Sensitivity analysis of EAC's robustness.

INTRODUCTION

Analyzing mutually exclusive alternatives with differing lives typically requires (1) estimating the salvage value at a selected horizon or (2) calculating an equivalent annuity [1, 5, 6, 7, 9, 10]. Most other approaches can be shown to be analogous to one of these,

The equivalent annuity method is used most often because of its conceptual familiarity and arithmetic simplicity. When the project has costs only, or the alternatives provide identical benefits, then the equivalent annuities may be simplified to EACs. This method compares the annualized cost for each alternative's least cost life.

When the EACs of alternatives with unequal lives are compared and an incremental difference is calculated, an inherent assumption is made - the alternatives under consideration repeat identically until the least common multiple of their lives is reached. This assumption is often not stated explicitly (and we believe often not recognized). This implicit assumption rarely "squares with reality," since advancing technology, changing markets, and fluctuating prices continually modify the EACs of existing alternatives and supply new alternatives.

Critical to the success of EAC as an indicator is its behavior when the identical repetition assumption is violated. We began our analysis of this in an earlier presentation [4], which included the first two of the eight figures in this paper. In the next section we describe our example problem, including a measure, fraction of asymptotic EAC, which supports easy extension of our results to real world problems. We also discuss why we have chosen year of termination as the variable for the x-axis of our graphs. The bulk of the paper is the presentation and analysis of six graphs focused on the discount rate; on the alternative's life, salvage value, annual cost amount, and annual cost profile; and on subsequent cost changes in repeated lives. The paper concludes with two tables and two figures that examine the relative sensitivity of these variables.

A GENERAL MODEL OF THE EAC

Rather than creating an example with a fixed first cost, specified annual expenses for operations and maintenance, or an assumed salvage value, we state the annual expenses and salvage values as fractions of the first cost. We also define an asymptotic EAC, which is the EAC calculated at the optional economic life of that alternative, which also corresponds to calculations for least common multiple and infinite life horizons. Consequently, this asymptotic value is the minimum EAC (unless costs decrease with subsequent repetitions) and it is also the lower limit of the EAC calculated over other time periods.

To create results that can be easily generalized, we divide all of the EACs we calculated by the asymptotic EAC. This creates a measure, fraction of asymptotic EAC, which is independent of the first cost's magnitude. Thus if a graphed function has a value of 1.2 at a point, the EAC is 20% larger than the asymptotic value. Note: this independence requires that all costs be stated as a fraction of the first cost. Mathematically, the cases with constant costs are stated as: [Mathematical Expression Omitted] where:

[F.sub.r] = fraction of asymptotic EAC

FC = first cost

MC = annual maintenance costs

i = interest or discount rate

N = full alternative life

n = termination year

R = number of repetitions

= n/N (rounded up) SV = salvage value at early termination.

A key element of this model is the termination year. If a fixed horizon is assumed, or the project's horizon does not equal the least common multiple of the alternatives' lives, or there is a limited horizon over which this data is valid, then one or more alternatives is in reality terminated prematurely during a repetition. The critical question is how different is this "true" EAC from the value calculated using a single life. Based on our experience with real world analysis, we concluded that values within 10% of the asymptotic minimum were not significantly different than that absolute minimum.

Before we detail the specific assumptions of our example, there are some general expectations we can state, quantifying these later. First, the later that a premature termination occurs the less important it will be. Second, premature termination will be most important for low interest rates and alternatives with no annual costs nor salvage values. Third, alternatives with longer lives can be more heavily affected as they might "waste" a larger fraction of their life. These generalizations imply that EAC is likely to produce less valid results when comparing alternatives at low interest rates where at least one is relatively lengthy, and where expected maintenance expense and salvage value are small.

Our example(s) typically assume a seven year life and a 15% interest rate. Generally, the y-axis is the fraction EAC, and the x-axis is the termination year. This often begins with year 5 or year 8, because termination early in the first life is unlikely. If it does occur, the effects are much larger and are likely to require an analysis of the specific case to be useful.

INTEREST RATE IMPACT

The impact on EAC is maximized when an alternative consisting only of a first cost is abandoned with no salvage value. Large first costs (typical in capital projects), allocated by EAC over the alternative's life, are now forced into a smaller number of time periods, thus inflating the EAC. This impact is diminished as the interest rate increases, because the present value of later cash flows decrease.

Figure 1 shows the fraction EAC for a seven year alternative that consists solely of a first cost terminating from years 8 until 42 (2 to 6 repetitions) at discount rates of 5%, 15%, and 25%. At i = 25%, EAC is within 10% of asymptotic value after 9 years whereas it takes 11 and 24 years at i = 15% and 5% respectively. Figure 1 also illustrates the importance of the discount rate for more stringent measures (< 10%) of significant difference. At i = 25%, termination during the third life makes less than a 4% difference, and it is less than 1% for the fourth life. At i = 5%, even the sixth life has a maximum increase over the asymptotic value of about 6%.

EFFECT OF THE TERMINATION YEAR

The value of the termination year in absolute measure and in terms of the number of repetitions which have occurred impacts EAC. Measured absolutely, the further out in time the termination, the smaller is its impact on the alternative's present value.

