*3.3. Evolution of Return on Equity*

In our model, shareholders use the return on equity (ROE) recorded by the company in their analyses regarding the quality of the management's decision (decisions supported by Classes A and D of shareholders) (see Section 3.2). We consider that the company is 100% equity-financed, so assets = equity. Retained earnings are the sole source for financing the investment projects (Walter 1956). ROE, as a proxy for the company's level of performance, is determined by different factors, more or less anticipable. In an over-simplified model (but useful for our application), we can consider ROE as a cumulative effect of: (i) the normal (usual, unaffected by the decider's bad decision) ROE; (ii) the decrease in ROE determined by making the bad decision; and (iii) the impact of other factors, which are acting independently by the first two components. In other words, ROE is affected by bad decisions, but also by bad forecasts.

The initial level of *ROE0* can be considered as a benchmark. It is determined by the ratio between net earnings recorded in the year 0 (*NE*0) and the level of equity in the previous year:

$$ROE\_0 = \frac{NE\_0}{TE\_{-1}} = \frac{NE\_0}{TA\_{-1}}\tag{5}$$

We consider the period [−t, 0] as being one characterized by making good decisions. As such, in this study, these historical levels of performance are important only for comparing the actual results with them. Moreover, we can assume that all the quantity of information is incorporated in this initial level of ROE, *ROE*0. The level of total assets at the moment 0, *TA*<sup>0</sup> is considered as an initial stock of shareholders' wealth and is another input in the model.

However, *ROE*<sup>0</sup> is only a punctual benchmark. Some aleatory factors can influence even this normal ROE. Even if the company maintains constant its level of assets, most probably *ROE*<sup>1</sup> will be different than *ROE*0. Even if it is considered that no bad decisions are made, it can be assumed that *ROEt* is a random variable. Each year, *NEt* is determined by the *ROEt* (the return at which the capital is invested) and the stock of capital in the previous year, by the accounting identity:

$$NE\_t = \, ROE\_t \cdot TA\_{t-1} \tag{6}$$

The level of total assets on one moment t (*TAt*) can be determined as a function by the level of the total assets in the previous year (*TAt*−1), net earnings (*NEt*) and dividend payments (*DIVt*):

$$TA\_t = TA\_{t-1} + NE\_t - DIV\_t \tag{7}$$

In another form:

$$TA\_t = TA\_{t-1}(1 + ROE\_t) - DIV\_t \tag{8}$$

<sup>7</sup> Individuals prefer in many situations the status quo (the "anchoring" effect) (Samuelson and Zeckhauser 1988). As an effect, in this paper, we considered that Class C agents do no change their voting preference instantly. In a different context, Harari (2015, pp. 264–67) provides historical evidence regarding this "anchoring" effect and explains it by a necessity of human beings to make sense of their decisions. If they should accept the fact that their past decision was wrong, they should accept that their past "sacrifices" were unuseful.

As an effect, *NEt* has *t* components: (1) a component resulted as the return of equity at which initial stock of assets is invested; (2) the result of the investments made in the first financial exercise (*t* = 1); and (3) the result of the investments made in the second financial exercise (*t* = 2), ... , (*t*) the result of the investments made in the last financial exercise (*t* = *t* − 1). The components (2), (3), ... (*t*) are a function of the internal rates of returns (IRR) corresponding to the invested capital in each financial exercise.

In our model, shareholders decide to invest in the company if the invested capital will determine an increase of their wealth higher than investing on the financial market in projects of similar risk (Walter 1956; Ross et al. 2010) 8. In this study, we assume that the decisions at the AGM are made comparing the expected IRR, respectively, *Et*−*1(IRRt),* with the expectations regarding the capital market return, *Et*−1(*kMt*).

It can be noticed that *IRRt* - *Et*−1(*IRRt*). *IRRt* = *Et*−1(*IRRt*) only in the case of a perfect forecast. Since management is making a bad decision, *IRRt* can be considered as a random variable, normally distributed, with a mean [*Et*−1(*IRRt*)·(<sup>1</sup> <sup>−</sup> *bd*)] and a finite standard deviation9,10.

As an observation, in this equation, *Et*−1(*IRRt*) is the expectation made by the Class A agents, and it is not optimal (it can be assumed by Class B shareholders make a better prediction).

The bad decision is modelled through a coefficient, *bd*, which is applied to the estimated rate of internal return of the new projects. For instance, 0.1 means that the expected internal rate of return is over-valuated by 10% by the deciders.

Similarly, market return can be considered as a random variable, normal distributed, with a mean [*Et*−1(*kMt*)] and a finite standard deviation11.

Table 3 presents *TA*, *NE*, and *ROE* in some financial exercises and their rules of evolution12. It can be noted that the bad decisions have impact only if the net earnings are invested in the company. For a numerical simulation of the financial indicators, see Table 4.
