Formula Sheet: Present Worth and Future Worth

Table of Contents
1. Time Value of Money Fundamentals
2. Single Payment Formulas
3. Uniform Series Formulas
4. Gradient Series Formulas
5. Nominal and Effective Interest Rates
6. Present Worth Analysis Methods
7. Future Worth Analysis Methods
8. Special Cases and Applications
9. Key Relationships and Identities
10. Cash Flow Diagrams and Sign Conventions
11. Common Applications in Engineering Economics
12. Analysis Period Considerations
13. Important Assumptions and Limitations
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Time Value of Money Fundamentals

Basic Definitions and Concepts

• Present Worth (PW or P): The equivalent value of all cash flows at time zero (present)
• Future Worth (FW or F): The equivalent value of all cash flows at a specified future point in time
• Interest Rate (i): The rate of return or discount rate per compounding period (expressed as a decimal)
• Number of Periods (n): The total number of compounding periods
• Cash Flow (CF): Money flowing in (positive) or out (negative) at specific time periods

Simple vs. Compound Interest

Simple Interest

• Formula: \[F = P(1 + ni)\]
• Where:
  • F = future worth
  • P = present worth (principal)
  • n = number of periods
  • i = interest rate per period
• Note: Interest is calculated only on the principal amount

Compound Interest

• Formula: \[F = P(1 + i)^n\]
• Where:
  • F = future worth
  • P = present worth (principal)
  • n = number of periods
  • i = interest rate per period
• Note: Interest is calculated on principal plus accumulated interest

Single Payment Formulas

Single Payment Compound Amount Factor (F/P)

• Formula: \[F = P(1 + i)^n\]
• Factor Notation: \[(F/P, i\%, n)\]
• Factor Value: \[(F/P, i, n) = (1 + i)^n\]
• Purpose: To find future worth F given present worth P
• Variables:
  • F = future worth
  • P = present worth
  • i = effective interest rate per period
  • n = number of compounding periods

Single Payment Present Worth Factor (P/F)

• Formula: \[P = F(1 + i)^{-n}\]
• Alternative Form: \[P = \frac{F}{(1 + i)^n}\]
• Factor Notation: \[(P/F, i\%, n)\]
• Factor Value: \[(P/F, i, n) = (1 + i)^{-n} = \frac{1}{(1 + i)^n}\]
• Purpose: To find present worth P given future worth F
• Note: This factor is the reciprocal of (F/P, i, n)
• Relationship: \[(P/F, i, n) = \frac{1}{(F/P, i, n)}\]

Uniform Series Formulas

Uniform Series Compound Amount Factor (F/A)

• Formula: \[F = A\left[\frac{(1 + i)^n - 1}{i}\right]\]
• Factor Notation: \[(F/A, i\%, n)\]
• Factor Value: \[(F/A, i, n) = \frac{(1 + i)^n - 1}{i}\]
• Purpose: To find future worth F given uniform series A
• Variables:
  • F = future worth at end of period n
  • A = uniform payment amount per period
  • i = effective interest rate per period
  • n = number of periods
• Assumptions: First payment occurs at end of period 1; last payment at end of period n

Sinking Fund Factor (A/F)

• Formula: \[A = F\left[\frac{i}{(1 + i)^n - 1}\right]\]
• Factor Notation: \[(A/F, i\%, n)\]
• Factor Value: \[(A/F, i, n) = \frac{i}{(1 + i)^n - 1}\]
• Purpose: To find uniform series A given future worth F
• Note: This factor is the reciprocal of (F/A, i, n)
• Relationship: \[(A/F, i, n) = \frac{1}{(F/A, i, n)}\]
• Application: Used to determine periodic deposits needed to accumulate a future amount

Uniform Series Present Worth Factor (P/A)

• Formula: \[P = A\left[\frac{(1 + i)^n - 1}{i(1 + i)^n}\right]\]
• Alternative Form: \[P = A\left[\frac{1 - (1 + i)^{-n}}{i}\right]\]
• Factor Notation: \[(P/A, i\%, n)\]
• Factor Value: \[(P/A, i, n) = \frac{(1 + i)^n - 1}{i(1 + i)^n}\]
• Purpose: To find present worth P given uniform series A
• Assumptions: Present worth is one period before the first payment

Capital Recovery Factor (A/P)

• Formula: \[A = P\left[\frac{i(1 + i)^n}{(1 + i)^n - 1}\right]\]
• Alternative Form: \[A = P\left[\frac{i}{1 - (1 + i)^{-n}}\right]\]
• Factor Notation: \[(A/P, i\%, n)\]
• Factor Value: \[(A/P, i, n) = \frac{i(1 + i)^n}{(1 + i)^n - 1}\]
• Purpose: To find uniform series A given present worth P
• Note: This factor is the reciprocal of (P/A, i, n)
• Relationship: \[(A/P, i, n) = \frac{1}{(P/A, i, n)}\]
• Application: Used to calculate loan payments and capital recovery costs

