Lecture 4: Repeated Cash Flows

Introduction

Most engineering projects involve repeated costs and benefits. This lecture develops formulas and techniques for handling repeated cash flow patterns: uniform series and gradient series.

I. Uniform Series Compound Interest Formulas

A uniform payment series consists of equal cash flows (\(A\)) at the end of each period for \(n\) periods.

A. Defining the Uniform Series (\(A\))

  • \(A\) is an end-of-period cash receipt/disbursement in a uniform series, equivalent to a present sum \(P\) or future sum \(F\).
  • Ordinary Annuity: All \(A\) occur at end of period; number of \(A\) matches number of periods \(n\).

B. Uniform Series Factors

  1. F/A (Uniform Series Compound Amount): Find \(F\) given \(A\) \[F = A \left[ \frac{(1 + i)^n - 1}{i} \right]\]
  2. A/F (Sinking Fund): Find \(A\) given \(F\) \[A = F \left[ \frac{i}{(1 + i)^n - 1} \right]\]
  3. P/A (Present Worth): Find \(P\) given \(A\) \[P = A \left[ \frac{(1 + i)^n - 1}{i(1 + i)^n} \right]\]
  4. A/P (Capital Recovery): Find \(A\) given \(P\) \[A = P \left[ \frac{i(1 + i)^n}{(1 + i)^n - 1} \right]\]

II. Arithmetic Gradient Series

An arithmetic gradient series (\(G\)) changes by a constant amount each period. Useful for modeling predictable linear cost changes.

A. Resolving the Cash Flow

  • Separate into:
    1. Uniform Series (\(A\) portion): Base cash flow in Period 1
    2. Gradient Series (\(G\) portion): Increase/decrease \(G\) starting in Period 2
  • Standard: First term in gradient series (end of Period 1) is zero: \(0, G, 2G, ...\)

B. Arithmetic Gradient Factors

  1. P/G (Present Worth): \[P = G \left[ \frac{(1 + i)^n - in - 1}{i^2(1 + i)^n} \right]\]
  2. A/G (Uniform Series): \[A = G \left[ \frac{1}{i} - \frac{n}{(1 + i)^n - 1} \right]\]
  • Total present worth: \(P_{Total} = A(P/A, i, n) + G(P/G, i, n)\)

III. Geometric Gradient Series

A geometric gradient series (\(g\)) changes by a uniform rate/percentage each period. Used for inflation, growth, etc.

  • \(A_t = A_1 (1 + g)^{t-1}\)

A. Geometric Gradient Present Worth Factor

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

IV. Advanced Equivalence Concepts

A. Moment Diagram Analogy

  • Cash flows as forces; time periods as distances
  • Sum of moments (discounted cash flows) must be zero at the pivot point
  • Moment arm for \(F\) at \(T\) periods: \((1+i)^{-T}\)

B. Relationships Between Compound Interest Factors

  • Single Payment: \((F/P, i, n) = 1/(P/F, i, n)\)
  • Uniform Series: \((A/P, i, n) = 1/(P/A, i, n)\); \((F/A, i, n) = 1/(A/F, i, n)\)
  • Capital Recovery: \((A/P, i, n) = (A/F, i, n) + i\)

C. Compounding/Payment Period Differences

  • Adjust for mismatched payment/compounding frequency using effective interest rate
  • Continuous Compounding:
    • \(A = P \left[ \frac{e^{rn}(e^r - 1)}{e^{rn} - 1} \right]\)
    • \(P = A \left[ \frac{e^{rn} - 1}{e^{rn}(e^r - 1)} \right]\)

V. Spreadsheets for Economic Analysis

Spreadsheets eliminate manual interpolation and simplify modeling.

A. Spreadsheet Annuity Functions

To Find Function Role
\(P\) =PV(rate, nper, pmt, [fv], [type]) Present Value
\(A\) =PMT(rate, nper, pv, [fv], [type]) Payment
\(F\) =FV(rate, nper, pmt, [pv], [type]) Future Value
\(n\) =NPER(rate, pmt, pv, [fv], [type]) Number of Periods
\(i\) =RATE(nper, pmt, pv, [fv], [type]) Interest Rate
  • type: 0 for end-of-period, 1 for beginning-of-period

B. Spreadsheet Block Functions

  • NPV(rate, values): Present worth of cash flows from Period 1 to \(n\) (add Time 0 manually)
  • IRR(values, guess): Interest rate for net present worth zero (include Time 0)

Example Usage

  • NPV: =NPV(0.05, 1000, 2000, 3000, 4000, 5000)
  • IRR: =IRR(A1:A5) (A1:A5 includes initial investment)