coparison of capital budgeting.docx

7/27/2019 coparison of capital budgeting.docx

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A Comparison of Capital Budgeting Techniques

Capital budgeting deals with setting the criteria and prescribing the process required for making capitalinvestment choices. Choosing an investment project, that is, making a capital investment choice isultimately a cost/benefit analysis. It requires valuing the project by comparing the payoff to its costs.

Problem

Value, rank and select investment projects

Example 1.

Project A

Required rateyear 1:year 2year 3year 4year 5Initial Cost

7.7%$400$1,250$900.00$3,000.00$1,000$5,045

Project B

3%$100.00$200$150.00$100$50$490.67

Project C

6%$5,200$4,000$1,000$200$100$9,687.23

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Capital Budgeting Techniques

A collection of methods allowing the manager to choose among a variety of investment projects.

Methods:

Average Accounting ReturnPaybackDiscounted paybackInternal Rate of ReturnModified Internal Rate of ReturnNet Present ValueProfitability Index

Average Accounting Return (AAR)

AAR is the ratio of the Average Net Income to the Average Book Value.Decision rule: Take the project if AAR is greater than some target ratio set by accountants.Disadvantages: It has too many flaws, don't ever use it.

Payback period

Payback is the time it takes to recover the initial cost of the investment. Payback is usually measured inyears.

Decision rule: Take the project with the shortest payback period

DisadvantagesIt ignores time value of moneyIt ignores riskIt ignores cash inflows beyond the cutoff point

Project A: Payback calculation

Period

0

1

2

3

0.83

Cash flow

-$5,045.00

$400

$1,250

$900.00

$3,000.00

Amount left to recover

-$5,045.00

$4,645.00

$3,395.00

$2,495.00

$0.00

Notice that in the fourth year, there is $2,495 left to recuperate, and the annual cash flow equals $3,000

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Obviously, $2,495/$3,000 = 0.83Payback here is interpreted as follows: It takes between three and four years to recuperate the initialcost of the project.

Project B: Payback calculation

Period

0

1

2

3

0.41

Cash flow

-$490.67

$100.00

$200

$150.00

$100


-$490.67

$390.67

$190.67

$40.67

$0.00

The payback period of project B is also between three and four years (3.41 years)

Notice that $40.67/$100 = 0.41 (approximately)

Project C: Payback calculation

Period

0

1

2

0.49

Cash flow

-$9,687.23

$5,200

$4,000

$1,000


-$9,687.23

$4,487.23

$487.23

$0.00

The payback of project C is 2.49 years. Notice that $487.23/$1,000 = 0.49 (approximately)

Ranking:1. project C: 2.49 years2. project B: 3.41 years3. project A: 3.83 years

All three projects are viable, but project C is the first to recover its initial cost.

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Discounted payback period (DPB)

DPB is the time it takes to recover the initial cost of the investment.Payback uses nominal CF; DPB uses discounted CF

Decision rule: Take the project with the shortest discounted payback period.

Disadvantages: DPB ignores cash inflows beyond the cutoff point

The calculation of discounted payback is exactly the same as that of payback, except that instead ofusing nominal cash flow, we use present values.

Project A: Discounted payback calculation

Period

0

1

2

3

4

0.94

Cash flow at 7.7%

-$5,045.00

$371.40

$1,077.65

$720.44

$2,229.76

$690.12


-$5,045.00

$4,673.60

$3,595.95

$2,875.51

$645.75

0

The discounted payback of project A is just under 5 years (4.94)Notice that $645.75/$690.12 = 0.94

Project B: Discounted payback calculation

Period

0

1

2

3

0.76

Cash flow at 3%

-$490.67

$97.09

$188.52

$137.27

$88.85


-$490.67

$393.58

$205.06

$67.79

$0.00

The discounted payback of project B is just under 4 years (3.76)Notice that $67.79/$88.85 = 0.76

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Project C: Discounted payback calculation

Period

0

1

2

3

4

5

Cash flow at 6%

-$9,687.23

$4,905.66

$3,559.99

$839.62

$158.42

$74.73


-$9,687.23

$4,781.57

$1,221.58

$381.96

$223.55

$148.82

Notice that the project never recovers its initial cost. At the end of year 5 there is still $148.82 left.

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Internal Rate of Return

IRR is the discount rate that makes the present value of the project equal to its initial cost.

