2. Course outline
ROR
Analysis
Incremental ROR Analysis
Benefit-Cost Ratio (BCR) Analysis
Present Value Ratio (PVR)
Payback Period
Breakeven Analysis
Choosing Analysis Method
3. Rate of Return Analysis
Rate of return:
the interest rate paid on the unpaid balance of a loan such
that the payment schedule makes the unpaid loan balance
equal to zero when the final payment is made
interest rate earned on the unrecovered investment such
that the payment schedule makes the unrecovered
investment equal to zero at the end of the life of the
investment
is the interest rate at which the benefits are equivalent to
the costs
4. Calculating Rate of Return
PW of benefits - PW of costs = 0
PW of benefits
PW of cost = 1
Net Present Worth = 0
PW of costs = PW of benefits
5. Calculating Rate of Return
Example:
An $8200 investment returned $2000 per year over a 5-year
useful life. What was the ROR on the investment?
PW of benefits
PW of cost
(P/A,i,5) =
2000(P/A,i,5)
8200
=1
8200
2000
i
(P/A,i,5)
6%
i = 7%
= 4.1
4.212
7%
4.100
8%
3.993
=1
6. Trial-and Error-Approach
• Iterative procedures require an initial guess for i*
• One makes an educated first guess and calculates the
resultant PV at the guess rate.
• If the NPV is not = 0, then another i* value is evaluated….
Until NPV “close” to “0”.
• The objective is to obtain a negative PV for an i* guess
value then.
• Adjust the i* value to obtain a positive PV given the
adjusted i* value.
• Then interpolate between the two i* values
7. Trial and Error Approach – Bracket “0”
• If the NPV is not = 0, then another i* value
is evaluated.
• A negative NPV generally indicates the i*
value is too high.
• A positive NPV suggests that the i* value
was too low.
8. ROR Criteria
• Determine the i* rate
• If i* => MARR, accept the project
• If i* < MARR, reject the project
9. Cautions when using the ROR Method No.1
• Many real-world cash flows may possess
multiple i* values
• More than one i* value that will satisfy the
definitions of ROR
• If multiple i*’s exist, which one, if any, is
the correct i*???
10. Cautions when using the ROR Method. 2.
Reinvestment Assumptions
• Reinvestment assumption for the ROR
method is not the same as the reinvestment
assumption for PW and AW
• PW and AW assume reinvestment at the
MARR rate
• ROR assumes reinvestment at the i* rate
• Can get conflicting rankings with ROR vs.
PV and AW
11. Cautions when using the ROR Method: 3.
Computational Difficulties
• ROR method is computationally more
difficult than PW/AW
• Can become a numerical analysis problem
and the result is an approximation
• Conceptually more difficult to understand
12. Cautions when using the ROR Method: 4.
Special Procedure for Multiple Alternatives
• For analysis of two or more alternatives,
when using ROR one must resort to a
different analysis approach as opposed to
the PW/AW methods.
• For ROR analysis of multiple alternatives,
one must apply an incremental analysis
approach.
13. Cautions When Using the ROR Method:
ROR Is More Difficult!
• ROR is computationally more difficult
• But is a popular method with financial
managers
• ROR is used internally by a substantial
number of firms
• Suggest using PW/AW methods where
possible
14. Valid Ranges for Usable i* Rates
Mathematically, i* rates must be:
−100% < i ≤ +∞
*
• If an i* <= -100% this signals total and
complete loss of capital.
• i*’s < -100% are not feasible and not
considered
• One can have a negative i* value (feasible),
but not less than -100%!
15. Incremental Analysis
is
the examination of the differences between
alternatives
Relationship between two alternatives:
Higher-cost alt. = lower-cost alt. + difference
between them
Graphical
analyisis
solution vs Incremental ROR
17. Multiple Alternative Problems
Example:
Consider the 3 mutually alternatives below
A
Initial cost
Uniform annual benefit
B
C
$2000
$4000
$5000
410
639
700
Each alternative has a 20-year life and no salvage
value. If the MARR is 6%, which alternative should
be selected
19. Incremental ROR Analysis
Alternative A:
2000 = 410 (P/A,i,20)
(P/A,i,20) = 2000 / 410 = 4.878
i = 20%
Alternative B:
4000 = 639 (P/A,i,20)
(P/A,i,20) = 4000 / 639 = 6.259
i = 15%
20. Incremental ROR Analysis
Alternative C:
5000 = 700 (P/A,i,20)
(P/A,i,20) = 5000 / 700 = 7.143
i = is between 12% and 15%
i = 12% + [(7.469-7.143) / (7.469 - 6.259)]
i = 12.8%
As the 3 alternatives exceed the MARR of 6%
therefore they are all acceptable.
