AlllOD - PUBLICATIONS

mi ff .

U.S. Department of Commerce

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Washington, D.C. 20234

Harold E. Marshall

Rosalie T. Ruegg

November 1981

NBSIR 81-2397

Recommended Practice

for Measuring Benefit/Cost

and Savings-to-Investment

Ratios for Buildings and

Building Systems
!

'i
NBSIR 81-2397

RECOMMENDED PRACTICE FOR MEASURING

BENEFIT/COST AND SAVINGS-TO-INVESTMENT RATIOS FOR

BUILDINGS AND BUILDING SYSTEMS

MATIUNAL tlUKliAi OF STAiNOA&UjB

UMMAUr

2 1982

ric>£ Cuff - Cutx. Gcif .

Harold E. Marshall

Rosalie T. Ruegg

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Washington, DC 20234

November 1981

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
,> t* I\ « '

'^1

^'i I

V’

'V

.i'
ABSTRACT

This report describes how to calculate a benefit-cost ratio (B/C) and a savings-toinvestment ratio (SIR) and how to use them in selecting building designs and building systems that will be cost effective in the long run. The B/C relates positive benefits, such

as revenues, to project costs in the form of a ratio. The SIR, a variation of the B/C,

relates project savings (i.e., cost reductions) to project costs in a ratio. It is used

when there are few if any positive cash flows from a project. The B/C and SIR can be used

to help answer such questions as: "Is a project cost effective?" "Which size and/or design

of a project is most cost effective?" "What priorities should be given individual projects

competing for a limited budget?" The report addresses different formulations of the ratios

and their implications for selecting cost-effective projects.

iii
PREFACE

The benefit-cost ratio (B/C) and a variation thereof, the savings-to-investment ratio

(SIR), are economic evaluation tools that compare discounted positive and negative cash

flows in the form of a ratio. These tools are useful in determining if projects are economically worthwhile, in designing and sizing projects, and in deciding project priorities.

Decreasing productivity in the construction industry coupled with rising costs of labor,

material, and energy have prompted builders, architects, engineers, building owners and

operators, and code writers to turn increasingly to economic evaluation techniques, including the B/C or SIR method, to identify building designs and building systems that will be

cost effective in the long run. A practical, standardized approach for calculating the

B/C and SIR for building decisions is needed.

This report has been prepared by the National Bureau of Standards (NBS) in support of an ongoing standards development activity in the American Society of Testing and Materials

(ASTM E-6, 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. This document has been submitted to ASTM E-6. 81, the Building Economics

Subcommittee, for its consideration in the development of a "Recommended Practice for

Calculating Benefit-Cost and Savings-to-Investment Ratios for Buildings and Building Systems." It is the second NBS report to be submitted to ASTM E-6. 81, and builds in part upon

the definitions and techniques (such as discounting) described in the first report, entitled

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

This report describes how to calculate the B/C and SIR and illustrates their uses. A

detailed discussion of alternative formulations of the ratios and of their respective

strengths and weaknesses is presented because the report is intended in part as a working

aid for the subcommittee members developing the standard. Specifically, it provides the

technical base for the development by the subcommittee of a standard method or recommended

practice for calculating B/C and SIR for investments in buildings and building systems.

Applying a standardized B/C and SIR method to building design and investment decisions will

help to assure that the cost effectiveness of alternative building projects can be compared

in a consistent and technically correct manner.

^ 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.

iv
The preparation of this report is one of a series of steps towards producing a comprehensive

set of standard recommended economic practices that will meet the diverse needs of the building community for measures of economic performance. Future reports will address other

economic evaluation techniques used in building decisions, including the internal rate of

return technique and the payback technique.

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. Thanks

are also due our colleagues at the National Bureau of Standards who assisted in the preparation of this report. Special appreciation is extended to Stephen Weber, Fred Stahl,

James Pielert, Robert Chapman, Stefan Leigh, Barbara Lippiatt, Laurene Linsenmayer, and

Ulesia Gray.

V
TABLE OF CONTENTS

Page

ABSTRACT iii

PREFACE iv

LIST OF FIGURES vii

LIST OF TABLES vii

1. INTRODUCTION 1

1.1 Scope and Organization 2 1.2 Procedures for B/C and SIR Evaluation 2

2. IDENTIFY OBJECTIVES, ALTERNATIVES, AND CONSTRAINTS 3

3. ESTABLISH ASSUMPTIONS A 4. COMPILE DATA 5

5. CONVERSION OF CASH AJIOUNTS TO A COMMON TIME BASIS 6 6. COMPUTE B/C OR SIR AND COMPARE ALTERNATIVES 8

7. RECOMMENDED APPLICATIONS 19

APPENDICES

A. SIR EVALUATIONS OF ENERGY CONSERVATION INVESTMENTS A-1

B. SELECTED B/C AND SIR REFERENCES B-1

vi
LIST OF FIGURES

Page

7.1 ALLOCATING THE BUDGET AMONG ALTERNATIVE PROJECTS RANKED BY B/C OR SIR 20

7.2 NETWORK DIAGRAM USING INCREMENTAL B/C ANALYSIS FOR PROJECT SIZING 25

7.3 EFFICIENT PROJECT SIZES 27

LIST OF TABLES

5.1 DISCOUNT FORMULAS 7

6.1 DIRECTION OF CHANGE IN B/C OR SIR INDUCED BY ALTERNATIVE PLACEMENT OF

OPERATING (0), MAINTENANCE (M), AND REPLACEMENT (R) COSTS, AND SALVAGE (V)

VALUE 10

6.2 HYPOTHETICAL PROJECTS EVALUATED WITH ALTERNATIVE FORMULATIONS OF THE B/C 13

6.3 ECONOMIC EFFICIENCY COMPARISON OF TWO B/C FORMULATIONS: PUBLIC INVESTMENT

DECISION 16

6.4 ECONOMIC EFFICIENCY COMPARISON OF TWO SIR FORMULATIONS: PRIVATE SECTOR

INVESTMENT DECISION 18

7.1 SENSITIVITY OF THE B/C AND NET BENEFITS TO VARIATIONS IN THE DISCOUNT RATE ... 22

7.2 PROJECT DATA 23

7.3 B/C's FOR PROJECT SIZE CHANGES 24

A.l EVALUATION OF WINDOW REPLACEMENT, ELECTRIC HEATING A-8

A. 2 EVALUATION OF WINDOW REPLACEMENT, GAS HEATING A-8

A. 3 EVALUATION OF STORM WINDOWS, ELECTRIC HEATING A-9

A. 4 EVALUATION OF STORM WINDOWS, GAS HEATING A-IO

A. 5 PROJECT SELECTION A-11

vii
#

’ v:T,,

V K"‘0. M

^Ilr T>f#'#%T4 i

s

( < • (> < ao. w. aV|^»^'8TjA tf

'.*'1 *;|-'';'(vrii

» > »•. •^ » • • . » • • T '•.*•..*-< f t I. ii - V»% 1 »‘» «v» \ r. * .

' fyBSi^- -•' * ^ Cl ' - iVf R*<

. r i

« a ^ t • fV « t • ^ ,^ -• 1 » # "f^ ’

f£t

’ if i^

‘ »

.'W

• « ‘V*’ • » • -

^1. /V) .3T30O fl> ‘'%)tS^rjU *

4 1 Vifc'i 4 «* - » -r ., i’i*< ...» i i .«_*» i .' - t ' * .«iw4 . . ri.iAvtl

Eiv-

\f

-** v.TR^jit23^n<.f :)j.»iri . « . :)X«ancj^

r _ '.‘i;

'-' 'i'*: V tsf* .'*'. v» » wfii . . , . . |(' /» ^ .^i f

-? » » » •_. j tit ut > . .» • » M*~ '“^'

(’55

• •^' *m^ O'#-? h=4k

-

f3# ^

> •<

:

T’ ^ A

' '

- •, ^ •

:fr4i!i I - ^'V'»'<** «•»<•#»»»•» t T-

- h\ .>

L

ii.L,' 'I c^vi>, Af* n ^‘xm * • •*

^m»$$»04* «»«» •• f * ' • \r-A • * 4 « •

v^-i

'•1

- » 4 (r> -4*^v . ^ • 4 * * •

E!

1 r 0 t '

: y 4 . ?|f :t a)i' ,f?)omuv
1. INTRODUCTION

The benefit-cost ratio (B/C) and a variation thereof, the savings-to-investment ratio

(SIR), are part of a family of economic evaluation techniques that provide measures of

economic performance of an investment over some period of time extending into the future.

This family of techniques includes, in addition to B/C and SIR, the techniques of net

benefits analysis, internal rate-of-return analysis, life-cycle cost analysis, and payback

analysis .

The B/C and SIR are the focus of this report, the second in a series of National Bureau of

Standards (NBS) reports on recommended practices for applying economic evaluation techniques

to building decisions. The life-cycle cost technique was the topic of the first report in

the NBS series, entitled Recommended Practice for Measuring Life-Cycle Costs of Buildings

and Building Systems, NBSIR 80-2040. Future publications are planned for the internal rateof-return and payback techniques to complete a comprehensive package of recommended practices

for economic analysis of buildings and building systems.