Decision rule: Take the project if the IRR exceeds the required rate of return

Disadvantages: Reinvestment rate assumption is unrealistic Multiple IRR IRR cannot rank mutually exclusive projects

Calculation

Project A:Set: Initial cost (A) = PV(project A):$5,045 = $400/(1+IRR) + $1,250/(1+IRR)2 + $900/(+IRR)3 + $3,000/(1+IRR)4 + $1,000/(1+IRR)5IRR(A) = 8%

Since IRR(A) > 7.7%, accept project (A)

Project B:

$490.67 = $100/(1+IRR) + $400/(1+IRR)2 + $150/(1+IRR)3 + $100/(1+IRR)4 + $50/(1+IRR)5IRR(B) = 8%Since IRR(B) > 3%, accept project B

Project C:$9,687.23=$5,200/(1+IRR) + $4,000/(1+IRR)2 + $1,000/(1+IRR)3 + $200/(1+IRR)4 + $100/(1+IRR)5IRR(C) = 5%.Since IRR(C) < 6%, reject project C

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Modified Internal Rate of Return

MIRR is the discount rate that makes the future value of the project equal to the future value of theinitial cost. MIRR requires a reinvestment rate.

Decision rule: Take the project if MIRR is larger than the required rate.

Disadvantages: MIRR cannot rank mutually exclusive projects.

MIRR calculation

Project A: Let us assume the reinvestment rate is 5%.Set the future value of project's cash flow at 5%. The time horizon, i.e. number of periods, will equalthe duration of the project.

FV(CF at 5%) = $400(1.05)4 + $1,250(1.05)3 + $900(1.05)2 + $3,000(1.05) + $1,000

On the other hand, the FV of the initial cost compounded at MIRR is:

FV (initial cost) = $5,045(1+MIRR)5

Obviously, MIRR is unknown at this stage, and this is precisely the quantity we want to find. To do sowe need to solve the equation:

$400(1.05)4 + $1,250(1.05)3 + $900(1.05)2 + $3,000(1.05) + $1,000 = $5,045(1+MIRR(A))5

Using elementary algebra, we find that:MIRR(A)= ([ $400(1.05)4 + $1,250(1.05)3 + $900(1.05)2 + $3,000(1.05) + $1,000]/$5,045)1/5 - 1MIRR(A) = 7%Since MIRR is less than 7.7% (the required rate of return for A), we must reject project A.

For project B:FV(CF at 5%) = $100(1.05)4 + $200(1.05)3 + $150(1.05)2 + $100(1.05) + $50FV (initial cost) = $490.67(1+MIRR(B))5$100(1.05)4 + $200(1.05)3 + $150(1.05)2 + $1001.05) + $50 = $490.67(1+MIRR(B))5MIRR(B) = 6.54%Since MIRR is more than 3% (the required rate of return for B), we should accept project B.

Project C:FV(CF at 5%) = $5,200(1.05)4 + $4,000(1.05)3 + $1,000(1.05)2 + $200(1.05) + $100

FV (initial cost) = $9,687.23(1+MIRR(C))5$5,200(1.05)4 + $4,000(1.05)3 + $1,000(1.05)2 + $2001.05) + $100 = $9,687.23(1+MIRR(C))5MIRR(C) = 5%Since MIRR is less than 6% (the required rate of return for C), we must reject project C.

Only is B viable, given a reinvestment rate of 5%

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Net Present Value

Net Present Value is the difference between the present value of a project and its initial cost:NPV= Present value - Initial Cost

Decision rule: If NPV is positive, take the project.

Disadvantages: Very complex analysis, too many variables to forecast, as it will be seen later.

Calculation:In order to calculate NPV, we must first estimate the PV of total cash flows; then we subtract the initialcost of the project.

Project A:PV(A) =$400/(1.077) + $1,250/(1.077)2 + $900/(1.077)3 + $3,000/(1.077)4 + $1,000/(1.077)5 Initial cost (A) = $5,045NPV(A) = $44.36

Project B:PV(B) =$100/(1.03) + $400/(1.03)2 + $150/(1.03)3 + $100/(1.03)4 + $50/(1.03)5Initial cost (B) = $490.67NPV (B) = $64.2

Project C:PV(C) =$5,200/(1.06) + $4,000/(1.06)2 + $1,000/(1.06)3 + $200/(1.06)4 + $100/(1.06)5 Initial cost (C) = $9,687.23NPV (C) = -$148.81

Corollary: When Required Rate = IRR, then NPV = 0Remember that IRR is the discount rate that makes the PV of the project equal its initial cost. In otherwords, IRR is the rate that makes the NPV of the project equal to zero. If still not convinced, considerthe present value of our projects at the IRR.

