Cheatsheet: Rate of Return

1. Rate of Return Fundamentals

1.1 Core Definitions

1.2 Basic ROR Equation

2. ROR Calculation Methods

2.1 Trial and Error Method

• Select two trial interest rates
• Calculate NPV at each rate
• Interpolate linearly between rates where NPV changes sign
• Formula: i* = i_L + [(NPV_L)/(NPV_L - NPV_H)] × (i_H - i_L)
• Where i_L gives positive NPV and i_H gives negative NPV

2.2 Direct Solution Methods

2.3 Spreadsheet Functions

3. ROR Analysis for Single Projects

3.1 Decision Criteria

• If ROR ≥ MARR: Accept project
• If ROR < marr:="" reject="">
• ROR must be calculated before comparison with MARR

3.2 Multiple ROR Values

3.3 Simple vs Non-Simple Investments

4. Incremental ROR Analysis

4.1 Two-Alternative Comparison

4.2 Incremental Analysis Formula

• 0 = PW_B-A = Σ[(CF_B,t - CF_A,t)/(1+Δi*)^t]
• Solve for Δi* and compare to MARR
• Must evaluate incremental investment, not individual RORs

4.3 Multiple Alternative Analysis

1. Eliminate alternatives with ROR < marr="" (if="" individual="" rors="">
2. Order remaining alternatives by increasing initial cost
3. Compare defender (lower cost) vs challenger (next higher cost) incrementally
4. Winner becomes new defender; repeat until all alternatives evaluated
5. Do-nothing is default defender if included

4.4 Common Errors to Avoid

• Do not select alternative with highest individual ROR
• Do not compare individual RORs directly to each other
• Must perform incremental analysis between alternatives
• Incremental investment must be justified by ΔROR ≥ MARR

5. Special ROR Cases

5.1 Bond ROR (Yield to Maturity)

• Bond premium: Purchase price > face value; yield < coupon="">
• Bond discount: Purchase price < face="" value;="" yield=""> coupon rate
• Bond at par: Purchase price = face value; yield = coupon rate

5.2 External Rate of Return (ERR)

• External rate e = MARR or other specified reinvestment rate
• Solves multiple ROR problem
• More realistic assumption about reinvestment

5.3 Modified Internal Rate of Return (MIRR)

• MIRR = [(FV_positive/PV_negative)^1/n] - 1
• FV_positive = future value of positive cash flows at reinvestment rate
• PV_negative = present value of negative cash flows at finance rate
• Eliminates multiple IRR problem
• Finance rate ≠ reinvestment rate in general

6. ROR vs Other Methods

6.1 Comparison with NPV Method

6.2 Relationship Between Methods

• When ROR = MARR, NPV = 0
• When ROR > MARR, NPV > 0
• When ROR < marr,="" npv=""><>
• For simple investments, both methods yield same accept/reject decision
• For mutually exclusive alternatives, incremental ROR = rate where NPVs are equal

7. Practical Considerations

7.1 ROR Calculation Tips

• Start trial rates far apart (e.g., 5% and 50%) to bracket the true ROR
• Narrow range once sign change is detected
• Check for multiple RORs by plotting NPV vs interest rate
• Use spreadsheet tools for complex cash flows
• Verify solution by substituting back into NPV equation

7.2 Setting MARR

7.3 Common Applications

• Capital budgeting decisions
• Equipment replacement analysis
• Lease vs buy decisions
• Bond investment analysis
• Project portfolio selection
• Make vs buy decisions

8. Key Formulas Summary

9. Decision Framework

9.1 Single Project Evaluation

1. Identify all cash flows and timing
2. Check for simple vs non-simple investment pattern
3. Calculate ROR using appropriate method
4. Compare ROR to MARR
5. Accept if ROR ≥ MARR; reject if ROR <>

9.2 Mutually Exclusive Alternatives

1. Ensure all alternatives have same service life (adjust if necessary)
2. Order alternatives by increasing initial investment
3. Calculate incremental cash flows between consecutive alternatives
4. Determine ΔROR for each increment
5. Select alternative where all incremental investments are justified (ΔROR ≥ MARR)

9.3 Quick Decision Rules

• Independent projects: Evaluate each against MARR separately
• Mutually exclusive projects: Must use incremental analysis
• Budget constraints: Use incremental analysis and select combination maximizing total NPV
• When in doubt: NPV method is simpler and more reliable than ROR