The number of repetitions of an alternative occurring prior to termination is determined by the alternative's life and by the termination year. Alternatives which have shorter lives reach asymptotic values of EAC in less absolute years because the last "terminated" repetition represents a smaller fraction of the total costs for all repetitions. They also take more repetitions to reach asymptotic value because the discount rate's effect is less over short time periods.

Figure 2 shows the fraction EAC for alternatives consisting only of a first cost with no salvage value with lives of 2, 6, and 10 years at a discount rate of 15%. The shortest alternative is within 10% of the asymptotic value within 5 years (3 placements) and the 10 year alternative within 14 years during the second life.

Alternatives are not normally abandoned early in any life because of the attractiveness of regaining sunk first costs [8]. However, this may be more of a problem for short lived alternatives such as the one here with a two year life, where a one-year extension equals fifty percent of the projected life. If terminations within one or two years of placement were prohibited (as is likely in practical situations), then asymptotic values would be "reached" far sooner.

IMPACT OF SALVAGE VALUE

Our previous examples have been worst cases with no salvage values. Realistically, alternatives that are liquidated early can expect to achieve more positive cash flow through asset disposal. Moreover, this mitigation of the harm of premature termination is normally most substantial when the termination is most premature.

Figure 3 shows projected salvage values of (1) none (same as before), (2) fifty percent of book value with straight line depreciation, and (3) book value with straight line depreciation for a 7 year alternative with no annual costs. The greater the assumed salvage value, the less sensitive EAC is to early termination. In fact, for any termination after the first life with salvage approaching book value based on straight line depreciation, EAC is virtually at asymptotic value.

ANNUAL COSTS

Just as most alternatives can expect salvage inflows, they also incur annual operation and maintenance costs. These costs are part of the EAC but they can be avoided in the event of early termination, so that alternatives with a greater proportion of costs spent annually are less sensitive to premature terminations. This is easily illustrated in the extreme; if all expenditures are annual, then the EAC is always at asymptotic value regardless of the termination year. Figure 4 illustrates that an alternative with even modest annual costs (12.5% of first costs) is significantly desensitized to early termination.

IMPACT OF COST CHANGES IN SUBSEQUENT REPETITIONS

An engineering economic analysis spring from imperfect knowledge and thus contains uncertainty [2, 8]. Cash flows of later repetitions are especially prone to forecast error since they are implicitly assumed to match cash flows of the first life, rather than being estimated explicitly. Future costs may unexpectedly decrease as new models become more reliable or less expensive, or the decrease may be attributed to learning curve effects. Costs may increase because of added complexity or differential inflation.

Figure 5 shows a seven year alternative at i = 15% with no projected salvage value where annual costs are assumed to rise or fall with each repetition. Annual costs are level throughout each life, and begin at 25% of first costs. This increase or decrease of maintenance costs results in a long term EAC of [+ or -] 7 to [+ or -] 8 percent of the originally calculated EAC.

Altering first costs was analyzed in [4] resulting in the same pattern as Figure 5, but its extreme values at year 8 range from 1.2 to 1.36, rather than from 1.13 to 1.15. However, for both annual and first costs, may significant unexpected variation will probably happen in the long term lessening the effect.

ANNUAL COST PROFILE

Figure 6 shows the effects of several annual cost profiles on fraction EAC. The relative importance of annual costs in each asymptotic value is equivalent, there is no assumed salvage value, and the cost profiles reinitialize at each placement. The profiles are increase - annual costs increasing from 22% of first cost in year 1 to 55% in year 7; decrease - annual costs decreasing from 55% to 22% (the reverse of 1); steady - annual costs constant at 39% of first cost; and none - no annual costs.

The most likely scenario is where maintenance costs increase steadily (due to aging equipment) through the life of the alternative, and it is this scenario where EAC is most robust. The decrease profile might come about if a learning curve was being conquered, while the last two profiles are simplistic but are often made for study convenience.

RELATIVE VARIABLES SENSITIVITY

Up to this point the various factors have been analyzed in isolation, in that no salvage value or no annual costs were often assumed. Figures 7 and 8 address a more realistic alternative with the properties shown in table 1. The first five variables are graphed as a percentage of their base case values. (The annual cost gradient has a base case of slope 0, so in Figure 8 its variations were placed at [+ or -] 30% of the base case). Note that each variable is subject to different sources of uncertainty, and thus the maximum percentage increase and decrease differ for each [3]. These graphs isolate the relative effective of changes in each variable, for this example. [Tabular Data 1 Omitted]

If Figures 7 and 8 are considered together, alternative life and termination year are the most sensitive variables, assuming more relative importance when they are decreased, because of the discount rate's effect. The other four variables have nearly linear sensitivity graphs and the impact of each is quite similar, since the ones with steeper slopes vary over narrower ranges. The discount rate's flatness, or insensitivity to change, is surprising considering Figure 1. However, here we have a typical rather than a worst case, and the discount rate is in effect for only ten years.

