Formula Sheet: Time Value of Money

Interest and Interest Rate Fundamentals

Simple Interest

Simple Interest Formula:

\[I = P \times i \times n\]

• I = total interest earned or paid ($)
• P = principal amount ($)
• i = interest rate per time period (decimal)
• n = number of time periods

Total Amount with Simple Interest:

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

• F = future amount ($)
• P = principal amount ($)
• i = interest rate per time period (decimal)
• n = number of time periods

Compound Interest

Compound Amount Formula:

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

• F = future worth ($)
• P = present worth ($)
• i = effective interest rate per compounding period (decimal)
• n = number of compounding periods

Present Worth from Future Amount:

\[P = F(1 + i)^{-n}\]

Nominal and Effective Interest Rates

Effective Interest Rate (Annual):

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

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

Continuous Compounding Effective Rate:

\[i_e = e^r - 1\]

• i_e = effective annual interest rate (decimal)
• r = nominal annual interest rate (decimal)
• e = Euler's number ≈ 2.71828

Future Worth with Continuous Compounding:

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

• F = future worth ($)
• P = present worth ($)
• r = nominal annual interest rate (decimal)
• n = number of years

Cash Flow Factors and Notation

Standard Factor Notation

General Factor Notation:

\[(X/Y, i, n)\]

• X = what you want to find (F, P, A, or G)
• Y = what you are given (F, P, A, or G)
• i = interest rate per period (%)
• n = number of periods

Single Payment Formulas

Single Payment Compound Amount Factor (F/P):

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

• Converts a present single sum to a future single sum
• F = future worth ($)
• P = present worth ($)

Single Payment Present Worth Factor (P/F):

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

• Converts a future single sum to a present single sum
• Also known as the discount factor

Uniform Series Formulas

Uniform Series Compound Amount Factor (F/A):

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

• Converts a uniform series of end-of-period payments to a future sum
• A = uniform end-of-period payment or receipt ($)
• F = future worth ($)
• First payment occurs at end of period 1; future worth is at end of period n

Uniform Series Sinking Fund Factor (A/F):

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

• Converts a future sum to a uniform series of end-of-period payments
• Used to determine the periodic deposit needed to accumulate a future sum

Uniform Series Present Worth Factor (P/A):

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

• Converts a uniform series of end-of-period payments to a present sum
• Present worth is located one period before the first payment

Uniform Series Capital Recovery Factor (A/P):

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

• Converts a present sum to a uniform series of end-of-period payments
• Used to determine loan payments or capital recovery amounts

Relationship Between Factors

Reciprocal Relationships:

• \((F/P, i, n) = \frac{1}{(P/F, i, n)}\)
• \((F/A, i, n) = \frac{1}{(A/F, i, n)}\)
• \((P/A, i, n) = \frac{1}{(A/P, i, n)}\)

Derived Relationships:

• \((P/A, i, n) = (P/F, i, n) \times (F/A, i, n)\)
• \((A/P, i, n) = (A/F, i, n) + i\)

Gradient Series

Arithmetic Gradient

Arithmetic Gradient Present Worth Factor (P/G):

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

• G = arithmetic gradient, the constant amount by which payments increase each period ($)
• P = present worth of the gradient series only ($)
• Gradient starts at end of period 2 (first payment = 0, second payment = G, third = 2G, etc.)
• Does not include base amount; combine with uniform series if base payment exists

Arithmetic Gradient Uniform Series Factor (A/G):

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

• Converts an arithmetic gradient series to an equivalent uniform series
• A = equivalent uniform amount for gradient only ($)

Combined Uniform and Gradient Series:

\[P = A_1(P/A, i, n) + G(P/G, i, n)\]

• A₁ = base payment at end of period 1 ($)
• Total present worth when a uniform series has a constant gradient increase

Geometric Gradient

Geometric Gradient Present Worth (Case 1: g ≠ i):

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

• A₁ = first payment at end of period 1 ($)
• g = rate of growth (or decline if negative) per period (decimal)
• i = interest rate per period (decimal)
• n = number of periods
• Each payment = previous payment × (1 + g)
• Valid only when g ≠ i

Geometric Gradient Present Worth (Case 2: g = i):

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

• Special case when growth rate equals interest rate
• Formula simplifies to avoid division by zero

Deferred Annuities (Delayed Uniform Series)

Present Worth of Deferred Annuity:

\[P_0 = A(P/A, i, n)(P/F, i, d)\]

