Formula Sheet: Cost-benefit Analysis

Time Value of Money - Fundamental Formulas

Interest Rate Conventions

• Nominal Interest Rate (r): Annual interest rate stated without considering compounding frequency
• Effective Interest Rate (i): Actual annual interest rate accounting for compounding periods

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

• i = effective annual interest rate (decimal)
• r = nominal annual interest rate (decimal)
• m = number of compounding periods per year

Single Payment Formulas

Future Worth from Present Worth (Compound Amount Factor)

\[F = P(1 + i)^n\] Or using standard notation: \[F = P(F/P, i, n)\] Where:

• F = future worth ($)
• P = present worth ($)
• i = effective interest rate per period (decimal)
• n = number of compounding periods
• (F/P, i, n) = single payment compound amount factor

Present Worth from Future Worth (Present Worth Factor)

\[P = F(1 + i)^{-n}\] Or using standard notation: \[P = F(P/F, i, n)\] Where:

• (P/F, i, n) = single payment present worth factor = \(\frac{1}{(1+i)^n}\)

Uniform Series Formulas

Future Worth from Uniform Series (Uniform Series Compound Amount Factor)

\[F = A\left[\frac{(1 + i)^n - 1}{i}\right]\] Or using standard notation: \[F = A(F/A, i, n)\] Where:

• A = uniform end-of-period payment or receipt ($)
• (F/A, i, n) = uniform series compound amount factor

Present Worth from Uniform Series (Uniform Series Present Worth Factor)

\[P = A\left[\frac{(1 + i)^n - 1}{i(1 + i)^n}\right]\] Or using standard notation: \[P = A(P/A, i, n)\] Where:

• (P/A, i, n) = uniform series present worth factor

Uniform Series from Present Worth (Capital Recovery Factor)

\[A = P\left[\frac{i(1 + i)^n}{(1 + i)^n - 1}\right]\] Or using standard notation: \[A = P(A/P, i, n)\] Where:

• (A/P, i, n) = capital recovery factor
• Used to determine uniform payment equivalent to a present cost

Uniform Series from Future Worth (Sinking Fund Factor)

\[A = F\left[\frac{i}{(1 + i)^n - 1}\right]\] Or using standard notation: \[A = F(A/F, i, n)\] Where:

• (A/F, i, n) = sinking fund factor
• Used to determine uniform payment needed to accumulate a future sum

Arithmetic Gradient Series Formulas

Present Worth from Arithmetic Gradient

\[P = G\left[\frac{(1 + i)^n - in - 1}{i^2(1 + i)^n}\right]\] Or using standard notation: \[P = G(P/G, i, n)\] Where:

• G = arithmetic gradient amount, uniform increment in receipts or disbursements from one period to the next ($)
• (P/G, i, n) = arithmetic gradient present worth factor
• First payment occurs at end of period 2
• Cash flow in period t = A + (t-1)G where A is base amount

Uniform Series from Arithmetic Gradient

\[A = G\left[\frac{1}{i} - \frac{n}{(1 + i)^n - 1}\right]\] Or using standard notation: \[A = G(A/G, i, n)\] Where:

• (A/G, i, n) = arithmetic gradient uniform series factor

Geometric Gradient Series Formulas

Present Worth from Geometric Gradient (g ≠ i)

\[P = A_1\left[\frac{1 - (1 + g)^n(1 + i)^{-n}}{i - g}\right]\] Where:

• A₁ = first payment in geometric series at end of period 1 ($)
• g = geometric gradient rate of increase per period (decimal)
• i = interest rate per period (decimal)
• Valid only when g ≠ i

Present Worth from Geometric Gradient (g = i)

\[P = A_1 \cdot \frac{n}{1 + i}\] Where:

• Special case when geometric gradient rate equals interest rate

Economic Analysis Methods

Present Worth (PW) Method

Net Present Worth (NPW)

\[NPW = \sum_{t=0}^{n} \frac{R_t - D_t}{(1 + i)^t}\] Where:

• NPW = net present worth ($)
• R_t = revenues or benefits in period t ($)
• D_t = disbursements or costs in period t ($)
• t = time period
• n = total number of periods
• i = minimum attractive rate of return (MARR) (decimal)

