Reference Pfj,;

SIMPLIFIED

ENERGY 07S7t3

DESI

EC

PRINCIPlE

CONSER

INVEST

-QO

100

.U57

NO . 54^

1980

NBS SPECIAL PUBLICATION 544 U S Department of Commerce

National Bureau of Standards

KATIONAL BUREAU

OP BTANDABDS

LIBRARY

SIMPLIFIED APR 17 1980

ENERGY '"^ ^ DESIGN

ECONOMICS

PRINCIPLES OF ECONOMICS APPLIED TO ENERGY /^^0>

CONSERVATION AND SOLAR ENERGY

INVESTMENTS IN BUILDINGS

HAROLD E. MARSHALL and ROSALIE T. RUEGG

Edited and illustrated by Forrest Wilson

Center for Building Technology

National Engineering Laboratory

National Bureau of Standards

Washington, D.C. 20234

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued January 1980
LIBRARY OF CONGRESS CATALOG NUMBER: 79-600196

National Bureau of Standards Special Publication 544

Nat. Bur. Stand. (U.S.), Spec. Publ. 544, 54 pages (Jan. 1980)

CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

Washington: 1980

For sale by the Superintendent of Documents, U.S. Government Printing

Office, Washington, D.C. 20402

Stock No. 003-003-02156-3 Price $3.50

(Add 25 percent for other than U.S. mailing.)
PREFACE

One of the most powerful determinants of building form, as we approach the end of this century, is the cost of energy

and energy conservation. Clients expect architects,

engineers and all those associated with the building design

professions to make economically intelligent and informed

energy design decisions on their behalf. The creative

challenge posed by this powerful design determinant and

the legal consequence of improper energy design decisions

are vital presences in today's design practice and

compelling reasons for professionals to continue to enhance their design related skills.

This publication is a joint product of the Design and Construction Technology Application Program (DACTAP) and

the Building Economics and Regulatory Technology

Division, both in the Center for Building Technology. The

report is a tool to be used by the design community in making energy decisions. It will aid you in making economic

evaluations and logical assessments of costs and benefits

inherent in design decisions over time and thus enhance un- derstanding between client and designer. It can serve as a

text for classes and self-instruction, as a reference for the

drafting table, and as a concise group of problem-solving

formats to outline the economic parameters of energy

design decisions.

As a handbook it provides the information you need to analyze straightforward economic problems, which

comprise perhaps 90 percent of those you will encounter. It

will also aid your understanding and facilitate your cooperation with experts retained to conduct more complex

economic analysis. This publication will provide you the

soundly based self-confidence in economics that you now

possess in structural, mechanical and other engineering dis- ciplines related to the design of buildings.

I commend this publication to you for these reasons and

sincerely hope that you will use it to expand your analytical

skills for your benefit and for that of the design professions.

DACTAP is a new program at the Center for Building Technology, a part of the National Engineering Laboratory in the

National Bureau of Standards. DACTAP seeks to

understand the technical information needs of designers

and builders and to provide useful technical information for application in the design and construction process. The

authors, the editor and I will appreciate your comments and

reaction to this publication and your suggestions for future

work.

Porter Driscoll, AIA

DACTAP Coordinator

Center for Building Technology

ACKNOWLEDGMENTS

The authors wish to thank Carol Chapman Rawie and

Stephen Weber, both from the National Bureau of

Standards, for their valuable technical and editorial

comments. Porter Driscoll, John Holton, and George

Turner, all from NBS were also most helpful in reviewing draft reports.

The authors are also indebted to the architects, engineers, urban and regional planners, and others who

provided field testing of this report as a teaching

manual by attending an extension course, "Design

Economics for Energy Conservation in Buildings," presented at the University of California, Berkeley. Steve

Petersen, from NBS, joined the authors in teaching that course and thereby contributed to this report.

Special credit is due Vladimir Bazjanac of the University of California, Berkeley, who gave particularly

useful insight into making the report suitable for stu- dents of architecture and engineering. The NBS Electronic Typesetting staff, directed by Rebecca Morehouse, provided generous assistance in preparing the

report for printing. Finally, thanks are due the many

persons at NBS who have spent time with the authors

discussing energy issues and suggesting significant

problems that this report should address.

iii
CONTENTS

INTRODUCTION

Firmness, Commodity, Delight, and Energy

Purpose of the Book

How the Book is Arranged

CONCEPTS OF ECONOMIC EFFICIENCY

The Objective

Graphical Illustrations MEASURING BENEFITS AND COSTS

Kinds of Benefits and Costs

Five Economic Tools

Life-Cycle Costs (LCC)

Net Benefits or Savings (B-C)

Savings-to-lnvestment Ratio (SIR)

Internal Rate of Return (IRR)

Discounted Payback (DPB)

ADVANTAGES, DISADVANTAGES,

RECOMMENDED APPLICATIONS

Common Economic Questions and

Recommended Tools for Solution

DISCOUNTING

The Method

Sample Discounting Problem

Problem Assumptions

Problem Solution

A PROBLEM IN FIVE SOLUTIONS

Problem Assumptions

Table A: Life-Cycle Costing

LCC Problem Solution

Table B: Net Benefits or Savings

B-C Problem Solution

Table C: Savings-to-lnvestment Ratio

SIR Problem Solution

Table D: Internal Rate of Return

IRR Problem Solution

Table E: Discounted Payback

DPB Problem Solution

2

2

4

5

6

6

7

9

9

10

10

11

11

12

12

13

13

16

16

17

17

18

21

21

22

22

23

23

24

24

24

25

26

26

FACTORS AFFECTING BENEFITS AND COSTS

Discount Rate

Incentives

Grants

Taxes

Government Cost Sharing

Loan Interest Subsidies and Guarantees

Inflation Salvage Value

Taxes

Property Taxes

Income Taxes

Time Horizon

Uncertainty

Sensitivity Analysis

Probability Analysis

SUMMARY

FOOTNOTES

APPENDICES

A Glossary of Economic Terms

B Discount Formulas

C Discount Factors

C-1 Single Compound Amount (SCA)

C-2 Single Present Worth (SPW)

C-3 Uniform Sinking Fund (USF)

C-4 Uniform Capital Recovery (UCR)

C-5 Uniform Compound Amount (UCA)

C-6 Uniform Present Worth (UPW)

D Uniform Present Worth Factors Modified for Energy Price Escalation

D-1 8% Discount Rate

D-2 10% Discount Rate

D-3 12% Discount Rate

27

27

29

29

29

29

29

29

30

31

31

31

31

32

33

34

35

36

38

39

41

42

42

43

44

45

46

47

48

48

49

50

IV
Fish are to bait as savings are to energy conservation costs

1
I

INTRODUCTION

Firmness, Commodity, Delight, and Energy

The fuel shortage has added a fourth element to the three

the Vitruvian view of architecture saw as essential to all buildings if they were to accommodate the physical and

aesthetic needs of humankind. Energy is now, more than at any time in recent history, a crucial element of the art of

building.

The critical need for energy conserving design stems from

the rapid increase in energy costs. A continued scarcity of

fuels, increased cost of energy production and delivery,

and the mounting demands for energy in an energyintensive society have combined to render energy a

primary economic concern of all those involved in the act

of building. The chart (fig. 1)^ on the opposite page

dramatically records energy consumption for building in terms of Btu and dollar expenditures during the past 26

years. During this period physical units of energy consumption increased over 134%. The unit price per 1,000,000 Btu

of energy vacillated, declining half the years, but overall

increased 44%. The major portion of this increase, 39%,

took place during the period 1970-76, and the total dollar

expenditure for energy increased by $26 billion. These

statistics clearly indicate why energy conservation is crucial to the design community.

Alarming conditions exist in all areas of energy consumption in America as energy prices rise and shortages

threaten. Governmental programs, ranging from research

and development in renewable energy sources, such as

solar energy, to legislation that encourages and sometimes

mandates energy conservation, have been instituted to counter these circumstances.^

Spurred by the overall rise in building costs, architects, civil and mechanical engineers, builders, their clients, and

building owners are rigorously examining all phases of

building energy consumption. We find a concentrated examination of the benefits of energy conservation taking place on an unprecedented scale.
Figure 1

ENERGY CONSUMPTION AND PRICE 1950-1976

1

1 ENERGY CONSUMPTION 10'^ Btu ENERGY CONSUMPTION 10^ 1977 $

PRICE PER 10^ Btu 1977$

•

•

•

• * "4 1

« * #

•

•

• # #

1 ••

t j— /' 7^

• .*

•

• M

•

•

•

•

•

•• —/- \-

•

•

• ^

\

•• f •

• /— • • A

/

• ^

^^=__v

f• A• 1§

• • ^^^^ • ^^^^

•

^^^^ •

• •

1950 1955 I960 1965 1970 1975

•

•

•

•

1

1

1

•

1

1

•

20.5 85.0 5.00

20.0 82.5 4.90

19.5 80.0 4.80

19.0 77.5 4.70

18.5 75.0 4.60

Ib.O 72.5 4.50

17.5 70.0 4.40

17.0 67.5 4.30

16.5 65.0 4.20

16.0 62.5 4.10

15.5 60.0 4.00

15.0 57.5 3.90

14.5 55.0 3.80

14.0 52.5 3.70

13.5 50.0 3.60

13.0 47.5 3.50

12.5 45.0 3.40

12.0 42.5 3.30

11.5 40.0 3.20

11.0 37.5 3.10

10.5 35.0 3.00

10.0 32.5 2.90

9.5 30.0 2.80

9.0 27.5 2.70

8.5 25.0 2.60
Purpose of the Book

Traditional microeconomic theory"*" and engineering

economics have long offered useful instruction in the

general principles of economic evaluation that could be

applied to building. However, there has been little material

available to satisfy the specific demands for economic

guidelines that present circumstances demand. The

purpose of this brief work is to help meet this urgent

demand in a simple, comprehensible fashion.

This book is designed as an adaptable instrument for the

"design community": architects, engineers, builders,

building owners and managers, codes and standards

writers, government policy makers, and students of these

disciplines.

We recognize that it is probably impossible to provide, in a

single brief volume, a guide that will address adequately

the range of economic problems faced by each of these

members of the design community. Decisions that affect

the energy consumption of a building begin with the earliest

conceptualization of its basic form and continue throughout

the design, financing, construction, and operational phases

of the building. The nature of the decisions and the

environment in which they must be made vary widely. For

example, an architect grappling with the basic configuration

of a building on its site, a mechanical engineer selecting an

energy control system, and a builder making envelope modifications in a retrofit context must each deal with a different

set of problems.

We have adopted a modest goal: to provide a guide to basic economic concepts and tools for solving simple

economic problems in energy conservation and for understanding complex problems more clearly. You will find here

economic principles and step-by-step examples to aid you

in determining the economic efficiency of specific energy in- vestments.

The focus is on economic analysis and choice. The

emphasis is on practical method rather than theoretical dis- cussion. Simplified formulas are presented and their applications illustrated. Mathematical derivations and complicated formulas are left to the traditional textbooks in economic

theory and engineering economics which you will find referenced in the footnotes. A knowledge of simple mathematics is all that is required to follow the discussion and

solve the problems.

"•"Economic terms are defined in appendix A. 4
How the Book is Arranged

The concepts of economic efficiency are discussed first, followed by a discussion of the basic principles of

economics required to comprehend the economic evaluations of alternative investment decisions. Step-by-step pro- cedures for evaluating the economic desirability of alternative investments are presented to illustrate and reinforce an

understanding of the principles.

Five tools of economic analysis are then introduced to equip the reader with the ability to measure economic

efficiency in a variety of ways:

1. Life-Cycle Costs (LCC)

2. Net Benefits or Savings (B-C)

3. Savings-to-lnvestment Ratio (SIR)

4. Internal Rate of Return (IRR)

5. Discounted Payback (DPB)

The advantages and disadvantages of each of these

economic tools are discussed and guidelines are given for selecting the appropriate one for dealing with specific types

of investment problems.

Discounting, the technique for assessing the time value of

money, is then explained and illustrated in a problem

analyzing the economics of heat pumps. Following this dis- cussion, the five tools are individually applied with the

discounting procedures to solve a problem in solar energy.

A discussion of the general economic factors that affect

benefits and costs is presented to aid the reader in selecting the appropriate economic tools to apply to

different kinds of economic problems. This is followed by a

brief summary.

A series of appendices, including a glossary of economic

terms, discount formulas, and tables of compound interest

factors, conclude the book. The tables are sufficiently

comprehensive to allow problem solving with the economic

tools presented here.
CONCEPTS OF ECONOMIC EFFICIENCY

The Objective

Economic efficiency is not the same as engineering

efficiency. For example, one furnace may be more

"efficient" than another in the sense that it delivers more

units of heat for a given quantity of fuel. Yet, it may not be

economically efficient if the first cost of the higher-output

furnace outweighs its savings in reduced fuel consumption.

To achieve economic efficiency in energy conservation, it is necessary to determine the most profitable levels of energy

conservation. To determine this optimal level of conservation requires the identification of the most advantageous

tradeoff between conserving and supplying energy.

In this section the fundamental principles of economic

analysis used to make economically efficient investments in energy conservation are described and illustrated graphically.

Economic analysis is a tool which can be used to determine the tradeoffs necessary (1) to find how much to spend on energy conservation to lower life-time building

costs, including investment costs, energy costs, and other

recurring and nonrecurring costs; (2) to find the largest

possible savings in energy costs for a given conservation

budget; or (3) to achieve a targeted reduction in energy

costs for the lowest expenditure in conservation.

The second and third applications of economic analysis

obtaining the largest savings in energy costs for a fixed conservation budget and attaining a targeted savings in energy

costs for the lowest conservation budget—are more limited

in terms of achieving economic efficiency than the first application, which seeks to minimize total building costs or maximize the net benefits from energy conservation.

In the first application, designers or builders may be asked

by their clients to include those energy conservation

features that will "pay off" in terms of lower life-cycle

building costs. In the second application, a building owner

may budget a specific sum of money for the purpose of retrofitting a building for energy conservation. In the third application, a designer may be required by State or Federal

building standards to reduce the design energy loads of a new building to a specified level.

