Formula Sheet: Cost Estimation

Present Worth and Future Worth

Present Worth (PW)

Present Worth Formula:

\[PW = F \times (P/F, i, n)\]

• PW = Present Worth ($)
• F = Future amount ($)
• i = Interest rate per period (decimal)
• n = Number of periods
• (P/F, i, n) = Single Payment Present Worth Factor

Present Worth Factor:

\[(P/F, i, n) = \frac{1}{(1 + i)^n}\]

Future Worth (FW)

Future Worth Formula:

\[FW = P \times (F/P, i, n)\]

• FW = Future Worth ($)
• P = Present amount ($)
• i = Interest rate per period (decimal)
• n = Number of periods
• (F/P, i, n) = Single Payment Compound Amount Factor

Future Worth Factor:

\[(F/P, i, n) = (1 + i)^n\]

Uniform Series

Uniform Series Present Worth

Present Worth of Uniform Series:

\[PW = A \times (P/A, i, n)\]

• PW = Present Worth ($)
• A = Uniform annual amount ($)
• i = Interest rate per period (decimal)
• n = Number of periods

Uniform Series Present Worth Factor:

\[(P/A, i, n) = \frac{(1 + i)^n - 1}{i(1 + i)^n}\]

Capital Recovery

Uniform Series from Present Worth:

\[A = P \times (A/P, i, n)\]

• A = Uniform annual amount ($)
• P = Present amount ($)
• (A/P, i, n) = Capital Recovery Factor

Capital Recovery Factor:

\[(A/P, i, n) = \frac{i(1 + i)^n}{(1 + i)^n - 1}\]

Uniform Series Future Worth

Future Worth of Uniform Series:

\[FW = A \times (F/A, i, n)\]

• FW = Future Worth ($)
• A = Uniform annual amount ($)

Uniform Series Compound Amount Factor:

\[(F/A, i, n) = \frac{(1 + i)^n - 1}{i}\]

Sinking Fund

Uniform Series from Future Worth:

\[A = F \times (A/F, i, n)\]

• A = Uniform annual amount ($)
• F = Future amount ($)
• (A/F, i, n) = Sinking Fund Factor

Sinking Fund Factor:

\[(A/F, i, n) = \frac{i}{(1 + i)^n - 1}\]

Gradient Series

Arithmetic Gradient

Present Worth of Arithmetic Gradient:

\[PW = G \times (P/G, i, n)\]

• PW = Present Worth of gradient ($)
• G = Gradient amount ($/period)
• (P/G, i, n) = Gradient Present Worth Factor

Arithmetic Gradient Present Worth Factor:

\[(P/G, i, n) = \frac{(1 + i)^n - in - 1}{i^2(1 + i)^n}\]

Uniform Series Equivalent of Arithmetic Gradient:

\[A = G \times (A/G, i, n)\]

• A = Uniform annual equivalent ($)
• G = Gradient amount ($/period)

Arithmetic Gradient Uniform Series Factor:

\[(A/G, i, n) = \frac{1}{i} - \frac{n}{(1 + i)^n - 1}\]

Geometric Gradient

Present Worth of Geometric Gradient (when g ≠ i):

\[PW = A_1 \times \frac{1 - (1 + g)^n(1 + i)^{-n}}{i - g}\]

• PW = Present Worth ($)
• A₁ = First period payment ($)
• g = Growth rate per period (decimal)
• i = Interest rate per period (decimal)
• n = Number of periods

Present Worth of Geometric Gradient (when g = i):

\[PW = \frac{A_1 \times n}{1 + i}\]

Annual Worth and Equivalent Uniform Annual Cost

Annual Worth (AW)

Annual Worth from Present Worth:

\[AW = PW \times (A/P, i, n)\]

• AW = Annual Worth ($/year)
• PW = Present Worth ($)

Annual Worth from Future Worth:

\[AW = FW \times (A/F, i, n)\]

