NAT L INST. OF STAND & TECH

a 1 1 1 o b 2bcnm

P6fH ^i~?o

t

U.S. Department of Commerce

National Bureau of Standards

National Engineering Laboratory

Center for Applied Mathematics

Washington, D.C. 20234

March 1984

NBSIR 84-2850

Recommended Practice

for Measuring Simple

and Discounted Payback

for Investments in Buildings

and Building Systems

NATIONAL BUREAU

OF STANDARDS

LIBRARY

NBSIR 84-2850

Q < RECOMMENDED PRACTICE FOR MEASURING

SIMPLE AND DISCOUNTED PAYBACK

FOR INVESTMENTS IN BUILDINGS

AND BUILDING SYSTEMS

Harold E. Marshall

March 1984

Sponsored By:

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Building Technology

and

Center for Applied Mathematics

Washington, DC 20234

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
t*

•r, , i V
ABSTRACT

This report describes how to calculate simple and discounted payback measures

of economic performance of buildings and building systems. Formulas for

calculating payback, applications for evaluating and selecting projects, and

limitations in the use of payback analysis are discussed. The simple payback

method measures the time between the date of initial project investment and

the date when cumulative future earnings or savings on that investment, net of

cumulative future costs, just pay off the investment. The discounted payback

method measures the time between the date of initial project investment and

the date when the present value of future earnings or savings, net of the

present value of future costs, just equals the initial investment. This

recommended practice will assist the private and public building communities

in making cost-effective decisions in the design, operation, maintenance, and

retrofit of buildings.

iii
PREFACE

Rising costs of materials, labor, energy, and construction loans have caused

architects, builders, engineers, building owners and operators, and code

writers to look more closely at economic evaluation methods in selecting

building designs and systems that are cost effective. Simple payback (SPB)

and discounted payback (DPB) methods for measuring economic performance of

buildings and building systems are commonly used because they are in many

instances relatively simple to compute and easy to understand. Standardized

approaches for calculating and applying the SPB and DPB methods are needed

because traditional applications of these methods sometimes lead to uneconomic

building choices.

The SPB method measures the time between the date of initial project

investment and the date when cumulative future earnings or savings on that

investment, net of cumulative future costs, just pay off the investment. The

DPB method measures the time between the date of initial project investment

and the date when the present value of future earnings or savings, net of the

present value of future costs, just equals the initial investment. The SPB

and DPB methods are used generally to decide if a single project is

economically feasible (i.e., if it pays off within a predetermined maximum

allowable payback period) and to choose among projects competing for the same

purpose or the same budget (i.e., to choose projects with a short SPB before

those with a long SPB). However, both payback methods have shortcomings which

sometimes result in inefficient project choices. Each method ignores benefits

and costs beyond the payback period, and the SPB method ignores the time value

of money, treating, for example, a dollar saved five years from today as the

equivalent to a dollar saved today.

iv
This report describes how to calculate SPB and DPB and describes the

circumstances under which each might be correctly applied in economic

evaluations of buildings and building components. Project examples are also

provided to illustrate the economic efficiency losses from using SPB and DPB

incorrectly in making investment choices. Developing standardized SPB and DPB

methods and providing guidelines for their application will help prevent

misuse of these methods and contribute to improved cost effectiveness in

building-related decisions.

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-6 , Performance of Building Constructions) and

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

economic anaylsis 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 basis for the development of an ASTM standard practice for

measuring SPB and DPB for building investments. It is the fourth in a series

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

the previous three reports (NBSIR 83-2657, on net benefits and the internal

rate of return; NBSIR 81-2397, on benefit-to-cost and savings-to-investment

ratios; and NBSIR 80-2040, on life-cycle costing). 1 The series of NBS reports

* Harold E. Marshall, Recommended Practice for Measuring Net Benefits and

Internal Rates of Return for Investments in Buildings and Building Systems , National Bureau of Standards Interagency Report 83-2657, October, 1983;

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

Recommended Practice for Measuring Life-Cycle Costs of Buildings and HniM i / Systems , National Bureau of Standards Interagency Report 80-2040, June 1980 .

v
and ASTM recommended practices is aimed at producing a comprehensive set of

measures of economic performance that will meet the needs of the private and

public building sectors.

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, Joseph H. Engel, James G. Gross, Barbara C. Lippiatt, Stephen R.

Petersen, Rosalie T. Ruegg, Larry Schindler, and Stephen F. Weber for their

helpful technical and editorial comments on the paper; to Janet M. Cassard and

Stefan D. Leigh for their assistance in generating some of the tabular and

graphical material in the report; and to Laurene B. Linsenmayer for typing the

manuscript.

vi
TABLE OF CONTENTS

Page

ABSTRACT iii

PREFACE iv

LIST OF FIGURES ix

LIST OF TABLES ix

1 . INTRODUCTION 1

2. PROCEDURES IN PAYBACK ANALYSIS 4

2.1 Objectives, Alternatives, and Constraints 4

2.2 Economic Evaluation Method 5

2.3 Data and Assumptions 5

2.4 Conversion of Cash Amounts to Present Values 7

3. COMPUTATION OF SIMPLE PAYBACK (SPB) 10

3.1 Uniform Net Cash Flows 10

3.2 Unequal Net Cash Flows 12

4. COMPUTATION OF DISCOUNTED PAYBACK (DPB) 15

4.1 Mathematical Solution 15

4.1.1 Uniform Net Cash Flows 15

4.1.2 Unequal Net Cash Flows 17

4.1.3 DPB With Escalation 21

4.2 Graphical Solution 24

4.2.1 Uniform Net Cash Flows 24

4.2.2 DPB With Escalation 29

vii
TABLE OF CONTENTS (CONTINUED)

Page

5. RECOMMENDED APPLICATIONS AND LIMITATIONS 40

5.1 SPB or DPB 40

5.2 Payback Versus Other Economic Methods 41

5.3 Payback As a Supplementary Method 48

APPENDIX A Derivation of DPB and DPBg A-l

APPENDIX B Selected Bibliography B-l

viii
LIST OF FIGURES

Page

4.1 Discounted payback for SPB = 0 to 4 years, i = 1 to

25% (odd) 25

4.2 Discounted payback for SPB = 0 to 4 years, i = 2 to

24% (even) 26

4.3 Discounted payback for SPB = 4 to 12 years, i = 1 to

23% (odd) 27

4.4 Discounted payback for SPB = 4 to 12 years, i = 2 to

24% (even) 28

4.5 Discounted payback with escalation for SPB 3 0 to 4

years, k = .77 to 1.17 (odd).... 31

4.6 Discounted payback with escalation for SPB = 0 to 4

years, k = .78 to 1.16 (even) 32

4.7 Discounted payback with escalation for SPB = 4 to 12

years, k = .81 to 1.17 (odd) 33

4.8 Discounted payback with escalation for SPB = 4 to 12

years, k = .82 to 1.16 (even) 34

4.9 Discounted payback for SPB = 0 to 10 years, e 3 0 to

10 years, e = 0 to 10% (even), i = 7% 37

4.10 Discounted payback for SPB 33 0 to 10 years, e 3 0 to

10% (even), i = 10% 38

LIST OF TABLES

2.1 Discounting Equations... 9

3.1 SPB Problem 13

4.1 DPB Problem With Unequal Annual Cash Flows 18

4.2 Matrix of k Values for Combinations of e and i 35

5.1 Economic Questions and Usefulness of Payback 41

5.2 Using Payback to Accept or Reject Projects 43

5.3 Project Sizing With Payback and Net Benefits 45

5.4 Project Acceptance, PVNB Versus SPB or DPB 46

5.5 Ranking Different-Purpose Projects 47

ix
.

.

-

'

. I
1 . INTRODUCTION

The simple payback (SPB) method and discounted payback (DPB) method are part

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

performance of an investment. 1 Included in this family of evaluation methods

are life-cycle costing, benefit-to-cost and savings-to-investment ratios, net

savings, and internal rates of return.

This is the fourth 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-toInvestment Ratios for Buildings and Building Systems , NBSIR 81-2397. The third

was Recommended Practice for Measuring Net Benefits and Internal Rates of Return

for Investments in Buildings and Building Systems , NBSIR 83-2657.

The SPB method measures the time between the date of initial project investment

and the date when cumulative future earnings or savings on that investment, net

of cumulative future costs, just pay off the investment. 2 in practice the SPB

method is used sometimes as the single criterion for judging economic efficiency

and sometimes as a supplementary criterion. One common use of the SPB method is

to decide if a single project is economically feasible. If the time required to

pay off the investment is less than some predetermined maximum allowable payback

period (MAPP) , then the project is presumed to be economical. Another common

^The terms payoff and payout are sometimes used instead of payback.

