ASTM E1057-15(2020)e1

Standard Practice for Measuring Internal Rate of Return and Adjusted Internal Rate of Return for Investments in Buildings and Building Systems

SIGNIFICANCE AND USE
5.1 The IRR method has been used traditionally in finance and economics to measure the percentage yield on investment.  
5.1.1 The IRR method is appropriate in most cases for evaluating whether a given building or building system will be economically efficient, that is, whether its time-adjusted benefits will exceed its time-adjusted costs over the period of concern to the decision-maker. However, it has deficiencies that limit its usefulness in choosing among projects competing for a limited budget.  
5.2 The AIRR method is a measure of the overall rate of return that an investor can expect from an investment over a designated study period. It is appropriate both for evaluating whether a given building or building system will be economically efficient and for choosing among alternatives competing for a limited budget.  
5.2.1 The AIRR method overcomes some, but not all, of the deficiencies of the IRR. The AIRR is particularly recommended over the IRR for allocating limited funding among competing projects.
SCOPE
1.1 This practice covers a procedure for calculating and interpreting the internal rate of return (IRR) and adjusted internal rate of return (AIRR) measures in the evaluation of building designs, systems, and equipment.  
1.2 The values stated in inch-pound units are to be regarded as standard. The values given in parentheses are mathematical conversions to SI units that are provided for information only and are not considered standard.  
1.3 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.  
1.4 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.

General Information

• Status: Published
• Publication Date: 31-Mar-2020
• ICS: 91.010.20 - Contractual aspects
• Technical Committee: E06 - Performance of Buildings
• Drafting Committee: E06.81 - Building Economics

Relations

• Replaces: ASTM E1057-15 - Standard Practice for Measuring Internal Rate of Return and Adjusted Internal Rate of Return for Investments in Buildings and Building Systems
• Effective Date: 01-Apr-2020
• Refers: ASTM E964-15(2020)e1 - Standard Practice for Measuring Benefit-to-Cost and Savings-to-Investment Ratios for Buildings and Building Systems
• Effective Date: 01-Apr-2020
• Refers: ASTM E917-17 - Standard Practice for Measuring Life-Cycle Costs of Buildings and Building Systems
• Effective Date: 01-Sep-2017
• Refers: ASTM E1765-16 - Standard Practice for Applying Analytical Hierarchy Process (AHP) to Multiattribute Decision Analysis of Investments Related to Projects, Products, and Processes
• Effective Date: 01-Mar-2016
• Refers: ASTM E1369-15 - Standard Guide for Selecting Techniques for Treating Uncertainty and Risk in the Economic Evaluation of Buildings and Building Systems
• Effective Date: 01-Oct-2015
• Refers: ASTM E1121-15 - Standard Practice for Measuring Payback for Investments in Buildings and Building Systems
• Effective Date: 01-Oct-2015
• Refers: ASTM E1185-15 - Standard Guide for Selecting Economic Methods for Evaluating Investments in Buildings and Building Systems
• Effective Date: 01-Oct-2015
• Refers: ASTM E917-15 - Standard Practice for Measuring Life-Cycle Costs of Buildings and Building Systems
• Effective Date: 01-Oct-2015
• Refers: ASTM E2204-15 - Standard Guide for Summarizing the Economic Impacts of Building-Related Projects
• Effective Date: 01-Aug-2015
• Refers: ASTM E1074-15 - Standard Practice for Measuring Net Benefits and Net Savings for Investments in Buildings and Building Systems
• Effective Date: 01-May-2015
• Refers: ASTM E964-15 - Standard Practice for Measuring Benefit-to-Cost and Savings-to-Investment Ratios for Buildings and Building Systems
• Effective Date: 01-May-2015
• Refers: ASTM E631-15 - Standard Terminology of Building Constructions
• Effective Date: 01-Mar-2015
• Refers: ASTM E631-14 - Standard Terminology of Building Constructions
• Effective Date: 01-Nov-2014
• Refers: ASTM E833-13b - Standard Terminology of Building Economics
• Effective Date: 01-Nov-2013
• Refers: ASTM E833-13a - Standard Terminology of Building Economics
• Effective Date: 15-Oct-2013

Overview

ASTM E1057-15(2020)e1 - "Standard Practice for Measuring Internal Rate of Return and Adjusted Internal Rate of Return for Investments in Buildings and Building Systems" - is an internationally recognized standard developed by ASTM International. This standard provides guidelines and procedures for calculating and interpreting the internal rate of return (IRR) and the adjusted internal rate of return (AIRR) specifically in the context of building designs, systems, and equipment investments.

