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Chapter6DiscountingBenefitsandCosts.pdf

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Chapter

Cost-Benefit Analysis Concepts and Practice

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition Boardman • Greenberg • Vining • Weimer

FOURTH EDITION

Discounting Benefits and Costs in Future Time Periods

Six

DISCOUNTING BENEFITS AND COSTS IN FUTURE TIME PERIODS

CHAPTER 6

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Purpose: This chapter deals with the practical issues one must know in order to compute the net present value of a project.

• It assumes the social discount rate is given, which is reasonable as the rate is often set by an oversight agency, such as the Office of Management and Budget.

• Appendix 6A provides shortcut formulas for calculating the present value of annuities and perpetuities.

DISCOUNTING BENEFITS AND COSTS IN FUTURE TIME PERIODS

THE BASICS OF DISCOUNTING Projects with Lives of One Year

Technically, discounting takes place over periods not years. However, for expositional simplicity, we assume that each period is a year. This section discusses projects that last one year.

Example: A city government has the opportunity to buy a parcel of land for $10 million . Also, suppose that if it buys the land, then the land will be sold for $11 million one year from now.

Should the city buy the land now?

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THE BASICS OF DISCOUNTING

Projects with Lives of One Year

• It is often useful to lay out the annual benefits and costs of a project on a time line (Figure 6-1).

• The horizontal axis represents time measured in years. Benefits appear above the time line and costs are below it.

• A time line is particularly useful when the timing of impacts is more complicated.

Time line

Figure 6-1 A Time Line Diagram for City Land Purchase Example

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THE BASICS OF DISCOUNTING Projects with Lives of One Year

To decide whether to buy the land, the city should compare the land purchase project, which has a cost of $10 million now with a value of $11 million in one year to the best alternative – in this case, the status quo (not buy the land and invest the money).

There are three possible methods to evaluate potential projects. Each gives the same answer.

(1) Future Value Analysis

(2) Present Value Analysis

(3) Net Present Value Analysis

THE BASICS OF DISCOUNTING Projects with Lives of One Year

(1) Future Value Analysis – Choose the project with the largest future value, FV, where the future value in one year of an amount X invested at interest rate i is:

FV = X (1 + i) (6.1)

Example: If the city does not buy the land and invest the money at an interest rate of 5%, then it will have $10.5 million in one year (the future value, FV).

If it buys the land, the future value is $11 million. In this case, the city should buy the land.

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THE BASICS OF DISCOUNTING

Projects with Lives of One Year

(2) Present Value Analysis – Choose the project with the largest present value, PV, where the present value of an amount Y received in one year is:

PV = Y/(1 + i) (6.2)

Note that if the PV of a project equals X, and the FV of a project equals Y, both equations (6.1) and (6.2) imply:

This equation shows that discounting (the process of calculating the present value of future amounts) is the opposite of compounding (the process of calculating future values).

i) + (1

FV = PV

THE BASICS OF DISCOUNTING

Projects with Lives of One Year

(2) Present Value Analysis

Example: The present value of the land that will be worth $11 million in one year is PV = $11,000,000/1.05 = $10,476,190

In contrast the present value of the best available alternative is $10 million. Therefore, the city is better off in present value terms if it buys the land.

i) + (1

FV = PV

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(3) Net Present Value Analysis – Choose the project with the largest net present value, which calculates the sum of the present values of all the benefits and costs of a project (including the initial investment):

NPV = PV(benefits) – PV(costs) (6.3)

Example: NPV = $10,476,190 - $10,000,000 = $476,190

Figure 6-2 shows that as the NPV of buying the land is positive, the city should buy the land.

THE BASICS OF DISCOUNTING Projects with Lives of One Year

Figure 6-2 NPV of Buying the Land

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• Usually projects are evaluated relative to the status quo. If there is only one new potential project and its impacts are calculated relative to the status quo, it should be selected if its NPV > 0, and should not be selected if its NPV < 0.

• If the impacts of multiple, mutually exclusive alternative projects are calculated relative to the status quo, one should choose the project with the highest NPV, as long as this project’s NPV > 0. If the NPV < 0 for all alternative projects to the status quo, one should maintain the status quo.

THE BASICS OF DISCOUNTING

Projects with Lives of One Year

COMPOUNDING AND DISCOUNTING OVER MULTIPLE YEARS

(1) Future Value over Multiple Years

(2) Present Value over Multiple Years

(3) Net Present Value of a Project

We now generalize these results to apply to projects with impacts that occur over many years.

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COMPOUNDING AND DISCOUNTING OVER MULTIPLE YEARS

(1)Future Value over Multiple Years – Interest is compounded when an amount is invested for a number of years and the interest earned each period is reinvested.

• Interest on reinvested interest is called compound interest. The future value, FV, of an amount X invested for n years with interest compounded annually at rate i is:

FV= X (1+i)n (6.4)

• The term (1+i)n is called the compound interest factor.

• If there is simple interest then interest iX is paid each year. Unless stated to the contrary we will assume interest is compounded once per year.

