Cheatsheet: Time Value of Money

1. Fundamental Concepts

1.1 Core Definitions

1.2 Cash Flow Conventions

• End-of-Period Convention: Cash flows occur at the end of each period (standard assumption)
• Beginning-of-Period: Cash flows occur at the start of each period (annuity due)
• Inflows: Positive cash flows (receipts, revenues, salvage values)
• Outflows: Negative cash flows (costs, expenses, investments)
• Net Cash Flow: Inflows minus outflows for a given period

2. Single Payment Formulas

2.1 Compound Amount Factor (F/P)

2.2 Present Worth Factor (P/F)

2.3 Relationship

• (F/P, i, n) and (P/F, i, n) are reciprocals: (F/P) × (P/F) = 1
• Both factors depend on interest rate i and number of periods n

3. Uniform Series Formulas

3.1 Compound Amount Factor (F/A)

3.2 Sinking Fund Factor (A/F)

3.3 Present Worth Factor (P/A)

3.4 Capital Recovery Factor (A/P)

3.5 Uniform Series Relationships

• (F/A, i, n) and (A/F, i, n) are reciprocals
• (P/A, i, n) and (A/P, i, n) are reciprocals
• (P/A, i, n) = (P/F, i, n) × (F/A, i, n)
• (A/P, i, n) = (A/F, i, n) + i

4. Gradient Formulas

4.1 Arithmetic Gradient

4.1.1 Gradient Present Worth Factor (P/G)

4.1.2 Gradient Uniform Series Factor (A/G)

4.1.3 Combined Uniform Series and Gradient

• Cash flow at period t: CF_t = A₁ + (t - 1)G, where A₁ is the first payment
• Present value: P = A₁(P/A, i, n) + G(P/G, i, n)
• Equivalent uniform: A = A₁ + G(A/G, i, n)

4.2 Geometric Gradient

4.2.1 When g ≠ i

4.2.2 When g = i

5. Interest Rate Concepts

5.1 Nominal vs. Effective Interest Rates

5.2 Continuous Compounding

5.3 Common Compounding Frequencies

• Annual: m = 1
• Semiannual: m = 2
• Quarterly: m = 4
• Monthly: m = 12
• Daily: m = 365
• Continuous: m → ∞

6. Economic Equivalence

6.1 Equivalence Principle

• Two cash flows are equivalent if they have the same value at a common point in time using the same interest rate
• Equivalence depends on both interest rate and timing
• Different cash flow patterns can have identical present or future values

6.2 Converting Between Time Points

• Use compound interest factors to move values forward or backward in time
• Moving forward: multiply by (F/P) or (F/A) factors
• Moving backward: multiply by (P/F) or (P/A) factors
• Can convert to any common time point (present, future, or intermediate)

7. Special Cases and Applications

7.1 Deferred Annuities

• Annuity with first payment occurring after more than one period
• Method 1: Find P at one period before first payment, then discount back to time 0
• Method 2: P = A(P/A, i, n) × (P/F, i, d), where d is deferral periods

7.2 Perpetuity

7.3 Capitalized Cost

• Present value of an infinite series of cash flows
• Used for permanent installations or indefinite service life
• P_capitalized = Initial cost + (Recurring cost)/i

7.4 Bond Valuation

• Bond value = Present value of interest payments + Present value of face value
• P = (Face value × Interest rate)(P/A, i, n) + Face value(P/F, i, n)
• Interest rate is coupon rate; i is market yield rate

8. Problem-Solving Strategies

8.1 Standard Approach

1. Draw a cash flow diagram with time on horizontal axis, cash flows as arrows
2. Identify given values: P, F, A, G, i, n
3. Identify unknown variable to solve for
4. Select appropriate formula or factor combination
5. Substitute values and calculate
6. Check units and reasonableness of answer

8.2 Cash Flow Diagram Conventions

• Time periods marked on horizontal axis (0, 1, 2, ..., n)
• Upward arrows represent inflows (positive)
• Downward arrows represent outflows (negative)
• Arrow length proportional to magnitude (optional)
• Label all known values and the unknown

8.3 Common Pitfalls

• Mismatching payment periods with compounding periods
• Forgetting that (P/A) assumes first payment at end of period 1
• Incorrect handling of gradient starting point (gradient starts at 0)
• Using nominal rate when effective rate is required
• Sign errors in cash flows (mixing inflows and outflows)

9. Factor Relationships Summary

9.1 Reciprocal Pairs

9.2 Useful Combinations

• (P/A, i, n) = (P/F, i, n) × (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) × (P/A, i, n)

9.3 Limit Cases

• When n = 1: (F/P, i, 1) = (1 + i); (P/A, i, 1) = (P/F, i, 1)
• When n → ∞: (P/A, i, ∞) = 1/i; (A/P, i, ∞) = i; (A/F, i, ∞) = 0
• When i = 0: (F/P, 0, n) = 1; (P/A, 0, n) = n; (F/A, 0, n) = n

10. Quick Reference Tables

10.1 All TVM Factors

10.2 Variable Notation