Engineering Economy - Cheat Sheet
Ultra-condensed last-30-minutes revision for time value of money, annuities, depreciation, and economic comparison. Every formula, every pitfall, every exam trigger — zero fluff.
Time Value of Money — Fundamentals
Formulas
F = P(1 + i)^n
Meaning
F = future worth; P = present worth; i = interest rate per period; n = number of periods
Watch out
CRITICAL: i and n MUST use the SAME TIME PERIOD. If i is monthly, n must be in months. Mixing annual rate with monthly periods = automatic failure.
When to use
Single lump-sum payment with compound interest; any time you need to find future value from a present amount.
P = F(1 + i)^{-n} = F / (1 + i)^n
Meaning
Discounting: bring future cash back to present value; (1 + i)^{-n} is the discount factor.
Watch out
The exponent is NEGATIVE or use division. Students often write (1 + i)^n in denominator and forget the negative sign.
When to use
When you have a future amount and need its present worth; used in NPV and cost-benefit analysis.
i_{eff} = (1 + r/m)^m - 1
Meaning
i_eff = effective annual interest rate; r = nominal annual rate; m = number of compounding periods per year
Watch out
NEVER use nominal rate directly for calculations if compounding ≠ annual. Always convert to effective rate first or adjust i and n to match compounding frequency.
When to use
When compounding frequency ≠ annual (monthly, quarterly, semi-annual). REQUIRED to compare rates with different compounding periods.
Meaning
Simple interest: F = future amount; i = interest rate; n = number of periods (rarely used in engineering economy)
Watch out
DO NOT use simple interest unless explicitly told. Board exams assume compound interest.
When to use
Only when problem explicitly states 'simple interest'; almost never in PRC exams. Default is COMPOUND interest.
Common values
Value
Bank savings 2–4%, bonds 4–6%, stocks/investments 8–15%, credit cards 12–24%
Symbol
i or r
Quantity
Typical interest rates (annual)
Value
2.71828
Symbol
e
Quantity
Euler's number
Important facts
- Compound interest ALWAYS applies in engineering economy unless problem says 'simple interest'.
- Effective rate is ALWAYS ≥ nominal rate (equal only if compounding is annual).
- i and n must match in time units; if i is monthly, n counts months.
- Present worth is ALWAYS less than future worth for positive interest rates (time value is positive).
- Discount factor (1 + i)^{-n} is always less than 1 for positive i; it 'shrinks' future values to present.
- Continuous compounding: i_eff = e^r − 1 (rare in PRC but possible; e ≈ 2.71828).
Key definitions
Present Worth (PW)
Example
₱10,000 five years from now at 10% interest has PW = ₱10,000 / (1.10)^5 = ₱6,209.
The equivalent value of all future cash flows expressed in today's pesos (discounted to time zero).
Future Worth (FW)
Example
₱10,000 invested today at 10% compounded annually for 5 years grows to FW = ₱16,105.
The equivalent value of all present (or past) cash flows expressed at a specified future time.
Interest Rate per Period (i)
Example
12% annual compounded monthly → i = 0.12 / 12 = 0.01 (1% per month).
The fractional return on money per compounding period (decimal form: 5% = 0.05).
Nominal Rate (r)
Example
Bank advertises '12% nominal compounded monthly'; actual effective rate is (1 + 0.12/12)^12 − 1 = 12.68%.
The stated annual interest rate before adjusting for compounding frequency; NOT the rate to use in calculations unless compounding is annual.
Effective Annual Rate (i_eff)
Example
12% nominal compounded monthly → i_eff = 12.68%; use this to compare against other investment options.
The equivalent annual interest rate that accounts for the effect of compounding within the year.
Diagrams to know
- Cash flow diagram (CFD): horizontal time axis, arrows up (inflows) and down (outflows).
- Compound interest growth curve (exponential shape, steeper with higher i).
- Present vs future worth trade-off graph (PW decreases, FW increases along time).
Annuities & Uniform Series
Formulas
Meaning
F = future worth of annuity; A = uniform payment per period; [(1+i)^n − 1]/i = future worth factor (FWF) or 'F/A factor'
Watch out
This assumes payments at the END of each period. If payments are at the START (annuity due), multiply result by (1+i). Misidentifying timing = wrong answer.