21. Incremental ROR Analysis
Arrange the 3 alternatives in order of increasing initial cost
A
Initial cost
Uniform annual benefit
Rate of Return
B
C
$2000
410
20%
$4000
639
15%
$5000
700
12.8%
Using A as the baseline, calculate the incremental cost,
incremental uniform annual benefit, and then incremental ROR
Increment B-A
Incremental cost
$4000 - $2000 = $2000
Incremental Uniform annual benefit
639 – 410 = 229
22. Incremental ROR Analysis
Incremental ROR for B-A:
2000 = 229 (P/A,i,20)
(P/A,i,20) = 2000 / 229 = 8.734
∆ROR = 9.6%
As ∆ROR > MARR, therefore A is discarded and B is selected,
and then is used as baseline for comparing alternative C
Increment C-B
Incremental cost
$5000 - $4000 = $1000
Incremental Uniform annual benefit
700 – 639 = 61
23. Incremental ROR Analysis
Incremental ROR for C-B:
1000 = 61 (P/A,i,20)
(P/A,i,20) = 1000 / 61 = 16.393
∆ROR = 2.0%
As ∆ROR < MARR, therefore C is discarded and B is still a
better alternative, which means the best of the 3 alternatives
24. Elements in Incremental ROR Analysis
1 Identify all alternatives
2
Compute ROR for each alternative
•
rejects any alternative with ROR < MARR
3 Arrange accepted alternatives in ascending order
of investment
4 Make a two-alternative analysis for the first two
alternatives
•
•
If DROR > MARR, retain the higher-cost alternative
If DROR < retain the lower-cost alternative
25. Elements in Incremental ROR Analysis
5 Take the preferred alternative from Step 4, and
the next alternative from the list created in Step 3,
then perform the two-alternative analysis
6 Continue until all alternatives have been examined
and the best of multiple alternatives has been
identified
26. Benefit-Cost Ratio Analysis
Based on the ratio of benefits to costs using either
present worth or annual cash flow analysis
PW of benefits - PW of costs > 0
EUAB - EUAC > 0
Could be stated as a ratio of benefits to costs
PW of benefits
Benefit-cost ratio (BCR) =
=
PW of cost
EUAB
>1
EUAC
27. Benefit-Cost Ratio Analysis
Sit uat ion
Cr it er ion
Fix input
Amount of money or other
input resources is fixed
Maximize B/ C
Fix output
Fixed task, benefit, or
other output to be
accomplished
Maximize B/ C
Neither input
nor output
fixed
Neither amount of money, I ncremental analysis
or other inputs, nor
amount of benefit, or other
outputs, is fixed
28. Benefit-Cost Ratio Analysis
Example:
Two mutually exclusive alternative as follows
Year
0
1
Alternative 1
-$10
+15
Alternative 2
-$20
+28
Any money not invested here may be invested
elsewhere at the MARR of 6%. If you can only
choose one alternative one time, which one you
select
29. Benefit-Cost Ratio Analysis
Alternative 1:
PW of cost
= $10
PW of benefit = $15(P/F,6%,1) = 15(0.9434) = $14.15
PW of benefits
BCR =
= 1.415
PW of cost
Alternative 2:
PW of cost
= $20
PW of benefit = $28(P/F,6%,1) = 28(0.9434) = $ 26.42
PW of benefits
BCR =
= 1.321
PW of cost
30. Benefit-Cost Ratio Analysis
Incremental BCR
Year
0
1
Alternative 1
-$10
+15
Alternative 2
-$20
+28
Alt. 2 – Alt. 1
-$20 – (-$10 ) = -$10
+$28 – (+$15) = $13
∆PW of cost = $10
∆PW of benefit = $13(P/F,6%,1) = 13(0.9434) = $12.26
PW of benefits
∆BCR =
= 1.226
PW of cost
∆BCR (alt. 2 - alt. 1) > 1
choose alt. 2
32. Payback Period
Is the period of time required for the profit or other
benefits of an investment to equal the cost of the
investment
is an approximate, rather than exact, economic analysis
calculation
all cost and all savings, or savings of investment, prior to
payback are included without considering differences in
their timing
all the economic consequences beyond the payback
period are completely ignored
33. Payback Period
Example:
two alternatives have following cash flows
Year
0
1
2
3
4
5
A
-$1000
200
200
1200
1200
1200
B
-$2783
1200
1200
1200
1200
1200
Using payback period, which alternative should be
selected?
34. Payback Period
Alternative A
Alternative B
cost recovered for the 1st two years = $400 of the $1000
the remaining $600 is recovered in the first half of the 3rd
year
the payback period is 2.5 years
annual benefit uniform $1200
Payback period = $2783/$1200 per year = 2.3 years
To minimize payback period choose Alt. B
35. Breakeven Analysis
Example:
A project may be constructed to full capacity now
or in two stages
Two-stage construction:
construct 1st stage now
construct 2nd stage n years from now
$120,000
Full-capacity construction:
construct full capacity now
$100,000
$140,000
useful life of 40years from installation
same annual cost of operation & maintenance
8% interest rate
38. Choosing an Analysis Method
MARR should be set before choosing any analysis
method
PW analysis & Annual cash flow analysis often
require far less computation than ROR analysis
Depend on company policy
39. Economic Criteria
Sit uat ion
Crit erion
Fix input
Maximize output
Fix output
Minimize input
Neither input nor output fixed Maximize (output – input)
40. Annual Cash Flow Analysis
Economic Criteria
Sit uat ion
Cr it er ion
Fix input
Amount of money or other
input resources is fixed
Maximize equivalent
uniform benefits
(maximize EUAB)
Fix output
Fixed task, benefit, or
other output to be
accomplished
Minimize equivalent
uniform annual cost
(minimize EUAC)
Neither input
nor output
fixed
Neither amount of money, Maximize (EUAB –
or other inputs, nor
EUAC)
amount of benefit, or other
outputs, is fixed