The B/C and SIR are numerical ratios which indicate the economic value of a project by the

size of the ratio. A ratio of less than one indicates a project that is uneconomical; a ratio of one indicates a project whose benefits or savings just equal its costs; and a ratio greater than one indicates an economical project. The larger the ratio, the more the

dollar benefits or savings exceed project costs. The B/C is used when the focus is on positive benefits, such as revenues, relative to project costs. The SIR, a variation of

the B/C ratio, is used when the focus is on project savings (i.e., cost reductions) relative

to project costs. Issues to be addressed in this report in addition to the formulation of

^ For a general treatment of these techniques, see a benefit-cost or engineering economics

textbook, such as E. J. Mishan, Cost-Benefit Analysis; An Introduction (New York, New

York: Praeger Publishers, 1971), or Gerald W. Smith, Engineering Economy; Analysis of

Capital Expenditures (Ames, Iowa: The Iowa State University Press, 1973). For a specific

treatment of these techniques applied to energy conservation and renewable energy in

buildings, see one of the following: 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 Special Report 544, January

1980; Harold E. Marshall and Rosalie T. Ruegg, Energy Conservation in Buildings ; An Economics Guidebook for Investment Decisions , National Bureau of Standards Handbook 132, May

1980; Rosalie T. Ruegg, Life-Cycle Costing Manual for the Federal Energy Management Programs , National Bureau of Standards Handbook 135, December 1980; and Rosalie T. Ruegg ^

al . , Life-Cycle Costing; A Guide for Selecting Energy Conservation Projects for Public

Buildings , National Bureau of Standards Building Science Series 113, September 1978.

1
the ratios are their application to accept-re ject decisions, to sizing and design problems,

and to setting priorities among independent projects competing for limited resources.

1.1 SCOPE AND ORGANIZATION

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

calculating and interpreting the B/C and SIR of building designs and systems. Sections 2 through 6 set forth the procedural framework within which the B/C or SIR technique is applicable. Section 2 identifies the objectives, alternatives, and constraints for a B/C or SIR

evaluation. Section 3 lists assumptions typically required for calculating the B/C or SIR

and suggests guidelines for selecting parametric values to be used in the calculation. Section 4 identifies major categories of benefits, savings, and costs typically treated in B/C

and SIR evaluations. Section 5 introduces "discounting" for converting cash flows spread over time to their equivalent values at a common time, and provides discounting formulas.^

Section 6 discusses issues regarding the formulation of the B/C and SIR and gives recommended

calculation procedures. Special attention is given the placement of benefit and cost items

in the ratios because varied placement can alter the ratios and bias the results. Section

7 explains the use of the B/C or SIR in different kinds of applications, such as to accept

or reject a project, to allocate limited investment funds among competing projects, and to

size and design a project. Appendix A illustrates the solution to a sample building investment problem by the SIR technique. Selected references conclude the report in appendix B.

1.2 PROCEDURES FOR B/C AND SIR EVALUATION

The recommended framework for using the B/C and SIR techniques to evaluate a building

design, project, or investment decision can be summarized in five procedural steps:

1. Identify Objectives, Alternatives, and Constraints,

2. Establish Assumptions,

3. Compile Data,

4. Discount Cash Flows to a Comparable Time Basis, and

5. Compute B/C or SIR and Compare Alternatives.

In sections 2 through 6, each of these steps is addressed further.

^ The first report in this series. Recommended Practice for Measuring Life-Cycle Costs of

Buildings and Building Systems , discusses discounting, inflation, uncertainty, and other

topics that are equally relevant to the B/C and SIR techniques as to the life-cycle

costing technique. The earlier report is cited where appropriate.

2
2. IDENTIFY OBJECTIVES, ALTERNATIVES, AND CONSTRAINTS

First, the problem to be addressed by the B/C or SIR technique should be clearly specified.

This entails identifying the objective of the investor or decision maker, as well as any

constraints to the achievement of the objective. Alternative approaches to reaching that

objective within the designated constraints are then identified.

An example of a building investment problem that could be evaluated with the B/C technique

is choosing the locations for new stores in a national chain. The objective is to maximize

profits or net benefits from investments in new stores. The alternatives are the different

locations where stores might be built. The constraint is that there is only enough investment capital for a few stores, whereas stores in many locations would likely yield profits.

The B/C technique will guide the decision maker to those store locations where net benefits

per investment dollar will be maximized.

An example of a problem that could be evaluated with an SIR is selecting retrofit options

for conserving energy in an existing building. The objective is to select that combination

of options that maximizes net savings for the available budget. Alternatives might include

attic insulation, weatherstripping, installing a heat pump, adding storm windows, or a com- bination thereof. A limited dollar budget for energy conservation is the investment constraint. The SIR technique will guide the decision maker to that combination of retrofit

options that will maximize net savings from investing the conservation budget.

3
3. ESTABLISH ASSUMPTIONS

A number of assumptions usually must be made in order to perform an economic evaluation of

a design or investment problem. The solution to a problem may vary considerably depending

on the assumptions. To arrive at realistic solutions, it is important to select carefully

the assumed values for critical parameters. Sensitivity analysis can be used to identify

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

those parameters.

Assumptions that are often significant in applying the B/C or SIR technique concern the

study period over which the evaluation is to be made, the value of the discount rate, the

level of taxes, and the rate of Inflation. General guidelines for selecting values for

these parameters are found in Recommended Practice for Measuring Life-Cycle Costs of

Buildings and Building Systems and elsewhere,^ and will not be discussed here.

^ Ruegg, Petersen, and Marshall, Recommended Practice for Measuring Life-Cycle Costs , pp. 9-11; Marshall and Ruegg, Simplified Energy Design Economics , pp. 27-34; and Marshall

and Ruegg, Energy Conservation in Buildings; An Economics Guidebook, pp. 1-42.

4
4. COMPILE DATA

For calculating a B/C or SIR for alternative building designs, building systems, or building

practices, typical cost categories for which data will be needed are a) investment costs,

including the costs of planning, design, engineering, construction, purchase, installation,

and financing; b) non-fuel operating and maintenance costs, including the materials and

labor costs for routine upkeep and operation other than energy; c) repair and replacement

costs, including future costs to repair or replace a building system or component that wears

out, fails, or is damaged, and related costs, such as design costs for the replacement and

insurance costs less reimbursements; d) energy costs; and e) property and capital gains

taxes. Data will also be needed for any positive dollar benefits, such as income or revenue,

resale values, and cash grants, as well as for any cost-reducing items, such as energy or maintenance cost savings, tax deductible expenses, and tax credits. Other data requirements

are income tax rates, discount rates, depreciation methods and periods, and financing terms.

Data may be compiled from published and unpublished sources, they may be estimated, or they

may be assumed.^

^ Ruegg's Life-Cycle Cost Manual for the Federal Energy Management Program has a chapter on "Data and Assumptions" (pp. 37-48) and lists 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.

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

Once the pertinent data have been collected, all cash amounts must be converted to a common

time basis, i.e., discounted to time-equivalent present values or annual values. This con- version of amounts to time-equivalent values (often called "discounting") is performed by

applying discount formulas, or corresponding discount factors calculated from those formulas,

to the estimated benefit and cost data associated with a given design or investment alternative. The discount formulas incorporate the "discount rate," a rate of return which should

reflect the investor's opportunity cost; i.e., the rate of return available on the next best

alternative investment. Similarly, discount factors are based on specific discount rates.

Table 5.1 lists the most commonly used discount formulas, indicates their use, and gives

their algebraic form. Discount factors calculated for alternative discount rates are published in many textbooks on benefit-cost analysis and engineering economics.^

The first step in the time-equivalency conversion is to select a common time to which all

cash amounts are adjusted—either (1) the present, whereby all cash amounts are converted

to an equivalent value occurring now, i.e., to a present value , or (2) annually, whereby

all cash amounts are converted to a time-equivalent value occurring in a uniform amount each

year over the study period, i.e., to an annual value . Then the appropriate formula or factor, incorporating the investor's opportunity cost as indicated by the discount rate, is

applied to each cash amount to convert it to its equivalent value at a selected time. The

O case example in appendix A illustrates the process.'^

^ See, for example. Smith, Engineering Economy; Analysis of Capital Expenditures . A set of

discount factors based on the formulas shown in table 5.1 and including the UPW* factors

for a range of escalation rates can be found in Marshall and Ruegg, Energy Conservation in

Buildings; An Economics Guidebook , and a set of discount factors including UPW* factors

based on the U.S. Department of Energy energy price escalation rates by region of the U.S.

and by type of fuel can be found in Ruegg, Life-Cycle Cost Manual for the Federal Energy

Management Program , appendix B.

^ Step-by-step procedures for discounting are found in Marshall and Ruegg, Simplified Energy

Design Economics , pp. 16-20. For a discussion of how to handle inflation in discounting,

see Recommended Practice for Measuring Life-Cycle Costs, pp. 14-15.

