Problem 37 - Calculus and its applications

Question

The capitalized cost, \(c,\) of an asset for \(a n\) unlimited 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 \(c=c_{0}+\int_{0}^{\infty} m(t) e^{-k t} d t,\)where \(c_{0}\) is the initial cost of the asset, \(k\) is the interest rate (compounded continuously), and \(m(t)\) is the annual cost of maintenance. Find the capitalized cost under each set of assumptions. $$ c_{0}=\$ 500,000, \quad k=5 \%, \quad m(t)=\$ 20,000 $$

Step-by-Step Solution

Answer

The capitalized cost is $900,000.

Step 1: Convert Interest Rate to Decimal

The interest rate given is 5%. To convert it to a decimal, divide by 100. Thus, \(k = 0.05\).

Step 2: Substitute Values into the Formula

The capitalized cost formula is given by \(c = c_0 + \int_{0}^{\infty} m(t) e^{-kt} dt\). Substitute \(c_0 = 500,000\), \(m(t) = 20,000\), and \(k = 0.05\). The equation becomes \(c = 500,000 + \int_{0}^{\infty} 20,000 e^{-0.05t} dt\).

Step 3: Evaluate the Integral

Recognize that the integral \(\int_{0}^{\infty} e^{-kt} dt\) is a standard form. It is known that \(\int_{0}^{\infty} e^{-kt} dt = \frac{1}{k}\). Here, \(k = 0.05\) so the integral becomes \(\int_{0}^{\infty} e^{-0.05t} dt = \frac{1}{0.05}\).

Step 4: Calculate the Integral Result

Compute \(\frac{1}{0.05} = 20\). Therefore, the integral \(\int_{0}^{\infty} 20,000 e^{-0.05t} dt = 20,000 \times 20 = 400,000\).

Step 5: Sum Initial Cost and Present Value of Maintenance

Now find the total capitalized cost by adding the initial cost \(500,000\) to the present value of maintenance \(400,000\). Thus, \(c = 500,000 + 400,000\).

Step 6: Final Calculation

Calculate the sum: \(c = 500,000 + 400,000 = 900,000\).

Key Concepts

Present Value

The Present Value helps us understand how much a future expense or sum of money is worth in today's terms. It answers the question: "How much should we invest today to reach a certain amount in the future?"
At its core, calculating the present value means discounting a future amount using a specific interest rate. This is why the concept is often explained using the formula: \[ PV = \frac{FV}{(1 + r)^n} \] where:

• \(PV\) stands for Present Value.
• \(FV\) is the Future Value.
• \(r\) is the interest rate as a decimal.
• \(n\) is the number of periods.

In our context, we utilize Present Value to transform continuous future maintenance expenses into a lump sum that represents today's value. This helps in figuring out the capitalized cost of an asset.

Continuous Compounding

Continuous Compounding is a process where the interest earned on an investment is computed and added back into the investment balance constantly, leading to exponential growth. Unlike periodic compounding, where interest might be applied annually or monthly, continuous compounding calculates interest infinitely many times in any given period. This leads to the equation for continuous growth:\[ A = Pe^{rt} \] Here,

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (initial investment).
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
• \(r\) is the continuous compounding rate expressed as a decimal.
• \(t\) is the time the money is invested or borrowed in years.

In capitalized cost calculations, continuous compounding ensures that the interest rate is applied in the most precise way possible, giving an accurate present value of future expenses.

Maintenance Expenses

Maintenance Expenses refer to the routine costs incurred to keep an asset in good working condition. These are predictable expenses, often estimated per annum. In our scenario, the maintenance expense is considered a continuous stream of costs, represented by a function \(m(t)\).
This function reflects the regular outflow of money required to maintain the asset. When calculating the capitalized cost, the goal is to find the present value of all future maintenance expenses. By integrating the maintenance cost over an infinite period, and using continuous compounding, we derive a singular sum that makes planning and budgeting easier.
It ensures that all future expenses are factored in from the start, minimizing surprises.

Interest Rate

The Interest Rate is the percentage at which money can grow over time. It influences how present and future values differ from each other. When it comes to calculating the capitalized cost, the interest rate is crucial as it acts as a discount factor.
By converting the interest rate from a percentage to a decimal (e.g., 5% becomes 0.05), it can be used in various financial formulas, such as continuous compounding and present value calculations.
This rate helps in quantifying how much future maintenance costs are "worth" right now. Even minimal changes in the interest rate can lead to significant differences in the capitalized cost.

Definite Integral

The Definite Integral plays a vital role in calculus, particularly when calculating exact areas under curves. In the context of capitalized cost, it helps determine the total present value of continuous future maintenance expenses. For the formula \[ c = c_0 + \int_{0}^{\infty} m(t) e^{-kt} dt \]

• \(c_0\) represents the initial cost of purchasing the asset.
• The integral part \(\int_{0}^{\infty} m(t) e^{-kt} dt\) helps sum up the perpetual maintenance costs by using an exponential decay function \(e^{-kt}\), which reflects the time value of money.

The solution of this integral gives a finite number that, when added to the initial cost, results in the capitalized cost. It is an elegant way to compress an infinite series of expenses into one manageable and comparable number.