^Note that future earnings as well as future costs are often not known with

certainty and therefore, for purposes of analysis, will have to be estimated.

1
use of the SPB method is for choosing among projects competing for the same

purpose or the same budget. Projects with a low SPB are preferred to projects

with a high SPB because the investment is paid off more quickly. Choosing

projects on the basis of the SPB, however, may lead to inefficient choices and

fewer net benefits than might be earned following other economic evaluation

methods.

The DPB method measures the time between the date of initial project

investment and the date when the discounted present value of earnings or

savings, net of discounted costs, just equals the initial investment. The DPB

differs from SPB in that DPB accounts for the time value of money. The DPB

method is used for the same types of decisions as the SPB method. The DPB

also shares some of the shortcomings of the SPB method, and it, too, may lead

to project choices that are not cost effective.

The purpose of this report is to establish a technical basis for the

development of a recommended practice for calculating, interpreting, and

applying SPB and DPB methods in the evaluation of building designs and

systems. Special attention is given to applications, since the traditional

use of payback methods sometimes leads to the construction of inefficient

projects. All economic terminology used in the report is consistent with

"Standard Definitions of Terms Relating to Building Economics.

1 American Society for Testing and Materials, "Standard Definitions of Terms

Relating to Building Economics," E833-83a, Annual Book of ASTM Standards , Vol 04.07 (Philadelphia, PA: ASTM, 1983).

2
Section 2 discusses the general procedures in carrying out a payback analysis.

Objectives, alternatives, constraints, assumptions, data, and discounting are

discussed as they pertain to the two payback methods. Readers are assumed to

have a working knowledge of discounting and present value analysis. Section 3

describes the calculation procedures for measuring SPB, and section 4 does the

same for DPB. Problem examples are used to illustrate the mathematical and

graphical procedures for determining payback. Section 5 discusses under what

circumstances SPB and DPB measures are appropriate for evaluating typical

building investment problems. Recommendations of specific methods are made

for different investment objectives. Limitations of the two methods and their

economic efficiency implications are also discussed. Appendix A provides

derivations of DPB with and without escalation of net cash flows. Appendix B,

a bibliography, concludes the report.

3
2. PROCEDURES IN PAYBACK ANALYSIS

The recommended steps for making an economic analysis of buildings or building

components are summarized as follows:

o Identify Objectives, Alternatives, and Constraints

o Select an Economic Evaluation Method

o Compile Data and Establish Assumptions

o Convert Cash Flows to a Common Time Basis

o Compute the Economic Measure and Compare Alternatives

Each of these steps, except the last one, is examined briefly in this section

with respect to making a payback analysis. For elaboration on these steps,

the three earlier reports in this series should be consulted. How to compute

SPB and DPB is presented respectively in sections 3 and 4. How payback

measures are used in comparing alternatives is discussed in section 5.

2.1 Objectives, Alternatives, and Constraints

The kind of building decision to be made must first be specified. The

objectives of the decision maker must be made explicit. Alternative

approaches for reaching the objectives and any constraints to reaching the

objectives should be identified.

An example of a building investment problem that might be evaluated with payback methods is the decision on what heating system to install in an existing

building. The objective of the building owner is to provide the required

level of thermal comfort most efficiently. Feasible technical alternatives

for providing the required level of thermal comfort might be replacement of

the existing heating system with a high-efficiency oil furnace or an electric

4
heat pump. Constraints might be a limit on investment funds and time limits

by which the new heating equipment must be in operation.

2 .2 Economic Evaluation Method

If the high-efficiency oil furnace and heat pump are technically feasible and

meet all constraints, the building owner will want to evaluate the two

alternatives to see which is preferred on economic grounds. The SPB or DPB

method would indicate which of the two alternatives would pay off their

investment cost first through cumulative future energy savings as compared

with heating bills from the current heating system.

Note that other economic evaluation methods are appropriate for solving the

problem cited above and that in many cases these other methods will lead to

decisions that maximize the present value of net savings whereas application of

payback methods will not. For examples of investment problems and their

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

1

Other applications and limitations of the SPB and DPB methods will be

discussed in section 5.

2 .3 Data and Assumptions

Data are needed to make SPB or DPB calculations. Some data can be collected

from published and unpublished sources, some can be estimated, and some will

^See Harold E. Marshall, Recommended Practice for Measuring Net Benefits and

Internal Rates of Return ; Harold E. Marshall and Rosalie T. Ruegg , Recommende d Practice for Measuring Benefit/Cost and Savings-to-Investment Ratios ; and

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

Practice for Measuring Life-Cycle Costs of Buildings and Building System s

.

5
have to be assumed.! Both engineering data (e.g., heating loads, equipment

life, and equipment efficiencies) and economic data (e.g., tax rates,

depreciation rates and periods, system costs, energy costs, and financing

terms) will be needed.

Important values that must sometimes be assumed in using payback methods are

the discount rate, project life, and price escalation rates. A maximum

acceptable payback period (MAPP) must also be chosen if a cutoff period is

used. If the SPB or DPB is longer than the MAPP, the investment is considered

economically unattractive. The selection of a MAPP and its implications will

be discussed in section 5.

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.

lFor a more complete discussion of data and assumptions, see Rosalie T. Ruegg,

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

Measuring Life-Cycle Costs of Buildings and Building Systems, 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; and

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. For data sources, see 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; Carol

Chapman Rawie , Estimating Benefits and Costs of Building Regulations: A Stepby-Step Guide , National Bureau of Standards Report 81-2223, June 1981, the

the end of chapters 1-3, for listings of sources of information on data

relating to economic analyses of buildings; and Alphonse J. Dell’Isola and

Stephen J. Kirk, Life-Cycle Cost Data (New York: McGraw Hill, 1983) for

actual cost data on replacement, energy, and maintenance.

6
Calculating the DPB for the high-efficiency oil furnace and the electric heat

pump, as described in section 2.1, provides an example of the types of data

that might be needed. The annual heating requirements of the building, the

efficiencies of the two heating systems, and the per unit energy content of

the two energy sources are needed to calculate annual purchased energy

requirements with the two systems. The present and future prices of

electricity and oil as well as the discount rate are needed to convert the

annual purchased energy requirements into an annual energy cost. Investment,

maintenance, and perhaps replacement costs, as well as the resale value may be

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

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

taxes should be considered if the investment is made by a private building

owner.

2.4 Conversion of Cash Amounts to Present Values

To calculate DPB, all benefits and costs estimated for future years must be

converted to equivalent present values to account for the time value of money.

Converting these cash amounts to present 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 2.1 presents the most commonly used discounting

equations, how they are used, and their algebraic form. This report does not

Cash 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.

7
explain discounting since the procedure has been explained in numerous

publications. *

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

tables, see the ASTM Standard Practice E917, "Recommended Practice for

Measuring Life-Cycle Costs of Buildings and Building Systems, and the Adjunct

Discount Factor Tables that accompany it (Philadelphia, Pennsylvania: ASTM,

1983).

8
Table 2.1 Discounting Equations

Name Schematic Illustration Application Algebraic Form a,b

Single Compound-Amount

(SCA) Equation

I pp I To find F when

I r '1 P is known F = P • [(1 + i) n

]

Single Present-Value

(SPV) Equation

Uniform Sinking-Fund

(USF) Equation

To find P when

F is known

E To find A when

F is known

P = F1

(1 + i) n _

A = F

• i

_ (1 + i) n - 1

Uniform Capital-Recovery

(UCR) Equation

To find A when

P is known A = P

•

i (1 -t- i) n

(1 + i) n - 1 _ Uniform Compound-Amount

(UCA) Equation S + 0 + 0

To find F when

A is known F = A

(1 + i) n - 1

i

Uniform Present-Value

(UPV) Equation

[a] + [a] • • + [a] To find P when

A is known P = A • (1 + i) n - 1

i(1 + i) n

Modified Uniform

Present-Value

(UPV*) Equation

To find P when

P = A0 7

1+e Y

A 1 + a2

• • + ^n known A0

is escalating v

- e /

at rate e

1 + e

1 + i

where:P = present sum of money,

F = future sum of money equivalent to P at the end of n periods of time at i interest or discount rate, A = end-of-period payment (or receipt) in a uniform series of payments (or receipts) over n periods at i interest or discount rate, A0 = initial value of a periodic payment (receipt) evaluated at the beginning of the

study period,

At = A0

(1 + e)\ where t = 1, . .