• P₀ = present worth at time 0 ($)
• A = uniform payment amount ($)
• d = number of periods of deferral (delay before first payment)
• n = number of payments in the uniform series
• First payment occurs at end of period (d + 1)
• Alternative: \(P_0 = A \times [(P/A, i, n+d) - (P/A, i, d)]\)

Capitalized Cost

Capitalized Cost (Perpetual Life):

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

• P = capitalized cost, present worth of perpetual service ($)
• A = uniform annual payment or cost ($)
• i = interest rate per period (decimal)
• Represents present worth of an infinite series of equal payments
• As n → ∞, \((P/A, i, n)\) → \(\frac{1}{i}\)

Capitalized Cost with Initial Investment:

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

• CC = total capitalized cost ($)
• P₀ = initial investment or first cost ($)
• A = uniform annual cost ($)

Economic Analysis Methods

Present Worth Analysis (PW)

Present Worth Formula:

\[PW = \sum_{t=0}^{n} \frac{CF_t}{(1 + i)^t}\]

• PW = present worth of all cash flows ($)
• CF_t = cash flow at time t ($)
• i = interest rate or minimum attractive rate of return (MARR) (decimal)
• t = time period
• n = total number of periods

Decision Rule:

• Single project: Accept if PW ≥ 0
• Multiple alternatives: Select alternative with highest PW (or least negative if all are negative)
• Mutually exclusive alternatives must be compared over equal time periods

Future Worth Analysis (FW)

Future Worth Formula:

\[FW = PW(1 + i)^n\] \[FW = \sum_{t=0}^{n} CF_t(1 + i)^{n-t}\]

• FW = future worth of all cash flows ($)
• All cash flows compounded to future time n

Decision Rule:

• Single project: Accept if FW ≥ 0
• Multiple alternatives: Select alternative with highest FW
• Yields same decision as PW analysis

Annual Worth Analysis (AW)

Annual Worth from Present Worth:

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

Annual Worth from Future Worth:

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

Annual Worth Direct Calculation:

\[AW = \text{Annual Benefits} - \text{Annual Costs}\]

• AW = annual worth, equivalent uniform annual worth ($)
• Also called Annual Equivalent Worth (AEW) or Equivalent Annual Worth (EAW)

Decision Rule:

• Single project: Accept if AW ≥ 0
• Multiple alternatives: Select alternative with highest AW
• Advantage: alternatives with unequal lives can be compared directly

Capital Recovery Cost

Annual Capital Recovery Cost:

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

• CR = annual capital recovery cost ($)
• P = initial cost or first cost ($)
• S = salvage value at end of life ($)
• i = interest rate (decimal)
• n = useful life (years)

Alternative Form:

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

Rate of Return Analysis

Internal Rate of Return (IRR)

IRR Definition:

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

• IRR = internal rate of return, the interest rate that makes PW = 0 (decimal)
• Also called Rate of Return (ROR)
• Solve for i when PW = 0

Decision Rule (Single Project):

• Accept project if IRR ≥ MARR
• Reject project if IRR <>
• MARR = Minimum Attractive Rate of Return

Incremental Rate of Return (ΔIRR)

Incremental Analysis:

\[\sum_{t=0}^{n} \frac{(CF_B - CF_A)_t}{(1 + \Delta IRR)^t} = 0\]

• ΔIRR = incremental rate of return between alternatives (decimal)
• CF_B = cash flow of higher-cost alternative B ($)
• CF_A = cash flow of lower-cost alternative A ($)

Decision Rule (Two Alternatives):

• Rank alternatives by increasing initial cost
• If ΔIRR ≥ MARR: select higher-cost alternative (B)
• If ΔIRR < marr:="" select="" lower-cost="" alternative="">

External Rate of Return (ERR)

ERR Method:

\[PW(\text{investments}) \times (1 + ERR)^n = FW(\text{receipts at external rate})\]

• Assumes positive cash flows reinvested at external rate (usually MARR)
• Overcomes multiple IRR problem
• More realistic reinvestment assumption than IRR

Benefit-Cost Ratio Analysis

Conventional Benefit-Cost Ratio

Conventional B/C Ratio:

\[B/C = \frac{PW(\text{Benefits})}{PW(\text{Costs})}\]

Or using Annual Worth:

\[B/C = \frac{AW(\text{Benefits})}{AW(\text{Costs})}\]

• B/C = benefit-cost ratio (dimensionless)
• Benefits and costs calculated at same interest rate
• Commonly used for public sector projects