When to use
Finding total future value of equal periodic payments (ordinary annuity: payments at END of each period).
Meaning
P = present worth of annuity; [(1+i)^n − 1] / [i(1+i)^n] = present worth factor (PWF) or 'P/A factor'
Watch out
Denominator has i(1+i)^n, NOT just i. Students often drop the (1+i)^n in denominator. Also, this is ordinary annuity (end-of-period); adjust if payments are at period start.
When to use
Finding lump-sum value TODAY equivalent to equal periodic payments (loan repayment, pension present value).
Meaning
P = present worth of perpetuity (infinite annuity); A = constant periodic payment forever
Watch out
Only works if n → ∞. If n is finite (even very large like 100 years), use the standard P/A formula, NOT perpetuity. Perpetuity is strictly infinite.
When to use
Perpetual payments (e.g., endowments, government bonds, infrastructure maintenance funds lasting forever).
Meaning
A = uniform periodic payment to repay a loan of amount P; this is the capital recovery factor (CRF) or 'A/P factor'.
Watch out
This is the INVERSE of the P/A formula. Denominator is (1+i)^n − 1 (without the i multiplication in numerator). Used for loan amortization.
When to use
Finding monthly/annual payment for a loan, mortgage, or to accumulate a future amount via regular savings.
Meaning
A = periodic payment to accumulate a future amount F; i / [(1+i)^n − 1] = sinking fund factor (SFF) or 'A/F factor'.
Watch out
Similar to A/P but numerator is JUST i (not i(1+i)^n). This is the inverse of F/A formula. Easy to confuse A/F with A/P.
When to use
Finding deposit amount needed periodically to reach a target future goal (e.g., saving for equipment replacement).
Common values
Value
Annual (n in years) or monthly (n in months)
Symbol
1 year = 12 months
Quantity
Standard annuity period (engineering practice)
Important facts
- F/A factor: [(1+i)^n − 1] / i — use to find FW from series of equal payments.
- P/A factor: [(1+i)^n − 1] / [i(1+i)^n] — use to find PW from series of equal payments.
- A/P factor (CRF): [i(1+i)^n] / [(1+i)^n − 1] — use for loan repayment calculations (most common in practice).
- A/F factor (SFF): i / [(1+i)^n − 1] — use for sinking fund (savings) calculations.
- Perpetuity formula P = A/i is ONLY for infinite series; very high but finite n still requires standard annuity formula.
- Annuity due FW = ordinary annuity FW × (1+i); annuity due PW = ordinary annuity PW × (1+i).
- All annuity factors are dimensionless ratios (pure numbers), not rates.
Key definitions
Ordinary Annuity
Example
Monthly loan payments of ₱5,000 due on the last day of each month.
Equal periodic payments occurring at the END of each period (most common in engineering practice and PRC exams).
Annuity Due
Example
Apartment rent of ₱10,000 due on the 1st of each month (beginning of period).
Equal periodic payments occurring at the BEGINNING of each period (rent, lease).
Perpetuity
Example
Endowment fund yielding ₱500,000 annually in perpetuity for scholarship grants.
An annuity with infinite duration; payments continue forever at a constant amount.
Deferred Annuity
Example
Loan where payments start 6 months after disbursement (12-month deferment before first ₱5,000 payment).
An annuity whose first payment occurs more than one period away from the present.
Diagrams to know
- Annuity cash flow diagram: vertical bars of equal height spaced at regular intervals along time axis.
- Perpetuity CFD: infinite series of equal payments (shown as repeating pattern → ∞).
- Deferred annuity CFD: gap of k periods with no payments, then n equal payments (gap shows deferment).
Depreciation Methods
Formulas
Meaning
d = annual depreciation; C = initial cost; S = salvage value (book value at end of life); n = useful life in years
Watch out
MUST subtract salvage value S before dividing by n. Common error: forgetting to subtract S, or using S = 0 when it shouldn't be. Also, n and d timing must align (annual depreciation if n is years).
When to use
Straight-line (SL) depreciation — the simplest and most commonly used method in PRC exams. Assumes constant depreciation each year.