6
TABLE 5.1 DISCOUNT FORMULAS

Formula Name Illustration

Single Compound

Amount Formula (SCA) S _!il

Single Present Value

Formula (SPW) ra F

Uniform Sinking Fund

Formula (USF) 113 + mi • • A? —

Uniform Capital Recovery

Formula (UCR) m + IK|.... |T?]

Uniform Compound

Amount Formula (UCA) m +

• • F?

[ Uniform Present Value Formula (UPW) R - + E ••••[a:

Uniform Present Value

Formula Modified (UPW*) p? ^

m 3...i

Where

:

Use Algebraic Form

To find F when F = F • (1 + i)^

P is known

To find P when

F is known

P = F

(1 + i)‘

To find A when

F is known (1

To find A when

P is known A = P- i(l + i)^

(1 + i)"-l

To find F when

A is known

F = A • (1 + i)*^-!

i

To find P when

A is known

To find P when A is

6 escalating at rate e

P = A . (1 + i)^-i

i(l + i)

P = A + e\ r1 -

fi + e\

li - ej u + ij J

P = a present sum of money.

F = a future sum of money.

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

N = number of interest or discounting periods.

A = an end-of-period payment (or receipt) in a uniform series of payments (or receipts) over w periods at i interest or discount rate.

e = rate of escalation of A in each of N periods.

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

value to be found.

^ To find P when A is escalating at a different rate over each of k escalation periods,

P =

+

M=i

+ e .T-TT

1 +

1 +

"1

11^ P

1 + e- rrr

\l + i

where n^ = the length of the period for a given escalation rate in a given period, and the subscript

h = the escalation period, and

7
6. COMPUTE B/C OR SIR AND COMPARE ALTERNATIVES

After all cash amounts relevant to a given investment have been discounted to present or annual values, they can be used to calculate the B/C or SIR. Although the concept of the

B/C and SIR is simple, there is some controversy regarding their formulation. The difficulty arises when a project involves costs and benefits that are subject to varying interpretation regarding placement in the numerator or denominator of the ratios. The formulation

is important because changes in the ratios can be induced by changing the placement of cost

and benefit items, and biasing effects detrimental to economic efficiency can result.^

Changing the placement of items from the numerator to the denominator of the B/C or SIR

will not cause a project which appears cost effective by one version of the ratio to appear

uneconomical by a different version."^ The placement, however, can affect the numerical

value of a ratio for a given project. Moreover, the placement can affect the relative values

of ratios for different projects and, thereby, their priority ranking.

^ For projects that give rise to a single kind of cost, such as an initial investment for

purchase and installation, there is no choice in the formulation of the ratio and no biasing effects to consider.

The following proof shows that changing the placement of items from the numerator to the

denominator will not cause a project which is cost effective by one measure to be cost

ineffective by another.

Given: B-Q-M-R + V > ^ prove 1 > 1.

I I + O + M + R- V

Proof: If B-O-M-R+V > then

B-0-M-R+V>I,-

B>I+0+M+R-V, and therefore, since I+0+M+R-V> zero,

2 > 1. Hence, if benefits (or savings) exceed costs under one I+O+M + R- V

formulation of the ratio, they will also exceed them under the other formulation.

(Variables are defined in the text, under equation 6.1.)

8
Table 6.1 shows the direction of change of the B/C and SIR (i.e., whether the ratio increases

or decreases) as a function of placement of given cost or salvage items in the ratio formulations. The cost items—operation, maintenance, and replacement—and salvage value appear

in the left hand column. The items may increase or decrease as a result of an investment.

The second column of table 6.1 indicates all possible placements in the denominator or numerator of both positive and negative cost and salvage items. The third column Indicates by means of arrows and dashes whether the ratios increase (+), decrease (+), or remain

unchanged ( —) when the placement is changed for any given item; that is, when an item is

transferred from the denominator (numerator) to the numerator (denominator). Since the

direction of the change in the ratio varies depending on the value of the ratio before the

change in placement i.e., whether the ratio is >, =, or <1, subcolumns are provided for

each of the three possible cases.

An example from the table will help illustrate how a change in placement can affect the B/C

and SIR. For a ratio greater than one, the value of the ratio will be higher if a savings

in operating cost is subtracted from other costs in the denominator rather than added to

other savings or benefits in the numerator. Row three shows further that, if the ratio had

previously been equal to one, the ratio would have remained the same regardless of the

placement of the savings in operating costs, and if the ratio had been less than one, it

would be lower if the savings were subtracted from the denominator.^

Since the ratio values may vary depending on the placement of cost and salvage items,

thereby influencing the investment decision, it is important to select a formulation for

the ratio that best satisfies the investor's objective. The remainder of this chapter

identifies alternative formulations of the B/C and SIR that are commonly used, identifies

the biases that appear to be inherent in some of the formulations, and determines which

formulations are appropriate for different investment objectives.

Differences in B/C and SIR formulations involve differences in the placement of operating

costs (0), maintenance costs (M) , replacement cost (R), and salvage value (V). Equations

6.1 through 6.8, presented below, depict the most widely used versions of the ratios. In

equations 6.1 and 6.2 all costs and salvage appear in the denominator. Equations 6.3

through 6.8 differ in that items other than benefits and savings appear in the numerator.'^

^ Table 6.2 gives numerical examples that illustrate the direction of changes in the ratios

described in table 6.1.

2 A discussion of B/C formulations is contained in Smith, Engineering Economy , Chapter 11.

(Smith does not address the treatment of replacement costs and salvage value.)

9
TABLE 6.1 DIRECTION OF CHANGE IN B/C OR SIR INDUCED BY ALTERNATIVE PLACEMENT

OF OPERATING (0), MAINTENANCE (M) , AND REPLACEMENT (R) COSTS AND

SALVAGE (V) VALUE

Costs and Salvage Items

1 Direction of Change of Ratios

Placement of Cost Item | B/C B/C B/C

1 SIR > 1 SIR =1 SIR < 1

1

Increase in 0, M, and/or

R Cost and/or Decrease

in V Value

Denominator | I — +

1 Numerator 1 I — I

1

Decrease in 0, M, and/or

R Cost and/or Increase

in V Value

Denominator 1 i — 4-

1 Numerator 1 I — I

1

Notation; I indicates B/C or SIR is lower than it would be with the alternative

placement

.

i indicates B/C or SIR is higher than it would be with the alternative

placement

.

— indicates B/C or SIR is unchanged by the choice of placement.

10
The equations are presented in pairs, the odd numbered equations giving the B/C and the

sequential, even numbered equation, the counterpart SIR.^

B/C = B (6.1)

I + O + M + R- V

SIR = S

9 (6.2)

I + O + M + R- V

where B = present value benefits such as revenue or other positive effects,

I = present value investment costs of the project,

0 = present value operating costs, including energy costs associated with the project,

M = present value maintenance costs,

R = present value replacement costs,

V = present value resale or residual value resulting from disposal of assets,

S = cost reductions resulting from the project.

B/C = B + V

I + 0 + M + R

(6.3)

and

SIR S + V

I + 0 + M + R

’ (6.4)

where variables are as defined above.

and

B/C = B - 0 - M

I + R - V

9

SIR = S - 0 - M

I + R - V

where variables are as defined above.

B/C = B-O-M-R + V

I

and

SIR = S ~0~M~R+V

where variables are as defined above.

(6.5)

( 6 . 6 )

(6.7)

( 6 . 8 )

^ For the purpose of exposition, it is assumed that all amounts are already in present

value or annual value dollars. Thus the discounting operation is not shown in the

equations.

11
Most other versions of the B/C or SIR that one encounters in the financial analysis

literature and in application are equivalent to one of the above formulations.^

To analyze the equations for bias, it is necessary to have an efficiency criterion. The

criterion proposed in this study is the maximization of net benefits (NB) for the available

investment budget, where the budget can be specified for the current year or a multi-year

planning period. Net benefits are the difference between benefits and costs, evaluated in

present or annual value dollars. Equations 6.7 and 6.8 are the only equations that consistently satisfy this criterion. Calculations based on either of these equations will lead

to the selection of designs, sizes, or combinations of projects that satisfy the efficiency

criterion. Applying any of the other equations may under certain conditions result in a bias against cost-effective project selection. That is, their use may lead to the selection

of designs, sizes, or combinations of projects which do not maximize net benefits per investment dollar. The magnitude of the bias from using equations 6.1 through 6.6 is a function

of how much the B/C and SIR values are distorted from the values that would be calculated

from equations 6.7 and 6.8.2

Table 6.2 shows B/C's calculated for a series of seven hypothetical projects. (The SIR

calculations are not shown but would correspond to the counterpart B/C examples.) The

purpose of the table is to show how the B/C will differ for a given investment project,

depending on the formula that is used, and how the choice will affect project rankings.

