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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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definition
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
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
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
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
example 1 exact computation
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 %
example 1 trial and error
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%

linear interpolation
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

example 2
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
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%
example 3
Example 3
  • Find the ROR of an investment of $100 with a revenue of $16 a year for 10 years.
example 313
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
example 3 trial and error
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%
example 4
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
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%
example 5
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
example 6 complex example
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.
example 6 trial and error
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%
example 7 non simple investment
Example 7: Non-simple Investment
  • A0 = -100, A1 = 405, A2 = -500, A3 = 200,
  • A4 = -100, A5 = 100
example 7
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)
simple case 1
Simple Case 1
  • Total revenue = total cost
  • ROR = 0
simple case 2
Simple Case 2
  • Uniform inflow with Capital entirely recovered
  • ROR = Inflow/Investment = A/P
simple case 3
Simple Case 3
  • Uniform inflow lasting forever
  • ROR = inflow/Investment = A/P
simple case 4
Simple Case 4
  • One factor involved
  • Solve for factor value and use the tables
making decisions with ror
Making Decisions with ROR
  • When Investing
    • Accept the project if ROR ≥ MARR
  • When Borrowing
    • Accept the project if Rate of Borrowing ≤ MARB