Meaning
BV_t = book value after t years; C = original cost; d = annual depreciation; t = year of interest
Watch out
This is a LINEAR equation. BV decreases by the SAME amount d each year. If you calculate d wrong, all book values will be wrong.
When to use
Finding the carrying value (remaining undepreciated value) of an asset at any year t during its life.
Meaning
d_t = depreciation in year t (SYD method); sum of years' digits denominator = n(n+1)/2; numerator = remaining useful life fraction
Watch out
Denominator is sum of digits 1+2+...+n = n(n+1)/2, NOT just n. Year 1 depreciation is the LARGEST; depreciates in declining amounts. Easy to invert numerator/denominator.
When to use
Sum-of-years-digits (SYD): accelerated depreciation method favoring larger deductions early. Used for assets that lose value faster initially (vehicles, equipment).
Meaning
BV_t = book value in year t (declining balance method); k = depreciation rate (fixed percentage, typically 2/n for double-declining); C = original cost
Watch out
This is EXPONENTIAL (multiplicative), not linear. If k = 2/n (double-declining), depreciation is aggressive early. Rate k does NOT guarantee the asset reaches salvage value S; it may undershoot or overshoot.
When to use
Declining balance (DB) or double-declining balance (DDB): accelerated method with exponential decay; larger early deductions.
Meaning
First-year depreciation in DDB method: use rate 2/n (double the straight-line rate of 1/n) applied to original cost.
Watch out
Year 1 ONLY uses original cost C; years 2+ use the NEW book value. Students often apply it to C every year (which is wrong).
When to use
Double-declining balance (DDB) year 1 only; subsequent years apply the rate to declining book value, not original cost.
Common values
Value
5, 10, or 20 years depending on industry and asset type
Symbol
n
Quantity
Common useful life (machinery/equipment)
Value
2/n (e.g., for 5-year asset, rate = 40% per year on declining book value)
Symbol
k = 2/n
Quantity
Double-declining balance rate (DDB)
Important facts
- Straight-line: uniform depreciation d = (C − S) / n each year; simplest, most conservative.
- SYD: accelerated method; year 1 gets the biggest deduction; depreciates in declining amounts; good for 'fast' asset loss.
- Declining balance: exponential decay; can be single rate (e.g., 1/n) or double-declining (2/n); book value never quite reaches salvage.
- DDB year 1 = (2/n) × C; year 2 = (2/n) × BV₁ (NOT × C again); applies rate to declining base.
- Straight-line always reaches exact salvage value S at end of year n; accelerated methods may not align exactly with salvage.
- For tax purposes, Philippines allows straight-line and accelerated methods; check tax code (BIR guidelines) for allowable rates.
- Salvage value S can be zero (no residual value) or positive (scrap/trade-in); never negative.
Key definitions
Depreciation
Example
A ₱100,000 machine with ₱10,000 salvage over 5 years depreciates by ₱18,000/year (straight-line).
The systematic allocation of an asset's cost minus salvage value over its useful life for accounting and tax purposes.
Book Value (BV)
Example
After 3 years of ₱18,000/year depreciation, BV = ₱100,000 − 3(₱18,000) = ₱46,000.
The undepreciated remaining value of an asset on the balance sheet (original cost minus accumulated depreciation).
Salvage Value (S)
Example
A vehicle with salvage value ₱50,000 after 5 years can be sold for that amount at end of life.
The estimated value of an asset at the end of its useful life (scrap, resale, or trade-in value).
Useful Life (n)
Example
Heavy equipment with useful life of 10 years depreciates over that 10-year period.
The expected period (years, months, hours, units) during which an asset will be used profitably.
Accumulated Depreciation
Example
After 3 years of ₱18,000/year depreciation, accumulated depreciation = ₱54,000.
The sum of all depreciation charges from acquisition to the current date; deducted from cost to get book value.
Diagrams to know
- Straight-line depreciation graph: linear downward slope from C to S over n years.
- SYD depreciation timeline: year 1 has largest bar, decreasing each year, reaching S at year n.
- Declining balance graph: exponential curve starting at C, flattening as it approaches (but may not reach) S.