To focus on the change in the ratio that may be induced by simple changes in placement of

cost and benefit components, each of the projects is assumed to yield the same present

value net benefits of $50,000. Projects 1 and 2 both have investment costs (I) of $50,000,

but project 1 results in positive revenue (B) of $100,000, while project 2 yields $80,000

of revenue and $20,000 of reductions in operating (0) and replacement (R) costs. Projects

3 and 4 have equal B values of $100,000, but their $20,000 of maintenance (M) and replacement (R) costs are split differently. Project 5 has $95,000 of B and $5,000 of residual

value (V), with $50,000 of costs made up of I, 0, & R. Project 6 has the highest investment

cost, but $20,000 of it is offset by 0 and R cost reductions. Project 7 has the largest

positive and negative cash flows, but the same net benefits as the other projects.

^ For an industry version of the B/C equation, equivalent to equation 6.1 above, see American Telephone and Telegraph Company, Engineering Economy; A Manager’s Guide to

Economic Decision Making , 3rd Ed. (New York, New York; McGraw-Hill Book Company, 1977),

pp. 411-412.

2 Note that investors may prefer in some cases a formulation of the ratio that has a bias

as defined here, because they may wish to maximize the return on a particular type of

funds, such as current account expenditures which might be the constraining resource.

12
TABLE

6.2

HYPOTHETICAL

PROJECTS

EVALUATED

WITH

ALTERNATIVE

FORMULATIONS

OF

THE

B/C®

r
Comparing the project ratios in table 6.2 derived by each equation, one sees inconsistencies

in results. Equations 6.1 and 6.3, for example, would give priority to project 2 over the

other projects; equation 6.5, priority to project 3; and equation 6.7, equal priority to

projects 3, 4, and 5. Equation 6.3 gives lowest priority to project 7, while equations 6.5

and 6.7 give it higher priority than some of the other projects. Each of the equations

gives the same economic efficiency indicator for project 1 because it involves only one category of cost, 1.

How do these differences in results among the B/C’s affect the economic efficiency of

project selection? Comparing projects 1 and 2, one finds not only the same investment

requirements, but also the same present value of future net cash flows (i.e., revenues (B)

plus cost reductions in 0 and R). The use of equations 6.1, 6.3, or 6.5 to choose between

projects 1 and 2 would result in the selection of project 2 over 1. This occurs because

the future benefits accrue in the form of cost reductions rather than revenues, and, as was shown in table 6.1, subtracting values from the denominator of a cost-effective project

results in a higher ratio than adding the same values to the numerator.

Projects 3, 4, and 5 are identical to one another in terms of investment costs and benefits;

they differ only in the apportionment of their future amounts among operating, maintenance,

replacement costs, and salvage value. Yet only equations 6.3 and 6.7 give identical ratio

values and rankings for the three projects, and their ratio values differ one from the

other (2 versus 2.67).

To determine how the different formulas rate on efficiency grounds, the first three

equations can be compared against equation 6.7, which best satisfies the criterion of

maximizing net benefits per investment dollar. Guided by equation 6.7, the decision maker

would choose projects 3, 4, and/or 5 with low initial investment requirements in favor of

projects 1, 2, 6, and/or 7 with higher investment requirements. With a limited budget of

$100,000 for example—and assuming independent, non-mutually exclusive projects—net benefits would be maximized by investing $30,000 each in projects 3, 4, and 5, spending $90,000

of the $100,000, and obtaining $150,000 in net benefits. This is clearly more efficient

than spending the $100,000 in projects 1 and 2, as equation 6.3 might suggest, and obtaining only $100,000 in net benefits. In short, table 6.2 shows that equations 6.1, 6.3, and

6.5 (and their counterparts, equations 6.2, 6.4, and 6.6) provide unreliable indicators of

a project's return on investment, because their priority rankings of projects depend on the

cost composition of the projects rather than on their comparative economic efficiency.

Their use may cause a decision maker to select projects that together do not meet the

economic efficiency criterion of maximizing net benefits for the available investment

budget.

Detailed examples of the biasing effects of using the conventional B/C and SIR equations

and the economic inefficiencies that may result are illustrated further in tables 6.3 and

14
6.4. Table 6.3 compares equation 6.1 to equation 6.7 with respect to the economic efficiency

criterion of maximizing net benefits per investment dollar. The hypothetical problem is to

assign priority to the construction of three public recreation facilities designated Pro~

jects A, B, and C. For simplicity, the projects are assumed equal in their initial investment cost of $1,000 (col. 5), but different in the present value of their benefits and the

level of the operating costs incurred by the public authorities. The present value of their

future total cash flows, i.e., benefits less operating costs (col. 4) differ, as well as their total net benefits (col. 6). With the objective of obtaining the maximum net benefits

from the expenditure of limited investment funds, the public authorities will rank the

three potential recreation projects in declining order of their B/C values.

As may be seen by comparing columns 7 and 9, the B/C's for each project differ depending on which B/C formulation is used. As may be seen by comparing columns 8 and 10, the assigned

priorities also differ by B/C formulation. Equation 6.1 gives first priority to Project B,

while equation 6.7 indicates that Project A—the last ranked by equation 6.1—is preferred.

Looking to the net benefits column (col. 6), we can see that the project rankings provided

by equation 6.7 are consistent with maximizing total net benefits from the public recreation

facilities budget. For example, if a budget of only $2,000 were available, equation 6.7

would indicate Projects A and C for a total net benefits of $5,500, whereas equation 6.1

would indicate projects B and C for a total net benefits of only $4,000.^

^ In the examples of tables 6.3 and 6.4, investment costs are held constant across projects,

causing the project rankings produced by the net benefits measure (col. 6) to be consistent with the rankings provided by equation 6.7 and 6.8. Furthermore, with the small

number of projects given in the table, one could easily arrive at the economically efficient selection of projects in the face of a budget constraint without a formal ranking

device, simply by summing and comparing the net benefits of alternative combinations of

projects. It should be noted, however, that when investment costs differ among projects,

ranking projects in descending order of their individual net benefits may produce project

rankings that are inconsistent with the rankings provided by equation 6.7. More specifically, if investment costs vary among projects, selecting projects in descending order of

their individual net benefits until the budget is exhausted may not produce a selection

of projects which together yield maximum total net benefits, whereas selecting projects

in descending order of their ratios calculated by equations 6.7 and 6.8 will accomplish

this objective, provided "lumpiness” in project costs does not prevent spending all or most of the available budget. The shortcoming of the net benefits measure for project

ranking may be seen in its failure to distinguish projects that require different investment outlays but which yield the same dollar net benefits. For example, a project that

costs $10,000 and yields benefits of $20,000 yields the same net benefits as one which

costs $100,000 and yields benefits of $110,000. Hence, equations 6.7 and 6.8 are superior

both to other B/C and SIR measures and to the individual project net benefits measure for

ranking projects for investment priority.

15
TABLE

6.3

ECONOMIC

EFFICIENCY

COMPARISON

OF

TWO

B/C

FORMULATIONS:

PUBLIC

INVESTMENT

DECISION

CJ

4J W)

PQ •H C3 •H o O ^ iH CO CM

.H

C • V4 05 vD P- Co

•H

O

4J •H

O o O cd 4J o in m

05 ^ • • • cr pp: <r CM CO W

05O fo

4-J bO

CQ •H

a

l-l •H O ^ 00 CO rH CM

rH *H c ^ • V-i 05 VD Pn co

•H

O

4J •H CO o O

cd 4-) <5- o o

CJ 05 N—

• • • cr rH CM w

in

N—

4-> CO 1 <D 4-1 o o o

;s .H CO o o o U-l a w o in in

fH QJ 1 1 »> 0^ 05 3 CQ CO rH CM

4J CO

w CM </> </>

O PO 'w' H II

v£) >.- iH 4J

05 3

•H CL) o o O e U o o O

•H 4J CO o o O

CJ COO in

r #»

M 3 U

W rH rH rH > <o- <J> <J> > 3

Qu

M

CO

4-) CO

w

O CO

4^

O

w

O o o

•H CO 1 o o o

CO bO ^ o m in

<D <0

C CM •» r*

a hO •H 'w' CM CO

05 4-1 II </> <ry PQ

60 3 ^l-l -O- 3 w&o 3

•H

4-1 o O o 3 W o o o

l-l 4-1 ^ o in in

0)

W fO

A 1 a O w vO rH o u </> <j> <J> >PLI M

4-1 o o o

•H o o o > IM o o o

Pj 05 CM •* C w o CM in

05 rH

pa </> <J> </>

CO

05>

4-1 •H o 4J

<0 cd •1—1 C ^ < PO o 5-1 w 1-J 05

p-' 4-) c

CO<

Qi>

.H

tfl c

l-l (U

4-1 f-H«

4JO

QJ 1-) O

o.po

M-l

60

(0

vOeo

.H

4-1 CO3

O'

(Uo

4J

60c

.H

V4Ooo

CO T3

<0

c6 O

OO•

II oo00K vD

•co- 1 ooooI—I

</> ooo

</y

II O

P3

<c

0>>

COc

V4

0)

COy

<u •nO

V4&

V4o

<4-(

60

CO

rVDco

CO3

cr

0)o

4-1

603

.H•3

)-i OOO

CO T3

<0

4-1 CO

l-^3o

CO u

16
The hypothetical problem depicted in table 6.4 is one of choosing the most economically

efficient energy conservation project for a building given a budget of $1,000. The budget

allows only one of the three available projects (D, E, and F) to be undertaken. Project

D, an automatic environmental control system requiring regular inspection and adjustment

for satisfactory performance, saves the most in energy dollars, but gives rise to substantial maintenance cost. Project E is wall insulation requiring only an initial purchase and

installation charge. Project F entails replacement of windows with a type that, unlike the

old, does not require periodic recaulking and painting. SIR's calculated by equation 6.2

(col. 7) lead to the selection (col. 8) of Project F, which costs $1,000 (col. 5) and

saves $3,000 in energy (col. 2) and another $600 in reduced maintenance (col. 3), for a total net savings of $2,600 (col. 6). Use of the SIR's calculated by equation 6.8 (col. 9)

would result instead in the selection (col. 10) of Project E, costing $1,000 (col. 5) and

saving $3,800 (col. 2) in energy, for a total net savings of $2,800 (col. 6). Hence, use

of equation 6.8 to guide project selection would in this case revise net savings by $200.

