4.3 Engineering Economics
Key Takeaways
- The Time Value of Money principle dictates that a dollar today is worth more than a dollar tomorrow due to its potential earning capacity via interest.
- Single Payment Compound Amount finds a future worth (F) given a present worth (P). Capital Recovery finds an annual uniform series (A) given a present worth (P).
- Net Present Value (NPV) brings all future revenues and costs back to present-day dollars; a positive NPV indicates a financially viable project.
- Life Cycle Cost Analysis (LCCA) evaluates all costs incurred over a facility's lifespan, including initial construction, maintenance, operations, and salvage value.
- Break-even analysis determines the point at which total revenues equal total costs, allowing engineers to compare alternative equipment or construction methods.
Construction projects represent massive financial investments. Engineering economics provides the mathematical tools necessary to evaluate these investments over time, compare alternative design or equipment choices, and determine project feasibility. The PE Construction exam frequently tests your ability to navigate interest rates, cash flow diagrams, and cost-comparison calculations.
The Time Value of Money
The core concept of engineering economics is the Time Value of Money (TVM). Because money can earn interest, money received today is more valuable than the same amount of money received in the future. Conversely, future costs represent a smaller burden in today's dollars.
Key variables used in TVM calculations:
- P (Present Worth or Principal): The value of money at time zero (today).
- F (Future Worth): The value of money at a specific future time.
- A (Annual/Uniform Series): A series of equal, consecutive, periodic payments or receipts.
- i (Interest Rate): The discount rate or rate of return per compounding period.
- n (Number of Periods): The total number of compounding periods (usually years).
Standard Interest Formulas and Factors
Instead of deriving formulas, engineers use standard interest factors denoted by a specific notation: (Find / Given, i%, n). For example, (F/P, i, n) means "Find Future Worth given Present Worth at interest rate i for n periods."
1. Single Payment Compound Amount (Find F, Given P)
This calculates how much a present sum will grow to in the future.
- Formula: $F = P(1 + i)^n$
- Factor Notation: $F = P(F/P, i, n)$
2. Single Payment Present Worth (Find P, Given F)
This determines the present value of a future lump sum. It "discounts" the future money back to today.
- Formula: $P = \frac{F}{(1 + i)^n}$
- Factor Notation: $P = F(P/F, i, n)$
3. Capital Recovery (Find A, Given P)
This converts a present lump-sum investment into an equivalent uniform annual series over a set period. It is commonly used to calculate annual loan payments or equivalent annual equipment ownership costs.
- Formula: $A = P \left[ \frac{i(1 + i)^n}{(1 + i)^n - 1} \right]$
- Factor Notation: $A = P(A/P, i, n)$
4. Uniform Series Present Worth (Find P, Given A)
This calculates the present value of a series of equal future payments, useful for finding the present value of ongoing maintenance costs.
- Formula: $P = A \left[ \frac{(1 + i)^n - 1}{i(1 + i)^n} \right]$
- Factor Notation: $P = A(P/A, i, n)$
Net Present Value (NPV) and Life Cycle Cost Analysis (LCCA)
Net Present Value (NPV)
Net Present Value is a method used to evaluate the profitability of an investment. It involves converting all future cash flows (revenues and costs) back to their Present Worth ($P$) using a Minimum Attractive Rate of Return (MARR) as the interest rate, and then summing them up.
- Revenues/Savings are treated as positive cash flows (+).
- Initial Investments and Operating Costs are treated as negative cash flows (-).
Decision Rule: If NPV $\ge 0$, the project is economically viable. When comparing mutually exclusive alternatives, select the one with the highest NPV.
Life Cycle Cost Analysis (LCCA)
LCCA is a specific application of NPV used in construction to evaluate the total cost of ownership over a facility's entire lifespan. It compares alternatives that may have different initial construction costs and different long-term maintenance costs.
Components of LCCA typically include:
- Initial Capital Cost: Construction and design fees (Occurs at Year 0).
- Operations and Maintenance (O&M): Uniform annual costs for energy, cleaning, and routine maintenance (Treated as an $A$ series).
- Major Repairs/Replacements: Lump sum costs occurring at specific future years (Treated as an $F$ value).
- Salvage Value: The residual value of the facility at the end of the analysis period (Treated as a positive $F$ value).
All future costs and salvage values must be discounted back to Present Worth ($P$) to find the total Life Cycle Cost.
Break-Even Analysis
Break-even analysis determines the production volume or duration at which two alternatives cost exactly the same, or when total revenues exactly offset total costs.
In construction, it is frequently used to decide between owning versus renting equipment.
General Break-Even Formula: $\text{Fixed Cost}_A + (\text{Variable Cost}_A \times X) = \text{Fixed Cost}_B + (\text{Variable Cost}_B \times X)$
Where $X$ is the break-even point in units produced, miles driven, or hours operated. If the expected usage is greater than $X$, choose the alternative with the lower variable cost. If usage is less than $X$, choose the alternative with the lower fixed cost.
A contractor is considering purchasing a new excavator for $150,000. It is expected to generate $35,000 in net annual revenue for 6 years, after which it will have a salvage value of $20,000. If the contractor's minimum attractive rate of return (MARR) is 8%, which calculation correctly determines the Net Present Value (NPV) of this investment?
A company must choose between renting or buying a bulldozer. Buying costs $80,000 upfront with operating costs of $15 per hour. Renting costs $65 per hour with no upfront cost. At how many hours of operation do the two options break even?