Equivalent Uniform Annual Cost (EUAC)

EUAC Formula:

\[EUAC = (P - S) \times (A/P, i, n) + S \times i\]

• EUAC = Equivalent Uniform Annual Cost ($/year)
• P = Initial cost ($)
• S = Salvage value ($)
• i = Interest rate per period (decimal)
• n = Service life (years)

Alternative EUAC Formula:

\[EUAC = P \times (A/P, i, n) - S \times (A/F, i, n)\]

Capitalized Cost

Capitalized Cost (CC)

Capitalized Cost Formula:

\[CC = P + \frac{A}{i}\]

• CC = Capitalized Cost ($)
• P = Initial cost ($)
• A = Annual cost ($)
• i = Interest rate (decimal)

Note: Capitalized cost represents the present worth of a project with infinite life.

Capitalized Cost with Periodic Replacement

Capitalized Cost with Replacement:

\[CC = P + (P - S) \times (A/F, i, n) \times \frac{1}{i}\]

• P = Initial cost ($)
• S = Salvage value ($)
• n = Service life (years)
• i = Interest rate (decimal)

Depreciation Methods

Straight-Line Depreciation

Annual Depreciation:

\[D = \frac{P - S}{n}\]

• D = Annual depreciation ($)
• P = Initial cost ($)
• S = Salvage value ($)
• n = Useful life (years)

Book Value at Year t:

\[BV_t = P - t \times D\]

• BV_t = Book value at year t ($)
• t = Number of years elapsed

Declining Balance Depreciation

Depreciation Rate:

\[d = \frac{k}{n}\]

• d = Depreciation rate (decimal)
• k = Multiplier (1.5 for 150% DB, 2.0 for Double Declining Balance)
• n = Useful life (years)

Annual Depreciation:

\[D_t = d \times BV_{t-1}\]

• D_t = Depreciation in year t ($)
• BV_t-1 = Book value at beginning of year t ($)

Book Value at Year t:

\[BV_t = P \times (1 - d)^t\]

Sum-of-Years-Digits (SOYD)

Sum of Years:

\[SOYD = \frac{n(n + 1)}{2}\]

• SOYD = Sum of years digits
• n = Useful life (years)

Depreciation in Year t:

\[D_t = (P - S) \times \frac{n - t + 1}{SOYD}\]

• D_t = Depreciation in year t ($)
• t = Year number

Book Value at Year t:

\[BV_t = P - \sum_{j=1}^{t} D_j\]

Modified Accelerated Cost Recovery System (MACRS)

Annual Depreciation:

\[D_t = P \times R_t\]

• D_t = Depreciation in year t ($)
• P = Initial cost (basis) ($)
• R_t = MACRS percentage rate for year t (from IRS tables)

Note: MACRS ignores salvage value. Recovery periods include 3, 5, 7, 10, 15, 20, 27.5, and 39 years.

Break-Even Analysis

Break-Even Point

Break-Even Quantity:

\[Q_{BE} = \frac{FC}{P - VC}\]

• Q_BE = Break-even quantity (units)
• FC = Fixed costs ($)
• P = Price per unit ($/unit)
• VC = Variable cost per unit ($/unit)

Total Cost:

\[TC = FC + VC \times Q\]

• TC = Total cost ($)
• Q = Quantity (units)

Total Revenue:

\[TR = P \times Q\]

• TR = Total revenue ($)

Profit:

\[\text{Profit} = TR - TC = Q \times (P - VC) - FC\]

Break-Even Between Alternatives

Break-Even for Two Alternatives:

\[FC_1 + VC_1 \times Q = FC_2 + VC_2 \times Q\]

Solving for Q:

\[Q = \frac{FC_2 - FC_1}{VC_1 - VC_2}\]

• Q = Break-even quantity where alternatives are equivalent
• FC₁, FC₂ = Fixed costs for alternatives 1 and 2
• VC₁, VC₂ = Variable costs per unit for alternatives 1 and 2