BAIT PRO PISCIS
Figure 2a

$

LEVEL OF ENERGY CONSERVATION THAT

MINIMIZES TOTAL ENERGY RELATED COSTS

' CONSERVATION AND

CONSUMPTION COSTS

CONSERVATION COSTS

CONSUMPTION

COSTS

c LEVEL OF ENERGY CONSERVATION

CONSERVATION COSTS

CONSERVATION

BENEFITS (ENERGY

SAVINGS)

Figure 2c

$

'c LEVEL OF ENERGY CONSERVATION

LEVEL OF ENERGY CONSERVATION THAT

WILL YIELD THE LARGEST NET BENEFITS

MARGINAL COSTS

OF CONSERVATION

MARGINAL BENEFITS OF CONSERVATION

Qc LEVEL OF ENERGY CONSERVATION

Graphical Illustrations

In figures 2a, 2b, and 2c the physical quantity of inputs

used to conserve energy is measured on the horizontal

axis and dollar costs are measured on the vertical axis. In 2a the upward sloping line from left to right indicates the

rising total dollar costs of conservation as the physical

quantity of inputs to conserve energy is increased, and the

downward sloping line from left to right indicates the

declining total cost of consumption as conservation is increased. Initially the rise in conservation costs is shown

to be more than offset by the fall in energy consumption

costs, but eventually, as more conservation is undertaken,

the rise in conservation costs can be seen to become

greater than the fall in energy consumption costs. This is reflected in the combined cost curve (the upper U-shaped

curve), which falls to a minimum point and then rises. The

most economically efficient level of energy conservation is that for which the combined cost curve is at a minimum, as

indicated by "Qc."

Another way of describing this concept is in terms of

maximizing the net benefits from energy conservation as shown in 2b and 2c. Using this approach, the reductions in energy costs are the benefits, and the objective is to find

the level of inputs to conserve energy for which the

difference between the costs and the benefits of conservation is greatest.

Figure 2b indicates that the total costs of conservation tend

to rise slowly at first but then begin to rise sharply as more

and more inputs of conservation are acquired and

difficulties begin to be experienced in carrying out the conservation. The result is that the total cost curve typically

bends upward. Total energy savings (benefits), on the other

hand, tend to rise at a decreasing rate as more and more

inputs of conservation are added to a building. As long as

the benefits curve lies above the costs curve, the energy

conservation is profitable. The conservation level at which

the curves are most distant, with the benefits above costs,

is most profitable. The level at which the curves intersect

indicates a breakeven investment, i.e., benefits are fully

offset by costs. The cost curve rising above the benefits curve indicates that the energy conservation investment

loses money.

7
ECONOMIC RETURN ON INVESTMENT

Figure 2c shows how "marginal analysis" can be used to

find the level of conservation which will yield the largest net

benefits. Figure 2c depicts the changes in the total benefits

and costs curves of 2b as the inputs of energy conservation are increased. The level of energy conservation where

these marginal costs and benefits curves intersect is the

most profitable level of energy conservation as indicated in figure 2b. This is the level at which the costs of adding one more unit of conservation are just equal to the corresponding additional benefits in terms of energy savings; that is, the level at which "marginal costs" and "marginal benefits" are equal.

For lower levels of conservation, the additional benefits

from increasing conservation by another unit are greater

than the additional costs, and it pays to invest more. For

higher levels of conservation, the costs of additional conservation exceed its benefits and the level of total net benefits

begins to fall. The most economically efficient level of conservation is indicated in each of figures 2a, 2b, and 2c as

"Qc-"

Because savings and costs of alternative conservation

techniques tend to differ, it will usually be necessary to make tradeoffs among techniques, investing more in some

and less in others. For example, if another dollar invested

in insulation offers a larger return than a dollar invested in solar energy, it will pay to increase the amount of insulation

before investing in solar. Because the energy savings from

most conservation techniques will tend to decrease as the

investment in the technique is increased, it will usually pay

at some point to shift further investment to some other

technique.

The most profitable level of each technique in combination occurs when an additional dollar spent on one of the

techniques will yield the same dollar return as that spent on each of the other techniques. In economic terms, this is the

combination of techniques where the ratio of marginal

savings to marginal costs is equal for all of the techniques.

If the ratio is higher for one technique than another, it will pay to shift resources from the technique with the lower

ratio to the one with the higher ratio in order to increase

total net savings. Figures 2a, 2b, and 2c are constructed

under the assumption that the most profitable combination

of techniques is being considered for each level of energy

conservation.

UNECONOMIC RETURN ON INVESTMENT

8
MEASURING BENEFITS AND COSTS

Kinds of Benefits and Costs

To determine the economic attractiveness of an investment, it is necessary to measure the benefits and costs as- sociated with it. In broad terms, benefits from a conservation investment include both the monetary value of the

resulting energy savings and the nonmonetary value of

other beneficial effects of the investment. "Nonmonetary

benefit" is defined as a benefit to which it is difficult to assign a dollar value. It does not mean that the benefit has

no value.

There may be monetary and nonmonetary benefits that are enjoyed directly by the person or organization who invests

in energy conservation as well as monetary and nonmonetary benefits that extend beyond the investor. An example

of a monetary benefit that accrues directly to the investor in conservation is the dollar savings in fuel costs. An example

of a more difficult to quantify benefit to the investor is the

improvement in occupant comfort that may result when conservation tightens the building envelope and reduces

drafts. Other possible examples of nonmonetary benefits to the investor from conservation are the achievement of a degree of energy independence, the prestige that may

accompany the use of a new technology such as solar

energy, and the respect that may be accorded private

actions taken in the public interest.

An example of a monetary benefit that may accrue beyond

the investor, in this case to the nation as a whole, is the improvement in the U.S. balance of payments from reduced

imports of oil. Societal benefits that are more difficult to quantify are the impacts of conservation on U.S. strategic

vulnerability and on environmental air quality.

There may also be monetary and nonmonetary benefits

that extend beyond the investor but not to the entire nation. For example, the planting of trees adjacent to a building for landscaping purposes may improve the environmental

quality of the community, in addition to reducing the

investor's air conditioning costs by shading the structure. Public recognition of "spillover" effects of private investor

decisions may be reflected in political and legal requirements, such as a requirement for planting around a building as a condition for approval of building permits.

Although most costs of conservation tend to be

measurable in dollars and incurred directly by the investor,

there may also be costs that are nonmonetary and

extending beyond the investor. For example, some conservation investments might be felt to have a negative impact on building aesthetics that will adversely affect the owner,

the occupants, and the community.

While both monetary and nonmonetary benefits and costs

are important, whether accruing only to the investor or to

society, the focus of this report is on monetary benefits and

costs to the investor. Monetary values are emphasized

simply because the state of the art of measuring benefits

and costs makes it difficult to assign values to nonmonetary benefits.^ The focus on monetary values, however, is not to suggest that nonmonetary benefits should be given

any less consideration in decisionmaking. Quantifying as many benefits and costs as possible in dollar terms

reduces the guesswork and provides a single measure

against which the more subjective elements may be

compared.

The focus on benefits and costs to the investor, rather than

to the nation, reflects the assumption that most investors in the private sector make decisions primarily on the basis of

direct monetary benefits to them, although they are often

influenced by community and national concerns. As a

result, the five economic evaluation methods, presented

here as tools to aid the design community in determining

economically efficient investments in energy conservation,

describe dollar costs and benefits solely to the investor.

9
Five Economic Toois

The five economic toois described here are Life-Cycle

Costs (LCC), Net Benefits or Savings (B-C), Savings-toInvestment Ratio (SIR), Internal Rate of Return (IRR) and

Discounted Payback (DPB)."

The first four are comprehensive analytical tools that can be used to evaluate investments in energy conservation.

They consider both first costs and future costs and

savings. Because they all look at the significant costs and

benefits over the life of an investment, they are often

referred to collectively as life-cycle techniques. The fifth

tool, the discounted payback method (DPB), does not fully use the life-cycle approach. It nevertheless may be quite

useful to designers under certain circumstances, as, for example, when the client requires rapid recovery of investment funds or when the durability of investment assets is highly uncertain. Each of the five tools considers the timing

of cash flows and associated cost of money.

Life-Cycle Costs (LCC): Life-cycle costing sums the

energy costs of the building together with the net costs of purchase and installation (less any salvage value), maintenance, repair, replacement, and all other costs attributed to the conservation investment. This includes the cost of money over the life of the investment. The investment that has the lowest total life-cycle cost while meeting the

investor's objective and constraints is the preferred investment.

All cash amounts are generally converted to either present

value or annual value dollars. Present Value is defined as

the equivalent value of past and future dollars corresponding to today's values. Annual Value means that all past,

present, and future costs are converted to an equivalent

constant amount recurring annually over the evaluation

period. The conversion process for both present value and

annual value dollars is called discounting.

Following is a general formula for finding the total life-cycle

costs of an energy conservation investment:

/

or

cr ^ S' / # / ^

LCC = P- S + M-|-R-|-E,

where all costs are in life-cycle present value or annual

value dollars and adjusted for taxes and incentives. An

example using this tool appears on page 22.

10
Net Benefits or Savings (B-C): This tool finds the

difference between the life-time dollar energy savings and

life-time dollar costs of a conservation investment. Net

benefits or savings may be expressed in either present

value or annual value dollars. This tool applies to the same

types of investments as the life-cycle cost (LCC) tool, but is formulated somewhat differently as is shown below:+

Savings-to-lnvestment Ratio (SIR): Like the two

preceding tools, the SIR is based on discounted cash

flows. However, savings and investment costs are expressed as a ratio rather than a dollar amount. For

positive net savings, the ratio must be greater than one. The higher the ratio, the more dollar savings realized per

dollar of investment.

/

/

5-

,$>

<3" <3'

A.

/

(B-C) = E* - (P* - S* -t- M* -h R*),

where all costs and benefits are in present value or annual

value dollars and adjusted for taxes and incentives. An

example using this tool appears on page 23.

+The values of "E," "P," "S," "M," and "R" in this and subsequent equations, where accompanied by an asterisk (*), represent

the difference between the present value or annual value costs for an energy conserving investment and its alternative. While the

previous LCC formula must be applied to each of two investments

being compared, the (B-C) formula is applied directly to the

difference between two alternative investments. The LCC formula

corresponds to figure 2a, while the (B-C) formula corresponds to

figure 2b.

Following is a general formula for computing the savingsto-investment ratio:+

i

<<7

•5-

SIR = (E* - M*) ^ (P* - S* -I- R*),

where all costs are in present value or annual value dollars

and adjusted for taxes and incentives. An example using

this tool is given on page 24.

+The ratio is sensitive to whether cost elements are subtracted

from savings in the numerator or added to costs in the denominator. For ratios greater than one, adding costs to the denominator

will reduce the ratio more than subtracting the same costs from

the numerator. Placement of salvage values, maintenance costs, and replacement costs in the numerator or denominator of the

ratio sometimes varies in application, depending on the costs for which the investor is trying to otain the highest return. The

formulation shown above, however, is widely used.

11
Internal Rate of Return (IRR): This tool finds the rate of return on an investment. This is the interest rate, stated as

a percent, for which life-time dollar savings are just equal to

life-time dollar costs. The calculated IRR is compared to the investor's minimum acceptable rate of return to determine if the investment is desirable.

Discounted Payback (DPB): This tool measures the

elapsed time between the point of initial investment and

the point at which accumulated savings, net of other accu- mulated costs, are sufficient to offset the initial investment

cost. Costs and savings are adjusted to account for the

changing value of money over time. If a time adjustment is omitted, the tool is termed "simple payback."

The IRR is generally calculated by a structured process of

trial and error."*" Selected compound rates of interest are used to discount the cash flows until a rate is found for which the net value of the investment is zero or close to zero.

Following is a general formula for the internal rate of return:

such

S ^ /

,^ / #

<3^ <3^

that (E • a) - [ P-(S • b) -I- (M • c) -I- (R • b)] =0

The bar over the symbols, e.g., "E," indicates that the cost

differences have not yet been converted to present or annual values. The terms a, b, and c refer to discounting

factors that are explained on pages 16—20 and listed in appendices C and D. A sample problem is solved by this tool on page 25.

+IRR programs available for some programmable desk calculators may be helpful in solving for IRR. 12

For investors who seek a rapid turnover of investment

funds, the investment increases in desirability as the

payback period decreases. However, a shorter payback

time does not necessarily indicate the most economically

efficient investment. An investment with a longer payback

period may prove more profitable than an investment with a

shorter payback period if it continues to yield savings for a

longer period of time.

The following formula shows how to find the payback

period, y:

i

/ / ^

ss $ <J ^

/ C^' / *° /

i

/

Y I<c O

Find Y such that I (E*j - M*, - R* -h S*,) = P*

i=i

where all costs are in present value or annual value dollars and adjusted for taxes and incentives. A sample problem is solved by this tool on page 26.
Advantages, Disadvantages, Recommended Applications

COMMON ECONOMIC QUESTIONS AND RECOMMENDED TOOLS FOR SOLUTION

LCC B-C SIR IRR DPB

CAN SAVINGS BE COMPARED TO COSTS? 0 0 0 0

LARGE AN INVESTMENT TO MAKE? • •

TO FIND THE LEVEL OF MAXIMIZED DOLLAR BENEFITS? •

MUCH OVERALL COSTS WILL BE LOWERED BY INCREASED CONSERVATION? •

CAN PROJECTS DIRECTLY COMPETING FOR THE SAME PURPOSE

(e.g., R-10 INSULATION VERSUS R-19 INSULATION) BE COMPARED? •

CAN DIFFERENT PURPOSE PROJECTS COMPETING FOR THE SAME BUDGET

(e.g., INCREASING THE THERMAL MASS OF THE BUILDING WALLS VERSUS THE

USE OF EXTERIOR OVERHANGS TO SHADE THE WINDOWS) BE COMPARED?

• •

TO FIND THE RATE OF RETURN ON INVESTMENT? •

SOON WILL ENERGY INVESTMENTS BE PAID OFF BY SAVINGS? •

HOW

HOW

HOW

HOW

HOW

HOW

HOW

HOW

• HOW can savings be compared to costs?

The five tools described are not equally appropriate for evaluating all types of energy conservation investment

decisions. However, if a simple "accept-reject" investment

decision is all that is desired, then any of the tools

described can be used as described below:

LCC The total life-cycle costs must be lower with the in- vestment than without it.

B-C The net dollar benefits must be positive.

SIR The ratio of dollar benefits to dollar costs must be

greater than one.

IRR The internal rate of return must be greater than that

minimally acceptable to the investor.

DPB The payback period must be shorter than the

expected life of the investment and must meet the

investor's timing requirements for recovery of investment funds.

However, to maximize net savings from conservation investments, choices among competing investments should be

based on more sophisticated evaluation criteria than those

just stated, and the choice of tool becomes important.

Each tool can be used to attain specific objectives and

each has unique advantages, disadvantages, and recom- mended applications. • HOW large an investment to make? • HOW to find the level of maximized dollar benefits? • HOW much overall costs will be lowered by increased

conservation? • HOW can projects directly competing for the same

purpose (e.g., R-10 insulation versus R-19 insulation) be

compared?
LCC and B-C are the tools recommended to find the economically efficient size of a conservation investment.

Although the SIR and IRR techniqes may also be used to determine the most advantageous size of a project if applied correctly to increments in the investment, LCC and

B-C are less apt to be misapplied.

