An introduction by F1F9: We were approached by Thierry Renard and Carlo Alberto Magni and were intrigued by what they had to say: a new approach to calculating an internal rate of return. What’s more, they were illustrating the application of the Average Internal Rate of Return (“AIRR”) using Excel and the FAST Standard.

So while we are still evaluating what impact AIRR might have on our own financial modelling, we are very pleased to share their work as a guest blog.

A COMPREHENSIVE APPROACH TO VALUATION AND DECISION-MAKING

A view on the AIRR by Storewatt

Storewatt develops investments in renewable energy projects, especially self-consumption photovoltaic projects for large industrial customers. In the process of looking for the best methodology to assess these projects’ financial viability, we came across the AIRR and, after thorough investigation, decided to adopt this methodology. The AIRR enables us to analyze the project’s financial viability and efficiency directly using the Income statement, which is especially suitable for self-consumption projects where the main benefit is from avoided cost of purchasing electricity from the grid, i.e. a reduction in expenses in the Income statement. With the AIRR, we can analyze the value created by the investment as well as the efficiency of that investment, i.e. the excess return from the industrial customer’s cost of capital earned on the invested capital.

As an eager adopter of the FAST modeling standard, we would like to share the AIRR methodology with the FAST community so that they can also benefit from this new and highly effective way of analyzing the merit of investment projects.

Thierry Renard, General Manager – Storewatt

AIRR is a new approach for financial modelling. In particular, it enables measuring the economic profitability of any project and assess shareholders value creation while extracting several of pieces of economic information. The approach avoids the usual pitfalls associated with IRR and enriches the traditional NPV analysis. It has the following features:

- • It guarantees existence and uniqueness of the rate of return
- • It provides both the project rate of return (
*project AIRR*or*average ROI*) and the equity rate of return (*equity AIRR*or*average ROE*) - • It leads to economically rational decisions
- • It is consistent with the NPV and decomposes the latter into project scale and project’s economic efficiency
- • It is consistent with the input data and the estimates on incremental revenues and costs
- • It copes easily with time-varying cost of capital
- • It may be applied to financial assets as well as real assets

**There are 4 equivalent ways for computing the AIRR**

**AIRR 1: Total Income divided by Total Capital (****click here for related video****) **

Consider a single-period project and let *r* be the cost of capital (COC) or minimum attractive rate of return (MARR), representing the expected rate of return of an equivalent-risk asset.

AIRR may be viewed as a straightforward generalization of a single-period rate of return. By definition, a rate of return is the ratio of aggregate (i.e., total) return or total income to the aggregate capital. If the project lasts one year, the total capital (TC) invested is *C*_{0}, at the beginning of the first period, while the total income (TI) generated in the second period is *I*_{1}.

The rate of return is then

If the project lasts 2 years, it means that some capital *C*_{1} remains invested in the project at the beginning of the second period, generating an additional income *I*_{2} in the second period:

Now, just take (1) and add *I*_{2} to the numerator and *C*_{1} to the denominator (after due discounting by one period, given that there is a lag of one period between first income and second income as well as between first capital and second capital). The result is the two-period project’s rate of return:

If the project lasts 3 periods, it means that some additional capital *C*_{2 }remains invested at the beginning of the third period generating additional income *I*_{3} in the third period:

Just take (2) and add *I*_{3} to the numerator and *C*_{2} to the denominator (after due discounting by two periods) to get the project’s rate of return:

In general, for an *n*-period project, it suffices to add, for each period, the income and the associated capital (properly discounted) and the project’s rate of return is

**AIRR 2: Average ROI (weighted mean of single-period rates) (****click here for related video****)**

AIRR is an average return on investment. Let

be the single-period rates of return generated in each period by the project (often called Return On Investment, ROI). The project’s overall rate of return is the weighted mean of the ROIs, where the weights are the present values (PV) of the capitals invested in each period, PV[*C*_{t}] = *C _{t}*(1 + r) –

^{t}:

Therefore, the project AIRR is an average ROI (likewise, considering the equity capital one gest the equity AIRR; which is an average Return On Equity, ROE, expressing the shareholders overall rate of return).