.

,

n, n = number of interest or discount periods,

i = interest or discount rate per period, and

e = price escalation rate per period.

a Note that the USF, UCR, UCA, and UPV equations yield undefined answers when i = 0. The

correct 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 UPV* equation also

yields an undefined answer when e = i. In this case,P = A0

• n.

b The terms by which the known values are multiplied in these equations are the formulas for the

factors found in discount factor tables. Using acronyms to represent the factor formulas, the

discounting equations can also be written as F = P • SCA, P = F • SPV, A = F • USF,

A = P • UCR, F = A • UCA, P = A • UPV, and P = A0

• UPV*.
3. COMPUTATION OF SIMPLE PAYBACK (SPB)

The SPB method finds the amount of time (usually specified in years) between the

date of initial project investment and the date when cumulative future earnings

or savings on that investment, net of cumulative future costs, just pay off the

investment. The minimum solution value of Y in equation 3.1 is the SPB.l

l (Bt - Ct ) - CQ = 0, (3.1)

t=l

where Y = number of years required for the project cash flows to

offset the initial investment,

B t = dollar value of benefits (including earnings, cost reductions

or savings, and resale values, if any) in year t,

C t = dollar value of costs (including operation, maintenance, and

replacement) in year t,

B

t - Ct = net cash flows in time period t,2 and

CQ = initial project investment costs.

If the net cash flows are low, or the project life is short, or both, the project

may not have a SPB; that is, it is undefined and the project will never pay off.

(This is also true for DPB, as will be seen in section 4.)

3 . 1 Uniform Net Cash Flows

Direct solution of Y is not possible for equation 3.1. It is possible, however, for the case where the annual net cash flow is uniform; i.e., where

(B t - Ct ) is a constant, as shown in equation 3.2.

1 Benefits and costs in all computations should be in after-tax terras for

all organizations that pay taxes.

^Cash flows are generally assumed to be end of year payments. The only exception

is that cash flows are assumed to be spread evenly over the last year of payback

so that partial years can be included in the payback measure.

10
SPB = Y

= (3.2)

_Co

(B - (!)

where (B - C) = a uniform annual net cash flow.l

Another approach to solving for SPB when cash flows are uniform is to use the

savings-to-investment ratio (SIR). The SIR relates the present value of annual

savings to initial costs as shown in the general formula below:

2

N (B t - Ct ) l t=l (1 + i) c SIR = , (3.3)

C0

where N = the total number of discounting periods in the study period.

When the net cash flow is a uniform annual amount, the SIR can be written as

UPVN i

• (B - C)

SIR = : C0

(3.4)

Y

1 Assuming that (B t - Ct ) in equation 3.1 is a constant, £ ( B t “ Ct) becomes

t-1

Y (B - C) . Rearrangement of terms in equation 3.1 yields Y

Co

(B - C)

^For a description of the SIR, see Harold E. Marshall and Rosalie T. Ruegg

,

Recommended Practice for Measuring Benefit/Cost and Savings-to-Investment

Ratios

.

11
By rearranging terms in equation 3.4 and substituting into equation 3.2,

equation 3.5 is obtained for finding simple payback with the UPV factor and

the SIR.

uPVNji

SPB = — — . (3.5)

SIR

Using equation 3.5 would be efficient only when values of UPV and SIR have

already been obtained for other evaluation measures. Even then it involves at

least as much effort as using equation 3.2., and therefore it would rarely be

used.

3.2 Unequal Net Cash Flows

For problems with unequal annual cash flows, the most common approach to

calculating SPB is to accumulate net future cash flows year by year and

compare the cumulative amount at the end of each successive year to CQ

. The

payback year is the year in which the aggregate net cash flows just offset the

initial investment costs.

This approach is illustrated with the problem outlined in table 3.1. A

project with 5 years of unequal cash flows (column 2) is described in tabular

form to facilitate computation of the SPB. Each additional year's net cash

flow is added to the cumulative net cash flow of all preceding years (column

3). Investment costs (column 4) are subtracted from the cumulative net cash

flow in each year to arrive at cumulative net benefits (column 5).

12
Table 3.1 SPB Problem

Years

(t)

Net Cash Flow

($)

(B t - Ct )

Cumulative

Net Cash Flow3

($)

Investment

Cost

($)

(c0 )

Cumulative

Net Benefits*5 ($)

S

1 ( fi t “ ct ) t=l

S

l (B t - Ct ) - CG

t=l

(1) (2) (3) (4) (5)=(3)-(4)

0 10,000

1 4,000 4,000 -6,000

2 3,000 7,000 -3,000

3 2,500 9,500 -500

4 1,500 11,000 +1 ,000

5 1,000 12,000 +2,000

aS » the number of years over which net cash flows are accumulated.

^Net benefits in this column have not been discounted to account for the time

value of money.

Inspection of table 3.1 shows that the investment cost is recouped sometime in

the fourth year. Assuming that net cash flows are spread evenly over the

fourth year, the SPB can be calculated as follows:

500

SPB = 3 years + years = 3.33 years.

1,500

13
Since the SPB is less than the number of years over which the project earns

positive net benefits, we know that the project returns more undiscounted

dollars than it costs initially.

Two economic measures derived from the payback (PB) method that are sometimes

used are the "payoff period rate of return,"! calculated as (100/PB)%, and the

"average yearly cash recovery, "2 calculated as l/PB. The payoff rate of

return is equivalent to the percent form of the average yearly cash recovery.

Both measures can be computed from the SPB measure as defined above or from

the DPB measure to be explained in the next section. The shorter the payback,

and therefore the higher the percent value, the more desirable would be the

project.

!e. J. Mishan, Cost-Benefit Analysis (New York: Praeger Publishers, 1982),

p. 209.

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

1973), p. 34.
4. COMPUTATION OF DISCOUNTED PAYBACK (DPB)

The DPB method finds the amount of time between the date of initial project

Investment and the date when the discounted present value of future earnings

or savings, net of future discounted costs, just equals the initial

investment. The DPB differs from the SPB in that the DPB, through

discounting, accounts for the time value of money. The DPB can be determined

mathematically or graphically.

4 . 1 Mathematical Solution

The minimum solution value of Y in equation 4.1 is the DPB.

Y

I t*i (1 + i) c

where i = discount rate.

B,- ~ Ct ~ CQ - 0

,

(4.1)

Direct solution of Y is not possible using equation 4.1.

4.1.1 Uniform Net Cash Flows

For the case where (B t - Ct ) is a constant, Y can be isolated for direct

solution by using logarithms, as illustrated in equation 4.2. 1

DPB = Y -

log

1

1 - (SPB • i)

log (1 + i)

when i * 0, (4.2)

DPB = SPB when i = 0, and (4.3)

DPB is undefined when (SPB • i) > 1.

*See Appendix A for the derivation of equation 4.2.

15
When i = 0, the stream of constant cash flows is in effect simply added,

thereby yielding DPB = SPB. When (SPB • i) > 1 , equation 4.2

cannot be solved because SPB is undefined. 1 However, we can infer certain

values for DPB and for the present value of net benefits (PVNB) when (SPB • i)

> 1. For example, as (SPB • i) approaches 1, we can infer that the DPB approaches

infinity. Since lim log Z = 00

, and the lim

Z-^°° (SPB • i)-*l 1 - (SPB • i)

= 00

, then the

lim DPB = 00

. (SPB • i)-»l

We can also show that the PVNB = 0 when (SPB • i) = 1 and the study period is

infinite. Since SPB = C0 /(B - C) , as shown in equation 3.2, and

i = (B - C)/PVCF, where PVCF = present value of net cash flows, 2 then by

substitution

C 0 (B - C) C0

(SPB • i) = * • ——- = . (B - C) PVCF PVCF

When (SPB • i) = 1 , then CQ /PVCF also equals 1, and CD must equal the PVCF

that represents the present value of the (B - C) annuity in perpetuity. Thus

the PVNB (i.e., PVCF - C0 ) must equal zero.

When (SPB • i) > 1, no period of time is long enough for the project to pay

back, and the PVNB will be negative.

^When (SPB • i) = 1, the divisor in the numerator of equation 4.2 is a zero,

and when (SPB • i) >1, the numerator becomes a negative number.

2 The formula for an annuity is (P»i) = A, where P is a present value, i is the

interest rate, and A is the uniform annual amount that will be earned in

perpetuity. Letting P = PVCF and A = (B - (?) , by rearrangement of terms we get i = (B - d)/PVCF.

