Lecture 7: Rate of Return Analysis

Introduction

Rate of Return (ROR) analysis is the third major technique, following Present Worth (PW) and Annual Cash Flow (A) analysis. ROR offers a single figure of merit (IRR) that is widely understood by engineers and business leaders. Its key advantage: no need to select an interest rate before calculation—the project’s IRR is compared to the Minimum Attractive Rate of Return (MARR) to determine acceptability.

I. Internal Rate of Return (IRR)

A. Definitions

Fundamental Concept: IRR is the interest rate (\(i\)) at which total benefits equal total costs.
Present Worth Basis: IRR is the rate where Net Present Worth (NPW) = 0.
Annual Worth Basis: IRR is the rate where Equivalent Uniform Annual Worth (EUAW) = 0.
Unrecovered Investment Basis: IRR is the rate earned on unrecovered investment, making it zero at the end of the project.

B. Cash Flow Equations

PW of benefits - PW of costs = 0
PW of benefits / PW of costs = 1
NPW = 0
EUAB - EUAC = 0
PW of costs = PW of benefits

C. Calculating IRR

Trial & Error: Guess \(i\), compute NPW, adjust until NPW = 0, interpolate if needed.
Graphical: Plot NPW vs. \(i\); IRR is where NPW crosses zero.
Spreadsheet Functions:
- RATE: For standard annuities
- IRR: For irregular cash flows
- IRR block: For a range of values

D. Decision Criterion

If IRR \(\ge\) MARR: Accept the project
If IRR \(<\) MARR: Reject the project

II. Incremental Analysis for Mutually Exclusive Alternatives

When comparing mutually exclusive alternatives, simply choosing the highest IRR is incorrect. The additional investment must be justified by its own rate of return.

A. Steps for Incremental Analysis

Calculate incremental cash flow (\(\Delta\)): Higher-cost minus lower-cost alternative
Calculate incremental IRR (\(\Delta\)IRR): Find \(i\) where NPW of \(\Delta\) = 0
Decision Rule:
- If \(\Delta\)IRR \(\ge\) MARR: Increment justified; choose higher-cost alternative
- If \(\Delta\)IRR \(<\) MARR: Increment not justified; choose lower-cost alternative
For 3+ alternatives: Use sequential challenger–defender comparisons

B. Solved Example

Example 1: IRR for investment annuity
Example 2: Incremental analysis for two alternatives

III. Difficulties and Advanced Techniques

A. Multiple Rates of Return

Multiple sign changes in cash flow can yield multiple IRRs; standard IRR may be ambiguous.

B. Modified Internal Rate of Return (MIRR)

Discount outflows to present (financing rate)
Compound inflows to future (investing rate)
MIRR equates present worth of expenses to future worth of receipts, ensuring a unique solution

IV. Spreadsheets in Rate of Return Analysis

Function	Purpose	Usage Context
RATE	Solves for \(i\) for simple annuities	RATE(nper, pmt, pv, fv)
IRR	Solves for \(i\) for complex cash flows	IRR(values, [guess])
GOAL SEEK	Solves for \(\Delta\)IRR for unequal lives	Varies \(i\) until EUAW\(_A\) = EUAW\(_B\)
XIRR	Solves for \(i\) for non-uniform dates	XIRR(values, dates, [guess])

V. Choosing the Best Alternative

A. Graphical Approach

Plot NPW or EUAW vs. \(i\) for each alternative
Crossover points: Where alternatives are equivalent
For any MARR, choose the alternative with highest NPW (or lowest EUAC for cost minimization)
Choice table: Defines optimal alternative for each interest rate range

B. Incremental ROR Analysis (Defender/Challenger)

Order alternatives by increasing cost
Initial defender: First acceptable alternative (IRR \(\ge\) MARR)
Challenger: Next higher-cost alternative
Calculate \(\Delta\) cash flow and \(\Delta\)IRR
If \(\Delta\)IRR \(\ge\) MARR: Challenger replaces defender
Repeat until all compared; final defender is best

C. Solved Example: Incremental ROR

Example with three packaging machines, step-by-step incremental comparisons

D. Comparison of Methods

ROR (IRR) is popular for its intuitive percentage measure
PW/EUAW often require less computation
Corporate policy may dictate preferred method
All methods yield the same decision if used correctly