In summary, equations 6.7 and 6.8 provide a consistent weighting of all future cash flows,

provided investment cost (I) is an initial outlay, with no financing over time. In the case where I is funded out of equity, equations 6.7 and 6.8 will give highest ranking to

cost-effective projects which have the largest return relative to the initial investment.

Thus, these equations would be best for project selection when the investor desires to

maximize the return on equity investment funds.

To the extent that investment costs are financed over time, equations 6.7 and 6.8 will give

greater weight to future investment costs (because they are placed in the denominator) than

to other future cash flows, ^ an effect which may or may not be desired by the investor. If

a project is financed from a combination of equity and borrowed funds, for instance, the

investor whose aim is to achieve the greatest return from equity funds may choose to formulate the B/C or SIR to place the downpayment in the denominator and to treat the present

value of investment financing as an operating cost in the numerator. A ratio formulated in

this manner would give greater priority to projects requiring less equity, other things

being the same.

In calculating B/C's and SIR's, it is also important to take into account the nature of the

problem to which they are applied. If used to compare projects that are not mutually

exclusive, the B/C and SIR are calculated on project totals for the relevant data. But if

used to compare alternative designs and/or sizes of a given project, i.e., mutually exclusive decisions, the B/C and SIR are calculated on incremental data corresponding to each

step towards a larger project cost. These two approaches are demonstrated in the next

section on applications.

1 This applies to projects for which the B/C or SIR is greater than one.

17
TABLE

6.4

ECONOMIC

EFFICIENCY

COMPARISON

OF

TWO

SIR

FORMULATIONS:

PRIVATE

SECTOR

INVESTMENT

DECISION

rO

00co

•H

4-1

Cfl 3

cr W

cr

4-1 00

•H (C

t-l -H

O 0^1-4

•H

C

>-4 CO Pj do

•H

4-) a^

cd ^

4-» <5C •H c

V4 tH ^ O ^ 00

•H

C

W v-i cd

4-J r--

cd

(D

Cd

4-J oH >

P-.

<voc

cdc

Q)

4-J c>>CL,

<D>

4-> ‘H U 4-J 0) CdC tH O V-i w Vh 0)

D-, 4-1

O o

00 ovO

csi 00 o

00 oin w

CN

00 o

r-- m

II oooA

</> oo

cn

OA

</> oo

CO

CM

</> ooo

v£) </> 1 II II o o o

CO o o o oi > ^ 00 M

M

CL 1 #» CO

y«-N CM CM CM

CM </> </> </>

Vw''

II P D

<u (U vO > >

Sw/ •H •H

4J 4-4 Cd n) C Cu

O o o q; 0) o o o 4.) 4-1 OD o o o rH r—1 o m A Cd tC O w rH 1—1

rH

</y <J> <fy 4J 4-1 O o

<D OJ

•r-) o o u u ^ CO c ^ O o o

CL

a

rH 1 O o o

Cch r-- 00 vO

O o

CM

AA «s M-4 14-1

X CO CO CO

CO II </> </> </>

cd • • O <t 00 00 • •

0) OJ • #, CM 00 • • o o o 4-> o o C W CO sC> O o O CO •H •H O w CM 4J 4J

<n- </> <J> Cd Cd + I 0 d

cr cr

0) (D o o

4-» 4J

cn o o o 00 o o oo 00 C ^ o 00 o c c

•H CM »% «K •H •H > w CO CO no

cd </> <J> <J>

U ooo

cd

0)

Cd o ooo

cd T3

a;

4-) cdDO

iH

Cd o

18
7. RECOMMENDED APPLICATIONS

The B/C or SIR technique can be used to indicate whether a given proiect is cost effective.

If the B/C or SIR is less than one, the investor's minimum required rate of return, as reflected in the discount rate, is not being achieved and the project is not cost effective.

If it is greater than one, the project exceeds the minimum required rate and is cost effective .

A second use of the B/C or SIR technique is to choose among non-mutually exclusive projects

competing for a limited budget. For example, if a firm with extensive facilities to retrofit for energy conservation had only enough money to carry out one-third of the potentially

cost-effective projects, net savings from the available budget could be maximized by undertaking projects in descending order of their SIR' s until the budget is exhausted. (See

appendix A and examples in tables 6.3 and 6.4 with accompanying text.)

During the initial budget period, all evaluated projects which best satisfy the costeffectiveness criterion and which together exhaust the first year's budget would be

selected. In subsequent budget periods all projects not previously selected should be

reanalyzed if their SIR's are expected to have changed. They can then be ranked together

with any new projects which have been identified.

The ranking and selection procedure over two one-year budget periods is illustrated

graphically in figure 7.1. Projects are arranged in order of their priority ranking and a selection of projects is made in accordance with a limited budget. There are six candidate

projects depicted in the first year as meeting the minimum cost-effectiveness criterion by

having a B/C or SIR of one or greater. However, the budget in that year only allows for

the first three to be funded. In the second year the budget allows for the remaining three

projects. A fourth new candidate project in that year is omitted because of the budget

constraint

.

In using the B/C or SIR for capital rationing, it is important that the ratios for all

projects being compared reflect the same opportunity cost of capital as incorporated in

the discount rate. If the B/C or SIR values are calculated with different discount rates

for different projects, the resulting rankings will not necessarily lead to the maximization of net benefits. Furthermore, it is important that the opportunity cost of capital

be used as the basis for the discount rate.^

^ The opportunity cost of capital is the rate of return available on the next best

available investment.

19
Figure 7.1 Allocating the Budget Among Alternative Projects Ranked by B/C or SIR

BUDGET

B/C

or

SIR

1.0

Year One Projects $

BUDGET

Pi0|cct At |Pro|cct B| Project Project El Project Fj

jsj^swp:

20
Table 7.1 illustrates the sensitivity of the B/C and project net benefits to variations in

the discount rate. For example, the B/C values in col. (7) show that project 1 receives

first priority, project 3, second, and project 2, third, when calculated at a 6 percent

discount rate. If, on the other hand, a higher opportunity cost indicated a 25 percent

discount rate as shown in col. (8), the second project would receive first priority, the

first project second priority, and the third project last priority. Using the Incorrect

discount rate may lead to fewer net benefits per investment dollar, as would be the case in

table 7.1 if, for example, ' pro jects were chosen on the basis of a 6 percent discount rate

when the correct rate was 25 percent.

A third application of the B/C or SIR technique is to determine which project size or design

is most efficient (i.e., which maximizes net benefits or net savings). If there is no budget

limitation for a given project, the most efficient size or design occurs when the ratio of

incremental benefits or savings to incremental cost is equal to one for the last unit of

investment (i.e., where marginal benefits equal marginal costs). With a budget constraint,

however, it pays to restrict project size or design to that investment for which the ratio

of incremental benefits or savings to incremental costs for the last unit of investment in

the project is just equal to the incremental ratio on the next best available investment.

In other words, an incremental B/C or SIR greater than one might be adopted as a "cutoff

ratio” beyond which it would not pay to increase size or alter the design. If B/C analysis

is to be used for project design or sizing, it must be on an incremental basis.

Table 7.2, table 7.3, and figure 7.2 together illustrate how project size can be selected on

the basis of incremental B/C analysis. Table 7.2 presents five size alternatives (zero and

A through D) and their corresponding benefits and costs. Alternative C results in the maxi- mum net benefits and, hence, would be the economically efficient choice if there are no budget constraints.

Table 7.3 shows the B/C's for all possible size changes for the alternatives described in

table 7.2. Table 7.3 is read by row from left to right. The top row gives, in effect, the

B/C’s on total investment. Although size A has the highest B/C, it is not the size that is

shown in table 7.2 to give the highest net benefits. (Table 7.2 shows that net benefits

from project size C are $55,000 more than net benefits from project A.)