Economic Comparison & Decision Methods
Formulas
Meaning
NPV = net present value; CF_t = cash flow (inflow positive, outflow negative) in year t; i = discount rate (MARR); n = project life
Watch out
ALL cash flows must be discounted using the SAME discount rate (MARR). Sign convention: inflows are +, outflows are −. Year 0 (present) is NOT discounted.
When to use
Comparing alternatives on a present-worth basis. Choose the alternative with the HIGHEST (least negative) NPV.
Meaning
AW = annual worth (equivalent annual cost or EAC); converts NPV into a uniform annual series over n years.
Watch out
AW 'spreads' the NPV evenly across n periods. If comparing assets of different lives, AW handles the discrepancy better than NPV. Must use same discount rate i.
When to use
Comparing alternatives with DIFFERENT life spans or timescales. Choose the alternative with the HIGHEST AW (or LOWEST annual cost if all are costs).
Meaning
B/C = benefit-cost ratio; PW_benefits = present worth of all benefits; PW_costs = present worth of all costs (including initial investment)
Watch out
B/C ≥ 1 means the project is justified (benefits ≥ costs). B/C < 1 rejects the project. Different B/C formulations (e.g., net B/C) exist; stick to the definition given in the problem.
When to use
Public projects (infrastructure, dams, highways) mandated to justify with B/C ≥ 1. Highest B/C is preferred among alternatives.
Meaning
ROR (or IRR) = rate of return that makes NPV = 0; solve for ROR by trial-and-error or interpolation.
Watch out
Setting NPV = 0 and solving for i = ROR is non-linear; use trial-and-error or interpolation (linear approximation between two interest rates). Multiple roots possible with non-conventional cash flows (multiple sign changes).
When to use
Finding the intrinsic return rate of an investment. Compare ROR to MARR: if ROR > MARR, accept; if ROR < MARR, reject.
Meaning
Set the costs (or present worths) of two alternatives equal to find the interest rate or quantity at which they have equal worth.
Watch out
Break-even may occur at a specific i, n, or production volume. The 'break-even quantity' is different from 'break-even interest rate'; problem context determines which.
When to use
Finding the decision point: at what i, n, or volume do two alternatives break even?
Common values
Value
8–15% annually depending on risk and industry
Symbol
i or MARR
Quantity
Typical MARR (private sector, Philippines)
Value
4–8% annually (lower due to social benefit)
Symbol
i
Quantity
Typical MARR (public sector, infrastructure)
Important facts
- NPV > 0: project is profitable (accept); NPV < 0: project loses money (reject); NPV = 0: indifferent (break-even).
- AW is preferred for comparing alternatives with different life spans; avoids the need to adjust lives to a common multiple.
- B/C ≥ 1: project is justified in public sector; must compare to budget and other constraints.
- ROR (IRR) = the discount rate that sets NPV = 0; compare ROR to MARR as the decision criterion.
- If ROR > MARR: accept (return exceeds minimum threshold); if ROR < MARR: reject.
- Present-worth method and annual-worth method give SAME ranking of alternatives (if MARR is constant).
- Multiple sign changes in cash flows can lead to multiple IRRs (rare but possible); use NPV instead in such cases.
- Discount rate i (MARR) must be consistent for all cash flows; different rates require separate analysis.
Key definitions
Net Present Value (NPV)
Example
Project costs ₱100,000 today, generates ₱30,000/year for 5 years at 10% discount. NPV = ₱113,723 − ₱100,000 = ₱13,723 (positive = good).
The difference between present worth of benefits and present worth of costs; measures net gain/loss in today's pesos.
Annual Worth (AW)
Example
NPV of ₱13,723 over 5 years at 10% = AW of ₱3,623/year (using capital recovery factor).
The equivalent uniform annual series value of all cash flows over the project life; also called equivalent annual cost (EAC) if only costs.
Minimum Attractive Rate of Return (MARR)
Example
If MARR = 12%, any project with ROR < 12% is rejected regardless of positive NPV at other rates.
The discount rate (hurdle rate) used by the organization; investments must exceed MARR to be acceptable.
Internal Rate of Return (IRR) or Rate of Return (ROR)
Example
A ₱100,000 investment returning ₱30,000/year for 5 years has IRR ≈ 15.24% (the rate at which NPV becomes zero).
