INCREMENTAL ANALYSIS

Engineering Economy

Monday, April 10, 2023

Course outline

ROR Analysis Incremental ROR Analysis Benefit-Cost Ratio (BCR) Analysis Present Value Ratio (PVR) Payback Period Breakeven Analysis Choosing Analysis Method

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

Calculating Rate of Return

PW of benefits - PW of costs = 0

PW of benefitsPW of cost = 1

Net Present Worth = 0

PW of costs = PW of benefits

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 benefitsPW of cost

= 12000(P/A,i,5)

8200= 1

82002000

= 4.1(P/A,i,5) = i (P/A,i,5)

6% 4.212

7% 4.100

8% 3.993

i = 7%

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

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.

ROR Criteria

• Determine the i* rate

• If i* => MARR, accept the project

• If i* < MARR, reject the project

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*???

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

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

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.

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

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%!

Incremental Analysis

is the examination of the differences between alternatives

Relationship between two alternatives: Higher-cost alt. = lower-cost alt. + difference

between them Graphical solution vs Incremental ROR

analyisis

Better

Multiple Alternative Problems

A

B

C

Better

Two-Alternative Analysis

D

Best of 4 Alternatives

Multiple Alternative Problems

Example:Consider the 3 mutually alternatives below

A B C

Initial cost $2000 $4000 $5000

Uniform annual benefit 410 639 700

Each alternative has a 20-year life and no salvage value. If the MARR is 6%, which alternative should be selected

Multiple Alternative Problems

Initial cost:- A = $2,000- B = $4,000- C = $5,000

Annual Benefits:- A = $ 410- B = $ 639- C = $ 700

0 20

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%

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.

Incremental ROR Analysis

Arrange the 3 alternatives in order of increasing initial cost

A B C

Initial cost $2000 $4000 $5000

Uniform annual benefit 410 639 700

Rate of Return 20% 15% 12.8%

Increment B-A

Incremental cost $4000 - $2000 = $2000

Incremental Uniform annual benefit 639 – 410 = 229

Using A as the baseline, calculate the incremental cost, incremental uniform annual benefit, and then incremental ROR

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%

Increment C-B

Incremental cost $5000 - $4000 = $1000

Incremental Uniform annual benefit 700 – 639 = 61

As ROR > MARR, therefore A is discarded and B is selected, and then is used as baseline for comparing alternative C

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

Elements in Incremental ROR Analysis

1 Identify all alternatives2 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

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

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

Benefit-cost ratio (BCR) =PW of benefits

PW of cost=

EUAB

EUAC> 1

Benefit-Cost Ratio Analysis

Situation Criterion

Fix input Amount of money or otherinput resources is fixed

Maximize B/C

Fix output Fixed task, benefit, orother output to beaccomplished

Maximize B/C

Neither inputnor outputfixed

Neither amount of money,or other inputs, noramount of benefit, or otheroutputs, is fixed

Incremental analysis

Benefit-Cost Ratio Analysis

Example: Two mutually exclusive alternative as follows

Year Alternative 1 Alternative 2

0 -$10 -$20

1 +15 +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

Benefit-Cost Ratio Analysis

Alternative 1:

PW of cost = $10PW of benefit = $15(P/F,6%,1) = 15(0.9434) = $14.15

BCR =PW of benefits

PW of cost = 1.415

Alternative 2:

PW of cost = $20PW of benefit = $28(P/F,6%,1) = 28(0.9434) = $ 26.42

BCR =PW of benefits

PW of cost = 1.321

Benefit-Cost Ratio Analysis

Year Alternative 1 Alternative 2 Alt. 2 – Alt. 1

0 -$10 -$20 -$20 – (-$10 ) = -$10

1 +15 +28 +$28 – (+$15) = $13

Incremental BCR

PW of cost = $10

BCR =PW of benefits

PW of cost = 1.226

PW of benefit = $13(P/F,6%,1) = 13(0.9434) = $12.26

BCR (alt. 2 - alt. 1) > 1 choose alt. 2

Present Value Analysis

PVR = NPV @ i*

PW cost @ i*=

PW revenue @ i* - PW cost @ i*

PW cost @ i*

PVR = BCR - 1

PVR > 0

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

Payback Period

Example:two alternatives have following cash flows

Year A B0 -$1000 -$27831 200 12002 200 12003 1200 12004 1200 12005 1200 1200

Using payback period, which alternative should be selected?

Payback Period

Alternative A 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

Alternative B annual benefit uniform $1200 Payback period = $2783/$1200 per year = 2.3 years

To minimize payback period choose Alt. B

Breakeven Analysis Example:

A project may be constructed to full capacity now or in two stages Two-stage construction:

construct 1st stage now $100,000construct 2nd stage n years from now

$120,000 Full-capacity construction:

construct full capacity now $140,000

useful life of 40years from installation same annual cost of operation & maintenance 8% interest rate

Breakeven Analysis

Construct full capacity now:PW of cost = $140,000

Two-stage construction:PW of cost = 100,000 + 120,000(P/F,8%,n)

n= 5 PW = 100,000 + 120,000(0.6806) = $181,700

n= 10 PW = 100,000 + 120,000(0.4632) = $155,600

n= 20 PW = 100,000 + 120,000(0.2145) = $125,700

n= 30 PW = 100,000 + 120,000(0.0994) = $111,900

Breakeven Chart

$0

$50.000

$100.000

$150.000

$200.000

$250.000

0 3 6 9 12 15 18 21 24 27 30

Year

PW

of

Co

st

Full capacity now 2-stage construction

Bre

akev

en p

oin

t

14 -

15

year

s

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

Economic Criteria

Situation Criterion

Fix input Maximize output

Fix output Minimize input

Neither input nor output fixed Maximize (output – input)

Annual Cash Flow Analysis

Situation Criterion

Fix input Amount of money or otherinput resources is fixed

Maximize equivalentuniform benefits(maximize EUAB)

Fix output Fixed task, benefit, orother output to beaccomplished

Minimize equivalentuniform annual cost(minimize EUAC)

Neither inputnor outputfixed

Neither amount of money,or other inputs, noramount of benefit, or otheroutputs, is fixed

Maximize (EUAB –EUAC)

Economic Criteria