Subsequent rows of table 7.3 give the incremental B/C's calculated on the differences

between project sizes other than zero. For example, the incremental B/C associated with

expanding project size from A to B is 3.0; from A to C, 2.2;^ from A to D, 1.9; and from B to C, 1.3.

^ The calculation of B/C from A to C, for example, is 600,000 - 500,000 = 2.2.

1A5,000 - 100,000

21
SENSITIVITY

OF

THE

B/C

AND

NET

BENEFITS

TO

VARIATIONS

IN

THE

DISCOUNT

RATE

II

pP3O

CJ 0) on o

CO P o o

•H 3 00 • • CO P pi ^ cn CM u m

CQ

4Ju

0)

p

•r-) 3 O 3

ON V-< O 3 CM O P ^ on cn CO P 3 t'' • • • •p e«i o

r- NO vO

p3

<J> 3

N_X

O o un un o cu rH m

CO CO P ^ cn ON

4J •H rt ^ •» •» •H O pi m o ON

CNJ cn

0)c lO

(U cvl ea

4J

0) P 2 3

un 3 o o

4-1

O CU i o CO CM

a

O P ^ m f-4 00

0) CO 3 LO •» 0>> •r“T .p Oi w cn cn 00 O o o m un u

Pm 6^

v3

4JC CU CO (U M-l P o un o

•'-) -P CC) 1-H cn c ^ Q) ^

V. p1—1 4-

03 i CO

O o o

rt c.) 4-J

O o o 3 0) tH ^ o o o

•!—) 4-1 </> 0-) •k •*

C

O cu

w o m un < U C r—<

rH

p. 0)

p:^

AJC T3

CU QJ o O o = o o o p ip o o o

CO 3 <c> csi * o o o ' > CD rH 1^ M

4-1

•V 0 CO 00 0) 4J •J3 '3 CJ p 3 0) /-V u 3 >P rH CM cn

a; P. o ^

P. 3 P e T3 CL, O 3 O M

ui •H<4-1 (UC

a;C

cfl c

(UBw

(U>cco

0)

05 •H

V4P.Bo

CJo

4-1 T3

OJBS

CO

CO

cc)

(U

cc)

CO5o

CO

cc) o4-> •Ho

p.e

•H

COPo

CP

22
TABLE 7.2 PROJECT DATA

Project

Size

Alternatives

Total

Investment

Required

($)

Project

Life

(Years)

Total

Benefits

($)

Net

Benefits

($)

0 0 0 0 0

A 100,000 20 500,000 400,000

B 125,000 20 575,000 450,000

C 145,000 20 600,000 455,000

D 155,000 20 605,000 450,000

23
TABLE 7.3 B/C's FOR PROJECT SIZE CHANGES

^ Based on data presented in table 7,2.
Figure

7.2

Network

Diagram

Using

Incremental

B/C

Analysis

for

Project

Sizing CD

O

U)

o

a

25
Figure 7.2 shows in a network diagram the matrix of B/C's for all potential sizing options.

Each line in the network connects either 0 and a lettered point or two lettered points, and

depicts a potential project sizing decision. The number appearing by each connecting line

is the B/C calculated on the benefits and costs of the represented decision. One example

is the line connecting 0 and D. It indicates that one could potentially choose the largest

project, size D, at the outset, with a B/C on total investment of 3.9. A second example is

the line connecting B and D which shows that project size B could be expanded to size D,

attaining a B/C on the incremental investment of 1.0. In both cases the B/C's are one or greater, indicating project cost effectiveness. However, a more detailed inspection of the

increments comprising project D shows that project D is not the most cost-effective size.

That is, size D is comprised of increment 0 to A (B/C = 5.0), plus increment A to B (B/C =

3.0), plus increment B to C (B/C = 1.3), plus increment C to D (B/C = 0.5). Even with no budget constraint, the last increment to project size D (i.e., C to D) is not cost effective

as indicated by the incremental B/C of 0.5. Thus, to find the most efficient project size, one increases the size by each separable increment until the incremental B/C equals or just exceeds the cutoff B/C. A larger size would reduce total net benefits from the project. For example, if the cutoff B/C were 1.8, increments A and B would be efficient, but

anything beyond would not. If there were no budget constraint, or if the cutoff B/C were

less than 1.3, then increment C would also be desirable, because it brings more benefits

than it costs. The diagram shows by means of arrows the path along which project size would

be expanded to C. This is consistent with what was found in table 7.2 where C was found to

be the size that maximized net benefits when there is no budget constraint.

The determination of the efficient project size is further illustrated by figure 7.3. Any

size (Q) between and Qg would be economically efficient in the general sense of having

benefits or savings in excess of costs. That is, the benefits (savings) function lies

above the cost function as shown on the upper graph, or, expressing the same concept in

another way, the B/C (SIR) is greater-than-or-equal-to one over this size range, as shown

on the bottom graph.

The most efficient size of a project would be Q2 in figure 7.3. This is the size for which

the difference in benefits (savings) and costs is greatest, i.e., where net benefits or net

savings are maximized, as shown in the top graph. It is also the size for which the incremental B/C (SIR), as measured by AB/AC (AS/AI) in the lower graph, is equal to one. Note

that the B/C (SIR) based on total benefits (savings) and costs does not indicate the most

26
Figure 7.3 Efficient Project Sizes

r CO

o ^

J

o

<a

CQ

<

(/) O

c5

cc

27
efficient size, either at the point that it is equal to one or at its maximum point.

Rather, it is the incremental ratio that indicates the most efficient size.^

The most efficient size would be smaller than Q 2 if the budget were limited and if other

projects with a higher incremental ratio than one were available. In the case of a cutoff

ratio of 2.3, for example, a project size between and Q2 would be selected where the

incremental ratio is just equivalent to the cutoff ratio.

Where there are several non-mutually exclusive projects with positive net benefits and

there is an insufficient budget to fund all of them, the theoretically correct approach

would be to size each project such that the incremental B/C ratios would be equal for all

projects and equal to the ratio available on the last increment of the next best investment

(i.e., equal to the opportunity cost). Then projects would be selected on the basis of

descending B/C’s or SIR's computed on the total project costs and benefits (savings) until

the budget is exhausted. However, due to the difficulty of simultaneously equating the

incremental ratios on all projects, a second best approach often used is to size each

^ It can be proved that if B/C is at its maximum, then B'/C = B/C; i.e., the locus of

points defining the ratio of marginal benefits to marginal costs intersects the ratio of

total benefits to total costs at its maximum.

Given B,C > 0, B',C > 0, B" < 0, and C" > 0,

2 Maximizing B/C implies that d(B/C) _ q <0.

dQ dp2

d(B/C) = 0 implies CB*-BC* = q implies CB'“BC* = 0.

dQ C2

Thus B'/C = B/C.

d^(B/C) = C^ (CB’’+C'B'-B'C'-B'C") - (CB'-BC')C^ , dq2 C^

which upon simplification reduces to

d^(B/C) ^ CB"-BC"

dq2 c2

2 Since C and B are greater than zero and by assumption C">0 and B"<Q, then d (B/C) ^

q. dq^

28
project so that the incremental ratio is equal to one. Projects are then selected as before

in descending order of B/C's or SIR's until the budget is exhausted.

A further limitation on the application of the SIR technique should be noted. In evaluating

candidate projects for a particular building or facility, the problem of interdependency

among projects may arise; that is, undertaking one project may affect the relative lifecycle costs and savings of remaining projects. Thus the value of adding an automatic environmental control system will be different depending on the level of insulation in the

building envelope and vice versa. Undertaking one will tend to diminish the value of the

other. Often a practical approach to this problem is to evaluate each of the candidate projects independently of one another, select the one with the highest B/C or SIR, and then

adjust the B/C or SIR of any remaining projects that are expected to be substantially altered

by the first, higher priority selection. The selection process can then be continued, with

necessary adjustments to remaining projects being made as each project is chosen.

Appendix A illustrates the application of the SIR technique to a sample problem in energy

conservation involving both decisions in sizing and in the selection of an efficient

combination of conservation options. Appendix B follows with some selected references on

the B/C and SIR techniques.

29
,1 •• » “ ' , » 2

' ^ ,'M. 'If'j'

'. V ••'v>tVudfc'r»3«-f 'loi »<;i " 'lo '•r»Sit<3M7 “a'iU yVf J.:JK,r--jt »irI:X r';iia'5^.C •'iuj} ttesfi Ij&aXa;* |f -ivY. ;/ y'S -i(;E

’ ?V'''' s/fa Xo 1

|

:J,)’i.s

' OiA' k tt yO' -J',

Xay.j.’/l'i*'' ';;’r m?

•‘ -

^ w ' ,,'&tsYi;Y;'r:'fi ‘ty ;'s''» 4;!

V •-;'• Y ''/: .(:)* i Si* *"-

.. f.’ViA'Crr''"' j'-i i^oiic ®A grtlM »5pAi„v7v.

