Convex-envelope method of optimal capital budgeting.

Capital budgeting is a process designed to achieve the greatest profitability and cost effectiveness in the private and public sectors of the economy. Because measurements of profitability and cost effectiveness are not universal, engineers, accountants, and economists use multiple criteria and constraints to account for risk, inflation, and investor expectations. Owing to the lack of uniformity, multiple criteria and constraints often give conflicting results.

Engineers, accountants, and economists need a common language for maximizing profitability and cost effectiveness in private enterprise, financial institutions, government agencies, and nonprofit organizations. Major problems with current methods could be resolved by focusing on the production objective of making the biggest pie, rather than the distribution objective of dividing the pie. All organizations seek to maximize output value for a given input cost and to minimize input cost for a given output value. Disagreements in carrying out the production objective arise when:

* alternatives differ in input costs, output values, or both;

* determining interest rates for discounting future cash flows of project alternatives; and

* measuring the effectiveness of capital constraints.

Financial and managerial accounting 

Financial accounting deals with the past. All past revenues and expenses of failed projects are recorded "as is" in periodic income statements without adjusting for the time value of money. The accounting period of an income statement reflects the amounts of cash flow between two points of time where balance sheets of an organization's financial positions are stated in double-entry bookkeeping. The synchronization of single-entry income statements with double-entry balance sheets at the beginning and end of accounting periods is essential for auditing purposes. The revenues and expenses of all funded projects are aggregated within accounting periods, and the resulting profits or losses may be causally connected to revenues and expenses in other accounting periods. The costs of borrowed money and paid taxes are not allocated to individual projects because these expenses are incurred by the organization as a whole.

Managerial accounting, also called cost accounting, plans for the future from forecasts of revenues and expenses of project alternatives over their lifespans. The alternative ways of doing each project may differ in scale, lifespan, quality of output, and financing. Even though the costs of borrowing money and paying taxes will be incurred by the organization as a whole, they must be allocated to each project alternative so that the total costs of borrowed money and paid taxes of all funded project alternatives will be the same as those that will be reported in future financial statements of the organization. In order to develop a provable method of optimal capital budgeting, we assume conditions of economic certainty in which accurate forecasts are made for each project alternative with respect to the timing of all its revenues and expenses.

In order to allocate future costs of borrowing money by an organization to individual project alternatives, the input and output cash flows of a project alternative are defined as the change of cash flows of the organization as a whole if the project alternative is accepted, as opposed to the cash flows if the project alternative is rejected. If a project alternative is accepted, the organization as a whole has less money to lend, or more money to borrow at the costs of borrowing money. If a project alternative is rejected, the organization as a whole has more money to lend, or less money to borrow at the costs of borrowing money. In either case, the discount rates used to calculate the net present values added by future cash flows of an individual project should be based on expected costs of borrowing money by the organization as a whole. Likewise, in order to allocate the costs of paying taxes to individual project alternatives on a zero-base budgeting basis, the input and output cash flows of a project alternative are defined as the change of cash flow of the organization as a whole between accepting and rejecting the project alternative.

Current methods of capital budgeting do not require the compatibility of financial and managerial accounting for determining discount rates. In practice and theory, engineering and financial management use discount rates that are much higher than costs of borrowing money. The discount rate commonly used by engineering management is Minimum Attractive Rate of Return (MARR), which is determined by ranking projects in descending order of their Internal Rates of Return (IRR). MARR is defined as the IRR of the project with the lowest rank at the cutoff of investment opportunities or available funds. The discount rate often used by financial management is Weighted Average Cost of Capital (WACC), which is determined by averaging interest rates on short- and long-term debt and opportunity costs of equity capital.

Capital constraints are more sensitive to MARR or WACC discount rates than they are to the lower costs of borrowing money. More specifically, investments can be funded only if their net present values are greater than zero or their benefit/cost ratios are greater than one. For example, suppose a $100 investment opportunity has a 12 percent/year IRR and yields $20.13 each year for 8 years. This investment cannot be funded when discounting at MARR or WACC of 12 percent/year because its net present value would equal zero and its benefit/cost ratio would equal one. But even discounting at the lower 8 percent/year cost of borrowing money, the same $100 investment opportunity may be funded because its net present value would equal $15.68 and its benefit/cost ratio would equal 1.1568, which exceeds the capital constraint requirements.