If the discount rate is equal to 8%,NPV(A) = $400/(1.08) + $1,250/(1.08)2 + $900/(1.08)3 + $3,000/(1.08)4 + $1,000/(1.08)5 - $5,045NPV(A) = 0

When the discount rate equals 8%:NPV(B) = $100/(1.08) + $400/(1.08)2 + $150/(1.08)3 + $100/(1.08)4 + $50/(1.08)5 - $490.67

NPV(B) = 0

When the discount rate equals 5%:NPV(C) = $5,200/(1.05) + $4,000/(1.05)2 + $1,000/(1.05)3 + $200/(1.05)4 + $100/(1.05)5 - $9,687.23NPV(C) = 0

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The Profitability Index

The profitability index is the ratio of project PV to initial cost

PI = PV/Initial cost

Decision rule: Take the project if PI > 1

DisadvantagesPI cannot rank mutually exclusive projects.

PI calculation

Project A:PI(A) = PV(A)/Initial cost = 5,089.36/$5,045PI(A) = $400/(1.077) + $1,250/(1.077)2 + $900/(1.077)3 + $3,000/(1.077)4 + $1,000/(1.077)5/$5,045 PI(A) = 1.0088

Project B:PI(B) = $100/(1.03) + $400/(1.03)2 + $150/(1.03)3 + $100/(1.03)4 + $50/(1.03)5 /$490.67PI(B) = 1.131

Project C:PI(C) = $5,200/(1.06) + $4,000/(1.06)2 + $1,000/(1.06)3 + $200/(1.06)4 + $100/(1.06)5 /$9,687.23PI(C) = 0.9846

Only A and B are viable

Why cant PI rank the projects?

Consider the following example:

Project x

Present valueInitial costPINPV

$25,000,000$24,000,0001.042$1,000,000

Project y

$3,000$1,0003$2,000.00

PIx PIybutPx PyIn project valuation, measures of absolute wealth are more appropriate than measures of relativeefficiency.

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Summary:

The results of our capital budgeting techniques are summarized below.

Project A:

Required rateCostPresent valueFuture value (5%)Payback periodDiscounted paybackIRRMIRRNet present valueProfitability index

7.7%$5,045$5,089.36$7,075.453.83 years4.94 years8.00%7.00%$44.361.01

Project B:

3%$490.67$554.84$673.453.41 years3.76 years8.00%6.54%$64.201.13

Project C:

6%$9,687.239,538.42$12,363.52.49 yearsN/A5%5%-$148.810.9846

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NPV profiles

NPV profiles are graphs showing the relationship between discount rates and NPV. For projects with conventional cash flows, this relationship is usually downward sloping; that is, as the discount rateincreases, NPV is decreasing.

NPV profiles are especially useful when all projects have the same required rate of return, or when thediscount rate is hard to ascertain.

NPV Profiles

$2,000.00

$1,500.00

$1,000.00

$500.00

$0.00

-$500.00

NPV ($) -$1,000.00

-$1,500.00

-$2,000.00

-$2,500.00

-$3,000.000.00% 5.00% 10.00% 15.00% 20.00% 25.00%

Discount rate (%)30.00%

NPV (A)

NPV (B)

NPV (C)

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What are the discount rates at which we are indifferent among those three projects? We have to find thepoint where all three profiles intersect at the same time. Following a casual inspection of the NPVprofiles, it is not hard to see that they do not intersect. There is no rate at which the NPV of all three isthe same.

At least, we can search for the discount rate that makes us indifferent between project A and B. Thisamounts to finding the rate at which NPV(A) = NPV(B)

Set:NPV(A) = NPV(B) and solve for the discount rate, that is:

$400/(1+r) + $1,250/(1+r)2 + $900/(+r)3 + $3,000/(1+r)4 + $1,000/(1+r)5 - $5,045 = = $100/(1+r) + $400/(1+r)2 + $150/(1+r)3 + $100/(1+r)4 + $50/(1+r)5 - $490.67

Using elementary algebra, or trial-and-error (more likely), we find that the NPVs of the two projectsare equal when r = 8% (approximately). At any rate below 8%, project A is better. Beyond 8% both projects have negative NPVs.

