a future stream of income or cash flows is worth in today's terms. Imagine you're promised $10,000 five years from now. It wouldn't be worth the same as $10,000 today due to the time value of money.
The time value of money principle states that money available now can be invested to earn returns. Therefore, its worth in the future will be different.
The present value is calculated by discounting future cash flows at a specific interest rate. In this exercise's case, the present value of receiving $8,000 annually forever is determined using integration, which accounts for every dollar that will be earned in the future unless the asset is sold, hence making it extremely valuable for calculations of indefinite returns.
By understanding present value, you can make more informed financial decisions and better evaluate investment opportunities.

Continuous Interest Rate

A continuous interest rate is a concept that applies exponential growth or decay over time. Unlike simple or compound interest, continuous interest compounds continuously at every possible moment. This is characterized by the formula involving the constant \(e\), where exponential decay or growth is smooth and uninterrupted.
In our scenario, the continuous interest rate is used to discount future cash flows to their present value. For instance, an interest rate of 5% implies the exponent \(e^{-0.05t}\) in the present value formula.
This rate is crucial as it influences how quickly an investment grows or how significantly future values decrease over time.
In practice, continuous compounding can lead to slightly higher interest accrued than regular compounding, making it important for precise financial calculations and theoretical modeling.

Perpetual Income

Perpetual income refers to a stream of cash flows that indefinitely continue over time. This is common in assets like bonds or real estate, where the income is expected to persist forever or for an undefined long period.
When valuing perpetual income, as with assets in real estate or investments, the present value formula assumes you will receive this income endlessly.
Assets with perpetual income can be particularly appealing since they provide a steady, ongoing revenue stream. The present value integrates these endless payments, calculating the total worth based on a set continuous interest rate.

• This allows investors to determine how much they should pay for a perpetual income-generating asset or compare different investments.
• Understanding perpetual income helps investors make decisions around inheritances, long-term investments, and asset purchases.

It is a core concept in determining the capital value of assets that generate continuous earnings.

Indefinite Integral

An indefinite integral represents the antiderivative of a function, which is essentially the reverse process of differentiation. For evaluating the present value of perpetual income, the integral captures all future cash flows discounted back to today's dollars.
In our exercise, we set up an indefinite integral to find the present value of incomes \( ext{indefinitely}\).
The formula provided for capital value is an integral from zero to infinity, \(\int_{0}^{\infty} C(t) e^{-rt} \, dt\). This integral computes the present value of every infinitesimal increment of time into the future.

• The process involves finding the antiderivative of \(e^{-0.05t}\), which is \(-\frac{1}{0.05} e^{-0.05t}\), and then evaluating the limit as time approaches infinity.
• This is crucial for accurately determining the long-term value of investments with perpetual incomes.

Understanding indefinite integrals is fundamental in many areas of mathematics and finance, allowing for calculations involving continuous change and infinite time horizons.