If the life-cycle costs fall with added investment, it is profitable to increase the investment. Or if net savings increase,

it is profitable to increase the investment.

By measuring the change in costs and the amount of

savings from energy conservation investments, the LCC

and B-C tools also provide an answer to the fourth

question, "How much will overall costs be lowered?"

Additionally, either tool will provide an answer to the fifth question, "How can directly competing projects for the

same purpose be compared?" This question is answered

by determining the project that minimizes life-cycle costs or maximizes net savings. If, for example, total costs are lower or net savings are higher with R-19 insulation in the

attic than with R-10 insulation, it pays to use R-19 insulation, other things equal. These investments are often

termed "mutually exclusive" because undertaking one

generally precludes undertaking the other.

Despite the fact that the LCC and B-C tools are effective in addressing the above five questions, they are not always

effective for solving other kinds of investment problems,

such as comparing projects that have different purposes

but that compete for a limited budget.

Although the life-cycle measures of LCC and B-C indicate

whether total building costs are higher or lower with or without the conservation investment, they do not

distinguish between large and small investments that result

in the same net dollar energy savings. That is, they do not

provide an indication of the return on the investment dollar.

+Because the SIR and IRR measures may begin to fall before the

economically efficient size of an investment is reached, they can be used to size an investment efficiently only if used to measure

the economic efficiency of each increment of the investment rather than of the total. The LCC and B-C measures, on the other hand, can be applied to the total investments for purpose of sizing.

LCC and B-C both have essentially the same advantages

and disadvantages. The two are generally interchangeable. • HOW can different purpose projects competing for the

same budget (e.g., increasing the thermal mass of the

building walls versus the use of exterior overhangs to shade the windows) be compared?

SIR or IRR are recommended for determining the ranking

to be given to competing investments which are not direct

substitutes for one another, where it is assumed that the

optimal size of each project has already been determined.

These investments are often termed "nonmutually

exclusive" because, aside from budget limitations, undertaking one investment does not necessarily preclude also un- dertaking the other. SIR and IRR are recommended

because both reflect the return on investment dollar and can be used to rank investment projects to determine the

combination that will result in the largest total return for a given budget.

LCC and B-C are not recommended for ranking competing

nonmutually exclusive investments because investments

selected on the basis of their LCC or B-C ranking may not

yield the highest total net benefits for a limited budget. For

example, suppose the investor has $12,000 to be spent on an energy conservation retrofit package and the following

four projects have been identified as possible candidates:

(1) replacement of all windows at a cost of $2,000 and an expected life-cycle savings of $6,000,

(2) a new energy control system with a cost of $12,000

and an expected life-cycle savings of $20,000,

(3) adding thermal insulation to the walls at a cost of $4,000 and an expected savings of $9,000, and

+For maximum economic efficiency, the sizing of individual projects and the allocation of limited funds among competing

projects cannot be independent decisions. A condition for achieving the economically efficient combination of projects is that

to the extent possible the last dollar spent on each project yields the same benefits as that spent on all the other competing

projects. However, in practice, potential projects are often compared based on predetermined sizes for each.
(4) an improved maintenance routine that is expected to cost $6,000 and save $12,000.

The following table shows the comparative ranking of the

four projects, first according to the B-C tool, and second

according to the SIR tool (the LCC rankings would be the

same as the B-C, and IRR rankings would be the same as

the SIR).

Project

Investment Investment Expected Net- B-C SIR

Alternatives Cost Savings Benefits Ranking SIR Ranking

(S) ($) ($)

(1)

1 2,000 6,000 4,000 (4) 3 2 12,000 20,000 8,000 (1) 1 2/3 (4)

3 4,000 9,000 5,000 (3) 2 1/4 (2)

4 6,000 12,000 6,000 (2) 2 (3)

Ranking by the B-C tool would suggest the selection of

project 2 for a cost of $12,000 and a net savings of $8,000,

while ranking by the SIR tool would indicate the selection

of projects 1 , 3, and 4 for a total cost also of $12,000 and a

total net savings of $15,000. Selecting the projects

according to their SIR (or IRR) results in the realization of an extra $7,000 in net savings from the fixed budget of

$1 2,000 over the net savings from the projects selected by

their B-C ranking.

Of course for a simple example like this one, the combination of projects with the highest total net benefits can be

found simply by trying different combinations of projects

costing $12,000 and adding their net benefits. When there

are many projects under consideration, however, the use of

the SIR or IRR ranking to guide project selection can be an

easier, more direct approach.

• HOW to find the rate of return on investment?

The IRR tool is the only one of the five tools that measures

economic efficiency in terms of the rate of return on investment. It is also unique among the tools discussed in that it does not require the specification of a discount rate. However, it does require an estimation of the minimum

acceptable rate of return against which the calculated rate

of return is compared to determine the desirability of an in- vestment. And, estimating the minimum acceptable rate of

return is similar to estimating the discount rate. This tool,

like the SIR, has the advantage of indicating the relative

economic efficiencies of alternative investments and can be used to rank competing projects in descending order of

their IRR's.

The IRR has several disadvantages. It is subject to misinterpretation in sizing projects because as an investment is expanded, the rate of overall return may fall while the rate on additional investment may continue to exceed the

investor's minimum attractive rate of return. It may prove

cumbersome to calculate and can, under certain

conditions, result in indeterminant or multiple solutions. • HOW soon will energy investments be paid off by

savings?

Because it provides a measure of the time period

necessary to recover funds, the DPB tool is recommended

when a fast turnaround on investment funds is required.

This is often a critical factor to speculative investors. DPB

is also recommended when the principal assets have highly

uncertain life expectancies and the economic viability of

the investment hinges on a minimum life.

The feature of rapid recovery of funds, however, may be

overemphasized with the result that less efficient, shortterm ventures are favored over more efficient, long-term in- vestments. The principal disadvantage of DPB is that, even based on discounted benefits and costs, it does not

provide a full measure of an investment's profitability

because it does not include benefits and costs that occur

after the payback date is reached. This problem can be

averted by supplementing DPB with one of the four comprehensive life-cycle evaluation tools.

15
The Method

Energy conservation costs often consist primarily of first costs, i.e., costs incurred during the time of initial design

and construction or retrofitting. The benefits, on the other

hand, typically accrue over the life span of the project in the form of yearly energy savings. To compare benefits and

costs that accrue at different points in time, it is necessary

to put all cash flows on a time equivalent basis. The

method for accomplishing this is called discounting.

The value of money is time dependent for two reasons:

first, inflation erodes the buying power of the dollar, and

second, money can be invested over time to yield a return over and above inflation. While it is possible that deflation

might also occur, inflation is stressed because it expresses

the more common condition in recent times. For these

reasons, a given dollar amount today will be worth more

than that same dollar amount in a year's time.

For example, if there were no inflation and investors could

at best earn 10% interest per annum in a risk free savings

account, they would find a given dollar amount this year

equivalent in value to that amount plus 10% a year hence.

They could be expected to be indifferent between $100

now and $110 a year from now unless they had better in- vestment opportunities available. A 10% rate of interest

would indicate the investor's time preference for money. If there were additionally a 5% rate of inflation, investors

would require $115 a year from now in order to be

indifferent between that future amount and $100 today. The

DIS COUNTING

higher the time preference, the stronger the desire for money now rather than in the future and the higher the rate of interest required to increase future cash flows sufficiently to make them equal to a given value today. The rate of interest at which an investor feels adequately compensated

for trading money now for money in the future is the appropriate rate to use for converting present sums to future equivalent sums and future sums to present equivalent

sums, i.e., the rate for discounting cash flows for that

particular investor. This rate is often called the discount

rate.

For a simplified treatment of the basic procedures of cash

flow analysis, all costs incurred during the planning, design,

and construction phases are here termed "first costs" and

treated as though they are all incurred at the beginning of the investor's time horizon. In practice, of course, the

budgeting, time scheduling, and financing of costs during

planning, design and construction can be extremely

complex and crucial to a project's success.^

To evaluate the economic efficiency of an energy conservation investment correctly, it is necessary to estimate the

values of the various expenditures and savings that accrue over time and convert them all to values in a common base

year. Usually, all past and future values are converted to equivalent present values, or all past, present, and future

values are converted to equivalent annual values.

16
Sample Discounting Problem

The remainder of this section illustrates how to derive an in- vestment's total life-cycle costs by putting all cash flows on

a time equivalent basis, i.e., by discounting. The

discounting procedure is illustrated in a sample problem of

purchasing, installing, maintaining, and operating a heat

pump. This type of cost analysis would be required, for example, if the life-cycle costs of a heat pump were to be

compared to those of an alternative heating/cooling

system to determine the most cost-effective system.

The life-cycle costs in the sample problem are shown for the heat pump alone, and not for alternative

heating/cooling systems. Although the example is intended

only to clarify the discounting process, and not to compare

alternative heating/cooling systems, realistic estimates are used for the heat pump costs.

The life-cycle cost calculations are shown alternatively for two reference times. The first is the present, called

"present value." The second is a yearly time scale, called

"annual value," whereby all costs are expressed as though

they occur in uniform yearly amounts over the study period.

These two reference points are the most common in economic evaluations of investments.

The future is a third reference point in time that can be

used in discounting. Appendices B and C, respectively,

show discount formulas and factors for discounting cash

flows to present, annual, and future values.

The life-cycle costs of an investment of more than 1 year's

duration will necessarily be lower in absolute dollars when

expressed as an annual value than when expressed as a

present value; nevertheless, they are equivalent values in time and will both give the same relative ranking of conservation investment priorities. This equivalence is demonstrated in the following heat pump illustration.

Problem Assumptions: A residential heat pump, not

including the duct system, costs $1,500 to purchase and

install. The heat pump has a useful life of 15 years and

incurs annual maintenance costs of $50 a year over its useful life. A compressor replacement is required in the

eighth year at a cost of $400.

The yearly electricity cost for using the heat pump is $425

based on the price of electricity at the beginning of the

investor's time horizon. Electricity prices are projected to escalate at a rate of 7% compounded annually.

The discount rate is 10% including inflation. No salvage

value is expected from the heat pump at the end of 15

years.

It should be noted that to focus on the discounting procedures, we accept these assumptions as given. In practice,

there may be uncertainty as to what assumptions

reasonably describe an investment. The problem of

uncertainty is discussed later in this handbook.
Problem Solution: The total costs of the heat pump

system include costs of purchase, installation, maintenance, replacement, and electricity for operation. Using the

present as the base time reference point, we must convert

the above listed costs to their present values before adding

them. If we assume that the purchase and installation cost

is incurred at the base reference point, the present, the

$1,500 is already in present value terms.

Figure 3 diagrams the conversion of the other cash flows

to present values. The first task is to convert the stream of

yearly maintenance costs to its present value. The mainte- nance costs, as shown in the cash flow diagram of figure 3, are $50 per year, measured in dollars of the years in which

they occur."*"

We follow the practice here of compounding interest at the

end of each year, and the costs and benefits in the future are always considered to occur at the end of the year in which they occur. Alternatively, cash flows can be assumed

to occur at the beginning or the middle of each year, as

well as continuously throughout the year. The present

refers to the beginning of year one.

The discounting operation for calculating the present value

of maintenance costs is to multiply the yearly maintenance

costs times the uniform present worth (UPW) factor. The

UPW is a multiplicative factor taken from the Table of

Discount Factors in appendix C, table C-6.

"•""'"

Discount factors allow one to calculate present values for both uniformly recurring values and one-time future values, as well as recurring values that increase over time. In the case of maintenance costs, the costs are uniform and

recurring and, therefore, the UPW factor is the appropriate

discount factor to use.

"•"For purpose of this illustration, maintenance costs are assumed

to remain the same in current dollars as might be the case if they were fixed by a long-term contractual agreement.

Discount formulas, described in appendix B, can also be used

to obtain present or annual values. Multiplicative discount factors,

described in appendix C, are easier to use, however, and we

therefore emphasize their use in this paper.

Table C-6 is searched for the column that provides the

discount factors for a 10% discount rate. We find a factor

of 7.606 for 15 years (N = 15) in the 10% column.

Multiplying this factor times $50 gives a present value of

maintenance equal to $380. Note that the $380 present

value of maintenance costs is much less than the sum of

$50 for 15 years (i.e., $750). This illustrates the importance

of discounting to compensate for time differences of cash

flows.

The second task is to convert the one-time future cost of

the compressor replacement, $400, to its present value.

The operation for calculating the present value of compressor replacement is to multiply the future value of the compressor replacement times the single present worth factor

(SPW). Finding the SPW table (C-2) in appendix C, we

move down the first column to N = 8 and across to the

column with 10% discount factors to find a value of .467. Multiplying this factor times $400 gives a present value

cost of the compressor replacement of $187, as shown in

figure 3. Again note that discounting makes a significant

difference in the measure of costs. Failing to discount the

$400 would result in an overestimate of cost of $213

($400-$187).

The third task is to convert the yearly electricity cost of

heating and cooling to its present value. The electricity

costs for a year, evaluated at the base year or reference

point, is shown in figure 3 to be $425. A price escalation

rate for electricity of 7% per annum is assumed. This is reflected yearly in the cash flow diagram in terms above

the line. For example, $425 (1.07)^ is the predicted cost of

energy in the fifth year after energy inflation has been

18
Figure 3

DETERMINATION OF PRESENT VALUE, LIFE-CYCLE COSTS OF A HEAT PUMP FOR HEATING AND

COOLING

TASK DESCRIPTION ^ CASH FLOW DIAGRAM DISCOUNTING OPERATION^

LFIND

P, GIVEN A

Find the present value (P„) of the $50 annual maintenance costs (A„) over 1 5 years.

$50 $50 $50

YEAR 1

$50 $50 $50

5 15 ^

TIME

INYEARS

P„ = (AJ (UPW)

P„ = ($50) (UPW, 10%, 15 yrs.)

P„ = ($50) (7.606) = $380.

LFIND

P, GIVEN F

Find the present value (P,) of the $400 future

cost of replacing compressor (FJ at the end

of 8 years

I I YEAR 1

$400 TIME

INYEARS

P, = (FJ (SPW)

Pc = (FJ (SPW. 10%, 8 yrs.)

P, = ($400) (.467) = $187.

LFIND

P, GIVEN A WITH ESCALATION

Find the present value (P^) of the annual electhcity

costs (AJ over 15 years, beginning with an annual

cost of $425, and with electricity cost escalation

of 7%.

$425

X

1.07'

$425

X

1.07'

$425

\

1.07^

YEAR 1

$425X

1.07"

$425

1.07'

$425

.07'= TIME

INYEARS

P. = (AJ (UPW)

P, = ($425) (UPW*, 10%, 15 yrs., 7% escalation)

P, = ($425) (12.109) = $5,146.