Gradient Series Formulas

Arithmetic Gradient Series

Arithmetic Gradient to Present Worth Factor (P/G)

• Formula: \[P = G\left[\frac{(1 + i)^n - in - 1}{i^2(1 + i)^n}\right]\]
• Alternative Form: \[P = G\left[\frac{1}{i}\left(\frac{(1 + i)^n - 1}{i(1 + i)^n}\right) - \frac{n}{(1 + i)^n}\right]\]
• Factor Notation: \[(P/G, i\%, n)\]
• Factor Value: \[(P/G, i, n) = \frac{(1 + i)^n - in - 1}{i^2(1 + i)^n}\]
• Purpose: To find present worth P of an arithmetic gradient series
• Variables:
  • P = present worth
  • G = constant arithmetic gradient (change per period)
  • i = effective interest rate per period
  • n = number of periods
• Cash Flow Pattern: Period 1 = 0, Period 2 = G, Period 3 = 2G, ..., Period n = (n-1)G
• Note: The base amount A must be handled separately; gradient starts at zero in period 1

Arithmetic Gradient to Uniform Series Factor (A/G)

• Formula: \[A = G\left[\frac{1}{i} - \frac{n}{(1 + i)^n - 1}\right]\]
• Factor Notation: \[(A/G, i\%, n)\]
• Factor Value: \[(A/G, i, n) = \frac{1}{i} - \frac{n}{(1 + i)^n - 1}\]
• Purpose: To convert arithmetic gradient G to equivalent uniform series A
• Relationship: \[(A/G, i, n) = (P/G, i, n) \times (A/P, i, n)\]

Combined Uniform Series and Gradient

• Present Worth: \[P = A_1(P/A, i, n) + G(P/G, i, n)\]
• Where:
  • A₁ = base uniform amount in period 1
  • G = constant gradient increase per period
• Cash Flow: Period 1 = A₁, Period 2 = A₁ + G, Period 3 = A₁ + 2G, etc.

Geometric Gradient Series

Geometric Gradient Present Worth (i ≠ g)

• Formula: \[P = A_1\left[\frac{1 - (1 + g)^n(1 + i)^{-n}}{i - g}\right]\]
• Alternative Form: \[P = \frac{A_1}{i - g}\left[1 - \left(\frac{1 + g}{1 + i}\right)^n\right]\]
• Purpose: To find present worth of geometric gradient series when i ≠ g
• Variables:
  • P = present worth
  • A₁ = first period payment amount
  • g = rate of change (growth rate) per period, expressed as decimal
  • i = effective interest rate per period
  • n = number of periods
• Cash Flow Pattern: Period 1 = A₁, Period 2 = A₁(1 + g), Period 3 = A₁(1 + g)², etc.
• Condition: Valid only when i ≠ g

Geometric Gradient Present Worth (i = g)

• Formula: \[P = \frac{nA_1}{1 + i}\]
• Purpose: To find present worth when interest rate equals growth rate
• Condition: Valid only when i = g
• Note: This is a special case derived from the limit as i approaches g

Geometric Gradient Future Worth (i ≠ g)

• Formula: \[F = A_1\left[\frac{(1 + i)^n - (1 + g)^n}{i - g}\right]\]
• Alternative Derivation: \[F = P(F/P, i, n)\]
• Purpose: To find future worth of geometric gradient series
• Condition: Valid only when i ≠ g

Geometric Gradient Future Worth (i = g)

• Formula: \[F = nA_1(1 + i)^{n-1}\]
• Purpose: To find future worth when interest rate equals growth rate
• Condition: Valid only when i = g

Nominal and Effective Interest Rates

Effective Interest Rate per Payment Period

• Formula: \[i_{\text{eff}} = \left(1 + \frac{r}{m}\right)^m - 1\]
• Variables:
  • i_eff = effective annual interest rate
  • r = nominal annual interest rate (APR)
  • m = number of compounding periods per year
• Purpose: To convert nominal interest rate to effective annual rate

Effective Interest Rate per Compounding Period

• Formula: \[i = \frac{r}{m}\]
• Variables:
  • i = effective interest rate per compounding period
  • r = nominal annual interest rate
  • m = number of compounding periods per year
• Use: This is the rate used in standard compound interest formulas