**AIRR 3: Starting from instantaneous rate of return ****(****click here for related video****)**

One may compute the AIRR by using the instantaneous project’s rate of return and converting it to an annual rate of return. Let PV[*C*_{t}] = *C _{t}*(1 +

*r*) –

^{t}and PV[

*I*

_{t}] =

*I*(1 +

_{t}*r*) –

^{t }be the present value of the capital invested and the present value of the income generated by the project. Dividing the present value of all profits by the present value of all capitals, one gets the instantaneous AIRR

It is *as if* the investor invested a total capital of TC = PV[C] at time 0 while generating an immediate return equal to PV[I]. Multiplying by (1 + *r*), the instantaneous AIRR is converted into an annual AIRR:

For example, consider a two-year project and let *I*_{1} = 225, *I*_{2} = 375 be the profits and *C _{0}* = 600,

*C*= 400 the capital at the beginning of each period. If the MARR is

_{1}*r*= 15%, then PV[I] =

and , whence

As 60.2% > 15% the project is worth undertaking and the NPV is 947.8 · (60.2% – 15%) / 1.15 = 372.6.

**AIRR 4: Starting from cash flows (****click here for related video****)**

The AIRR can be computed via cash flows as well. First, one needs compute the cash flows from incomes and capitals: *F _{t}* =

*I*– (

_{t}*C*–

_{t}*C*). Then, consider the following index, obtained as the discounted sum of the cash flows divided by the total capital:

_{t-1}The numerator is the so-called net present value (NPV). The above ratio measures the NPV per unit of invested capital. It is an excess (instantaneous) rate of return. It measures the net gain over and above the cost of capital referred to time 0. Multiplying it by (1 + *r*) one gets the annual excess rate of return. Then, adding the cost of capital, one gets the project’s rate of return:

**Value creation and decision-making**

**AIRR decision criterion. **The project is worth undertaking (i.e., it creates value) if and only if AIRR > *r*.

**NPV consistency. **The net present value (NPV) may be obtained as

which implies that NPV > 0 if and only if AIRR > *r*. This establishes the link of AIRR with NPV and, therefore, the NPV-consistency of AIRR. In addition, the above equation shows a significant decomposition of NPV into two fundamental value drivers:

• the project scale (how much capital is invested)

• the economic efficiency (how the project’s rate of return compares with the MARR rate of return)

In such a way, the analyst is informed about whether a given NPV is the result of a large-size investment at a small rate of return or a small-scale investment at a high rate of return (or anything in between these extremes).

**DOWNLOADS**

– Example AIRR calculation in a model following the FAST Standard

About AIRR

The AIRR approach has been devised, conceptualized, and developed by Carlo Alberto Magni in Magni (2010, *The Engineering Economist*). The paper has received the “Eugene L. Grant” Award in 2011 from the Engineering Economy Division of ASEE (American Society for Engineering Education) as best paper published in *The Engineering Economist* in 2010. A vast array of papers has appeared in the literature since then which expanded the approach in various directions. Here is a list of papers on AIRR updated at March 23 2018 (ascending chronological order).

Magni CA. 2010. Average Internal Rate of Return and investment decisions: a new perspective.

*The Engineering Economist*, 55(2), 150‒181.

[Ranked n. 1 in the list of most read papers and ranked n. 10 in the list of most cited papers of *The Engineering Economist*]

Magni CA 2010. On the long-standing issue of the Internal Rate of Return: A complete resolution. *Proceedings of XXXIV AMASES Conference*, Macerata.

Magni CA. 2011. Addendum to “Average Internal Rate of Return and investment decisions: A new perspective”. *The Engineering Economist*, 56(2), 140–169.

Magni CA 2011. Using Average Internal Rates of Return for investment performance measurement and attribution. *Proceedings of XXXV AMASES Conference*, Pisa.

Guerra ML, Magni CA, Stefanini L 2011. Average Internal Rate of Return with interval arithmetic, *Proceedings* *of XXXV AMASES Conference*, Pisa.

Magni CA 2011. Return On Equity, Internal Rate of Return and shareholder value creation*. **Proceedings of EAA 2011 Annual Congress*, Rome 20-22 April, European Accounting Association.

Guerra ML, Magni CA, Stefanini L. 2012. Average rate of return with uncertainty, Advances in Computational Intelligence, part IV, Berlin-Heidelberg: Springer-Verlag, pp. 64-73, *Proceedings of 14th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems *(IPMU 2012), Catania, July 9-13.

Altshuler D, Magni CA 2012. Why IRR is not the rate of return on your investment: Introducing the AIRR to the Real Estate community. *Journal of Real Estate Portfolio Management*, 18(2), 219‒230.