FIND THE TOTAL PRESENT VALUE OF THE HEAT PUMP

PRESENT VALUE = Installation Cost + P„ + P, + P,

PRESENT VALUE = $1,500 + $380 + $187 + $5,146

PRESENT VALUE = $7,213

a P = present value b UPW = uniform present worth factor Subscript A = annual value SPW = present worth factor nn = maintenance costs

F = future value UPW* = uniform present worth factor c = compressor cost

with energy escalation e = electricity costs
added. Appendix D, table D-2, provides a table of modified

uniform present value factors (UPW*) for a discount rate of

10% and energy escalation rates from 1% to 10%. The

modified uniform present value discount formula that

includes energy price escalation is given in appendix B. The asterisk following UPW (i.e., UPW*) means that a term

for energy price escalation is included.

The discounting operation (shown in fig. 3) for finding the

present value of electricity costs is to multiply the base

year electricity cost times the appropriate UPW* factor in appendix D. Locating year 15 in the first column of

appendix D-2 and looking across the table for the UPW*

factor under the escalation rate of 7%, the value 12.109 is found. Multiplying this factor by $425 gives a present value

of electricity costs of $5,146. Note once again that failing to discount would overestimate costs by $1,229 ($6,375-

$5,146). Discounting with a UPW factor that does not

incorporate energy price escalation would underestimate

costs by $1,913 ($5,146-$3,233).

The final operation described in figure 3 is to sum the

purchase and installation costs and the present values of

maintenance, compressor replacement, and electricity

costs. Total life-cycle costs of the heat pump in present

value terms is found to be $7,213. This is the cost figure

that a designer would need to compare the cost effective- ness of this heat pump with alternative heating/cooling

systems.

Figure 3 provides a model for the designer who must

calculate present values from a number of benefit and cost

streams. It can be seen that many different kinds of

benefits and costs occurring in future years can be handled

either with the SPW, the UPW factor, or the UPW* factor

with price escalation.^

Only one discounting operation is required for converting

the present value costs of the heat pump to annual value ! terms. The total present value amount is multiplied by the

'

uniform capital recovery factor (UCR) for 10% and 15

years. The UCR factor, found in table C-4, is .131. Multiplying this factor by the total present value of $7,213

gives the cost of the heat pump as $945 in annual value terms.

Note that annual values are not the same as average

yearly values. For example the installation cost of $1,500,

divided by 15 years, or $100 per year, will not be the

annual value of installation costs. Because average yearly

values do not include discounting, they give erroneous

estimates of benefits and costs. The two figures, $945 in annual value terms over 15 years and $7,213 in present

value terms, are time equivalent values, made consistent

through the application of discounting.

Using the discounting procedures described above,

together with appendices B, C, and D, the building decision

maker can formulate and solve many conservation investment problems.
A PROBLEM IN FIVE SOLUTIONS

To illustrate the five tools described earlier, a solar energy

system with a conventional backup system is evaluated for cost effectiveness against a conventional energy system

used alone.

Problem Assumptions

For the purpose of this illustration all costs are evaluated

with a discount rate of 10%, a fuel price escalation rate of

5%, and a time horizon of 20 years.

The combined solar/conventional energy system is assumed to cost $20,000 which is $8,000 more than the

conventional system alone.

Major components of the solar/conventional system are

anticipated to last 20 years and be without value after that

time.

Figure 4

The conventional system if used alone is estimated to require a major replacement at the end of the 15th year, calculated to cost $6,000, and to have a salvage value of

$4,000 at the end of 20 years, the end of the time horizon.

Maintenance costs for purposes of this illustration are assumed to be no higher for the combined solar/conventional system than for the conventional system alone.

The combined system is assumed to result in substantial

fuel savings, reducing base year energy costs from $2,000

to $500. A fuel price rise of 5% per year and a discount

rate of 10% are assumed.

Figure 4 shows two cash flow diagrams for this problem,

the upper part for the combined system, the lower part for the conventional system. Upward pointing arrows indicate

expenditures; downward pointing arrows, receipts.

The values for both systems are purely hypothetical and for the purpose of illustration only.

CASH FLOW DIAGRAM FOR SOLAR ENERGY PROBLEM

$20,000 $500(1+ .05)' $500(1 +.05)^ $500(1 +.05)= $500(1 + .05)' $500(1 +.05)' ^ 1 2 3 4 5

$500(1 + .05)"

20 4

Combined Solar/Conventional System

$6,000 +

$12,000 $2,000(1 +.05)' $2,000(1 +.05)' $2,000(1 +.05)' $2,000(1 +.05)" $2,000(1 +.05)' $2,000(1 +.05)" $2,000(1 +.05)"

± 15 20

Conventional System Only

$4,000

21
Table A: Life-Cycle Costing

TYPE OF ENERGY SYSTEM

COMBINED SOLAR/

CONVENTIONAL

CONVENTIONAL

ONLY

1 PERIOD OF ANALYSIS 20 Yrs. 20 Yrs.

2 P INITIAL INVESTMENT COSTS $20,000 $12,000

3 (S)' SALVAGE 0 $4,000

20th Yr.

4 S PRESENT VALUE OF SALVAGE 0 $596

5 (M) MAINTENANCE + REPAIR COSTS 0 0

6 M PRESENT VALUE OF MAINTENANCE + REPAIR 0 0

7 (R) REPLACEMENT COSTS 0 $6,000

15th Yr.

8 R PRESENT VALUE OF REPLACEMENT COSTS 0 $1,434

9 (E) BASE-YEAR ENERGY COSTS $500 $2,000

10 E PRESENT VALUE OF ENERGY COSTS $6,359 $25,436

11 LCC PRESENT VALUE OF TOTAL COSTS $26,359 $38,274

LCC = P- S + M + R + E

For COMBINED SOLAR/CONVENTIONAL LCC = 20,000 - 0 + 0 + 0 + 6,359 = $26,359

For CONVENTIONAL ONLY LCC = 12,000 - 596 + 0 + 1,434 + 25,436 = $38,274

'Present values are used In the LCC formula. Symbols for numbers expressed in other terms are enclosed In parentheses to indicate they cannot be inserted directly into the formula.

LCC Problem Solution: The relevant information from tine problem assumptions are entered in table A in lines 1 , 2, 3,

5, 7, and 9. Lines 4, 8, and 10 are filled in after the

follov\/ing calculations:

Line 4 lists the present value of the salvage value of the

conventional system as $596. This figure was obtained by

multiplying the $4,000 of estimated salvage value in the

20th year times the SPW factor in appendix C-2 for a 10%

discount rate, i.e., [($4,000)(.149) = $596].

Line 8 gives the present value of replacement costs as $1,434 for the conventional system. This figure was

obtained by multiplying the $6,000 in replacement costs

times the single present worth factor (SPW) in appendix

C-2 for 10% and the 15th year, i.e., [($6,000){.239) =

$1 ,434].

Line 10 gives the present value of fuel costs for the

combined system calculated from the base-year cost of $500. With an expected fuel price rise of 5% per year and

a discount rate of 10%, the present value of energy costs

for the combined system over a 20-year time horizon

amounts to $6,359. This figure was derived using the

UPW* factor from appendix D, table D-2, for 20 years at 5% price escalation and a 10% discount rate, i.e.,

[($500)(1 2.71 8) = $6,359].

The present value of fuel costs for the conventional system

used alone, also given in line 10, is $25,436. This figure is calculated by multiplying the base-year fuel cost of $2,000

times the same UPW* factor for 20 years and 5% price escalation that was used to calculate present value fuel costs

for the combined system, i.e., [($2,000)(1 2.71 8) = $25,436].

22
Line 1 1 gives the total life-cycle cost for the solar/conventional system as $26,359. This is the sum of lines 2, 6, 8, and 10 and the subtraction of line 4. The total life-cycle cost for the conventional system is $38,274, obtained in the same way.

A comparison of the total costs of the two systems over

the 20-year time horizon indicates that the cost of the

combined solar/conventional system is $11,915 less than

the costs of the conventional system alone and is therefore

the preferred investment.

Table B: Net Benefits (Savings)

1 PERIOD OF ANALYSIS 20 Yrs.

2 E BASE-YEAR SOLAR ENERGY SAVINGS $1,500

3 E* PRESENT VALUE OF ENERGY SAVINGS $19,077

4 P* DIFFERENTIAL SOLAR INVESTMENT COSTS $8,000

5 S* PRESENT VALUE OF DIFFERENTIAL SALVAGE VALUE -$596

6 M* PRESENT VALUE OF DIFFERENTIAL MAINTENANCE COSTS 0

7 R* PRESENT VALUE OF DIFFERENTIAL REPLACEMENT COSTS -$1,434

8 B-C* NET PRESENT VALUE SAVINGS $11,915

(B-C*) = E* - (P* - 8* + M* + R*) =

$19.077 -($8,000 -(-$596) + 0 + (-$1,434)] = $11,915

'(Indicates the differences between tfie present value costs for the connbined solar/conventional

system and for the conventional system alone) E (Indicates the differences between costs prior to discounting)

B-C Problem Solution: The above net-benefits solution is based on the same costs and assumptions used in the

preceding example. However, most of the values in table B

differ from those in table A because they are shown for differences in the two heating systems. Using differences

requires applying the formula only once for comparing the

economic efficiency of the two heating systems.

The combined solar/conventional system is assumed to cost $8,000 more than the conventional system (Line 4), to save $1,500 in energy costs in the base year (Line 2), and

to last 20 years without a major replacement cost.

Line 3 gives the present value of energy savings over the

20-year period of analysis as $19,077. This figure was

derived using the UPW* factor for 20 years at 10%

discount and a yearly 5% escalation rate, i.e., [(12.718)($1,500) = $19,077].

Line 7 shows that the combined system requires $1,434

less in replacement costs in present value terms than the

conventional system used alone.

Line 5 shows that the combined system results in less

salvage value than the conventional system alone due to the replacement for the conventional system late in the

study period which has remaining life at the end of the

study period. The net effect of the lower replacement cost

and lower salvage value is a cost advantage for the

combined solar/conventional system.

The total net savings are found by subtracting the differential costs of the system from the savings. Table B shows

the present value of net savings to be $11,915, the same

results obtained by finding the difference in the present

value of total life-cycle costs of the two alternatives in line

1 1 of table A. 23
Table C: Savings-to-lnvestment Ratio

1 E BASE-YEAR SOLAR ENERGY SAVINGS $1,500

2 E* PRESENT VALUE OF ENERGY SAVINGS $19,077

3 P* DIFFERENTIAL SOLAR INVESTMENT COSTS $8,000

4 S* PRESENT VALUE OF DIFFERENTIAL SALVAGE VALUE -$596

5 M* PRESENT VALUE OF DIFFERENTIAL MAINTENANCE COSTS 0

6 R* PRESENT VALUE OF DIFFERENTIAL REPLACEMENT COSTS -$1,434

7 E* - M* SIR NUMERATOR $19,077

8 P* - S* + R* SIR DENOMINATOR $7,162

9 SIR RATIO 2.66

SIR = (E* - M-) ~ (P* - S* + R*) =

($19,077 - 0) ^ |$8,000 - (-$596) + (-$1,434)] = 2.66

SIR Problem: This Savings-to-lnvestment ratio solution uses the same hypothetical cost data presented in the

preceding two illustrations.

Line 2 shows the present dollar value of energy savings over the 20-year period of analysis (this was calculated in the previous example) as $19,077. Since there are no differ- ential maintenance costs, the numerator of the SIR is $19,077.

Lines 3, 4, and 6 combine to form the denominator of the

SIR. The lower replacement costs and the lower salvage

value together offset somewhat the higher solar investment

costs, and the denominator of the SIR is $7,162.

The Savings-to-lnvestment Ratio of 2.66 indicates an

average gross return per investment dollar of $2.66.

Table D: Internal Rate of Return

TRIAL INTEREST RATES 20% 25%

1 El* PRESENT VALUE OF ENERGY SAVINGS $9,773 $7,634

2 P* DIFFERENTIAL SOLAR INVESTMENT COSTS $8,000 $8,000

3 SI* PRESENT VALUE OF DIFFERENTIAL SALVAGE VALUE -$104 -$48

4 Mi* PRESENT VALUE OF DIFFERENTIAL MAINTENANCE COSTS 0 0

5 Rl* PRESENT VALUE OF DIFFERENTIAL REPLACEMENT COSTS -$390 -$210

6 Bi-Ci NET PRESENT VALUE BASED ON THE TRIAL INTEREST

RATES (1)- [(2) -(3) + (4) -I- (5)]

$2,059 -$204

24
(E a) - IP -(S • b) + (M • c) + (R . b)l = 0

For i = 20%

($1,500) (6.515) - |($8,000) - (-$4,000) (.026) + (0) + (-$6,000) (.065)] =

($9,773) - [($8,000) - (-$104) + 0 + (-$390)) = $2,059

For i = 25%

($1,500) (5.089) - (($8,000) - (-$4,000) (.012) + (0) + (-$6,000) (.035)) =

($7,634) - [($8,000) - (-$48) + 0 + (-$210)) = -$204 r t ^ 1

Therefore, 20% <IRR< 25% By interpolation, IRR = .20 +l.05[ ) 100l= 24.5% L $2,059 + $204 J

(Note: "E," "P," "S," "M," and "R." are symbols Indicating ttie difterences between ttie energy conserving investment and its alternative that have not yet been converted to present values. "El*," "P*," "SI*," "Mi*," and "Rl*," are symbols

indicating the present values of the amounts based on a discount rate I, v»here I is first assumed equal to 20% and then

to 25%.)

IRR Problem Solution: The above Internal Rate of Return

solution also uses the hypothetical cost data employed in the previous three examples.

Visual inspection is used to identify trial interest rates that

might balance benefits and costs in terms of present value

dollars. The columns indicate the selected trial interest rates of 20 and 25%.

Line 1 shows the present value of energy savings

calculated at 20 and 25%. The amount of $9,773 for the

present value of energy savings is found by multiplying the

UPW* factor for 20 years at 20% by the base-year energy

savings of $1,500. The amount of $7,634 is found by multiplying the UPW* factor for 20 years at 25% by the base

year energy savings.

Line 2 lists the solar/conventional differential investment

cost. This cost which occurs initially is already in present

value dollars.

Line 3 gives the present value of differential salvage.

Recall that salvage is assumed to be less for the combined

solar/conventional system than for the conventional

system alone; therefore, the value is negative. The present

value amount of -$104 is found by multiplying the SPW

factor for 20 years and 20% by the estimated $4,000 in salvage foregone. The present value amount of -$48 is found by multiplying the SPW factor for 20 years at 25% by

the amount of salvage foregone.

Line 4 shows that no difference in maintenance costs is assumed for the two alternatives.