Understanding economic efficiency for buildings and building systems is critical for owners, investors, and facility managers. Using methods such as IRR and AIRR allows for the assessment of whether time-adjusted benefits of an investment will exceed its time-adjusted costs over a specified period. The AIRR method also facilitates more accurate decision-making when allocating limited funding among competing projects, overcoming some limitations inherent in the traditional IRR method.

Key Topics

• Internal Rate of Return (IRR): A widely used financial metric to determine the percentage yield on investment by equating the present value of expected benefits and costs. Useful for evaluating project feasibility but has limitations when comparing multiple projects with budget constraints.
• Adjusted Internal Rate of Return (AIRR): Measures the overall return, accounting for earnings from reinvested receipts over the study period. Especially recommended for comparing alternative projects when resources are limited.
• Comparison with Minimum Acceptable Rate of Return (MARR): Both IRR and AIRR results are compared to the investor’s benchmark rate to gauge economic viability.
• Limitations and Best Practices: AIRR is preferred for budget allocation decisions, while both IRR and AIRR should be applied carefully to avoid misleading conclusions, especially with design sizing or varying cash flow profiles. The importance of incremental analysis is emphasized.
• Analysis and Reporting: The standard outlines steps for economic evaluation, including precise identification of objectives, assumptions, data compilation, calculation processes, and comprehensive reporting guided by ASTM protocols.

Applications

ASTM E1057-15(2020)e1 is indispensable for professionals involved in the economic analysis of buildings and building systems, offering practical benefits for:

• Project Feasibility Studies: Determine whether new building projects, renovations, or equipment upgrades are financially justified over intended timeframes.
• Capital Budget Allocation: Prioritize projects and allocate limited funding based on comparative AIRR results, ensuring maximum economic benefit from investment portfolios.
• Design and Sizing Decisions: Support decisions about the extent of investment in systems (such as insulation thickness or energy efficiency measures) using incremental IRR or AIRR methods.
• After-Tax Analyses: Adjust for tax effects, grants, and subsidies to evaluate actual returns from building investments.
• Comprehensive Reporting: Use the prescribed format for clear communication of analysis, results, and recommendations to stakeholders, including non-technical audiences.
• Risk Assessment: Incorporate uncertainty analysis and risk considerations in investment decision-making using referenced ASTM guides.

Related Standards

To ensure a robust economic evaluation process, ASTM E1057-15(2020)e1 references and aligns with the following related standards:

• ASTM E917: Practice for Measuring Life-Cycle Costs of Buildings and Building Systems
• ASTM E964: Practice for Measuring Benefit-to-Cost and Savings-to-Investment Ratios
• ASTM E1074: Practice for Measuring Net Benefits and Net Savings
• ASTM E1121: Practice for Measuring Payback for Investments
• ASTM E1185: Guide for Selecting Economic Methods for Evaluating Investments
• ASTM E1369: Guide for Treating Uncertainty and Risk in Economic Evaluations
• ASTM E1765: Analytical Hierarchy Process for Multiattribute Decision Analysis
• ASTM E1946: Practice for Measuring Cost Risk
• ASTM E2204: Guide for Summarizing Economic Impacts of Building-Related Projects

These referenced documents provide further detail on life-cycle cost analysis, benefit-to-cost ratios, payback calculations, uncertainty, and risk analysis, supporting a holistic approach to economic evaluation in the built environment.

Keywords: internal rate of return, IRR, adjusted internal rate of return, AIRR, building economics, investment analysis, life-cycle cost, economic evaluation, capital budgeting, ASTM standards.