Decision Rule:

• Accept project if B/C ≥ 1.0
• Reject project if B/C <>

Modified Benefit-Cost Ratio

Modified B/C Ratio:

\[B/C_{modified} = \frac{PW(\text{Benefits}) - PW(\text{Operating Costs})}{PW(\text{Initial Investment})}\]

• Subtracts operating and maintenance costs from benefits
• Only initial capital costs in denominator

Incremental B/C Analysis

Incremental B/C Ratio:

\[\Delta(B/C) = \frac{PW(\text{Benefits}_B) - PW(\text{Benefits}_A)}{PW(\text{Costs}_B) - PW(\text{Costs}_A)}\]

• Compare higher-cost alternative (B) to lower-cost alternative (A)
• If Δ(B/C) ≥ 1.0: select higher-cost alternative
• If Δ(B/C) < 1.0:="" select="" lower-cost="">

Payback Period

Simple Payback Period

Simple Payback (Uniform Cash Flows):

\[n_p = \frac{P}{A}\]

• n_p = payback period (years)
• P = initial investment ($)
• A = uniform annual net cash flow ($)
• Does not consider time value of money
• Does not consider cash flows after payback

Simple Payback (Non-uniform Cash Flows):

• Sum annual cash flows until cumulative cash flow equals initial investment
• Payback period is when: \(\sum CF_t = P\)

Discounted Payback Period

Discounted Payback:

• Find n_p when: \(\sum_{t=1}^{n_p} \frac{CF_t}{(1+i)^t} = P\)
• Accounts for time value of money using interest rate i
• Discounted payback is always longer than simple payback

Decision Rule:

• Accept if payback period ≤ maximum acceptable period
• Not a complete measure of profitability
• Should be used as supplementary criterion with other methods

Break-Even Analysis

Break-Even Point

Break-Even Equation:

\[\text{Total Revenue} = \text{Total Cost}\] \[p \times Q = FC + VC \times Q\]

• p = price per unit ($/unit)
• Q = quantity produced and sold (units)
• FC = fixed costs ($)
• VC = variable cost per unit ($/unit)

Break-Even Quantity:

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

• Q_BE = break-even quantity (units)
• Quantity at which profit = 0

Break-Even Interest Rate

Definition:

• Interest rate at which two alternatives have equal economic value
• Solve for i when: PW_A(i) = PW_B(i)
• Also equals incremental rate of return (ΔIRR)

Inflation

Inflation Adjustment

Actual (Inflated) Dollar Amount:

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

• F_n = actual dollar amount at year n ($)
• P = present amount ($)
• f = inflation rate per period (decimal)
• n = number of periods

Real vs. Actual Interest Rates

Relationship Between Real and Actual Rates:

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

Solving for Real Interest Rate:

\[i_r = \frac{1 + i_f}{1 + f} - 1 = \frac{i_f - f}{1 + f}\]

Approximate Relationship (small rates):

\[i_f \approx i_r + f\]

• i_f = actual (market, inflated) interest rate (decimal)
• i_r = real interest rate (decimal)
• f = inflation rate (decimal)
• Use i_f with actual dollars or i_r with real (constant) dollars
• Never mix actual dollars with real interest rate or vice versa

Bonds

Bond Valuation

Bond Present Value (Price):

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

• P = bond price or present value ($)
• C = periodic coupon payment ($)
• F = face value (par value) of bond ($)
• i = market interest rate per period (yield rate, required rate) (decimal)
• n = number of periods to maturity

Coupon Payment:

\[C = F \times c\]

• c = coupon rate (decimal)
• F = face value ($)

Bond Price Relationships:

• If market rate (i) = coupon rate (c): bond sells at par (P = F)
• If market rate (i) > coupon rate (c): bond sells at discount (P <>
• If market rate (i) < coupon="" rate="" (c):="" bond="" sells="" at="" premium="" (p=""> F)

Depreciation (For Tax Calculations)

Straight-Line Depreciation

Annual Depreciation:

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

• D = annual depreciation charge ($)
• B = initial cost basis ($)
• S = salvage value at end of useful life ($)
• n = useful life or recovery period (years)

Book Value at Year t:

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

Declining Balance Depreciation

Depreciation in Year t:

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

• D_t = depreciation in year t ($)
• d = depreciation rate (decimal)
• BV_t-1 = book value at beginning of year t ($)

Book Value at Year t:

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

Common Declining Balance Rates:

• Double Declining Balance (DDB): d = 2/n
• 150% Declining Balance: d = 1.5/n

Sum-of-Years'-Digits (SYD) Depreciation

Depreciation in Year t:

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

Sum of Years' Digits:

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

• SYD = sum of years' digits
• t = year number (1, 2, 3, ..., n)

Modified Accelerated Cost Recovery System (MACRS)

Annual Depreciation:

\[D_t = B \times r_t\]

• D_t = depreciation in year t ($)
• B = cost basis (initial cost) ($)
• r_t = MACRS depreciation rate for year t (from IRS tables)
• Salvage value is ignored (assumed to be zero)
• Half-year convention: half-year of depreciation in first and last years
• Recovery periods: 3, 5, 7, 10, 15, 20, 27.5, 39 years depending on asset class

After-Tax Cash Flow Analysis

Taxable Income and Income Tax

Taxable Income:

\[TI = GI - E - D\]

• TI = taxable income ($)
• GI = gross income or revenues ($)
• E = operating expenses (deductible) ($)
• D = depreciation ($)

Income Tax:

\[T = TI \times t\]

• T = income tax ($)
• t = effective tax rate (decimal)

After-Tax Cash Flow

After-Tax Cash Flow (ATCF):

\[ATCF = BTCF - T\] \[ATCF = GI - E - T\] \[ATCF = (GI - E - D)(1 - t) + D\]

• ATCF = after-tax cash flow ($)
• BTCF = before-tax cash flow = GI - E ($)
• T = income tax ($)
• Depreciation is added back because it is not a cash expense

Simplified Form:

\[ATCF = BTCF(1 - t) + D \times t\]

• D × t = tax savings due to depreciation (depreciation tax shield)

After-Tax Rate of Return

Relationship Between Before-Tax and After-Tax ROR:

\[i_{AT} = i_{BT}(1 - t)\]

• i_AT = after-tax rate of return (decimal)
• i_BT = before-tax rate of return (decimal)
• t = effective tax rate (decimal)
• Approximation valid for investment income and simple tax situations

Replacement Analysis

Defender vs. Challenger

Defender:

• Currently owned asset
• First cost = current market value (opportunity cost)
• Sunk costs are ignored

Challenger:

• Proposed replacement asset
• First cost = purchase price of new asset

Economic Service Life

Economic Service Life (ESL):

• Number of years at which equivalent annual cost (EAC) is minimized
• Calculate EAC for n = 1, 2, 3, ... years
• ESL occurs at year with minimum EAC

Equivalent Annual Cost:

\[EAC = (P - S_n)(A/P, i, n) + S_n \times i + AOC\]

• EAC = equivalent annual cost ($)
• P = first cost or current market value ($)
• S_n = salvage value at end of year n ($)
• AOC = annual operating cost ($)
• n = service life being evaluated (years)

Replacement Decision Rules

Decision Criterion:

• Calculate EAC of defender at its ESL
• Calculate EAC of challenger at its ESL
• If EAC_challenger <>_defender: replace now
• If EAC_challenger > EAC_defender: keep defender

Comparing Alternatives with Unequal Lives

Least Common Multiple of Lives Method

Method:

• Find least common multiple (LCM) of service lives
• Repeat each alternative over LCM period
• Calculate PW, FW, or AW over LCM period
• Select alternative with best economic measure

Annual Worth Method

Method:

• Calculate AW for each alternative over its own life
• Compare AW values directly
• Select alternative with highest AW (or least negative cost)
• Assumes alternatives can be repeated indefinitely
• Most common method for unequal lives

Study Period Method

Method:

• Define common study period (analysis period)
• Truncate longer-lived alternatives (estimate market value at end of study period)
• Repeat shorter-lived alternatives as needed
• Calculate PW or FW over study period

Lease vs. Purchase Analysis

Lease Analysis

Present Worth of Leasing:

\[PW_{lease} = L(P/A, i, n)\]

• PW_lease = present worth of lease payments ($)
• L = periodic lease payment ($)
• May need to adjust for timing (beginning vs. end of period payments)
• For beginning-of-period payments: multiply by (1 + i)

Present Worth of Purchase:

\[PW_{purchase} = P - S(P/F, i, n) + AOC(P/A, i, n)\]

• P = purchase price ($)
• S = salvage/resale value ($)
• AOC = annual operating cost ($)

Decision Rule:

• Compare PW_lease vs. PW_purchase
• Select option with lower present worth of costs
• After-tax analysis may be required considering depreciation and tax deductibility of lease payments