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Chapter 4. DISCOUNTING AND ALTERNATIVE INVESTMENT CRITERIA. Contents . Discounting Alernative Investment Criteria - Net Present Value Criteria - Benefit-Cost Ratio Criteria - Pay Back Period Criteria - Internal Rate of Return Criteria. 1.1 Discounting and Compounding.

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### Chapter 4

DISCOUNTING AND ALTERNATIVE INVESTMENT CRITERIA

Contents
• Discounting
• Alernative Investment Criteria

- Net Present Value Criteria

- Benefit-Cost Ratio Criteria

- Pay Back Period Criteria

- Internal Rate of Return Criteria

1.1 Discounting and Compounding
• Same amount of money "today" and "in future" do not have the same value.
• Finding the “today” value of a future amount is DISCOUNTING.
• Finding the “future” valuea current amount is COMPOUNDING.
1.1 Discounting and Compounding(cont’d)

An investment of \$1, with a discount rate = r

ValueValue

PresentAfter One YearPresentAfter n-Years

B/(1+r) B B/(1+r)n B

r is the discount rate

1/(1+r) is the discount factor

(1+r) is the compound factor

Ex: Present value of 250\$ for 10 years at 6% discount rate

4PV = \$250/ (1+0.06)10

PV = \$250/1.79= \$139.6 or \$250* 1/(1+0.06)10 = 139.6

Example of Discounting

Year 0 1 2 3 4

Net Cash Flow -1000 200 300 350 1440

1.1Discounting and Compounding(cont’d)
• Discounting Net Benefits

- the period (number of years)

- the size of the discount (r)

NPVr0= (B0-C0)/(1+r)0 + (B1-C1)/(1+r)1 +---+

(Bn-Cn)/(1+r)n

1.1Discounting and Compounding(cont’d)
• In ranking projects, the particular point in time to which all the net benefits in each period are discounted does not matter.
• Instead of discounting all the net benefit flows to the initial year (year 0), they can be dicounted to year “k” NPV at year k will be a constant multiple of NPV in year 0. It will be multiplied with (1+r). This will not change the ranks of the projects, as all projects’ NPVs in year 0 will be multiplied with the same constant (1+r).

Table 4-1. Calculating the present value

of net benefits from an investment project

1.2 Variable Discount Rate
• Discount rates may vary through time.
• Funds may be very scarce at the beginning of the project and the discount rates may be very high.
• This will fall in the following years as the supply and demand for funds return to normal
1.2 Variable Discount Rates

Adjustment of Cost of Funds Through Time

r0

r1

If funds currently are abnormally scarce

r2

r3

r4

r5

Normal or historical average cost of funds

r

*4

r

*3

If funds currently are abnormally abundant

r

*2

r

*1

r

*0

Years from present period

0

1

2

3

4

5

• For variable discount rates r1, r2, & r3 in years 1, 2, and 3, the discount factors are, respectively, as follows:

1/(1+r1), 1/[(1+r1)(1+r2)] & 1/[(1+r1)(1+r2)(1+r3)]

1.3 Factors Affecting the Discount Rates for Public Projects
• Discount rate for a private investment is the weighted average of:
• Rate of return from the sale of equity
• The cost of borrowed funds
• Discount rate for a public sector investment is the Economic Opportunity Cost of Capital (EOCK). This rate reflects the opportunity forgone for not using the funds in an alternative public project. It considers the lost opportunity for the whole economy. It reflects the true cost of the resources (funds) used.
• EOCK is taken as 12% by the World Bank for developing countries.
1.3 Factors Affecting the Discount Rates for Public Projects (con’t)
• If significant market distortions exits i.e. domestic distortions (taxes and subsidies), and external distortions (tariffs and subsidies for exports), the market price of inputs and outputs of the projects do not reflect the true costs of the resources. Their economic (shadow) prices should be used in public projects.
• For public sector projects, economic analysis (rather than financial analysis) is more relevant. Economic analysis uses the EOCK (rather than the discount rate) and the economic values (rather than the market prices).

Financial AnalysisEconomic Analysis

* Financial discount rates * EOCK

* Market prices * Economic values

* More relevant to private sector * More rel. to public sector

* Owner’s and banker’s point of view * Economy point of view

2. Alternative Investment Criteria

First Criterion: Net Present Value (NPV)

• What does net present value mean?
• Measures change in wealth
• Use as a decision criterion to answer following:

a. When to reject projects?

b. Select project(s) under a budget constraint?

c. Compare mutually exclusiveprojects?

2.1 Net Present Value Criterion

a. When to Reject Projects?

Rule:“Do not accept any project unless it generates a positive net present value when discounted by the opportunity cost of funds”

Examples:

Project A: Present Value Costs \$1 million, NPV + \$70,000

Project B: Present Value Costs \$5 million, NPV - \$50,000

Project C: Present Value Costs \$2 million, NPV + \$100,000

Project D: Present Value Costs \$3 million, NPV - \$25,000

Result:

Only projects A and C are acceptable. The country is made worse off if projects B and D are undertaken.

