8. Average Internal Rate of Return

Actually, the adjective “internal” is somewhat redundant. It refers to the fact that the income rates \(i_t\) are internal to the project (they do not depend on r) and that AIRR is an average of those rates (note that AIRR is not internal, that is, it does depend on r). See also Sect. 9.​2 on this point.

In spreadsheet modelling, (8.14)–(8.15) are particularly easy to use, especially when the COC is time-varying (see Sect. 8.3): One just has to use the already available SUM PRODUCT for calculating the present value of incomes (income stream \(\varvec{I}\) times discount-factor stream \(\mathbf {d}\)) and then copy the same formula for capitals.

See also Magni (2010b, 2013a).

For tutorials see related YouTube Channel: https://​www.​youtube.​com/​channel/​UC9hfAZ8Sj5DwxUO​_​gIZUwxg.

See also Magni and Martin (2018).

Unlevered asset BAIRR might also be labeled financially Unlevered BAIRR (see Definition 2.​2).

We assume \(\text {PV}[C^e]>0\) (i.e., equity is invested). In case where \(\text {PV}[C^e]<0\), the equity capital is not raised from equityholders but, rather, lent to equityholders (i.e., equity is financed); then, the sign of the inequality is reversed.

For example, see Krugman (1979, p. 320), Shemin (2004, p. 15), Nosal and Wang (2004, p. 23), Smithers (2013, p. 66).

We have already computed the project NPV in Example 8.1 as \(V_0(\text {RI})\). If the project were unlevered, then project NPV and equity NPV would coincide: \(\text {NPV}=\text {NPV}^e=173.1\). Accordingly, the cost of equity and the WACC would coincide: \(r^{uo}_t=r^e_t\). Since the project is levered, the cost of equity is not equal to the WACC.

If, conversely, the cost of debt were equal to the ROD, then the equity NPV would be equal to the project NPV.

Numbers are rounded for illustrative purposes, as usual. Therefore, the manual computations do not produce an exact result. For a better approximation, one should consider more decimal digits. For example, the total capital is, more precisely, \(\$\)1,444,195.889, the average ROI is 11.29158%, the average WACC is 9.36347%, the first-period WACC is 8.8053897%, so that multiplying 1,444,195.889 by \((11.29158\%-9.36347\%)\) and dividing by \((1+8.8053897\%)\) one indeed gets 25,592.3. The use of a spreadsheet (e.g., Excel) makes this issue an idle one.

\(C_{t\!-\!1}\text {d}^{u}_{t, 1}\) and \(\text {PV}[C^u]\) are calculated by discounting capital values at WACC (\(r^{u}_t\)), not at \(r_t\) (in a levered project, \(r_t\ne r^{u}_t\)).

This does not diminish the practical usefulness of (8.19)–(8.20) for computing the rate of return and of (8.31)–(8.34) in case of time-varying COCs.

See details in Magni (2010b, 2013a, 2016b), Magni et al. (2018).

In a thoughtful paper, Keane (1979) realized that the ratio of NPV to the overall capital employed brings about an economically significant measure of efficiency: “If any ratio can be perceived as being the correct ‘cost benefit’ index it would be the NPV expressed as a ratio of the total number of units of capital employed” (Keane 1979, p. 54, footnote 7).

Do not confuse with \(\varvec{V}=(V_0,V_1, \ldots , V_{n-1}, 0)\).

Lindblom and Sjögren (2009) call this pattern of capital depreciation “strict market depreciation schedule”. See also Magni (2013a, 2014a), Magni et al. (2018).

See also the use of EAIRR in Magni (2013a, b, 2014a), Cuthbert and Magni (2016), Bosch-Badia et al. (2014), Barry and Robison (2014).

It is easy to verify that the BAIRR is \(\bar{\imath }\Bigl (\text {PV}[C]\Bigr )=\bar{\imath }(275.12) =7.71\%\), slightly smaller than the EAIRR. This is due to the fact that, for \(t\ge 1\), every book value is slightly smaller than the respective intrinsic value (with a negative numerator, a smaller denominator makes the ratio smaller).

As we know, the vector of incomes rates is such that \(-2+20 \cdot 1.16^{-1} -5\cdot (1.16 \cdot 1.13)^{-1} -75\cdot (1.16\cdot 1.13\cdot 1.08)^{-1} +5 \cdot (1.16\cdot 1.13\cdot 1.08\cdot 1.19)^{-1}=0\).

Capital and income are “kings” for a further reason: The estimates of cash flows in a capital asset investment or in a firm depend on the determination of the capital’s depreciation, according to the law of motion \(F_t=I_t-\varDelta C_t\): “the more general point is all changes in the balance sheet are relevant in the determination of cash flows because they come, in an important sense, logically prior to the estimate of cash flows.” (Magni and Peasnell 2012, p. 14). One may object that, if liquid assets are present, CFLs are not zero for all t. which means that the latter depend on decisions about payout policy as well as on accounting magnitudes. Furthermore, when dealing with financial investments, the principal (client) has full control on cash drawdowns and injections (as well as the length of the investment); in all these cases, accounting magnitudes are not at stake.

We sustain this objection. In actual facts, the project system consists of a network of projected accounting magnitudes and a set of decisions regarding the financing and distribution policy, which jointly determine the distributed cash flow (and, therefore, the value created). We then finally claim that the whole basic trinity \((\varvec{C}, \varvec{I}, \varvec{F})\) is the only true “king”.