There is considerable room for questioning the magnitude of capital constraints generated by MARR or WACC discount rates because they are independent of the relative scales of investment and borrowing opportunities. For example, suppose a $100 investment opportunity is offered which has a 12 percent/year IRR and yields $20.13 each year for 8 years. If money could be borrowed at 8 percent/year, it would be better to borrow as much of the $100 investment cost as is needed to earn the 4 percent/year spread between the 12 percent/year investment and the 8 percent/year borrowing cost plus the 12 percent/year return on the equity investment, than it would be to forego the investment because its IRR was not greater than a MARR or WACC of 12 percent/year.

Moreover, rate of return is independent of the lifespan of an investment. When comparing a $100 investment which yields $112 one year later to the previous $100 investment which yields $20.13 per year for 8 years, both investments have a 12 percent/year IRR and a zero net present value at a 12 percent/year MARR discount rate. But discounting at the 8 percent/year cost of borrowing money gives a net present value of $3.70 for investing one year, and $15.68 for investing 8 years. This indicates MARR discount rates would value short- and long-term returns equally, whereas borrowing cost discounts would favor long-term returns.

Convex-envelope method of optimal capital budgeting

Let us divide an economic organization into non-overlapping projects that compete for funds from a common capital constraint. Each project may have any number of mutually exclusive and indivisible alternatives that differ in scale, lifespan, quality of output, or financing. We now ask three questions:

* What is the best way of doing each project?

* Which are the best projects to do?

* Which projects should be funded?

Each way of doing a project is represented by a two-dimensional vector whose components are present value input costs [Delta]C, and present value output revenues [Delta]R, that would be added to the overall input costs and output revenues of the organization as a whole. Present values are defined as the equivalent worths at the present time of future cash flows discounted at the expected costs of borrowing money. The vectors which represent alternative ways of doing each project are formed into a bundle with a common initial point and distinct terminal points as shown in Figure 1.

The equation [Delta]R-[Delta]C [equivalent to] [Delta]NPV measures the net present value added of a vector, which is also called its absolute profitability. The ratio [Delta]R/[Delta]C [equivalent to] [null set] measures the capital efficiency of a vector to transform [Delta]C into [Delta]R, which is also called its relative profitability. Project X has three mutually exclusive alternatives, [X.sub.1](5,9), [X.sub.2](4,9), and [X.sub.3](2,9), which represent the problem of minimizing input costs, [Delta]C, for given output revenues, [Delta]R = 9. Since [X.sub.3] has the smallest [Delta]C and the greatest absolute and relative profitabilities, that is, [Delta]NPV([X.sub.3]) = 7 and [null set]([Delta][X.sub.3]) = 4.5, [X.sub.3] is the best way of doing project X.

Project Y has three alternatives, [Y.sub.1](5,9), [Y.sub.2](5, 11), and [Y.sub.3](5, 13), which represent the problem of maximizing output revenues, [Delta]R, for given input costs, [Delta]C = 5. Since [Y.sub.3] has the greatest [Delta]R and the greatest absolute and relative profitabilities, that is, [Delta]NPV([Y.sub.3]) = 8, and [null set]([Y.sub.3]) = 2.6, [Y.sub.3] is the best way of doing project Y.

Project Z has three alternatives, [Z.sub.1](5,9), [Z.sub.2](8, 14.8) and [Z.sub.3](6, 12.4), none of which have the same [Delta]C input costs or [Delta]R output revenues. In the absence of a capital constraint, the process of capital budgeting would be complete - that is, the alternative with the largest net present value added, [Delta]R-[Delta]C [equivalent to] [Delta]NPV, should be funded. But owing to the natural scarcity of capital, absolute profitability cannot be the sole criterion of capital budgeting. Relative profitability or capital efficiency, [Delta]R/[Delta]C [equivalent to] [null set], is also a necessary criterion.