16
An application of equation 4.2 is presented for the following investment

problem. What would be the DPB for a project investment of $12,000, earning

uniform annual net cash flows of $4,500 for 6 years? A 10% discount

rate applies. Equation 4.2 would yield the following:

Y

log

1

$ 12,000 • .10

1 $4,500

log 1.10

log 1.3636 .1347

= = 3.25.

log 1.1000 .0414

Since the DPB is less than the number of years over which the project earns

constant net benefit returns, we know that the project is cost effective.

If the DPB were longer than the period over which the project earns positive

net benefits, then it would not be cost effective because the PVNB would be

negative.

4.1.2 Unequal Net Cash Flows

For problems with unequal annual net cash flows, the most common approach to

calculating DPB is to accumulate the present value of net cash flows yearby-year until the sum just offsets the original investment costs. The

year in which the two become equal will be the DPB.

This approach is illustrated with the problem outlined in table 4.1. a

project with 7 years of unequal cash flows (column 2) is evaluated at a

17
discount rate of 12%. The net cash flow in each year is discounted at 12% to

present value (column 3). Each year’s addition to the present value is

accumulated in column 4. The present value of net benefits (column 6) is

derived by subtracting the investment costs from the cumulative, discounted,

future net cash flows. The present value of net cash flows just offsets

investment costs in the fifth year. The DPB can be calculated as follows:

3,011

DPB = 4 years + year = 4.38 years.

7,944

Since the DPB is less than the period over which the project earns positive

net benefits, we know it is cost effective in the sense that the PVNB > 0.

Table 4.1 DPB Problem With Unequal Annual Cash Flows

Cumulative

Discounted Discounted

Net Net Cash Flows 3 Net Cash Flows Invest- Cumulative PVNB

Cash Flows ($) ($) ment ($)

Years

(t)

($)

(B t - Ct )

-B t - Ct

- S

v Bt - Ct Cost

($)

(c0 )

S

V

-B t - Ct

-

L t=l ,(l+i)t

Lt=l . ( 1+i) c

_

(1) (2) (3) (4) (5) (6) = (4) - (5)

0 50,000

1 10,000 8,929 8,929 -41,071

2 20,000 15,944 24,873 -25,127

3 15,000 10,677 35,550 -14,450

4 18,000 11,439 46,989 -3,011

5 14,000 7,944 54,933 +4,933

6 12,000 6,080 61,013 +11,013

7 8,000 3,619 64,632 +14,632

aThe discount rate = 12%.

18
An alternative approach to calculating DPB for a project with unequal annual

cash flows is to find the time period for which the SIR = 1.0. * The DPB is the

number of years required for the present value of future net savings to just

cover initial costs. The procedure is to select a trial payback (P)

,

calculate the SIR, and if the SIR is not close to one, select another P

calculated from equation 4.4 and repeat the process. When the SIR

approximates one, the last trial payback is in fact the true payback.

Basing the P on previous results helps reduce the iterations required to find

the P for which SIR = 1.0. In most cases, the following approach will provide

P's that will converge on DPB within three or four iterations. Equation 4.4

yields the trial payback values.

?i

SIRi-! - 1

SIRi-j - SIRj-2

• Pi-2 ~ Pi-1 + pi-l (4.4)

where Pi = ith trial value of DPB,

i = index starting with the value one for the first calculation of Pi

,

SIRi = SIR computed for a study period of value Pi,

SIR0 = SIR computed for the original study period,

SIR_i = 0,

P

0 = original study period, and

P_i = 0.

^Note that this approach will also work for projects with uniform net cash

flows.

^Note that the trial P's are in fact the values of N used to calculate the SIR

in equations 3.3 and 3.4.

19
The general approach can be summarized as follows:

o The project SIR is calculated.

o If the SIR is not close to 1.0, try another P calculated from

equation 4.4.

o Continue until the SIR is close to 1.0. The P for that SIR will

approximate the DPB.

The method can be illustrated with the problem described in table 4.1.

Assuming an intial study period of seven years, the SIRj_j will be 1.29 (i.e.,

$64 ,632/$50 ,000) . Applying equation 4.4, the first trial value of the DPB

would be calculated as follows:

1.29 - 1.0

p „ — 1 F o 1 - 1.29 - 0

L J L

Substituting the value 5.43 for the study period N in equation 3.3 yields a

new SIR of 1.15, ($54,933 + .43 ($6 ,080) ) /$50 ,000. Since the SIR value

exceeds 1.0, a second trial period is computed with equation 4.4. Using 1.15

as SIR^_i and 5.43 as P^= j

,

P

2 = 1.15

1.15

- 1.0

- 1.29

• 7 - 5.43 + 5.43 3.75.

Substituting the value of 3.75 for the study period N in equation 3.3 yields a

new SIR of .88, (($35,550) + .75 ($11 ,439) )/$50 ,000. Since the SIR is still

not close to 1.0, a third trial period is computed.

.88 - 1.0

P 3 = 5.43 3.75 + 3.75 = 4.50.

.88 - 1.15

20
Substituting 4.50 into equation 3.3 yields a SIR of 1.02, (($46,989) +

.5 ($7 ,944))/$50,000. This SIR is very close to 1.00 and the corresponding

trial payback, 4.50, is close to the value computed earlier of 4.38.

The SIR-based approach is most useful when mathematical solutions of closed

form are not available and when it is used either (1) with a standardized

worksheet in a step-by-step procedure that simplifies the calculations or (2)

in a computer application.

4.1.3 DPB With Escalation

The solution value of Y in equation 4.5 is the DPB with escalation (DPBg)

.

Y / 1 + e\ c

I ( “ ^t^ ” C0

** 0, (4.5)

t=l \1 + i /

where e = constant escalation rate of net cash flows (e.g., energy savings).

It accounts for annual net cash flows that escalate at a constant rate e.*

Direct solution of Y is not possible using equation 4.5.

By using logarithms, however, as was done in equation 4.2, Y can be isolated

for direct solution, as illustrated in equation 4.6.2

^In recent years investments in energy conservation projects typically involved

escalating savings from reduced energy consumption due to energy prices rising

faster than other prices.

2For a derivation of equation 4.6, see Appendix A.

21
log 1 + (SPB) /l -

1 + e

1 + i

DPBp = Y = when e * i. (4.6)

1 + e log

1 + i

DPBg = SPB when e = i, and

DPBg is undefined when (SPB) / 1 -

(4.7)

1 + e

< - 1 ,

1 + i

where DPBg = discounted payback year with cash flow escalation.

1

When e > i, the term (1 + e)/(l + i) > 1.0, and therefore DPBg < SPB. That

is , the stream of net cash flows will be inflated more by escalation than it

is reduced by discounting, resulting in a shorter DPBg than SPB.^ When e < i,

the term (1 + e)/(l + i) < 1.0, and thus DPBE > SPB. That is, the discounting

outweighs the escalation, resulting in a longer DPBg than SPB. When e = i,

the discounting just cancels out the escalation effects, and DPBg = SPB.

As long as the DPBg is less than the period over which the project yields

returns, the project is cost effective in the general sense that the PVNB > 0.

1 A variation of equation 4.6 sometimes used is the following:

log

(B - C) (1 + e)

C0 (e - i) + (B - C) (1 + e)

log

"1 + i“

_1 +

when e * i.

^This assumes that cash flows are not inflated by the escalation factor when

SPB is calculated.

22
When (SPB) 1

1 < -1.0, then DPBj? is undefined because the logarithm

1 + e

1 + i

of either zero or a negative number is undefined. If the value is -1.0, however,

the DPBe can be inferred to be approaching infinity, following the logic

described in section 4.1 for DPB. When this term is -1.0, it can also be

inferred that the PVNB = 0 when the study period is infinity. If

(SPB) 1 -

1 + e

,< -1.0, the project never pays for itself, even if it

1 + i

continues to earn net benefits forever, and the PVNB will be negative.

Equation 4.6 can be illustrated with the following problem. An energy

conservation investment of $40,000 yielding energy savings initially worth

$8,000 annually is to be evaluated with an 8% energy price escalation and a

12% discount rate. Applying equation 4.6 yields the following:

dpbe =

log

$40,000

1 + I |/ 1 $ 8,000

1.08

1.12

log [1 + 5 ( -.0370))

log .9643

log .8150

log .9643

= 5.63 years.

If the conservation project yields energy savings for a period longer than

5.63 years, the PVNB of the project will be positive.

