NBSIR 83-2657

PGfrH HH* 7S~

A

October 1983

Recommended Practice

for Measuring Net Benefits

and Internal Rates of Return

for Investments in Buildings

and Building Systems

NATIOHAI. BUBIAq

OP STANDARDS UBRART ^-6

NBSIR 83-2657 Quoo

RECOMMENDED PRACTICE

FOR MEASURING NET BENEFITS

AND INTERNAL RATES OF RETURN

no.

1423

FOR INVESTMENTS IN BUILDINGS

AND BUILDING SYSTEMS

Harold E. Marshall

October 1983

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

ABSTRACT

This report describes how to calculate net benefits (savings) and internal

rates of return and how to use them in selecting building designs and building

systems that will be cost effective over time. Net benefits are the

difference between positive benefits or cost reductions (savings) and project

costs, measured in present value or annual value dollars. The net benefits

method is used to decide if a given project is cost effective and which size

or design for a given purpose is most cost effective when no budget constraint

exists. The internal rate of return on an investment is the compound rate of

interest that, when used to discount the streams of dollar benefits (savings)

and dollar costs over a defined study period, will make the two equal. The

internal rate of return is used to determine if a given project is cost

effective, to compare the relative cost effectiveness of different purpose

projects competing for a limited budget, and, when calculated on incremental

changes in benefits and costs, to evaluate which size or design for a given

purpose is most cost effective. The report describes formulas for calculating

net benefits and the internal rate of return, their applications in selecting

cost-effective projects, and limitations in their use. This recommended

practice for measuring net benefits and internal rates of return will assist

the private and public building communities in making cost-effective decisions

in the design, operation, maintenance, and retrofit of buildings.

iii
PREFACE

Decreasing productivity in the construction industry coupled with high costs

of construction loans, labor, material, and energy have prompted builders,

architects, engineers, building owners and operators, and code writers to look

more closely at economic evaluation methods to identify building designs

and systems that are most cost effective. Practical, standardized, economic

methods for evaluating building alternatives are needed. The net benefits

(NB) method and the internal rate of return (IRR) method are economic

evaluation tools that can be used to assess the cost effectiveness of a given

project and to compare the cost effectiveness of alternative projects. The NB

method (also called the net present value method) computes the difference of

discounted benefits (or savings) and discounted costs as a measure of the cost

effectiveness of a project. The IRR method computes the compound rate of

interest which equates the stream of dollar benefits or savings to dollar

costs over a defined study period. The NB method is used to decide if a given

project is cost effective (net benefits greater than zero) and which size

or design competing for a given purpose is most cost effective where no budget

constraint exists. The IRR is used to determine if a given project is cost

effective, to compare the relative cost effectiveness of different purpose

projects competing for a limited budget, and, when calculated on incremental

changes in benefits and costs, to evaluate which size or design for a given

purpose is most cost effective.
This report describes how to calculate NB and illustrates use of the NB method

in measuring the cost effectiveness of building-related decisions. The

calculation procedures for the IRR are also described and illustrated. The NB

and IRR methods are compared, and guidelines for their use are given. Applying

standardized methods will help assure that the cost effectiveness of

alternative building designs and projects can be determined and compared in a

consistent and technically correct manner.

This report was prepared by the National Bureau of Standards (NBS) in support

of an ongoing standards development activity by the American Society of

Testing and Materials (ASTM E.06, Performance of Building Constructions) and

in response to requests from the building community for assistance in applying

economic analysis in a uniform and practicable manner. The report has been

submitted to ASTM E06.81, the Building Economics Subcommittee, to be used as

the technical base for the development of an ASTM standard practice for

measuring NB and IRR's for building investments. It is the third in a

series of reports to be submitted by NBS to ASTM E06.81, and it builds in part

upon both the first report (NBSIR 80-2040, on life-cycle costing) and the

second report (NBSIR 81-2397, on benefit-to-cost and savings- to-investraent

ratios). 1 The series of NBS reports and ASTM recommended practices are aimed

at producing a comprehensive set of measures of economic performance that will

meet the needs of the private and public building sectors.

^Rosalie T. Ruegg, Stephen R. Petersen, and Harold E. Marshall, Recommended

Practice for Measuring Life-Cycle Costs of Buildings and Building Systems , National Bureau of Standards Interagency Report 80-2040, June 1980; and Harold

E. Marshall and Rosalie T. Ruegg, Recommended Practice for Measuring

Benefit/Cost and Savings-to-Investment Ratios for Buildings and Building

Systems , National Bureau of Standards Interagency Report 81-2397, November

1981.

v
Thanks are due the members of ASTM who have participated in the Building

Economics Subcommittee meetings and thereby have helped determine the

framework of this paper. Special appreciation is extended to Robert E.

Chapman, James G. Gross, Barbara C. Lippiatt, Stephen R. Petersen, Rosalie T.

Ruegg, and Stephen F. Weber for their helpful comments on the paper.
TABLE OF CONTENTS

Page

ABSTRACT iii

PREFACE iv

LIST OF FIGURES viii

LIST OF TABLES viii

1. INTRODUCTION 1

1.1 Scope and Organization 2

1.2 Procedures for NB and IRR Evaluations 3

2. OBJECTIVES, ALTERNATIVES, AND CONSTRAINTS 4

3. ASSUMPTIONS 6

4. DATA 7

5. CONVERSION OF CASH AMOUNTS TO A COMMON TIME BASIS 10

6. COMPUTE NB 13

7. COMPUTE IRR AND ADJUSTED IRR 16

7.1 IRR 16

7.1.1 Trial and Error 17

7.1.2 Graph 20

7.1.3 Target UPV Factor 23

7.2 Adjusted IRR..... 24

7.3 Other Return-on-Investment Measures 29

8. RECOMMENDED APPLICATIONS AND LIMITATIONS 31

8.1 IRR or AIRR Measure 31

8.2 NB or IRR Method 35

APPENDIX A Discount Factor Tables A-l

APPENDIX B Selected Bibliography B-l

vii
LIST OF FIGURES

Page

7.1 Graphical Solution of IRR 21

8.1 Level of Energy Conservation That Maximizes the PVNB 39

8.2 Network Diagram Using Incremental AIRR Analysis for Project

Sizing 40

8.3 Graphical Illustration of Inconsistent Project Rankings by

NB and IRR Methods 45

LIST OF TABLES

4.1 Benefits and Costs 8

5.1 Discount Formulas 11

6.1 Calculation of Net Benefits 15

7.1 Trial-and-Error Solution for Internal Rate of Return 19

7.2 Data for Graphical Solution of IRR 22

7.3 Investment Project With Different Reinvestment Rates 28

7.4 Measures of Percentage Return on Investments 30

8.1 Economic Questions and Recommended Methods 36

8.2 AIRR's for Project Size Changes 39

8.3 Project Data for Comparing Projects by NB and IRR Methods. 44

8.4 Optimal Ranking Using AIRR as a Guideline 48

8.5 NB Versus IRR Method for Ranking Projects 51

viii
1 . INTRODUCTION

The net benefits (NB) method and internal rate-of-return (IRR) method are part

of a family of economic evaluation methods that provide measures of economic

performance of an investment over some period of time. Included in this

family of evaluation methods are life-cycle cost analysis, benefit-to-cost and

savings-to-investment ratios, and payback analysis.

This report is the third in a series of National Bureau of Standards (NBS)

reports on recommended practices for applying economic evaluation methods to

building decisions. The first report in the series was Recommended Practice

for Measuring Life-Cycle Costs of Buildings and Building Systems , NBSIR 80-

2040. The second was Recommended Practice for Measuring Benefit/Cost and

Savings-to-Investment Ratios for Buildings and Building Systems , NBSIR 81-

2397. A future publication on payback methods is planned to complete the

basic package of recommended practices.

The NB method (sometimes called the net present value method) calculates the

difference between discounted benefits (or savings) and discounted costs as a

measure of the cost effectiveness of a project. The NB method is used to

decide if a project is cost effective (net benefits greater than zero) or

which size or design competing for a given purpose is most cost effective

(the one with the greatest net benefits).

1
The IRR provides the compound raL^ of interest which equates the stream of

dollar benefits or savings to dollar costs over some defined study period.

It is used to determine if a given project is cost effective, to compare the

relative cost effectiveness of different purpose projects competing for a

limited budget, and, when calculated on incremental changes in benefits and

costs, to evaluate which size or design for a given purpose is most cost

effective.

1 . 1 Scope and Organization

This report establishes a technical base for the development of a recommended

practice for calculating and interpreting the NB and IRR's for building

designs and systems. ^ Section 2 discusses the formulation of objectives,

alternatives, and constraints for NB and IRR analyses. Examples are given of

problems that might be evaluated with each of the economic methods. Section 3

lists assumptions that must be made to perform the economic calculations and

cites sources of information on how to make those assumptions. Section 4

identifies data needs under benefit and cost categories typically used in

calculating NB or the IRR. Section 5 presents an overview of "discounting"

and "compounding" for converting cash flows spread over time to their

equivalent values at a common time. Discounting formulas are also provided.

Section 6 describes the calculation procedures for measuring NB.

iThe study period (sometimes called time horizon) is the length of time over which an investment is analyzed.

^Terminology in this report is consistent with American Society for Testing

and Materials, "Standard Definitions of Terms Relating to Building Economics,

E833-83a," Annual Book of ASTM Standards (Philadelphia, Pennsylvania: ASTM,

1983).

2
Section 7 describes alternative calculation procedures for computing IRR and

adjusted IRR measures. Trial-and-error, graphical, direct solution, and

discount-factor-table approaches to finding the rate of return are illustrated

with hypothetical investment problems. Section 8 describes the controversy

regarding which measure (IRR or adjusted IRR) and which method (NB or IRR) are

appropriate for evaluating typical investment problems facing the building

community. Recommendations of specific methods and measures are made for

selected problems. The economic consequences of applying other methods or

measures than those recommended are described. Limitations of the two methods

are also discussed. Appendix A, a selected set of discount factor tables, and

Appendix B, a bibliography, conclude the report.

1.2 Procedures for NB and IRR Evaluations

The recommended steps for applying the NB or IRR method to an investment

decision are summarized as follows:

o Identify Objectives, Alternatives, and Constraints

o Establish Assumptions

o Compile Data

o Convert Cash Flows to a Common Time Basis

o Compute NB or IRR and Compare Alternatives

Sections 2 through 7 treat these steps in detail.

3
2. OBJECTIVES, ALTERNATIVES, AND CONSTRAINTS

The problem to be addressed by the NB or IRR method should first be

clearly specified. The objectives of the investor or decision maker must be

made explicit. Alternative approaches for reaching the objective should be

identified. Any constraints to reaching the objective should also be

identified.

An example of a building investment problem that could be evaluated with the

NB method is the sizing of a manufacturing plant in a given location.

Constraints of building site and size of market would limit the choice to a

range of size alternatives. The objective of the investor is to select that

plant size which maximizes profits (i.e. , net benefits) while at the

same time satisfying all of the constraints. The NB method can be used to

guide the investor to the most efficient size.

An example of a building investment problem that could be evaluated with the

IRR is selecting among alternative energy conservation retrofit projects for a

corporation or government agency with many buildings and a limited budget.

The objective is to minimize the costs of owning and operating the buildings

(i.e., maximize net benefits from the retrofits). Feasible technical

alternatives might be roof and wall insulation, improvements in the efficiency

of heating and cooling equipment, passive or active solar systems, and

improved controls. Constraints might include a given level of thermal comfort

and a minimum hot water temperature requirement in addition to a maximum first

cost budget for the overall program of conservation retrofits. The IRR method

4
provides a ranking tool that can be used to select the set of projects which

maximizes net benefits given the constraints.

1

Note that other economic evaluation methods are also appropriate for solving

the problems cited above and that the NB and IRR methods are also appropriate

for other types of building investment problems. For examples of problems and

their solutions by other methods, see the first two reports in this series.

2

Other applications and limitations of the NB and IRR methods will be discussed

in section 8.

sequential approach to project economic analysis is assumed here. That is,

the appropriate size or scale of each of the technical alternatives is first

determined, and then the individual projects are chosen in descending order

of their IRR's until the budget is exhausted.

^See Rosalie T. Ruegg, Stephen R. Petersen, and Harold E. Marshall,

Recommended Practice for Measuring Life-Cycle Costs , and Harold E. Marshall

and Rosalie T. Ruegg, Recommended Practice for Measuring Benefit/Cost and

Savings-to-Investment Ratios .

5
3. ASSUMPTIONS

Assumptions about the values of parameters generally are required to perform

economic evaluations of design and investment problems. Because the economic

measure of a project's worth varies considerably depending on the assumptions,

it is important to use care in making them and to consider the effects of different parameter values. Sensitivity analysis can be used to identify

critical parameters and to test the outcome for a range of values in any given

problem.

Policy dictates parameter values in some economic evaluations. In others,

parameter values are assumed on the basis of judgment or market analyses.

Important values that must sometimes be assumed in applying the NB method are

the discount rate, length of study period, price escalation rates, and

salvage/resale values. ^ For the IRR method, these same values are

significant, with one exception. A minimum acceptable rate of return (MARR)

instead of a discount rate must be chosen. 2 If the IRR is larger than the

MARR, the investment is considered economically attractive.

Guidelines for selecting these critical parameters will not be discussed here

because they are provided elsewhere.

lln this report an asset's salvage value is defined as that value assigned for

tax computation purposes that is expected to remain at the end of the

depreciation period. Resale value is the monetary sum expected from the

disposal of an asset at the end of the study period.

^The MARR is the minimum economic return an investor will accept for making an investment.

3Rosalie T. Ruegg, Stephen R. Petersen, and Harold E. Marshall, Recommended

Practice for Measuring Life-Cycle Costs ; Harold E. Marshall and Rosalie T.

Ruegg, Simplified Energy Design Economics; Principles of Economics Applied to

Energy Conservation and Solar Energy Investments in Buildings , National Bureau

of Standards Report 544, January 1980; and Harold E. Marshall and Rosalie T.

Ruegg, Energy Conservation in Buildings: An Economics Guidebook for

Investment Decisions , National Bureau of Standards Handbook l32. Mav 1980.

6
4 . DATA

Data must be collected before NB or IRR calculations can be made for an

investment or design alternative. Data can be collected from published and

unpublished sources, they can be estimated, or they can be assumed. * Both

engineering data (e.g., heating loads and equipment efficiencies) and economic

data (e.g., tax rates, discount rates, MARR's, depreciation rates and

periods, and financing terms) will be needed.