For example, let us first compare [Z.sub.1](5,9) to [Z.sub.2](8, 14.8). The absolute and relative profitabilities of [Z.sub.1] are both smaller than those of [Z.sub.2], that is, [Delta]NPV([Z.sub.1],) = 4 [less than] [Delta]NPV([Z.sub.2]) = 6.8 and [null set]([Z.sub.1]) = 1.8 [less than] [null set]([Z.sub.2]) = 1.85. Assuming there is sufficient capital to undertake the larger capital cost of either [Z.sub.1] or [Z.sub.2], then [Z.sub.1] is eliminated because its profitability is worse both absolutely and relatively than [Z.sub.2] for doing project Z.

Let us now compare [Z.sub.2](8, 14.8) to [Z.sub.3](6, 12.4). Alternative [Z.sub.2] has greater absolute but smaller relative profitability than [Z.sub.3], that is, [Delta]NPV([Z.sub.2]) = 6.8 [greater than] [Delta]NPV([Z.sub.3]) = 6.4 but [null set]([Z.sub.2]) = 1.85[Delta] [null set]([Z.sub.3]) = 2.07. Also, [Z.sub.2] has a greater capital cost than [Z.sub.3], that is, [Delta]C([Z.sub.2]-[Z.sub.3]) = 8-6 = 2. This raises the question of whether the increase [Delta]C([Z.sub.2]-[Z.sub.3]) = 8-6 = 2 of capital cost is worth the increase [Delta]NPV {[Z.sub.2]-[Z.sub.3]} = 6.8-6.4 = 0.4 of net present value added. If another project V has a [V.sub.1] (2,2.6) alternative with [Delta]NPV{[V.sub.1]} = 0.6 and [null set]{[V.sub.1]} = 1.3, then [Z.sub.3]+[V.sub.1] would have a greater absolute profitability than [Z.sub.1] for the same capital cost, that is, [Delta]NPV([Z.sub.3] + [V.sub.1],) = 7.0 [greater than] [Delta]NPV([Z.sub.2]) = 6.8 and [Delta]C([Z.sub.3] + [V.sub.1]) = [Delta]C([Z.sub.2]) = 8.

The last example illustrates the economic significance of capital constraints. The best project alternative may not have the greatest net present value added because other projects compete for the same capital. We may want to fund an alternative with a smaller net present value added but which uses less capital more efficiently, so that other projects can better utilize the capital released for maximizing overall [Delta]NPV under the capital constraint. A simple method of determining the optimal mix of project alternatives is to compute the overall [Delta]NPV of all possible combinations. However, this may be impractical even for small sets of projects due to the exponential increase in the number of possible combinations that must be examined. For example, if 100 non-overlapping projects have only two alternatives each, there would be [3.sup.100] [approximately equal to]5[multiplied by][10.sup.46] budget choices because each project may be performed in either one of two ways, or not at all.

The convex-envelope method of optimal capital budgeting permits a rapid scanning of the best way of doing each project and the best projects to do within a planned range of capital constraints without exhaustively evaluating every possible combination. The vector bundles of each project are first ranked in descending order of their steepest-slope vectors, which are added geometrically to form a convex envelope as shown in Figure 2. It is possible for steepest-slope vectors of the convex envelope to be intersected by steeper-slope vectors from bundles of higher ranking steepest-slope vectors.

The vector of the convex envelope whose terminal point touches the capital-constraint line has coordinates ([Sigma][Delta]C, [Sigma][Delta]R) whose distance above the 45 [degrees] line through the origin represents the sum [Sigma][Delta]NPV of vectors in the convex envelope, and whose distance along the [Delta]C-axis represents the sum [Sigma][Delta]C of vectors in the convex envelope. In order to maximize [Sigma][Delta]NPV subject to a capital constraint [Sigma][Delta]C, vectors in the convex envelope could be replaced or eliminated by other vectors in the bundles that have greater [Delta]C-input costs and smaller [Delta]R/[Delta] C-capital efficiencies. For this purpose, optimal capital budgeting uses a slope, [[null set].sub.m], that represents the marginal capital efficiency, which is determined initially by the slope of the last vector of the convex envelope whose terminal point touches the capital-constraint line. The marginal comparison slope, [[null set].sub.m], is translated parallel to itself near the terminal side of each vector bundle until it first encounters:

* The steepest-slope vector in a bundle that was already in the convex envelope;

* Another vector with a larger [Delta]C-input cost and a smaller [Delta]R/[Delta]C-capital efficiency; or

* Two or more vectors with larger [Delta]C-input costs and smaller [Delta]R/[Delta]C-capital efficiencies.