Line 5 gives the present value of differential replacement

costs as a negative amount because the combined

solar/conventional system is assumed to require less re- placement than the conventional system alone. The

present value amount of -$390 is found by multiplying the

SPW factor for 15 years and 20% by the estimated $6,000

replacement costs saved in the 15th year. The present

value amount of -$210 is found in a similar way by using

the SPW factor for 15 years and 25%.

Line 6 gives the net present value of the combined components of costs evaluated first at 20% and then at 25%. It indicates a saving of $2,059 for the 20% trial interest rate and a loss of $204 for the 25% trial interest rate. Thus the

internal rate of return on the investment lies between 20%

and 25%. By interpolation it can be found that 24.5% is the

internal rate of return on investment.
Table E: Discounted Payback

Y Y Y Y Y Y 1 E*, Z M*| X R*, 1 s*, p* X B-C 1=1 1=1 1=1 )=1 1=1

PRESENT PRESENT

PRESENT VALUE OF VALUE OF PRESENT

VALUE OF CUMULATIVE CUMULATIVE VALUE OF DIFFERENTIAL YEARS INTO CUMULATIVE DIFFERENTIAL DIFFERENTIAL DIFFERENTIAL SOLAR PRESENT

THE ENERGY MAINTENANCE REPLACEMENT SALVAGE INVESTMENT VALUE OF

INVESTMENT SAVINGS COSTS COSTS VALUE COSTS NET SAVINGS

(1) (2) (3) (4) (5) (6) (7)

1 $1,433 0 0 0 $8,000 -$6,567

2 2,801 0 0 0 -5,199

3 4,103 0 0 0 -3,897

4 5,349 0 0 0 -2,651

5 6,537 0 0 0 -1,463

6 7,673 0 0 0 -327

7 , 8,756 0 0 0 +756

Find Y such that 1 (E*, - M*, - R'i + S*,) = P* i=i

For Y = 6, ($7,673 - 0 - 0 + 0) < $8,000

For Y = 7, ($8,756 - 0 - 0 + 0) > $8,000

Therefore 6 < Y < 7

DPB Problem Solution: The above discounted payback

solution is calculated for the same hypothetical investment

problem used to illustrate the four previous economic tools.

Cumulative discounted savings and recurring costs are compared each year with the initial cost until the present

value of net savings becomes positive. Given energy price

escalation at a rate of 5% and discounting at 10%, we find

that energy savings offset investment costs in the seventh

year.

26

Column 2 lists the present value of cumulative energy

savings. Figures in Column 2 are found by multiplying the

base-year energy savings of $1,500 by the UPW* factor

from table D-2 for a 10% discount rate, a 5% escalation

rate, and the stated number of years into the investment as | indicated by Column 1. For the first year, 0.955 x $1,500= '

$1,433, and for the seventh year 5.837 x $1,500 = $8,756.

The seventh year is the first year the cash flow becomes

positive. This indicates that the time of payback of the

initial investment of $8,000 occurs in the seventh year, , before year end.
FACTORS AFFECTING BENEFITS AND COSTS

To this point we have discussed economic efficiency,

described discounting, and demonstrated the application of

five economic tools as well as the advantages and disadvantages of their use. The purpose of this section is to

clarify elements that affect the use of these tools to measure benefits and costs.

The elements to be discussed are discount rates, incentives, inflation, salvage values, taxes, time horizons and

uncertainty. They are arranged in alphabetical order. Their

relative importance depends on the characteristics of a

particular investment.

Discount Rate

The discount rate is a rate of interest used to convert

benefits and costs occurring at different times to a common time. The discount rate is selected to reflect the

investor's time preference for money. Once a discount rate has been chosen it can be used either in discount formulas

or to select discount factors to carry out various

discounting procedures.

The selection of a discount rate may be guided by the level of return on alternative investment opportunities, on the

cost of borrowing money, or, in the case of public organizations, on legislative or mandated requirements. The United

States Office of Management and Budget (0MB) currently

requires Federal agencies to use a discount rate of 10% in excess of the rate of inflation for evaluating most government investments.®
If an investor is unsure of the potential return on an alternative investment, the cost of borrowing can be used as the

discount rate. However, the earning rate available on alter- native investments should take precedence over the

borrowing rate as an indicator of the appropriate discount

rate and should be taken into account regardless of

whether the money is borrowed or not. In selecting the appropriate discount rate, the client might be asked for the

after-tax rate of return on other investments, in the case of a homeowner, for example, the rate of interest received on

savings accounts. These may vary considerably by firm and

industry. If the evaluation is being made for a government

client, it is important to determine if there are legislative or mandated requirements for the discount rate.

Clients may request that high risk projects be evaluated

with higher discount rates than those with low risk.^ Risk can also be treated in other ways, such as basing benefit

and cost estimates on probabilities of occurrence,

incorporating contingency estimates of cash flow into the

calculations, or by employing sensitivity analysis to assess

the impact of different time horizons or of different amounts

of energy savings on the profitability of the investment.

(Probability and sensitivity analyses are discussed in more

detail later in this section.)

Discount rates may be expressed in either "nominal" or

"real" terms. Nominal rates include the effects of inflation and the real earning power of money invested over time,

"fvlarket rates" are nominal rates because they reflect both

inflation and real earning power. A real rate reflects only

the real earning power of money, and therefore is lower

than a nominal rate, given the same conditions.

Ensuring that dollar estimates of benefits and costs and ( discount values are compatible with one another by either

including inflation or excluding inflation from all values is a

basic principle to be followed in discounting. A real discount rate is appropriate if inflation is removed from the

cash flows prior to discounting. A nominal rate is appropri- ate if cash flows are inflated.

For example, the fuel escalation rate must be consistent

with the discount rate in terms of either including or excluding inflation. If a nominal rate of discount is used,

then the projected rate of total change in energy prices

must be used. This was the approach used in the problem

in the earlier section on discounting. On the other hand, if a

real rate of discount is used, the differential rate of energy

price change—that is, the projected escalation rate of energy prices minus the average escalation rate of prices

in general—is appropriate.

The discount rate is one of the most dramatic factors

affecting the net benefits of conservation investments. The

rate selected may make a project seem either economic or uneconomic. For example, a project which has positive net savings when evaluated using a 6% discount rate might

yield negative net savings when evaluated at 7%.

As the discount rate is increased, the present value of any

given future stream of costs or benefits becomes smaller.

High discount rates tend to favor projects with quick

payoffs over projects with deferred benefits.

COST ADJUSTMENT
INTERNAL RATE OF RETURN

An incentive is a positive inducement to encourage a

particular type of behavior or action. Incentives are considered in economic evaluations of conservation and

solar energy investments because they can affect the

economic viability of an investment and its optimal size. Their cash values should be discounted to present values

just as are other cost savings. Following are examples of in- centives for energy conservation and solar energy provided

by the government.

preciation allowances for energy conservation and solar

energy investments reduce taxable income. The imposition

of higher taxes on nonrenewable energy sources raises

their prices and encourages investments in conservation

and renewable energy. The elimination of tax deductibility

for business fuel expenses would further encourage energy

conservation and solar energy investments in commercial

buildings.

Government Cost Sharing: Another form of governmental incentive is provided when one unit of government

bears a specified percentage of the costs of an investment

made either by another unit of government or by an

investor in the private sector. The National Energy Act, for example, provides Federal cost sharing for energy conservation in schools, hospitals, local government, and public care buildings.

Loan Interest Subsidies and Guarantees: Loan

subsidies provide for loans at rates below the market rate and reduce borrowing costs to make energy conservation

and solar energy more economical. Loan guarantees may

induce lending institutions to provide loans on more liberal terms or to make loans when they otherwise would not. Conservation designs that are uneconomic without

subsidies may in fact be cost effective if subsidies are included in the economic evaluation. Federal and State

energy offices and associations such as the National

Conference of State Legislatures are potential sources of

information on available subsidies for energy conservation.

Grants: Cash subsidies in specified amounts are sometimes made to purchasers of energy conservation

equipment. The National State/Federal Combined Program

for Providing Subsidies to Residential Users of Solar Hot

Water Heaters is an example. The cost of a solar hot water

heater to the recipient of a grant is the life-cycle cost of the

system minus the government grant.

Taxes: Taxes may be used as a means of providing

several types of incentives. Income tax credits for conservation and solar expenditures provide a subsidy by allowing

specific deductions from the investor's taxes owed.

Property tax exemptions for energy conservation and solar

energy capital equipment eliminate or reduce property

taxes that would otherwise add to annual costs. Liberal deInflation

Inflation is a rise in the general price level. Although all prices cannot be expected to rise or fall together and in the

same amount, average price increases in specific and

general categories of goods and services can be

measured.^" In making economic evaluations of energy conservation and solar energy investments, it is important that

price inflation as indicated by average price increases in the economy be eliminated from estimates of benefits and

costs.

As was shown in figure 1, fuel prices have increased

dramatically in recent years—so rapidly that they should be

given special attention in evaluating investments to 29
conserve nonrenewable energy. Since benefits fronn conservation vary directly with fuel prices, assumptions regarding

the change in fuel prices over time have a major impact on

the predicted benefits of a conservation project. Projected

energy prices are usually based on contractually stated

prices, extrapolated trends from historical prices, or government/industrial predictions of future prices. The Department of Energy's price projections, for example, provide

estimates of prices of energy for residential and commercial users in 1978 constant dollars for the period of 1977 to 1990.^^ These government price projections are updated

periodically.

Other prices affecting the benefits and costs of conservation investment over time are those related to operation,

maintenance, and replacement. The possibility of inflation affecting these prices, as well as energy prices, should be

considered.

There are several basic methods of handling inflation reflected in future prices. One is to eliminate it from inflated

cash flows by applying a price deflator index.^^ Future

prices may be expressed in constant dollars in some base

year, as in the case of the Department of Energy's

projections described above. The prices in constant dollars

may rise, but they will reflect real price changes (i.e., increases over and above the average inflation rate for all goods and services) rather than changes due to inflation. The constant dollar prices must then be discounted with a

real discount rate to arrive at present or annual values.

As indicated earlier, a second way of handling inflation is to discount cash flows that contain inflation with a nominal

discount rate that reflects both real changes in value and

the expected inflation rate.

The impact of inflation on the economic viability of conservation and solar investments depends on which prices are

inflated most, as well as on institutional arrangements such

as taxes. For example, the higher the escalation rate of energy relative to other prices, the more economical investments to conserve nonrenewable energy will be. On the

other hand, for commercial properties depreciation writeoffs for tax purposes will become less significant with

higher rates of general inflation because the depreciation is based on the original "book value" of the investment.

Salvage Value

For the purpose of evaluating the economic feasibility of an

investment, its salvage value is defined broadly here to encompass (1) its residual value, net of the cost of disposal, whenever it is removed or replaced during the

study period; (2) the value remaining at the end of the

study period; or (3) the value recovered through resale at the end of the study period. If an existing investment is being compared with a new alternative, the current salvage

or resale value of the existing investment is used to compare against the first cost of the new alternative. The

present value of salvage value can generally be expected

to decrease, other things equal, as the discount rate rises,

the equipment deteriorates, and the time horizon

lengthens.

A measure of salvage value is the amount that could be

added to the selling price of the building due to the energy

conservation or solar investment. It might be assumed that,

with perfect information, a building buyer would be willing to pay an additional amount for an energy conserving or solar- equipped building equal to the capitalized value of fuel savings, net of costs, over the remaining life of the

conserving investment. If the investor's time horizon is the

same as the useful life of the investment, there will be no

salvage value; if the time horizon is shorter, there may be

salvage. Even if potential energy savings remain, there will be no salvage value to the current owner unless these

savings can be expected to be reflected in the resale value

of the building or unless the energy conservation devices can be removed and sold or used in another application.

These possibilities are very uncertain. Thus estimating

salvage value with any reliability can be very difficult.

30
Time Horizon

Taxes

Taxes may have an impact on the economic viability of conservation investments in two ways. One way is as a mechanism for providing direct financial subsidies; the

other way is through regular tax laws, such as existing

property tax laws, sales taxes, and income tax laws.

Property Taxes: Because many energy conservation in- vestments are capital intensive, they tend to increase the

cost of a building and raise the value of the property tax. This effect reduces the net savings from capital-intensive

conservation investments.

Income Taxes: The deductions from taxable income of

interest on loans and depreciation on capital investments

have positive effects on conservation and solar investments. The deduction from taxable income of fuel expenses as a business expense, on the other hand, has a

negative effect on conservation and solar investments

because it effectively reduces the cost of fuel to business.

Conserving fuel may result in after-tax dollar savings for businesses that are less than the before-tax value of the

fuel saved. For homeowners, dollars saved from fuel conservation are savings which are not taxed.

The time horizon is simply the period of economic analysis,

measured usually in years. For a conservation investment,

it is the length of time over which costs and benefits from

conservation are calculated. The time horizon is the "life cycle" in life-cycle cost analysis.

The selection of a time horizon or life cycle is based on a concept of investment life or on the personal time

perspective of the investor. There are often no fixed time

horizons for investment projects. The life of the building or the particular conservation investment may be used as the

time horizon, as well as the investor's planned time of

occupancy of the building.

Two concepts of investment life may be considered in an

evaluation. One is "useful life," the other, "economic life." Useful life is the period over which the investment has

value in conserving energy. Economic life is the time period over which the investment is the least-cost means of

providing a specified type of conservation. The economic

life, in most instances, will be shorter than the useful life. In theory, evaluations of investments should be based on

their economic life. In practice, the economic life is probably more difficult to predict than useful life, and the

concept of useful life more often enters analyses.

NET BENEFITS

31
The personal time perspective of the building owner will depend on the objective and circumstances of the owner. A speculative builder planning for immediate sale, for example, may view the relevant time horizon as that period

of ownership from the acquisition of property to the first sale of the building. Thus, although the useful life of a solar

domestic hot water heating system might be 20 years, the

speculative builder might project his economic time horizon

for only 1 year. If the speculator does not anticipate gain

through a higher selling price of the building due to the

potential savings to the buyer, then the solar investment

with its high first cost is unlikely to be economic in the

speculator's evaluation. Such conditions might also be

anticipated by the owner of a residence who expects to move before the end of the payback period for conservation investments.

If the client that requests an economic evaluation of energy

conservation investments does not specify a period ot analysis, the designer or analyst must select one. Federal,

State, and local guidelines may provide an answer for public buildings. Mortgage lending periods for buildings,

normally ranging from 20 to 30 years, may provide an index

of building life. The time horizon for government buildings is usually longer than private buildings because they tend to have one owner, are built to rigid specifications, and are

generally well maintained. Information from research

reports and literature from manufacturers of energy conservation materials, as well as warranties and guarantees may

also provide guidelines.