2.1 Net Present Value Criterion (Cont’d)

b. When You Have a Budget Constraint?

Rule: “Within the limit of a fixed budget, choose that subset of the available projects which maximizes the net present value”

Example:

If budget constraint is \$4 million and 4 projects with positive NPV:

Project E: Costs \$1 million, NPV + \$60,000

Project F: Costs \$3 million, NPV + \$400,000

Project G: Costs \$2 million, NPV + \$150,000

Project H: Costs \$2 million, NPV + \$225,000

Result:

Combinations FG and FH are impossible, as they cost too much. EG and EH are within the budget, but are dominated by the combination EF, which has a total NPV of \$460,000. GH is also possible, but its NPV of \$375,000 is not as high as EF.

2.1 Net Present Value Criterion (Cont’d)

c. When You Need to Compare Mutually Exclusive Projects?

Rule:“In a situation where there is no budget constraint but a project must be chosen from mutually exclusive alternatives, we should always choose the alternative that generates the largest net present value”

Example:

Assume that we must make a choice between the following three mutually exclusive projects:

Project I: PV costs \$1.0 million, NPV \$300,000

Project J: PV costs \$4.0 million, NPV \$700,000

Projects K: PV costs \$1.5 million, NPV \$600,000

Result:

Projects J should be chosen because it has the largest NPV.

2.1 Net Present Value Criterion (Cont’d)

Constraints of Using NPV

• NPV, not only tells you whether the project will be accepted or rejected but also gives you the present value of the surplus or the deficit of the project. This is an advantage for NPV.
• If the life periods of the strickly alternative projects are not the same, adjustments have to be made, so that the projects will be compared for the same lenght of lives.
• It is not correct to compare the NPV of a gas turbine plant with a life of 10 years, to a coal plant with a life of 30 years. Their lenght of lives should be equated by repeating the gas plant for three times.

2.2 Benefit-Cost Ratio

It is a widely used by the analysts. Should be very careful, otherwise incorrect and misleading decisions can be made.

Benefit-Cost Ratio (R) = Present Value Benefits/Present Value Costs

Basic rule:

If benefit-cost ratio (R) >1, then the project should be undertaken.

Problems?

Sometimes it is not possible to rank projects with the Benefit-Cost Ratio

• Mutually exclusive projects of different sizes
• Mutually exclusive projects and recurrent costs subtracted out of benefits or benefits reported gross of operating costs
• Not necessarily true that RA>RB that project “A” is better
2.2 Benefit-Cost Ratio (Cont’d)

First Problem:

The Benefit-Cost Ratio Does Not Adjust for Mutually Exclusive Projects of Different Sizes.

For example:

Project A: PV0of Costs = \$5.0 M, PV0 of Benefits = \$7.0 M

NPVA = \$2.0 M RA = 7/5 = 1.4

Project B: PV0 of Costs = \$20.0 M, PV0 of Benefits = \$24.0 M

NPVB = \$4.0 M RB = 24/20 = 1.2

According to the Benefit-Cost Ratio criterion, project A should bechosen over project B because RA>RB, but the NPV of project B is greater than the NPV of project A.

So, project B should be chosen

2.2. Benefit-Cost Ratio (Cont’d)

Second Problem:

The Benefit-Cost Ratio Does Not Adjust for MutuallyExclusive Projects where the Costs are treated in different ways.

Project AProject B

PV of gross benefits 2,000 2,000

PV of operating Costs 500 1,800

PV of capital costs 1,200 100

• B/C Ratio (Operating costs netted out of the benefits)

RA = (2000-500)/1200 = 1.15 RB = (2000-1800)/100 = 2.0

Project B is preferred to Project A (RB > RA ).

2. B/C Ratio (Operating costs added to capital costs) RA = 2000/(1200+500)= 1.18 RB = 2000/(1800+100) = 1.05

Project A is preferred to Project B (RA > RB ).

NPV of a project is not sensetive to the way the acountants treat costs. Thus NPV is far more reliable than B/C ratio as a criterion for project selection.

Conclusion:The Benefit-Cost Ratio CANNOT be used to rank projects

2.3 Pay-Out or Pay-Back Period
• It is widely used criterion as it is very easy to apply. Unfortunately it can give misleading results especially in cases of investment with a long life.
• In a simplest form, it measures the number of years it will take for the undiscounted net benefits (positive net cash flow) to repay the investment. If the number is greater than an arbitrary chosen year, the project is accepted.
• In more sophisticated form, it divides the discounted net benefits over a given year with the discounted investment costs. If the number is greater than 1, the project is accepted. One assumes that after the chosen net benefits are so uncertain (war and political inability) that they can be neglected. This assumption is not realistic in cases of bridges and roads, etc.
2.3 Pay-Out or Pay-Back Period
• Project with shortest payback period is preferred by this criteria

There is no reason to expect that quick yielding projects are superior to long term invetments.