:‘‘

;

,... to l-M

••.;• r:^'

'jii. i.cfo iq s.Cqwjia' s .d'i"’

••<; '^-.ij ?2y ,>_'> Ai; (Tol;^.'>'ii's« aX >'n« ^R'isliS .Jii feWiV^ ’ -. ,

jA/syy?)?.''. ;,' bif,? s .{w«C"S-. s’l-i*-* .•••oX'^Oi- 0., sri’feiistli^A' V^;'.«'i-.‘»<f-^v '<»•'. ;l

,(

•: it"' .'..'.'.'Bi, 41

' ^>t>Y-.-j: f v'i i 4

" 'V I

'’<i.('i®i,> “',

' (<:m vV '. <•' J

. '.[‘'[im

•:nr B«*n* j -', '

/'.A

•,t i)

t;,2i!'A.- ''

Eli

V

."^3

. '<‘'V

""V /Ail', i '^'A7^^;";v,;;,'' ., ^

'

' ' .1

' :, j

' -I- "

• •>* •'"• . y ,; . ..-;-w,^tf ,, J.' ,;;; 'i!,t,v,;' : ^«»*lHSie8Sro2M!W.fc : • -.;> Yi->V ;

•' :•". - ' Y''- r; tY'-Y

, * -C • %'*1H

i' j

'iJ 1 ^ „

• AAA B.WBuiifii& l!. , ''r. 'itt^''* .' Vvr'' /^Vdwk^alKllS

'C'

...i"

• 'i • •• A •sivr.-ijBp

.fcj. '-jffti’ JiisaM
APPENDIX A

SIR EVALUATIONS OF ENERGY CONSERVATION INVESTMENTS 1

A home improvements firm has been contracted to plan and install an energy conservation

package for the owner/occupant of the house described below. Candidate retrofit projects

have been proposed, and the owners want to know what combination of those projects would

maximize net savings to them for their conservation budget of $1,500. For each of the

candidate projects an SIR is computed along with the corresponding net savings of that

project.

The house has been previously weatherstripped and caulked. It has R-11 insulation in the

attic, as well as all the insulation that can be accommodated in the floors and walls

without making major structural modifications. A jacket has already been added to the

domestic water heater, thermal draperies have been added to the windows, and the family is

practicing energy conservation in using lighting, appliances, and nighttime set-back of the

thermostat during the heating season.

The house is currently heated by an electric resistance system that is in good condition

and could reasonably be expected to last over the remaining life of the house with only

negligible maintenance and repair. For purposes of illustration, the efficiency of the

system is assumed to be 100 percent.

The annual space heating load is 83 x 10^ Btu (88 GJ). The owners now pay $16.89 per 10^

Btu ($16.01/GJ) of electricity and expect that price to escalate at an average annual

compound rate of 9 percent, including inflation, over the next 15 years. The house does

not have an air conditioning system.

The annual domestic hot water load is 22 x 10^ Btu (23 GJ). It is currently supplied by an electric water heater. The efficiency of the existing hot water system is assumed to be

100 percent.

The owners expect to occupy the house for at least another 15 years, and would like to base

their energy conservation investment decisions on a 15-year time horizon, neglecting possible resale effects at the end of that time. They have a limited budget of $1,500 to spend

on the house and would like to obtain the largest possible return on their conservation

1 This problem is adapted from problem 7.14 in Marshall and Ruegg, Energy Conservation in

Buildings; An Economics Guidebook , pp. 100-110.

A-1
budget. Their best alternative return on the $1,500 is an 8 percent market rate from

tax-exempt municipal bonds.

The following options are being considered for retrofit to the house:

(A) Addition of a solar domestic water heater. The system that has been recommended is

reliable and sufficiently durable to last the 15 years without major maintenance or repair, costs $1,500, and is expected to meet 80 percent of the annual hot water load.

(B) Replacement of the existing electric resistance space heating system with a relatively

high efficiency (0.7 efficiency) gas furnace. The replacement of the existing system

with the gas furnace will cost $1,000. No net salvage value is expected from disposal

of the existing system. The gas furnace is expected to have about the same maintenance

and repair costs and life expectancy as the existing system. The price of gas is now

$4.70 per 10^ Btu ($4.45/GJ) and is expected to escalate at an average annual compound

rate of 10 percent, including inflation, over the next 15 years.

(C) Addition of attic insulation to raise the current resistance (R) level from R-11 to

R-19. The insulation will cost $225 to purchase and install and is expected to reduce

the annual energy consumption for space heating by 12 percent.

(D) Conditional on Alternative (C), the addition of attic insulation to raise the R-value

from R-19 to R-30. Increasing insulation from R-19 to R-30 will cost $100 over the

cost of the R-19 addition, and is expected to reduce energy consumption by 5 percent of

the heating costs at R-19.^

(E) Conditional on Alternatives (C) and (D), the addition of attic insulation to raise

the R-value from R-30 to R-38. This will cost $75 more than raising the value to R-30

and is expected to save 2 percent of the heating cost at R-30.

^ The options selected are purely for illustrative purposes and are not intended as an endorsement or recommendation of these particular investments.

It is assumed that all increases in insulation would be made during the same visit from

the contractor. Therefore, the fixed costs which were incorporated into the cost of

Alternative (C) do not apply to the R-30 and R-38 applications.

A-2
(F) Replacement of from one to five existing north-facing single-glazed windows with

double-glazed windows. Each window will cost $200 and each is expected to reduce the

energy consumption for space heating by 2 percent, for a total of 10 percent if all

five are replaced.^

(G) Addition of from one to five storm windows to north-facing windows (instead of

replacing the windows as described in (F)) and/or the addition of up to three storm

windows to east-facing windows. The storm windows will cost $50 each. They are expected to reduce the energy consumption for space heating by 9 percent if all five

of the north-facing windows are retrofitted, or 1.8 percent per north-facing window.

They are expected to reduce the energy consumption by 0.7 percent per east-facing

window, for a total reduction of 2.1 percent if storm windows are added to all three of

the east-facing windows.

In evaluating the alternatives it is assumed that there are no available grants or tax

credits and that property taxes are not expected to be affected by the retrofit investments.

Solution

Compute the present value (PV) of costs, savings, net savings (NS), and the

savings-to-investment ratio (SIR) for each alternative, taking into account where necessary

the interdependencies between those investments that improve the shell of the house and

those that affect the heating system. 2 Rank projects in descending order of their SIR*s

until the budget is exhausted. Then sum the net savings of those projects selected.

Begin by evaluating each candidate project as follows;

^ Interdependencies between shell modifications (i.e., wall insulation and windows) are not

treated here. Slight changes in actual energy savings might be expected from one of these

modifications depending on whether the other one was undertaken.

2 Equation 6.8 is used to compute the SIR's.
(A) Addition of a Solar Domestic Water Heater

PV Cost = $1,500

S' S'

A/ > *S'

PV Savings = 22 x 10^ Btu

o/<&

0.80

Qj A. .O O

^(0 <7

/?

«/ oV

//a /b'

C} $16.89/10^ Btu X 16.1606 $4,803.96

where UPW* is taken from a table of discount factors.

NS = $4,803.96 - $1,500 = $3,303.96

SIR = $4,803.96 = 3.20

$1,500

(B) Replacement of Existing Space Heating System with Gas Furnace

PV Cost = $1,000

PV Savings =

^‘o ^

K ^

•S'A ^

83 X 10^ Btu X $16.89/10^ Btu

<J S

J'O-

(V o

*v

<Jr 0} S f'O Ci

- 83 X 10^ Btu

.7 A

$4.70/10^ Btu

«/

iV

•V'

fb

16.1606

<ij /

>'

1 17.4264

= $22,655.06 - $9,711.48 = $12,943.58

NS = $12,943.58 - $1,000 = $11,943.58

SIR = $12,943.58 = 12.94

$ 1,000

A-4

t
(C) Addition of Attic Insulation, R-11 to R-19

(C-1) With Existing Electric Space Heating System

PV Cost = $225

*V PV Savings = $22,655.06 x 0.12 = $2,718.61

NS = $2,718.61 - $225 = $2,493.61

SIR = $2,718.61 = 12.08

$225

(C-2) With Replacement of Existing System with Gas Furnace

PV Cost = $225

PV Savings = $9,711.48 x 0.12 = $1,165.38

NS = $1,165.38 - $225 = $940.38

SIR = $1,165.38 = 5.18

(D) Addition of Attic Insulation, R-19 to R-30

Alternative (D) is conditional on Alternative (C) being undertaken. It can be evaluated in terms of the incremental costs and savings over and above those associated

with raising the R value from R-11 to R-19.

$225
(D-1) With Existing Electric Space Heating System

PV Cost = $100 /

<h

A <f

PV Savings = ($22,655.06 $2,718.61)

/

<\o/

0.05 = $996.82

NS = $996.82 - $100 = $896.82

SIR = $996.82 = 9.97

$100

(D-2) With Replacement of Existing System with Gas Furnace

PV Cost = $100 /

e/ ^

PV Savings = ($9,711.48 - $1,165.38) x 0.05 = $427.31

NS = $427.31 - $100 = $327.31

SIR = $427.31 = $4.27

$100

A6
(E) Addition of Attic Insulation, R-30 to R-38

Alternative (E) is conditional on Alternatives (C) and (D) being undertaken. It can be

evaluated in terms of the incremental costs and savings over and above those associated

with raising the R value to R-30.

