Economic Analysis Methods

Present Worth (PW) Analysis

Discount every cash flow to time 0 using the MARR (Minimum Attractive Rate of Return), then sum: $PW = \sum_{t=0}^{n} CF_t\,(1+i)^{-t}$

Decision rules. For a single project, accept if PW ≥ 0. Among mutually exclusive alternatives, choose the highest PW (least negative cost, or greatest net benefit). A subtle but heavily tested rule: when alternatives have different service lives, a direct PW comparison is invalid. You must evaluate over a common horizon — the least common multiple (LCM) of the lives (e.g. a 3-year and a 4-year option are compared over 12 years, repeating each project) — or switch to annual worth, which sidesteps the problem.

Annual Worth (AW) Analysis

Convert the project's entire cash-flow stream to one equivalent end-of-year amount: $AW = PW \times (A/P, i\%, n)$

AW is often the cleanest method because it automatically handles unequal lives — comparing $/year requires no LCM study period, provided each alternative repeats indefinitely. Decision rule: pick the alternative with the highest AW (for costs, the least-negative AW). AW also reads naturally to a client: "this design costs $8,400 per year."

Rate of Return (ROR / IRR)

The rate of return i* is the interest rate that makes present worth exactly zero: $0 = \sum_{t=0}^{n} \frac{CF_t}{(1+i^*)^t}$ For a single project, accept if ROR ≥ MARR. The trap: you cannot simply pick the alternative with the highest individual ROR. A small project can show a huge ROR yet leave money on the table.

Incremental (ΔROR) Analysis

To rank mutually exclusive alternatives correctly:

1. Order them by increasing first cost.
2. Compute the incremental cash flow of the larger-cost option minus the smaller (defender vs. challenger).
3. Find the ROR on that increment, Δi*.
4. If Δi* ≥ MARR, the extra investment earns its keep — keep the higher-cost option; otherwise keep the lower-cost one. The same incremental logic applies to ΔB/C ratios for public projects.

Benefit-Cost (B/C) Analysis

Standard for government/public-sector work: $B/C = \frac{PW(\text{benefits})}{PW(\text{costs})}$

Watch placement: disbenefits may be subtracted from benefits (numerator) or added to costs (denominator) — the two conventions give different ratios, so follow the problem's instruction.

Break-Even Analysis

Find the quantity Q where two alternatives cost the same (or revenue equals cost). Example: Machine A costs $10,000 with $5/unit variable cost; Machine B costs $20,000 with $3/unit. Setting total costs equal: 10,000 + 5Q = 20,000 + 3Q → 2Q = 10,000 → Q = 5,000 units. Below 5,000 units choose A (lower fixed cost dominates); above 5,000 units choose B (lower variable cost wins).

Capitalized Cost

For assets assumed to last forever (dams, perpetual endowments, roadways with indefinite service), the present worth of a perpetual annual cost A at rate i is the capitalized cost: $CC = \frac{A}{i}$ This is the limit of (P/A) as n → ∞. Example: maintaining a structure at $8,000/year forever, at i = 8%, has a capitalized cost of 8,000/0.08 = $100,000. Capitalized cost is the standard way to compare very-long-life public alternatives without an LCM study period.

Choosing the Right Method

All four worth-based methods give the same accept/reject decision for a single project when applied consistently at the same MARR — PW ≥ 0 is equivalent to AW ≥ 0, to ROR ≥ MARR, and to B/C ≥ 1. They can disagree only when misused for ranking mutually exclusive options, which is exactly why incremental analysis exists.

Cost Terminology

The most-tested item is sunk cost: money already paid for a machine you now own does not change which future option is best — only present and future cash flows (and opportunity costs) do.

Depreciation

Depreciation spreads an asset's cost over its useful life for tax accounting.

Straight-Line (SL)

$D = \frac{\text{Cost} - \text{Salvage}}{\text{Useful life}}$ Equal depreciation each year. Example: a $40,000 asset with $4,000 salvage over 6 years gives D = (40,000 − 4,000)/6 = $6,000/year.

MACRS (Modified Accelerated Cost Recovery System)

The U.S. tax-default method. It applies fixed IRS percentages to the original basis for each year of a stated recovery period (3, 5, 7, … years), ignores salvage value, and front-loads deductions (accelerated). The FE Reference Handbook tabulates the MACRS factors. Year-n depreciation is D_n = (rate_n) × (initial cost).

Book Value

$BV_n = \text{Initial cost} - \sum_{k=1}^{n} D_k$ Book value is what remains on the books after accumulated depreciation; under SL it declines linearly toward salvage, while under MACRS it drops faster early.