Chisini means and rational decision making: equivalence of investment criteria

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.

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

$$\begin{aligned} \bar{x}=q^{-1}(g(x_1,\ldots ,x_n)). \end{aligned}$$

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

$$\begin{aligned} NPV=\sum _{k=0}^n(c_{k-1}(1+i_k)-c_k)v_{k,0}=\sum _{k=0}^n (c_{k-1}(1+\bar{\imath })-c_k)v_{k,0}, \end{aligned}$$

(36)

where $c_{-1}=0$. Simplifying the constant and solving for $\bar{\imath }$, one gets

$$\begin{aligned} \bar{\imath }=\frac{\sum _{k=1}^n i_{k} c_{k-1} v_{k,0}}{\sum _{k=1}^n c_{k-1}v_{k,0}}=\frac{\sum _{k=1}^nC_k i_k}{\sum _{k=1}^nC_k} \end{aligned}$$

(37)

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

$$\begin{aligned} \begin{aligned} NPV&=\sum _{k=0}^n (c_{k-1}v_{k,0} - c_{k}v_{k,0}) +\sum _{k=0}^n i_k c_{k-1} v_{k,0}\\&=-\sum _{k=1}^n r_kc_{k-1}v_{k,0}+\sum _{k=1}^n i_k c_{k-1} v_{k,0}, \end{aligned} \end{aligned}$$

(38)

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

$$\begin{aligned} \sum _{k=1}^nr_kc_{k-1}v_{k,0}=\sum _{k=1}^n\bar{r} c_{k-1}v_{k,0}, \end{aligned}$$

we have

$$\begin{aligned} \bar{r}=\frac{\sum _{k=1}^n r_{k} c_{k-1} v_{k,0}}{\sum _{k=1}^n c_{k-1}v_{k,0}}=\frac{\sum _{k=1}^nC_k r_k}{\sum _{k=1}^nC_k}, \end{aligned}$$

(39)

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

$$\begin{aligned} NPV=\sum _{k=0}^n (i_k-r_k)c_{k-1}v_{k,0}=\sum _{k=0}^n \pi _k c_{k-1}v_{k,0}. \end{aligned}$$

(40)

Thus, the mean of the $\pi _k$’s that leaves unchanged NPV is:

$$\begin{aligned} \bar{\pi }=\frac{\sum _{k=1}^n C_k \pi _k}{\sum _{k=1}^n C_k}, \end{aligned}$$

and, using (37) and (39), $\bar{\pi }=\bar{\imath }-\bar{r}.$

From (40),

$$\begin{aligned} NPV=\sum _{k=1}^n (i_k -r_k)\cdot c_{t-1}v_{k,0}=\sum _{k=1}^n (i_k \cdot C_k -r_k \cdot C_k)=\sum _{k=1}^n (I_k-R_k)=I-R, \end{aligned}$$

and thus, since $\bar{\pi }$ is the Chisini mean that leaves unchanged NPV, Eq. (12) is proved. $\square$