Decision Rule:

• If NPW > 0, project is economically acceptable
• If NPW < 0,="" project="" is="" not="" economically="">
• For mutually exclusive alternatives, select the one with highest NPW (if positive)

Annual Worth (AW) Method

Annual Worth Calculation

\[AW = NPW \cdot (A/P, i, n)\] Or equivalently: \[AW = NPW \cdot \left[\frac{i(1 + i)^n}{(1 + i)^n - 1}\right]\] Where:

• AW = annual worth, equivalent uniform annual value ($)
• NPW = net present worth ($)

Decision Rule:

• If AW > 0, project is economically acceptable
• If AW < 0,="" project="" is="" not="" economically="">
• For mutually exclusive alternatives with different lives, AW is preferred over NPW

Future Worth (FW) Method

Future Worth Calculation

\[FW = NPW \cdot (1 + i)^n\] Or: \[FW = NPW \cdot (F/P, i, n)\] Where:

• FW = future worth at end of analysis period ($)

Decision Rule:

• If FW > 0, project is economically acceptable
• If FW < 0,="" project="" is="" not="" economically="">

Rate of Return Analysis

Internal Rate of Return (IRR)

\[NPW = \sum_{t=0}^{n} \frac{R_t - D_t}{(1 + IRR)^t} = 0\] Where:

• IRR = internal rate of return (decimal)
• IRR is the interest rate that makes NPW = 0

Decision Rule:

• If IRR > MARR, project is economically acceptable
• If IRR < marr,="" project="" is="" not="" economically="">
• For mutually exclusive alternatives, incremental analysis is required

External Rate of Return (ERR)

\[FW_{receipts}(1 + ERR)^{-n} = FW_{disbursements}\] Or: \[\sum R_t(1 + \epsilon)^{n-t} = \sum D_t(1 + ERR)^{n-t}\] Where:

• ERR = external rate of return (decimal)
• ε = external reinvestment rate for positive cash flows (decimal)
• Addresses reinvestment assumption limitation of IRR

Benefit-Cost Ratio (BCR) Analysis

Conventional Benefit-Cost Ratio

\[BCR = \frac{PW_{benefits}}{PW_{costs}}\] Or: \[BCR = \frac{\sum_{t=1}^{n} B_t(P/F, i, t)}{\sum_{t=0}^{n} C_t(P/F, i, t)}\] Where:

• BCR = benefit-cost ratio (dimensionless)
• B_t = benefits in period t ($)
• C_t = costs in period t ($)
• Initial investment is included in denominator

Decision Rule:

• If BCR ≥ 1.0, project is economically acceptable
• If BCR < 1.0,="" project="" is="" not="" economically="">

Modified Benefit-Cost Ratio

\[BCR_{modified} = \frac{PW_{benefits} - PW_{operating\,costs}}{PW_{capital\,costs}}\] Where:

• Operating costs (maintenance, operations) are subtracted from benefits
• Only capital costs appear in denominator

Incremental Benefit-Cost Ratio

\[\Delta BCR = \frac{PW_{benefits,B} - PW_{benefits,A}}{PW_{costs,B} - PW_{costs,A}}\] Where:

• ΔBCR = incremental benefit-cost ratio
• B = higher-cost alternative
• A = lower-cost alternative
• Used to compare mutually exclusive alternatives

Decision Rule for Incremental Analysis:

• If ΔBCR ≥ 1.0, select higher-cost alternative B
• If ΔBCR < 1.0,="" select="" lower-cost="" alternative="">

Payback Period Analysis

Simple Payback Period

\[n_p = \frac{Initial\,Investment}{Uniform\,Annual\,Net\,Cash\,Flow}\] Where:

• n_p = payback period (years)
• Does not consider time value of money
• Valid only for uniform annual cash flows

Discounted Payback Period

Find smallest n_p such that: \[\sum_{t=1}^{n_p} \frac{NCF_t}{(1 + i)^t} \geq Initial\,Investment\] Where:

• NCF_t = net cash flow in period t ($)
• Accounts for time value of money using discount rate i