Benefit-Cost Analysis

Benefit-Cost Ratio (BCR)

Conventional Benefit-Cost Ratio:

\[BCR = \frac{PW_B}{PW_C}\]

• BCR = Benefit-Cost Ratio (dimensionless)
• PW_B = Present worth of benefits ($)
• PW_C = Present worth of costs ($)

Decision Rule: Accept project if BCR > 1.0

Modified Benefit-Cost Ratio:

\[BCR_{mod} = \frac{PW_B - PW_{O\&M}}{PW_I}\]

• BCR_mod = Modified Benefit-Cost Ratio
• PW_O&M = Present worth of operating and maintenance costs ($)
• PW_I = Present worth of initial investment ($)

Incremental Benefit-Cost Ratio

Incremental BCR:

\[\Delta BCR = \frac{PW_{B,2} - PW_{B,1}}{PW_{C,2} - PW_{C,1}}\]

• ΔBCR = Incremental Benefit-Cost Ratio
• Subscripts 1, 2 = Alternatives being compared (2 has higher cost)

Decision Rule: If ΔBCR > 1.0, choose alternative 2; otherwise, choose alternative 1

Rate of Return Analysis

Internal Rate of Return (IRR)

IRR Equation:

\[0 = -P + \sum_{t=1}^{n} \frac{CF_t}{(1 + IRR)^t}\]

• IRR = Internal Rate of Return (decimal)
• P = Initial investment ($)
• CF_t = Cash flow in period t ($)
• n = Number of periods

Decision Rule: Accept project if IRR > MARR (Minimum Attractive Rate of Return)

External Rate of Return (ERR)

ERR Formula:

\[0 = -PW_{\text{costs}} + PW_{\text{receipts}}\]

Negative cash flows discounted at MARR; positive cash flows compounded at external rate.

Inflation and Price Indices

Inflation Adjustment

Future Amount with Inflation:

\[F = P \times (1 + f)^n\]

• F = Future amount in inflated dollars ($)
• P = Present amount ($)
• f = Inflation rate per period (decimal)
• n = Number of periods

Real vs. Nominal Interest Rates

Relationship Between Real and Nominal Rates:

\[1 + i_n = (1 + i_r)(1 + f)\]

• i_n = Nominal (market) interest rate (decimal)
• i_r = Real interest rate (decimal)
• f = Inflation rate (decimal)

Simplified Formula (when rates are small):

\[i_n \approx i_r + f\]

Price Index

Price Index Ratio:

\[I_t = \frac{C_t}{C_0} \times 100\]

• I_t = Price index at time t
• C_t = Cost at time t ($)
• C₀ = Cost at base time ($)

Cost Estimation Using Index:

\[C_t = C_0 \times \frac{I_t}{I_0}\]

Cost Indices and Estimation

Cost Capacity Factor

Cost Scaling Equation:

\[C_2 = C_1 \times \left(\frac{Q_2}{Q_1}\right)^x\]

• C₁, C₂ = Costs for capacity Q₁ and Q₂ ($)
• Q₁, Q₂ = Capacities (units)
• x = Cost capacity factor (typically 0.6 to 0.8)

Note: The "six-tenths rule" uses x = 0.6

Learning Curve

Unit Time Learning Curve:

\[T_n = T_1 \times n^b\]

• T_n = Time to complete nth unit (hours)
• T₁ = Time to complete first unit (hours)
• n = Unit number
• b = Learning curve exponent

Learning Curve Exponent:

\[b = \frac{\ln(LC)}{\ln(2)}\]

• LC = Learning curve percentage (decimal, e.g., 0.80 for 80% learning curve)

Cumulative Average Time:

\[T_{avg,n} = T_1 \times \frac{n^{b+1} - 1}{n(b + 1)}\]

• T_avg,n = Average time per unit for first n units (hours)

Total Time for n Units:

\[T_{total} = n \times T_{avg,n}\]

Life-Cycle Costing

Life-Cycle Cost Components

Total Life-Cycle Cost:

\[LCC = C_I + C_{O\&M} + C_R - S\]

• LCC = Life-Cycle Cost in present worth ($)
• C_I = Initial (capital) cost ($)
• C_O&M = Present worth of operating and maintenance costs ($)
• C_R = Present worth of replacement costs ($)
• S = Present worth of salvage value ($)

Replacement Analysis

Economic Service Life:

Minimize EUAC over all possible service lives to find optimal replacement time.

