BASIC EQUATIONS FOR THE SIX METHODS OF ANALYSIS
The equations to be given for solving for the numerical solution of the six methods of analysis for
economy are written on the concept of the cash-flows diagrams wherein the outward flows of
investments and expenses (disbursements) are negative and-the inward flows of income or
benefits are positive.
NOTATION SCHEME
The factors mid terms in the several equations correspond to the following notations in addition
to those gives in Chapter 6 for compound interest term:
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EUAC
PWOC
EUANR
NPV
B/C
ROR
MARR
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T
K
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KG
KE
Equivalent uniform annual cost
Present worth of costs and expenses
Equivalent uniform annual net return
Net present value
Benefit/cost ratio
Rate of return
Minimum attractive rate of return
Original, or initial, investment, or equivalent investment at time
zero including discounted investments subsequent to time zero
Terminal value at end of analysis period
Total uniform annual expense administration A. Traffic services
and highway operations J, and highway _maintenance M
Equivalent uniform annual K when K grows as a uniform gradient
Equivalent uniform annual K when K grows exponentially
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U
Uniform annual road-user costs, exclusive of road-user taxes.
inclusive of travel time value, and accident costs when so
designated
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UD
Equivalent uniform annual road-user benefits, being the difference
in road-user costs between a pair of alternative.
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UG
Equivalent uniform annual road-user costs under a gradient
growth of traffic volume
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UE
Equivalent uniform annual road-user costs under an exponential
growth of traffic volume
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UT
Equivalent uniform annual road-user tax payment or tax revenue
from road users.
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R
Uniform annual gross income from rules revenue, receipts or their
equivalent. or gross benefits. R is inclusive of return of
investment (depreciation) and return on investment (net profit).
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RG & RE
Equivalent uniform annual R when either a gradient or exponential
increase is present
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RD
Difference in the equivalent uniform annual receipts of a pair of
alternatives.
EQUATIONS FOR THE SIX METHOODS OF SOLUTION FOR
ECONOMY
The six methods of solution for the economy of alternative proposals may be
written is more than one form and by using different compound interest factors.
Further, the particular cash flows involves will call for different compositions of
the basic equation. For instance more than one investment may be involved.
Annual expenses and road-user costs may be stated in terms gradient or
exponential increases (or decreases) and there may or may not be a terminal
value.
following equations do not include any capital investment capital investments
put in place subsequent to the initial. or first, investment at n = 0. Subsequent
capital investments, such is for stage construction. may be easily handled in
the solutions by treating them in the same manner is the initial investment, just
taking care to see that the money sums involved .are transferred to the
same point in time. This transfer is accomplished by applying the appropriate
vestcharge factor.
The following twelve equations illustrate the general forms of the equations to
use: where the gradient and exponential factors are shown it is for the
purpose of illustrating one possible form oft bee equation.
CHARACTERISTICS & LIMITATIONS
OF THE
METHODS OF ECONOMIC ANALYSIS
Economic analysis of proposed investments
/improvements deals with the subject on the basis that
economic analysis (analysis to define the degree of
economic feasibility) is the only application of economic
analysis. Yet test up to the alternative in design (project
formulation) is the object more often than in economic
evaluation.
Although the principles of analysis hold for both
objectives, the required data, quantification of factors
& choice of method often differ. Many of the details in
designs when the alternatives are considered offer a
choice only on the basis of cost, there being no income
or benefits applicable except the diff is costs, Ex. Costs
materials, geometrics, density, basic designs & methods
& requirements of contract whereas vehicular traffic is
affected by the choice of designs.
1. UNIVALENT UNIFORM ANNUAL COST METHOD
 EUAC method requires as input data.
1. The investment costs-- initial & subsequent
2. Year by year annual expenses for overheads, operations, maintenance &
production.
 The item of cash inflows, or benefits is not considered.
 Terminal values of the investments are included, but although +ve in sign, are
not considered as an income or benefit, but a reduction in capital cost.
 The method is popular because of its simplicity & because most people
understand the meani9ng of av. Annual costs.
 This method is applied mainly to those project formulations proposals for
which there is no ready or usable way to establish monetary measure of
benefits.
 The method is widely used in industry as a means of selecting preferable m/c
process, or design feature.
THE FACTOR OF ECONOMIC EVALUATION
• The method can not be used to measure the economic desirability of
proposals. The difference in annual ,cost does not mean incremental
investment & annual expenses contributing to the annual cost diff are
economic warrants.
• EUAC method is consistent & reliable as compared to ROR method & B?C
method.
Level - of – service factor
• For mutually exclusive alternatives, they should be compared under
conditions of equality of service performed.
2. PRESENT-WORTH-OF-COST METHOD
 EUAC & PWOC methods are numerically equal by multiplying either
answer by the appropriate compound interest factor.
 The present-worth-cost-method is not as well understood as the EUAC
method, especially by public officials.
 The discussions & explanations given for EUAC hold good for PWOC
also.
