Rate of Return

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# Rate of Return - PowerPoint PPT Presentation

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

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 = -100 + 250 (P/F, i, 10) = 0
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
• 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
• 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
• 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 = -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
• 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)10 - 1)] = 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 = 0.418

• Try 10%:

NAW = - 100(A/P, 0.10, 10) + 16 = -0.275

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