Depreciation Methods

Common Depreciation Terms

• B = initial cost basis or unadjusted basis ($)
• S = salvage value or market value at end of useful life ($)
• N = depreciable life or recovery period (years)
• D_t = depreciation charge in year t ($)
• BV_t = book value at end of year t ($)

Straight-Line Depreciation

\[D_t = \frac{B - S}{N}\] \[BV_t = B - t \cdot D_t = B - \frac{t(B - S)}{N}\] Where:

• Depreciation is constant each year
• Valid for t = 1, 2, ..., N

Declining Balance Depreciation

Double Declining Balance (DDB)

\[D_t = BV_{t-1} \cdot \frac{2}{N}\] \[BV_t = B\left(1 - \frac{2}{N}\right)^t\] Where:

• BV₀ = B (initial cost basis)
• Depreciation rate = 2/N
• Salvage value is not considered in calculation but book value cannot go below salvage value

Declining Balance with Rate R

\[D_t = BV_{t-1} \cdot R\] \[BV_t = B(1 - R)^t\] Where:

• R = depreciation rate (decimal), commonly 1.5/N or 2/N

Sum-of-Years-Digits (SOYD) Depreciation

\[D_t = (B - S) \cdot \frac{N - t + 1}{SOYD}\] \[SOYD = \frac{N(N + 1)}{2}\] \[BV_t = B - \sum_{j=1}^{t} D_j\] Where:

• SOYD = sum of years digits
• Accelerated depreciation method
• Largest depreciation in first year, declining each subsequent year

Units of Production Depreciation

\[D_t = (B - S) \cdot \frac{Units\,Produced\,in\,Year\,t}{Total\,Lifetime\,Units}\] Where:

• Depreciation based on actual usage rather than time
• Useful for equipment with measurable output

Modified Accelerated Cost Recovery System (MACRS)

\[D_t = B \cdot r_t\] Where:

• r_t = MACRS percentage rate for year t (from IRS tables)
• Salvage value S = 0 for MACRS
• Recovery periods: 3, 5, 7, 10, 15, 20 years
• Half-year convention applies (½ year depreciation in first and last years)
• Most common method for US federal income tax purposes

Replacement and Retention Analysis

Economic Service Life

Annual Equivalent Cost (AEC)

\[AEC_n = \frac{P - S_n(P/F, i, n)}{(P/A, i, n)} + AOC_n\] Where:

• AEC_n = annual equivalent cost if asset is kept n years ($/year)
• P = initial purchase price ($)
• S_n = salvage/market value after n years ($)
• AOC_n = average annual operating cost over n years ($/year)
• n = analysis period (years)

Economic Service Life:

• The value of n that minimizes AEC_n
• Optimal time to replace an asset from purely economic perspective

Defender-Challenger Analysis

Defender (Existing Asset)

\[AW_{defender} = (MV_0 - MV_N)(A/P, i, N) + AOC_{defender}\] Where:

• MV₀ = current market value of defender ($)
• MV_N = estimated market value at end of retention period ($)
• AOC_defender = annual operating cost of defender ($/year)
• Past costs (sunk costs) are not relevant

Challenger (Replacement Asset)

\[AW_{challenger} = (P - S_N)(A/P, i, N) + AOC_{challenger}\] Where:

• P = purchase price of challenger ($)
• S_N = salvage value of challenger at end of period ($)
• AOC_challenger = annual operating cost of challenger ($/year)

Decision Rule:

• If AW_challenger <>_defender, replace now
• If AW_challenger > AW_defender, keep defender
• Use economic service life for both defender and challenger

Inflation and Price Changes

Inflation-Adjusted Interest Rates

\[1 + i_f = (1 + i_r)(1 + f)\] Or approximately: \[i_f \approx i_r + f\] Where:

• i_f = market (inflation-adjusted) interest rate (decimal)
• i_r = real (inflation-free) interest rate (decimal)
• f = inflation rate (decimal)

Real vs. Actual Dollar Analysis

Converting Actual to Real Dollars

\[Real\,Dollars_t = \frac{Actual\,Dollars_t}{(1 + f)^t}\] Where:

