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Rate of Return

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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

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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

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

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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

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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 %

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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%

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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) xy A B i 0

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

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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.

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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 175x 2 + 150x -200 = 0 So x = 1/(1+i) = 0.72318 => i = 0.3828 or 38.28%

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

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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

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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%

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

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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%

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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

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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.

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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%

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Example 7: Non-simple Investment A 0 = -100, A 1 = 405, A 2 = -500, A 3 = 200, A 4 = -100, A 5 = 100

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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)

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Example 7: Graphically

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Simple Case 1 Total revenue = total cost ROR = 0

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

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

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

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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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