3. EQUIVALENT UNIFORM ANNUAL NET RETURN METHOD
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EUANR method has the some general characteristics as does the EUAC
method.
Difference: The final answer is in terms of net return rather than annual
gross costs. Economy or productivity can be compared.
Since min. attractive rate of return [MARR] is used in the calculations, the
final EUANR is a direct measure of economic desirability.
Inclusion of the benefits or income factor limits application of the method to
those proposal producing annual income.
The method is applicable to alternatively, each separately, and the comparison
is made by the magnitude of the answer.
The method is equally applicable to economic evaluation & to project
formulation.
The method is reliable & consistent.
Since it measures consequential benefits against costs, L-O-S rendered is not a
critical factor.
EUANR & net present value methods are equivalents. They merely express
answer in different terms.
4. NET PRESENT VALUE METHOD
• The NPV & EUANR methods may be made numerically identical by
multiplying by appropriate compound interest factor.
• The NPV method favorite of economists. It is highly useful method in
business application. Where in there results a stream of income rupees
from an investment in production property or processes – at assumed
cost of capital rate or MARR.
GENERAL APPLICABILITY
• NPV method is applicable to single proposal wherein other alternative
is do nothing.
• Reversals of the net accumulated cash flows will lead to misleading
conclusions with NPV methods, PW methods & ROR method.
• With NPV method it is no use to incremental analysis when each alt.
has its individual income or benefits.
• NPV method is applicable to pairs of alternatives in the same way that
EUANR method must be applied.
GENERAL LIMITATION
• It can not be applied to single alternatives except when the benefits or
incomes can be estimated.
• It does not express the profitability as a rate (it is expressed as a lump
sum for the period of analysis) so it is not as enlightening as ROR
method.
• The L-O-S in NPV method is not factor in comparing the alternatives,
but in interpreting the results.
• The method may be used either for project evaluation or project
formulation.
• In project formulation care should be exercised in comparing the
alternatives which do not render the same service.
APPLICATION TO HIGHWAY
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As a whole, the NPV method has no particular advantage in economy studies
of highway. It can be used for estimating the economy of alternatives which
reduce the motor vehicle operating costs.
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The decision to use the NPV is preference to the B/C method or ROR method
is simply what qualitative form is the answer more meaningful to decision
maker & whether the favorable consequences are priceable.
5. BENEFIT COST RATIO METHOD
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The B/C ratio method is one found mainly in public works field, though it is as
applicable to industrial application. It has been used since 1952 in highway
field.
CHARACTERISTICS OF METHOD
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The data require for b/C ratio method. Here only the answer is an abstract
number.
In application, the B/C ratio must have a measure of benefits of particular or
alternative being analyzed.
B/C ratio method is applicable to a group of more than two alternatives & to
both project economic evaluation & to project formulation.
It is particularly adopted for to the public work on the basis that the method
places emphasis on the benefits received instead of on profitability & because
the benefit may accrue to a group different from investors.
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POSITION OF ANNUAL EXPENSE FACTOR
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The basic ratio have been used:Annual road user benefits-annual highway maint. expense
1. B/C Ratio= --------------------------------------------------------------------------Annual highway capital cost
Annual road-user benefits
2. B/C Ratio= ---------------------------------------------------------------------------annual highway capital costs plus annul highway maint.
expense
The second one has been popular only because of AASHTO report on Road User
Benefits in this form.
• Numerator Vs Denominator: The logical consideration
1. Conceptual logic-what is being sought.
2. Normal handling of expense & income in profit
OBJECTIVE OF ANALYSIS:
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All six methods of economic analysis have as their basic objective the search
for the efficient use of money.
• EUAC & PWOC methods compare alternatives on total outward flow of cash,
• But NPV , B/C ratio & ROR methods compare alternatives on the rate of
inflow of returns or benefits.
• The real objectives of the outward flow of cash is the investment or initial cost.
• The real objective of three methods is to see whether the original & subsequent
investments, will return their investments with sufficient vest charge to make
the investment economically desirable.
• The NPV,B/C & ROR of any pair of alternatives in terms of objective, They
are best expressed by using the investment cost as the basis upon which the net
benefit is to be measured. In terms of B/C ratio this means that the ratio
should be expressed as
Gross benefit - Annual expenses
-----------------------------------------Investment
Putting, the annual maintenance expense in the denominator is illogical &
conceptually unsound in terms of objective.
• Cost accounting procedure: In ordinary cost account procedure &
profit & loss statement the concept most often used is that operating
expenses is a deduction from income before gross profits are state or
Income- Expense = Profit
• Consistency with other Method of analysis
Normal eqns. of NPV& ROR utilize the annual maintenance expense
as a deduction to capital costs.
Since all the six method indicate the same alternative as the first choice,
it is logical to handle the terms in all the eqns. of solution in
consistent manner.
Simplified ROR eqns.
0 = - I (CR-i-n)+UP-K
CR-i-n = UP-K
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And it is logical to use the same form for B/C rates.