The discount rate that makes NPV = 0; the intrinsic return rate of an investment.
Benefit-Cost Ratio (B/C)
Example
Dam project with PW of benefits ₱500M and PW of costs ₱300M has B/C = 500/300 = 1.67 (justified; B/C > 1).
The ratio of present worth of benefits to present worth of costs; used for public project justification.
Diagrams to know
- Cash flow diagram for comparison: show all inflows and outflows for both alternatives aligned on same time axis.
- NPV profile: graph NPV (y-axis) vs discount rate i (x-axis); shows how NPV changes with i; intersects x-axis at IRR.
- Decision tree: branches for 'accept' (if criterion met) and 'reject' (if not); shows decision logic for NPV, AW, B/C, ROR.
Interest & Compounding Adjustments
Formulas
i_{eff} = (1 + r/m)^m - 1
Meaning
i_eff = effective annual rate; r = nominal annual rate; m = number of compounding periods per year
Watch out
Nominal rate is NOT the rate to use in calculations unless m = 1 (annual compounding). Must convert to effective rate or adjust i and n to match period.
When to use
Converting between nominal and effective rates. ALWAYS convert before comparing rates with different compounding frequencies.
i_p = r / m
Meaning
i_p = interest rate per compounding period; r = nominal annual rate; m = compounding periods per year (1 year = 12 months = 4 quarters = 2 semi-annual)
Watch out
If nominal rate is 12% compounded monthly, i_p = 0.12 / 12 = 0.01 (1% per month); then use this with n in months, NOT years.
When to use
Use i_p in F = P(1 + i_p)^{n_p} formulas; i_p must match the period of n_p.
Meaning
For continuous compounding at nominal rate r, effective annual rate is e^r − 1; rare but sometimes tested.
Watch out
e ≈ 2.71828; continuous rate is slightly higher than any discrete m. Almost never used in practice but appears in some advanced problems.
When to use
Only if problem explicitly states 'continuous compounding'; not common in PRC exams but good to know.
Common values
Value
m = 1 (annual), m = 2 (semi-annual), m = 4 (quarterly), m = 12 (monthly), m = 365 (daily)
Symbol
m
Quantity
Common compounding frequencies
Important facts
- If m = 1 (annual compounding), nominal and effective rates are identical.
- If m > 1, effective rate ALWAYS > nominal rate (compounding effect).
- As m → ∞ (continuous compounding), i_eff → e^r − 1.
- Periodic rate i_p = r / m; use with n in matching units (e.g., n_months with i_p per month).
- Mixing units = INSTANT WRONG ANSWER: e.g., using annual rate with monthly n, or vice versa.
Key definitions
Nominal Rate (r)
Example
Bank advertises '12% nominal compounded monthly'; r = 0.12.
The stated annual interest rate without adjusting for compounding frequency within the year.
Effective Annual Rate (i_eff or APY)
Example
12% nominal compounded monthly has i_eff = (1 + 0.12/12)^12 − 1 = 0.1268 or 12.68%.
The true annual interest rate accounting for within-year compounding; always ≥ nominal for m > 1.
Compounding Period
Example
Monthly compounding means interest is calculated and added 12 times per year (m = 12).
The time interval at which interest is calculated and added to the principal (annual, semi-annual, quarterly, monthly, daily, continuous).
Diagrams to know
- Nominal vs effective rate comparison chart: shows i_eff increases with m for fixed r; approaches e^r − 1 as m → ∞.
Must remember
- 1. PERIOD MATCHING (MOST CRITICAL): Interest rate i and period count n MUST use the SAME time unit. Monthly rate with annual n = automatic failure. Convert nominal to effective or adjust both to match.
- 2. PRESENT WORTH METHOD (CORE TOOL): PW = F / (1+i)^n discounts ALL future cash flows to time zero at MARR. This is the #1 method for comparing alternatives; highest PW wins.
- 3. ANNUITY FORMULAS (4 CORE): F/A factor, P/A factor, A/P factor (CRF—most used), A/F factor (SFF). Know when each applies; A/P = loan payment = most common in engineering practice.
- 4. STRAIGHT-LINE DEPRECIATION (SIMPLEST): d = (C − S) / n; book value drops by SAME amount each year; reaches EXACT salvage S at year n. Forgetting to subtract S = wrong answer.