What is the discount rate that makes us indifferent between project A and C? This amounts to findingthe rate at which NPV(A) = NPV(C)$400/(1+r) + $1,250/(1+r)2 + $900/(+r)3 + $3,000/(1+r)4 + $1,000/(1+r)5 - $5,045 = = $5,200/(1+r) + $4,000/(1+r)2 + $1,000/(1+r)3 + $200/(1+r)4 + $100/(1+r)5 -$9,687.23

It is not hard to see that for any discount rate within a reasonable range (that is 0 to 100%) there is nosolution to this equation. The NPV profile of project A and C never meet. We will always prefer A overC regardless of the discount rate, because the profile of A is always above that of C.

What is the discount rate that makes us indifferent between project B and C? This amounts to findingthe rate at which NPV(B) = NPV(C)

$100/(1+r) + $400/(1+r)2 + $150/(1+r)3 + $100/(1+r)4 + $50/(1+r)5 - $490.67 = $5,200/(1+r) + $4,000/(1+r)2 + $1,000/(1+r)3 + $200/(1+r)4 + $100/(1+r)5 -$9,687.23

Using elementary algebra, or trial-and-error, we find r = 4.75% (approximately). Between 0 and 4.75% for both projects, C is always preferred over B. For discount rates above 4.75%, up to 8%, B is alwayspreferred over C.

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A summary of NPV profiles

Range of discount Significance ofratesupper bound

0 to 4.75%

4.75% to 5%

5% to 8%

Crossover betweenB&C

IRR(C)

IRR (A), IRR (B)and crossoverbetween A & B

Project A

Accept

Accept

Accept

Project B Project C Ranking

A, C, B

A, B, C

A, B, C

AcceptAccept(marginally)

AcceptAccept(marginally)

AcceptReject(marginally)

Reject reject8% and beyond Reject B, A, C

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Example 2

Consider the following cash flows:

Period

01

234567

Discount rateReinvestment rate

PaybackDiscounted payback

IRRMIRRNPV

PI

Projects A and B are comforting, nice, clean, and conventional projects; project C has alternating signcash flow. As we will see, the estimation and interpretation of its IRR is more problematic; luckily wecan always fall back on MIRR and NPV, especially NPV.

Cash flow A

-$1,600.00$0.00

$0.00$400.00$400.00$400.00$400.00$400.00

4.60%3.00%

Cash flow B

-$451.00$100.00

$200.00$150.00$100.00$50.00

Cash flow C

-$503.99$2,862.00

-$6,070.00$5,700.00

-$2,000.00

3.00%10.00%

66.90%10.00%

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Payback

Project A: Payback calculation

Period

0

1

2

3

4

5

6

Cash flow

-$1,600.00

$0.00

$0.00

$400.00

$400.00

$400.00

$400.00


-$1,600.00

$1,600.00

$1,600.00

$1,200.00

$800.00

$400.00

$0.00

The payback of project A is exactly 6 years

Project B: Payback calculation

Period

0

1

2

3

0.01

Cash flow

-$451.00

$100.00

$200.00

$150.00

$100.00


-$451.00

$351.00

$151.00

$1.00

$0.00

The payback of project B is 3 years (3.01)

Project C has cash flow with alternating signs. The initial cost is recovered after the first year, butsubsequent negative cash outflows make the estimation of payback less meaningful.

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Discounted payback

Project A: Discounted payback calculation

Period

0

1

2

3

4

5

6

7

Cash flow at 4.60%

-$1,600.00

$0.00

$0.00

$349.51

$334.14

$319.45

$305.40

$291.97


-$1,600.00

$1,600.00

$1,600.00

$1,250.49

$916.34

$596.89

$291.49

$0.00

The discounted payback of project A is exactly 7 years

Project B: Discounted payback calculation

Period

0

1

2

3

0.32

Cash flow at 3%

-$451.00

$97.09

$188.52

$137.27

$88.85


-$451.00

$353.91

$165.39

$28.12

$0.00

The discounted payback of project B is years (3.32). notice that $28.12/$88.85 = 0.32

Project C has cash flow with alternating signs. Again, estimating the discounted payback is not aninsightful exercise.

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Internal Rate of Return

Project A:

$1,600 = $400/(1+IRR)3 + $400/(1+IRR)4 + $400/(1+IRR)5 + $400/(1+IRR)6 + $400/(1+IRR)7

Using trial and error we find that IRR(A) is approximately 4.6%, virtually equal to the required rate.

Project B

$451 = $100/(1+IRR) + $200/(1+IRR)2 + $150/(1+IRR)3 + $100/(1+IRR)4 + $50/(1+IRR)5Using trial and error we find that IRR(B) is approximately 11.7%, larger than the required rate.