Defender-Challenger Analysis:

• Compare EUAC of existing asset (defender) vs. new asset (challenger)
• Replace if EUAC_challenger <>_defender

Construction Cost Estimation

Types of Estimates

Order of Magnitude Estimate:

• Accuracy: ±30% to ±50%
• Based on minimal information

Preliminary (Budget) Estimate:

• Accuracy: ±15% to ±30%
• Based on preliminary design

Detailed (Definitive) Estimate:

• Accuracy: ±5% to ±15%
• Based on complete plans and specifications

Estimating Methods

Unit Cost Method:

\[C = Q \times U\]

• C = Total cost ($)
• Q = Quantity (units)
• U = Unit cost ($/unit)

Parametric Estimating:

\[C = a + b \times X\]

• C = Estimated cost ($)
• a, b = Regression coefficients
• X = Parameter (e.g., area, volume, capacity)

Contingency

Total Project Cost with Contingency:

\[C_{total} = C_{base} \times (1 + r_c)\]

• C_total = Total project cost including contingency ($)
• C_base = Base estimate ($)
• r_c = Contingency percentage (decimal)

Bonds and Financing

Bond Valuation

Bond Value:

\[V = C \times (P/A, i, n) + F \times (P/F, i, n)\]

• V = Bond value ($)
• C = Periodic coupon payment ($)
• F = Face (par) value ($)
• i = Market interest rate per period (decimal)
• n = Number of periods to maturity

Coupon Payment:

\[C = F \times r_c\]

• r_c = Coupon rate per period (decimal)

Loan Payments

Loan Payment Amount:

\[A = P \times (A/P, i, n)\]

• A = Periodic payment ($)
• P = Loan principal ($)
• i = Interest rate per period (decimal)
• n = Number of payments

Interest Portion of Payment t:

\[I_t = i \times B_{t-1}\]

• I_t = Interest in payment t ($)
• B_t-1 = Outstanding balance before payment t ($)

Principal Portion of Payment t:

\[PP_t = A - I_t\]

• PP_t = Principal portion of payment t ($)

Outstanding Balance After Payment t:

\[B_t = B_{t-1} - PP_t\]

Effective Interest Rates

Nominal and Effective Rates

Effective Annual Interest Rate:

\[i_e = \left(1 + \frac{r}{m}\right)^m - 1\]

• i_e = Effective annual interest rate (decimal)
• r = Nominal annual interest rate (decimal)
• m = Number of compounding periods per year

Continuous Compounding

Effective Rate with Continuous Compounding:

\[i_e = e^r - 1\]

• e = Base of natural logarithm (≈ 2.71828)
• r = Nominal annual rate (decimal)

Future Worth with Continuous Compounding:

\[F = P \times e^{rn}\]

Present Worth with Continuous Compounding:

\[P = F \times e^{-rn}\]

Sensitivity Analysis

Parameter Variation

Percent Change in Output:

\[\%\Delta_{output} = \frac{\Delta_{output}}{output_{base}} \times 100\%\]

Sensitivity Coefficient:

\[S = \frac{\%\Delta_{output}}{\%\Delta_{input}}\]

• S = Sensitivity coefficient (dimensionless)
• %Δ_output = Percent change in output metric
• %Δ_input = Percent change in input parameter

Note: Higher absolute value of S indicates greater sensitivity.