Standards Content (Sample)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
ϵ1
Designation: E1057 − 15 (Reapproved 2020)
Standard Practice for
Measuring Internal Rate of Return and Adjusted Internal
Rate of Return for Investments in Buildings and Building
Systems
This standard is issued under the fixed designation E1057; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
ε NOTE—Adjunct title and stock number in 2.2 were updated editorially in April 2020.
INTRODUCTION
The internal rate-of-return (IRR) and adjusted internal rate-of-return (AIRR) methods are members
of a family of economic evaluation methods that provide measures of economic performance of an
investmentovertime.Othermethodsinthisfamilyofevaluationmethodsarelife-cyclecostanalysis,
net benefits and net savings analysis, benefit-to-cost and savings-to-investment ratio analysis, and
payback analysis.
The IRR andAIRR methods are the topic of a single standard practice because they both measure
economic performance as a compound yield on investment.The IRR is the compound rate of interest
that,whenappliedasadiscountratetoaproject’sstreamofdollarbenefitsandcosts,willequatethem.
The AIRR is the overall yield taking into account earnings on receipts reinvested to the end of the
study period.The IRR orAIRR is compared against the investor’s minimum acceptable rate of return
(MARR), and the investment is considered economically attractive if the calculated yield exceeds the
MARR. If an investment entails an initial outlay and a single receipt at the end of the study period,
there is no difference between the IRR and the AIRR. But if cash flows occur over multiple time
periods, the two will normally be different. This arises because theAIRR includes in its measure the
return on reinvestment of receipts, whereas the IRR does not.
The AIRR is recommended for most applications in which a measure of yield is desired. Caution
is recommended in applying either measure, however, because problems arise under certain
conditions.
1. Scope responsibility of the user of this standard to establish appro-
priate safety, health, and environmental practices and deter-
1.1 This practice covers a procedure for calculating and
mine the applicability of regulatory limitations prior to use.
interpreting the internal rate of return (IRR) and adjusted
1.4 This international standard was developed in accor-
internal rate of return (AIRR) measures in the evaluation of
dance with internationally recognized principles on standard-
building designs, systems, and equipment.
ization established in the Decision on Principles for the
1.2 Thevaluesstatedininch-poundunitsaretoberegarded
Development of International Standards, Guides and Recom-
as standard. The values given in parentheses are mathematical
mendations issued by the World Trade Organization Technical
conversions to SI units that are provided for information only
Barriers to Trade (TBT) Committee.
and are not considered standard.
2. Referenced Documents
1.3 This standard does not purport to address all of the
safety concerns, if any, associated with its use. It is the
2.1 ASTM Standards:
E631Terminology of Building Constructions
E833Terminology of Building Economics
This practice is under the jurisdiction of ASTM Committee E06 on Perfor-
mance of Buildings and is the direct responsibility of Subcommittee E06.81 on
Building Economics. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Current edition approved April 1, 2020. Published May 2020. Originally contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
approved in 1985. Last previous edition approved in 2015 as E1057-15. DOI: Standards volume information, refer to the standard’s Document Summary page on
10.1520/E1057-15R20E01. the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
ϵ1
E1057 − 15 (2020)
E917Practice for Measuring Life-Cycle Costs of Buildings 4.1.9 Section 9, IRR Calculation—Describes the third step,
and Building Systems performing calculations, as it applies to the IRR.
E964Practice for Measuring Benefit-to-Cost and Savings- 4.1.10 Section 10, AIRR Calculation—Describes the third
to-Investment Ratios for Buildings and Building Systems step, performing calculations, as it applies to the AIRR.
E1074Practice for Measuring Net Benefits and Net Savings 4.1.11 Section 11, Choosing Between the IRR and AIRR—
for Investments in Buildings and Building Systems Discusses how to choose between the IRR and the AIRR.
