Rate of Return Analysis: Single Alternative

1
Chapter 7

• Rate of Return Analysis Single Alternative

2
Recall

• ROR is the interest rate at which the benefits
  are equivalent to the costs.
• PW (Benefits) PW (Costs)
• NPW (i) 0
• AW (i)0

3
How to find ROR

• A- One factor
• -100 (P/A, i, 5) 0
• Look for the value in the table. If it is in the
  table, that is
• your i.
• If the value is not in the table, it is between
  two other
• values (one less, one more). Use interpolation to
  find i.
• B- More than one factor
• -100 (P/A, i, n) (P/F, i, n)0
• Find the value than make NPW( i)0.Use trial
• and error.

4
Problems with ROR

• 1- Multiple Values In some cash flows, when
  solving for i NPW (i) 0, there is more than
  one answer for i.
• 2- Computational Difficulty Trial-and Error

5
Multiple Rate of Return

• Conventional (simple cash flow)
• There is only one sign change in conventional
  (simple) cash flow.
• Example
• Year Cash Flow
• 0 -10,000
• 1 3,000
• 2 3,000
• 3 3,000
• 4 3,000
• There is always one answer for i in a
  conventional cash
• flow.

6
Non-conventional Cash Flow

• There is more than one sign change in the cash
  flow.
• Example
• Year Cash Flow
• 0 -10,000
• 1 3,000
• 2 3,000
• 3 3,000
• 4 3,000
• 5 -1,000
• 6 6,000
• There might be more than one answer for i in a
  non-
• conventional cash flow.

7
Check for Multiple Answers

• To see whether a non-conventional cash flow has
• more than one answer or not we perform the
• following tests
• 1- Descartes Rule of Signs
• The total number of real-number roots is always
• less than or equal to the number of sign changes
• in the cash flow.
• Year Cash Flow Year Cash Flow
• 0 -10,000 3 5000
• 1 20,000 4 -2000
• 2 30,000 5 3000

8
Cumulative Cash Flow Sign Test

• Norstroms criterion If only one sign change
• occurs in the cumulative cash flow, then there is
• one answer for the i. The cash flow should start
  
• negatively.
• Year Cash Flow Year Cumulative Cash Flow
• 0 -2,000 0 -2,000
• 1 6,000 1 4,000
• 2 -2,000 2 2,000
• 3 1,000 3 3,000
• 4 -2,000 4 1,000
• 5 3,000 5 4,000
• of sign change5 of sign change1

9
Example

• Arc-bot Technologies, manufacturer of six-axis,
• electric servo-driven robots, has experienced the
• cash flows below in a shipping department. (a)
• Determine the number of possible rate of return
• values. (b) Find all i values between 0 and
• 100.
• Year Expenses, Savings,
• 0 - 33,000 0
• 1 - 22,000 19,000
• 2 - 30,000 97,000
• 3 - 40,000 25,000
• 4 - 20,000 29,000
  

10
Solution

• Year Expenses, Savings, Net Cash Flow,
• 0 - 33,000 0
  -33,000
• 1 - 22,000 19,000 -3,000
• 2 - 30,000 97,000 67,000
• 3 - 40,000 25,000 -15,000
  
• 4 - 20,000 29,000 9,000
• There is three sign change in the net cash flow.
• Therefore, there is three possible answers for i.

11
Solution

• Net Cash Flow, Cumulative Cash Flow,
• 0 -33,000 -33,000
• 1 -3,000 -36,000
• 2 67,000 31,000
• 3 -15,000 16,000
• 4 9,000 25,000
• There is one sign change in cumulative cash
• flow. Therefore, there is only one answer for i.

12
Solution

• NPW (i) 0
• NPW (i) -33000 -3,000 (P/F, i,1) 67,000
  (P/F, i,2)
• - 15,000 (P/F, i,3) 9,000 (P/F,
  i,4) 0
• Solve with trial and error
• i20
• NPW - 33,000 -3,000 (P/F, 20,1) 67,000(P/F,
  20, 2) -
• - 15,000 (P/F, 20,3) 9,000 (P/F,
  20,4)
• NPW -33,000 -3,000 (0.8333) 67,000 (0.6944) -
  
• 15000 (0.5787) 9000 (.4823)
• NPW 6685.1

13
Solution

• i30
• NPW - 33,000 -3,000 (P/F, 30,1) 67,000 (P/F,
  30, 2)
• 15,000 (P/F, 30,3) 9,000 (P/F,
  30,4)
• NPW -33,000 -3,000 (0.7692) 67,000 (0.5917) -
  
• 15000 (0.4552) 9000 (0.3501)
• NPW 590.08
• i35
• NPW - 33,000 -3,000 (P/F, 35,1) 67,000 (P/F,
  35, 2)
• 15,000 (P/F, 35,3) 9,000 (P/F,
  35,4)
• NPW -33,000 -3,000 (0.7407) 67,000 (0.5487) -
  
• 15000 (0.4064) 9000 (0.3011)
• NPW - 1845.3

14
Solution

• i30 ? NPW 590.08
• i ??? ? NPW 0
• i35 ? NPW -1845.3
• 5 (increase in i) 590.08 (- 1845.3)
  decrease
• ---------------------- --------------------------
  -----------
• X 590.08 - 0
• X 0.012
• 1.2
• i 30 1.2
• 31.2

15
Example

• For the net cash flows shown below, the maximum
  number of
• possible rate of return solutions is
• Year Net Cash Flows,
• 0 -60,000
• 1 -20,000
• 2 22,000
• 3 15,000
• 4 35,000
• 5 13,000
• 6 -2,000
• (a) 0
• (b) 1
• (c) 2
• (d) 3

16
Example

• A bulk materials hauler purchased a used
• dump truck for 40,000.The operating cost
• was 5000 per month, with revenues of 7500
• per month. After 2 years, the truck was sold
• for 12,000. The rate of return was closest to
• (a) 4.6 per year
• (b) 4.6 per month
• (c) 8.2 per year
• (d) 8.2 per month

17
Solution

• NPW (i) 0
• -40,000 (7,500 5,000) (P/A, i, 24) 12,000
  (P/F, i, 24) 0
• -40,000 2,500 (P/A, i, 24) 12,000 (P/F, i,
  24) 0
• Solve with trial- and-error
• i5
• NPW40,000 2,500 (P/A, 5, 24) 12,000 (P/F,
  5, 24)
• NPW - 40,000 2,500(13.7986) 12,000
• (0.3101)
• -1782.3

18
Solution

• i10
• NPW40,000 2,500 (P/A, 10, 24)
• 12,000 (P/F, 10, 24)
• NPW - 40,000 2,500(8.9847) 12,000
• (0.1015)
• -16,320.25
• i4
• NPW40,000 2,500 (P/A, 4, 24)
• 12,000 (P/F, 4, 24)
• NPW - 40,000 2,500(15.240) 12,000
• (0.3901)
• 2781.2

19
Solution

• NPW (4) 2781.2
• NPW (i) 0
• NPW (5) -1782.3
• 1 (increase in i) 2781.2 (- 1782.3)
• ---------------------- --------------------------
  
• X 2781.2 0
• X 0.006
• i 4 0.6
• i 4.6