Project C$503.99 = $2,862/(1+IRR) - $6,070/(1+IRR)2 + $5,700/(1+IRR)3 - $2,000/(1+IRR)4Using trial and error we find two internal rates:

One at approximately 24.1%, and the other one at 66.9%, virtually equal to the required rate. There are two more rates, but we do not bother with them since they must be outside any reasonable range. Whatsignificance would a negative, or a positive but large rate could add to an already ambiguous situation?How do we interpret the results? The answer will become clear when plotting NPV profiles.

Modified internal rate of return

Project A:

$1,600(1+MIRR)7 = $400(1.03)4 + $400(1.03)3 + $400(1.03)2 + $400/(1.03) + $400We find that the MIRR(A) is approximately 4.13%

Project B

$451(1+MIRR)5 = $100(1.1)4 + $200(1.1)3 + $150/(1.1)2 + $100(1.1) + $50We find that MIRR(B) is approximately 10.8%, larger than the required rate, yet smaller than the IRR,simply because reinvestment was made at a rate lower than 11.7%

Project C$503.99(1+MIRR)4 = $2,862(1.1)3 - $6,070(1.1)2 + $5,700(1.1) - $2,000

We find that MIRR(C) is approximately 9.9%

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Net Present Value

Project A

NPV(A) = -$1,600 + $400/(1.046)3 + $400/(1.046)4 + $400/(1.046)5 + $400/(1.046)6 + $400/(1.046)7

NPV(A) = $0.48

Project BNPV(B) = -$451 + $100/(1.03) + $200/(1.03)2 + $150/(1.03)3 + $100/(1.03)4 + $50/(1.03)5NPV(B) = $103.86

Project CNPV(C) = -$503.99 + $2,862/(1.67) - $6,070/(1.67)2 + $5,700/(1.67)3 - $2,000/(1.67)4NPV(C) = 0

Profitability index

PI(A) = [$400/(1.046)3 + $400/(1.046)4 + $400/(1.046)5 + $400/(1.046)6 + $400/(1.046)7 ]/$1,600PI(A) = 1

PI(B) = [$100/(1.03) + $200/(1.03)2 + $150/(1.03)3 + $100/(1.03)4 + $50/(1.03)5]/$451PI(B) = 1.23

PI(C) = [$2,862/(1.67) - $6,070/(1.67)2 + $5,700/(1.67)3 - $2,000/(1.67)4]/$503.99PI(C) = 1

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Summary of several capital budgeting methods.

Period

01234567

Discount rateReinvestment rate

PaybackDiscountedpayback

IRRMIRRNPVPI

Cash flow A

-$1,600.00$0.00$0.00

$400.00$400.00$400.00$400.00$400.00

4.60%3.00%

67

4.60%4.13%$0.481.00

Cash flow B

-$451.00$100.00$200.00$150.00$100.00$50.00

Cash flow C

-$503.99$2,862.00

-$6,070.00$5,700.00

-$2,000.00

3.00%10.00%

3.013.32

11.70%10.83%$103.861.23

66.90%10.00%

na.na.

24.10% & 67.00%9.90%$0.001.00

In this example, project B clearly dominates the other two projects. Although project A's NPV ismarginally positive, the project represents a questionable choice. The same goes for project C. Zero netpresent value makes us indifferent between accepting and rejecting the project (that is, in theory).Without other extra-monetary considerations, it is hard to justify taking on a project that offers noobvious benefits.

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NPV Profiles

$600.00

$400.00

$200.00

$0.00

-$200.00

-$400.00NPV ($)

-$600.00

-$800.00

-$1,000.00

-$1,200.000.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00%

Discount rate (%)

NPV (A) NPV (B) NPV (C)

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Crossover rate between Projects A and B

Period

01234567

Cash flow A

-$1,600.00$0.00$0.00

$400.00$400.00$400.00$400.00$400.00

Cash flow B

-$451.00$100.00$200.00$150.00$100.00$50.00$0.00$0.00

A-B

-$1,149-$100-$200$250$300$350$400$400

In order to find the crossover rate between projects A and B, we need to estimate the discount rate thatmakes the present value of the difference in the cash flow between the two projects equal to zero. Inother words, we take the difference in cash flow between the two projects and we estimate the IRR.