Data requirements for evaluating the NB of replacing a conventional oil

furnace with an electric heat pump in a given climate provide an example of

the types of data that might be needed for conducting economic analyses. The

annual heating requirements of the building and the efficiencies of the two

heating systems are needed to calculate annual purchased energy requirements

with the two systems. The annual purchased energy requirements, the present

and future prices of electricity and oil, the discount rate, and the study

period are needed to calculate the costs of energy with the two systems.

Investment, maintenance, and replacement costs, as well as the resale value

are needed both for the oil system and for the heat pump to complete the data

requirements to find the NB from the heat pump substitution. Financing and

taxes should be considered if the investment is made by the private sector.

ISee in Rosalie T. Ruegg, Life-Cycle Cost Manual for the Federal Energy

Management Program , National Bureau of Standards Handbook 135, December 1980,

the chapter on "Data and Assumptions" (pp. 37-48), and some projections of

energy prices in appendix C. See Carol Chapman Rawie, Estimating Benefits and

Costs of Building Regulations: A Step-by-Step Guide , National Bureau of

Standards Report 81-2223, June 1981, the end of chapters 1-3, for listings of

sources of information on data relating to economic analyses of buildings.

7
Some of the benefits and costs that may be relevant when calculating NB and

IRR measures are listed in table 4.1. The time schedule of their occurrence

is also needed to convert them to a common time basis. These benefits and

costs may be available directly or only after computations are performed using

engineering and economic data, as would be required for determining the dollar

value of energy savings in the heat pump example given above. If private

sector investments subject to taxes are evaluated, then benefit and cost data

should be adjusted for taxes so that the NB

tax" effects.

Table 4.1 Benefits

and IRR methods describe "after

and Costs

Benefits (Savings) Costs

Revenues Investment

Resale/Salvage Planning

Tax Credits Design

Subsidies (e.g., grants) Engineering

Energy Cost Reductions Construction

Operating Cost Reductions Purchase

Replacement Cost Reductions Installation

Financing

Operation and Maintenance

Fuel

Repairs and Replacement

Property and Capital Gains Taxes

8
Risky and uncertain investments may require additional data and analysis to

evaluate their worth. One approach is to calculate expected values of

benefits and costs when probability data exist. However, since probability

data are often difficult to acquire, two common approaches to account for more

risk are to reduce the time period (study period) over which a project is

evaluated and to raise the discount rate or MARR.l Both of these common

approaches have shortcomings. Reducing the study period ignores net benefits

late in the project life and fails to account for risk early in the project

life. Raising the discount rate biases project choice against projects

yielding most of their benefits late in life and fails to account for risk

early in the project life.

iFor a description of how to apply these and other techniques, and of what

data are needed, see Lee G. Anderson and Russell F. Settle, Benefit-Cost

Analysis: A Practical Guide (Lexington, Massachusetts: Lexington Books,

1977), pp. 99-106.

9
5. CONVERSION OF CASH AMOUNTS TO A COMMON TIME BASIS

To make economic evaluations, all benefits and costs that accrue over the

study period must be converted to equivalent values on a common time basis,

usually to present values or annual values. Converting these cash amounts to

equivalent values (generally called discounting) is performed by applying

discount formulas, or corresponding discount factors, to benefit and cost data

associated with a given investment or design alternative. ^ Table 5.1 presents

the most commonly used discount formulas, how they are used, and their

algebraic form.^

In calculating NB, all cash amounts are converted to equivalent values at the

present, at some future or past date, or on an annual basis. Cash amounts are

converted typically to a present value (i.e., an equivalent value occurring

now) or an annual value (i.e., a time-equivalent value occurring in a uniform

amount each year over the study period). ^ The resulting discounted costs are

subtracted from discounted benefits to arrive at the NB measure in present

value or annual value terms. Some examples of the discounting procedure are

shown in section 6.

ICash amounts refer to both the annually recurring benefits and costs as well

as the non-annually recurring benefits and costs that make up the cash flow

associated with an investment.

2For an explanation of discounting and a complete set of discount factor

tables, see the ASTM Adjunct Discount Factor Tables , to be published with

"Recommended Practice for Measuring Life-Cycle Costs of Buildings and Building

Systems" (Philadelphia, Pennsylvania: ASTM, 1983).

3Cash amounts converted to equivalent values at some future time are called

future values. The process of calculating future values is often called com- pounding because present values or annual values are compounded by some rate

of interest to determine future values.

10
Table 5.1 Discount Formulas

Formula Name Illustration Use

Single Compound

Amount (SCA) Formula

Single Present Value

(SPV) Formula

Uniform Sinking Fund

(USF) Formula

To find F when

P is known

To find P when

F is known

To find A when

F is known

Uniform Capital Recovery

(UCR) Formula

Uniform Compound

Amount (UCA) Formula

Uniform Present (UPV) Value

Q- ED

EJ + 0 -

F?

To find A when

P is known

To find F when

A is known

To find P when

A is known

Algebraic Form

F - P • (1 + i) N

~a~t ET

p , F

(1 + i)F

A " F> 1 N

(1 + i) -l

A - P- l(l + i) “

(1 + i) n-l

F - A • (1 + i) N-l

i

P - A, (1 + i) w-i

i(l + i)*“

where:

P = a present sum of money,

F = a future sum of money.

i = an interest or discount rate for the study period being considered.

N = number of interest or discounting intervals in the study period.

A - an end-of-period payment (or receipt) in a uniform series of payments

(or receipts) over N periods at i interest or discount rate.

F? - a future value to be found; P?, a present value to be found; and A?, an annual value to be found.

Note that the USF, UCR, UCA, and UPV formulas yield undefined answers when

i=0. Alternative algebraic forms for this special case would be as follows:

USF formula, A = F/N; UCR formula, A = P/N; UCA formula, F = A • N; and UPV

formula, P = A • N.

^The terms by which the known values are multiplied in these equations are the

formulas for the factors calculated in Appendix A. Using acronyms to represent

the factor formulas, the discount formulas can also be written, in the order as listed above, as F = P • SCA, P = F • SPV, A = F • USF, A = P • UCR, F = A* UCA, and

P = A • UPV

11
Discounting is also a procedure used in calculating the IRR. For example, to

find the IRR with the trial-and-error method, trial interest rates are used to

discount benefit and cost cash flows to a base time until a rate is found

which will make the two streams equal. Thus the discounting process must be

repeated with trial rates until the IRR is found for which net benefits are

zero, or until interpolation can be used to find the IRR between two trial

rates for which NB are above and below zero. Examples of this and related

procedures are given in section 7.

12
6 COMPUTE NB

.

The NB method finds the difference between benefits and costs, where both are

discounted to present or annual value dollars. Equation 6.1 is the formula

for finding the present value of net benefits for a project with benefits and

costs accruing in the future. (The discounting operation is performed by

using the SPV formula in table 5.1.)

N

PVNB = £

t=l

Bt- ~

(i + i)M

- cO » ( 6 . 1 )

where PVNB = present value of net benefits,

Bt - the dollar value of benefits (including savings and resale

values) in time period t,

C t = the dollar value of costs in time period t,

B t - Ct = net cash flows in time period t,*

i = the discount rate,

t = a discounting period,

N = the total number of discounting periods in the study period, and

CQ = initial project costs as of the base time.

Note that the NB method assumes implicitly that all net cash flows over the

study period are reinvested at rate i.2

leash flows are assumed to accrue at the end of each period.

^An adjusted PVNB could be calculated where the net cash flows are compounded

to a terminal value at some reinvestment rate (r) different from the discount

rate and then are brought back to the present with the discount rate. The

formula would appear as:N

l (Bt - Ct ) (1 + r t )N-t

t=l

Adjusted PVNB — - CQ . (1 + i)N

Since, in a profit-maximizing context, reinvestment is likely made at the

highest return available (the rate represented by i) , r t will probably equal

i, and equation 6.1 applies. Thus the adjusted PVNB is not discussed

further.

13
Equation 6.2 (the UCR formula in table 5.1) can be used to convert the present

value of net benefits to annual value terms, where N is the number of years in

the study period.

i(l + i) N

AVNB = PVNB . , (6.2)

(1 + i) N-l

where AVNB = annual value of net benefits,

i( 1 + i) N

= the formula for the uniform capital recovery (UCR) factor.*

(1 + i) N-l

Equations 6.1 and 6.2 provide no particular computational difficulties.

Furthermore, for a given problem and data set, solutions in present value

and annual value terms will be time equivalent values (although different in

nominal dollar values) and will support the same investment or design

decisions.

A simple application of equation 6.1 is presented in table 6.1 for an initial

investment of $10,000 that yields an uneven yearly cash flow over four years. 2

*The UCR factor is the term by which a present value is multiplied to compute an annual value. Using precalculated multiplicative factors instead of the

whole discount formulas is often more convenient in discounting operations.

Appendix A shows how to calculate these factors and has tables of values for

the UCR and other factors. Only those tables whose values are needed for

problems described in the report are included in Appendix A.

2lt is assumed in all illustrations of discounting in this report that net

cash flows accrue annually at the end of the year.

14
Table 6.1 Calculation of Net Benefits

Year

(t)

(1)

Benefits

(B t

) (2)

Costs

(c

t

) (3)

Net

Cash Flow

(Bt - Ct

) (4)=(2)-(3)

SPV Factor3

for

i=15%

(5)

PVNB

(6)=(4) • (5)

0 0 $10,000 $-10,000 1.000 $-10,000

1 $4,000 3,000 +1,000 .8696 +870

2 11,500 4,500 +7,000 .7561 +5,293

3 10,000 4,000 +6,000 .6575 +3,945

4 8,000 5,000 +3,000 .5718 +1,715

TOTAL $33,500 $26,500 $+7,000 $1,823

aTo find the PVNB of the net cash flow for each discounting period, the single

present value (SPV) factor in column 5 is multiplied times the net cash flow

in column 4. The SPV factor, calculated as 1 , is found for each of

(1 + i)N

various i and N combinations in Table 15 in Appendix A.

Assuming a discount rate of 15%, the discounted cash flows yield a PVNB of

$1,823. The larger the PVNB for a given project, the more economically

attractive it will be, other things being equal. (Note that the sum of net

cash flows, $7,000, does not reflect the time value of money.)

To find the AVNB that is time equivalent to $1,823, equation 6.2 can be used.

The UCR factor for four years and 15% from table 15 in Appendix A is .3503.

Multiplying it times the PVNB of $1,823 yields an AVNB of $639.

15
7. COMPUTE IRR AND ADJUSTED IRR

Two measures of the rate of return on investment are discussed in detail in

this section. One is the IRR and the other is the adjusted IRR. The IRR

method is used throughout this report as a general term to include both the

unadjusted and adjusted rate of return measures. Several approaches are

described for calculating the two measures. The section concludes with a

brief listing of other rate-of-return measures used in making economic

evaluations of investments.

7.1 IRR

The IRR is the compound rate of interest that, when used to discount a

project's cash flows, will reduce the PVNB to zero. The solution value

of i in equation 7.1 is the IRR.l

where Bt = the dollar value of benefits (including savings and resale values)

in time t,

Ct = the dollar value of costs in time t,

CQ = initial project costs as of the beginning of the base time, and

i = the rate of interest which discounts net cash flows to zero.

An implicit assumption in solving for the IRR is that net cash flows are

reinvested over the remainder of the study period at the derived IRR.

Allowing for different rates of reinvestment requires an adjustment to

equation 7.1 that is presented in section 7.2.

iNote that the IRR will be equal to the user's discount rate as used in

equation 6.1 only when the PVNB is zero at that discount rate.

16
Direct solution of i is not possible with equation 7.1, so calculation methods

have been developed to approximate the IRR. Three commonly used methods to be

examined in this section are the trial-and-error, graphical, and target UPV

factor approaches.

7.1.1 Trial and Error

One technique for calculating the IRR is the trial-and-error approach. A trial

rate of return is chosen that would be expected to approximately balance benefits and costs over the project study period. Then present value

calculations are made for that discount rate. If the PVNB is zero, then the

trial rate is the internal rate of return. If the PVNB is negative, the trial

rate is too high, and a second, lower trial rate should be used. If the PVNB

is positive for the original trial rate, then that rate is lower than the IRR,

and a second, higher trial rate should be used. When two trial rates are

found such that one yields a PVNB greater than zero and the other a PVNB less

than zero, the IRR will lie between those rates and can be approximated by

linear interpolation using the PVNB figures.

Considerable time can be saved in the trial-and-error approach if the first

trial rate is close to the true rate. One approach is to start with the MARR.

If the PVNB is negative with the MARR, then the project will not be

economically feasible, and no further calculations are necessary. If the PVNB

is positive, then higher trial rates are selected in an attempt to bound the

true rate.

If by inspection of project cash flows it is clear that the IRR will exceed

the MARR, then other approaches may be more appropriate for finding the

17
first trial rate. For example, when cash flows are relatively uniform

throughout the study period, the UPV factor tables can be used to find the

first trial rate. The first step is to add the undiscounted net cash flows,

divide the sum by the number of years in the study period, and then divide the

initial project cost by this average annual net cash flow to obtain a rough

estimate of the number of years the project would take to pay off the

investment. This crude approximation of "simple payback" is also an

approximation of the UPV factor, P/A, as shown in footnote b of table 5.1.

The second step is to find in the UPV factor tables of Appendix A the UPV

factor value at that project's study period which is closest to the calculated

payback figure. The corresponding discount rate then becomes the first trial

rate. The more uniform are annual net cash flows, the better this approach

generally works in finding a first trial rate close to the true rate.l

A third approach is to calculate simple payback directly by adding annual net

cash flows until they just cover initial costs. The first trial rate will be

the discount factor that corresponds to the UPV factor closest to that payback

value.

Table 7.1 illustrates the trial-and-error approach for calculating the IRR for

the same investment problem that was described in table 6.1. Columns 2 and 3

iSection 7.3 describes how the IRR can be approximated directly through

interpolation of UPV factors when annual net cash flows are constant.

2For extended discussions of how to select trial rates of return that bracket

the true rate, see Eugene L. Grant and W. Grant Ireson, Principles of

Engineering Economy , 5th ed. (New York: The Ronald Press Co., 1970), pp.

122-123; and Maurice G. Wright, Discounted Cash Flow (New York: McGraw Hill,

1967), pp. 29-32. For a numerical procedure that converges on the IRR within

.001 percent in two or three trials, see Dennis Eisen, "Computer Connection:

Program Description of a Fast, Efficient, Algorithm for the IRR," The

Appraisal Journal (January 1982), pp. 127-134.

