Problem 48 - Calculus and its applications

Question

The capitalized cost, \(c,\) of an asset over its lifetime is the total of the initial cost and the present value of all maintenance expenses that will occur in the future. It is computed with the formula $$ c=c_{0}+\int_{0}^{L} m(t) e^{-k t} d t $$ where \(c_{0}\) is the initial cost of the asset, \(L\) is the lifetime (in years), \(k\) is the interest rate (compounded continuously), and \(m(t)\) is the annual cost of maintenance. Find the capitalized cost under each set of assumptions. $$ c_{0}=\$ 500,000, k=5 \%, m(t)=\$ 20,000, L=20 $$

Step-by-Step Solution

Answer

The capitalized cost is \( \$752,840 \).

Step 1: Identify Constants and Functions

Firstly, identify the given values for the constants and the function for the exercise. We have: - \( c_0 = \\(500,000 \) (the initial cost of the asset) - \( k = 0.05 \) (interest rate in decimal form) - \( m(t) = \\)20,000 \) (annual maintenance cost which is constant) - \( L = 20 \) years (lifetime of the asset).

Step 2: Set Up the Integral

The formula for capitalized cost is given by:\[ c = c_0 + \int_{0}^{L} m(t) e^{-k t} dt \]Substitute the known values into the integral:\[ c = 500,000 + \int_{0}^{20} 20,000 \cdot e^{-0.05 t} dt \]

Step 3: Integrate the Maintenance Cost

To solve the integral, integrate the function \( 20,000 e^{-0.05 t} \) from \(0\) to \(20\):\[ \int 20,000 e^{-0.05 t} dt = 20,000 \cdot \left[ -\frac{1}{0.05} e^{-0.05 t} \right] = -400,000 e^{-0.05 t} \]

Step 4: Evaluate the Integral

Evaluate \(-400,000 e^{-0.05 t}\) at the bounds of \(0\) and \(20\): \[ \left[-400,000 e^{-0.05 \times 20}\right] - \left[-400,000 e^{-0.05 \times 0}\right] \]Simplify:\[ = -400,000 \cdot e^{-1} + 400,000 \cdot e^{0} \]\[ = -400,000 \cdot \frac{1}{e} + 400,000 \cdot 1 \]Calculate the expression:\[ \approx -400,000 \cdot 0.3679 + 400,000 \cdot 1 \]\[ \approx -147,160 + 400,000 \]\[ \approx 252,840 \]

Step 5: Compute Total Capitalized Cost

Add the result of the integral to the initial cost \(c_0\):\[ c = 500,000 + 252,840 \]\[ c = 752,840 \]

Key Concepts

Present Value

Present value is a key component in evaluating financial decisions, particularly in capitalized cost calculations. It represents the current worth of a future stream of cash flows, determined by discounting them using a specific interest rate.
In capitalized cost scenarios, the present value helps you understand how much future maintenance expenses are worth in today's money.

• The present value formula considers the interest rate and the number of periods until payment.
• Higher interest rates result in a lower present value, as they devalue the future cash flows more significantly.

In our original exercise, the present value is utilized to find what future maintenance costs are equivalent to today, making it easier to combine these costs with the initial asset cost when calculating the total capitalized cost.

Continuous Compounding

Continuous compounding is a concept used within finance where interest is applied at every possible instant. Unlike traditional compounding (annually, semi-annually, etc.), it assumes that money grows exponentially thanks to constant interest application.
In capitalized cost problems like this:

• Continuous compounding is denoted by the exponential function \(e\), combined with the negative exponent of the interest rate \(k\).
• The function \(e^{-kt}\) is used to calculate how maintenance costs diminish over time in present value terms.

Understand that in terms of present value calculations for the capitalized cost, continuous compounding ensures that the calculations account for every moment’s growth, leading to more precise results.

Maintenance Cost

Maintenance cost refers to any expenses needed for keeping an asset in operational condition. In the context of capitalized cost calculations, this cost needs to be evaluated over the asset's lifetime to understand its total impact.
With capitalized costs, understanding maintenance costs aids in making comprehensive financial decisions. Consider:

• In a constant scenario like our exercise, the same amount of maintenance cost \(m(t)\) is repeated yearly.
• This cost affects the integral calculation within the capitalized cost formula, forming the basis of the expense stream being discounted into present value.

In calculating capitalized cost, it's crucial to acknowledge how maintenance cost over time, when adjusted by present value and continuous compounding, impacts the overall investment in the asset.