- 5. ACCELERATED VS STRAIGHT-LINE: SYD and DDB depreciate MORE in early years, LESS in later years (tax advantage). Straight-line is uniform. Choose based on asset loss pattern.
- 6. NOMINAL vs EFFECTIVE RATE (CRITICAL): 12% nominal compounded monthly ≠ 12% annual. Convert: i_eff = (1 + 0.12/12)^12 − 1 = 12.68%. ALWAYS compare on effective basis.
- 7. ANNUAL WORTH (MOST PRACTICAL): Convert all PW to equivalent annual series (AW). Use for alternatives with different life spans; AW method is preferred for that case.
- 8. BENEFIT-COST RATIO FOR PUBLIC PROJECTS: B/C = benefits / costs. B/C ≥ 1 justifies the project (public sector mandate). B/C < 1 = reject. Compare B/C ratios for ranking.
- 9. BREAK-EVEN ANALYSIS: Set two alternatives equal (same cost or same worth) to find the decision point (interest rate, quantity, or time). Often used for 'buy vs lease' or 'make vs buy'.
- 10. CASH FLOW SIGN CONVENTION: Inflows (revenues, salvage) = POSITIVE; outflows (costs, investments) = NEGATIVE. Mixing signs up = wrong answer. Use CFD (cash flow diagram) to track timing.
Last minute tips
- TIP 1 — ALWAYS Draw a Cash Flow Diagram: A 30-second sketch of inflows/outflows on a time axis prevents sign errors and period confusion. More PRC failures come from misread timing than bad math.
- TIP 2 — Verify Unit Consistency FIRST: Before any calculation, check i and n have matching periods. If i = 10% annual and n = 60 months, STOP: convert n to 5 years or convert i to 10%/12 = 0.833% monthly. Do this ritual for every problem.
- TIP 3 — Know the 'Decision Shortcut': For comparing 2+ alternatives, PW is quickest (discount all at MARR); if lives differ, use AW (avoids common-multiple hassle). For public projects, B/C ≥ 1 is the gate; for private, use NPV or ROR > MARR.
- TIP 4 — Depreciation 'Reach Test': Straight-line ALWAYS reaches exact salvage S at year n. SYD also reaches S at year n. Declining balance (DDB) may NOT. If problem asks book value in year n and method is DDB, be careful—it might not equal salvage.
- TIP 5 — Perpetuity = Infinite Annuity: P = A / i is ONLY valid if n → ∞. If n = 50 or 100 years (very large but finite), use standard P/A annuity formula, NOT perpetuity. This is a common trap.
Comparison tables
Future Worth vs Present Worth — Quick Comparison
| Method | Formula | Direction | When to Use | |
|---|---|---|---|---|
| Future Worth (FW) | F = P(1+i)^n | Time → moves forward (single sum) or F = A[(1+i)^n − 1]/i (annuity) | From present TO future | Finding amount available at end of project or savings goal |
| Present Worth (PW) | P = F(1+i)^{−n} | Time ← moves backward (single sum) or P = A[(1+i)^n − 1]/[i(1+i)^n] (annuity) | From future TO present | Comparing alternatives, NPV analysis, loan valuation (most common in engineering economy) |
Depreciation Methods — Side-by-Side Comparison
| Method | Year t Depreciation Formula | Pattern | When Used | Key Trait |
|---|---|---|---|---|
| Straight-Line (SL) | d = (C − S) / n (constant) | Linear (uniform each year) | Default; simple accounting; conservative | Book value reaches EXACT salvage S at year n |
| Sum-of-Years-Digits (SYD) | d_t = [(n − t + 1) / (n(n+1)/2)] × (C − S) | Declining (larger early, smaller late) | Accelerated; for faster asset loss; vehicles, equipment | Year 1 depreciation is LARGEST; reaches salvage S at year n |
| Declining Balance (DB) | d_t = k × BV_{t−1} (rate k fixed, e.g., k = 1/n or 2/n) | Exponential decay (curved, flattening) | Accelerated; aggressive early deductions; tech, machinery | Book value may NOT reach salvage S; depends on rate k |