Continuous Compounding

• Effective Annual Rate: \[i_{\text{eff}} = e^r - 1\]
• Future Worth: \[F = Pe^{rn}\]
• Present Worth: \[P = Fe^{-rn}\]
• Variables:
  • e = Euler's number (approximately 2.71828)
  • r = nominal annual interest rate (continuous)
  • n = number of years
• Note: Represents the limiting case as compounding frequency approaches infinity

Interest Rate Conversions

• Converting between different compounding frequencies: \[\left(1 + \frac{r_1}{m_1}\right)^{m_1} = \left(1 + \frac{r_2}{m_2}\right)^{m_2}\]
• Where:
  • r₁, r₂ = nominal rates for different compounding frequencies
  • m₁, m₂ = number of compounding periods per year

Present Worth Analysis Methods

Net Present Worth (NPW)

• Formula: \[NPW = \sum_{t=0}^{n} \frac{CF_t}{(1 + i)^t}\]
• Alternative Notation: \[NPW = \sum_{t=0}^{n} CF_t(P/F, i, t)\]
• Variables:
  • NPW = net present worth (also called NPV, net present value)
  • CF_t = cash flow at time t (positive for inflows, negative for outflows)
  • i = discount rate (minimum attractive rate of return, MARR)
  • t = time period index
  • n = total number of periods
• Decision Rule:
  • If NPW > 0: Project is economically acceptable
  • If NPW = 0: Project breaks even
  • If NPW < 0:="" project="" is="" not="" economically="">

Present Worth of Costs (PWC)

• Formula: \[PWC = \sum_{t=0}^{n} \frac{C_t}{(1 + i)^t}\]
• Variables:
  • PWC = present worth of all costs
  • C_t = cost at time t
• Purpose: Used for comparing alternatives with equal benefits
• Decision Rule: Select alternative with lowest PWC

Capitalized Cost (CC)

• Formula for Perpetual Uniform Series: \[CC = \frac{A}{i}\]
• General Formula: \[CC = P_0 + \frac{A}{i} + \sum \frac{F_k}{(1+i)^k - 1}\]
• Variables:
  • CC = capitalized cost (present worth of infinite series)
  • A = uniform annual cost
  • i = interest rate per period
  • P₀ = initial cost
  • F_k = periodic renewal/replacement cost at interval k
• Application: Used for projects with infinite life (e.g., endowments, perpetual infrastructure)

Future Worth Analysis Methods

Net Future Worth (NFW)

• Formula: \[NFW = \sum_{t=0}^{n} CF_t(1 + i)^{n-t}\]
• Alternative Notation: \[NFW = \sum_{t=0}^{n} CF_t(F/P, i, n-t)\]
• Relationship to NPW: \[NFW = NPW(F/P, i, n) = NPW(1 + i)^n\]
• Variables:
  • NFW = net future worth
  • CF_t = cash flow at time t
  • i = interest rate per period
  • n = analysis period (total number of periods)
• Decision Rule:
  • If NFW > 0: Project is economically acceptable
  • If NFW = 0: Project breaks even
  • If NFW < 0:="" project="" is="" not="" economically="">
• Note: NFW and NPW yield identical accept/reject decisions

Future Worth of Single Cash Flow

• Formula: \[F_n = CF_t(1 + i)^{n-t}\]
• Purpose: To find the future worth at time n of a cash flow occurring at time t

Special Cases and Applications

Deferred Annuity (Delayed Uniform Series)

• Present Worth Formula: \[P = A(P/A, i, n)(P/F, i, d)\]
• Variables:
  • d = number of periods of deferral (delay before first payment)
  • n = number of payments in the uniform series
  • A = uniform payment amount
• Note: First payment occurs at end of period (d + 1)

Shifted Uniform Series

• Present Worth at Time t: \[P_t = A(P/A, i, n)\]
• Present Worth at Time 0: \[P_0 = A(P/A, i, n)(P/F, i, t)\]
• Where:
  • P_t = present worth one period before first payment
  • t = time period where P_t is located

Multiple Payment Series

• Present Worth: \[P = \sum_{j=1}^{m} A_j(P/A, i, n_j)(P/F, i, t_j)\]
• Where:
  • m = number of different uniform series
  • A_j = uniform amount for series j
  • n_j = number of periods in series j
  • t_j = starting period for series j

Non-Standard Compounding Periods

When Payment Period ≠ Compounding Period

• Method 1: Convert to effective interest rate per payment period
• Method 2: Determine equivalent payment per compounding period
• Effective rate per payment period: \[i_p = \left(1 + \frac{r}{m}\right)^{m/c} - 1\]
• Where:
  • i_p = effective interest rate per payment period
  • m = number of compounding periods per year
  • c = number of payment periods per year
  • r = nominal annual interest rate