18
Table 7.1 Trial--and-Error Solution for Internal Rate of Return

Year

(t)

Benefits

(B

t

) Costs

(c

t

>

Net

Cash Flow

(Bv - C ) t t

SPV

Factor

for i=25%

PVNB

at 25%

SPV

Factor

for i=22%

PVNB

at 22%

(1) (2) (3) (4)=(2)-(3) (5) (6)= (4)x(5) (7) (8)=(4)x(7)

0 0 $10,000 $-10,000 1.000 $-10,000 1.000 $-10,000

1 $4,000 3,000 +1,000 0.8000 +800 0.8197 +820

2 11,500 4,500 +7,000 0.6400 +4 , 480 0.6719 +4,703

3 10,000 4,000 +6,000 0.5126 +3,076 0.5507 +3,304

4 8,000 5,000 +3,000 0.4096 +1,229 0.4514 +1,354

Total $+7,000 $-415 $+181

list the dollar values of benefits and costs that accrue in eat h of the four

years, and column 4 shows the net cash flow for each of those years, including

the initial investment. By inspection of column 4, the analyst might expect a relatively high return given the net cash flow beyond the $10,000 investment

of $7,000 over four years. Using the second or third approaches described

above for selecting a trial rate, the calculated UPV value for four years

would correspond in Appendix A most closely to a discount rate of 25%.

Multiplying yearly net cash flows by the corresponding single present value

(SPV) factor for a 25% discount rate (column 5) converts them to equivalent

present values (column 6). Summing the values in column 6 produces a PVNB of

$-415.

Since the PVNB is less than zero, the IRR must be lower than the 25% trial

rate. Thus another, lower rate is chosen, and the corresponding SPV factors

19
are multiplied times yearly net cash flows. Using a trial rate of 22% (as

illustrated in columns 7 and 8) yields a positive PVNB of $181. Therefore

the IRR must lie between 22% and 25%. Linear interpolation yields an IRR of

22.9% as follows:

$181 - $0

IRR = 22% + (25% - 22%) = 22.9% . $181 + $415

7.1.2 Graph

Another approach in approximating the IRR is to use a graphical technique. The

profile of the PVNB for a given investment is plotted for different discount

rates. The IRR will be that rate where the PVNB curve intersects with the

discount rate axis (i.e., where the PVNB is zero). Figure 7.1 illustrates

such a graph for the project described in table 7.2.

Given a discount rate of zero, the PVNB would be the arithmetic sum over time

of the net cash flows (i.e., $1,300). Thus the function intersects the

vertical axis at $1,300. Using each discount rate on the horizontal axis of

figure 7.1 as a trial discount rate, and substituting into equation 6.1, the

PVNB values are calculated as shown in table 7.2 and then plotted in figure

7 . 1

.

1

iFor the function shown in figure 7.1, only a few of the PVNB points about the

horizontal axis would be required to approximate reasonably well the IRR. For

other curvilinear functions, more points are needed.

20
PVNB

( $ 100)

An IRR of 27.2% is found by visual inspection of the intersection of the

profile of PVNB with the horizontal axis. Note that the graphical method can

be expected to give approximations closer to the true IRR than the linear

interpolation illustrated earlier with the trial-and-error approach, assuming

that enough PVNB points above and below the horizontal axis are plotted to

accurately form the curve. However, if two trial rates are chosen very close

to the true value, the linear interpolation approach may approximate the true

value closely enough without having to calculate the PVNB for many discount

rates. In this problem, for example, the trial-and-error approximation for

trial rates 27% and 28% is 27.2%, the same as the graphical solution. The

true value of the IRR is in fact 27.2%. 1 The trial-and-error approximation

for trial rates of 25% and 30% is 27.3%, being only .1% off.

J-The 27.2% value of the IRR can be confirmed by substituting 27.2% for i in

equation 7.1.

21
8M

4-1Ocoo

C/D xCX

COuopo

4-1

Cd u

cO Q

csj•

r>»

0) *H•O

CO H o CO NO o o ^3- CM o CO

co CM r^. oo >3- CM z * 1 > CM </> cx

</>

m o O oo nO co o <r O > o in >3- Cx • • • • CO o on 00 in oo o vO 00 m 00

CQ CM n- 00 *3 1 z •» cry > CM

cx 1 CO- CM CM O on r-H in

CO o nO ON

m > o in *3- cx • • • • CO •H

m o O

O co co

CM o o NO p-H PQ CM 00 ON in cry Z> CM

(X 1 </> o o NO o o o CM o o »—H > o 00 vO m

cx • • • • CO r—4

CD o o CO CM ON <t

<V CM o CO

m

4-1 CQ CM 00

O

m CM

CO 2 •* • cry Pi > CM t—i CX 1 4-1 cry co co *3* o o CO 00 o o co ON r*. CD > o oo NO m

•rH CX • • • • O CO —Ho o 00 CM o CO in NO

CQ CM 00 NO ^3- 2 A • cry > CM •H

CX 1 cry nO r-H in

m o on NO n-

*—< o nO in

m > o 00 NO

CX • • • • CO o On o O o O >3* m o

CQ CM ON CM rv r- 2 p> cry > CM pH

CX 1 <oi-H co o o on NO ^H

•—H o o CM in > o on oo r*. CX • • • • CO o CM r-H Mfr CQ o in NO NO z CM ON CO 00 ON > r * cry CX CM|

r-H

m -3- o 00 > O CM CO

CX o in

O NO

CO o ON ON 00 • • ^H &o 4-1 iH a o o o o o

Cx o o o o o

4J i CM o m o CO

0) -C

r * •* *> z CD 4-1 CM »”H H ^H r-H CO CQ 1 + 4" 4- 4* CJ </> cry

CO

O o o o 4-1 4J o

CO CJ CM o N_X •» o CM

cry

CD 4J

O o o o i* o o o 4-4 4J o m o

0) CQ *> •» * c rH rH

<D </> CQP

CO /*s

<U 4-1 o r-H CM CO

•O

CO Nm

co

•OC

CO o

roPo

iw

CD0/ rHX

COPouo

CO

4-1

a>acH

CD

<y0op

4-4 •o

0)

**

incoc

CO o

co

i-i o

4-4

CD

I-I oUc

CO

4-4 >

CXCO

0) x:HCO i

22
7.1.3 Target UPV Factor

A third technique for calculating the IRR is to use the UPV factor tables.

This approach works only when the annual net cash flows are constant. The

procedure is to compute the target Uniform Present Value (UPV) factor that

equates benefits and costs, and then to search the UPV tables for the values

closest on either side to the computed UPV target value to see what discount

rates those factors represent. * Linear interpolation using the UPV figures

yields an approximation of the IRR between the two discount rates. Note in

this approach that the IRR is interpolated directly from the UPV factor

values. In the trial-and-error method described in section 7.1.1, the IRR is

interpolated from PVNB figures. The UPV tables are used there only to find

the first trial rate for calculating the PVNB.

Equation 7.2 shows how to calculate the target UPV factor that embodies the

IRR (i.e., the UPV factor that makes the present value of net benefits equal

zero) .

iThe term CQ is derived from the UPV formula P = A • UPV in footnote b of

(BHS)

table 5.1 for the special case where the present value of net benefits is

, c0

equal to zero; that is, (B-C)(UPV) - C0

= 0, and therefore UPV =

, (B-C)

where P = CQ and A = (B-C).

iThe UPV factor in this case of constant annual cash flows is identical to

simple payback.

23
target UPV factor = (7.2)

(B-C)

where target UPV factor = uniform present value factor where the present

value of net benefits equals zero,

CQ

= initial project costs as of the base time, and

(B-C) = constant annual net cash flow.

This approach can be illustrated with the following problem. Find the IRR for

a project with a one-time initial cost of $1,000 and benefits that accrue in a

uniform stream of $200 per year for 16 years. Dividing the one-time initial

cost of $1,000 by the constant annual net cash flow of $200 yields a target

UPV factor of 5. Searching the rows where N = 16 in the UPV factor tables of

Appendix A, two UPV factors are found that most closely bracket the value 5.

One factor is 5.162 for a discount rate of 18%, and the other is 4.938 for a

discount rate of 19%. Having bracketed the IRR between 18% and 19%, a single

value of 18.7% can be approximated through linear interpolation as follows:

5.162-

5.000

IRR = 18% + (19% - 18%) = 18.7%. 1

5.162-

4.938

7 .2 Adjusted IRR

The approaches to approximating the IRR described above assume that returns or

benefits from the investment are reinvested at a rate of return equal to the

IRR. If the net cash generated by the investment is not reinvested at a rate

equal to the IRR, equation 7.1 will not give the true IRR. An adjusted IRR

^Although the relationship between the table values is not linear, a solution

based on a linear relationship will give a reasonable approximation. The

closer the table values are to the true value, the closer will be the

interpolated IRR to the true IRR.

24
(AIRR) is needed to find the true return. The solution value of i in

equation 7.3 is the AIRR.

N N-t

I (Bf- ~ Ct ) (1 + r t

) tzi -

" C

o = °>

(1 + i)^

(7.3)

where r t = the rate of return on reinvestment of cash flows realized in year t.

Net cash flows (B t-Ct

) are carried forward (compounded) at the appropriate

reinvestment rates (r t ) to equivalent values at the end of the study period (end

of the Nth period) and summed. The value of i that discounts the resulting

future value to a present value equal to CQ is the AIRR. The higher the assumed

reinvestment rate, the higher will be the calculated value of the AIRR.

Equation 7.3 is the same as equation 7.1 if the assumed reinvestment rate is the

same for each reinvestment opportunity and equal to the unadjusted IRR (i.e.,

when r^ = the value of i found in equation 7.1)1. Thus, given a constant

reinvestment rate, the AIRR is equal to the IRR when rt = i. If rt < i, then

the AIRR < IRR. If rt > i, on the other hand, then AIRR > IRR. If rt changes

from one period to the next, and if some values are larger and others smaller

than i, then the AIRR and IRR measures would have to be computed for each

project to determine which measure was larger.

lit can be proved that equations 7.1 and 7.3 are the same when rt = i by

showing that the term 0 + rt)H-t ln equation 7>3 becomes i as shown

(1 + i) w (1 + i)t

in equation 7.1 when r t = i. Substituting i for r t into (1 + r 1-) N~ t yields

(1 + i) N

(l+i)N-t = (1 + i)N (1 + i) -t , which upon simplification reduces to 1 (l+i) N

(1 + i) w

(1 + i)t

25
The i in equation 7.3 can be solved for directly if the equation is rearranged

as shown in equation 7.4.1

AIRR = -1.0 + (TV/C0 ) 1/n

, (7.4)

N

where TV = £ (Bf- “ Ct ) (1 + r)N~t

, which is the terminal (future) value

t=l

at the end of the study period of a net cash flow in time t.

Using equation 7.4, the TV must first be computed by applying the single

compound amount factor to each net cash amount and then summing them.

Substitution of known values for C G and N is then all that is required to solve

directly for the AIRR.

Once the terminal value is computed, an alternative approach to equation 7.4,

utilizing the discount factor tables, can be used if calculation aids are not

available for taking roots. Since the single compound amount (SCA) factor (as

shown in table 5.1) is multiplied times a present value to calculate a terminal

(future) value, we know that TV = C0• SCA when the terminal value of net cash

flows equals the future value of initial costs. (This is equivalent to the

condition that the PVNB equals zero.) Thus we know that the value of i that

equals the AIRR will be the rate that corresponds to the SCA factor equal to

TV/C0

. To find the AIRR, the SCA factor is computed as TV/C0. Then the SCA

N TV

Ilf we let TV = y (B

t - Ct ) (1 + r t )N-t equation 7.3, then — - CQ = 0.

t=l (1 + i) N

TV

Rearrangement of terms yields (1 + i)^ = —— , and therefore i = -1 + (TV/C0 )^/^

c0 = AIRR.

26
factor tables are searched for the values on either side that are closest to the

computed SCA factor. The AIRR is interpolated by using those SCA values and

their corresponding discount rates.

To illustrate the calculation of the AIRR when different reinvestment rates

apply, equation 7.4 will be used. In section 7.1.2, an approximate IRR of 27.2%

was calculated for the investment data in table 7.2. Now an AIRR will be

computed for the modified case where reinvestment rates are lower than 27%, and

where those reinvestment rates change during the project life.l

Table 7.3 illustrates the calculations of terminal values from a reinvestment

of the first year's net cash flow at 20% and a reinvestment of the second

year's cash flow at 15%. Note that net cash flows are identical to those for

the investment described in table 7.2.

^Reinvestment rates that change over the project's life are quite likely with

changing economic conditions. Although an analyst might prefer to use a single rate for an ex ante analysis due to the difficulty in predicting how

rates are going to change, an AIRR computed with changing rates seems appropriate when an analyst is making an ex post evaluation and knows the

different rates at which net cash flows were reinvested.

27
Table 7.3 Investment Project With Different Reinvestment Rates

Year

(t)

Net

Cash Flow

(B t - Ct

)

Reinvestment Rate

(r t

> SCA

Factor

Terminal

Value

0 $-2 ,200a

1 +1,000 20% 1.440 $1,440

2 +1,500 15% 1.150 1,725

3 oo o +r-H 1,000

$+1,300 $4,165

aThis is equal to -C0 in equation 7.3.

The future value of $1,000 earning 20% for two years is $1,440, the future

value of $1,500 earning 15% for one year is $1,725, and the future value of

$1,000 earned in that same year is $1,000. The terminal value (combined

future values of the positive net cash flows) is $4,165. Solving the problem

in table 7.3 by using equation 7.4 would yield the following:

AIRR

r$4,

= -l.o M —

165 1/3 = .237 or 23.7%.

I $2,200

As would be expected, reinvestment rates lower than the earlier computed IRR

result in an AIRR lower than the IRR. This example shows that two investments

with identical net cash flows may yield an AIRR that differs from the IRR,

depending on how those net cash flows are reinvested over the project life.

The more concentrated the net cash receipts are early in the study period, and

the greater the difference between reinvestment rates and the IRR, the larger

will be the difference between the AIRR and the IRR for any given investment,

and the more important it will be to use the AIRR to find the true return on

an investment.

28
Equation 7.5, which is equivalent to equation 7.4, can be used when the

reinvestment rate is constant from year to year and the PVNB has already

been computed.

AIRR = -1.0 + (1 + r) 1 +

PVNB 1/N

(7.5)

Not having to derive the TV simplifies the calculation of the AIRR. (Note

that r is a constant reinvestment rate and that the PVNB has been discounted

at r.

)

A major computational advantage of the AIRR over the IRR is that the AIRR can

be computed directly without using the awkward approximation techniques

required for the IRR.

7.3 Other Return-on-Investment Measures

Numerous names exist for measures of the percentage return on investments.

Some are equivalent to the IRR and AIRR as defined above, whereas others are

different measures. Two reasons that account for many of the differences are

that some measures use discounting and others do not, and some use different

^Equation 7.5 is derived as follows: To have the PVNB term replace TV, we define TV in relation to PVNB as TV = (PVNB + CQ ) (1 + r)N. Since PVNB has

subtracted CQ from the present value of the net cash flows, CQ has to be

added back to PVNB to find the present value of net cash flows which are then

compounded at (1 + r) for N periods to find the terminal value. Thus TV as defined above is substituted for TV in equation 7.4, yielding

AIRR = -1.0 +

of terms yields equation

(PVNB + C0 ) (1 + r) N — C0 —

7.5.

