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 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

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

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

Example 1: Exact Computation Set NPW = 0 NPW = (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) = Therefore, the ROR = %

Example 1: Trial and Error Set NPW = 0 NPW = (P/F, i, 10) = 0 Try 9% : (P/F, 0.09, 10) = Try 10%: (P/F, 0.10, 10) = Linearly Interpolating: ROR = [(5.603)/( (-3.614))]( ) = or 9.608%

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

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

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.

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

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

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

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 = Try 10%: NAW = - 100(A/P, 0.10, 10) + 16 = Linear Interpolating: ROR  9.604%

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

Example 4: Trial and Error Set FW = 0 FW = -16 (F/A, i, 10) = 0 Try 8% : -16 ( ) = Try 10%: -16 ( ) = Interpolating: ROR = [ /( )]( ) = or 9.569% ROR is approximately 9.569%

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 = (P/F, i, 10) = 0

Example 6: Complex Example  A machine costs  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.

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

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

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 = (P/F,i,1) - 500(P/F,i,2) + 200(P/F,i,3) - 100(P/F,i,4) (P/F,i,5)

Example 7: Graphically

Simple Case 1 Total revenue = total cost ROR = 0

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

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

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

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