In cases 1 and 2, the vectors first touched by the marginal comparison slope are ranked again in descending order of their slopes and then added geometrically to form a new convex envelope of vectors. The vectors from case 3 need to be merged with vectors from cases 1 and 2 to obtain the greatest [Delta]NPV.

Hence, if the marginal comparison slope is rotated continuously in the region 90 [degrees] [greater than] [[null set].sub.m] [greater than] 45 [degrees] , the proposed solution will provide optimal capital budgets consisting of vectors that form convex envelopes. Each vector in the convex envelope represents the best way of doing that project, and vectors that compose the convex envelope up to the capital constraint represent the best projects to do. Since the range of capital constraints has finite upper and lower bounds, only a finite number of marginal comparison slopes are needed to exhaust the optimal capital budgets of all engineering and financial alternatives.

Proof of optimal capital budgeting

In order to prove that the net present value added is maximized for an organization as a whole under an effective capital constraint, the marginal comparison slope, [[null set].sub.m], is drawn as a dashed line in Figure 3. Three types of changes in the convex-envelope solution are possible:

* The removal of the convex-envelope vectors that appear in the solution;

* The vector differences between bundle vectors that are and are not in the solution; or

* The introduction of vectors from bundles that are not in the solution.

From the geometry of the proposed solution, the three possible types of change in the solution would be represented by dashed vectors, which are pointed in the direction of the plane below the dashed line of the marginal comparison slope, [[null set].sub.m]. Therefore, the resultant of these possible changes cannot be in the portion of the plane above the line of the marginal comparison slope. Hence, no change of the proposed solution could increase overall [Delta]NPV under the capital constraint.

What is the best way of doing each project?

The best way of doing each project depends upon the capital constraint. The marginal capital-efficiency slope, [[null set].sub.m], can be determined in the range of the capital constraint after which it can be translated near the vector bundle of each project. The ongoing alternative of a project may be replaced by cost-increasing alternatives that increase [Delta]R more than [Delta]C. A classic example of cost-increasing alternatives is that of replacing equipment that costs more, but has greater net present values added because of longer lifespans, lower operating and maintenance costs, better output quality, or higher salvage values. Cost-decreasing alternatives may replace the ongoing alternative by decreasing [Delta]C more than [Delta]R. Examples of cost decreasing alternatives involve problems of economic downsizing, maintenance versus replacement, make versus buy, leasing versus purchasing, debt versus equity financing and asset sales.

The geometric method of determining the best way of doing each project is carried out algebraically in Figure 1 by drawing a line with the marginal comparison slope, [[null set].sub.m] through the terminal point, [Delta][C.sub.D], [Delta][R.sub.D], of defending alternative D, and determining whether terminal point, [Delta][C.sub.C], [Delta][R.sub.C], of challenging alternative C lies above, on, or below the marginal comparison line. The worse alternative is eliminated and the better alternative is retained as a defender against another challenger. This method of sequential elimination, also called dynamic programming, enables one to determine the best of N alternatives from N-1 binary comparisons. At each binary comparison, the worse alternative is known not to be best and can be eliminated. After eliminating N-1 worse alternatives, only the best one is left. It is important to note that the determination of the best mutually exclusive alternative is independent of the sequence of binary comparisons and the initial choice of defender and challenger. The capital budgeting procedure is the same for every project of an organization, but the costs of borrowing money, the capital constraint, and the marginal comparison slope, [[null set].sub.m] are parameters that must be evaluated specifically for each organization.

Which are the best projects to do?