1/N

which with rearrangement

29
categories of costs and earnings in computing the percentage returns. Table

7.4 lists some of the common names of measures of the percentage return on

investments.^ These terms are not always used consistently. This is due in

part to their evolving from several different fields, including economics,

business finance, management science, and real estate. 2 The IRR method

discussed in this paper deals exclusively with the IRR as found using equation

7.1 and with the AIRR discussed in section 7.2.

Table 7.4 Measures of Percentage Return on Investments

accounting rate of return

after-tax equity yield

discounted cash flow (DCF) rate

of return

discounted internal rate of return

discounted rate of return

external rate of return

financial-management rate of return

interest rate of return

internal rate of discount

internal rate of interest

investor’s method

project rate of return

rate of return

rate of return over cost

return on investment (ROI)

time-adjusted rate of return

unadjusted rate of return

yield

yield to maturity

lit is recognized that measures included in table 7.4 that are computed with

other formulas than equations 7.1 and 7.4 are also helpful in making certain

types of decisions not addressed in this paper.

2For a comprehensive survey of the IRR literature evolving from these fields, see Austin J. Jaffe, "Is There a New Internal Rate of Return Literature?,”

Journal of the American Real Estate and Urban Economics Association (Vol. 5,

No. 4, Winter 1977).

30
8. RECOMMENDED APPLICATIONS AND LIMITATIONS

This section makes a case for using the AIRR measure instead of the IRR

measure for those economic evaluations where an IRR method is appropriate.

Recommendations for the application of the NB and IRR methods are made for

cases where there is some controversy as to the appropriate use of the two

methods. Calculation limitations of the two methods are also described.

8.1 IRR or AIRR Measure

Controversy persists over whether the IRR measure or the AIRR measure is a

more accurate index of an investment’s return. This question is complicated by

the conceptual question of what analysts should be measuring; namely, the

return solely on the original investment (which implicitly suggests the same

return on reinvestments as the original investment), or the combined return

on the original investment and on the reinvested earnings (at possibly

differential rates).

Some authors accept the unadjusted IRR as technically correct. 1 Among them is

Kerr who rejects reinvestment models (which include the AIRR) in general with

the following statement:

Reinvestment rate models ..., which have been developed to avoid

the ambiguities of the IRR yet retain the flavor of a yield, are incorrect because the measure of the desirability of a proposed

investment is contaminated by expected yields on other investments; ....2

lSee Geraldine Montag, "Commercial Building Ownership Energy Conservation

Cost Analysis Model—A Follow-Up," ASHRAE Journal (September 1981), p. 45, and

William R. Park, Cost Engineering Analysis (New York, New York: John Wiley and

Sons, 1973), pp. 42-48.

^H. S. Kerr, "A Final Word on FMRR," The Appraisal Journal (January 1980), p.

102 .

31
Other authors acknowledge that both the AIRR and IRR measures have certain

advantages and disadvantages, and that each measure might be appropriate under

certain circumstances.

The finding of this paper is that the AIRR measure may be preferred over the

IRR measure for the following three reasons: the AIRR is a more accurate

measure of project returns; it is generally easier to compute; and it does not

present the problem of yielding multiple solutions as does the IRR measure.

The AIRR is a more accurate measure because a direct solution is possible,

whereas approximating techniques are generally used for the IRR. It is also

more accurate in the sense that the AIRR incorporates the expected real

earnings of reinvested net cash flows whereas the IRR measure assumes

reinvestment at the IRR rate. There is no reason to expect in general that

each project's reinvestment rates will be the same as the IRR for that

project. ^ Furthermore, when comparing two competing projects with different

IRR's, different reinvestment rates are implicitly assumed when in fact one

would expect the same reinvestment rates to apply to earnings on those two

projects.

ISee Eisen, "Computer Connection," p. 127, and J. Thomas Montgomery and

Charles F. Raper, "Equity Yields and the Reinvestment Issue," The Appraisal

Journal (October 1981), pp. 519-521.

^If r = IRR, then the IRR gives the same answer as the AIRR as long as a unique IRR can be computed.

32
The reinvestment rate r needed in solving equation 7.3 for the AIRR might be

specified as part of the investment project, or, if the project were already

built and operating, the return on reinvestments would have been established

historically. If r is not known, however, the discount rate (i.e., the rate of

return available on the next best investment) would appear to be the best

approximation for r. Having an r higher than the discount rate would suggest

that the discount rate is too low; that is, it does not reflect the next best

investment opportunity. If r were lower than the discount rate, it would

suggest that the discount rate is not high enough to reflect the next best

investment opportunity. Thus in absence of information on the rate r, the

discount rate seems the appropriate value to substitute for r in using

equations 7.3, 7.4, and 7.5.

So far the AIRR has been recommended over the IRR as the more accurate economic

measure when evaluating one investment. However, the AIRR is also preferred

when choosing among a group of non-mutually-exclusive projects, since the AIRR

and IRR will not always yield the same rankings of a set of projects. That is,

the AIRR will be a more accurate guide than the IRR to the set of projects that

will maximize the aggregate PVNB when a budget constraint exists.

The AIRR is easier to compute than the IRR because a direct solution of the

AIRR is possible, using equations 7.4 and 7.5, whereas iterative procedures

such as the trial-and-error approach are required to arrive at an

approximation for the IRR. (Computers, however, make the computational

differences of the two measures less important.)

33
A limitation to the IRR is that multiple measures may result. That is, there

may be no unique IRR value that serves as a solution value for i in equation

7.1. This situation is likely to occur when the cash flow shows reversals in

sign over the study period. ^ Finding more than one solution value results in

confusion, for example, when the MARR is between two solution values of the

IRR. Furthermore, trial-and-error solutions may find only one answer and fail

to indicate the existence of other mathematically correct solution values. The

AIRR, as calculated in equations 7.4 and 7.5, avoids this problem.

2

A second limitation arises with both the IRR and the AIRR measures when the

MARR is expected to change over the lifetime of the investment. For example,

if the MARR were expected to be 17% for the first four years of a 10 year

project and 9% for the last six years, it is not clear how efficient that

project would be if it had an IRR or AIRR of 15%.

^Another way of describing such a cash flow is that over the project life some periods are revenue earning (positive cash flow) periods and others are investment or cash disbursement (negative cash flow) periods.

^The modified rate of return (MROR) , a more complicated method than the IRR,

provides a percentage measure of economic worth while avoiding some of the

problems inherent in the IRR. For a discussion of the method and flow chart of

the procedures for calculating it, see American Telephone and Telegraph

Company, Engineering Economy; A Manager's Guide to Economic Decision Making

(New York, New York: McGraw Hill, 1977), pp. 317-321.

34*
8.2 NB or IRR Method

Another controversial issue is which of the two methods, NB or IRR, gives the

proper answer for specific investment questions. Some authors suggest the NB

method for some investment decisions and the IRR method for others. ^ A few

authors recommend the NB method exclusively over the IRR method.

2

The position taken in this

typical economic questions

NB method, the IRR method,

report is summarized by table 8.1, which lists

asked by the building community and indicates if the

or both are appropriate for answering those

questions 3

lSee J. Fred Weston and Brigham, Managerial Finance , 7th ed. (Hinsdale, Illinois

The Dryden Press, 1981), p. 413 and p. 423, and Marshall and Ruegg, Simplified

Energy Design Economics , p. 13.

^Kerr, "A Final Word on FMRR," p. 102, asserts that "... it is correct to use

the NB method in all investment decisions." Lee G. Anderson and Russel F.

Settle, Benefit-Cost Analysis: A Practical Guide (Lexington, Massachusetts:

D.C. Heath and Company, 1977), pp. 96-98, and Peter G. Sassone and William A.

Shaffer, Cost-Benefit Analysis: A Handbook (New York, New York: Academic

Press, 1978), p. 28, agree with the exception that they recommend the benefitto-cost ratio method for choosing among independent projects with a budget

constraint.

^The NB and IRR methods are not the only economic methods for solving

these questions. The savings-to-investment ratio and benefit-cost ratio

methods (see Harold E. Marshall and Rosalie T. Ruegg, Recommended Practice

for Measuring Benefit/Cost and Savings-to-Investment Ratios ), for example, can

be used to answer most of the questions for which the IRR is identified in

table 8.1. Also, the life-cycle cost method (see Rosalie T. Ruegg,

Stephen R. Petersen, and Harold E. Marshall, Recommended Practice for

Measuring Life-Cycle Costs ) is substitutable in most cases for the NB

method.

35
Table 8.1 Economic Questions and Recommended Methods*

NB IRR

1. HOW can benefits be compared to coats? • •

2. HOW large an investment to make? • A

3. HOW can projects directly competing for the same • purpose be compared?

4. HOW can different-purpose projects competing for • the same budget be compared?

5. HOW to find the rate of return on investment? •

• Based on total, present value cost and benefit figures.

A Based on incremental, present value cost and benefit figures.

aThe table is adapted from Rosalie T. Ruegg and Harold E. Marshall,

"Economics of Building Design," Solar Age (July 1981), p. 27.

Table 8.1 shows in the first row that both the NB and IRR methods can be used

to indicate whether a given project is cost effective (i.e., if benefits exceed

costs). If the PVNB is greater than zero, the investor's minimum acceptable

rate of return (MARR) , as reflected in the discount rate, has been achieved,

and the project is cost effective. ^ If the PVNB is less than zero, then the

project is not cost effective. If the AIRR is greater than the MARR for a

given project, then the project is cost effective (i.e., benefits exceed costs

in present value terms). If the IRR measure is greater than the MARR, the

project may also be cost effective, but the calculated IRR value will be

correct (i.e., equal to the AIRR value) only when r IRR.

1-The MARR and the discount rate can be assumed to be equal when each is based

on the anticipated return on the next best investment.

36
The second question, how large an investment to make, is generally answered

with NB analysis, but can be answered with the IRR method if the IRR or AIRR

is calculated for marginal investment changes. Unless rate of return data are

available for other reasons, however, increasing scale until the PVNB is

maximized (i.e., using the NB method) is probably a more efficient approach

from the standpoint of making the calculations.

Typical size or scale examples from the building industry are as follows:

o How large a building should be constructed?

o How large a dam should be constructed?

o How much insulation should be put in a house?

o How many square feet of collector area should be installed in a solar energy system?

Taking the example of how large a building to construct, if the objective of

an owner/builder is to maximize profits (net benefits) from a building, the

builder can be expected to increase the square feet, number of floors, and

height variables as long as the increase in scale yields more new benefits

(income) than costs. The size at which the PVNB is maximized will be the most

economically efficient building scale.

Ilf computer assistance is available, the choice for computational efficiency

is less important.

37
Figure 8.1 Illustrates graphically how the NB method Is used to choose the

economically efficient level of energy conservation. Conservation costs are

shown to Increase at an Increasing rate as the physical quantity of Inputs to

conserve energy (Q^) Is Increased (e.g., Increased Insulation). Conservation

benefits, as measured by energy savings, also Increase with additional Inputs

to energy conservation, but at a decreasing rate. The difference between

dollar conservation benefits and costs at any given level of conservation

inputs is the PVNB. The level of energy conservation where the PVNB is

maximized is Qe

. Any smaller (Q^) or larger investments (Q2 or Q3 ) than Qe would be economically inefficient, because the potential PVNB (profits) is

greatest at Qe

.* To find the economically efficient level of insulation for a

residence, therefore, the PVNB is computed for increasing applications of

insulation until the level is found where the PVNB Is maximized.

Table 8.2 and figure 8.2 together illustrate how project size can be selected

on the basis of incremental AIRR analysis. Table 8.2 presents five size

alternatives (A, B, C, D, and E) with corresponding AIRR's for all possible

size changes. Reading table 8.2 from left to right, the top row gives the

AIRR’s on total investment at each of those size levels (size A is assumed

to be a project size of zero). Although size B has the largest AIRR (35%), it

is not necessarily the one that yields maximum net benefits, which is the

objective in selecting the most efficient size.

iNote that the efficient size could be smaller if the investment budget

were limited and if other projects were available with incremental

benefit-to-cost ratios greater than one. For a discussion of these

conditions, see Harold E. Marshall and Rosalie T. Ruegg, Recommended Practice

for Measuring Benefit/Cost and Savlngs-to-Investment Ratios , pp. 28-29.

38
Figure 8.1 Level of Energy Conservation That Maximizes the PVNB

Table 8.2 AIRR's For Project Size Changes

C 7 4

D 3

Subsequent rows of table 8.2 give the incremental AIRR's calculated on the

differences between project sizes. Thus the incremental AIRR from expanding

size from B to C is 13%; from B to D, 11%; and from C to E, 4%.

39
Figure 8.2 Network Diagram Using Incremental AIRR Analysis for Project Sizing

D 7% C

40
Figure 8.2 shows all of the potential sizing options in a network diagram.

Each line in the network connects two lettered points and represents a

potential sizing decision. The number by each connecting line is the AIRR for

that sizing increment. For example, the line from A to D shows size D with a

return of 19%. Looking at the successive increments up to D, however, shows A

to B earning an AIRR of 35%; B to C 13%, and C to D 7%. Only if the MARR is

less than 7% will an investment in size D be the economically efficient

decision. This is the case because the last increment returns only 7%, even

though the overall AIRR is 19%. If the MARR were 12%, the efficient size

would be C, because the increment B to C is the last one to earn more than the

12% MARR. Thus the rule for selecting the efficient size is to increase size

by each separable increment until the incremental AIRR equals or just exceeds

the MARR.

Note that failing to expand size whenever the incremental AIRR exceeds the

MARR or failing to reduce size whenever the incremental AIRR is less than the

MARR will result in potential PVNB foregone. This is equivalent to choosing a

Qi in figure 8.1 that is respectively less than or higher than Qe. Note

further that choosing a scale where the AIRR on the total investment is just

equal to the MARR (say choosing D when the MARR is 19%) would result in PVNB

being equal to zero (Q3 in figure 8.1). Clearly this is not the most economic

choice.

41
As long as net benefit! ere calculated with a discount rate equal to the MARR,

the efficient eicee indicated by the NB and Incremental AIRR meaeuree will be

the same.l

The NB method Is recommended for answering the third question in table 8.1,

how to compare projects or designs competing for the same purpose to see which

is most efficient (i.e., cost effective). 2 Typical examples from the building

industry include:

o How to select between single, double, or triple glazing?

o How to choose between a solar energy system and a conventional

energy system?

o How to choose between a large dam and a small dam with levees to

provide flood control?