E1121Practice for Measuring Payback for Investments in 4.1.12 Section 12, Limitations—Discusses limitations and
Buildings and Building Systems shortcomings of the IRR and AIRR.
E1185Guide for Selecting Economic Methods for Evaluat- 4.1.13 Section 13, Analysis of IRR or AIRR Results and the
ing Investments in Buildings and Building Systems Decision—Discussesthedecisioncriterionandthetreatmentof
E1369Guide for Selecting Techniques for Treating Uncer- uncertainty, risk, and unqualified effects.
tainty and Risk in the Economic Evaluation of Buildings 4.1.14 Section 14, Applications—Describes the types of
and Building Systems decisions to which the IRR and AIRR are applicable.
E1765Practice for Applying Analytical Hierarchy Process 4.1.15 Section 15, Report—Identifies information that shall
(AHP) to Multiattribute DecisionAnalysis of Investments be included in a report of an IRR or AIRR application.
Related to Projects, Products, and Processes
5. Significance and Use
E1946Practice for Measuring Cost Risk of Buildings and
Building Systems and Other Constructed Projects
5.1 The IRR method has been used traditionally in finance
E2204Guide for Summarizing the Economic Impacts of
and economics to measure the percentage yield on investment.
Building-Related Projects
5.1.1 The IRR method is appropriate in most cases for
2.2 ASTM Adjunct:
evaluating whether a given building or building system will be
Discount Factor Tables - Adjunct to E917 Practice for
economically efficient, that is, whether its time-adjusted ben-
Measuring Life-Cycle Costs of Buildings and Building
efits will exceed its time-adjusted costs over the period of
Systems - Includes Excel and PDF Files
concern to the decision-maker. However, it has deficiencies
that limit its usefulness in choosing among projects competing
3. Terminology
for a limited budget.
3.1 Definitions—For definitions of general terms related to
5.2 The AIRR method is a measure of the overall rate of
building construction used in this practice, refer to Terminol-
return that an investor can expect from an investment over a
ogyE631;andforgeneraltermsrelatedtobuildingeconomics,
designated study period. It is appropriate both for evaluating
refer to Terminology E833.
whether a given building or building system will be economi-
cally efficient and for choosing among alternatives competing
4. Summary of Practice
for a limited budget.
4.1 This practice is organized as follows:
5.2.1 TheAIRRmethodovercomessome,butnotall,ofthe
4.1.1 Section 1, Scope—Identifies coverage.
deficiencies of the IRR. The AIRR is particularly recom-
4.1.2 Section 2, Applicable Documents—Lists ASTM stan-
mended over the IRR for allocating limited funding among
dards that are referenced.
competing projects.
4.1.3 Section 3, Terminology—Addresses definitions of
terms. 6. Procedure
4.1.4 Section 4, Summary of Practice—Outlines the con-
6.1 The recommended steps for applying the IRR or the
tents.
AIRR method to an investment decision are summarized as
4.1.5 Section 5, Significance and Use—Explains the rel-
follows:
evance of the IRR and AIRR and indicates their appropriate
6.1.1 Identify objectives, constraints, and alternatives.
uses.
6.1.2 Compile data and establish assumptions.
4.1.6 Section 6, Procedure—Summarizes the steps in IRR
6.1.3 Compute IRR orAIRR based on a comparison of two
and AIRR analysis.
alternatives (one of which may be to do nothing).
4.1.7 Section 7, Objectives, Constraints, and Alternatives—
6.1.4 Compare the computed IRR or AIRR against the
Discusses the first step in an analysis, that is, the identification
MARR to determine the acceptability of the alternative with
of the objectives of the analysis, any constraints that must be
the higher investment cost.
taken into account in finding a solution, and technically
6.1.5 Ifalimitedbudgetistobeallocatedamongcompeting
feasible project alternatives.
alternatives, select alternatives in descending order of their
4.1.8 Section 8, Data and Assumptions—Discusses the sec-
IRR or AIRR measures until the budget is exhausted.
ondstepinananalysis;thatisthedataandassumptionsthatare
6.1.6 Report the results.
typically required for calculating the IRR and AIRR, and, in
particular, the requirement of the AIRR for specification of a
7. Objectives, Constraints, and Alternatives
reinvestment rate.
7.1 Specify clearly the objective of the economic analysis.
7.1.1 Suppose, for example, an individual or organization
