Abstract
We present the Internal Average Rate of Return (IARR), and the Aggregate Return on Investment (AROI). We show that AIRR, IARR, and AROI are three consistent approaches which are associated, respectively, with incomes, cash flows, and capitals. The three methodologies are logically equivalent; they produce the same value and the same decision. As a result, the analyst has a complete toolkit of approaches for appraising projects, conducting refined economic analyses, and making rational decisions. This unified theory, based on an explicit link between project scale and financial efficiency, provides a complete reconciliation between absolute valuation methods and relative valuation methods.
Accounting measurements are, at this writing, the only basic sources of data which establish (however imperfectly) the income for a period, the amount of investment, and the bases of classification and matching which establish the rate of return currently being realized by operations or projects in which we are interested.
Vatter (1966, p. 682)
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Notes
- 1.
This is also true whenever fair-value accounting is applied. Fair-value accounting means that assets are evaluated at their fair (market) value, \(C_t=V_t\), and, therefore, the capital amounts (and the relative income rates) are, at the same time, internal and external.
- 2.
Interestingly, (10.12) holds even if there exists some t for which \(V_{t\!-\!1}=0\). The reason is that \(r_t\) is determined by demand and supply in the market, not by \(V_{t\!-\!1}\). In mathematical terms, \(r_t\) is not a function of \(V_{t\!-\!1}\) (in contrast, \(i_t\) is a function of \(C_{t\!-\!1}\) so it is not defined if \(C_{t\!-\!1}=0\)).
- 3.
More precisely (and with a pinch of pedantry): \(V_t=V_{t\!-\!1}(1+r_t)-F^V_t\) for \(t\in \mathbb {N}^0_n\) where \(F^V_t=F_t\) for \(t\in \mathbb {N}^1_n\) and \(F^V_0=-V_0\).
- 4.
For this project, if one uses the AIRR approach, one gets
$$ \bar{\imath }=\frac{90/1.1+80/1.1^2}{300+150/1.1}\cdot 1.1=37.3\%. $$The economic efficiency is \(\xi =(37.3\%-10\%)/1.1=24.82\%\) and the economic value created is \(\text {NPV}=436.36\cdot 24.82\%=108.3\).
- 5.
Unlevered asset BIARR might also be said to called financially unlevered BIARR (see Definition 2.2).
- 6.
In the case where \(C^e<0\), that is, the equity capital is not raised from equityholders but, rather, lent to equityholders, the sign of the inequality is reversed.
- 7.
A more precise value of the tax shield contribution is 0.273%. The error is due to rounding the average ROI and its MARR to the second decimal. More precisely, rounding to the third decimal, one gets \(\bar{\jmath }=12.417\%\), \( \bar{\rho }=12.144\%\) and their difference is precisely 0.273%.
- 8.
If one assumes, ceteris paribus, \(r^d=i^d=2\%\), then \(C^d_t=V^d_t\) for \(t=0, 1,2\) so that \(\text {NPV}^d=0\) and \(\text {NPV}=\text {NPV}^e=171.28\): Equityholders do not capture any additional value from debtholders.
- 9.
See Gray and Dewar (1971) for an axiomatization of the Time-Weighted Rate of Return.
- 10.
It has been introduced in Magni (2009e) and developed in Magni (2011a, 2015b, 2016a) (see an application to real estate assets in Althsuler and Magni 2015).
- 11.
It is worth reminding that, if \(I_0\ne 0\), the first internal capital does not comply with the mechanics of an economic system, since \(-F_0=C_0 (\sigma )\ne C_0=\varDelta C_0=I_0-F_0\).
- 12.
The three methodologies, as well as the three basic notions, are three in one: They are the same concept that may take on three personas (see also Remark 1.4 on Sect. 1.3). Thus, a debate about which of the three methodologies best serves the needs of practitioners boils down to debating whether capital or income or cash flow is preferable. From a logical point of view, they are equivalent; whether cash flows or incomes or capitals should be used in real life to make decisions is a practical issue. All six methods are now embodied in a unified logically consistent framework. Learning about similarities and differences between the approaches and their links with the three basic notions may bolster the understanding of the mechanics of the project/firm and the appreciation of six perspectives which provide different pieces of economic information but are naturally reconciled one another.
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Magni, C.A. (2020). Internal Average Rate of Return and Aggregate Return on Investment. In: Investment Decisions and the Logic of Valuation. Springer, Cham. https://doi.org/10.1007/978-3-030-27662-1_10
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DOI: https://doi.org/10.1007/978-3-030-27662-1_10
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