The most efficient project in each case would be the one with the greatest

PVNB . Applying equation 6.1, for example, to the selection of a flood control

project, if PVNB is greater for the small dam and levees than for the large

dam, then the small dam and leeves are the economically preferred system.

In comparing projects competing for the same purpose, the analyst must sometimes normalize the PVNB with respect to time in order to have a valid

economic comparison. The PVNB of projects with identical expected lives can be

compared directly. If the expected lives are different, however, adjustments

iThis assumes the same reinvestment rates for net cash flows are used in both

methods

2Under certain circumstances the incremental IRR method can also be used to

compare projects or designs competing for the same purpose. For example, one case of comparing mutually exclusive designs is comparing scale or size as was discussed under question 2 in table 8.1.

42
are required. A common adjustment is to convert each project’s life to the

least common multiple of the lives of all projects under consideration. By

making assumptions about reinvestment costs and earnings, a time-normalized

PVNB can then be calculated for each project for comparison over the common

study period.

A second approach in comparing projects of unequal lives is to convert the

PVNB calculated on the basis of each project's life to an annual value of net

benefits (AVNB). (See equation 6.2.) The AVNB figures will yield a valid

economic comparison if replacement assets in the short-lived projects repeat

the costs and earnings of the original assets.

A third approach is to assume a perpetual or infinite period of earnings

through continual reinvestment. ^ The PVNB of each project is converted to

an AVNB. Dividing the AVNB by the discount rate for each competing project

yields PVNB measures for projects with infinite lives. This normalization

with respect to lives results in PVNB figures that provide valid comparisons

among projects.

Note that an implicit assumption typically made in all of these approaches is

that the replacement investment is identical to the original investment and

will repeat the returns of the original investment. 2

iFor further descriptions of these three approaches, see Grant and Ireson,

Principles of Engineering Economy , pp. 77-81 and 92-98; Gerald W. Smith,

Engineering Economy; Analysis of Capital Expenditures , 2nd ed. (Ames, Iowa;

Iowa State University Press, 1973), pp. 107-108; and Weston and Brigham,

Managerial Finance , pp. 436-438.

2This same assumption is implicit in the IRR method. However, the assumption

may not reflect actual reinvestment opportunities.

43
A mistake analysts sometimes make is to use the IRR measure, based on total

present value cost and benefit figures, for selecting among projects competing

for the same purpose. Table 8.3 helps illustrate how using the unadjusted IRR

measure instead of the NB method to select between projects competing for the

same purpose will lead to inefficient choices under certain circumstances.

Cash flows for projects H and I are given in table 8.3. The two example

projects are designed to reflect high payoffs early in one project and late in

the other so as to highlight the inconsistent answers that occur when the cash

flows differ among projects. Figure 8.3 illustrates the potential

inconsistency of the two methods by comparing plots or profiles of the PVNB

for the two projects, H and I, over a range of discount rates. The profiles

are derived the same way as the graph shown in figure 7 .1.

Table 8.3 Project Data for Comparing Projects by NB and IRR Methods

Year Project H Project I (t) Bt C

t <Bt - Ct

) B

t c

t (Bt - Ct

)

0 $0 $1,000 $-1,000 $0 $1,000 $-1,000

1 100 0 100 1,100 0 1,100

2 125 0 125 200 0 200

3 1 ,300 0 1,300 50 0 50

$525 $350

iThe circumstances that lead to different answers are typically that the costs

and benefits of one project far exceed those of the other project or that the

cash flows have a different timing sequence.

44
Figure 8.3 Graphical Illustration of Inconsistent Project Rankings by NB

and IRR Methods

PVNB

Given a discount rate of zero, the PVNB would be the arithmetic sum over time

of the net cash flows—$525 for project H, and $350 for project I. Thus the

function for H intersects the vertical axis at $525 and for I at $350.

Discount rates in 5% increments are substituted into equation 6.1 to find the

PVNB at those rates. Connecting the PVNB values for each project for the

assumed discount rates gives a function approximating the PVNB for a wide

range of discount rates. The function for project H crosses the horizontal

axis at an IRR of approximately 16.5%. For project I, the unadjusted IRR

appears to be approximately 28.5%. Thus project I appears to be the more

attractive investment. This is consistent with the results from applying the

NB method for all discount rates greater than the crossover rate (in this case

about 8%) where the two functions cross. That is, for all discount rates

45
greater than 8%, the PVNB is greater for I than H. This is what would be

expected in that I earns benefits early during the project life and is not

affected by high discount rates as much as H which earns benefits late in the

project life.

The inconsistency between the two methods occurs whenever the discount rate

used to compute the PVNB is less than the crossover rate. Thus in figure 8 . 3 ,

for example, project H yields higher PVNB than project I for every discount

rate below 8%. Again this is expected because project H's major benefits

accrue late in the project life. But the IRR measure still favors project I

while the NB method favors H. Therefore, as long as the objective is

maximization of the PVNB, the NB method is preferred, since the unadjusted

IRR will not consistently lead to the efficient project.

The AIRR, if computed with a reinvestment rate equal to the discount rate,

would yield similar results to the NB method for the problem described in

table 8.3. Under certain circumstances, however, such as one project having

benefits and costs that far exceed those of the other project, the AIRR will

indicate the incorrect alternative (i.e., it will be higher for the project

with a lower PVNB). Thus the NB method is preferred to both the IRR and AIRR

measures.

Figure 8.3 also shows that different conclusions result within the NB

method itself as the discount rate changes. For example, at rates less

than 8%, project H is preferred, whereas at rates greater than 8%, project I

is preferred. This does not indicate any limitation in the method. It

46
shows only that, as the opportunity cost of funds changes, so does the

relative worth of projects with different payouts over time.

The AIRR measure is used to answer the fourth question, how can differentpurpose projects competing for the same budget be compared. Typical examples

from the building industry are as follows:

o Which conservation retrofits are most cost effective for the

homeowner when the budget permits only some of the economically

feasible projects to be undertaken?

o Which packages of energy conservation projects for its buildings

should a government agency select, given a limited budget, among

the many opportunities identified by field offices?

o Which water resource construction projects should an agency select

(e.g., among flood control, irrigation, and recreation projects) when

the construction budget will not cover all the economically feasible

alternatives?

Each of the examples above describes a set of non-mutually-exclusive,

independent projects competing for a limited budget. 1 Given that the budget

cannot fund all of the projects for which the AIRR exceeds the MARR, the

aggregate PVNB from the limited budget will be maximized generally by undertaking projects in descending order of their AIRR's until the budget is

exhausted. If the AIRR's fall below the MARR before the available budget is

exhausted, then project acceptance should terminate with the last project

whose AIRR exceeds the MARR. It is economically efficient to hold back part

^For non-mutually-exclusive, independent projects with no budget constraint,

the appropriate selection criterion is simply to take every project where the

PVNB exceeds zero. If one or more non-mutually-exclusive projects were

dependent (i.e., the PVNB of any one project depends on whether or not another

is built), then the set of projects that would maximize the PVNB in

combination would be economically efficient. If a budget constraint were present, then there would be fewer sets to choose from.

For a description of how to choose combinations of interdependent projects,

see Sassone and Schaffer, Cost-Benefit Analysis , pp. 25-29.

47
of the budget if the remaining available projects have AIRR's less than the

MARR. If projects with high AIRR's cost more than the available budget, lower

cost projects with lower AIRR's (but AIRR's greater than the MARR) may be substituted until the budget is exhausted.

The AIRR rankings as described above will not indicate the optimal project mix

in every case. For that reason, the set of projects selected by the AIRR

rankings should be used only as a guideline and checked against other project

mixes to make sure that no other combination will yield a higher aggregate

PVNB for the limited budget. An example of a readjustment of the project mix

that would maximize the total PVNB for a limited budget is illustrated in

table 8.4.

Table 8.4 Optimal Ranking Using AIRR as a Guideline3

Allocation of

$10K Budget

Projects

Initial

Costs

(C0 ) PVNB

AIRR

(%)

AIRR

Ranking

AIRR

Project

Set

Optimal

Project

Set

M $4,000 $5,222 30 1 M M

N 1,000 895 25 2 N

0 6,000 4,488 23 3 0

P 2,000 391 14 4 P

Q 3,000 283 12 5 Q

Aggregate PVNB = $6,791 $9,710

aThe five hypothetical projects used to illustrate optimal ranking are evaluated over a 5-year study period at a 10% reinvestment rate.

48
Five projects, each yielding earnings for five years and having expected reinvestment rates of 10%, are ranked according to their AIRR's. Selecting

projects in descending order of the AIRR’s yields a project mix of M, N, P,

and Q, given a $10,000 budget. Project 0 is passed by since inadequate funds

remain after choosing M and N. An aggregate PVNB of $6,791 results from the

four projects chosen. Alternative project mixes including project 0 should

be examined, however, to be certain that the aggregate PVNB is maximized at

$6,791. Inspection of table 8.4 shows that spending the $10,000 budget on M

and 0 would yield an aggregate PVNB of $9,710, almost $3,000 more than the

project set indicated by the AIRR ranking. ^ Thus to assure selection of the

optimal set of projects, alternative mixes should be compared against the

initial AIRR ranking whenever the AIRR ranking passes over cost-effective

projects more costly than the remaining budget.

The description of table 8.4 above explains in part the confusion arising from

some textbooks recommending IRR methods while others recommend NB methods for

choosing among non-mutually-exclusive, independent projects. Both

recommendations are correct under certain circumstances. Project selection

according to the AIRR rankings, for example, will generally lead to the mix

that maximizes the aggregate PVNB, with the exception illustrated in table

8.4. The NB method, on the other hand, is correct the aggregate PVNB of a

group of projects is maximized for a limited budget. Selection on the basis

lit is assumed for this problem that all or most of the budget must be spent.

That is, remaining budget funds are not retained by the investor. However, if

the remaining budget could be retained, and the objective were to maximize the sum of the aggregate PVNB and the remaining budget, then projects M and N

would be preferred to M and 0, because the aggregate PVNB plus remaining

budget for M and N ($11,117) exceed that of M and 0 ($9,710).

49
of individial project net benefits, however, will not necessarily lead to the

optimal mix.* Yet project analysts sometimes rank projects according to the

PVNB of individual projects. The following case example is provided to show

the economic implications of such rankings.

Table 8.5 lists seven non-mutually-exclusive, independent projects (column 1).

They are ranked by the AVNB (column 7) and by the AIRR (column 9). (The AVNB

measure is used here instead of the PVNB measure to account for the unequal

project lives.) AIRR's (column 8) are calculated on the basis of uniform net

cash flows per year (column 3) and the initial project cost (column 4). The

study period, determined by the number of earning years for each project,

ranges from 3 to 6 years. The AVNB (column 6) for each project is calculated

by subtracting the annual value of initial costs (column 5) from the uniform

annual net cash flows (column 3). A 12% discount rate was used to find annual

values of cash flows, and 12% was also assumed to be the MARR and the reinvestment rate for net cash flows.

lA computer program that calculates aggregate PVNB for every possible project

mix for a given budget would be the most efficient approach to project

selection.

50
Table 8.5 NB Versus IRR Method for Ranking Projects

Projects

Earning

Years

Annual

Net Cash

Flows

(B

t - V

C

o

Annual Value

of

CD AVNB

Ranking

by

AVNB AIRR

Ranking

by

AIRR

(1) (2) (3) (4) (5) (6)=( 3)-(5) (7) (8) (9)

A 3 395 1,000 $416 $-21 7 10.1% 6

B 4 1,094 3,000 987 107 3 14.9% 3

C 6 162 500 122 40 6 17.5% 1

D 4 1,401 4,000 1,316 85 4 13.7% 5

E 3 4,007 9,000 3,744 263 1 14 . 5% 4

F 5 350 1,000 277 73 5 17.3% 2

G 4 1,641 4,500 1,481 160 2 14 . 9% 3

a A 12% discount rate is assumed for calculating the PVNB and the AVNB. A 12% rate is also

assumed to be the MARR and the reinvestment rate for net cash flows.

Note that project A is clearly inefficient by both methods because the AVNB is

negative and the AIRR is less than the MARR. However, rankings for the rest

of the projects are inconsistent between the two methods and may lead to

different packages of selected projects, depending on the budget constraint.

For example, if the budget were $9,000, the AIRR ranking would indicate projects C, F, and G and B, in that order, for an aggregate AVNB of $380.1 For

the same budget, the AVNB ranking, on the other hand, would indicate project E

alone, with an AVNB of $263. Ranking with the AVNB measure in this case

^Projects G and B have the same AIRR and are therefore equally ranked.

51
example results in annual-value, economic efficiency losses of $117 ($360 -

$263). 1 Thus the table shows how ranking with the AVNB measure may bias

project selection to one or a few large projects with a high AVNB per project

but a relatively low return on the limited budget.

The last question of table 8.1, finding the rate of return on an investment,

can be answered only with the IRR or AIRR measures. Neither the NB method nor

any of the other economic evaluation tools for capital investment expresses

the "value" of an investment in percentage terms.

The guidelines for applying the NB and IRR methods described above cover many

of the questions that the private and public building communities must answer

in making economically efficient decisions in the design, operation,

maintenance, and retrofit of buildings. While other methods could be used

successfully in answering most of these questions, the focus of this report is

on the NB and IRR methods. For an alternative to the NB method, see the

National Bureau of Standards (NBS) report Recommended Practice for Measuring

Life-Cycle Costs of Buildings and Building Systems . For an alternative to the

IRR method, see the NBS report Recommended Practice for Measuring Benefit/Cost

and Savings-to-Investment Ratios for Buildings and Building Systems.

^Ranking these seven projects with the unadjusted IRR would give exactly the same priority as with the AIRR, but the spread of returns would be much wider

(9% to 23%). However, under different cash flow assumptions the AIRR and IRR

measures might yield different rankings.

