ᎏᎏ X CA/U ⫻ X CO/EXCA/E/XCA/U
6.5 Economic Methods of Comparing Alternatives
G2⫽gradient series of the first half of the tent
⇒increasing G2is a positive value and decreasing G2is a negative value
The equivalent AW, A, and FW, F, values can be calculated by multiplying the P amount by the A/P or F/P factor, respectively.
6.4.10 Geometric Gradient Series Factor
In the case of geometric series cash flow, the amounts in the cash flow increase or decrease by a constant percentage from one period to the other. This type of cash-flow series is common in operations involv- ing operating costs, construction costs, and revenues. The uniform rate of change is the geometric gradi- ent series of the cash flows, defined as g. Therefore, g is the constant rate of change by which amounts increase or decrease from one period to the next expressed in decimal form. Figure 6.9 presents CFDs for geometric gradient series with increasing and decreasing constant rates. The series starts in year 1 with an initial amount A1; however, A1is not considered a base amount as it is in the arithmetic gradient series cash-flow equation.
The formula for calculating the PW of the increasing geometric series cash flow is
P⫽A1冤 冥 i⫽g and g is increasing
P⫽ i⫽g and g is increasing
(6.29)
However, the formula for calculating the PW of the decreasing geometric series cash flow is (Badiru, 1996)
P⫽A1冤 冥, when g is decreasing (6.30)
The equivalent AW, A, and FW, F, values can also be derived; however, it is usually easier to determine the P amount and multiply it by the A/P or F/P factor, respectively.
project may be selected if the investment opportunities are independent; hence, the alternatives do not compete with one another in the evaluation.
There are several methods for comparing investment alternatives: PW analysis, AW analysis, ROR analysis, and B/C ratio analysis. In using these methods, the do-nothing (DN) option is a viable alterna- tive that must be considered unless otherwise stated. The selection of DN as the accepted alternative means that no new investment will be initiated, that is, the current state of affairs continues. Three dif- ferent analysis periods are used for evaluating alternatives: equal service life for all alternatives, different service life for all alternatives, and infinite service life for all alternatives. The kind of analysis period may influence the method of evaluation used.
6.5.1 Present Worth Analysis
The present worth analysis is the application of some of the engineering economic analysis factors in which the present amount is unknown. It is used for projects with equal service life and can be used for evaluating one alternative, two or more mutually exclusive opportunities, and independent alternatives.
Present worth analysis evaluates projects by converting all future cash flows into present amount. The guidelines for using PW analysis projects are:
● For one alternative: Calculate PW at the minimum attractive rate of return (MARR). If PWⱖ0, the requested MARR is met or exceeded and the alternative is economically viable.
● For two or more alternatives: Calculate the PW of each alternative at the MARR. Select the alterna- tive with the numerically largest PW value. Numerically largest indicates a lower PW of cost cash flows (less negative) or larger PW of net cash flows (more positive).
● For independent projects: Calculate the PW of each alternative. Select all projects with PWⱖ0 at the given MARR.
The general equation for PW analysis is
PW⫽A0⫹A(P/A, i, n)⫹F(P/F, i, n)⫹[A1(P/A, i, n)⫹G(P/G, i, n)]⫹A1(P/A, g, i, n) (6.31) This equation reduces to a manageable size depending on the cash-flow profiles of the alternatives. For example, for bonds, the PW analysis equation becomes
PW⫽A(P/A, i, n)⫹F(P/F, i, n)⫽rZ(P/A, i, n)⫹C(P/F, i, n) (6.32) where
Z⫽face, or par, value of the bond
C⫽redemption or disposal price (usually equal to Z) r ⫽bond rate (nominal interest rate) per interest period n⫽number of periods before redemption
i ⫽bond yield rate per period
Two extensions of the PW analysis are capitalized cost and discounted payback period (Badiru, 1996).