Magni CA 2013. The Internal-Rate-of-Return approach and the AIRR paradigm: A refutation and a corroboration. *The Engineering Economist*, 58(2), 73‒111

[Ranked n. 2 in the list of most cited papers of *The Engineering Economist*]

Magni CA 2013. Generalized Makeham’s formula and economic profitability*. **Insurance: Mathematics and Economics*, 53(3) (November), 747‒756.

Magni CA 2014. Arithmetic returns for investment performance measurement. *Insurance: Mathematics and Economics*. 55 (March), 291‒300.

Magni CA. 2014. Mathematical analysis of average rates of return and investment decisions: The missing link. *The Engineering Economist*, 59(3), 175‒206.

Guerra ML, Magni CA, Stefanini L. 2014. Interval and fuzzy Average Internal Rate of Return for investment appraisal, *Fuzzy Sets and Systems*, 257,217-241.

Pressacco F, Magni CA, Stucchi P. 2014. A quasi-IRR for project without IRR. *Frontiers in Finance and Economics*, 11(2), 1-23.

Magni CA 2015. Investment, financing and the role of ROA and WACC in value creation*, **European Journal of Operational Research*, 244(3) (August), 855-866.

Magni CA, Veronese P, Graziani R 2016. Chisini mean and a unified approach to capital budgeting criteria. *Nineteenth International Working Seminar on Production Economics*, Innsbruck, February 22-26.

Cuthbert J, Magni CA. 2016. Measuring the inadequacy of IRR in PFI schemes using profitability index and AIRR. *International Journal of Production Economics*, 179, 130-140.

Magni CA 2016. Capital depreciation and the underdetermination of rate of return: A unifying perspective*. **Journal of Mathematical Economics*, 67 (December), 54-79.

Magni CA, Veronese P, Graziani R 2018. Chisini means and rational decision-making: Equivalence of investment criteria. *Mathematics and Financial Economics*, 12(2), 193-217.

Marchioni A, Magni CA 2018. Investment decisions and sensitivity analysis: NPV-consistency of rates of return, *European Journal of Operational Research*, 268, 361-372.

Cuthbert J, Magni CA 2018. Some problems of the IRR in measuring PEI performance and how to solve it with the pure-investment AIRR*. **Journal of Performance Measurement*, Winter 2017/2018, 39-50.

Magni CA (in progress). *Project Appraisal and the Logic of Valuation. Linking Finance, Accounting, and Engineering Economics*. Springer UK.

Carlo Alberto Magni’s Personal Webpage

Carlo Alberto Magni’s SSRN Author Page

Carlo Alberto Magni’s You Tube Channel

Carlo Alberto Magni’s Email

## Comments

It would be interesting to compare IRR with CAGR in the article. CAGR seems to be used more often in my experience, but often not for the right reasons. Part of the reason is that IRR cannot be computed analytically: it requires successive approximations until a value can be calculated: something some online calculators can do for you (e.g. https://www.gigacalculator.com/calculators/irr-calculator.php) or you can code yourself in Excel if you know macros. Regardless, it is harder for people to understand and since people are generally less likely to use something they don’t understand well or can implement easily in their tools internal rate of return sees less usage than it should.

Thanks for your comment. I am not sure what definition of CAGR (Compound Annual Growth Rate) you refer to and whether your refer to a capital budgeting project or a financial investment. Assuming it is the former, if the investment is made of multiple cash flows, a CAGR may be obtained by making some assumptions about reinvestment of cash flows. The resultant rate is known as Modified Internal Rate of Return. However, there are several pitfalls of MIRR which I have described in my papers (as well as other authors). One of these is that it is not clear what the reinvestment rate should be. Another is that, if the investment is described as a stream of multiple cash flows, it means that cash flows are periodically distributed to the investors, which precisely means that they are not reinvested by the firm, so the assumption of reinvestment is not acceptable. A third pitfall is that the implied capital amounts of MIRR are not tied to the actual transactions made by the firm and assumes that the growth of the invested capital is constant through time.

As a result, the MIRR may be interpreted as the hypothetical project’s rate of return that would arise if (i) the project’s cash flows were reinvested, (ii) the reinvestments were made at some (not well specified) reinvestment rate, and (iii) the growth of rate of the capital were constant through time.

The IRR is better than the MIRR, for it does not suffer from the first two pitfalls mentioned above.

The AIRR avoids all problems of IRR and MIRR.

Carlo Alberto Magni