INFLATION

As public awareness of the cost of energy and potential

savings from conservation increases and as conservation

becomes increasingly capitalized in the resale prices of

buildings, time horizons for conservation investments in the

private sector are likely to increase. From the perspective

of national economic efficiency, a time horizon based on a concept of life rather than brief, speculative periods is more

appropriate. As buyers become aware of the potential

benefits from energy conservation design, the responsive- ness of the speculative market will predictably improve and

considerations of the long-term net benefits from conservation will begin to take precedence.

The selection of a time horizon can affect the value of the

net benefits from conservation and thereby can affect in- vestment decisions. The impact of varying time horizons

depends in part upon three related factors: the discount

rate, the rate of fuel price escalation, and salvage value.

Uncertainty

Evaluations of benefits and costs from energy conservation

design are only as good as the values in formulating the

analysis. Some of the life-cycle costs and most of the life- cycle benefits from conservation design accrue in the

future. The design community will therefore experience

uncertainty as to the correct values to use in predicting

future benefits and costs.

Two analytical techniques that can be used to help make

decisions about conservation investments whose economic

payoffs are uncertain are sensitivity analysis and probability

analysis.''^

32
Figure 5

.10,E = .15

D = O.E = 0 D = .10,E = .10

D=.10,E = .05 D = .05,E = 0

D=.10,E = 0 D = .15,E = 0

YEARS

SENSITIVITY OF FUEL SAVINGS TO TIME HORIZONS, DISCOUNT RATES AND ENERGY

ESCALATION RATES.

D = Discount Rate E = Energy Escalation

Sensitivity Analysis: This technique involves a test of the

responsiveness of life-cycle costs, net benefits, or other

economic measures to alternative values of key factors

about which there is uncertainty. It shows decision makers

how the economic viability of a conservation project

changes as, for example, fuel price escalation, discount

rates, time horizons and other critical factors vary.

To illustrate, figure 5 shows the sensitivity of fuel savings

by a solar heating system to three critical factors: time

horizons (0 to 25 years), discount rates (D equals 0, 5, 10

and 15%), and energy escalation rates (E equals 0, 5, 10,

and 15%). The present value of savings in each case is based on a base-year fuel savings of $1,000.

Note that, other things equal, savings increase over time,

but more slowly with higher discount rates and more

quickly with higher escalation rates. The impact of fuel price escalation is most apparent when comparing the top curve of the graph (D=10, E = .15) with one close to the

bottom (D=.10, E=0). The present value of savings at the

end of 25 years is about $50,000 for a fuel price escalation

rate of 15%, and only about $8,000 for an escalation rate

of 0%, other things equal. Whereas the Btu savings and

initial prices are the same, the present value of the Btu

savings varies widely depending on the selection of the escalation rate of fuel prices and the discount rate.

Although impact scenarios such as these illustrated in figure 5 do not show the analyst what parametric values to choose, they do. show decision makers the impact of alter- native assumptions. Knowing the consequences of error may help analysts make better decisions about conservation investments with uncertain outcomes.
Table F

EXPECTED VALUE OF COMPRESSOR REPLACEMENT

Year of Replacement

6

7

8

9

Probability Cost ($)

SPW

10% Discount Rate

Expected

Present

Value Cost ($)

.1

.2

.6

.1

400

400

400

400

.565

.513

.467

.424

23

41 112

17 Expected value of compressor replacement: $193

Expected Value of Cost = Cost x Probability x SPW

Probability Analysis: This technique is sometimes called

"expected value analysis." It is used to evaluate the

benefits and costs of an event whose expected chance of occurrence can be predicted.^'' Often historical data are

available for generating probability data for existing

technologies. But for many new conservation technologies,

probability data may not be available. Computer simulation

is sometimes used to provide probability data on innovative

technologies.

An example of using probability analysis is shown in table

F. The heat pump case described earlier in figure 3 as an

illustration of discounting is used once again.

The heat pump case treated the replacement of a compressor. The expected value of the compressor replacement, as measured in present value dollars using probability

analysis, is shown in table F to be $193. Note that this differs from the $187 estimated in figure 3. While it is unlikely that the exact cost of replacing the compressor will be predicted using a probabilistic approach, generally, over a large number of applications, the difference between the

actual cost and the predicted cost will be less than in the case when a single point estimate is employed.

HIGH RISK

34
ENERGY

SUMMARY

To the average member of the design community, the

problem of carrying out an economic analysis of a

particular type of investment problem often appears to be a

difficult, even an incomprehensible task. Most of the

available books on economic analysis assume that the

reader has previously acquired a knowledge of fundamental principles, and thus they proceed to the more complex

aspects of analysis.

It is true that some economic problems are exceedingly

difficult, but it is also true that many of the problems that

frequently occur are surprisingly simple in their solution.

This modest handbook was compiled with this fact in mind.

With knowledge of the various economic terms and

techniques explained here, the reader should be more able

to understand economic analysis and to carry out simple

economic evaluations.

We have tried to explain economic principles without the use of advanced mathematics or the derivation of formulas.

Complex economic problems are not treated, although now

the reader will have a better comprehension of them. We

have furnished a kit of tools for design professionals to help them solve simple economic problems and better

appreciate those of greater complexity.

We do not believe that a little knowledge is a dangerous

thing, but that it is rather the best means of whetting the

appetite for more. We hope that this introductory handbook

will encourage our readers to plan their energy investments

with as much care and understanding as Vitruvius

counseled in designing buildings for firmness, commodity,

and delight.

35
Footnotes

(1) Chart figures for Energy Consumption 10'^ Btu were obtained

from Buildings' Energy Facts and Trends (Draft in Preparation),

Oak Ridge National Laboratory, Oak Ridge, Tennessee,

November, 1977, pp. 7-10. Data for 1976 were obtained from Mr.

Chiarles Reading, U.S. Department of Interior.

Figures chiarted for Energy Consumption 1(f in 1977 Dollars were

obtained by multiplying thie annual energy consumption by tfie appropriate fuel price. Prices for natural gas, electricity distributed,

and fuel oil were taken from Buildings' Energy Facts and Trends

(Draft). Coal prices were calculated by extrapolation using the Anthracite Coal (stove size) Historical Price Index taken from the

Historical Statistics of the United States, Department of Commerce, September, 1975.

Chart figures for Price Per 1(f Btu in 1977 Dollars reflect a weighted average of the four energy sources (gas, electricity,

petroleum, and coal) converted from current year dollars to 1977

dollars by the Implicit Price Deflator for Personal Consumption

Expenditures. The deflator index was taken from the Survey of

Current Business, U.S. Department of Commerce, July, 1977,

Table 8-8.

(2) For example, see State of California, Energy Resources

Conservation and Development Commission, Conservation

Division, Energy Conservation Design Manual for New NonResidential Buildings, October, 1977, a document that describes

the requirements for energy conservation that must be met prior to the application for a nonresidential building permit in California.

(3) Often, however, dollar values are assigned to effects which are

difficult to measure. See, for example, the measurement of the cost of airport noise pollution on an urban environment and of congestion costs on wilderness recreation in Theory and

/Measurement of Economic Externalities, ed. Steven A. Y. Lin (New

York; Academic Press, 1976).

(4) Our emphasis is on a simplified treatment of these methods.

For a more in-depth description of these methods, see Eugene L. Grant and W. Grant Ireson, Principles of Engineering Economy, 5th

ed. (New York; The Ronald Press Co., 1970); E. J. Mishan, CostBenefit Analysis: An Introduction (New York; Praeger, 1971); A/it K. Dasgupta and D. W. Pearce, Cost-Benefit Analysis: Theory and

Practice (New York; Harper and Row, 1972); and Peter G. Sassone and William A. Scheffer, Cost-Benefit Analysis: A

Handbook (New York; Academic Press, 1 978).

(5) For a discussion of the budgeting of probable costs, scheduling of disbursements and control of costs during

programming, master planning, preliminary drawings, working

drawings, specifications and construction; see Creative Control of

Building Costs, ed. Wilhan-Dudley Hunt, Jr., AIA (New York; McGraw Hill Book Co., 1967).

(6) To treat future energy costs that are expected to escalate at changing rates over time, several additional steps, beyond those il- lustrated here, are required.

The procedure and worksheets for calculating the present value of energy savings when multiple escalation rates are us^d are provided in Rosalie T. Ruegg's, et al., Life-Cycle Costing, National

Bureau of Standards Science Series 113, September 1978, pp. 18-

21. UPW* factors incorporating multiple energy price escalation

rates as projected by the Energy Information Administration of the U.S. Department of Energy are provided in Rosalie T. Ruegg and

John S. McConnaughey's Manual for the Federal Energy

Management Program, National Bureau of Standards Special

Report (In Press), 1979, Appendix B.

(7) Here the problem is presented as one of determining whether

it will pay to have a solar energy system of a specified design, size, and cost. In practice a more common problem will be to determine

the optimal design and size for a system and the cost effective- ness of that system. For a more in-depth treatment of the economics of solar energy systems, including optimization

analysis, see Rosalie T. Ruegg and G. Thomas Sav, Microeconomics of Solar Energy, National Bureau of Standards Special Report,

1980, in press.

(8) 0MB Circular A-94. Other rates are required for different types

of investment. For example, for decisions on leasing or purchasing

real property, 0MB requires a discount rate of 7%.

(9) For a discussion of investment decisions under high risk, see Henry A. Latane, Criteria for Choice Among Risky Ventures; John

W. Pratt, Risk Aversion In The Small and In The Large; Jack

Hirshleifer, Risk, The Discount Rate, and Investment Decisions; ed. Stephen H. Archer and Clarke A. D'Ambrosio, The Theory of

Business Finance, 2nd ed. (New York; MacMillan Publishing Co.,

Inc., 1976).

(10) For a description of price indexes and how to use them, see

U.S. Department of Labor, Bureau of Labor Statistics, Monthly

Labor Review, any issue.
(11) Department of Energy; Historic and Forecasted Energy Prices

by U.S. Department of Energy; Region and Fuel Types for Three

fi/licroeconomic Scenarios and One Imported Oil Price Escalation

Scenario, DOE/ETA-01 02/27, December, 1978, Energy

Information Administration Clearinghouse, 1726 M Street, N.W.,

Room 210, Washington, D.C. 20461.

(12) For past and present price indices, see U.S. Department of Labor, Monthly Labor Review, Current Issue. For a description of how to construct a constant dollar price deflator based on past

inflation rates for use in adjusting future current dollar payments, see U.S. Office of Management and Budget Circular No. A-104,

June 14, 1972.

(13) For a general text on decisionmaking in the design process

under conditions of uncertainty, see Myron Tribus, Rational

Descriptions, Decisions and Designs (New York; Pergamon Press,

1969). For an overview of alternative approaches to dealing with uncertainty and criteria for decisionmaking under uncertainty, see

Ajit K. Dasgupta and D. W. Pearce, Cost-Benefit Analysis: Theory

and Practice (New York; Harper and Row, 1972, pp. 174-178).

(14) Probabilities may be "objective" based on frequency of occurrences information or "subjective," based on intuition. For a more in-depth discussion of probability analysis see ibid., pp. 1 74-

198.
APPENDICES

A GLOSSARY OF ECONOMIC TERMS

B DISCOUNT FORMULAS

C DISCOUNT FACTORS

C-1 Single Compound Amount (SCA)

C-2 Single Present Worth (SPW)

C-3 Uniform Sinking Fund (USF)

C-4 Uniform Capital Recovery (UCR)

C-5 Uniform Compound Amount (UCA)

C-6 Uniform Present Worth (UPW)

D UNIFORM PRESENT WORTH FACTORS MODIFIED

FOR ENERGY PRICE ESCALATION

D-1 8% Discount Rate

D-2 10% Discount Rate

D-3 12% Discount Rate

38
APPENDIX A

GLOSSARY OF ECONOMIC TERMS

Annual Value: Benefits or costs occurring in uniform amounts annually, or the uniform annual equivalents of past,

present or future benefits.

Benefit-Cost Analysis: A means of evaluating alternative projects or investments by comparing the discounted value

of total expected benefits with the discounted value of total expected costs for each alternative.

Benefit-Cost Ratio: Benefits expressed as a ratio to costs, where both are discounted to a present or annual value;

the ratio must be greater than one for an investment to be economically efficient.

Constant Dollars: Values expressed in terms of the general purchasing power of the dollar in the base year. Constant

dollars do not reflect price inflation.

Current Dollars: Values expressed in terms of actual prices of each year. Current dollars reflect price inflation.

Discount Rate: The rate of interest reflecting the time value of money that is used to convert benefits and costs

occurring at different times to equivalent values at a common time.

Discounted Payback Period: The time required for the cumulative net benefits derived from an investment to pay

back the investment cost, considering the time value of money.

Discounting: A technique for converting cash flows that occur over time to equivalent amounts at a common point in time.

Economic Efficiency: Maximizing net benefits or minimizing costs for a given level of benefits.

Economic Life: The period of time over which an investment is considered to be the least-cost alternative for meeting

a particular objective.

Future Value (Worth): The value of a present dollar amount at some point in the future, considering the time value of

money.

Incentive: A positive inducement to encourage a particular type of behavior or action.

Inflation: A rise in the general price level resulting from a decline in the purchasing power of the dollar.

Internal Rate of Return: The interest rate for which the total discounted benefits from an investment equal its total discounted costs.

Grants: Cash payments for the purpose of encouraging a particular practice or the use of a good or service by

reducing its net cost to the owner or user.
Investment Cost: The sum of the planning, design, and construction costs necessary to provide a finished project

ready for use.

Life Cycle: The period of time between the starting point and cutoff date of analysis, over which the costs and

benefits of a certain alternative are incurred.

Life-Cycle Cost: The total of all relevant costs associated with an activity or project during the time it is analyzed. For

buildings, life-cycle costs include all costs of owning, operating, and maintaining a building over its period of analysis,

including its energy costs.

Marginal Analysis: Evaluating incremental changes in costs and benefits resulting from incremental changes in an in- vestment.

Monetary Benefits: Benefits assigned a dollar value.

Net Benefits: The difference between benefits and costs, evaluated in present or annual value dollars.

Nonmonetary Benefits: Benefits to which it is difficult to assign dollar values.

Operation and Maintenance Costs: The costs associated with the normal operation and maintenance of a system,

often accumulated on a recurring basis.

Payback: (See Discounted and Undiscounted Payback Period)

Present Value (Worth): Past and future cash flows expressed in time-equivalent amounts as of the present time,

adjusted for inflation and the time value of money.

Probability Analysis: A technique, also called expected value analysis, used to evaluate the dollar value of an event

whose expected chance of occurrence can be predicted.

Salvage Value: The net sum to be realized from disposal of an asset, net of disposal costs, at the time of its replacement, resale, or at the end of the study period.

Sensitivity Analysis: Testing the outcome of an evaluation by altering the values of key factors about which there is uncertainty.

Time Horizon (Study Period): A period of economic analysis over which time the costs and benefits of an investment

are calculated.