hasfundsonhandtoinvestinrealestateprojects.Theproblem
Available from ASTM International Headquarters. Order Adjunct No.
ADJE091717-EA. Original adjunct produced in 1984.Adjunct last revised in 2003. is which projects to choose from potential candidates. The
ϵ1
E1057 − 15 (2020)
N
objectiveoftheeconomicanalysisinthiscaseistoidentifythe
t
¯
PVNB 5 ~B 2 C !/ 11i* 50 (1)
~ !
( t t
project or set of projects within the budget that is expected to
t50
maximize profits over the long run.
where:
7.2 Identify any constraints that narrow the field of candi-
PVNB = present value of net benefits (or, if applied to a
dates.
cost-reducing investment, present value of net
7.2.1 Constraints, for example, might include a budget of
savings (PVNS)),
$1000000; a geographical limitation to buildings located
N = number of discounting periods in the study period,
within 100 km from downtown; and a strong preference for
B = dollar value of benefits in period t for the building
t
nonresidential property.
or system evaluated less the counterpart benefits in
period t for the mutually exclusive alternative
7.3 Identify feasible alternatives.
against which it is compared,
7.3.1 Feasible alternatives include an office building in the
¯
C = dollar costs, including investment costs, in period t
suburbs costing $1000000; convenience shopping strips in t
for the building or system evaluated less the
nearby towns costing a total of $900000; two medical/dental
counterpart costs in period t for the mutually
offices costing $500000 each; and a $1000000 investment
exclusive alternative against which it is compared,
share in a downtown shopping complex.
and
i* = interest rate for which PVNB=0, that is, the IRR
8. Data and Assumption
measure expressed as a decimal.
8.1 To calculate the IRR or AIRR, data are needed.
9.2 An algebraic solution of i* is not possible with Eq 1 for
8.2 Benefit and cost data that are often relevant when
allvaluesofN.Useacomputerprogramwithbuilt-informulas
calculating the IRR or AIRR are revenues, resale or salvage
to calculate IRR and AIRR. Or, use a manual approach to
value, subsidies (for example, grants), and costs of planning,
approximate the IRR such as the trial-and-error approach, the
design,engineering,construction,purchase,installation,opera-
graphical approach, and an approach that uses simple payback
tion and maintenance, utilities, and repairs and replacement.
and uniform present value (UPV) factor tables. (See Practice
E1121 for a description of payback and the Adjunct on
8.3 The time of occurrence of each benefit and cost is also
Discount Factor Tables for UPV factors.)
needed.
9.2.1 Trial and Error Solution:
8.4 Taxes such as tax credits, property taxes, and income
9.2.1.1 The trial-and-error approach to calculating the IRR
taxes are also often relevant because they affect benefits and
entails choosing a trial rate of interest that is expected
costs. If benefits and costs are adjusted for taxes, the IRR or
approximately to balance benefits and costs over the project
AIRR measure gives the after-taxrate of return.
studyperiod.Thenpresentvaluecalculationsaremadeforthat
8.5 If the terms of financing are unique to each alternative,
trial rate. (For an illustration of discounting calculations, see
financingcosts(andassociatedtaxeffects)shouldalsobetaken
Practice E917.) If the PVNB is zero, then the trial rate is the
into account.
solution value of the internal rate of return. If the PVNB is
negative,thetrialrateistoohigh,andasecond,lowertrialrate
8.6 Choose a minimum acceptable rate of return (MARR)
is then used. If the PVNB is positive for the original trial rate,
for comparison against the calculated IRR or AIRR.
then the IRR is higher than the trial rate, and a second, higher
8.6.1 TheappropriateMARRindicatestheinvestor’soppor-
trial rate is used. When two trial rates are found such that one
tunity cost of foregoing the return on the next best investment
yields a PVNB greater than zero and the other a PVNB less
opportunity in order to invest in the project in question.
than zero, the IRR lies between those rates and can be
8.7 If the AIRR is used, a reinvestment rate is needed.
approximated by interpolation, provided the investment has a
8.7.1 The reinvestment rate is usually set equal to the
unique IRR. Considerable time is saved in the trial-and-error
MARR; hence, it equals the discount rate. This is because the
approach if the first trial rate is close to the true rate. One