(E-1) With Existing Electric Space Heating System

PV Cost = $75

PV Savings = ($22,655.06

/

<b ^ S' /

^ s

$2,718.61 $996.82) X

<\o 0.02 = $378.79

NS = $378.79 - $75 = $303.79

SIR = $378.79 = 5.05

$75

(E-2) With Replacement of Existing System with Gas Furnace

PV Cost = $75

PV Savings = ($9,711.48

SO/

$1,165.38 - $427.31) x 0.02 = $162.38

NS = $162.38 - $75 = $87.38

SIR = $162.38 = 2.17

$75

A7
(F) Replacement of From One to Five Existing North-Facing Single-Glazed Windows with

Double-Glazed Windows

(F-1) With Existing Electric Space Heating System

TABLE A.l EVALUATION OF WINDOW REPLACEMENT, ELECTRIC HEATING

No. Windows PV Cost PV Savings NS SIR

1 $ 200 $22,655.06 X 0.02 =

$ 453.10 $ 253.10 2.27

2 400 22,655.06 X 0.04 = 906.20 506.20 2.27

3 600 22,655.06 X 0.06 = 1,359.30 759 .30 2.27

4 800 22,655.06 X 0.08 = 1,812.40 1,012.40 2.27

5 1,000 22,655.06 X 0.10 = 2,265.51 1,265.51 2.27

(F-2) With Replacement of Existing System With Gas

TABLE A. 2 EVALUATION OF WINDOW REPLACEMENT,

Furnace

GAS HEATING

No. Windows PV Cost PV ;Savings NS SIR

1 $ 200 $9,711.48 X .02 = $194.23 $ - 5.77 0.97

2 400 9,711.48 X .04 = 388.46 -11 .54 0.97

3 600 9,711.48 X .06 = 582.69 -17.31 0.97

4 800 9,711.48 X .08 = 776.92 -23.08 0.97

5 1,000 9,711.48 X .10 = 971.15 -28.85 0.97

A8
(G) Addition of From One to Five North-Facing Storm Windows (Instead of Replacing

Windows) and From One to Three East-Facing Storm Windows

(G-1) With Existing Electric Space Heating System

TABLE A. 3 EVALUATION OF STORM WINDOWS, ELECTRIC HEATING

No. Windows PV Cost

North-Facing Windows

PV Savings NS SIR

1 $ 50 $22,655.06 X 0.018 = $ 407.79 $ 357.79 8.16

2 100 22,655.06 X 0.036 = $ 815.58 715.58 8.16

3 150 22,655.06 X 0.054 = 1,223.37 1,073.37 8.16

4 200 22,655.06 X 0.072 = 1,631.16 1,431.16 8.16

5 250 22,655.06 X 0.090 = 2,038.96 1,788.96 8.16

East-Facing Windows

1 $ 50 $22,655.06 X 0.007 = $158.59 $108.59 3.17

2 100 22,655.06 X 0.014 = 317.17 217.17 3.17

3 150 22,655.06 X 0.021 = 475.76 325.76 3.17

A9
(G-2) With Replacement of Existing System With Gas Furnace

TABLE A.

4

EVALUATION OF STORM WINDOWS, GAS HEATING

No. Windows PV Cost

North-Facing Windows

PV Savings NS SIR

1 $ 50 $9,711.48 X 0.018 = $174.81 $124.81 3.50

2 100 9,711.48 X 0.036 = 349.61 249.61 3.50

3 150 9,711.48 X 0.054 = 524.42 374.42 3.50

4 200 9,711.48 X 0.072 = 699.23 499.23 3.50

5 250 9,711.48 X 0.090 = 874.03 624.03 3.50

East-Facing Windows

1 $ 50 $9,711.48 X 0.007 = $ 67.98 $17.98 1.36

2 100 9.711.48 X 0.014 = 135.96 35.96 1.36

3 150 9,711.48 X 0.021 = 203.94 53.94 1.36

Now select projects from among the candidates in descending order of their SIR' s until the

$1,500 budget is exhausted. Table A. 5 lists that set of projects.

The project given the highest priority on the basis of its SIR is (B), replacement of the

electric resistance heating system with a gas furnace. Acceptance of that project means

that, thereafter, projects which improve the thermal Integrity of the shell of the house,

such as the attic insulation and storm windows, must be evaluated on the basis of reductions

in heating costs.

A-10
TABLE A. 5 PROJECT SELECTION

Priority

Ranking

Investment

Alternative SIR PV Cost

PV

Savings

Net

Savings

1 (B) Replace Space

Heating System 12.94 $1,000 $12,944 $11,944

2 (C-2) Add R-11 to R-19

Attic Insulation 5.18 225 1,165 940

3 (D-2) Add R-19 to R-30

Attic Insulation 4.27 100 427 327

4 (G-2) Add 3 Storm Windows

on North 3.50 150 524 374

Totals 4 projects n.a.* $1,475 $15,060 $13,585

* not applicable

Given that project costs occur in varied amounts, depending on the project, the full $1,500

is not allocated, and $25 remains unallocated. In this case the net savings from undertaking

these projects are greater than from any other combination of projects which would exhaust

the total budget. However, under some circumstances, selecting a lower ranked project that

exhausts the budget might increase total net benefits.

In summary, the package of energy conservation projects which in this illustrative example

will maximize net savings from the limited conservation budget of $1,500 consists of replacing the electric resistance space heating system with a gas furnace, increasing the attic

insulation from R-11 to R-30, and outfitting 3 windows on the north side of the house with

storm windows. If the budget were not limited, it would pay to undertake all of the

candidate projects except replacement of the north-facing windows with double-glazed

windows.

A-11

APPENDIX B

SELECTED B/C AND SIR REFERENCES

1. 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),

2. 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.

3. 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.

4. Mishan, E. J., Cost-Benefit Analysis: An Introduction (New York, New York: Praeger

Publishers, 1971).

5. 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,

6. 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.

7. Ruegg, Rosalie T., Life-Cycle Costing Manual for the Federal Energy Management Program , National Bureau of Standards Handbook 135, December 1980.

8. Ruegg, Rosalie T. et al., Life-Cycle Costing: A Guide for Selecting Energy

Conservation Projects for Public Buildings , National Bureau of Standards Building

Science Series 113, September 1978.

9. Smith, Gerald W., Engineering Economy: Analysis of Capital Expenditures , 2nd Edition,

(Ames, Iowa: The Iowa State University Press, 1973).

B-1
'•.r 'flRac

„ V.;

,/„.

'ill'

r m.

.4 ..

,;vf W

' . ';,/ •' '' - . ^ •Jj''l'' :: '"T'^'— ;.

' „r

-‘' 'V.

.;''— - ::|

{\s ;,:'viwC

7",

..-'?.'S!l

„

' .aaiJi; af=^w».

.,M hat.x<y^af» .. .a bfaju.)! . IWW^Ws

’(7 JB.CS'SSf-i/S’

' "'3

' 7 ' ^ t ,f#"

,. ^. i ‘3 , V .

^
U.S. DEPT. OF COMM. 1. BIBLIOGRAPHIC DATA

SHEET (See instructions)

PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3.

NBSIR 81-2397

Publication Uate

4. TITLE AND SUBTITLE

RECOMMENDED PRACTICE FOR MEASURING BENEFIT/COST AND SAVINGS-TO-INVESTMENT

RATIOS FOR BUILDINGS AND BUILDING SYSTEMS

5. AUTHOR(S)

Harold E. Marshall and Rosalie T. Ruegg

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, State, ZIP)

National Bureau of Standards

Department of Commerce

Washington, D.C. 20234

10. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT fA 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 a benefit-cost ratio (B/C) and a savingsto-investment ratio (SIR) and how to use them in selecting building designs and

building systems that will be cost effective in the long run. The B/C relates

positive benefits, such as revenues, to project costs in the form of a ratio.

The SIR, a variation of the B/C, relates project savings (i.e., cost reductions)

to project costs in a ratio. It is used when there are few if any positive cash

flows from a project. The B/C and SIR can be used to help answer such questions

as; "Is a project cost effective?" "Which size and/or design of a project is

most cost effective?" "What priorities should be given individual projects

competing for a limited budget?" The report addresses different formulations of

the ratios and their implications for selecting cost-effective projects.

12. KEY WORDS (Six to twelve entries; aiphabetical order; capitalize only proper names; and separate key words by semicolons)

benefit-cost analysis; building economics; cost effective; economic analysis;

energy conservation; investment analysis; life-cycle cost; recommended practice;

savings-to-investment ratio

13. AVAILABILITY 14. NO. OF

PRINTED PAGES

Uni imited

I I For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

48

15. Price

Q3! Order From National Technical Information Service (NTIS), Springfield, VA. 22161 7.50

USCOMM-DC 6043-P80
I

I it %

«

J

I

1

1

i