23
4.2 Graphical Solution ^

The DPB for projects with uniform annual net cash flows or flows that

increase at a constant rate can be found by using graphs. Since equation 4.2

for DPB and equation 4.6 for DPBg are functions in part of SPB, and since the

value of SPB is often desired along with that of DPB, the DPB graphs used here

are presented as functions of SPB.

4.2.1 Uniform Net Cash Flows

Figure 4.1 plots DPB up to 10 years as a function of SPB for SPB values of

zero to 4 years and discount rate values of 1% to 25% in 2% increments.

Figure 4.2 is similar to figure 4.1 except that even values of the discount

rate, from 2% to 24%, are used to plot DPB. 2 Figures 4.3 and 4.4 are the same

respectively as figures 4.1 and 4.2, except that SPB values range from 4 to 12

years and DPB values range from 4 to 24 years. All of the curves are derived

from equation 4.2. The procedure for finding DPB is to solve first for SPB

using equation 3.2, and then to find the corresponding value of DPB on the

curve for the given discount rate.

Taking the DPB problem from section 4.1.1, we can use the graphical approach

to find the DPB for a $12,000 investment earning uniform annual net cash flows

of $4,500 for 6 years. A 10% discount rate is used. The SPB is 2.7 (i.e.,

$12,000 t $4,500). Thus figure 4.2 is used. Finding the value 2.7 on the

^Graphs can be constructed that provide values of SPB for different combinations of initial costs and uniform cash flows. But given the ease with

which SPB can be calculated from equation 3.2, there is no need for such

graphs

.

2 The odd/even designation of the curves has no significance beyond making the

values along the curves easier to read.

24
25

23

K IO CO tCO

GQ

C\J Q_

CO

O

CQ

CLO

25

Figure

4.1.

Discounted

payback

for

SPB=0

to4

years, i=1

to

25%

(odd)
24

22

00 CD CM

CO

GQ

CM CL

CO

GQ

CLO

26

Figure

4.2.

Discounted

payback

for

SPB=0

to4

years, i=2

to

24%

(even)
27

Figure

4.3.

Discounted

payback

for

SPB=4

to

12

years, i=1

to

23%

(odd)
28

Figure

4.4.

Discounted

payback

for

SPB=4

to

12

years, i

=2

to

24%

(even)
horizontal SPB axis, a vertical line is drawn from that point to find its

intersection with the DPB curve for 10% • Extending a line horizontally from

that intersection to the vertical axis indicates a DPB value of approximately

3.3 corresponding to the SPB value 2.7.

The DPB > SPB for all values of i > 0. Thus the functions in figures 4.1

through 4.4 will all lie above the straight line function (not shown on the

graphs) that would equate SPB to DPB when i = 0. Furthermore, due to the

cumulative discounting effect, the DPB functions will increase at an increasing

rate.

When the DPB value is greater than the limit of the vertical axis, such as is

the case for SPB * 9 and i = 11% in figure 4.3, then the DPB value cannot be

read from the graph and would have to be computed from equation 4.2.

As explained in section 4.1, any project for which (SPB • i) > 1 will have an

undefined DPB. Thus the project never pays off and there is no DPB. For

example, in figure 4.3, a project evaluated with a SPB = 4.8 and a discount

rate of 21% would never pay off, and consequently the DPB curve is truncated

before it reaches a DPB value corresponding to an SPB of 4.8 on the horizontal

axis.

4.2.2 DPB With Escalation

Two types of graphs are presented here for finding DPBE . The first shows DPB F

as a function of the constant k, where k = (1 + e)/(l + i). The second type

of graph relates DPBE directly to SPB for specific escalation rates and

discount rates. Each type of graph is presented because each has advantages

over the other under certain circumstances.

29
Figures 4.5 through 4.8 present a family of DPBE curves labeled with their k

values. Each curve is derived from equation 4.6, where k has been substituted

for (1 + e)/(l + i) . Figures 4.5 and 4.6 present respectively DPBg for odd

and even k values over the range k = .77 through 1.17, for SPB's up to 4

years. Figures 4.7 and 4.8 are the same respectively as figures 4.5 and 4.6,

except that SPB values range from 4 to 12 years, and k values have a lower

bound of .81.

The major advantage of plotting DPBg for each value of k is that few graphs

are required to describe many combinations of e and i. The use of figures 4.5

through 4.8 can be simplified by finding the value of k in table 4.2, which

provides a matrix of k values for all likely combinations of e and i.

Using again the problem example for DPBE in section 4.1.3, we illustrate the

graphical approach in finding the DPBE for a $40,000 investment initially

yielding annual net savings of $8,000, with an energy price escalation rate

of 8% and a discount rate of 12%. Since the SPB is 5 (i.e., $40,000 *

$8,000), we know the DPBg will be found either in figure 4.7 or 4.8, which

cover the SPB range 4 to 12 years. By consulting the matrix of table 4.2, we

find a k value of .96 in the cell intersection for e = 8 and i = 12. Since

the last digit of k is an even number, we look to figure 4.8 (even) for the

DPBg for 5 years and k = .96. The answer is approximately 5.7.

The DBPg = SPB whenever k = 1, since the escalation and discounting effects

cancel each other. When k < 1, DPBE > SPB, and the curves will increase at an

increasing rate. When k > 1, DPBE < SPB, and the curves will increase at a

decreasing rate.

30
CO

CM

CD

CL

CO

lu CD

CL

Q

31

Figure

4.5.

Discounted

payback

with

escalation

for

SPB=0

to

4

years,

k

=

.77

to

1.17

(odd)
32

Figure

4.6.

Discounted

payback

with

escalation

for

SPB=0

to4

years,

k=

.78

to

1.16

(even)
85|

.87|

.89|

i.91 ii

.93

33

Figure

4.7.

Discounted

payback

with

escalation

for

SPB=4

to

12

years,
© O OJ t ®

o T” r- 9- 'flu CD

0. O

34

Figure

4.8.

Discounted

payback

with

escalation

for

SPB=4

to

12

years,

k=.82

to

1.16

(even)
Table

4.2.

Matrix

of

k

values

for

combinations

of

e

and i

X X if> -r m ru *-• o CD N -0 n rt rs-j ru T-H IJ X X r-v r~- X m rr r»

nr. rc to rH t-H t-H *H T-H -1 T-H t-h O o o o o o d o O o X cr X X o X X c^ X X

t-H t-H t-H t-J -4 >—

I

T—1 *—* T-H rH t-H t-H t-H tH O G d o o o o d o o

0 m rr CD M t-h O O' CO >0 in rr 0) r-j rH O O' X X r- X o CD CD ru T-l

t-H t-H t-H t-H t-h *-h o o o o o o o o C o X X X X X X X X X X X

t-H tH t-H T-H t-H t-h H rH T-H rH T-H rH rH o o o o o o o o o o c

in rr CD ru t~« c O' CD N <

i

rr CD ru r-H o X X X X HD in fn CD ru T-H c

00 t-H t-H t~* t-h — O O o o G o O o O O X X X X X X X X O' X X O'

f-i -rH t*H - t-H H t-4 rH T-H T-H rH T-H T-H o o 0 0 o G o o' o o O o

CM

rr n ru »-H O O' CD X >0 in rr CD ru rH O O' X X X HD in rr CD n ru T-H o o

— o c o o O o o O'- X X X X X X O' X X Ch 0 Cr - - - - t-* - - - v~1 t-H rH t-H rH G o o o o o G rx c c o o’ cco ru — O o CD x x to rr

rr^ ru T-H O X fv X x in rT m m ru T-H o o O' T—

tH ri t-H t-H o O o o o O o o O o 0- PK X X 0- X X X X X X X X X

T-H t-H t-H t—H •H t-H ^—4 rH rH t-H rH o O o o 0 0 O o o o o’ o o o

ru — o O' CD N -o in r}- CD ru T-H O 0- X hx <3 H3 m rr X X ru t-H O X X X O r>* C o o o O O o O O' O' X X X X X X X X X X X X X

- ^ - - - - t-h *-* r-J - - c d «w- 6 c

0. 0. V5 d o o G o o c

H o o- CD N. x m rr fn ru r—

H

o O' X -0 HD m rr X ru ru G X X X X

o> -1 — 0 o o o o o d o O O 0- 0- O' rr. o XXX X X X X X X X X

t-H t-H t*H T-H -rH H rH o d d o O o 0 0 o o o’ o o o o o

O O' X rv X LO rr cd ru O O' X x <3 <3 in rr x ru ru t-H o X X X N X

CO c

r-S o o c

*—* C o O' 0- o- o- X X X X X o O' X X Gj X X X - - - - *-* - - T~i r- c c o c o o’ o o G o b o c r

r> rr s X | 1*N — cd ru •—

o 0- X' r- >0 m rr X ru - T—5 o o- X X r- X X K o o c o O o o o o o O' o O' i> O' O' X O' X X X O' X CD X X X X - -H -T T-H t-H T-H H »H r4 t-4 o o o o o o o o o c O o o O d G o

