6.4 Present Worth, Life-Cycle Cost, and Sustainability

Put Alternatives on One Time Basis

The April 2024 WRE Project Planning specification includes economic and sustainability analysis: present worth, life-cycle costs, and comparison of alternatives. The exam usually gives the interest rate and cash flows, so the real challenge is building the timeline correctly. Never compare a low-bid pump station to a high-efficiency one on first cost alone when one option burns far more energy for 20 years.

Present worth (PW) converts every cost and benefit to equivalent dollars at time zero. The two factors you will use constantly are:

• Single payment: P = F(P/F, i, n) = F / (1 + i)^n
• Uniform series: P = A(P/A, i, n), where (P/A, i, n) = [(1 + i)^n − 1] / [i(1 + i)^n]

In a cost comparison, costs are positive and salvage or residual value is subtracted because it offsets cost at the end of the study period.

Life-Cycle Cost Workflow

Use a repeatable setup so timing errors do not creep in:

1. Confirm the study period (n) and the discount rate (i).
2. Draw the cash-flow timeline for each alternative.
3. Put all amounts on one dollar basis: constant dollars with a real rate, or inflated dollars with a nominal rate — never mixed.
4. Discount each cost to present worth with the correct factor.
5. Subtract salvage or residual value at the end of the period.
6. Choose the alternative with the lower equivalent cost when service and non-cost requirements are equal.
7. State non-economic (sustainability) considerations separately when the question asks for them.

Worked example — two pumps, 20-year period, i = 5%. Alternative A: $600,000 initial, $58,000/yr energy and maintenance, $90,000 overhaul in year 10, $50,000 salvage in year 20. Alternative B: $720,000 initial, $41,000/yr, $70,000 overhaul in year 10, $70,000 salvage in year 20. Factors: (P/A, 5%, 20) = 12.462; (P/F, 5%, 10) = 0.6139; (P/F, 5%, 20) = 0.3769.

• PW(A) = 600,000 + 58,000(12.462) + 90,000(0.6139) − 50,000(0.3769) = $1,359,301.
• PW(B) = 720,000 + 41,000(12.462) + 70,000(0.6139) − 70,000(0.3769) = $1,247,546.

Even though B costs $120,000 more up front, its present worth is about $111,800 lower because reduced energy, maintenance, overhaul, and higher salvage dominate over 20 years. The lowest-first-cost option (A) is the wrong economic choice.

Equivalent Annual Cost and Sustainability Interpretation

When alternatives have unequal service lives, do not compare raw present worths over different periods. Either analyze over a common period (often the least common multiple of the lives) or convert each present worth to an equivalent uniform annual cost (EUAC) using EUAC = PW × (A/P, i, n). Comparing EUAC values is the cleanest way to rank options with mismatched lives, such as a 15-year pump versus a 30-year gravity alternative.

Sustainability is not a slogan in an exam calculation; it becomes decisive only when tied to measurable outcomes:

• Lower annual energy use and greenhouse-gas emissions.
• Fewer chemical deliveries and reduced hauling miles.
• Reduced sanitary sewer overflows (SSOs) and improved permit compliance.
• More reliable drought supply, beneficial reuse, or recharge.
• Smaller disturbed area and better flood resilience.

The decision logic the exam expects: when two alternatives have similar present worth, sustainability criteria may justify the option with lower life-cycle emissions or better service continuity. But if an alternative fails a permit limit, a level-of-service target, or a public-safety requirement, a lower present worth does not make it acceptable — economics never override a hard constraint. A frequent trap is forgetting to subtract salvage, discounting a periodic overhaul with a P/A factor instead of P/F, or mixing a real discount rate with inflated future dollars.

Confirm the dollar basis and the timing of every cash flow before you compute the factor.

Factor Selection, Benefit-Cost, and Real vs. Nominal Rates

Pick the time-value factor by matching it to the cash-flow shape — this single habit prevents most economics errors:

The real vs. nominal rule is non-negotiable: a real (constant-dollar) discount rate pairs only with constant dollars, and a nominal rate pairs only with inflated (current) dollars. The relationship is (1 + nominal) = (1 + real)(1 + inflation), so with 5% real and 3% inflation the nominal rate is about 8.15%. Mixing the two systematically biases long-horizon WRE comparisons.

For publicly funded projects, the exam may ask for a benefit-cost ratio (BCR) = present worth of benefits ÷ present worth of costs. A BCR ≥ 1.0 indicates an economically justified project; for flood-control or water-supply alternatives, incremental B/C analysis ranks mutually exclusive options. Worked logic: if a $2.0M levee yields $2.6M present-worth flood-damage reduction, BCR = 1.30 and the project is justified; a competing $3.0M alternative adding only $0.5M more benefit fails the incremental test (ΔB/ΔC = 0.5/1.0 = 0.5 < 1.0) and is rejected even though its total BCR may still exceed 1.0.

Always anchor the analysis to the stated study period and discount rate, hold service level equal, and present sustainability findings as a separate, supporting layer rather than a substitute for the economic ranking.