52
APPENDIX A

DISCOUNT FACTOR TABLES*

*Tables needed for problems illustrated in this report were taken from

American Society for Testing and Materials, Discount Factor Tables , to be

published as an Adjunct to "Recommended Practice for Measuring Life-Cycle

Costs of Buildings and Building Systems" (Philadelphia, Pennsylvania: ASTM,

1983 ).
Table 5 DISCRETE RATE-OF-RETURN FACTORS*

Single

Compound

Amount

Factor

SCA

Single

Present

Value

Factor

SPV

Uniform

Capital

Recovery

Factor

UCR

Uniform

Present

Value

Factor

UPV

Uniform

Sinking

Fund

Factor

USF

Uniform

Compound

Amount

Factor

UCA

Formula

(l+i)

n

1 l(l+i) n

(l+i) n-l i (l+i) n

-l

(1+D n

(l+i) n-l i(l+i) n

(l+i) n-l 1

Given P F P A F A

To Find F P A P A F

Period

n

i - 5%

1 1.050 .9524 1.050 .9524 1.000 1.000

2 1.103 .9070 .5378 1.859 .4878 2.050

3 1.158 .6638 .3672 2.723 .3172 3.153

4 1.216 .8227 .2820 3.546 .2320 4.310

5 1.276 .7835 .2310 4.329 .1810 5.526

6 . 1.340 .7462 .1970 5.076 .1470 6.802

7 1.407 .7107 .1728 5.786 .1228 8.142

8 1.478 .6768 .1547 6.463 .1047 9.549

9 1.551 .6446 .1407 7.108 .0907 11.03

10 1.629 .6139 .1295 7.722 • .0795 12.58

11 1.710 .5847 .1204 8.306 .0704 14.21

12 1.796 .5568 .1128 8.863 .0628 15.92

13 1.886 .5303 .1065 9.394 .0565 17.71

14 1.980 .5051 .1010 9.899 .0510 19.60

15 2.079 .4810 .0963 10.38 .0463 21.58

16 2.183 .4581 .0923 10.84 .0423 23.66

17 2.292 .4363 .0887 11.27 .0387 25.84

18 2.407 .4155 .0355 11.69 .0355 28.13

19 2.527 .3957 .0827 12.09 .0327 30.54

20 2.653 .3769 .0802 12.46 .0302 33.07

21 2.786 .3589 .0780 12.82 .0280 35.72

22 2.925 .3419 .0760 13.16 .0260 38.51

23 3.072 .3256 .0741 13.49 .0241 41.43

24 3.225 .3101 .0725 13.80 .0225 44.50

25 3.386 .2953 .0710 14.09 .0209 47.73

26 3.556 .2812 .0696 14.38 .0196 51.11

27 3.733 .2678 . .0683 14.64 .0183 54.67

28 3.920 .2551 .0671 14.90 .0171 58.40 1 29 4.116 .2429 .0660 15.14 .0160 62.32

30 4.322 .2314 .0651 15.37 .0151 66.44

31 4.538 .2204 .0641 15.59 .0141 70.76

32 4.765 .2099 • .0633 15.80 .0133 75.30

33 5.003 .1999

’ .0625 16.00 .0125 80.06

34 5.253 .1904 .0618 16.19 .0117 85.07

35 5.516 .1813 .0611 16.37 .0111 90.32

36 5.792 .1727 .0604 16.55 .0104 95.84

37 6.081 .1644 .0598 16.71 .0098 101.6

38 6.385 .1566 .0593 16.87 .0093 107.7

39 6.705 .1491 .0588 17.02 .0088 114.1

40 7.040 .1420 .0583 17.16 .0083 120.8

* All formulae assume end-of-period payments.

P = a present sum of money; F = a future sum of money, equivalent to P at the

end of n periods of time at discount rate of i; A = an end-of-period

payment (or receipts) in a uniform series of payments (or receipts) over

n periods at i interest rates.
Table 10 DISCRETE RATE-0F-RETURN FACTORS*

Single Single Uniform Uniform Uniform Uniform

Compound Present Capital Present Sinking Compound | Amount Value Recovery Value Fund Amount

Factor Factor Factor Factor Factor Factor

SCAF SPVF UCRF UPVF USFF UCAF

( For- (l+i)

n

1 i(l+i)

n

(l+i)

n-l i (l+i) n

-l i mula

(l+i) n

(l+i) n-l i(l+L ) n (l+i)

n

-l

1

‘ ! Civen P F P A F A

To Find F P A P A F

Period

n i=10%

_ - i

1 1.100 .9091 1.100 .9091 1.000 1.000

2 1.210 .8264 .5762 1.736 .4762 2.100

3 1.331 .7513 .4021 2.487 .3021 3.310

4 1.464 .6820 .3155 3.170 .2155 4.641

5 1.611 .6209 .2638 3.791 .1638 6.105

6 1.772 .5645 .2296 4.355 .1296 7.716

7 1.949 .5132 .2054 4.868 .1054 9.487

8 2.144 .4665 .1874 5.335 .0874 11.44

9 2.358 .4241 .1736 5.759 .0736 13.58

10 2.594 .3855 .1627 6.145 .0627 15.94

11 2.853 .3505 .1540 6.495 .0540 18.53

12 3.138 .3186 .1468 6.814 .0468 21.38

13 3.452 .2897 .1408 7.103 .0408 24.52

14 3.798 .2633 .1357 7.367 .0357 27.98

15 4.177 .2394 .1315 7.606 .0315 31.77

16 4.595 .2176 .1278 7.824 .0278 35.95

17 5.054 .1978 .1247 8.022 .0247 40.54

18 5.560 .1799 .1219 8.201 .0219 45.60

19 6.116 .1635 .1195 8.365 .0195 51.16

20 6.728 .1486 .1175 8.514 .0175 57.28

21 7.400 .1351 .1156 8.650 .0156 84.00

22 8.140 .1228 .1140 8.772 .0140 71.40

23 8.954 .1117 .1126 8.883 .0126 79.54

24 9.850 .1015 .1113 8.984 .0113 88.50

25 10.83 .0923 .1102 9.077 .0102 98.35

26 11.92 .0839 .1092 9.161 .0092 109.2

27 13.11 .0763 .1083 9.237 .0083 121.1

28 14.42 .0693 .1075 9.307 .0074 134.2 • 29 15.86 .0630 .1067 9.370 .0067 148.6

30 17.45 .0573 .1016 8.427 .0061 164.5

31 19.19 .0521 .1055 9.479 .0055 181.9

32 21.11 .0474 .1050 9.526 .0050 201.1

' 33 23.23 .0431 .1045 9.569 .0044 222.3

34 25.55 .0391 .1041 9.609 .0001 245.5

35 28.10 .0356 .1037 9.644 .0037 271.0

36 30.91 .0323 .1033 9.676 .0033 299.1

37 34.00 .0294 .1030 9.706 .0030 330.0

38 37.40 .0267 .1027 9.733 .0027 364.0

39 41.14 .0243 .1025 9.757 .0025 401.4

40 45.26 .0221 .1023 9.779 .0023 442.6

* All formulae assume end-of-period payments.

P = a present sum of money; F = a future sum of money, equivalent to P at the

end of n periods of time at discount rate of i; A = an end-of-period

payment (or receipts) in a uniform series of payments (or receipts) over

n periods at i interest rates. over
Table 14 DISCRETE RATE-OF-RETURN FACTORS*

Single Single Uniform Uniform Uniform Uniform

Compound Proaont Capital Prasunt Sinking Compound

Amount Value Recovery Value Fund Amount

Factor Factor Factor Factor Factor Factor

SCAF SPVF UCRF UPVF USFF UCAF

For- (l+i)

n

1 i (l+i)

n (l+i)

n-l 1 (l+i)

n

-l

roula (l+i)n

(l+i)

n-l i(l+i)

n (l+i)n-l 1 Given P F P A F A

To Find F P A P A F

Period

n i-14*

1 1.140 .8772 1.140 .8772 1.000 1.000

2 1.300 .7695 .6073 1.647 .4673 2.140

3 1.482 .6750 .4307 2.322 .2907 3.440

4 1.689 .5921 .3432 2.914 .2032 4.921

5 1.925 .5194 .2913 3.433 .1513 6.610

6 2.195 .4556 .2572 3.889 .1172 8.536

7 2.502 .3996 .2332 4.288 .0932 10.73

8 2.853 .3506 .2156 4.639 .0756 13.23

9 3.252 .3075 .2022 4.946 .0622 16.09

10 3.707 .2697 .1917 5.216 .0517 19.34

11 4.226 .2366 .1834 5.453 .0434

’ 23.04

12 4.818 .2076 .1767 5.660 .0367 27.27

13 5.492 .1821 .1712 5.842 .0312 32.09

14 6.261 .1597 .1666 6.002 .0266 37.58

15 7.138 .1401 .1628 6.142 .0228 43.84

16 8.137 .1229 .1596 6.265 .0196 50.98

17 9.276 .1078 .1569 6.373 .0169 59.12

18 10.58 .0949 .1546 6.467 .0146 68.39

19 12.06 .0829 .1527 6.550 .0127 78.97

20 13.74 .0728 .1510 6.623 .0110 91.02

21 15.67 .0638 .1495 6.687 .0095 104.8

22 17.86 .0560 .1483 6.743 .0083 120.4

23 20.36 .0491 .1472 • 6.792 .0072 138.3

24 23.21 .0431 .1463 6.835 .0063 158.7

25 26.46 .0378 .1455 6.873 .0055 181.9

26 30.17 .0331 .1448 6.906 .0048 208.3

27 34.39 .0291 .1442 6.935 .0042 238.5

28 39.20 .0255 .1437 6.961 .0037 272.9

29 44.69 .0224 .1432 6.983 .0032 312.1

30 50.95 .0196 .1428 7.003 .0028 356.8

31 58.08 .0172 .1425 7.020 .0025 407.8

32 66.21 .0151 .1421 7.035 .0021 465.8

33 75.48 .0132 .1419 7.048 .0020 532.0

34 86.05 .0116 .1416 7.060 .0016 607.5

35 98.10 .0102 .1414 7.070 .0014 693.6

36 111.8 .0089 .1413 7.079 .0013 791.7

37 127.5 .0078 .1411 7.097 .0011 903.5

38 145.3 .0069 .1410 7.094 .0010 1031

39 165.7 .0060 .1409 7.100 .0009 1176

40 188.9 .0053 .1407 7.105 .0007 1342

* All

P = a formulae assume present sum of

end-of-period payments,

money; F = a future sum of money, equivalent to P at the

end of n periods of time at discount rate of i; A = an end-of-period

payment (or receipts) in a uniform series of payments (or receipts) over

n periods at i interest rates.
Table 15 DISCRETE RATE-OF-RETURN FACTORS*

Single Single Uniform Uniform Uniform Uniform

Compound Present Capital Present Sinking Compound

Amount Value Recovery Value Fund Amount

Factor Factor Factor Factor Factor Factor

SCAF SPVF UCRF UPVF USFF UCAF

. For- (l+i) n

1 Ki+i)n

(l+i)

n-l i (l+i)

n

-l

inula (1+1)" (l+i) n-l ia+4 n (l+i)"-l i

Given P F P A F A

To Find F P A P A F ! Period

n 1=15?------ |

1 1.150 .8696 1.150 .8696 1.000 1.000

2 1.323 .7561 .6151 1.626 .4651' 2.150

3 1.521 .6575 .4380 2.283 .2880 3.473

4 1.749 .5718 .3503

' 2.855 .2003 4.993

5 2.011 .4972 .2983 3.352 .1483 6.742

6 2.313 .4323 .2642 3.784 .1142 8.754

7 2.660 .3760 .240 4.160 .0904 11.07

8 3.059 .3269 .2230 4.487 .0729 13.73

9 3.518 .2843 .2096 4.772 .0596 16.79

10 4.046 .2472 .1993 5.019 .0493 20.30

11 4.652 .2149 .1911 5.234 .0411 24.35

12 5.350 .1869 .1845 5.421 .0345 29.00

13 6.153 .1625 .1791 5.583 .0291 34.35

14 7.076 .1413 .1747 5.724 .0297 40.50

15 8.137 .1229 .1710 5.847 .0210 47.58

16 9.358 .1069 .1679 5.954 .0180 55.72

17 10.76 .0929 .1654 6.047 .0154 65.08

18 12.38 .0808 :1632 6.128 .0132 75.84

19 14.23 .0703 .1613 6.198 .0113 88.21

20 16.37 .0611 .1598 6.259 .0098 102.4

21 18.82 .0531 .1584 6.312 .0084 118.8

22 21.64 .0462 .1573 6.359 .0073 137.6

23 24.89 .0402 .1563 6.399 .0063 159.3

24 28.63 .0349 .1554 6.434 .0054 184.2

25 32.92 .0304 .1547 6.464 .0047 212.8

26 37.86 .0264 .1541 6.491 .0041 245.7

27 43.54 .0230 .1535 6.514 .0035 283.6

28 50.07 .0200 .1531 6.534 .0031 327.1

29 57.58 .0174 .1527 6.551 .0027 377.2

30 66.21 .0151 .1523 6.566 .0023 434.7

31 76.14 .0131 .1520 6.579 .0020 501.0

32 87.57 .0114 .1517 6.591 .0017 577.1

33 100.7 .0099 .1515 6.600 .0015 664.7

34 115.8 .0086 .1513 6.609 .0013 765.4

35 133.2 .0075 .1511 6.617 .0011 881.2

36 153.2 .0065 .1510 6.623 .0010 1014

37 176.1 .0057 .1509 6.629 .0009 1168

38 202.5 .0049 .1507 6.634 .0007 1344

39 232.9 .0043 .1506 6.63S .0006 1546

40 267.9 .0037 .1506 6.642 .0006 1779

* All formulae assume: end-of-period payments » P = a present sum of money; F = a future sum of money