CD X in cr CD ru *h O O' CO N J3 in in rr n ru tH t-H o X X X K. X in i

n

(0 o o o o o O o o O O' O' O' O' O' O' X X X X X X X X X X X X X

"*4 •"* —* th T-H tH t-h t-h »H o o d d o o o G o O G o o t-H d o’ o 0 c

N U3 UO rr CD ru - c O' CD N -0 in in «u- X ru tH O X X X N X in m rr in O O O O O o o c O' O' O' O' O' 0- O' X X XXX X X X X X X X X

t-H t-H -H *•1 t-h o o o o o o d o o o 0 0 O o o d o o o c

0 tO rr n ru _ o 0- CD N. •-0 !fj ID rr pn rj ^H r\ O X CO N X in m Ph X COO c o o O O' > O'- o O' O' O' Cr X X- X O'- X CO X X X X X X cr

t-H T-H t-H t*H T-H t-H - o O o o o o o O G o o 0 0 O o o o o G o o

m rr x ru 'w' O' cr- fv c tn rr X ru •H G o X X N X X »r*» 's' X ru CO o c o o o O 0- O' O' O' O' O' O' 0- O' O' X X X X X X UJ X X X X X

—t —H -> t-H o o 0 o o o r*j -r-x o o o 0 0 O o c G o o o

rr CD CU t-H o o- CD x -0 m «T rr CD ru o X X X X X X tn rr p?* X ru ru CM C C 0 o 0* !> O' o- o O' 0 o- O' X I- X X X rp X X X rr< X X X - - - - T*H o o c d £. o o o o c o o o G C o G rx O o o r~

r

0 CO r " -c in •r

nr, ru i—' o X X X f' -0 < in rH- T, m ru T- —

T- o o o 6 r> O' O' O' O' O' O' O' O' 0- O' X X X X X X ro X X X CO X X

t-H --H t-H o o 0 0 o o o o o o o o o o O O o o O o o o o o

ru —* o cr CD X D in rr ro n ru ^H o O' X X X x m UD rr CD X ru T-H T-H o

o o o o O' O' Ch O' O' O' O' O' O' 0- O' X X X X X X X X X X X X X X

tH t—

<

o O O 0 0 o o o o d d o G d o 0 0 o o o o o o o G — C X £C fx -0 in rr CD CD ru T-H o O' X X X HD in in rr CD ru ru o o X

r- O O X 0- O' O' O' 0- O' O' O' O' 0- X CO X X X X X X X X X X X X X

1 *-« —« o o o o 0 0 o d o O o o o o o o 0 0 o o o o o o o o

O X X X in rr n ru ru rH O O' X X K HD in rr rr CD ru ru ^H o o X X

CM C X O O' O 0 O O' O' i*V O- Cr X X X X X X X X X X CC X X X I'

1 - o o o o o G O o o o o o o G G o o 0 0 o o /“N o o G o c

XXX x in «T cd ru ru «r-4 o O' CD N N O in rr rr X ru ru T-H G o X X X

CO XXX O' O' O' O' O' cr O' O' CD CD X X X X X X X X X X X X X X X

1 o o o o o d 0 0 o o o O O o o d o o 0 0 o o O o o o o o

CD -0 to r? CD ru -i *-i o O' PQ N 0 >0 m rf X x ru H o X X X X fv O' O' O' O' x O' O' o O' O' CD CD X X X X X X X X X X X N V f'

1 o o c Q o o o >-/ d d 6 3 c G G o C c c

/<D/ CM Y c T— CMCOTtmcON-OOOO CMCOH-inCDfx-OOCXO T- CM CO tT lO 7* 1 1 T“ T“ CM CM CM CM CM CM

35
When the DPBg value Is greater than the limit of the vertical axis, such as

would be the case for a SPB = 8. A and a k = ,90 in figure 4.8, then the DPB

value cannot be read from the graph and would have to be computed from

equation 4.6.

When the DPB]? is undefined, i.e. , when the term (SPB)(1 - 1/k) < -1, the

project never pays off, so there will be no reading from the graph and

there will be no solution to equation 4.6.

The second type of DPBg graph is illustrated with figures 4.9 and 4.10. Each

figure presents a family of DPBg curves for a given rate of discount.

Equation 4.6 is used to plot the curves. Figure 4.9, for example, yields the

DPBE curves for a discount rate of 7% and for escalation rates of 0%, 2%,

4%, 6%, 8%, and 10%. Each curve is labeled with its escalation rate. Figure

4.10 does the same for a discount rate of 10%. An advantage of this approach

over the k graphs in figures 4.5 through 4.8 is that no k factor has to be

determined and the DPBg can be found directly from the graph once the SPB is

calculated. However, the disadvantage is that many graphs would be required

to cover the feasible combinations of e and i that are covered just with the

figures 4.5 through 4.8.

To read the graphs in figures 4.9 and 4.10, find the figure for the given

discount rate, extend a vertical line for the given SPB value, and find the

corresponding DPBE where the vertical line intersects the DPBE curve for the

given escalation rate. For example, if we had a SPB = 10, an i = 7%, and an

escalation rate of zero, we would find a DPBE of about 17.8 years.

36
37

Figure

4.9.

Discounted

payback

for

SPB

=0

to

10

years,

e=0

to

10%

(even),

i=7%
oo o

<

o

O

CO

Q.

CO

in CO

CL

Q

38

Figure

4.10.

Discounted

payback

for

SPB

=0

to

10

years,

e

=0

to

10%

(even),

i

=1

0%
For e < i, DPBe > SPB, and the curves will increase at an increasing rate and

lie above the straight-line function (not shown in the figures) where

DPBjr = SPB and e = i. When e > i, DPBE < SPB, and the curves will increase at

a decreasing rate and lie beneath the straight-line function where

DPBe = SPB.

Figures 4.9 and 4.10 will give slightly different answers for the same SPB’s

than will figures 4.5 through 4.8 because the value of k in table 4.2 on which

the k plots are based is rounded to 2 decimal places whereas the corresponding

value of (1 + e)/(l + i) in figures 4.9 and 4.10 is computed to 8 decimal

places in making the plots. These differences in answers will become greater

as k decreases and the curves in figures 4.5 through 4.8 increase at an

increasing rate. If a greater degree of accuracy is required using the k

matrix, the k values in table 4.2 could be carried out to more decimal places

and more k plots at these intermediate values could be generated.

Alternatively, equation 4.6 could be used to solve directly for DPBE with k

calculated to whatever number of decimal places is desired.

39
5. RECOMMENDED APPLICATIONS AND LIMITATIONS

This section discusses whether the SPB or DPB is a better measure of a project's

economic worth. Recommendations regarding the applicability of payback methods

are made for typical investment problems where there is some controversy as to

which economic evaluation method is appropriate. Limitations of the payback

method in general are described. The use of payback as a supplementary method

of economic evaluation is discussed.

5.1 SPB or DPB

Because SPB is quick and easy to compute, it is appropriately used as a

screening "first measure" of project worth when time, resources, or expertise

are not available for calculating a DPB measure. The SPB measure also may have

some value as an accounting measure in that it is computed in current,

undiscounted dollars and indicates when actual net cash flows just balance

investment costs.

Time and resources permitting, the DPB measure will be preferred to the SPB

measure because DPB accounts for the time value of money. During a period when

inflation, real earning opportunities for capital, or both result in a positive

discount rate, a given cash amount in the future will be worth less than that

same amount today. Thus the DPB measure will be longer than the SPB measure for

the same set of cash flows over time. Using a SPB measure under these

circumstances suggests a shorter payback than the project really has. The DPB

measure rather than the SPB measure is required to determine when the present

value of future net cash flows equals the initial investment.

40
5 . 2 Payback Versus Other Economic Methods

Table 5.1 lists some typical economic questions pertaining to investment projects

and indicates whether payback measures are appropriate for answering those

questions. Payback information may be helpful in answering the first two

questions, but will generally lead to economically poor investment decisions if

used to answer the last three questions.

Table 5.1 Economic Questions and Usefulness of Payback

Use Payback?

Yes No

1

.