Bt - Ct

Ba

Bb

ta

0

tb

Time

Ca = Cb

Payout period for project a

Payout period for project b

Figure 4.2 Comparison of Two Projects With Differing Lives Using Pay-Out Period Criterion
2.3 Pay-Out or Pay-Back Period (Cont’d)
• In such situations pay-back period criterion gives wrong recommendation for choice among investments.
• Assumes all benefits that are produced by in longer life project have an expected value of zero after the pay-out period.
• The criteria may be useful when project subject to high level of political risk.

t

i=0

2.4 Internal Rate of Return Criterion
• IRR is the discount rate (K) at which the present value of benefits are just equal to the present value of costs for the particular project.....(IRR= k) wkich equates the net benefits tı zero.
• NPVr 0= 0 = (B0 - C0 ) + (B1 – C1 )/((1+k)1 + (B2 – C2 ) / (1+k)2

Bt - Ct

(1 + K)t

Note: the IRR is a mathematical concept, not an economic or financial criterion

Common uses of IRR:

(a). If the IRR is larger than the cost of funds then the project should be undertaken

(b). Often the IRR is used to rank mutually exclusive projects. The highest IRR project should be chosen

• An advantage of the IRR is that it only uses information from the project
• Another advantage of IRR is that it does not require the calculation of EOCK. With NPV one has to calculate the EOCK in the economic analysis.

= 0

Bt - Ct

+300

Time

-100

-200

2.4 Difficulties With the Internal Rate of Return Criterion

First Difficulty: Multiple rates of return for project

Solution 1: K = 100%; NPV= -100 + 300/(1+1) + -200/(1+1)2 = 0

Solution 2: K = 0%; NPV= -100+300/(1+0)+-200/(1+0)2 = 0

2.4.1 Difficulties With the Internal Rate of Return Criterion(IRR Makes Misleading Choice under following conditions)
• For Single Projects
• If the net cash flow is negative in the initial year (due to initial investment) but all positive in the following years, then IRR has a unique solution i.e. One solution
• If negative net cash flows take place after the negative net cash flow in intial year, we cannot have a unique solution for the IRR. You will have two values for IRR (figure 4.3)
• If there is a large negative benefit in the final year of the project, there will not be a unique solution for IRR again.
Figure 4.3 Time Profiles of the Incremental Net Cash Flows for Various Types of Projects

Bt - Ct

+

time

-

Bt - Ct

+

time

-

(Cont’d)

Ґ

Year 0

1

2

3

...

...

Project A

-

2,000

+600

+600

+600

+600

+600

+600

Project B

-

20,000

+4,000

+4,000

+4,000

+4,000

+4,000

+4,000

NPV and IRR provide different Conclusions:

Opportunity cost of funds = 10%

0

NPV : 600/0.10

-

2,000 = 6,000

-

2,000 = 4,000

A

NPV : 4,000/0.10

-

20,000 = 40,000

-

20,000 = 20,000

0

B

Hence, NPV > NPV

0

0

B

A

IRR

: 600/K

-

2,000 = 0 or K

= 0.30

A

A

A

IRR

: 4,000/K

-

20,000 = 0 or K

= 0.20

B

B

B

Hence, K

>K

A

B

Second difficulty: Projects of different sizes and also strict alternatives

Third difficulty:

Projects of different lengths of life and strict alternatives

Opportunity cost of funds = 8%

Project A: Investment costs = 1,000 in year 0

Benefits = 3,200 in year 5

Project B: Investment costs = 1,000 in year 0

Benefits = 5,200 in year 10

0

5

NPV :

-

1,000 + 3,200/(1.08)

= 1,177.86

A

10

0

NPV :

-

1,000 + 5,200/(1.08)

= 1,408.60

B

0

0

Hence, NPV > NPV

B

A

IRR

:

-

1,000 + 3,200/(1+K

)

= 0 which implies that K

= 0.262

5

A

A

A

IRR

:

-

1,000 + 5,200/(1+K

)

= 0 which implies that K

= 0.179

10

B

B

B

Hence, K

>K

A

B

2.4.3 Difficulties With The Internal Rate of Return Criterion (Cont’d)

Fourth difficulty:

Same project but started at different times

Project A: Investment costs = 1,000 in year 0

Benefits = 1,500 in year 1

Project B: Investment costs = 1,000 in year 5

Benefits = 1,600 in year 6

NPV

:

-

1,000 + 1,500/(1.08) = 388.88

A

6

5

NPV

:

-

1,000/(1.08)

+ 1,600/(1.08)

= 327.68

B

0

0

Hence, NPV > NPV

A

B

IRR

:

-

1,000 + 1,500/(1+K

) = 0 which implies that K

= 0.5

A

A

A

6

IRR

:

-

1,000/(1+K

)

+ 1,600/(1+K

)

= 0 which implies that K

= 0.6

5

B

B

B

B

Hence, K

>K

B

A

2.4.4 Difficulties With The Internal Rate of Return Criterion (Cont’d)