Normalized profitability metrics provide a basis for comparing the efficiency of capital investments, but they are often misunderstood. New interpretations of these metrics can help engineers to make more informed decisions
In a financial sense, a large capital project can be viewed as a series of cash flows into and out of an enterprise. The cash flows vary in timing and size, depending on the project and its characteristics. Cost engineers mathematically reduce these cash-flow patterns into discrete profitability metrics, such as net present value (NPV) and internal rate of return (IRR) so they can be more easily compared. However, such metrics are often misunderstood.
This article uses graphs to illustrate practical interpretations of three different metrics — two standard and one recently developed. Figure 1 presents a typical series of cash flows, which are the basis for the examples given in all other graphs in the article. As shown in Figure 1, cash flow for a typical capital project has four stages:
- One to two years of negative cash flow while capital is committed
- One or more years of subnormal cash flow during startup
- Some period of normal operation
- A final year with higher-than-normal cash flow, when working capital and salvage value are recovered
Two profitability metrics are needed to represent a capital project: an absolute metric and a normalized metric. The absolute metric represents the lifetime value that the project will add to the enterprise. The normalized metric represents the efficiency of the investment. While NPV is generally accepted as the industry standard for absolute metrics, there is still debate and confusion about the best choice of normalized metric.
Compounding investments versus capital projects
This article draws a distinction between compounding investments and capital projects. Your retirement account is a compounding investment. You invest an initial principal and the principal grows over time. As principal accumulates, it grows faster. A capital project is like opening a restaurant. You invest in a kitchen, dining room, initial stock of ingredients and so on, and then each year a certain amount of people pay to eat at your restaurant.
There are two important distinctions here:
- Unlike compounding investments, capital projects begin with an effectively non-recoverable commitment of funds
- For a capital project, a given year’s profit is effectively independent of the
previous year’s profit, not enhanced by it
For most practitioners, the default normalized metric is IRR [1, 2]. IRR is useful in certain situations, but its popularity is more likely due to organizational legacy, as well as a familiar name that suggests intuitive value. The primary alternative to IRR is modified internal rate of return (MIRR). While MIRR has gained enough traction to earn its own function in Microsoft Excel, it still mostly appears as a footnote to discussions of traditional IRR. This is unfortunate, because MIRR actually provides the intuitive value that many people attribute to traditional IRR.
This article first clarifies the practical significance of IRR and MIRR using graphs. It then examines a third metric that follows logically from this discussion, net present value percent (NPV%). NPV% was recently proposed by D.A. Mellichamp of the University of California Santa Barbara  to overcome the shortcomings of IRR and MIRR. The goal of this article is to help practitioners make informed decisions regarding profitability metrics and to give them the language to explain their results to others.
Internal rate of return
If the project was financed at a constant compound interest rate, IRR is the interest rate at which it would break even. IRR may be the most widely used profitability metric, but it may also be the most widely misunderstood. The name “internal rate of return” (as well as many sources) suggest that you might treat it as a rate of return on a compounding investment, like a retirement account. This comparison may be convenient, but it is not appropriate for capital projects. The major problem is not obvious from the mathematical definition, as shown in Equation (1):
In Equation (1), CFt is cash flow in year t and n is project length in years. The formula can only be solved numerically, and it can have multiple, zero, or even impossible solutions, in some cases. Simply put, IRR is the discount rate for which the NPV of the project is equal to zero. “Simply put” does not always mean “simply understood” though.
It is instructive to analyze what happens when IRR is used incorrectly as a compound interest rate to calculate the final value of an investment. Figure 2 shows the actual cumulative return on investment (ROI) for the example project, alongside that calculated using IRR as a rate of return. The actual project has a lifetime ROI of 4.8, while using IRR yields a lifetime ROI of 13.4. Treating IRR as a rate of return implicitly assumes that profits can be reinvested at an interest rate equal to IRR for the remaining life of the project. This so-called “reinvestment assumption” has been discussed widely in literature, as well as in informal online channels. The reality is that IRR itself does not assume reinvestment. However, if you treat IRR as a rate of return, then you are assuming reinvestment.
If IRR is not a rate of return, then what is it? It can be described like this: if the project were financed at a constant compound interest rate, IRR is the interest rate at which it would break even. This statement is mathematically equivalent to the definition given above, and it is represented graphically in Figure 3. The red and blue bars represent draws from and payments to the loan — note that they are the same as the cash flows in Figure 1. The striped bars represent financing interest accrued at a rate of IRR on the previous year’s balance. The grey line represents the balance on the loan. Note that in the final year, it drops to zero. This is the break-even point.
