Rate of Return

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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
• gt (P/F, i, 10) 100/250 0.4
• gt 1/(1i)10 0.4
• gt (1i)10 1/0.4 2.5
• gt (1i)10(0.1) (2.5)0.1
• gt 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)
• gt i-x A / (A-B) (y-x)
• gt i x A / (A-B) (y-x)

A
0
B
x
y
i
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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/(1i) 175/(1i)2 0
• Let x 1/(1i) and the expression becomes
• 175x2 150x -200 0
• So x 1/(1i) 0.72318 gt 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 /(18.21444.9984)(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

• A0 -100, A1 405, A2 -500, A3 200,
• A4 -100, A5 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