• Real dollars expressed in base-year (constant) purchasing power
• Actual dollars are then-current (inflated) amounts

Present Worth with Inflation

Using actual dollars and market interest rate: \[PW = \sum_{t=0}^{n} \frac{CF_t^{actual}}{(1 + i_f)^t}\] Or using real dollars and real interest rate: \[PW = \sum_{t=0}^{n} \frac{CF_t^{real}}{(1 + i_r)^t}\] Where:

• CF_t = cash flow in period t
• Both methods yield identical present worth values

Income Tax Effects

After-Tax Cash Flow Analysis

Taxable Income

\[TI_t = Revenue_t - Expenses_t - D_t\] Where:

• TI_t = taxable income in year t ($)
• D_t = depreciation deduction in year t ($)
• Depreciation is tax-deductible but not a cash expense

Income Tax

\[Tax_t = TI_t \cdot t_r\] Where:

• Tax_t = income tax in year t ($)
• t_r = effective income tax rate (decimal)

After-Tax Cash Flow (ATCF)

\[ATCF_t = BTCF_t - Tax_t\] Or equivalently: \[ATCF_t = Revenue_t - Expenses_t - Tax_t\] \[ATCF_t = (Revenue_t - Expenses_t - D_t)(1 - t_r) + D_t\] Where:

• ATCF_t = after-tax cash flow in year t ($)
• BTCF_t = before-tax cash flow in year t ($)

After-Tax Present Worth

\[PW_{after-tax} = \sum_{t=0}^{n} \frac{ATCF_t}{(1 + i_{after-tax})^t}\] Where:

• i_after-tax = after-tax MARR (decimal)

Capital Gains Tax

\[Capital\,Gain = Selling\,Price - Book\,Value\] \[Capital\,Gains\,Tax = Capital\,Gain \cdot t_{cg}\] Where:

• t_cg = capital gains tax rate (decimal)
• Applies when asset is sold for more than book value
• If selling price < book="" value,="" results="" in="" capital="">

Break-Even Analysis

Break-Even Quantity

Single Project Break-Even

\[Revenue = Total\,Cost\] \[p \cdot Q_{BE} = FC + VC \cdot Q_{BE}\] \[Q_{BE} = \frac{FC}{p - VC}\] Where:

• Q_BE = break-even quantity (units)
• p = selling price per unit ($/unit)
• FC = fixed costs ($)
• VC = variable cost per unit ($/unit)

Two-Alternative Break-Even

\[TC_1 = TC_2\] \[FC_1 + VC_1 \cdot Q_{BE} = FC_2 + VC_2 \cdot Q_{BE}\] \[Q_{BE} = \frac{FC_2 - FC_1}{VC_1 - VC_2}\] Where:

• Subscripts 1 and 2 denote two alternatives
• Valid when VC₁ ≠ VC₂

Break-Even Interest Rate

Find i such that: \[NPW(i) = 0\] Or for two alternatives: \[NPW_1(i) = NPW_2(i)\] Where:

• Interest rate at which decision changes
• Same as IRR for single project
• Incremental IRR for two alternatives

Break-Even Life

Find n such that: \[NPW(n) = 0\] Or: \[AW(n) = 0\] Where:

• Minimum life for project to be economically acceptable

Capitalized Cost

Capitalized Cost Formula

\[CC = P_0 + \frac{A}{i}\] Where:

• CC = capitalized cost, present worth of infinite uniform series ($)
• P₀ = initial investment at time zero ($)
• A = uniform annual amount ($/year)
• i = interest rate (decimal)

Capitalized Cost with Recurring Renewal

\[CC = P_0 + P_R \cdot (A/F, i, n) / i\] Or: \[CC = P_0 + \frac{P_R \cdot i}{(1+i)^n - 1}\] Where:

• P_R = renewal cost every n periods ($)
• n = renewal period (years)
• Used for perpetual service with periodic asset replacement

Sensitivity Analysis

Sensitivity to Single Parameter

Analyze how changes in one variable affect the decision measure: \[\Delta NPW = \frac{\partial NPW}{\partial x} \cdot \Delta x\] Where:

