Problem 47 - 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 \(5 \%\). $$ C(t)=50 \sqrt{t} \text { thousand dollars } $$

Step-by-Step Solution

Answer

The capital value of the property is 1000 thousand dollars.

Step 1: Understand the Formula

The formula for the capital value of an asset, given the continuous income stream, is \( V = \int_{0}^{\infty} C(t) e^{-r t} \, dt \). Here, \( C(t) \) represents the income function, and \( r \) is the continuous interest rate.

Step 2: Substitute the Given Function and Rate

We have \( C(t) = 50 \sqrt{t} \). The interest rate \( r \) is \( 5\% \) or 0.05 when expressed as a decimal. Substitute these into the formula:\[ V = \int_{0}^{\infty} 50 \sqrt{t} e^{-0.05t} \, dt \]

Step 3: Find the Indefinite Integral

To solve the integral \( \int 50 \sqrt{t} e^{-0.05t} \, dt \), apply integration techniques such as substitution. Let \( u = \sqrt{t} \) which means \( du = \frac{1}{2\sqrt{t}} \, dt \), hence \( dt = 2u \, du \). Substitute and simplify to get:\[ \int 50 \sqrt{t} e^{-0.05t} \, dt = \int 50 u e^{-0.05 u^2} 2u \, du \] which simplifies to:\[ 100 \int u^2 e^{-0.05 u^2} \, du \]

Step 4: Solve the Integral with Substitution

Let \( v = -0.05 u^2 \) which gives \( dv = -0.1 u \, du \). Substitute these into the integral:\[ 100 \int u e^{v} \frac{-dv}{0.1} \] Simplifying gives:\[ -1000 \int e^{v} \, dv \] which is simply \[-1000 e^v + C \] where \( C \) is the integration constant.

Step 5: Evaluate the Limits of Integration

Since this is a definite integral from 0 to infinity, evaluate it:Substitute back for \( v \): \(-1000 e^{-0.05 u^2} + C \). As we evaluate from 0 to infinity:\[ \lim_{{u \to \infty}} (-1000 e^{-0.05 u^2}) = 0 \] At \( u = 0 \): \(-1000 e^{-0.05 \cdot 0} = -1000 \). Thus, \[ V = 0 - (-1000) = 1000 \text{ thousand dollars} \]

Key Concepts

Present Value

Present Value is a fundamental concept in finance that represents the current worth of a sum of money that is expected to be received in the future. When calculating the present value, we utilize a discount rate to determine how much future cash flow is worth today. This is crucial as it accounts for the time value of money.

• It helps in understanding the true worth of an investment or an asset in today's terms.
• Present value is used to assess investment opportunities, indicating whether the returns meet a specified rate of interest.
• Investors use this figure to compare different financial benefits accruing over different time periods.

The present value of future cash flows can be calculated using the formula:\[PV = \int_{0}^{\infty} C(t)e^{-rt} dt\]where \( C(t) \) is the cash flow at time \( t \), and \( r \) is the interest rate. This formula provides a mathematical way to measure how much a stream of future income is worth in today's dollar value, which is critical in determining the capital value of assets like real estate or corporations.

Continuous Interest Rate

Continuous Interest Rate is used in calculations where the interest is compounded continuously, meaning that it is calculated at every instant. It's an essential concept, especially within the realm of finance for exponential growth or decay processes.

• Continuous compounding assumes that interest is being added at every possible moment, not just annually, quarterly, or monthly.
• It provides a more precise calculation than typical discrete compounding methods due to its prolific nature.
• Helps in calculating present and future values more accurately when payments or income streams are ongoing.

The formula for continuous compounding is:\[A = Pe^{rt}\]where \( A \) is the amount of money accumulated after time \( t \), \( P \) is the principal amount, \( r \) is the interest rate, and \( e \) is the base of the natural logarithm. In our problem, utilizing a continuous interest rate helps in determining the capital value of an asset.

Indefinite Integral

An Indefinite Integral, often denoted as the antiderivative, is a fundamental concept in calculus. It functions as the reverse process of differentiation, providing the original function given its derivative.

• Used to model the accumulation of a variable, such as income or costs over time.
• Helps in finding the area under the curve of a graph of a function.
• In economic and financial models, it calculates total income, cost, or value over time.

The indefinite integral is represented as:\[\int f(x) \, dx = F(x) + C\]where \( F(x) \) is the antiderivative, and \( C \) is the constant of integration. In the problem presented, to solve for capital value, the integral \[ \int 50\sqrt{t} \cdot e^{-0.05t} \, dt \] is computed. Leveraging knowledge of indefinite integrals allows us to find the accumulated effect of a continuously changing function, such as the generation of income over time.