The marginal comparison slope, [[null set].sub.m], enables the best way of doing each project to be determined independently of all other projects of an organization. But this cannot be determined until the best ways of doing every project of an organization have been determined. The projects are ranked in descending order by capital efficiency or relative profitability, [Delta]R/[Delta]C, to form a convex envelope from which the sequence of best projects are selected.

Which projects should be funded?

The projects should be funded in descending order of the slopes of the vectors that form the convex envelope. If funding for those projects exceeds the range of feasible capital constraints, then the magnitude of the marginal comparison slope should be increased to form a convex envelope whose vectors have a smaller capital cost, [Sigma][Delta]C, but a greater average capital efficiency, [Sigma][Delta]R/[Sigma][Delta]C. Conversely, if funding for projects of vectors that form the convex envelope is less than the range of feasible capital constraints, the magnitude of the marginal comparison slope should be decreased to form a convex envelope whose vectors have a greater total capital cost, [Delta]C, but a smaller average capital efficiency, [Sigma][Delta]R/[Sigma][Delta]C.

Comparisons to current methods of capital budgeting

Current methods of capital budgeting are based on multiple criteria such as net present values, rates of return, benefit/cost ratios, and payback periods. If multiple criteria always resulted in the same choices of alternatives, they would be regarded as a single objective. But multiple criteria often point to different alternatives in important practical situations. Consequently, current methods of capital budgeting use weighted systems of multiple criteria, which readily give way to subjective analyses.

Many capital budgeting difficulties stem from differences between managerial and financial accounting. Costs of borrowed money in financial accounting are strongly grounded in fact. Discount rate forecasts in managerial accounting of the MARR and the WACC are much higher than the costs of borrowing money that would be predicted from market rates of interest. As a result, managerial accounting forecasts of discount rates have practically no chance of matching costs of borrowed money in future financial accounting statements of an organization.

The conceptual differences between discount rate forecasts and future costs of borrowed money spill over to forecasts of cash flows from project alternatives. Risk and inflation are incorporated discount rates for groups of projects instead of tailoring those estimates for the cash flows of individual project alternatives. Moreover, opportunity cost expectations of equity investors further inflate discount rate forecasts. The problem with artificially high discount rates is that they obscure the future by making long-term investment opportunities look unprofitable. In contrast, discount rates based on the costs of borrowing money enable one to make the best use of available engineering and financial information.

The method of convex-envelope capital budgeting provides a practical means of comparing all project alternatives due to the uniformity of its implementation and the economy of its computational cost. The method is applicable to a variety of organizational structures, and it serves to unify the language of capital budgeting so that all parts of an organization can analyze the future with a single decision-making tool.

For further reading

Goldberg, B., "Economic Decision-making for Engineering and Financial Management - A Cause-Effects Approach," (in preparation).

Littleton, A.C., "Structure of Accounting Theory," Monograph No. 5. Thirteenth Printing, American Accounting Association, 1985.

Paton, W.A., and A.C. Littleton, "An Introduction to Corporate Accounting Standards," Monograph No. 3, Twenty-second Printing, American Accounting Association, 1991.

Canada, J.R., W.G. Sullivan, and J.A. White, Capital Investment Analysis for Engineering and Management, Second Edition, Prentice-Hall, 1996.

Clark, J.J., Hindelang, T.J., and Pritchard, R.E., Capital Budgeting - Planning and Control of Capital Expenditures, Third Edition, Prentice-Hall International, 1989.

DeGarmo, E.P., W.G. Sullivan, J.A. Bontadelli, and E.M. Wicks, Engineering Economy, Tenth Edition, Prentice-Hall, 1997.

Grant, E.L., W.J. Ireson, and R.S. Leavenworth, Principles of Engineering Economy, Eighth Edition, John Wiley & Sons, 1990.

Riggs, J.L., D.D. Bedworth, and S.U. Randhawa, Engineering Economics, Fourth Edition, McGraw-Hill, 1996.

Thuesen, G.J., and Fabryckv, W.J., Engineering Economy, Eighth Edition, Prentice-Hall, 1993.

Bernard Goldberg, Ph.D., teaches engineering economy courses in the Industrial Engineering Department at the University of Houston in Houston, Texas and is a senior member of lIE.