How to compare benefits with costs? X

2. How to find the break-even year? X

3. How large an investment to make? X

4. How can projects competing for the same purpose be compared? X

5. How can different-purpose projects competing

for the same budget be compared? X

The first question, how can benefits be compared with costs, can be reliably

answered under some circumstances with payback measures. For example, if a project

is being examined to determine if it is cost effective in the general sense that

PVNB > 0, the DPB can be used to test for cost effectiveness. That is, If the DPR

is less than the period over which positive net benefits are earned, then the

project is cost effective. If negative cash flows occur beyond the payback year,

however, one of the following methods would be needed to determine if the pro] err

were cost effective: net benefits analysis (to see if the present value

41
of net benefits exceeds zero), the adjusted internal rate of return, AIRR (to

see if the AIRR is greater than a minimum acceptable rate of return) , or the

savings-to-investment ratio (SIR) or benefit-to-cost ratio (BCR) method (to see

if the SIR or BCR is greater than 1.0).l Thus payback as the sole measure of a

project's worth is limited in usefulness to those cases where the stream of

benefits clearly dominates all costs over the project's life.

Another approach sometimes used in payback analysis is to establish a maximum

acceptable payback period (MAPP) as a cutoff value for accepting or rejecting

projects. For example, an arbitrary time of 3 years might be determined by the

management of a firm as the MAPP they will allow. Thus all projects with

paybacks greater than 3 years, regardless of their cost effectiveness as

measured by the PVNB, will be rejected.

Table 5.2 illustrates the PVNB implications of using a MAPP to decide whether a

project is to be accepted or rejected. If the cutoff payback value were 2.5,

for example, no project with a payback longer than 2.5 years would be accepted.

Thus project A, with SPB of 3.1 and DPB of 3.9, would be rejected. Yet project

A is cost effective because it has a positive PVNB. Rejecting it would result

in an economic efficiency loss of $30. Project B, on the other hand, meets

the MAPP criterion in that its paybacks are less than 2.5 years. However,

*For a description of net benefits analysis and the AIRR, see Harold E.

Marshall, Recommended Practice for Measuring Net Benefits and Internal Rates of

Return . For a description of the SIR and BCR, see American Society for Testing

and Materials, "Standard Practice for Measuring Benefit-to-Cost and Savings-toInvestment Ratios for Buildings and Building Systems," E-964 , Annual Book of

ASTM Standards (Philadelphia, PA: ASTM, 1984).

42
Table 5.2 Using Payback to Accept or Reject Projects3

(Bt - Ct ) ($) Co PVNB

Projects Years/

1

2 3 4 ($) ($) SPB DPB

A 325 325 325 325 1,000 30 3.1 3.9

B 800 500 -100 -300 1,000 -140 1.4 1.7

N

aA 10% discount rate is used to compute PVNB = £

t=l

(B t - Ct )

(1 + i) c Co*

accepting project B will result in an economic efficiency loss of $140 in PVNB

due to negative cash flows beyond the payback year.l Since accept/reject

decisions made with the payback method may lead to such efficiency losses, the

other evaluation methods listed above are recommended for making these

decisions whenever benefits do not obviously dominate costs over the project's

life.

The second question, how to find the break-even year, can be answered

correctly by using the payback method. The number of years required for a

^We recognize that a manager concerned with efficiency would not implement

project B as described in this simple example because losses in the final two

years make the project cost ineffective. However, the manager roust look

beyond the payback year to see this, and that is why the other methods that

look at the whole study period are recommended over payback.

43
project to just break even (i.e., the number of years for earnings or savings

to pay off all project costs) is called the break-even year.l Hence it is the

same as the payback year. Although the break-even (payback) year is helpful to

an investor in deciding whether or not to undertake a project, it does not show

that a project is cost effective except under the special conditions described

above.

The third question in table 5.1, how large an investment to make, cannot be

answered properly using payback measures. That is, sizing a project so as to

minimize payback would lead to inefficient projects (i.e., projects that do not

maximize the PVNB). The net benefits method, on the other hand, is an

appropriate method.

Table 5.3 illustrates the efficiency implications of using payback instead of

the net benefits method in choosing project size. Both payback measures favor

size C, requiring a $1,000 investment and yielding a PVNB of $751. However,

the PVNB is maximized at size D for an investment of $2,000 which yields a PVNB

of $ 1 , 073.2 Choosing size C on the basis of payback would result in an

economic loss of $322 ($1,073 -$751).

^Note that break-even analysis in general can be applied to any variable that

affects the net benefits of a project. For example, the break-even wage rate

might be computed for a labor-intensive construction job to determine the wage

rate at which the builder would just break even.

^It is assumed for this example that there is no budget limitation regarding

initial project investment costs.

44
Table 5.3 Project Sizing With Payback and Net Benefits 3

(Ten Year Project Life)

Size

(1)

(®t - Ct)

Each Year

(%)

(2)

(cG ) ($)

(3)

PVNB

($)

(4)

SPB

(5)

DPB

(6)

C 285 1,000 751 3.5 4.5

D 500 2,000 1,073 4.0 5.4

E 1,000 8,000 -1,855 8.0 16.9

a The discount rate used is 10%.

The fourth question, how can projects competing for the same purpose be compared,

again is best answered by the net benefits method. Since benefits and costs

beyond the payback year are ignored, selecting the project with the minimum

payback biases project selection against projects with a high proportion of net

benefits accruing in the future. Thus the project with the shortest payback

period may not be the one that maximizes the PVNB.

Table 5.4 illustrates how payback can lead to selection of an inefficient project

when two projects are competing for the same purpose. Project F earns all of its

benefits in the last 3 years, whereas project G earns most of its benefits in the

first 2 years. Both projects are cost effective in the sense that they earn in

present value terms more than they cost (i.e., the PVNB is greater than zero).

However, project F is the more efficient choice since it yields a larger PVNB

than project G.l If minimum payback were the criterion for choosing, project G

*In this and subsequent illustrations risk is assumed to be the same among

project alternatives and not to increase as a function of time.

45
would be preferred, because its payback is shorter than F’s. The economic

loss from choosing according to the minimum payback would in this case be

$402 ($555 - $153).

Table 5.4 Project Acceptance, PVNB Versus SPB or DPBa

(Bt - Ct ($)

Proj ects Years/

1

2 3 4 5 CQ PVNB SPB DPB

F 0 0 1,000 1,000 1,000 1,500 555 3+ 5+

G 700 600 10 10 10 1,000 153 2+ 2+

N

aA 10% discount rate is used to compute PVNB = £

t=l

"(B t - Ct

)‘

.(1 + i) C

.

Co*

The last question in table 5.1, how can different-purpose projects competing

for the same budget be compared, can be answered properly with the

savings-to-investraent ratio (SIR) method or adjusted internal rate of return

(AIRR) method. Either of these methods when correctly applied will generally

lead to a set of projects that maximizes the PVNB for the limited budget.

Table 5.5 illustrates the losses in net benefits that would result if a limited

budget of $1,800 were allocated among a set of different-purpose projects on

the basis of minimum paybacks. Four projects with earnings from 2 to 8 years

are considered. Uniform annual cash flows are shown in column 3, initial

investment costs are shown in column 4, and the annual values of initial costs

(converted at a 10% discount rate) are shown in column 5.1

^The equation for converting present values (P) to annual values (A) is the

Uniform Capital-Recovery (UCR) Equation in table 2.1.

46
Table 5.5 Ranking Different-Purpose Projects

Annual

Years Value of

Projects (N) (Bt- ~ Cf-) Co Co AVNB

($) ($) ($) ($) SIR SPB DPBa

(1) (2) (3) (4) (5) (6M3)-(5) (7)=(3)t (5) (8)=(5)t (3) (9)

H 8 750 1,000 187 563 4.0 1.3 1.5

I 5 308 500 132 176 2.3 1.6 1.9

J 3 643 800 322 321 2.0 1.2 1.4

K 2 864 1,000 576 288 1.5 1.2 1.3

a DPB is computed from equation 4.2.

Given the budget of $1,800, projects J and K would be chosen if minimum

paybacks were the criterion for choosing projects. The AVNB (column 6) for J

and K would be $609. At first glance, projects J and K might seem attractive

in their quick recovery of first cost. But two other combinations would yield

greater returns. For example, if the same $1,800 were allocated to H and I,

on the basis of the highest SIR’s, an AVNB of $739 would result, with $300

left in the budget. Payoff of the initial costs would take a little longer,

but the extra benefits in subsequent years make selection of projects H and I

worth $739 as compared to the J and K combined worth of only $609 on an annual

basis. Following the SPB or DPB method here would result in a net efficiency

loss of $130 ($739 - $609) per annum. Alternatively, to get the most from the

$1,800, projects H and J could be selected by moving to the project with the

next lowest value of SIR which exhausts the budget. 1 The AVNB would then be

$884. Choosing J and K instead of H and J would result in a net efficiency

loss of $275 ($884 - $609).

