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Rate of Return

Rate of Return. Definition. The Rate of Return (ROR) is: A percentage (or interest rate) that describes the merit of an investment. (Return on investment during a year)/(Amount Invested) The interest rate than makes the cash flows of income equivalent to the cash flows of cost. Usage.

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Rate of Return

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  1. Rate of Return

  2. Definition The Rate of Return (ROR) is: • A percentage (or interest rate) that describes the merit of an investment. • (Return on investment during a year)/(Amount Invested) • The interest rate than makes the cash flows of income equivalent to the cash flows of cost

  3. Usage • We use the ROR to evaluate investments because • percentage rates are familiar • percentage rates are dimensionless • they are commonly used as business measures • Synonyms • ROR: Rate of Return • ROI: Return on Investment • IRR: Internal Rate of Return

  4. Single Project • The ROR is the interest rate that makes • NPW = PW Benefits - PW Costs = 0, or • NAW = AW Benefits - AW Costs = 0

  5. Example 1: • Find the ROR of an investment of 100 at time 0 and a return of 250 at time 10. • NPW = -100 + 250 (P/F, i, 10) = 0

  6. Example 1: Exact Computation • Set NPW = 0 • NPW = -100 + 250 (P/F, i, 10) = 0 • => (P/F, i, 10) = 100/250 = 0.4 • => 1/(1+i)10 = 0.4 • => (1+i)10 = 1/0.4 = 2.5 • => (1+i)10(0.1) = (2.5)0.1 • => i = (2.5)0.1 - 1 = 0.09595 • Therefore, the ROR = 9.595 %

  7. Example 1: Trial and Error • Set NPW = 0 • NPW = -100 + 250 (P/F, i, 10) = 0 • Try 9% : -100 + 250 (P/F, 0.09, 10) = 5.6063 • Try 10%: -100 + 250 (P/F, 0.10, 10) = -3.614 • Linearly Interpolating: ROR = 0.09 + [(5.603)/(5.603- (-3.614))](0.1-0.09) = 0.09608 or 9.608%

  8. Linear Interpolation • Shape ratio of pale rectangle: (A-B) / (y-x) • Shape ratio of smaller rectangle: (A-0) / (i-x) • Since shapes are the same: • (A-B) / (y-x)) = (A) / (i-x) • => i-x = [ A / (A-B) ] (y-x) • => i = x + [ A / (A-B) ] (y-x) A 0 B x y i

  9. NPW (or NAW) as a function of i • For an investment

  10. Example 2 • Find the ROR of an investment of $200 at time 0 and returns of $150 at time 1 and $175 at time 2.

  11. Example 2: Exact Computation • Set NPW = 0 • NPW = -200 + 150/(1+i) + 175/(1+i)2 = 0 • Let x = 1/(1+i) and the expression becomes 175x2 + 150x -200 = 0 • So x = 1/(1+i) = 0.72318 => i = 0.3828 or 38.28%

  12. Example 3 • Find the ROR of an investment of $100 with a revenue of $16 a year for 10 years.

  13. Example 3: • NAW = - 100(A/P, i, 10) + 16 = 0 • (A/P, i, 10) = 0.16 • or [ i (1 + i)10]/[(1 + i)10 - 1)] = 0.16 • Difficult to solve for i using because of the nonlinear factor

  14. Example 3: Trial and Error • Use trial and error • NAW = - 100(A/P, i, 10) + 16 • Try 9%: NAW = - 100(A/P, 0.09, 10) + 16 = 0.418 • Try 10%: NAW = - 100(A/P, 0.10, 10) + 16 = -0.275 • Linear Interpolating: ROR » 9.604%

  15. Example 4 • Find the ROR an investment of $16 a year for 10 years with a return of $250 at year 10

  16. Example 4: Trial and Error • Set FW = 0 • FW = -16 (F/A, i, 10) + 250 = 0 • Try 8% : -16 (14.4866) + 250 = 18.2144 • Try 10%: -16 (15.9374) + 250 = -4.9984 • Interpolating: ROR = 0.08 + [18.2144 /(18.2144+4.9984)](0.1-0.08) = 0.09569 or 9.569% • ROR is approximately 9.569%

  17. Example 5 • Find the Rate of Borrowing associated with borrowing 100 and paying back 250 after 10 years. • ROR here is approximately 9.6% • ROR of return is actually the cost borrowing. • NPW = 100 - 250 (P/F, i, 10) = 0

  18. Example 6: Complex Example • A machine costs 2000. • We expect a return of $600 per year for ten years. • The machine is then sold with a salvage of $400. • Operating cost is 100 in the first year and increases by $50 per year thereafter.

  19. Example 6: Trial and Error • NPW = -2000 + 500(P/A, i, 10) + 400(P/F, i, 10)- 50(P/G, i, 10) • Try i = 0.05, NPW = 523.83 • Try i = 0.1, NPW = 81.93 • Try i = 0.12, NPW = -58.8 • Use linear interpolation to compute a value between 10% and 12%

  20. Example 7: Non-simple Investment • A0 = -100, A1 = 405, A2 = -500, A3 = 200, • A4 = -100, A5 = 100

  21. Example 7: • This is an example of a non-simple investment since • the initial cash flow is negative, but • more than one sign change occurs in the net cash flow series. • NPW = -100 + 405(P/F,i,1) • - 500(P/F,i,2) • + 200(P/F,i,3) • - 100(P/F,i,4) • + 100 (P/F,i,5)

  22. Example 7: Graphically

  23. Simple Case 1 • Total revenue = total cost • ROR = 0

  24. Simple Case 2 • Uniform inflow with Capital entirely recovered • ROR = Inflow/Investment = A/P

  25. Simple Case 3 • Uniform inflow lasting forever • ROR = inflow/Investment = A/P

  26. Simple Case 4 • One factor involved • Solve for factor value and use the tables

  27. Making Decisions with ROR • When Investing • Accept the project if ROR ≥ MARR • When Borrowing • Accept the project if Rate of Borrowing ≤ MARB

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