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.

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.

00 1 2 3 4 5 6 7 8 109

-100

250

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)

x y

A

B

i

0

NPW (or NAW) as a function of i

• For an investment

NPW

($100.00)

($50.00)

$0.00

$50.00

$100.00

$150.00

0

0.0

2

0.0

4

0.0

6

0.0

8

0.1

0.1

2

0.1

4

0.1

6

0.1

8

0.2

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.

0

150

0 1 2

-200

175

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%

150 1502 4(175)( 200)2(175)

150 403.113350

Example 3

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

0

16

0 1 2 3 4 5 6 7 8 109

-100

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

0

-16

0 1 2 3 4 5 6 7 8 109

250

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.

01 2 3 4 5 6 7 8 109

100

-250

• 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.

0

600

0 1 2 3 4 5 6 7 8 109

-2000

400

-100 -150 -200

-550

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

00 1 2 3 4 5

-100

405

-500

-100

100

200

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)

Example 7: Graphically

($3.00)

($2.00)

($1.00)

$0.00

$1.00

$2.00

$3.00

$4.00

$5.00

$6.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20

NPW

Simple Case 1

• Total revenue = total cost • ROR = 0

100

-100

Simple Case 2

• Uniform inflow with Capital entirely recovered• ROR = Inflow/Investment = A/P

15050 50 50

-100

Simple Case 3

• Uniform inflow lasting forever• ROR = inflow/Investment = A/P

-100

50 50 50 50 50

Simple Case 4

• One factor involved• Solve for factor value and use the tables

-100

50 50 50 50 50

Making Decisions with ROR

• When Investing– Accept the project if ROR ≥ MARR

• When Borrowing– Accept the project if Rate of Borrowing ≤ MARB