| Double-Declining Balance (DDB) | d_1 = (2/n) × C; d_t = (2/n) × BV_{t−1} for t > 1 | Exponential decay (even steeper than DB) | Accelerated; maximum early tax deductions | Year 1 uses ORIGINAL cost C; year 2+ use declining BV; may under/overshoot salvage |
Annuity Factors — Formula & Application Matrix
| Factor Name | Abbreviation | Formula (if per period = i, periods = n) | Converts | Used For |
|---|---|---|---|---|
| Future Worth of Annuity | F/A | [(1+i)^n − 1] / i | Uniform payment series A → Future amount F | Savings accounts, accumulation funds, retirement planning |
| Present Worth of Annuity | P/A | [(1+i)^n − 1] / [i(1+i)^n] | Uniform payment series A → Present amount P | Loan valuation, pension present value, bond pricing |
| Capital Recovery Factor | A/P (CRF) | [i(1+i)^n] / [(1+i)^n − 1] | Present amount P → Uniform payment series A | Loan payment calculation, mortgage, equipment financing (MOST COMMON in practice) |
| Sinking Fund Factor | A/F (SFF) | i / [(1+i)^n − 1] | Future amount F → Uniform payment series A | Savings plan to reach future goal, equipment replacement fund |
Decision Criteria Comparison — NPV vs AW vs B/C vs ROR
| Criterion | Formula/Method | Decision Rule | Best For | Common Pitfall |
|---|---|---|---|---|
| Net Present Value (NPV) | NPV = Σ CF_t / (1+i)^t | NPV > 0 → accept; NPV < 0 → reject; highest NPV wins | Private projects, single-criterion comparison | Confusing sign convention (inflows +, outflows −); discount rate must match |
| Annual Worth (AW) | AW = NPV × [i(1+i)^n / ((1+i)^n − 1)] | AW > 0 → accept; higher AW wins; AW < 0 → reject (cost problems) | Comparing alternatives with DIFFERENT life spans; most practical | Forgetting to convert NPV to AW; using different discount rates for alternatives |
| Benefit-Cost Ratio (B/C) | B/C = PW(benefits) / PW(costs) | B/C ≥ 1 → justified; B/C > 1 → more benefit than cost; highest B/C wins | Public infrastructure projects (mandated by law); water, roads, dams | Mixing up numerator/denominator; forgetting that B/C < 1 rejects the project |
| Rate of Return (ROR or IRR) | Solve NPV = 0 for i = ROR (trial-and-error or interpolation) | ROR > MARR → accept; ROR < MARR → reject; highest ROR wins | Comparing investments; checking internal return; intuitive 'percent return' | Non-linear equation; multiple roots possible with non-conventional cash flows; interpolation error |
Ordinary Annuity vs Annuity Due vs Perpetuity
| Type | Payment Timing | FW Adjustment | PW Adjustment | Common Example |
|---|---|---|---|---|
| Ordinary Annuity | Payments at END of each period | F = A[(1+i)^n − 1] / i (standard formula) | P = A[(1+i)^n − 1] / [i(1+i)^n] (standard formula) | Loan repayment (payment on last day of month); most common |
| Annuity Due | Payments at BEGINNING of each period | F = A[(1+i)^n − 1] / i × (1+i) = (standard) × (1+i) | P = A[(1+i)^n − 1] / [i(1+i)^n] × (1+i) = (standard) × (1+i) | Rent/lease paid on 1st of month (beginning of period) |
| Perpetuity | Payments forever (n → ∞) | F → ∞ (infinite accumulation) | P = A / i (perpetual present worth, FINITE) | Endowment funds, government bonds, 'forever' maintenance (rare in practice) |
Diagrams
Engineering Economy Master Map — Formulas & Concepts
Complete hierarchical view of all major topics, formulas, and methods in engineering economy. Use this to orient yourself at the start of revision.
Economic Comparison Decision Tree — Which Method to Use?
Step-by-step logic tree to select the correct comparison method based on project type, life spans, and criteria. Saves time and reduces method confusion on exam.
Depreciation Method Selection & Calculation Flow
Decision and calculation flow for selecting the correct depreciation method and computing book value. Highlights the key check: does book value at year n equal salvage?