In the equation below, r is the crossover rate between projects A and B:-$1,149-$100/(1+r) -$200/(1+r)2+$250/(1+r)3 + $300/(1+r)4 +$350/(1+r)5 +$400/(1+r)6 +$400(1+r)7 =0Using trial and error we find that r is approximately equal to 3.35%. At rates between 0 and 3.35%,project A ranks better than project B, that is, it has a higher net present value. Beyond 3.35%, project Branks better than project A.

Crossover rate between projects A and CPeriodCash flow A

01234567

-$1,600.00$0.00$0.00

$400.00$400.00$400.00$400.00$400.00

Cash flow C

-$503.99$2,862.00

-$6,070.00$5,700.00

-$2,000.00$0.00$0.00$0.00

A-C

-$1,096-$2,862.00$6,070.00

-$5,300.00$2,400.00$400.00$400.00$400.00

In the equation below, r is the crossover rate between projects A and C:-$1,096-$2,862/(1+r)+$6,070/(1+r) 2-$5,300/(1+r)3 +$2,400/(1+r)4 +$400/(1+r)5 +$400/(1+r)6 +$400/

(1+r)7 =0Using trial and error, we find the crossover rate between projects A and B approximately equal to 4.67%. Between zero and 4.67%, Project A ranks better than project C, that is, it has a higher netpresent value. Beyond 4.67%, project C ranks better than A.

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Crossover rate between projects B and CPeriodCash flow B

012345

-$451.00$100.00$200.00$150.00$100.00$50.00

Cash flow C

-$503.99$2,862.00

-$6,070.00$5,700.00

-$2,000.00$0.00

B-C

$52.99-$2,762.00$6,270.00

-$5,550.00$2,100.00$50.00

In the equation below, r is the crossover rate between projects B and C:$52.99-$2,762/(1+r)+$6,270/(1+r) 2-$5,550/(1+r)3 +$2,100/(1+r)4 +$50/(1+r)5 =0Using trial and error, we find the crossover rate between projects B and C approximately equal to11.8%. Between zero and 11.8%, Project B ranks better than project C, that is, it has a higher netpresent value. Beyond 11.8%, project C ranks better than B.



0 to 3.35%

3.35% to 4.6%

4.6% to 4.67%

4.67% to 11.7%

11.7% to 11.8%

11.8% to 24.1%

24.1% to 67%

67% and beyond

Crossover betweenA&B

IRR(A)

Crossover betweenA&C

IRR(B)

Crossover betweenB&C

IRR(C)1

IRR(C)2

Project A

Accept

Accept

Reject

Reject

Reject

Reject

Reject

Reject

Project B

Accept

Accept

Accept

Accept

Reject

Reject

Reject

Reject

Project C

Reject

Reject

Reject

Reject

Reject

Reject

Accept(marginally)

Reject

Ranking

A, B, C

B, A, C

B, A, C

B, C, A

B, C, A

C, B, A

C, B, A

C, B, A

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Example 3

Consider the following three projects:Period0

12345

Discount rateReinvestment

PaybackDiscounted paybackIRRMIRRNPV

PI

Cash flow A-$1,600.00

$1,600.00

Cash flow B$500

$0-$400

Cash flow C$500

-$500$500-$500$500-$500

30.00%10.00%

30.00%3.00%

11.00%0.00%

The three projects have again different lives and unconventional cash flows, which makes comparisonand evaluation a little trickier.

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Project A

Payback:It is obviously one year, no need to engage in redundant calculations.

Discounted payback:As long as the discount rate is larger than zero, the project will never pay back, since the PV of $1,600will always be less than $1,600

IRR:$1,600 = $1,600/(1+IRR)The PV of $1,600 is equal to $1,600 only when the discount rate is zero, hence IRR = 0%

MIRRMIRR = $1,600(1+MIRR) = $1,600Obviously, MIRR has to equal zero.

NPVNPV = $1,600/(1.3) - $1,600 = -$369.23

PIPI = [$1,600/(1.3)]/$1,600 = 0.77

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Project B

Project B has unconventional cash flow. There is a cash inflow at the beginning, and a cash outflow inthe second period. It does not make sense to estimate either payback or discounted payback for thesimple reason that both require an initial cost.

IRR-$500 = -$400/(1+IRR)2IRR = ($500/$400)1/2 -1IRR = -10.56%Obviously, this number does not have any real meaning, hence we could safely conclude that there isno finite positive rate that satisfies the equation. The interpretation given to this measure is that the rateof return cannot be calculated, yet the project is valuable because the future cash outflow is smallerin magnitude then the present cash inflow.