^For an explanation of how to apply the SIR technique under limited budget

conditions, see ASTM Standard Practice for Measuring Benef it-to-Cost and Savlngs - to-Investment Ratios , section 10.

47
5.3 Payback As a Supplementary Method

Most of the questions in table 5.1 have been marked "No" regarding the

applicability of payback. It received an unqualified yes only in finding the

break-even value. In fact, much of the engineering economics literature

dismisses payback from serious consideration as a reliable economic decision

tool. For example, Park, using the term payout for payback, states the

following

In summary, the payout period is often used and referred to as a "quick-and-dirty" means of looking at the relative attractiveness

of investment proposals. But by itself the payout time is _so dirty

that the analyst cannot see what lies underneath—or worse still, sees the wrong thing. Therefore the payout period should never be

used as the sole criterion in investment evaluation and can seldom

be justified for any use other than as an interesting, even if

irrelevant, piece of information.!

Grant and Ireson also criticize the use of payback, as described in the

following statement:

Except for the special case where funds are so limited that _no outlay can be made unless the money can be recovered in an extremely short time ..., the payout time is never an appropriate way to compare a group of proposed investments.

2

However, the primary contribution of payback lies not in its use as a method

for making major decisions, but in its use as a supplementary method of

economic evaluation. That is, it gives one kind of information that, in

conjunction with other economic measures, helps determine the economic

desirability of one or more projects.

^William R. Park, Cost Engineering Analysis , pp. 35 and 36.

^Eugene L. Grant and W. Grant Ireson, Principles of Engineering Economy , 5th ed. (New York: The Ronald Press Co., 1970) p. 528.

48
As a supplementary method, payback is probably most helpful as a screening

method for evaluating investment candidates that have limited lives

beyond which potential returns may become irrelevant. The payback measure

helps define the feasible set of projects to which additional economic methods

can be applied. For example, an investor who is considering investments in a

foreign country might establish a MAPP of 2 years if nationalization,

revolution, political instability, or other conditions that might diminish

returns on the investment were likely to occur within several years. A

manufacturer of building components, for example, might establish a MAPP of

only 3 or 4 years for a product line that could potentially yield profits for

10 years, because the uncertainty and risk of such factors as obsolescence,

competitive products, and shifting market conditions might threaten the

manufacturer's profit potential after 3 or 4 years.

Numerous other reasons can be found for the widespread applications of payback

in practice. It is easy to determine and to understand intuitively. It can

be used to indicate how long an investor's capital is at risk in terras of how

many years are required before payoff. It serves as an index to short-run

earnings per share of stock. It helps to identify projects that will be

unusually profitable or unprofitable early in their life. And finally, with

tight capital conditions, investors often want to be assured of short paybacks

in addition to high rates of return or high PVNB before they will part with

their capital.

49
The basic limitations of payback remain, however. It ignores net benefits

beyond the payback year. This imposes a bias against long-term projects such

as public works or commercial projects with significant research and

development startups in favor of projects with quick payoffs. The SPB

measure further distorts economic evaluations by ignoring the time value of

money. Thus to make economically efficient choices among competing projects

and among designs/sizes for a single project, payback as an evaluation method

is generally appropriate only when used as a supplementary method with other

economic evaluation methods.

50
APPENDIX A

DERIVATION OF DPB AND DPBg 1

A. 1 DPB (Equation 4.2)

We know that the solution value of Y in equation 4.1 is the DPB. By

rearranging terms in equation 4.1 we get

Y 1

l (S t - ct> - c0 . t=l (1 + l) c

Given that B t - C t is a constant,

Y 1 (B - C) l - C0>

t=l (1 + i)t

t=l (1 + i) c (B - C)

Y 1 = SPB

.

t=l (1 + i)t

, and

Y 1 (1 + i) Y -1

Since — = ' , the UPV factor formula from table 2.1,

t=l (1 + i) c i (1 + i) Y

(1 + i) Y -1

SPB =

i (1 + i) Y

^•Thanks are due Robert E. Chapman and Stephen F. Weber for their help in the

derivation of these equations.

A-l
By rearranging terms.

(1 + i) Y -1

i • SPB

(1 + i) Y

i SPB = 1 and

(1 + i) Y —— = 1 - (i • SPB).

(1 + i) Y

Taking logarithms of both sides of the equation,

-Y log (1 + i) = log (1 - i • SPB),

Y log (1 + i) = -log (1 - i • SPB),

A. 2 DPBp (Equation 4.6)

We know that the solution value of Y in equation 4.5 is the DPBg. Given that

(B t - Ct ) is a constant, we can rearrange equation 4.5 as follows:

1 Y log (1 + i) = log and

(i • SPB)

1 log

1 - (SPB • i)

Y

log (1 + i)

t=l

Y

l (B - C)

and therefore

A2
1 + e\ t Y

l

t=l \ 1 + i

= SPB.

Since Y / 1 + e

1 — t=l \1 + i

1 + e

N 1 - e.

1 -

1 +

1 +

Che UPV* factor formula from table 2.1,

SPB

1 + e

i - e

1 + e\Y

1 -

1 + i

By rearranging terms

,

/ i - e \

1 + e SPB =

1 + e\

Y

1 -

1 + i

1 -I

i - e

1 + e

1 + e\

Y

SPB =

1 + i

'< *

Y log

1 + e'

1 + i

log 1 -

i - e

1 + e SPB

Y =

log "(tH)TM

log

1 + e

1 + i

Y =

log 1 + SPB

e - i\n

1 + e

log

1 + e

1 + i

, and

A3
Since

e - i (1 + e) - (1 + i)

1 + e 1 + e

= 1 -

log

Y =

1 + SPB / 1-

1 +

1 + i

log

1 + e

1 + i

A4
APPENDIX B

SELECTED BIBLIOGRAPHY

American Society for Testing and Materials, Adjunct Discount Factor Tables , (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 , Vol. 04.07 (Philadelphia, Pennsylvania: ASTM, 1983).

American Society for Testing and Materials, "Standard Practice for Measuring

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

Systems," E-964, Annual Book of ASTM Standards (Philadelphia, Pennsylvania:

ASTM, 1984).

'

Dell’Isola, Alphonse J. and Kirk, Stephen J. , Life-Cycle Cost Data (New York:

McGraw Hill, 1983).

Grant, Eugene L. and Ireson, W. Grant, Principles of Engineering Economy , 5th ed. (New York: The Ronald Press Company, 1970).

Marshall, Harold E. , Recommended Practice for Measuring Net Benefits and

Internal Rates of Return , National Bureau of Standards Interagency Report

83-2657, October 1983.

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. , 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. , 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.

Mishan, E. J., Cost-Benefit Analysis (New York: Praeger Publishers, 1982).

Park, William R. , Cost Engineering Analysis (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. , Life-Cycle Cost 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. , Recommend <-d Practice for Measuring Life-Cycle Costs of Buildings and Building Systems , National Bureau of Standards Interagency Report 80-2040, June 1980.

B-l
NBS-H4A (REV. 2-80

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

4. TITLE AND SUBTITLE

1. PUBLICATION OR 2 REPORT NO.

NBSIR 84-2850

. Performing Organ. Report NoJ 3. Publication Date

March 1984

Recommended Practice for Measuring Simple and Discounted Payback

for Investments in Buildings and Building Systems

5. AUTHOR(S)

Harold E. Marshall

6. PERFORMING ORGANIZATION (If joint or other than N BS, 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

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

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here)

This report describes how to calculate simple and discounted payback measures of

economic performance of buildings and building systems. Formulas for calculating

payback, applications for evaluating and selecting projects, and limitations in

the use of payback analysis are discussed. The simple payback method measures the

time between the date of initial project investment and the date when cumulative

future earnings or savings on that investment, net of cumulative future costs, just

pay off the investment. The discounted payback method measures the time between the

date of initial project investment and the date when the present value of future

earnings or savings, net of the present value of future costs, just equals the initial

investment. This recommended practice 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 semicolon s) benefit-cost analysis; building economics; cost effective; discounted payback; economic

analysis; investment analysis; life-cycle cost; payback; recommended practice; savingsto-investment ratio: simple payback

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