• x = parameter being varied (e.g., interest rate, cost, revenue)
• Identify most critical variables affecting decision
• Typically vary parameters by ±10%, ±20%, ±30%

Optimistic-Most Likely-Pessimistic Analysis

Evaluate economic measure under three scenarios:

• Optimistic: favorable parameter values
• Most Likely: expected parameter values
• Pessimistic: unfavorable parameter values

Comparison of Alternatives with Unequal Lives

Least Common Multiple (LCM) Method

• Extend analysis period to LCM of alternative lives
• Assume identical replacement at end of each alternative's life
• Calculate NPW over LCM period

\[NPW_{total} = NPW_1 + NPW_1(P/F, i, n_1) + NPW_1(P/F, i, 2n_1) + ...\] Where:

• n₁ = life of alternative 1 (years)
• Repeat until reaching LCM of all alternative lives

Annual Worth Method (Preferred for Unequal Lives)

• Convert each alternative to equivalent annual worth
• No need to find LCM
• Valid assumption: services required indefinitely
• Compare AW values directly regardless of life differences

Study Period Method

• Define common analysis period based on project requirements
• Estimate terminal values for alternatives at end of study period
• Calculate NPW over study period

Public Sector Economic Analysis

Benefit-Cost Analysis for Public Projects

User Benefits

• Direct benefits: measurable improvements to users
• Indirect benefits: secondary economic effects
• Intangible benefits: non-monetary improvements (safety, environment)

Discount Rate for Public Projects

• Social discount rate (typically lower than private sector MARR)
• Often set by government policy
• Reflects societal time preference

Cost-Effectiveness Analysis

\[CE = \frac{Cost}{Effectiveness\,Measure}\] Where:

• CE = cost-effectiveness ratio
• Effectiveness Measure = non-monetary output (e.g., lives saved, pollution reduced)
• Used when benefits cannot be monetized
• Select alternative with lowest CE ratio

Risk and Uncertainty

Expected Value Analysis

\[E(X) = \sum_{i=1}^{n} P_i \cdot X_i\] Where:

• E(X) = expected value of outcome X
• P_i = probability of outcome i
• X_i = value of outcome i
• ∑P_i = 1

Expected Net Present Worth

\[E(NPW) = \sum_{i=1}^{n} P_i \cdot NPW_i\] Where:

• E(NPW) = expected net present worth ($)
• NPW_i = net present worth for scenario i ($)

Variance and Standard Deviation

\[\sigma^2 = Var(X) = \sum_{i=1}^{n} P_i[X_i - E(X)]^2\] \[\sigma = \sqrt{Var(X)}\] Where:

• σ² = variance
• σ = standard deviation
• Measures uncertainty or risk

Bonds and Loans

Bond Valuation

\[PV_{bond} = \frac{C}{i}[1 - (1+i)^{-n}] + \frac{F}{(1+i)^n}\] Or: \[PV_{bond} = C(P/A, i, n) + F(P/F, i, n)\] Where:

• PV_bond = present value of bond ($)
• C = periodic coupon payment ($)
• F = face value (par value) of bond ($)
• i = market interest rate per period (yield) (decimal)
• n = number of periods until maturity

Loan Payments

Equal Principal Payment

\[Principal\,Payment_t = \frac{P}{n}\] \[Interest\,Payment_t = i \cdot Outstanding\,Balance_{t-1}\] \[Total\,Payment_t = Principal\,Payment_t + Interest\,Payment_t\] Where:

• Principal payment is constant each period
• Total payment decreases over time

Equal Total Payment (Amortized Loan)

\[A = P(A/P, i, n) = P \cdot \frac{i(1+i)^n}{(1+i)^n - 1}\] \[Interest\,Payment_t = i \cdot Outstanding\,Balance_{t-1}\] \[Principal\,Payment_t = A - Interest\,Payment_t\] Where:

• Total payment A is constant each period
• Principal portion increases over time
• Interest portion decreases over time

Outstanding Loan Balance

\[Outstanding\,Balance_t = A(P/A, i, n-t)\] Or: \[Outstanding\,Balance_t = P(1+i)^t - A(F/A, i, t)\] Where:

• Balance after t payments
• n-t = remaining number of payments