MIRRThe rate that makes the future value of the initial cost equal to the future value of the cash inflowreinvested at 0% can be found by solving the following equation:

-$500(1+MIRR)2 = -$400MIRR = -10.56%Again, it is hard to interpret this result. Suffice to say that common sense indicates the project is nodoubt valuable.

NPVIt is not hard to see that Net Present Value is positive for any positive discount rate.NPV = -$400/(1+ r)2 + $500 > 0NPV = -$400/(1.11)2 + $500 = $175.35

PIPI = [-$400/(1+ r)2 ]/-$500 = 0.65

For the uninitiated, this result will probably cast doubts over the viability of the project. Using a purelyprocedural approach based on the above calculation, we should reject the project. Rejecting would be abig mistake. This example represents a dream investment. Would you not like to borrow $500 million,and have to repay only $400 m in two years? Any common sense investor would jump at thisopportunity. A mathematician would probably explain that this amounts to a negative interest rate, butwe do not even need to do the calculation to grasp the concept. The benefits of such an investment areobvious.

We have here a glaring example of the limitations of quantitative performance measures. Performancemetrics are accurate in some instances, and dead wrong in other situations. In this particular situation, itis not hard to see that the PI is dead wrong. Unfortunately, there are other real life situations in whichan indicator might give us a wrong or conflicting answer, but we have no way of knowing it, becauseour intuition and common sense are overwhelmed by the complexity of the problem at hand. Of course,there are cases when the opposite is true: our common sense gives us false solutions, which are at oddswith numerical calculations.

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Project C

Project C boasts unconventional cash flow as well. Cash inflows and outflows alternate according to asimple pattern. It starts with a cash inflow, followed by a cash outflow of equal magnitude, followed byanother cash inflow, etc up to year five. As with the previous example, we will have to check ournumerical answers against our intuition and common sense .

Payback and discounted payback.The project starts with a cash inflow, hence, calculating a payback period is uninformative and difficultto interpret.

IRR

-$500 = -$500/(1+IRR) + $500/(1+IRR)2 - $500/(1+IRR)3 + $500/(1+IRR)4 - $500/(1+IRR)5

Since the sign of the cash flow is alternating, we expect multiple IRRs. One solution is immediatelyvisible through casual inspection. At 0%, the present value of all cash flow equals zero. Between zero

and 100% there is no other solution, and at this point we should give up trying to estimate IRR becauseit would not provide any helpful insights.

MIRR

-$500(1+MIRR)5 = -$500(1.1)4 + $500(1.1)3 - $500/(1.1)2 + $500(1.1) - $500MIRR is 4.5%

NPV

NPV = -$500/(1.3) + $500/(1.3)2 - $500/(1.3)3 + $500/(1.3)4 - $500/(1.3)5 +$500 = $224.06

PIPI = [ -$500/(1.3) + $500/(1.3)2 - $500/(1.3)3 + $500/(1.3)4 - $500/(1.3)5 ]/-$500 = 0.55

It appears that PI and IRR suggest we should reject the project. Yet again, this is another example ofconflicting results in which IRR and PI are dead wrong. NPV is a positive number. The solutionprovided by NPV is consistent with our common sense, which tells us that cash inflows are acquiredearlier than cash outflows. Again, this example amounts to a situation in which we borrow every otheryear, only to pay less the following year. If banks agreed to terms like these, that they would run out of

money fairly quickly. Evidently, we should accept the project.

If the three projects were mutually exclusive, we should pick C because it has the largest NPV. ProjectA is out of the question regardless of the situation.

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Summary

Period0123

45

Discount rateReinvestment

PaybackDiscounted paybackIRRMIRRNPV

PI

Cash flow A-$1,600.00$1,600.00

Cash flow B$500.00$0.00-$400.00

Cash flow C$500.00-$500.00$500.00-$500.00

$500.00-$500.00

30.00%10.00%

na.na0.00%4.45%$224.06

0.55

30.00%3.00%

1.00na.0.00%0.00%-$369.23

0.77

11.00%0.00%

na.na.-10.56%-10.56%$175.35

0.65

Our analysis yields interesting result. As already pointed out, some results are contradictory,. Here wehad clearly shown that some measures can be wrong at times, yet NPV is the only one that never failed.In this example we do not doubt our common sense, but in more complex situations common sense canplay tricks on us; we should always rely on the NPV beacon guiding our investment decision, as theultimate test, as the golden standard. (NPV is not without problems, but they are of a different nature,as it will be seen later).