,

equivalent to P at the

end of n periods of time at discount rate of i; A = an end-of-period

payment (or receipts) in a uniform series of payments (or rece ipts) over

n periods at i interest rates.
Table 18 DISCRETE RATE-OF-RETURN FACTORS*

Single

Compound

Amount

Factor

SCAF

Slnglo

Present

Value

Factor

SPVF

Uniform

Capital

Recovery

Factor

UCRF

Uniform

Prosent

Value

Factor

UPVF

Uniform

Sinking

Fund

Factor

USFF

Uniform

Compound

Amount ; Factor

UCAF

For- (l+i)

n

1 i (l+i) n

(l+i)

n

-l 1 (l+i)

n

-l

mula

(l+i)

n (l+i)n-l i(l+i) n (l+i)

n

-l 1 Given P F P A F A

To Find F P A P A F

Period

n 18*

1 1.180 .8475 1.180 .8475 1.000 1.000

2 1.392 .7182 .6387 1.566 .4587 2.180

3 1.643 .6086 .4600 2.174 .2799 3.572

4 1.939 .5158 .3717 2.690 .1917 5.215

5 2.288 .4371 .3198 3.127 .1398 7.154

6 2.700 .3704 .2859 3.498 .1059 9.442

7 3.185 .3139 .2624 3.812 .0824 12.14

8 3.759 .2660 .2452 4.078 .0652 15.33

9 4.435 .2255 .2324 4.303 .0524 19.09

10 5.234 .1911 .2225 4.494 .0425 23.52

11 6.176 .1619 .2148 4.656 .0348 28.76

12 7.288 .1372 .2086 4.793 .0286 34.93

13 8.600 .1163 .2037 4.909 .0237 42.22

14 10.15 .0985 .1997 5.008 .0197 50.82

15 11.97 .0835 .1964 5.091 .0164 60.97

16 14.13 .0708 .1937 5.162 .0137 72.94

17 16.67 . .0600 .1915 5.222 .0115 87.07

18 19.67 .0508 .1896 5.273 .0096 103.7

19 23.21 .0431 .1881 5.316 .0081 123.4

20 27.39 .0365 .1868 5.352 .0068 146.6

21 32.32 .0309 .1857 5.384 .0057 174.0

22 38.142 .0262 .1848 5.410- .0048 206.3

23 45.01 .0222 .1841 5.432 .0041 244.5

24 53.11 .0188 .1835 5.451 .0035 289.5

25 62.67 .01595 .1829 5.467 .0029 342.6

26 73.95 .0135 .1825 5.480 .0025 405.3

27 87.26 .0115 .1821 5.492 .0021 479:2

28 103.0 .0097 .1818 5.502 .0015 566.5

29 121.5 .0082 .1815 5.510 .0015 669.4

30 143.4 .0070 .1813 5.517 .0013 790.9

31 169.2 .0060 .1811 5.523 .0011 934.3

32 199.6 .0050 .1809 5.528 .0009 1103

33 235.6 .0042 .1808 5.532 .0008 1303

' 34 278.0 .0036 .1807 5.536 .0006 1538

35 328.0 .0030 .1806 5.539 .0006 1816

36 387.0 .0026 .1805 5.541 .0005 2144

37 456.7 .0022 .1804 5.543 . 0004 2531

38 538.9 .0019 .1803 5.545 .0003 2988

39 635.9 .0016 .1803 5.547 .0003 3527

40 750.4 .0013 .1802 5.548 .0002 4163

* All formulae assume end-of-period payments • P = a present sum of money; F = a future sum of money, equivalent to P at the

end of n periods of time at discount rate of i; A = an end-of-period

payment (or receipts) in a uniform series of payments (or receipts) over

n periods at i interest rates.
Table 19 DISCRETE RATE-0F-RETURN FACTORS*

Single

Compound

Amount

Factor

SCAF

Single

Present

Value

Factor

SPVF

Uniform

Capital

Recovery

Factor

UCRF

Uniform

Present

Value

Factor

UPVF

Uniform

Sinking

Fund

Factor

USFF

Uniform j Compound

Amount

Factor ^ UCAF

! For- (l+i) n

1 i(l+i)

n (l+i)"-l i (l+i)"-l I mula

<l+i) n

(l+i) n-l i(l+i)" (l+i)

n-l

J

* ! Given P F P A F A

To Find F P A P A F

Period

n 1=1,92

_ ,

• . i

1 1.190 .8403 1.190 .8403 1.000 1.000

2 1.416 .7062 .6467 1.547 .4567 2.190

3 1.685 .5934 .4673 2.140 .2773 3.606

4 2.005 .4987 .3790 2.639 .1890 5.291

5 2.386 .4190 .3271 3.058 .1371 7.297

6 2.840 .3521 .2933 3.409 .1033 9.683

7 3.379 .2959 .2699 3.706 .0799 12.52

8 4.021 .2487 .2529 3.954 .0629 15.90

9

• 4.785 .2190 .2402 4.163 .0502 19.92

10 5.695 .1756 .2305 4.339 .0405 24.71

11 6.777 .1476 .2229 4.487 .0329 30.40

12 8.064 .1240 .2169 4.611 .0269 37.18

13 9.596 .1042 .2121 4.715 .0221 45.24

14 11.42 .0876 .2082 4.802 .0182 54.84

15 13.59 .0736 .2051 4.876 .0151 66.25

16 16.17 .061? .2025 4.938 .0125 79.85

17 19.24 .0520 .2004 4.990 .0104 96.02

18 22.90 .0437 .1987 5.033 .0087 115.27

19 27.25 .0367 .1972 5.070 .0072 138.2

20 32.43 .0308 .1960 5.101 .0060 165.4

21 38.59 .0259 .1951 5.127 .0051 197.8

22 45.92 .0218 .1942 5.149 .0042 236.4

23 54.65 .0183 .1935 5.167 .0035 282.4

24 65.03 .015 4 .1930 5.182 .0030 337.0

25 77.39 .0129 .1925 5.195 .0025 402.0

26 92.09 .0109 .1921 5.206 .0021 479.4

27 109.6 .0091 .1918 5.215 .0017 571.5

28 130.4 .0077 .1915 5.223 .0015 681.1

29 155.2 .0064 .1912 5.229 .0012 811.5

30 184.7 .0054 .1910 5.235 .0010 966.7

31 219.8 .0046 .1909 5.239 .0009 1151

32 261.5 .0038 .1907 5.243 .0007 1371

33 311.2 .0032 .1906 5.246 .0006 1632

34 370.3 .0027 .1905 5.249 .0005 1943

35 440.7 .0023 .1904 5.251 .0004 2314

36 524.4 .0019 .1904 5.253 .0004 2754

37 624.1 .0016 .1903 5.255 .0003 3279

38 742.7 .0013 .1903 5.256 .0003 3903

39 893.8 .0011 .1902 5.257 .0002 4646

40 1051 .0009 .1902 5.258 .0002 5529

* AH formulae assume end—of—period payments.

P - a present sum of money; F - a future sum of money, equivalent to P at the end of n periods of time at discount rate of i; A = an end- of-period

payment (or receipts) in a uniform series of payments (or receipts) over

n periods at i interest rates.
Table 20 DISCRETE RATE-OF-RETURN FACTORS*

Single

r . Single Uniform Uniform Uniform Uniform

Compound Present Capital Present Sinking Compound

Amount Value Recovery Value Fund Amount

Factor Factor Factor Factor Factor Factor

SCAF SPVF UCRF UPVF USFF UCAF

For- (l+i)

n

1 1(1+1)" (l+i)

n-l i (l+i)

n

-l

mula

(l+i)

n

(l+i)

n-l i(l+i)

n (l+i) n

-l

...

| 1

I Civcn P F P A F A

To Find F P A P

' A F

Period

n •i=20

1 1.200 .8333 1.200 .8333 1.000 1.000

2 1.AA0 .69AA • 65A5 1.528 • . A5A5 2.200

3 1.728 .5787 . A7A7 2.106 • 27A7 3.6A0

A 2.07A . A823 .3863 2.589 .1863 5.368

5 2.A88 . A019 . 33AA 2.991 .13AA 7.AA2

6 2.986 .33A9 .3007 3.326 .1007 9.930

7 3.583 .2791 . 277A 3.605 .077A 12.92

8 A. 300 .2326 .2606 3.837 .0606 16.50

9 5.160 .1938 . 2A81 A. 031 .0A81 20.80

10 6.192 .1615 .2385 A. 192 .0385 25.96

11 7.A30 . 13A6 .2311 A. 327 .0311 32.15

12 8.916 .1122 .2253 A.A39 .0253 39.58

13 10.70 .0935 .2206 A. 533 .0206 A8.50

1A 12. 8A .0779 .2169 A. 611 .0169 59.20

15 15. A1 .06A9 .2139 A. 675 .0139 72. 0A

16 18. A9 .05A1 . 211A A. 730 .011A 87. AA

17 22.19 • .0A51 . 209A A. 775 .009A 105.9

18 26.62 .0376 .2078 A. 812 .0078 128.1

19 31.95 .0313 .2065 A.8AA .0065 15A.7

20 38. 3A .0261 .205A A. 870 .0057 186.7

21 A6.01 .0217 . 20AA A. 891 .00AA 225.0

22 55.21 .0181 .2037 A. 909 .0037 271.0

23 66.25 .0151 .2031 A. 925 .0031 326.2

2A 79.50 .0126 .2025 A. 937 .0025 392.5

25 95. A0 .010A .2021 A.9A8 .0021 A72.0

26 11A.A .0087 .2018 A. 956 .0018 567.

A

27 137.

A

.0073 .2015 A.96A .0015 681.9

28 16A.8 .0061 .2012 A. 970 .0012 819.2'

29 197.8 .0051 .2010 A. 975 .0010 98A.1

30 237.

A

• 00A2 .2008 A. 979 .0008 1181

31 28A.9 .0035 .2007 A. 982 .0007 1A19

32 3A1.8 .0029 .2006 A. 985 .0006 170A

33 A10.2 .002A .2005 A. 988 .0005 20A6

3A A92.2 .0020 . 200A A. 990 .000A 2A56

35 590.7 .0017 .2003 A. 992 .0003 29A8

36 708.8 .001A .2003 A. 993 .0003 3539

37 350.6 .0012 .2002 A.99A .0002 A2A8

38 1021 .0010 .2002 A. 995 .0002 5098

39 1225 .0008 .2002 A. 996 .0002 6119

AO 1A70 .0007 .2001 A. 997 ' .0001 73AA

* All formulae assume end-of-period payments>. P = a present sum of money; F = a future sura of money, equivalent to P at the

end of n periods of time at discount rate of i; A = an end-of-period

payment (or receipts) in a uniform series of payments (or receipts)

n periods at i interest rates. over
Table 22 DISCRETE RATE-OF-RETURN FACTORS*

Single

Compound

Amount

Factor

SCAF

Single

Present

Value

Factor

SPVF

Uniform

Capital

Recovery

Factor

UCRF

Uniform

Present

Value

Factor

UPVF

Uniform

Sinking

Fund

Factor

USFF

Uniform

Compound

Amount

Factor

UCAF

Formula

( 1+i ) (1+i)

i(l+i)

(l+i) n

-l

(l+i)

n

~l

i(l+i)

n (l+i)

n-l

(l+i)

n-l

Civen

To Find

P

F

F

P

P

A

A

P

F

A

A

F

Period

n

•1=22 %

1 1.220 .8197 1.220 .8197 1.000 1.000

2 1.A88 .6719 .6705 1.A92 . A505 2.220

3 1.816 .5507 •A897 2.0A2 .2697 3.708

A 2.215 . A51A . A010 2.A9A .1810 5.52A

5 2.703 .3700 .3A92 2.86A .1292 7.7A0

6 3.297 .3033 .3158 3.167 .0958 10. AA

7 A. 023 .2A86 .2928 3.A16 .0729 13. 7A

8 A. 908 .2038 .2763 3.619 .0563 17.76

9 5.987 .1670 .26A1 3.786 .0AA1 22.67

10 7.306 .1369 .25A9 3.923 .03A9 28.66

11 8.912 .1122 • 2A78 A. 035 .0278 35.96

12 10.87 .0920 . 2A23 A. 127 .0223 AA. 87

13 13.26 .075A .2379 A. 203 .0180 55.75

1A 16.18 .0618 .23A5 A. 265 .01A5 69.01

15 19. 7A .0507 .2317 A. 315 .0117 85.19

16 2A.09 .0A15 .2295 A. 357 .0095 10A.9

17 29.38 .03A0 .2278 A. 391 .0078 129.0

18 35.85 .0279 .2263 A.A19 .0063 158.

A

19 A3.7A .0229 .2251 A.AA2 .0051 19A.3

20 53.36 .0187 .22A2 A.A60 .00A2 238.0

21 65.10 .015A . 223A A.A76 .003A 291.3

22 79. A2 .0126 .2228 A.A88 .0028 356. A 23 96.89 .0103 .2223 A.A99 .0023 A35.9

2A 118.2 .0085 .2219 A. 507 .0019 532.6

25 1AA.2 .0069 .2215 A.51A .0015 651.0

26 175.9 .0057 .2213 A. 520 .0013 795.2

27 21A.6 .00A7 .2210 A.52A .0010 971.0

28 261.9 .0038 .2208 A. 528 .0008 1186

29 319.5 .0031 • .2207 A. 531 .0007 1AA8

30 399.8 .0026 .2206 A.53A .0006 1767

31 A75.5 .0021 .2205 A. 536 .0005 2157

32 580.1 .0017 .220A A. 538 ' .000A 2632

33 707.7 .001A .2203 A. 539 .0003 3212

3A 863.

A

.0012 .2203 A.5A0 .0003 3920

35 1053 .0005 .2202 A.5A1 .0002 A78A

36 1285 .0008 .2202 A.5A2 .0002 5837

37 1568 .0006 .2201 A.5A3 .0001 7122

38 1913 .0005 .2201 A.5A3 .0001 8690

39 233A .000A .2201 A.5AA .0001 10603

AO 28A7 .000A .2201 A.5AA .0001 12938

* All formulae assume end-of-period payments.

P = a present sum of money; F » a future sum of money , equivalent to P at the

end of n periods of time at discount rate of i; A = an end-of-period

n periods at i interest rates.
Table 23 DISCRETE RATE-OF-RETURN FACTORS*

Single

Compound

Amount

Factor

SCAF

Single

Present

Value

Factor

SPVF

Uniform

Capital

Recovery

Factor

UCRF

Uniform

Present

Value

Factor

UPVF

Uniform

Sinking

Fund

Factor

USFF

—t, Uniform i Compound

Amount

Factor

UCAF

For- (l+i)

n

1 Ki+i) n (l+i)"-l i (l+i)

n

-l

mula

(l+i) n

(l+i)

n-l i(l+i)

n (l+i)

n-l i Given P F P A F A

To Find F P A P A F

Period

n

1 = 23 %

1 1.230 .8130 1.230 .8130' 1.000 1.000

2 1.513 .6610 .6784 1.47*1 , .4484 2.230

3 1.861 .5374 .4972 2.011 .2672 3.743

4 2.289 .4369 .4085 2.448 .1785 5.604

5 2.815 .3552 .3567 2.803 .1267 7.893

6 3.463 .2888 .3234 3.092 . .0934 10.71

7 4.259 .2348 .3006 3.327 .0706 14.17

8 5.239 ‘ .1909 .2843 3.5l8 .05426 18.43

9 6.444 .1552 .2722 3.673: .0422 23.67

10 7.926 .1262 .2632 3.799 .0332 30.11

11 9.749 .1026 .2563 3.902 .026 38.04

12 11.99 .0834 .2509 3.985. .0209 47.79

13 14.75 .0678 .2467 4.053- .0167 59.78

14 18.14 .0551. .2434 4.108' .0134 74.53

15 22.31 .0448 .2408 4.153 .010 92.67

16 27.45 .0364 .2387 4.189 .0087 115.0

17 33.76 • .0296 ' .2370 4.219 .0070 142.4

18 41.52 .0241 .2357 4.243. .0057 176.2

19 51.07 .0196 .2346 4.263 .0046 217.7

20 62.82 .0160 .2337 4.279 .0037 268.8

21 77.27 .0130 .2330 4.29,2 .0030 331.6

22 95.04 .0105 .2324 4.302

* .0024 408.9

23 116.9 .0086 .2320 4.311 .0020 503.9

24 143.8 .0070 .2316 4. 3l8 .0016 620.8

25 176.9 .0057 .2313 4.323. .0013 764.6

26 217.6 .0046 .2311 4.328 .0011 941.5

27 267.6 .0037 .2309 4.33.2 . 000.9 1159

28 329.1 .0030 .2307 4.335 .0007 1427

29 404.3 .0025 .2306 4.337 .0006 1756

30 497.9 .0020 .2305 4.339 .0005 2160

31 612.4 .0016 .2304 4.341 ' .0004 2658

32 753.3 .0013 .2303 4.342 ' .0003 3271

33 926.6 .0011 .2302 4.342 .0002 4024

34 1140 .0010 .2302 4.344 .0002 4951

35 1402 .0007 .2302 4.345 .0002 6090

36 1724 .0006 .2301 4.345 .0001 7492

37 2121 .0005 .2301 4.34° • .0001 9216

38 2609 .0004 .2301 4.346 .0001 11337

39 3208 .0003 .2301 4.346 .0001 13946

40 3946 .0003 .2301 4.34 7 .0001 17154

* All formulae assume end—of—period payments#

P = a present sum of money; F = a future sum of money, equivalent to P at the

end of n periods of time at discount rate of i; A = an end-of-period

payment (or receipts) in a uniform series of payments (or receipts)