COMPOUNDING AND DISCOUNTING OVER MULTIPLE YEARS

(1) Future Value over Multiple Years

Example: Suppose the city could invest the $10 million for 5 years with 5% interest per annum.

Under simple interest, each year interest is paid only on the original principal amount, and the city would receive $500,000 per year. The future value would be $12.5 million.

Under compounded interest, interest is earned on the principal amount and on the interest that has been reinvested.

If $10 million is invested for 5 years with interest compounded annually at 5%, then the future value is:

FV = $10 million (1 + 0.05)5 = $12.763 million

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COMPOUNDING AND DISCOUNTING OVER MULTIPLE YEARS

(2) Present Value over Multiple Years – The present value, PV, of an amount Y received in n years, with interest compounded annually at rate i is:

(6.5)

The term 1/(1+i)n is called the present value factor or discount factor.

The present value of a stream of benefits or costs over n years is:

(6.6)

(6.7)

COMPOUNDING AND DISCOUNTING OVER MULTIPLE YEARS

(2) Present Value over Multiple Years

Example using Equation 6.6: Consider a government agency that has to choose between two alternative projects. Project I yields a benefit of $10,500 four years from now, whereas Project II yields $5,500 four years from now and an additional $5,400 five years from now. Assume that the interest rate is 8%.

Which is the better project?

The present values of the projects are:

PV(I) = $10,500/(1+0.08)4 = $7,718

PV(II) = $5,500/(1+0.08)4 + $5,400/(1+0.08)5 = $4,043 + $3,675 = $7,718

In this example, the present values of the two projects happen to be identical. Thus, one would be indifferent between them.

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COMPOUNDING AND DISCOUNTING OVER MULTIPLE YEARS

(3) Net Present Value of a Project – Inserting equations (6.6) and (6.7) into (6.3) gives the following useful expression for computing the NPV of a project:

Or, equivalently, the NPV of a project equals the present value of the net benefits (NBi = Bi - Ci):

(6.9)

(6.8)

COMPOUNDING AND DISCOUNTING OVER MULTIPLE YEARS

(3) Net Present Value of a Project

Example: Suppose a district library is considering purchasing a new information system that would give users access to a number of online databases for 5 years. The benefits are estimated to be $100,000 per annum, including both cost savings to the library and user benefits.

The information system costs $325,000 to purchase and set up initially, and $20,000 to operate and maintain each year. After 5 years, the system would be dismantled and sold, resulting in a net cash inflow of $20,000 (terminal value or liquidation value). Assume that the discount rate is 7 % and there are no other costs or benefits.

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COMPOUNDING AND DISCOUNTING OVER MULTIPLE YEARS

(3) Net Present Value of a Project

Table 6-2 contains the annual benefits, annual costs, and annual net benefits of the library information system project.

Using Equation 6.9, the present value of the net benefits of the project is $17,276, as shown in the last column of Table 6- 2.

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TIMING OF BENEFITS AND COSTS

• Thus far, we have assumed that impacts occur immediately, or at the end of the first year, or at the end of the second year, and so on.

• Time lines are very useful ways to specify exactly when benefits and costs do occur.

• If benefits arise throughout a year, rather than at the end as we have assumed , one possibility is to compute the NPV as if the benefits occurred in the middle of the year.

• Alternatively, one could compute the NPV under the assumption they occur at the beginning of the year and under the assumption that they occur at the end of the year and take the average.

COMPARING PROJECTS WITH DIFFERENT TIME FRAMES

Analysts should not choose one project over another solely based on the NPV of each project if the time spans are different. Such projects are not directly comparable.

Two appropriate methods to evaluate projects with different life spans are:

(1)Roll-Over Method

(2) Equivalent Annual Net Benefits (EANB) Method

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COMPARING PROJECTS WITH DIFFERENT TIME FRAMES

(1) Roll-Over Method

If project A spans n times the number of years as project B, then assume that project B is repeated n times and compare the NPV of n repeated project Bs to the NPV of (one) project A.

For example, if project A lasts 30 years and project B lasts 15 years, compare the NPV of project A to the NPV of 2 back-to-back project B’s, where the latter is computed:

NPV = x + x/(1+i)15

where, x = NPV of one 15-year project B.

COMPARING PROJECTS WITH DIFFERENT TIME FRAMES

(1) Roll-Over Method

Example: Suppose the utility decides to build the cogeneration power plant. Further suppose that in 15 years time it builds another new cogeneration plant; in 30 years it builds another one; and it builds another again in 45 and 60 years.

The NPV of back-to-back cogeneration power plants denoted 5CGP, is:

NPV(5CGP) = $24,000,000 +$24,000,000/(1+0.08)15

+$24,000,000/(1+0.08)30 +$24,000,000/(1+0.08)45

+$24,000,000/(1+0.08)60

= $34.94 million9

The utility should select the option with the higher NPV.

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(2) Equivalent Annual Net Benefits (EANB) Method

The EANB is the amount received each year for the life of the project that has the same NPV as the project itself.

It is computed by dividing the NPV of the project by the annuity factor ai

EANB= NPV / ai n (6.10) where

Where, ai n is the present value of an annuity of $1 for the life of the

project (n years), where i = interest rate used to compute the NPV.