Undiscounted Payback Period (Simple Payback): The length of time necessary for the cumulative benefits or savings resulting from an investment to recover the original cost of the investment, not considering the time value of

money.

Useful Life: The period over which an investment is considered to have value.

40
APPENDIX B

DISCOUNT FORMULAS

FORMULA NAME

Single Compound Amount

Formula (SCA)

Single Present Worth

Formula (SPW)

Uniform Sinking Fund

Formula (USF)

Uniform Capital Recovery

Formula (UCR)

Uniform Compound Amount

Formula (UCA)

Uniform Present Worth

Formula (UPW)

Uniform Present Worth

Formula Modified (UPW*)

To find F when P is known

To find P when F is known

To find A when F is known

To find A when P is known

To find F when A is known

To find P when A is known

To find P when A is escalating at rate e

F=P (1+i/

P=F

A = F

A = P

F=A

i (i+ir-i

i(1+f

(1+i)^-1

i(1+i)'

P = A

(1+e)

(i-e) L

1+1

Where:

P = a present sum of money.

F = a future sum of money, equivalent to P at the end of N periods of time at an interest or discount rate

of i.

i = an interest or discount rate. N = number of interest or discounting periods.

A = an end-of-period payment (or receipt) in a uniform series of payments (or receipts) over N periods at

i interest or discount rate.

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

SINGLE COMPOUND AMOUNT FORMULA F = P (1 +i)^

A present sum of money (P) times SCA factor for appropriate discount rate

(i) gives (F) future value of (P). N = Number of discount periods

DISCOUNT FACTORS N 6% O O/ O /O

DISCOUNT RATES

10% 12% 1 D /o tlU /o 25%

1 1.060 1.080 1.100 1.120 1.150 1.200 1.250

2 1 . 1 24 1 . 1 DD 1.210 1.254 1 .OC.C. 1 5g3

o 1.191 1 .dou 1.331 1.405 1 .oc. \ 1 953

4 1 .2bo 1 OCA 1 .464 1 574 1 74Q 2 074 2 441

cD 1 , JJo 1 .4Dy 1.611 1 .762 9 ni 1 2 488 3 052

6 1,419 1.587 1.772 1.974 2.313 2.986 3.815

1 1 .0U4 1 71 /I1 . / 1 4 1.949 2.211 2 660 3 583 4.768

QO 1 .oy4 1 Rc: 1 2.144 2.476 3 059 4 300 5.960

Q 1 .boy 1

1 .yyy QQQ 2.358 2.773 3.51

8

5 1 60 7.451

1 u 1 . / y 1 0 1 d. 1 oy 2.594 3.106 4 046 6.192 9.313

11 1.898 2.332 2.853 3.479 4.652 7.430 1 1.642

1 o1 iL -^i.U 1 d d.D \0 3.138 3.896 5 350 8.91

6

1 4.552

1 O 1 oo £1. 9 79n / dV 3.452 4.633 6.153 1 0.699 18.190

1 A1 4 ii.iiD 1 9 QTT 3.797 4.887 7 076 1 2.839 22.737

ic. jy / 9 179 4.1 77 5.474 8.137 1 5.407 28.422

16 2.540 3.426 4.595 6.130 9.358 18.488 35.527

1 7 ii-oy J o. / uu 5.054 6.866 1 0 761 22.186 44.409

1

1 o0 £i.O04 o.yyD 5.560 7.690 1 2.375 26.623 55.51

1

1 y 4 . 0 1 D 6.1 1

6

8.613 14 232 31 .948 69 389

on£L\J T on7 4.DD /I Pi(^11 6.727 9.646 1 D. OD / oo . ooo 86 736

21 3.400 5.034 7.400 10.804 18.822 46.005 108.420

o.uU4 R A 97 8.140 12.100 91 i^A^ c;c: one. 135 1 o o . 5?5 o^o

do c; Q71 O.O / 1 8.954 12.552 OA RQ1 fifi 947 1 59 407

OAd^ D. J4 I 9.850 15.179 9ft fi9c; 21 1 758

D.04o 10.835 17.000 '^9 Q1 Q

26 4.549 7.396 1 1.918 19.040 37.857 114.475 330.872

27 4 822 7 988 13.1 10 21.325 45.535 1 37.37

1

413.590

28 5.1 1

2

8.627 14.421 23.884 50.066 164.845 516.988 29 5.41

8

9.31

7

15.863 26.750 57.576 197.814 646.235 30 5.743 10.063 17.449 29.960 66.212 237.376 807.794

31 6.088 10.868 19.194 35.555 76.144 284.852 1009.742

32 6.453 1 1.737 21.1 14 37.582 87.565 341.822 1262.177 33 6.841 12.676 23.225 42.092 100.700 410.186 1577.722 34 7.251 13.690 25.548 47.143 115.805 492.224 1972.152 35 7.686 14.785 28.102 52.800 133.176 590.668 2465.190

40 10.286 21.725 45.259 93.051 267.864 1469.772 7523.164

45 13.765 31.920 72.890 163.988 538.769 3657.262 22958.874

50 18.420 46.902 117.391 289.002 1083.657 9100.438 70064.923

42
C-2

SINGLE PRESENT WORTH FORMULA P SPW

A future sum of money (F) times SPW factor for appropriate discount rate

(i) gives (P) present value of (F).

DISCOUNT FACTORS 6%

DISCOUNT RATES

8% 10% 12% 15%

1 0.943 0.926 0.909 0.893 0.870 0.833 0 800

2 .890 .857 .826 .797 .756 .694 0.640

3 .840 ,794 ,751 .712 ,658 ,579 0.512 4 .792 ,735 .683 .636 .572 ,482 0.410 5 .747 .681 .621 .567 .497 ,402 0.328

6 .705 .630 ,565 .507 .432 .335 0,262

7 ,665 ,584 .513 .452 .376 ,279 0.210

8 .627 ,540 .467 .404 .327 .233 0.168

9 ,592 ,500 .424 .361 .284 ,194 0.134 10 .558 ,463 ,386 .322 .247 ,162 0.107

11 .527 .429 .351 .288 .215 ,135 0.086 12 .497 .397 ,319 .257 .187 ,112 0.069 13 .469 .368 ,290 .229 .163 ,094 0.055 14 .442 .341 ,263 .205 .141 .078 0,044 15 .417 .315 ,239 .183 .123 .065 0.035

16 .394 .292 ,218 .163 .107 .054 0.028

17 .371 ,270 ,198 .146 .093 .045 0.022 18 .350 .250 ,180 .130 .081 .038 0.018 19 .331 .232 ,164 .116 .070 .031 0.014 20 .312 .215 ,149 .104 .061 .026 0.012

21 .294 .199 .135 .093 .053 .022 0.009 22 .278 .184 .123 .083 .046 .018 0.007 23 .262 .170 .112 .074 .040 .015 0.006 24 .247 ,158 .102 .066 .035 .013 0,005

25 .233 ,146 .092 .059 .030 .010 0.004

26 .220 ,135 .084 .053 .026 .009 0.003 27 .207 ,125 .076 .047 .023 .007 0.002 28 .196 .116 .069 .042 .020 .006 0.002 29 .185 .107 .063 .037 .017 .005 0.002 30 .174 .099 .057 ,033 .015 .004 0.001

31 .164 .092 .052 .030 .013 .004 0.001 32 .155 .085 .047 .027 .011 .003 0.001 33 ,146 .079 .043 .024 .010 .002 0.001 34 ,138 ,073 .039 .021 .009 .002 0.000 35 .130 .068 .036 .019 .008 .002 0.000

40 .097 .046 .022 .011 .004 .001 0.000 45 .073 .031 .014 .006 .002 .000 0.000 50 ,054 .021 .009 .004 .001 .000 0.000

43
C-3

USF UNIFORM SINKING FUND FORMULA A (1+0-1

A future sum of money (F) times USF factor for appropriate discount rate (i) gives (A) an end-of-period payment equivalent over (N) discount periods to (F).

DISCOUNT FACTORS N 6% 8%

DISCOUNT RATES

10% 12% 15% 20% 25°

1 1.000 1.000 1 .000 1 .000 1.000 1,000 1.000 2 0.485 0.481 0.476 0.472 .465 0,455 0.444 3 .314 .308 .302 .296 .288 ,275 0.262 4 .229 222 21

5

.209 .200 ,186 0.1 73

5 .177 .170 1 64 .1 57 .148 ,134 0, 1 22

6 .143 136 .130 .123 .114 .101 0,089

7 .119 .1 12 .105 .099 .090 .077 0,066

8 .101 .094 .087 .081 .073 .061 0,050 9 .087 .080 .074 .068 .060 .048 0.039 10 .076 .069 063 .057 .049 .039 0,030

11 .067 .060 .054 .048 .041 .031 0.023 12 .059 .053 .047 .041 .034 .025 0.01

8

13 .053 .047 .041 .036 .029 .021 0.01

5

14 .048 .041 .036 .031 .025 .017 0.01

2

15 .043 .037 031 .027 .021 .014 0,009

16 .039 .033 .028 .023 .018 .011 0,007 17 .035 .030 .025 .020 .01

5

.009 0,006 18 .032 .027 .022 .018 .013 .008 0.005 19 .030 .024 020 .016 .01

1

.006 0.004 20 .027 -022 .017 .014 .010 .005 0.003

21 .025 .020 .016 .012 .008 .004 0.002 22 .023 .018 .014 .011 .007 .004 0.002 23 .021 .016 .013 .010 .006 .003 0.001 24 .020 .01

5

.01

1

.008 .005 .003 0.001 25 .018 .014 .010 .008 .005 .002 0.001

26 .017 .013 .009 .007 .004 .002 0.001 27 .016 .01

1

.008 ,006 .004 .001 0,001

28 .01

5

.010 .007 .005 .003 .001 0.000 29 .014 .010 .007 .005 .003 .001 0.000 30 .013 .009 .006 .004 .002 .001 0.000

31 .012 .008 ,006 .004 002 .001 0.000 32 .011 .007 ,005 .003 .002 .001 0.000 33 .010 .007 .005 .003 .002 .000 0.000 34 .010 .006 .004 .003 .001 .000 0.000 35 .009 .006 .004 .002 .001 .000 0.000

40 .006 .004 .002 .001 .001 .000 0.000 45 .005 .003 ,001 .001 .000 .000 0.000 50 .003 .002 .001 .000 .000 .000 0.000

44
C-4

UNIFORM CAPITAL RECOVERY FORMULA A = P

(1+0-1 ,y , UCR

A present sum of money (P) times UCR factor for appropriate discount rate (i) gives

(A) an end-of-period payment equivalent over (N) discount periods to (P).

DISCOUNT RATES

6% 8% 10% 12% 15% 20% 25% DISCOUNT FACTORS

1 1 060 1.080 1.100 1.120 1.150 1.200 1.250

o

c. U.OD 1 61

5

0 655 0 694

3 .374 .388 .402 .416 .438 .475 0.512 4 .289 .302 .315 .329 .350 .386 0.423

5 .237 .250 .264 .277 .298 .334 0.372

6 .203 .216 .230 .243 .264 .301 0.339 7 17Q 1 92 .205 .219 .240 277 0.316 8 .161 .174 .187 .201 .223 .261 0.300 9 .147 .160 .174 .188 .210 .248 0.289 10 .136 .149 .163 .177 .199 .239 0.280

11 .127 .140 .154 .168 .191 .231 0,273 12 .119 .133 .147 .161 .184 .225 0.268 13 .113 .127 .141 .156 .179 .221 0.265 14 .108 .121 .136 .151 .175 .217 0.262 15 .103 .117 .131 .147 .171 .214 0.259

16 .099 .113 .128 .143 .168 .211 0.257 17 .095 .110 .125 .140 .165 .209 0.256

18 .092 .107 .122 .138 .163 .208 0.255 19 .090 .104 .120 .136 .161 .206 0.254 20 .087 .102 .117 .134 .160 .205 0.253

21 .085 .100 .116 .132 .158 .204 0.252 22 .083 .098 .114 .131 .157 .204 0.252 23 .081 .096 .113 .130 .156 .203 0.251 24 .080 .095 .111 .128 .155 .203 0.251 25 .078 .094 .110 .128 .155 .202 0.251 26 .077 .013 .109 .127 .154 .202 0.251 27 .076 .091 .108 .126 .154 .201 0.251 28 .075 .090 .107 .125 .153 .201 0.250 29 .074 .090 .107 .125 .153 .201 0,250

30 .073 .089 .106 ,124 .152 .201 0,250

31 .072 .088 .106 .124 .152 .201 0,250

32 .071 .087 .105 .123 .152 .201 0,250

33 .070 .087 .104 .123 .152 .200 0.250 34 .070 .086 .104 .123 .151 .200 0.250 35 .069 .086 .104 ,122 .151 .200 0.250

40 .066 .084 .102 .121 .151 .200 0.250 45 .065 ,083 .101 .121 .150 .200 0.250 50 .063 .082 .101 .120 .150 .200 0,250

45
C-5

UCA UNIFORM COMPOUND AMOUNT FORMULA F = A

An end-of-period payment (A) times UCA factor for appropriate discount rate

(i) gives (F) future value of (A) over (N) discount periods.