n periods at i interest rates. over
Table 24 DISCRETE RATE-0F-RETURN FACTORS*

Single

Compound

Amount

Factor

SCAF

Single

Present

Value

Factor

SPVF

Uniform

Capital

Recovery

Factor

UCRF

Uniform

Present

Value

Factor

UPVF

Uniform

Sinking

Fund

Factor

USFF

Uniform

Compound

Amount

Factor

UCAF

Formula

(l+i) n

1 Kl+i) n

(l+i)

n-l i (l+i)

n

-l

<l+i) n

(l+i) n-l i(l+i)

n (l+i)

n-l i Given

To Find

P F P A F A

F P A P

' A F

Period

n

• is 2k%

1 1.240 .8065 1.240 .8065 1.000 1.000 2 1.538 .6504 .6864 1.457 .4464 2.240 3 1.907 .5245 .5047 1.981 .2647 3.778 4 2.364 .4230 .4159 2.404 .1759 5.684 5 2.932 .3411 .3642 2.745 .1242 8.048 6

-f 3.635 .2751 .3311 3.020 .0911 10.98

7n 4.508 .2218 .3084 3.242 .0684 14.62 8 5.590 .1789 .2923 3.421 .0523 19.12

9

t A

6.931 .1443 .2805 3.566 .0405 24.71 10

4 -t 8.59 .1164 .2716 3.682 .0316 31.64 11 10.66 .0938 .2649 3.776 .0249 - 40.24 12 13.21 .0757 .2596 3.851 .0198 50.S0 13 16.39 .0610 .2556 3.912 .0156 64.11 14

1 p 20.32 .0492 .2524 3.962 .0124 80.50 15 25.20 .0397 .2499 4.001 • .0100 100.8 16 31.24 .0320 .2479 4.033 .0080 126.0 17

*1 o 38.74 .0258 .2464 4.059 .0064 157.3 18 48.04 .0208 .2451 4.080 .0051 196.0 19

O A

59.57 .0168 .2441 4.097 .0041 244.0 20

A 1 73.86 .0135 .2433 4.110 .0033 303.6 21

A A 91.59 .0109 .2426 4.121 .0027 377.5 22 113.6 .0088 .2421 4.130 .0021 469.1 23 140.8 .0071 .2417 4.137 .0017 582.6 24 174.6 .0057 .2414 4.143 .0014 723.5 25 216.5 .0046 .2411 4.147 .0011 898.1 26

A -J 268.5 .0037 .2409 4.151 .0009 1115 27

A A 333.0 .0030 .2407 4.154 .0007 1383 28

a 412.9 .0024 .2406 4.157 .0006 1716 29

A A

512.0 .0020 .2405 4.159 .0005 2129 30 634.8 .0016 .2404 4.160 .0004 2641 31 787.2 .0013 .2403 4.161 .0003 3276 32 976.1 .0010 .2402 4.162 .0002 4063 33 1210 .0008 .2402 4.163 .0002 5039 34 1501 .0007 .2402 4.164 .0002 6249 35 1861 .0005 .2401 4.164 .0001 7750

36 2308 .0004 .2401 4.165 .0001 9611 37 2862 .0003 .2401 4.165 .0001 11919 38 3548 .0003 .2401 4.165 .0001 14781

39 4400 .0002 .2401 4.166 .0001 18329

40 5456 .0002 .2400 4.166 .0000 22729

* All

P = a formulae assume present sum of

end-of-period payments,

money; F * a future sum of money, equivalent to P at the

end of n periods of time at discount rate of 1; A = an end—of—period

payment (or receipts) in a uniform series of payments (or receipts) over

n periods at i interest rates.
Table 25 DISCRETE RATE-OF-RETURN FACTORS*

Single

Compound

Amount

Factor

SCAF

Single

Present

Value

Factor

SPVF

Uniform

Capital

Recovery

Factor

UCRF

Uniform

Present

Value

Factor

UPVF

Unifom

• Sinking

TAnd

Factor

USFF

Uniform

Compound

Amount

Factor

UCAF

Formula(l+i)

n

1 i(l+i)

n

(l+i)

n

*-l i (l+i)

n-l

(l+i)

n

(l+i) n

-l i(l+i)

n (l+i)n-l i Given P F P A F A

To Find F P A P A F

Period

n 1* 257

1 1.2500 .8000 1.250 .8000 1.000 1.000

2 1.563 .6400 .6944 1.440 .4444 2.250

3 1.953 .5126 .5123 1.952 .2623 3.813

4 2.441 .4096 .4234 2.362 .1734 5.766

5 3.052 .3277 .3718 2.689 .1218 8.207

6 3.815 .2621 .3388 2.951 .0889 11.26

7 4.768 .2097 .3163 3.161 .0663 15.07

8 5.961 • .1678 .3004 3.329 .0504 19.84

9 7.451 .1342 .2888 3.463 .0388 25.80

10 9.313 .1074 .2801 3.570 .0301 33.25

11 11.64 .0859 .2735 3.656 .0235 42.57

12 14.55 .0687 .2684 3.725 .0184 54.21

13 18.19 .0550 .2645 3.780 .0145 68.76

14 22.74 .0440 .2615 3.824 .0115 86.95

15 28.42 .0352 .2591 3.859 .0091 109.7

16 35.53 .0281 .2572 3.887 .0072 138.1

17 44.41 .0225 .2558 3.910 . .0050 173.6

18 55.51 .0180 .2546 3.928 .0046 218.0

19 69.39 .0144 .2537 3.942 .0037 273.6

20 86.74 .0115 .2529 3.954 .0030 342.9

21 108.4 .0092 .2523 3.963 .0023 429.7

22 135.5 .0074 .2519 3.970 .0019 538.1

23 169.4 .0060 .2515 3.976 .0015 673.6

24 211.8 .0047 .2512 3.981 .0012 843.0

25 264.7 .0038 .2509 3.985 .0010 1055

26 330.9 .0030 .2508 3.988 .0008 1319

27 413.6 .0024 .2506 3.990 .0006 1650

28 517.0 .0019 .2505 3.992 .0005 2064

29 646.2 .0015 .2504 3.994 .0004 3581

30 207.8 .0012 .2503 3.995 .0003 3227

31 1010 .0010 .2502 3.996 .0002 4035

32 1262 .0008 .2502 3.997 .0002 5045

33 1578 .0006 .2502 3.997 .0002 6307

34 1972 .0006 .2501 3.998 .0001 7885

35 2465 .0004 .2501 3.998 .0001 9857

36 3081 .0003 .2501 3.999 .0001 12322

37 3952 .0003 .2501 4.000 .0001 15403

38 4815 .0002 .2501 4.000 .0001 19255

39 6019 .0002 .2500 4.000 .0000 24070

40 7233 .0001 .2500 4.000 .0000 30089

* All formulae assume end-of-period payments.

P “ a present sum of money; F a future sum of money, equivalent to P at the

end of n periods of time at discount rate of i; A - an end-of-period

payment (or receipts) in a uniform series of payments (or receipts) over

n periods at i interest rates.
APPENDIX B

SELECTED BIBLIOGRAPHY

American Society for Testing and Materials, Discount Factor Tables , to be

published as an Adjunct to '‘Recommended Practice for Measuring Life-Cycle

Costs of Buildings and Building Systems" (Philadelphia, Pennsylvania:

ASTM, 1983).

American Society for Testing and Materials, "Standard Definitions of Terms

Relating to Building Economics," E833-83a, Annual Book of ASTM Standards

(Philadelphia, Pennsylvania: ASTM, 1981).

American Telephone and Telegraph Company, Engineering Economy: A Manager*

s

Guide to Economic Decision Making , 3rd Edition (New York, New York:

McGraw-Hill Book Company, 1977).

Anderson, Lee G. and Settle, Russell F., Benefit-Cost Analysis: A Practical

Guide (Lexington, Massachusetts: D.C. Heath and Company, 1977).

Eisen, Dennis, "Computer Connection: Program Description of a Fast,

Efficient Algorithm for the IRR," The Appraisal Journal , January 1982,

pp. 127-134.

Grant, Eugene L. and Ireson, W. Grant, Principles of Engineering Economy , 5th Edition (New York, New York: The Ronald Press Company, 1970).

Jaffe, Austin J., "Is There a New Internal Rate of Return Literature?,"

Journal of the American Real Estate and Urban Economics Association , Vol. 5, No. 4, Winter 1977, pp. 482-502.

Kerr, H. S., "A Final Word on FMRR," The Appraisal Journal , January 1980,

pp. 95-103.

Marshall, Harold E. and Ruegg, Rosalie T., Energy Conservation in Buildings:

An Economics Guidebook for Investment Decisions , National Bureau of

Standards Handbook 132, May 1980.

Marshall, Harold E. and Ruegg, Rosalie T., Recommended Practice for Measuring

Benefit/Cost and Savings-to-Investment Ratios , National Bureau of Standards

Interagency Report 81-2397, November 1981.

Marshall, Harold E. and Ruegg, Rosalie T. , Simplified Energy Design Economics:

Principles of Economics Applied to Energy Conservation and Solar Energy

Investments in Buildings , National Bureau of Standards Special Report 544,

January 1980.

B-l
Mentag, Geraldine, "Commercial Building Ownership Energy Conservation Cost

Analysis Model—A Follow-Up," ASHRAE Journal . September 1981, pp. 43-47,

Montgomery, J, Thomas and Raper, Charles F., "Equity Yields and the

Reinvestment Issue," The Appraisal Journal . October 1981, pp. 509-521.

Park, William R., Cost Engineering Analysis (New York, New York: John Wiley

and Sons, 1973).

Rawie, Carol Chapman, Estimating Benefits and Costs of Building Regulations:

A Step-by-Step Guide, National Bureau of Standards Interagency Report

81-2223, June 1981.

\

•, <

• Ruegg, Rosalie T. and Marshall, Harold E., "Economics of Building Design,"

Solar Age , Vol. 6, No. 7, July 1981, pp. 22-27.

Ruegg, Rosalie T., Life-Cycle Costing Manual for the Federal Energy Management

Program , National Bureau of Standards Handbook 135, December 1980.

Ruegg, Rosalie T.; Petersen, Stephen R.; and Marshall, Harold E., Recommended

Practice for Measuring Life-Cycle Costs of Buildings and Building Systems . National Bureau of Standards Interagency Report 80-2040, June 1980.

Sassone, Peter G. and Schaffer, William A., Cost-Benefit Analysis: A

Handbook (New York, New York: Academic Press, 1978).

Smith, Gerald W., Engineering Economy: Analysis of Capital Expenditures

(Ames, Iowa: Iowa State University Press, 1973).

Weston, J. Fred and Brigham, Eugene F., Managerial Finance , 7th Edition

(Hinsdale, Illinois: The Dryden Press, 1981). T~

1 \ 5 j «i i Wright, Maurice G., Discounted Cash Flow (New York, New York: McGraw-Hill,

1967).

B-2
MBS.114A (REV. 2-80) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See In struct/on s)

i. publication or REPORT NO.

NBSIR 83-2657

2. Performing Organ. Report No. i. Publication Date October 1983

4. TITLE AND SUBTITLE

RECOMMENDED PRACTICE FOR MEASURING NET BENEFITS AND INTERNAL RATES OF RETURN

FOR INVESTMENTS IN BUILDINGS AND BUILDING SYSTEMS

5. AUTHOR(S)

Harold E. Marshall

6. PERFORMING ORGANIZATION (If joint or other than NBS. see Instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. Stole. ZIP)

National Bureau of Standards

Department of Commerce

Washington, D.C. 20234

10. SUPPLEMENTARY NOTES

I I Document describes uriuc& a compuicr computer program; program, SF-185, or-ioj, FIPS riro juuwmc Software Summary, nummary, ias auauica, attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

This report describes how to calculate net benefits (savings) and internal rates of

return and how to use them in selecting building designs and building systems that

will be cost effective over time. Net benefits are the difference between positive

benefits or cost reductions (savings) and project costs, measured in present value

or annual value dollars. The net benefits method is used to decide if a given

project is cost effective and which size or design for a given purpose is most cost

effective when no budget constraint exists. The internal rate of return on an

investment is the compound rate of interest that, when used to discount the streams

of dollar benefits (savings) and dollar costs over a defined study period, will make

the two equal. The internal rate of return is used to determine if a given project

is cost effective, to compare the relative cost effectiveness of different purpose

projects competing for a limited budget, and, when calculated on incremental changes

in benefits and costs, to evaluate which size or design for a given purpose is most

cost effective. The report describes formulas for calculating net benefits and

the internal rate of return, their applications in selecting cost-effective projects,

and limitations in their use. This recommended practice for measuring net benefits

and internal rates of return will assist the private and public building communities

in making cost-effective decisions in the design, operation, maintenance, and

retrofit of buildings.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Adjusted internal rate of return; benefit-cost analysis; building economics; building

economics standard; constrained optimization; cost effectiveness; discounting; economic

efficiency; internal rate of return; net benefits; net present value

13. AVAILABILITY

(x5fcUnlimited

| | For Official Distribution. Do Not Release to NTIS

I"”] Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

[Xj^Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

74

15. Price

$ 10.00

U S COMM** DC S043-P80
Jr.' •

;

1 -r,

-

- .

.

• -

• ' .-*1

'» ^ . . -(

•

• •'* o

"* • -

-

t : ob *»oJL /

....

•

.

v • • ••

'

.

*.

' ..

• 1 -

•

• *

: . ;u

•. . i o : i

• J |3| t ltf: 7 ~