Since IRR is effectively a finance rate, it can be quantitatively compared to other finance rates, like weighted average cost of capital (WACC). WACC is the blended cost of raising funds from debt and equity. If IRR is greater than WACC, then the project will be cash-flow positive. It is difficult to extract more meaningful information beyond this, though. In an analogous sense, evaluating a project using IRR can be likened to evaluating a job offer based on the highest mortgage rate it would allow you to afford without going bankrupt in the next ten years.
Modified internal rate of return
If the project capital were put in a compounding investment, MIRR is the interest rate that would provide the same lifetime ROI as the project. MIRR provides the intuitive value that is often attributed to IRR: it represents the interest rate on a compounding investment that would provide the same lifetime ROI as the project. Also, unlike IRR, the formula for MIRR can be solved analytically and it never has more than one solution, as shown in
In Equation 2, CFIt is cash flow in for year t, CFOt is cash flow out for year t, n is project length in years, k is reinvestment rate and r is finance rate.
For compounding investments, MIRR is the geometric mean rate of return, but it does not retain this meaning when used for capital projects. For capital projects, MIRR’s value is limited to providing a convenient way to compare compounding investments on an equal time scale. Figure 4 shows actual ROI and MIRR-estimated ROI for the example project. Both reach the same final value of 4.8, but the lines do not track one another. Actual ROI begins at –1, whereas the MIRR approximation begins at zero. Furthermore, as can be seen by comparing the slopes of the two lines in corresponding years, MIRR underestimates the yearly rate of return.
So, while the analogy is not perfect, MIRR is still a useful tool for conveying information in a world where investors often think in terms of compound interest rates. To make another analogy, evaluating capital projects using MIRR is like evaluating a job offer based on the interest rate that would earn the same amount over ten years in your retirement account.
Net present value percent
NPV% is the average percent of the investment that is added to an enterprise each year over the life of a project. Capital projects do not earn compound interest — the ROI for any given year is independent of the previous year’s ROI, not enhanced by it. It is therefore more appropriate to evaluate each year’s ROI in relation to the initial capital investment instead of the previous year’s cumulative cash flow, and likewise, to use the arithmetic mean instead of the geometric mean. The metric NPV% is derived from these principles and can be thought of as the average percent of the investment that is added to an enterprise each year over the life of a project, as given in Equation (3):
In Equation 3, PVTCI is the present value of the total capital investment and n is the length of the project in years. Figure 5 shows the graphical interpretation of NPV%. NPV% is a direct measure of the investment efficiency. Using NPV% to evaluate a capital project is like evaluating a job offer based on the average present value of your expected salary over the next ten years. Ref. 3 and 4 provide additional useful applications of NPV%. Normalized profitability metrics provide a basis for comparing the efficiency of investments — past, present, and future. So, it makes sense to use metrics that are familiar to your organization and management team. At the same time, though, consider experimenting with other metrics like NPV% or average internal rate of return (AIRR)  that might prove to be more meaningful. It only takes a few extra key-strokes and can provide significant value.
Edited by Mary Page Bailey
1. Graham, J.R. and Harvey, C.R., How Do CFOs Make Capital Budgeting and Capital Structure Decisions?, Journal of Applied Corporate Finance, Vol. 15, no. 1, March 2002, pp. 8–23.
2. Farragher, E.J., Kleiman, R.T., Sahu, A.P., Current Capital Investment Practices, The Engineering Economist, Vol. 44, no. 2, 1999, pp. 137–150.
3. Mellichamp, D.A., New Discounted Cash Flow Method: Estimating Plant Profitability at the Conceptual Design Level While Compensating for Business Risk/Uncertainty, Computers and Chemical Engineering, Vol. 48, Jan. 2013, pp. 251–263.
4. Mellichamp, D.A., Internal Rate of Return: Good and Bad Features, and a New Way of Interpreting the Historic Measure, Computers and Chemical Engineering, Vol. 106, Nov. 2017, pp. 396–406.
5. Magni, C.A., The Internal Rate of Return Approach and the AIRR Paradigm: A Refutation and a Corroboration, The Engineering Economist, Vol. 58, 2013, pp. 73–111.
Chris Burk (Email: firstname.lastname@example.org; Website: www.burkengineeringllc.com) is a chemical engineer and an independent consultant. He works with companies that are developing or investing in new chemical and bioprocess technologies, helping them use techno-economic modeling to better understand their economics at a commercial scale. His clients have included venture capital firms, universities, laboratories, independent startups and startup incubators. He is also a member of Lee Enterprises Consulting’s team of renewable-fuel and chemical experts. Prior to consulting, Burk spent twelve years in industry, where he gained diverse experience in research and development, pilot plant engineering, procurement, construction and startups. He is a licensed Professional Engineer and he holds B.S.Ch.E. and M.S.Ch.E. degrees from Cornell University.
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