Obviously, one would choose the project with the highest EANB.

COMPARING PROJECTS WITH DIFFERENT TIME FRAMES

(2) Equivalent Annual Net Benefits (EANB) Method

Example: EANB(HED) = $30/12.461 = $2.407 million EANB(CGP) = $24/8.559 = $2.804 million

EANB of the cogeneration project implies that this project is equivalent to an annuity of $2.804 million per year for 15 years.

In contrast, the net benefit of the hydroelectric alternative is equivalent to an annuity of $2.407 million per year for 75 years.

Consequently, the cogeneration alternative is preferable, assuming replacement of both types of plant is possible at the end of their useful lives.

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INFLATION AND REAL VERSUS NOMINAL DOLLARS

Conventional private sector financial analysis measures monetary amounts in nominal dollars (sometimes called current dollars). But, due to inflation, one cannot buy as many goods and services with a dollar today as one could one, two or more years previously–“a dollar’s not worth a dollar anymore”. It is important to control for inflation (i.e. general price increases).

We control for inflation by converting nominal dollars to real dollars (sometimes called constant dollars). We usually use the consumer price index (CPI) deflator, but sometimes use the gross national product (GNP) deflator.

INFLATION AND REAL VERSUS NOMINAL DOLLARS

Problems with the CPI

The CPI is the most commonly used measure of inflation. Most economists think that the CPI overstates inflation.

Two reasons for the overstatement are:

1) Commodity substitution effect: The CPI basket of goods does not accurately reflect consumers’ purchases because people quickly switch to lower-priced substitutes as prices rise. One variant is called the discount stores effect. Another variant is called the “new goods” problem: consumers switch to new, cheaper generic drugs.

2) Quality improvements: The CPI does not accurately reflect improvements in product quality to existing goods, e.g. cars are more safe or reliable.

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Analyzing Future Benefits and Costs in CBA

Analysts should either measure the benefits and costs in real dollars and discount using a real discount rate or measure the benefits and costs in nominal dollars and discount using a nominal discount rate. Both methods would result in the same numerical answer.

It is suggested to work in real dollars for public-sector project evaluation as it is usually easier and more intuitive.

APPENDIX 6A: SHORTCUT METHODS FOR CALCULATING THE PRESENT VALUE OF ANNUITIES

AND PERPETUITIES

• Annuity - is an equal, fixed amount received (or paid) each year for a number of years.

• A perpetuity - is an annuity that continues indefinitely.

Many CBAs contain annuities or perpetuities. Fortunately, there are some simple formulas for calculating their PVs.

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APPENDIX 6A: SHORTCUT METHODS FOR CALCULATING PV OF ANNUITIES & PERPETUITIES

Example: Suppose that in order to finance a new state highway, a state government issues $100 million worth of 30-year bonds with an interest rate of 7% paid annually.

The annual interest payments of $70,000 are an annuity. If at the end of each 30-year period the state government refinances the debt by issuing another 30-year bond that also has an interest rate of 7%, then the annual interest payments of $70,000 would continue indefinitely, which is a perpetuity.

Present Value of an Annuity Using equation (6.6), the present value of an annuity of $A per annum (with payments received at the end of each year) for n years with interest at i percent is given by:

This is the sum of n terms of a geometric series with the common ratio equal to 1/(1 + i). Consequently,

PV = A x (6A.1)

where , (6A.2)

The term , which equals the present value of an annuity of $1 per year for n years when the interest rate is i percent, is called an annuity factor.

n ia

APPENDIX 6A

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Present Value of an Annuity

APPENDIX 6A

Example: The library information system problem contains two annuities: the annual benefits of $100,000 per year for 5 years , which we refer to as annuity A1, and annual costs of $20,000 per year for five years, which we refer to as annuity A2.

From Figure 6-4 we see that the present value of A1 is $410,020 and the present value of A2 is $82,004. But the easier way is to use Equation 6A1.

The present value of annuity A1 is:

PV(A1) = $100,000 x 1-(1+0.07)-5/0.07 = $100,000 x 4.1002 = $410,020

and

PV(A2) = $20,000 x 4.1002 = $82,004

Present Value of a Perpetuity

Taking the limit of equation (6A.2) as n goes to infinity implies that the present value of an amount, denoted by A, received (at the end of) each year in perpetuity is given by:

if i > 0 (6A.3)

APPENDIX 6A: SHORTCUT METHODS

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APPENDIX 6A: SHORTCUT METHODS

Present Value of a Perpetuity

Example: Suppose that a municipality has an endowment of $10 million. If interest rates are 6%, then this endowment will provide an annual interest payments of $600,000 indefinitely.

More generally, if the municipality has an endowment of X and if the interest rate is i, then the perpetual annual income from the endowment, denoted by A, is given by A = iX. Rearranging this equation shows the present value of the perpetual annuity is given by X = A/i, which is equation 6.A3.

The present value of a perpetuity of $150,000 per year when interest rates are 8% is

PV = $150,000/0.08 = $1,875,000