In the example above, C dominates all other choices, but B is acceptable as well. A would not beacceptable in any case. If we do not face mutually exclusive choices, we should pick both B, and C;otherwise we should pick C, for it has the largest NPV.

One of the question we always like to ask is whether the cash flow of project B appears to be realistic;are there real-life projects resembling this pattern? The cash flow of project B is much more prevalentin real life than many people imagine. The simplest example that comes to mind is a (successful) shortsale. Imagine selling short the shares of some company for $400. After two days you buy back thestock (which meanwhile has dropped in price by 20%) at $400 and make over $100 profit; In our case,at 11%, you make exactly $175.35 in profit .

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NPV ($)NPV Profiles

$400.00

$300.00

$200.00

$100.00

$0.00

-$100.00

-$200.00

-$300.00

-$400.00

-$500.000.00% 5.00%

NPV (A) NPV (B)

10.00%

NPV (C)

15.00%

Discount rate (%)

20.00% 25.00% 30.00



0.00%

0% and beyond

Crossover betweenA&C

Project A

Reject

Reject

Project B

Accept

Accept

Project C

Accept

Accept

Ranking

B, C, A

B, C, A

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One last question: Why is NPV the golden standard in capital budgeting? Why not IRR, or PI, or thepayback period?

Payback is a fairly crude measure, and has too many flaws: it ignores cash flows after the cutoff point,and it ignores time value of money. IRR cannot rank projects, and is based on unrealistic assumptions(i.e., reinvestment rate). PI cannot rank projects, and is easily fooled by unconventional cash flow.

Remarkably, both payback and IRR are still widely used in capital budgeting. The reason for thisresilience is to be found in their simplicity: both can be conveyed in one simple number, easy to pitchto no-nonsense, straight-taking, business folk. NPV can also be conveyed in one number, but theconcept is more difficult to grasp than a simple rate of return, or a payback period. In addition, one hasto specify a myriad of other assumptions. Chief among those assumption is the required rate.

Imagine yourself trying to convince a bunch of old-fashioned, grudging, and reluctant accountants; or agroup of corporate directors in a hurry about the merits of such and such project. You can prepare anelaborate slide presentation with tables and neat formulas, or you can tell them the project will payback in three years, and will return 11%. Now, that is much more poignant than engaging in a long

discussion about what you think is an appropriate rate of return. Many will have the same reaction youhave when attending a lecture on NPV estimation: they will start yawning and rolling their eyes.

We must acknowledge, however, that NPV is better in theory, but its practical estimation is a headache,as it will be seen later. The first question remains: why is it considered beforehand superior to all othermethods?

Let us go back to project B in example nr. 1, the first in the series of three examples presented earlier.

Period

Required rate

Cost

Present value (at 3%)

Future value ( at 5%)

Payback period

Discounted payback

IRR

MIRR

Net present value

Cash flow

0.03%

-$490.67

$554.84

$673.45

3.41 years

3.76 years

8.00%

6.54%

$64.20

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Imagine an entrepreneur gearing up to start the project. At the very beginning, the balance sheet of theproject would look like this:

Assets

Cash

Liabilities and equity

Equity $490.67

$490.67

Remember that the cash flow of the project is only a projection at this stage (this is why the investmentis called project; however, we understand that this projection represents our best guess; in reality,things might turn out differently; however, on average our projection might still be realistic. At the endof the project, given a reinvestment rate of 5%, the balance sheet would look like this:

Assets

Cash


Equity $673.45

$673.45

In other words, the project would have created value for our entrepreneur. In a world of perfectinformation and rational expectations, everyone would know in advance the magnitude of the gain. Themoment the entrepreneur buys specific assets, everyone will foresee the cash flow and the gainsengendered by this investment. It follows that the entrepreneur could turn around and sell his equitybefore the project actually begins for more than the $490.67 initially invested. At the inception of the

project, just after the cash has been spent on firm-specific assets, the books (at fair market value)should look like this (assuming a 3% required rate of return):

Assets

Firm-specific assets at cost

Intangibles


Equity $554.87

$490.67

$64.20

In other words, by signalling the intention to engage in the project, the market value of the projectshould jump by $64.20, which is exactly the NPV of the project. Remember that in a world predicatedon rational behavior and rational expectations, the maximization of shareholders' wealth is the ultimategoal of business. NPV is the golden standard of investment because it estimates by how much thewealth of shareholders is expected to change. In a different cultural setting, in which business is gearedtowards attaining other goals, NPV would be of secondary importance.

coparison of capital budgeting.docx

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