reinvestmentrateisanindicatoroffutureopportunitycost,and
approach is to start with the MARR as the trial rate. If the
that is also the purpose of the discount rate. Setting the
PVNB is negative with the MARR, then the project is not
reinvestment rate and the discount rate equal makes the
economically feasible, and no further calculations are neces-
reinvestment rate assumption in the AIRR method consistent
sary.IfthePVNBispositive,thenselecthighertrialratesinan
withthereinvestmentrateassumptionthatisimplicitinthenet
attempt to bound the true rate.
benefits (net savings) method (Practice E1074).
9.2.1.2 The UPV factor tables are useful in finding a trial
rate. The first step is to sum the undiscounted cash flows (not
9. IRR Calculation
including the initial cost) and divide the sum by the number of
9.1 The IRR is the compound rate of interest that, when years in the study period (excluding any planning/design/
used to discount a project’s cash flows, will reduce the present
construction period) to obtain an average annual cash flow.
value of net benefits (PVNB) to zero. (See Practice E1074 for Then divide the initial project cost by the average to obtain a
adiscussionofhowtocomputethePVNB.)Thesolutionvalue
rough estimate of simple payback (SPB).The second step is to
of i*in Eq 1 is the IRR. It is computed as a decimal, then search the UPV discount factor tables in the row that corre-
expressed as a percent.
sponds to the study period for the UPVfactor that is closest to
9.1.1 Find the value of i* for which: the estimated SPB. (Again exclude any years in the planning/
ϵ1
E1057 − 15 (2020)
design/construction period.) The rate that appears at the top of solution IRR more closely than does the linear interpolation
the column in which the UPV factor is found is a promising illustrated earlier with the trial-and-error approach, provided
trial rate. The more uniform the annual cash flow, the more
that enough PVNB points above and below the horizontal axis
likely that this trial rate will be close to the solution rate. are plotted to form the curve accurately. However, if two trial
9.2.1.3 Table 1 illustrates the trial-and-error approach for
rates are chosen very close to the true value, the linear
calculating the IRR for an initial investment that yields an
interpolation approach can also approximate the true value
uneven yearly cash flow over four years. Columns 2 and 3 list
closely.
thedollarvaluesofbenefitsandcoststhataccrueineachofthe
9.2.3 Simple Payback-UPV Factor Table Solution:
four years, and Column 4 shows the net cash flow for each of
9.2.3.1 AthirdtechniqueforcalculatingtheIRRworksonly
those years, including the initial investment.
when the annual net cash flows are constant or change at a
9.2.1.4 From inspecting Column 4 in Table 1, one might
constant rate. It works best if there is no planning/design/
expect a relatively high return over four years. Using the
construction period. The procedure is first to compute simple
approach described in 9.2.1.2 to select a trial rate, the calcu-
payback (SPB). The next step is to search through the UPV or
lated UPV value for four years corresponds in the Adjunct
UPV* factors in the row corresponding to the study period for
discount tables most closely to a rate of 25%. Multiplying
the factors that most closely bracket the value of the computed
yearly net cash flows by single present value (SPV) factors for
SPB. (UPV* factors, available in the adjunct on Discount
each year based on a 25% discount rate (Column 5) converts
Factors, are identified by an asterisk, indicating that annual net
them to equivalent present values (Column 6). Summing the
cash flows change at a constant rate.) Then see what discount
values in Column 6 produces a PVNB of−$415. Since the
rates those factors represent. Linear interpolation using the
PVNB is less than zero, the IRR must be lower than the 25%
bracketing UPV or UPV* factors yields an approximation of
trial rate.Therefore another, lower rate is chosen, and the SPV
the IRR. Note in this approach that the IRR is interpolated
factors for the lower rate are multiplied times yearly net cash
directly from the UPV or UPV* factor values; whereas, in the
flows. Using a second trial rate of 22% (as illustrated in
trial-and-error method described in 9.2.1, the IRR is interpo-
Columns 7 and 8) yields a positive PVNB of $181. Therefore,
latedfromPVNBfigures.In9.2.1theUPVtablesareusedonly
