Abstract
A plethora of tools are used for investment decisions and performance measurement, including net present value, internal rate of return, profitability index, modified internal rate of return, average accounting rate of return. All these and other known metrics are generally considered non-equivalent and some of them are regarded as unreliable or even naïve. Building upon Magni (Eng Econ 55(2):150–180, 2010a, Eng Econ 58(2):73–111, 2013)’s average internal rate of return, we show that the notion of Chisini mean enables these tools to be used as rational decision criteria. Specifically, we focus on 11 metrics and show that, if properly used, they all provide equivalent accept–reject decisions and equivalent project rankings. Therefore, the intuitive notion of mean is the founding basis of investment decision criteria.
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Notes
It is worth noting that, while the NPV notion is gold standard for capital budgeting purposes, the NPV notion is of limited use (and hardly ever mentioned) in investment performance measurement, when ex post assessment is accomplished of a portfolio or a fund managers’ skill. In these cases, only relative measures of worth such as rates of return or excess returns are relevant.
See also Barry and Robison [2].
In the case of \(\mathbf {F}\), the MIRR-implied capital is \(c_t=c_{t-1}(1+MIRR)=c_0(1+MIRR)^t\) so the total capital is \(C^{M^{*}}=\left[ c_0\frac{1-q^n}{1-q}\right] \) where \(q=(1+MIRR)/(1+r)\).
See also Gray and Dewar [15] for an axiomatization of the time-weighted rate of return.
See also the average ROA in Magni [29].
Note that this class is analogous to the equivalence class implied by the SRR, where values are not discounted. Therefore, \(\bar{\jmath }(c_0)\) can be viewed as an SRR, where discounting takes no place.
MD is also used as an approximation of the so-called Time-Weighted Rate of Return by chain-linking a series of Modified Dietz returns [42, p. 92].
Note that \(\text {v}_0\) is the first term of the linear combination and its weight is 1.
If \(C^{h-k}<0\), then one can consider the incremental project \(k-h\).
See Ben-Horin and Kroll [3] for a different incremental analysis for ranking projects and its relation to AIRR approach.
If \(C^j<0\), then \(\min _{1\le j\le m} \bar{\imath }^j\) replaces \(\max _{1\le j\le m} \bar{\imath }^j\).
It can be shown that both \(\bar{\imath }^j(K)\) and \(\bar{\pi }^j (K)\) can be framed as weighted means of period rates \(i_t^j (K)\) and \(\pi _t^j (K)\) that can be derived from the equivalence class [K]. However, the presented shortcut expressed by Eq. (35) is much more frugal from a computational point of view.
Lima e Silva et al. [21] apply the direct method to so-called pre-purchase financing pools, which are loans where the financed amount is the same for all the members of the pool. The authors supply a correct ranking of the various loans by choosing, as benchmark capital, the very amount distributed to borrowers.
It can be shown that, formally, \(\bar{\pi }^j (K)\) can also be viewed as the weighted mean of the excess return rates \(\pi _t^j\).
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The authors wish to thank an anonymous reviewer, who supplied invaluable remarks for revising the paper.
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Appendices
Appendix A: Chisini mean
The Chisini mean of n homogeneous values \(x_1,\ldots ,x_n\), with respect to the invariance requirement g, is the number (if it exists) \(\bar{x}\) such that \(g(\bar{x},\bar{x},\ldots ,\bar{x})=g(x_1,\ldots ,x_n)\), which is Eq. (3). The latter may have no solutions, so that the mean does not exist, or it may have several solutions and each one may be used as a mean. Indeed the solution exists and is unique if \(g(x,\ldots ,x)=q(x)\), with \(q(\cdot )\) continuous and strictly monotone. In this case, the mean is given by
Note that the Chisini mean is always consistent, in the sense that when all values \(x_1,\ldots ,x_n\) are equal to a, then \(\bar{x}=a\). Due to its general definition, it has no other specific properties. In particular it is not necessarily internal, i.e. included between the minimum and the maximum values of the observations, or associative. The latter property which, roughly speaking, states that a mean of n-data can be computed from the means of partial non overlapping subgroups of values, is however satisfied when the function q meets suitable conditions. This happens, for example, when the Chisini mean generates an arithmetic or a geometric mean as it occurs in our paper.
Appendix B: Proof of Proposition 4
Proof
Since, from (6), \(f_k=c_{k-1}(1+i_k)-c_k\), it follows that NPV is a function of the \(i_k\)’s and we can look for a mean of them which leaves NPV unchanged, for fixed \(c_k\)’s and \(v_{k,0}\)’s. Thus, considering the NPV as the invariance requirement (3), we have
where \(c_{-1}=0\). Simplifying the constant and solving for \(\bar{\imath }\), one gets
which is the project AIRR, i.e., the capital-weighted arithmetic mean given in (7).
Notice now that, from (36), NPV can also be written as
where the first term in the last equality is obtained using the expression of the \(v_{k,0}\)’s obtained by (10). Solving the invariance condition
we have
which is the market AIRR, i.e. the weighted arithmetic mean of forward rates found in (9).
Finally, letting \(\pi _t=i_t-r_t\) be the project’s excess return (rate) in period \([t-1,t]\), NPV can be written, from (38), as
Thus, the mean of the \(\pi _k\)’s that leaves unchanged NPV is:
and, using (37) and (39), \(\bar{\pi }=\bar{\imath }-\bar{r}.\)
From (40),
and thus, since \(\bar{\pi }\) is the Chisini mean that leaves unchanged NPV, Eq. (12) is proved. \(\square \)
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Magni, C.A., Veronese, P. & Graziani, R. Chisini means and rational decision making: equivalence of investment criteria. Math Finan Econ 12, 193–217 (2018). https://doi.org/10.1007/s11579-017-0201-4
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DOI: https://doi.org/10.1007/s11579-017-0201-4
Keywords
- Value creation
- Accept–reject decisions
- Project ranking
- Equivalence class
- Net present value
- Average internal rate of return