Abstract
This chapter addresses the issue of choosing between mutually exclusive projects and ranking a bundle of projects. We show that ranking projects with RI and NPV is rational; in contrast, ranking projects with NFV is rational if and only if the competing projects have the same cost of capital. Then, we show that ranking projects with rates of return is not rational. Nevertheless, it is possible to obviate these limitations with two techniques: An incremental or differential analysis, whereby an iterated pairwise comparison of projects is accomplished, taking into account the differences between project’s incomes and projects’ capitals; the second method is a direct method, which standardizes the project scale so that the relative measures of worth become comparable. The incremental method informs about the sources of incremental value of one project over another, something that the absolute measures of worth cannot provide.
It can be shown that the scale effect is present in all project comparisons even between projects with identical outlay.
Keane (1979, p. 55)
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Notes
- 1.
If \(I^{e, j}_0 \ne 0\), \(\text {NPV}^{e,j}=V^{e, j}_0+I^{e, j}_0-C^{e, j}_0=\sum _{t=0}^n F^{e,j}_t \cdot \text {d}^{e,j}_{t, 0}\).
- 2.
The use of \(n=\max [n_h, n_k]\) as a common date for the computation of net future value does not solve the problem (possibly, it worsens it).
- 3.
In this particular case, NFV ranking is equivalent to NPV and RI ranking only because the magnitudes of the COCs and the number of periods are not sufficiently dissimilar to cause a ranking’s reversal.
- 4.
More precisely, remembering the arguments in Sect. 5.3:
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if the firm announces the undertaking of project A, a normal market will acknowledge that its value is \(\$130.2\) so the share price will increase by \((\$130.2-\$100)/n_0=\$30.2/n_0\). The firm’s new share price, after the equilibrium is restored, will be \(p_0+\$30.2/n_0\)
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if the firm announces the undertaking of project B, the market will acknowledge that its value is \(\$45.8\) so the share price increases by \((\$45.8-\$0)/n_0=\$45.8/n_0\). The firm’s new share price will be \(p_0+\$45.8/n_0\)
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if the firm announces the undertaking of project C, the market will acknowledge that its market value is \(\$\)57.9 so the share price will increase by \((\$57.9-\$30)/n_0=\$27.9/n_0\). The firms’ new share price will be \(p_0+\$27.9/n_0\).
In order to maximize shareholders’ wealth, managers must maximize the share price, which is equivalent to maximizing NPV.
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- 5.
The zero symbol here discriminates the region of value creation from the region of value destruction. Economically, it means that investing in C and B is better than investing in the respective null alternatives (assets equivalent in risk to C and B, respectively) whereas investing in A is worse than investing in its null alternative.
- 6.
Negative taxes for this project means that the firm will pay less taxes than it would pay if it rejected the project.
- 7.
Keane (1979) also realized that the project scale is expressed by the total capital: “Profitability Index fails because it incorrectly equates scale with initial outlay. 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). This is exactly what we have called, in this book, economic (or financial) efficiency. Interestingly, the proposal of the author for measuring the overall capital was to consider the discounted sum of \(\mathcal {C}_t= \mathcal {C}_{t\!-\!1}(1+r_t)-F_t\). The latter is exactly the capital base of the benchmark system \((\varvec{\mathcal {C}}, {\varvec{I}^{\varvec{\mathcal {C}}}}, {\varvec{F}^{\varvec{\mathcal {C}}}})\), associated with AROI (see also the RP-AIRR introduced in Magni 2016a).
- 8.
In this case, the project’s average ROIs, the investment scales, and the MARRs coincide in the three approaches (AIRR, IARR, and AROI) because we have assumed \(r=0\%\).
- 9.
- 10.
If the COC is constant across periods and equal across assets (i.e., \(r^h_t=r^k_t=r\)) then IARR and AROI coincide: \(\bar{\jmath }^{h-k}_\text {d}=I_\text {d}/C_\text {d}=I/C=\bar{\jmath }^{h-k}\).
- 11.
That is, one first sums and then discounts the amounts from n to 0, as opposed to the AIRR approach where amounts are first discounted from t to 0 and then summed. See also Sect. 10.5.
- 12.
Worshippers of IRR might be relieved in knowing that each of the three relative paradigms (AIRR, IARR, AROI) supply a lifeline for IRR to overcome this problem. Simply as it is, the incremental MARRs associated with incremental IRRs may be computed as weighted means of the market period returns, \(r_t\). We crossrefer the reader to Magni et al. (2018, Sect. 4.1) for details. Needless to say, all the other shortcomings of IRR may not be healed. (See also Ben-Horin and Kroll 2017 on NPV-consistent methods of project ranking. See Foster and Mitra 2003 on dominance conditions independent of the discount rate.)
- 13.
The two cash-flow streams of these projects have appeared in White et al. (2014, p. 223, Example 6.5).
- 14.
The rates \(\xi ^{A-B}\) and \(\epsilon ^{A-B}\) appear to be equal, for the numbers are rounded. More precisely, \(\xi ^{A-B}=-6.797\%\) and \(\epsilon ^{A-B}=-6.802\%\).
- 15.
This is equivalent to straight-line depreciation.
- 16.
See Marchioni and Magni (2016) on strong NPV-consistency of this rate of return in the AIRR approach.
- 17.
Note that \(\bar{\jmath }(K)=(2/(n+1))\cdot \bar{\jmath }(C_0)\).
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Magni, C.A. (2020). Ranking Projects. In: Investment Decisions and the Logic of Valuation. Springer, Cham. https://doi.org/10.1007/978-3-030-27662-1_11
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DOI: https://doi.org/10.1007/978-3-030-27662-1_11
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