6.5.2 Capitalized Cost
Capitalized cost is a special case of uniform-series PW factor when the number of periods is infinitely long. Therefore, it refers to the PW equivalent of a perpetual series of equal end-of-period amounts. The PW in the case of an infinitely long period is given as
C⫽PW⫽A冤 冥 (6.33)
where
C⫽present worth of the asset A⫽equal end-of-period amounts i ⫽investment interest rate
ᎏ1i
6.5.3 Discounted Payback Period Analysis
The discounted payback period, np, is the estimated number of years it will take for the estimated rev- enues and other economic benefits to recover the initial investment at a stated ROR (Badiru, 1996). This method is used as an initial screening technique for providing additional insight into the various invest- ment opportunities. It is not used as a primary measure of worth to select an alternative because it does not take cash flows after the payback period into consideration in the evaluation process (Blank and Tarquin, 2002). The payback period can be calculated at iⱖ0%. Calculations at i⬎0% take the time value of money into account and are more economically correct. This calculation is called discounted payback period and its general expression is given as
0⫽ ⫺P⫹冱np
t⫽1
NCFt(P/F, i, t)⫹NCFA(P/A, i, np) when i⬎0% (6.34) where
P⫽initial investment or first cost
NCFt⫽estimated net cash flow for each year t
NCFA⫽estimated equal amount net cash flow for each year np⫽discounted payback period
When i⫽0% in Equation (6.34), the expression becomes
np⫽ when i⫽0% (6.35)
Equation (6.35) is the simple (no-return) payback period. The simple payback period does not take the time value of money into consideration; however, it is a readily understood concept by people not famil- iar with economic analysis.
To facilitate the computation of payback and discounted payback periods using Equations (6.34) and (6.35), the following tabular headings can be used (Sullivan et al., 2003):
End of Net Cash Flow Cumulative PW at PW of Cash Flow Cumulative PW at i⬎0%/yr
Year j i⫽0%/yr through Year j at i⬎0%/yr through Year j
(A) (B) (C) (D)
Column A: Net cash flow for the alternative.
Column B: Cumulative of the net cash flow in column A. The jth year at which the cumulative balance in this column turns positive is the payback period for the alternative.
Column C: PW at the given interest rate of the respective net cash flow in column A.
Column D: Cumulative of the PW in column C. The jth year at which the cumulative balance in this column turns positive is the discounted payback period for the alternative.
6.5.4 Annual Worth Analysis
The AW method of evaluating investment opportunities is the most readily used of all the measures because people easily understand what it means. This method is mostly used for projects with unequal service life since it requires the computation of the equivalent amount of the initial investment and the future amounts for only one project life cycle. Annual worth analysis converts all future and present cash flows into equal end-of-period amounts. For mutually exclusive alternatives, AW can be calculated at the MARR and viable alternative(s) selected on the basis of the following guidelines:
● One alternative: Select alternative with AWⱖ0 since MARR is met or exceeded.
● Two or more alternatives: Choose alternative with the lowest cost or the highest revenue AW value.
ᎏNCFP
The AW amount for an alternative consists of two components: capital recovery for the initial invest- ment P at a stated interest rate (usually at the MARR) and the equivalent annual amount A. Therefore, the general equation for AW analysis is
AW⫽ ⫺CR⫺A
⫽ ⫺[P(A/P, i, n)⫺S(A/F, i, n)] ⫺A (6.36)
where
CR⫽capital recovery component
A⫽annual amount component of other cash flows P⫽initial investment (first cost) of all assets
S⫽estimated salvage value of the assets at the end of their useful life
Annual worth analysis is specifically useful in areas such as asset replacement and retention, breakeven studies, make-or-buy decisions, and all studies relating to profit measure. It should be noted that expen- ditures of money increase the AW, while receipts of money such as selling an asset for its salvage value decrease AW. The assumptions of the AW method are that (Blank and Tarquin, 2002) the following:
1. The service provided will be needed forever since it computes the annual value per cycle.
2. The alternatives will be repeated exactly the same in succeeding life cycles. This is especially impor- tant when the service life is several years into the future.
3. All cash flows will change by the same amount as the inflation or deflation rate.
The validity of these assumptions is based on the accuracy of the cash-flow estimates. If the cash-flow estimates are very accurate, then these assumptions will be valid and will minimize the degree of uncer- tainty surrounding the final decisions based on this method.
6.5.5 Permanent Investments
This measure is the reverse of capitalized cost. It is the AW of an alternative that has an infinitely long period. Public projects such as bridges, dams, irrigation systems, and railroads fall into this category. In addition, permanent and charitable organization endowments are evaluated using this approach. The AW in the case of permanent investments is given by
AW⫽A⫽Pi (6.37)
where
A⫽capital recovery amount P⫽initial investment of the asset i ⫽investment interest rate
6.5.6 Internal Rate of Return Analysis
Internal rate of return (IRR) is the third and most widely used method of measure in the industry. It is also referred to as simply ROR or return on investment (ROI). It is defined as the interest rate that equates the equivalent worth of investment cash inflows (receipts and savings) to the equivalent worth of cash outflows (expenditures); that is, the interest rate at which the benefits are equivalent to the costs. If i∗
denotes the IRR, then the unknown interest rate can be solved by using any of the following expressions:
PW(Benefits)⫺PW(Costs)⫽0
EUAB⫺EUAC⫽0 (6.38)
where
PW⫽present worth
EUAB⫽equivalent uniform annual benefits
EUAC⫽equivalent uniform annual costs
The procedure for selecting the viable alternative(s) is:
● If i∗ⱖMARR, accept the alternative as an economically viable project.