CONSISTENCY WITH OBJECTIVES OF ANALYSIS
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Based on the objective that the analysis is to indicate the profitability of the
invested capital, the annual expenses must be place in the denominator.
• But large annual expenses in proportion to the invested capital lead to
misleading ratio when the annual expenses place in the denominator.
Therefore logically it can be concluded that in B/C ratio method of analysis ,the
annual expenses of the highway maintenance & operation should be placed in
the denominator.
POSITION OF THE TERMINAL VALUE FACTOR
The terminal value is just a logical benefit as in any reduction in road-user costs.
Further the terminal value is a logically +ve benefit that is annual highway
expenses a –ve benefit.
Logical analysis will lead to the conclusion that terminal value is reduction from
capital cost and not to treat it as such leads to overstating capital costs upon
which the degree of profitability is measured.
6. RATE OF RETURN METHOD
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In other method a vest charge rate or discount rate is selected. In ROR ,the
selection of vest charge rate or MARR can be delayed sometimes clear to the
decision rates.
In ROR method algebraic discounted sum of the negative & positive cash flows
is zero.
Where reversal in the sign of the summation of the cash flow is indicated, there
may be more than one solution
No solution situation is also possible characteristics 3.& 4. are only objections
to only use.
Alternatives should have a Rs. Priced which can be actual cash income as in
business ventures or toll facilities, or it may be a cost reduction between a pair
of alternatives.
ROR method equally applicable to project evaluation & formulation as long
there are Rs priced associated with pair of alternatives.
Method is especially suited to project evaluation because the end product
directly indicates degree of profitability of the proposal . To the most decision
makers rate-of-return is more meaningful.
OTHER METHODS OF ANALYSIS
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Many methods have been developed for examining proposals for investments
of capital to bring to light their economic worth. They were variations of other
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These six methods are most important ones. For the sake of information three
other methods are mentioned.
1. Capitalized cost method
2. Payback period method
3. Break-even Analysis method
CAPITALIZED COST METHOD
• CC method is simply the present-worth-of- costs
method when the analysis period is assumed to be
50 yrs or very long .
• With high rates of vest charge, cc method will give
results comparable to PWOC method for analysis.
• CC method not used now ,but it was a popular
method in the early days of rail road cost.
• It is equally suited to project evaluation & project
formulation ,but poor choice of method in either
application.
2. PAYBACK PERIOD METHOD
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Payback period or pay out period have been used as an index for comparison
of alternatives. It does not recognize the time value of money.
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No value is given to the future possible operations after the end of payback
period.
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It has little or no application to public work unless the management desires
early return of its capital. The analysis is made for project formulation rather
than project evaluation. It may also be used when it is desired to know the
period of changing the users.
3. BREAK – EVEN ANALYSIS
• It is a process of analysis by controlling certain factors while one factor
is allowed to vary until a point of equality is reached between two
situations.
Ex.
Calculating ADT at which two EUAC’s are equal for gravel road Vs
low-cost bituminous road.
Ex.
To det. Required for specific which would equate the road –user
tax earnings & EUAC.
• It has many applications in highway managements in developing
criteria for decision making.
RANKING OF INDEPENDENT PROJECT
• Total highway program & independent projects
Application when capital budget is not restricted, the independent projects
may be programmed in order of their economic desirability as indicated by
NPV, B/C ratio or ROR method.
Application under capital budget constraint, The total construction
program for year may be restricted to max capital budget, under such
circumstances it becomes necessary to compare alternatives within given
project to alternative within other independent projects.
FACTORS FOR SPECIAL ATTENTION
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Unequal L.O.S
Alternatives with unequal analysis period
* Once the of cash flow is established, its EUAC, B/C ratio & ROR will
continue the same for any number of repetition of the whole cash flow cycle.
* The POWC & NPV, of course, will continue to increase with each added
cash flow cycle.
Adjusting the Analysis for unequal time periods
The adjustment in procedure to equalize the analysis period between the
alternatives is to use as an analysis period equal to the shorter lived alternative
and to allow a terminal value for the remainder of the life of longer lived
alternative.
Reveals in direction of cash flows.
hen the cash flow are irregular- that is fluctuating up & down as between the
alternatives being compared – diff vest charge rates may result in different
ranking of alternatives.
SENSITIVITY OF FACTOR
To understand how some factors affect the solution, A good practice is to
solve for economy using low, medium & high value of critical factors.
The sensitivity to the each factor in controlling the result should be
understood by each analyst & each decision maker.
For economic evaluation – Certain factors may be really critical.
For project formulation – factors may not be so critical
SENSITIVITY TO TERMINAL VALUE
 The effect of the recovery of terminal value at the end of years is –
reduce the capital cost.
 The effect of terminal value is = fn 1.The amt of terminal value T, as
% of investment ,I
2. The no. of years hence when the
T is to be recovered
3. The rate of vestcharge.
 The present worth Proportional to 1/n