the IRR must lie between 22 and 25%. Linear interpolation
to find the first trial rate for calculating the PVNB.
yields an IRR of 22.9% as follows:
9.2.3.2 Eq 3 shows how to calculate SPB, which is the first
$181 2$0
IRR 522%1 25%222% 522.9% (2)
S D ~ ! step in the procedure.
$1811$415
C
o
9.2.2 Graphical Solution:
SPB 5 (3)
¯
~ !
B 2 C
9.2.2.1 AnotherapproachtoapproximatingtheIRRistouse
a graphical technique. The profile of the PVNB for a given
where:
investment is plotted for a range of discount rates. The IRR is
SPB = simple payback time,
approximately that rate where the PVNB curve intersects the
C = the initial project costs as of the base time, and
o
discount rate axis, that is, where the PVNB is zero. Fig. 1
¯
= constant annual net cash flow, or initial annual
~B2C!
illustrates a graphical solution of the IRR for the project
valueofanetcashflowchangingataconstantrate.
described in Table 2.
9.2.3.3 Thisapproachisillustratedwiththefollowingprob-
9.2.2.2 Given a discount rate of zero, the PVNB is the
lem. Find the IRR for a project with a one-time initial cost of
arithmetic sum of the net cash flows over time (that is, $1300
$1000andbenefitsthataccrueinauniformstreamof$200per
for the project described in Table 2). Therefore the function
year for 16 years. Dividing the one-time initial cost of $1000
intersects the vertical axis at $1300. Using the discount rates
by the constant annual net cash flow of $200 yields an SPB of
labeledonthehorizontalaxisofFig.1astrialrates,thePVNB
5. By searching through the row where N=16 in the UPV
values are calculated as shown in Table 2 and then plotted in
columns of the adjunct factor tables, two UPV factors are
Fig. 1.
9.2.2.3 An IRR of 27.2% is found by visual inspection of foundthatmostcloselybracketthevalue5.Onefactoris5.162
foradiscountrateof18%,andtheotheris4.938foradiscount
theintersectionoftheprofileofPVNBwiththehorizontalaxis.
Note that the graphical method usually approximates the rate of 19%. Having bracketed the IRR between 18% and
TABLE 1 Trial-and-Error Solution for Internal Rate of Return
(1) (2) (3) (4) (5) (6) (7) (8)
Net
SPV PVNB SPV PVNB
Year Benefits Costs Cash Flow
Factor at 25 % Factor at 22 %
¯ ¯
(t) (B) (C) (B − C)
t t t t
for i=25% (6) = (4) × (5) for i=22% (8) = (4) × (7)
(4)=(2)−(3)
0 0 $10 000 $−10 000 1.000 $−10 000 1.000 $−10 000
1 $ 4 000 3 000 1 000 0.8000 800 0.8197 820
2 11 500 4 500 7 000 0.6400 4 480 0.6719 4 703
3 10 000 4 000 6 000 0.5126 3 076 0.5507 3 304
4 8 000 5 000 3 000 0.4096 1 229 0.4514 1 354
Total $7 000 $−415 $181
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E1057 − 15 (2020)
FIG. 1 Graphical Solution of IRR
TABLE 2 Data for Graphic Solution of IRR
Net Trial Rates
Year Benefits Costs Cash
5% 10% 15% 20% 25% 30% 35%
¯
(t) (B) (C) Flow
t t
A
¯
SPV PVNB SPV PVNB SPV PVNB SPV PVNB SPV PVNB SPV PVNB SPV PVNB
(B − C)
t t
0 0 $−2200 $−2200 1.000 $−2200 1.000 $−2200 1.000 $−2200 1.000 $−2200 1.000 $−2200 1.000 $−2200 1.000 $−2200
1 $1000 0 1000 0.9524 952 0.9091 909 0.8696 870 0.8333 833 0.8000 800 0.7692 769 0.7407 741
2 1500 0 1500 0.9070 1361 0.8264 1240 0.7561 1134 0.6944 1042 0.6400 960 0.5917 888 0.5487 823
3 1000 0 1000 0.8638 864 0.7513 751 0.6575 658 0.5787 579 0.5126 513 0.4552 455 0.4064 406
$1300 $977 $700 $462 $254 $73 $−88 $−230
A
TheSPVfactorsfor30 %and35 %werecalculatedfromtheequation SPV5 becausefactorsfor30 %and35 %wereunavailableintheAdjunctdiscountfactor
a
s11id
tables.
19%, a single value of 18.7% is approximated through linear where:
interpolation as follows:
Ĉ = dollar costs, excluding investment costs, in period t for
t
the building or system evaluated less counterpart costs
5.162 25.000
IRR 518%1 19%218% 518.7% (4)
~ !
inperiodtforthemutuallyexclusivealternativeagainst
5.162 24.938
which it is compared,
10. AIRR Calculation
r = prescribed rate of return on reinvestment of cash flows,
I = investment costs in period t on which return is to be
t
10.1 The solution value of ¯ı in Eq 5 is the AIRR.
maximized,
N N
ˆ N2t
~B 2 C !~11r!
t t
t
5 I / 11r (5)
~ !
N
( ( t
11¯ı
~ !
t50 t50
ϵ1
E1057 − 15 (2020)
the final year is $1000. The terminal value (combined future
¯ı = the interest rate that equates the two sides of the
values of the positive net cash flows) is $4048. Solving the
equation; that is, the AIRR measure expressed as a
problem in Table 3 using Eq 6 yields the following:
decimal, and
B = is as previously defined. 1/3
t
$4,048
AIRR521.01 50.225or22.5% (7)