DISCOUNT RATES

N 6% 8% 10% 12% 15% 20% 25%

1 1 UUU 1 .UUU 1 .UUU 1

1 ,uuu nnn 11 .UUU nnn 11 .UUU nnn 1 000

ii.UOU 9 1 fin £1 . 1 uu ^9 . 1

1 ^u9n ^.9 11 QURn 9 9nn 2 250

3 3,184 3.246 3.310 3.374 3.472 3.640 3.813 4 4.375 4.506 4.641 4.779 4.993 5.368 5.766 5 5.637 b.ob/ D. 1 UD D.OOO A 7y1 9 R0 £:U 9n7/

D D.y / 0 7 716 8 115 8.754 9.930 1 1 259

71 9 487 10 089 1 1 .067 1 2.91

6

1 5.073 8 9,897 10.637 1 1.436 12.300 13.727 16.499 19.842

9 11.491 12.488 13.579 14.776 16.786 20.799 25.802

10 1 3.1 81 1 4.487 1 c nov on TO/i

1 1

1 O. OO 1 20.655 24.349 32.150 42.566

1 0 1 A7n 1 ft Q77 24.1 33 29.002 39.581 54.208

13 18.882 21.495 24.523 28.029 34.352 48.497 68.760 14 21.015 24.215 27.975 32.393 40.505 59.196 86.949

1 O

0 1 779 07 9Dn H 1 . DOU 79 09^ 1 nQ fift7 1 uy .DO /

1

1 RO

oc; (:;7T on OQyi OO. PRO y ou 42 753 55.71

7

87.442 138 109

1 7 eL\j .c. \ o OO. / nJU 40 545 48.884 65.075 1 0S 9?1 1 73 636

18 30.906 37.450 45.599 55.750 75.836 128.117 218.045

19 33.760 41.446 51.159 63.440 88.212 154.740 273.556 on ob. /ob A a TCO 4D. /Oil C~7 07C 79 oc;9 1 09 AAA 1 AA RPft 1 oD.Doo 0/10 QA^

91C. \ cn /lOO ft1 fiQQ 118 810 225 026 429.681 00 cc /icy 71 40'^ / 1 .'+UO Q9y*!, ^OT ouo 1 37 632 271 031 S^R 101 23 46.996 60.893 79.543 104.603 159.276 326.237 673.626 24 50.816 66.765 88.497 118.155 184.168 392.484 843.033

dxs 54.865 73.106 98.347 1 1 3.334 9 1 0 700 4/1 .yo \ 1

1 OC/I 7Q1 uo4. / y I

26 59.156 79.954 109.182 150.334 245.712 567.377 1319.489 27 63.706 87.351 121.100 169.374 283.569 681.853 1650.361 28 68.528 95.339 134.210 190.699 327.104 819.223 2063.952

29 73.640 103.966 148.631 214.583 377.170 964.068 2580.939

30 79.058 113.283 164.494 241.333 434.745 1181.882 3227.174

31 84.802 123.346 181.943 271.293 500.957 1419.258 4034.968

32 90.890 134.214 201.138 304.848 577.100 1704.109 5044.710

33 97.343 145.951 222.252 342.429 664.666 2045.931 6306.887

34 104.184 158.627 245.447 384.521 765.365 2456.118 7884.609 35 111.435 172.317 271.024 431.663 881.170 2948.341 9856.761

40 154.762 259.057 442.593 767.091 1779.090 7343.858 30088.655

45 212.744 386.506 718.905 1358.230 3585.128 18281.310 91831,496

50

6

290.336 573.770 1163.909 2400.018 7217.716 45497.191 280255,693

DISCOUNT

FACTORS
UNIFORM PRESENT WORTH FORMULA P = A (l UPW + iT-i

An end-of-period payment (A) times UPW factor for appropriate interest rate (i) gives

(P) a present sum of money equivalent to (A) over (N) discount periods.

DISCOUNT FACTORS 6%

DISCOUNT RATES

8% 10% 12% 15% 20% 25%

1 0.943 0.926 0.909 0.893 0.870 0.833 0.800

2 1 .833 1 .783 1 ,736 1 .690 1 .626 1 .528 1 ,440

3 2.673 2.577 2.487 2.402 2,283 2.106 1.952

4 3.465 3.312 3.170 3.037 2.855 2.589 2.362

5 4.212 3.993 3.791 3.605 3.352 2.991 2 689

6 4.917 4.623 4.355 4.111 3.784 3.326 2.951 7 5.582 5 206 4.868 4.564 4,160 3.605 3-161 8 6.210 5.747 5.335 4.968 4.487 3.837 3,329

9 6.802 6.247 5.759 5.328 4.772 4,031 3.463

10 7.360 6.710 6.145 5.650 5 019 4,192 3.571

11 7.887 7.139 6.495 5.938 5.234 4.327 3.656

1

2

8.834 7.536 6.81

4

6.194 5.421 4.439 3 725

13 8.853 7.904 7.103 6.424 5.583 4.533 3.780

14 9.295 8.244 7.367 6.628 5.724 4,61

1

3.824

15 9.712 8.559 7.606 6.811 5.847 4.675 3.859

16 10.106 8.851 7.824 6.974 5.954 4.730 3.887

1

7

1 0.477 9.1 22 8.022 7. 1 20 6.047 4.775 3.910

18 10.828 9.372 8.201 7.250 6.128 4.812 3.928

19 11.158 9.604 8.365 7.366 6.198 4.843 3.942 20 11.470 9.818 8.514 7.469 6.259 4.870 3.954

21 11.764 10.017 8.649 7.562 6.312 4.891 3.963 22 1 2.042 1 0.201 8.772 7.645 6.359 4.909 3.970 23 12.303 10.371 8.883 7.718 6.399 4.925 3.976 24 12.550 10.529 8.985 7.784 6.434 4.937 3.981 25 12.783 10.675 9.077 7.843 6.464 4,948 3.985

26 13.003 10.810 9.161 7.896 6.491 4,956 3.988 27 13.21

1

10.935 9.237 7.943 6.514 4,964 3.990 28 13.406 11.051 9.307 7.984 6.534 4.970 3.992 29 13.591 11.158 9.370 8.022 6.551 4.975 3.994 30 13.765 11.258 9.427 8.055 6.566 4.979 3.995

31 13.929 11.350 9.479 8.085 6.579 4.982 3.996 32 14.084 11.435 9.526 8.112 6.591 4.985 3.997 33 14.230 11.514 9.569 8.135 6.600 4.988 3.997 34 14.368 11.587 9.609 8.157 6.609 4.990 3.998 35 14.498 11.655 9.644 8.176 6.617 4.992 3.998

40 15.046 11.925 9.779 8.244 6.642 4.997 3.999 45 15.456 12.108 9.863 8.283 6.654 4.999 4.000 50 15.762 12.233 9.915 8.304 6.661 7.999 4.000

(
UPW*

APPENDIX D

MODIFIED UNI ORM PRESENT WORTH FORMULA

p_A ^1±^^

^ - ^ (i-e)

An end-of-base-year payment (A) times UPW* factor for appropriate discount (i) and price escalation rate

(e) gives (P), a present sum of money equivalent to (A) escalating at rate e over (N) discount periods.

8% Discount Rate

UPW FACTORS MODIFIED FOR ENERGY PRICE ESCALATION

RATE OF ENERGY PRICE ESCALATION

N 1 % 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 .935 .944 .954 .963 .972 .981 .991 1.000 1.009 1.019

2 1.810 1.836 1.863 1.890 1.917 1.945 1.972 2.000 2.028 2.056

? 679 2.731 2.783 2.836 2.890 2.945 3.000 T 056 T 1 1 ?

A 3.474 3.558 3.643 3.730 3.818 3.908 4.000 4 093

c; 4 108 4.226 4.347 4.471 4.598 4.729 4.863 5.000 5.141 5 ?ft5

6 4.777 4.936 5.099 5.268 5.443 5.623 5.809 6.000 6.197 6.401

7 5.402 5.606 5.817 6.036 6.264 6.500 6.745 7.000 7.264 7.538 nyj 5.987 6.239 6.501 6.776 7.062 7.361 7.674 8.000 8.341 8.696

QC7 6 534 6.837 7.154 7.488 7.838 8.207 8.593 9.000 9.427 9.876

10 7 046 7.401 7.777 8.173 8.593 9.036 9.505 10.000 10.524 1 1 .077

11 7.525 7.935 8.370 8.834 9.326 9.850 10.407 1 1 .000 11.630 12.301

12 7.972 8.438 8.936 9.469 10.039 10.649 11.302 12.000 12.747 13.547

13 8.391 8.914 9.476 10.082 10.733 11.434 12.188 13.000 13.875 14.817

14 8.782 9.363 9.991 10.671 11.407 12.203 13.066 14.000 15.012 16.110

15 9.148 9.787 10.483 11.239 12.062 12.905 13.936 15.000 16.161 17.426

16 9.490 10.188 10.951 11.786 12.699 13.700 14.797 16.000 17.320 18.768

17 9.810 10.567 1 1 .398 12.312 13.319 14.428 15.651 17.000 18.489 20.134

18 10.110 10.924 1 1 .824 12.819 13.921 15.142 16.497 18.000 19.670 21.525

19 10.390 1 1 .262 12.230 13.307 14.507 15.843 17.335 19.000 20.861 22.942

20 10.651 1 1 .580 12.618 13.777 15.076 16.532 18.165 20.000 22.063 24.386

21 10.896 11.881 12.987 14.230 15.629 17.207 18.988 21.000 23.277 25.856

22 11.125 12.166 13.340 14.666 16.167 17.870 19.802 22.000 24.502 27.353

23 11.339 12.434 13.676 15.086 16.690 18.520 20.610 23.000 25.738 28.878

24 11.539 12.688 13.996 15.490 17.199 19.159 21.410 24.000 26.985 30.431

25 11.727 12.928 14.302 15.879 17.694 19.785 22.202 25.000 28.244 32.013

48
D-2

UPW*

MODIFIED UNIFORM PRESENT WORTH FORMULA

(i-e) [ M + i^J

An end-of-base-year payment (A) times UPW* factor for appropriate discount (!) and price escalation rate

(e) gives (P), a present sum of money equivalent to (A) escalating at rate e over (N) discount periods.

10% Discount Rate

UPW FACTORS MODIFIED FOR ENERGY PRICE ESCALATION

RATE OF ENERGY PRICE ESCALATION

N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 0.918 0.927 0.936 0.945 0.955

0c. 1 7R1 1 7fi7 11 .0an1

0

1 1 flfi7 0 ? RR4 1 00

A T 4m 0.*tOO 0T . Sfifi •iJ\J\J

\J 4 noQ 4 1?T 4 4 ^5fi

6 4.498 4.645 4.797 4.953 5.115

7 5 4?8 5 628 5.837

a0 5 781 fi 01Q 6.267 6.526

q fi 01

7

6 ?88 6 572 6.871 7.184

1 u fi 758 7 090 7.441 7.812

11 6.834 7.194 7.575 7.981 8.411

12 7.193 7.598 8.030 8.491 8.983

13 7.523 7.972 8.455 8.973 9.530

14 7.825 8.320 8.853 9.429 10.051

15 8.102 8.642 9.226 9.860 10.549

16 8.357 8.941 9.576 10.268 11.268

17 8.592 9.218 9.903 10.653 11.477

18 8.807 9.475 10.209 11.018 11.910

19 9.004 9.713 10.496 11.362 12.323

20 9.186 9.934 10.764 10.688 12.718

21 9.351 10.139 11.015 11.996 13.094

22 9.504 10.329 11.251 12.287 13.454

23 9.645 10.505 11.471 12.562 13.796

24 9.774 10.668 1 1 .678 12.822 14.124

25 9.892 10.819 11.871 13.068 14.437

0.964 0.973 0.982 0.991 1.000

1 .0\7^ 1 Q1 Q 11 .yiD Q4R 1 .9 / 0

£.,101 9 7ft7 0 ft^Q o.UUU

T fi4Q 0. f 0*T T0.0^ R911 o.y 1 u

4 4ftn 4 fi^f^ A. TiA / Of o.UUU

5.281 5.453 5.630 5.812 6.000

6.053 6.277 6.509 6.750 7.000

6.796 7.078 7.372 7.680 8.000

7.512 7.858 8.220 8.601 9.000

8.203 8.616 9.053 9.513 10.000

8.868 9.354 9.870 10.418 11.000

9.510 10.072 10.672 11.314 12.000

10.127 10.770 1 1 .460 12.202 13.000

10.723 1 1 .449 12.233 13.082 14.000

11.296 12.109 12.993 13.954 15.000

1 1 .849 12.752 13.738 14.818 16.000

12.382 13.377 14.470 15.674 17.000

12.896 13.984 15.189 16.523 18.000

13.390 14.576 15.895 17.363 19.000

13.867 15.151 16.588 18.196 20.000

14.326 15.711 17.268 19 022 21.000

14.769 16.255 17.936 19 840 22.000

15.196 16.784 18.591 20 650 23.000

15.607 17.299 19.235 21 454 24.000

16.003 17.800 19.867 22 250 25.000

49
RATE OF ENERGY PRICE ESCALATION

N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10

1 .902 .911 .920 .929 .938 .946 .955 .964 .973 .982

2 1.715 1.740 1.765 1.791 1.816 1.842 1.868 1.894 1.920 1.947

QO ? 44ft ? 4Q5 ? '543 2.591 2.640 2.690 2.740 2.791 2 842 ? ftQ4

A O. 1 1 \J 3 1ftT 3 258 3.335 3.413 3.492 3.573 3.655 3 739 3 R?5

O.T 7nfi / \J\J 3 810 3 916 4.025 4.137 4.252 4.369 4.489 4.612 4 738

6 4.244 4.380 4.521 4.666 4.816 4.970 5.129 5.293 5.462 5.636

7 4.729 4.900 5.078 5.262 5.452 5.650 5.856 6.068 6.289 6.517

oo 373 5 589 5.814 6.049 6.294 6.550 6.816 7.094 7.383

q ^ ft04 6 060 6.328 6.609 6.903 7.213 7.537 7.877 8.234

1 n fi 1Q7 fi 4Q? fi 804 7 133 7.480 7.846 8.232 8.639 9 069

1

1

6.237 6.554 6.890 7.247 7.625 8.026 8.451 8.902 9.381 9.889

12 6.526 6.880 7.256 7.658 8.086 8.542 9.029 9.549 10.103 10.694

13 6.787 7.176 7.593 8.039 8.518 9.031 9.581 10.172 10.805 11.486

14 7.022 7.446 7.902 8.394 8.923 9.494 10.109 10.773 11.489 12.263

15 7.234 7.692 8.187 8.723 9.303 9.931 10.613 1 1 .352 12.155 13.026

16 7.426 7.916 8.449 9.028 9.659 10.346 11.095 11.911 12.802 13.775

17 7.598 8.120 8.689 9.312 9.993 10.738 11.555 12.450 13.433 14.512

18 7.754 8.306 8.911 9.575 10.306 11.109 1 1 .994 12.970 14.046 15.234

19 7.894 8.475 9.114 9.820 10.599 11.461 12.414 13.471 14.643 15.945

20 8.020 8.629 9.302 10.047 10.874 11.793 12.815 13.954 15.224 16.642

21 8.134 8.769 9.474 10.258 11.132 12.108 13.199 14.420 15.789 17.337

22 8.237 8.897 9.632 10.454 1 1 .374 12.405 13.565 14.869 16.340 18.000

23 8.330 9.013 9.778 10.636 1 1 .600 12.687 13.914 15.302 16.875 18.660

24 8.414 9.119 9.912 10.805 11.813 12.954 14.249 15.720 17.396 19.309

25 8.489 9.216 10.035 10.961 12.012 13.207 14.568 16.123 17.904 19.947

tr U.S. GOVERNMENT PRIMING OFFICE : 1980—0-3 12-896 50

U.S. DEPARTMENT OF COMMERCE

IMational Bureau of Standards

Washington. D C 20234

OFFICIAL BUSINESS

Penalty for Private Use, $300

POSTAGE AND FEES PAID U S. DEPARTMENT OF COMMEHCE

COM-2 1 5 U.S.MAIL

SPECIAL FOURTH-CLASS RATE

BOOK

9721

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WERT

BOOKBINDING

MIODLETOV.N PA MAY 81

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