● If i∗⬍MARR, the alternative is not economically viable.
When applied correctly, IRR analysis will always result in the same decision as with PW or AW analy- sis. However, there are some difficulties with IRR analysis: multiple i∗, reinvestment at i∗, and computa- tional difficulty. Multiple i∗usually occurs whenever there is more than one sign change in the cash-flow profile; hence, there is no unique i∗value. In addition, there may be no real value of i∗that will solve Equation (6.38), but only real values of i∗are valid in economic analysis. Moreover, IRR analysis usually assumes that the selected project can be reinvested at the calculated i∗, but this assumption is not valid in economic analysis. These difficulties have given rise to an extension of IRR analysis called external rate of return (ERR) analysis (Sullivan et al., 2003).
6.5.7 External Rate of Return Analysis
The difference between ERR and IRR is that ERR takes into account the interest rate external to the proj- ect at which the net cash flow generated or required by the project over its useful life can be reinvested or borrowed. Therefore, this method requires the knowledge of an external MARR for a similar project under evaluation. The expression for calculating ERR is given by
F⫽P(1⫹i⬘)n (6.39)
where
P⫽the present value of all cash outflows at the MARR F⫽the future value of all cash inflows at the MARR i′⫽the unknown ERR
n⫽the useful life or evaluation project of the project
Using this method, a project is acceptable when the calculated i′is greater than the MARR. However, if i′is equal to the MARR (breakeven situation), noneconomic factors may be used to justify the final decision. The ERR method has two advantages over the IRR method: it does not result in trial and error in determining the unknown ROR and it is not subject to the possibility of multiple rates of return even when there are several sign changes in the cash-flow profile.
6.5.8 Benefit/Cost Ratio Analysis
The three methods of analysis described above are mostly used for private projects since the objective of most private projects is to maximize profits. Public projects, on the other hand, are executed to provide services to the citizenry at no profit; therefore, they require a special method of analysis. The B/C ratio analysis is normally used for evaluating public projects. It has its roots in The Flood Act of 1936, which requires that for a federally financed project to be justified, its benefits must exceed its costs (Blank and Tarquin, 2002). The B/C ratio analysis is the systematic method of calculating the ratio of project bene- fits to project costs at a discounted rate. For over 60 years, the B/C ratio method has been the accepted procedure for making go/no-go decisions on independent and mutually exclusive projects in the public sector.
The B/C ratio is defined as
B/C⫽ ⫽ (6.40)
where
Bt⫽is benefit (revenue) at time t Ct⫽is cost at time t
冱nt⫽0Bt(1⫹i)⫺t ᎏᎏ冱nt⫽0Ct(1⫹i)⫺t PW(Benefits)
ᎏᎏPW(Costs)
If the B/C ratio is⬎1, then the investment is viable; if the ratio is⬍1, the project is not acceptable. A ratio of 1 indicates a breakeven situation for the project and noneconomic factors may be considered to validate the final decision about the project.
6.5.9 Incremental Analysis
Under some circumstances, IRR analysis does not provide the same ranking of alternatives as do PW and AW analyses for multiple alternatives. Hence, there is a need for a better approach for analyzing multiple alternatives using the IRR method. Incremental analysis can be defined as the evaluation of the differ- ences between alternatives. The procedure essentially decides whether or not differential costs are justi- fied by differential benefits. Incremental analysis is mandatory for economic analysis involving the use of IRR and B/C ratio analyses that evaluate three or more mutually exclusive alternatives. It is not used for independent projects since more than one project can be selected. The steps involved in using incremen- tal analysis are:
1. If IRR (B/C ratio) for each alternative is given, reject all alternatives with IRR⬍MARR (B/C⬍1.0).
2. Arrange other alternatives in increasing order of initial cost (total costs).
3. Compute incremental cash flow pairwise starting with the first two alternatives.
4. Compute incremental measures of worth using the appropriate equations.
5. Use the following criteria for selecting the alternatives that will advance to the next stage of com- parisons:
(i) If∆IRR⬎MARR, select higher-cost alternative.
(ii) If∆B/C⬎1.0, select higher-cost alternative.
6. Eliminate the defeated alternative and repeat steps 3–5 for the remaining alternatives.
7. Continue until only one alternative remains. This last alternative is the most economically viable alternative.