Since alternative length and termination year are the two most sensitive variables, a closer look at their interaction is warranted. The number of repetitions is inherently tied to both values. Taking the above scenario of an early termination of one half of planned alternative life, fraction EAC was calculated for various discount rates and alternative lives. In this analysis, a worst case approach was again taken, with both salvage value and annual costs equal to zero.

Table 2 shows the minimum full alternative life (to the nearest even number) for fraction EAC to be within asymptotic value (i.e. 1.1 or less) when termination occurs at the mid-point of a given placement and at a given discount rate. If the first life is considered assured, then an alternative of 8 years or longer will be within 10% of its asymptotic EAC value if terminated mid-way through its second life at a discount rate of at least 15%. This table shows that except for extremely short projects, any termination after the first repetition will have little effect.

Table 2. Minimum Alternative Life (Years) Required to Reach Asymptotic EAC When
            Prematurely Terminated Halfway Through a Placement.
Placement         Discount Rate
            5%   10%   15%   20%   25%
First       92   48    32    26    20
Second      22   12    8     6     6
Third       8    4     4     4     2
Fourth      4    2     2     2     2

CONCLUSIONS

Fundamentally, equivalent annuity is a powerful measure. Although maximum deviations can be large when the assumptions are not met, most real world circumstances will quickly mitigate these extremes. Presence of salvage values and annual costs have been shown individually to increase the robustness of EAC, and in practice, their joint effect will reinforce their individual contributions. This analysis has not considered salvage values of normal termination and the opportunity to claim a tax loss on disposal, both of which would increase the potency of EAC.

Longer projects with severely shortened lives will experience the most radical deviation from originally calculated EAC. Small discount rates yield greater sensitivity to both premature termination and to subsequent cost change. In reality, projects often do not repeat, but are rarely divested during their first life while dramatic cost changes occur only in the long term. Similarly, annual costs are normally present, and generally increase over the alternative's life. Salvage can also be expected, especially in cases of early termination. Thus, EAC's robustness is strong in typical applications.

REFERENCES

[1]American Telephone and Telegraph Company, Engineering Economy; Third Edition, New York: McGraw-Hill Book Company, 1977. [2]Chandra, M.J. and R. Guild, "The Effect of Probability Distribution Functions of Economic Life on Retirement Alternatives," The Engineering Economists, Vol. 30, No. 4 (Summer, 1985), pp. 315-328. [3]Eschenbach, T.G. and L.S. McKeague, "Exposition on Using Graphs for Sensitivity Analysis," The Engineering Economist, Vol. 34, No. 4 (Summer 1989), pp. 315-333. [4]Eschenbach, T.G. and A.E. Smith, "Violating the Identical Repetition Assumption of EAC," Proceedings IIE (May 1990), pp. 99-104. [5]Grant, E.L., W.G. Ireson and R.S. Leavenworth, Principles of Engineering Economy, Eight Edition, New York: John Wiley & Sons, 1990. [6]Kulonda, D.J., "Replacement Analysis with Unequal Lives - The Study Period Method," The Engineering Economist, Vol. 23, No. 3 (Spring, 1978), pp. 171-179. [7]Seitz, N.E., Capital Budgeting & Long Term Financing Decisions, Hinsdale, Il.: Dryden Press, 1989. [8]Schwab, B. and P. Lusztig, "A Note on Investment Evaluations in Light of Uncertain Future Opportunities," Journal of Finance, Vol. 27, No. 5 (December, 1972), pp. 1093-1100. [9]Singhvi, S.S., "Discounted Cash Flow Analysis and the Reinvestment Assumption," Managerial Planning, Vol. 19, No. 4 (July/August, 1971), pp. 27-30. [10]Thuesen, G.J. and W.J. Fabrycky, Engineering Economy, Seventh Edition, Englewood Cliffs, N.J.: Prentice Hall Inc., 1989. Ted G. Eschenbach, P.E., when this paper was written, was the Robert B. Koplar Professor of engineering management at the University of Missouri-Rolla (paper revised at UAA). He is the founding and current editor of the Engineering Management Journal for the American Society for Engineering Management. He has served on the faculty of the University of Alaska Anchorage since 1975 with sabbaticals at the US General Accounting Office (faculty fellow), the Naval Postgraduate School, the UMR (1988-1990). His research areas include engineering economy, forecasting, and strategic management of technology. His Cases in Engineering Economy was published by Wiley in 1989. His second edition of Bussey's Economic Analysis of Industrial Projects will be available from Prentice Hall in late 1991. Dr. Eschenbach holds a B.S from Purdue University and the M.S. in operations research and Ph.D. in industrial engineering from Stanford University. Alice E. Smith, P.E., when this paper was written, was a Ph.D. student (and Chancellor's Fellow) in Engineering Management at the University of Missouri - Rolla. She is now an assistant professor at the University of Pittsburgh. Her research interests are engineering economy, project management, expert systems, and neural networks. Ms. Smith was employed for ten years at Southwestern Bell Telephone Company in various technical and management positions. She holds a B.S C E. (cum laude, 1979) from Rice University, an M.B.A. (1988) from Saint Louis University, and the Ph.D. from UMR (1991). She is a member of Tau Beta Pi, IIE, IEEE, NSPE, SWE, ASEM, and TIMS.