S D
$2,200
10.2 Net cash flows (B −Ĉ) are carried forward (com-
t t
pounded)atthespecifiedreinvestmentrate(r)totheendofthe
10.9 Eq8,whichisequivalenttoEq6,canbeusedwhenthe
study period (end of the Nth period) and summed. Future
reinvestment rate is constant from year to year and the PVNB
investment costs are discounted to present value using r as the
and PVI have already been computed.
discount rate. The value of i that discounts the resulting
1/N
PVNB
terminal value of net cash flows to a present value equal to
AIRR521.01~11r! 11 (8)
S D
PVI
present value investment costs is the AIRR.
10.3 The AIRR equals the IRR if the reinvestment rate, r,
10.9.1 Nothavingtoderivethe TVsimplifiesthecalculation
equals the IRR, that is, the solution value of i*. If, however, r of the AIRR. (Note that r is a constant reinvestment rate and
islessthan i*,theAIRRislessthantheIRR;andif risgreater
that the PVNB and PVI have been computed with a discount
than i*, the AIRR is greater than the IRR.
rate equal to r.)
10.4 Thesolutionvalueof¯ıinEq5,expressedasadecimal,
10.10 Eq 9 can be used when the reinvestment rate is
is the AIRR. It can be solved for directly if the equation is
constant from year to year and the savings-to-investment ratio
rearranged as shown in Eq 6.
(SIR) has already been computed (Practice E964). A compre-
1/N
hensiveexamplelinkingtheSIRmethodandtheAIRRmethod
AIRR521.01 TV/PVI (6)
~ !
applied to a building economics problem is provided in
where:
Appendix X1.
N
TV =
N2t
ˆ
1/N
~B 2C ! 11r , the terminal (future) value at
~ !
( AIRR521.01 11r SIR (9)
t t ~ !~ !
t50
the end of the study period of net cash flows
10.11 The appropriate value of the reinvestment rate, r,is
excluding investment costs, and
normally best approximated by the MARR, or discount rate,
N
PVI =
t
I / 11r , the present value of investment costs. rather than the return on the original investment. Setting r
~ !
( t
t50
higher than the discount rate suggests that the discount rate is
10.5 Once the TV and PVI are computed, the AIRR can be
too low, that is, it does not adequately reflect the next best
solved for directly by substituting TV, PVI, and N into Eq 6.
investment opportunity. Setting r lower than the discount rate
10.6 To illustrate the calculation of theAIRR, Eq 6 is used.
suggeststhatthediscountrateistoohightoreflectthenextbest
In 9.2.2, an approximate IRR of 27.2% was calculated for the
investment opportunity.Thus, the discount rate is normally the
investment data in Table 2. Now anAIRR is computed for the
appropriate value to substitute for r in using Eq 5, Eq 6, Eq 8,
modified case where the reinvestment rate is lower than
or Eq 9 to calculate the AIRR.
27.2%.
10.7 Table 3 illustrates the calculation of terminal values 11. Choosing Between the IRR and AIRR
from a reinvestment of the first and second years’cash flow at
11.1 ControversypersistsoverwhethertheIRRortheAIRR
15%. Note that net cash flows are identical to those for the
is a more accurate measure of an investment’s return. This
investment described in Table 2.
controversy centers on how reinvested earnings over the study
10.8 The future value of $1000 earning 15% for two years
period are treated by the two methods. The IRR does not
is $1323, the future value of $1500 earning 15% for one year
explicitly take into account the earnings on an investment’s
is $1725, and the future value of $1000 received at the end of
benefit stream or cash payouts during the study period. The
AIRR does.
The National Institute of Standards and Technology Building Life-Cycle Cost
11.2 By incorporating the expected earnings of reinvested
(BLCC) Computer Program helps users calculate measures of worth for buildings
net cash flows, the AIRR provides a measure of the rate of
and building components that are consistent withASTM standards. The program is
return based on the investor’s expected position at the end of
downloadable from: http://www.eere.energy.gov/femp/information/download_
the study period as compared with the initial position. The
blcc.html.
AIRR is a more accurate guide than the IRR for selecting that
set of projects that will maximize aggregate PVNB over the
TABLE 3 Calculation of Terminal Value
study period.
Net
Year Reinvestment SCA Terminal
Cash Flow
(t) Rate (r) Factor Value
11.3 For simple accept/reject decisions, either the IRR or
¯
(B − C)
t t
A
theAIRR will generally suffice, except in cases where there is
0 $−2200
1 1000
...