Key Relationships and Identities

Factor Relationships

• 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:
  • \[(P/A, i, n) = (P/F, i, n) \times (F/A, i, n)\]
  • \[(A/P, i, n) = (A/F, i, n) + i\]
  • \[(A/F, i, n) = (A/P, i, n) - i\]
  • \[(F/A, i, n) = (F/P, i, n) \times (P/A, i, n)\]

• Gradient Relationships:
  • \[(A/G, i, n) = (P/G, i, n) \times (A/P, i, n)\]
  • \[(P/G, i, n) = (A/G, i, n) \times (P/A, i, n)\]

Special Case: n = 1

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

Limiting Cases

• As n → ∞:
  • \[\lim_{n \to \infty} (P/A, i, n) = \frac{1}{i}\]
  • \[\lim_{n \to \infty} (A/P, i, n) = i\]
  • \[\lim_{n \to \infty} (P/F, i, n) = 0\]
  • \[\lim_{n \to \infty} (A/F, i, n) = 0\]

• As i → 0:
  • \[\lim_{i \to 0} (F/P, i, n) = 1\]
  • \[\lim_{i \to 0} (P/A, i, n) = n\]
  • \[\lim_{i \to 0} (F/A, i, n) = n\]

Cash Flow Diagrams and Sign Conventions

Sign Conventions

• Positive (+): Cash inflows, revenues, receipts, salvage value
• Negative (-): Cash outflows, costs, expenses, initial investment
• Perspective: All cash flows from viewpoint of the decision maker or investor

Time Period Conventions

• Time 0: Present time; beginning of period 1
• End-of-Period Convention: Cash flows occur at the end of the period unless stated otherwise
• Uniform Series: First payment at end of period 1, last payment at end of period n
• Gradient Series: Base amount in period 1, gradient starts in period 2

Common Applications in Engineering Economics

Loan Repayment Analysis

• Loan Payment: \[A = P(A/P, i, n)\]
• Remaining Balance after k payments: \[B_k = A(P/A, i, n-k)\]
• Principal Payment in period t: \[PP_t = A - I_t = A - B_{t-1} \times i\]
• Interest Payment in period t: \[I_t = B_{t-1} \times i\]
• Where:
  • B_k = remaining balance after k payments
  • PP_t = principal portion of payment in period t
  • I_t = interest portion of payment in period t

Bond Valuation

• Present Worth of Bond: \[P = I(P/A, i, n) + F(P/F, i, n)\]
• Variables:
  • P = present worth (purchase price) of bond
  • I = periodic interest payment (coupon payment)
  • F = face value (par value) of bond
  • i = yield rate (market interest rate)
  • n = number of periods until maturity
• Coupon Payment: \[I = F \times i_c\]
• Where i_c = coupon rate (stated rate on bond)

Depreciation Recovery

• Present Worth with Salvage Value: \[P = P_0 - S(P/F, i, n)\]
• Where:
  • P₀ = initial cost
  • S = salvage value at end of life
  • n = useful life

Analysis Period Considerations

Equal Life Alternatives

• Method: Compare NPW or NFW directly over the common life
• Condition: Both alternatives have same useful life n
• Decision: Select alternative with highest NPW (or NFW)

Unequal Life Alternatives

Least Common Multiple (LCM) Method

• Analysis Period: n = LCM of individual lives
• Assumption: Each alternative is repeated until reaching LCM
• Formula: \[NPW_{\text{total}} = NPW_1 + NPW_1(P/F, i, n_1) + NPW_1(P/F, i, 2n_1) + \ldots\]
• Where n₁ = life of first cycle

Study Period Method

• Approach: Define a common study period
• Treatment: Estimate salvage/market value at end of study period
• Application: Used when repeatability assumption is not valid

Infinite Analysis Period

• Present Worth: \[P = P_0 + \frac{P_1}{(1+i)^{n_1} - 1}\]
• Where:
  • P₀ = initial cost
  • P₁ = cost of first replacement
  • n₁ = life of asset
• Use: Long-lived infrastructure projects

Important Assumptions and Limitations

Standard Assumptions

• End-of-Period Cash Flows: Unless specified, all cash flows occur at the end of the period
• Constant Interest Rate: Interest rate i remains constant throughout the analysis period
• Deterministic Cash Flows: All cash flows are known with certainty
• Perfect Capital Markets: Unlimited borrowing and lending at rate i
• Reinvestment at i: All intermediate receipts are reinvested at the same rate i

Critical Conditions for Formula Application

• Uniform Series: All payments must be equal and occur at regular intervals
• Arithmetic Gradient: Change between consecutive payments must be constant
• Geometric Gradient: Percentage change between consecutive payments must be constant
• Present Worth Location: For uniform/gradient series, P is located one period before the first payment
• Future Worth Location: F is located at the same time as the last payment