Effects on value of one-time additional retained earnings

Equation (12) can be used to calculate the terminal value according to the free cash flow ap-proach. The numerator comprises the free cash flow, which is distributed to investors of a fully equity-financed firm, minus the personal tax rate on dividends. The capital gains taxation is included in the growth rate reduction w 1 s√ ,( _g) at the denominator. As standard in the free cash flow approach, the effects of debt financing are fully embodied in the cost of capital. It reflects two aspects. First, the cost of capital relates to the tax deductibility of interests, which increases the amount that can be distributed to equity investors. Specifically, tax deductibility is captured by the cost of debt after personal and corporate taxes,

( ) ( ) ( )

s d

kd √ , < ,1 1 s √ √ ,kd 1 . Second, debt issuance replaces a reduction in dividend pay-ments. While the latter is tax-relevant for equity investors, debt provision has no fiscal conse-quences for debt investors. The resulting effect is measured by the growth rate reduction

( _d)

w 1 s√ , . The advantage of determining the terminal value with the free cash flow approach in (12) is that the market value at the beginning of the steady state can be calculated without circularity problems.

2.3 Effects on value of one-time additional retained earnings

Based on the terminal value model of the previous chapter, we now investigate how results change if we assume an earnings-based dividend policy. To illustrate the overall effects of ad-ditional retained earnings, we firstly assume one-time adad-ditional earnings retention x_{T 1}¹_∗ in pe-riod T 1∗ . Thus, x_{T 1}¹_∗ adds up to the retained earnings RE_{T 1∗} in (7), which were held back because of the financing of inflation-based and real growth. Similarly to RE_{T 1∗} , x_{T 1}¹_∗ serves as an equity financial contribution for additional investments and is complemented by

addi-Notice that for value neutrality it is not necessary that the risk related to the additional invest-ments corresponds to that of the existing investment program. So for example retained earnings can be also invested in risk-free securities. Additionally, the value neutrality of additional in-vestments can be complemented with specific conditions. For example, it can be assumed that the additional expected operating profit [±OP¹t 1∗ ] reflects a constant return (ROE) on the additional invested equity E IE[±¹t]:⁷

± ±

[ ¹t 1] [ ¹t]

E OP ∗ <ROE E IE√ for t T 1 T 2< ∗ , ∗ .... (14) The distribution of a constant proportion q of the additional operating profit generates addi-tional flow to equity:

± ±

[ FTE¹t 1∗ ]< √q E OP[ ¹t 1∗ ] for t T 1 T 2< ∗ , ∗ .... (15) According to the retention of (1 q E OP, √) [± ¹t 1∗ ] additional equity is invested. The change in book value of the additional invested equity in period t 1∗ can be expressed as follows:

± ± ±

Finally, we assume that the additional investments have the same operating risk as the existing investment program. With cost of equity ke₁^κ,s, we obtain the value increase in period T 1∗ :

° ^{, ,} ° ^,

As to equation (13), the additional retained earnings are value neutral if the following holds:

7 The argumentation is analog to the previous chapter. However, it refers to the return on equity and not to the return on invested capital to make the following comparison with similar studies in the literature possi-ble.

,

By considering that w^z < , √(1 q ROE) , the return on equity under given value neutrality yields:

,

This result is in line with the explanations in the handbook of auditors (note 412), which refer to Tschöpel et al. (2010).⁸ The ROE, which in our setting depicts the return on the invested equity, is derived as the cost of equity before taxes in the handbook of auditors.It is shown that the value of an initially available financial amount remains unchanged as long as the return of additional investments complies with (21).⁹ Precisely this was shown in the above analysis for the case that the available amount at the beginning of the observation period equals the addi-tional retained earnings in period T 1∗ . Therefore, the example in the handbook of auditors serves as a clarification of the above mentioned relationships from which it becomes obvious that the same understanding underlies the handbook of auditors with regard to value neutrality.

So far, only the value effect of the additional retained earnings in period T 1∗ was subject of investigation. In order to determine the effect on value in period T, we need to consider that the change in distribution x_{T 1}¹_∗ and the value increase in equity E_{T 1}^κ_∗^,1 have an effect on per-sonal taxation of equity investors. If dividends are taxed at s_d and changes in market values at

sg, we obtain the following value effect in comparison to a residual dividend policy:

° _, ∃ ° ^, ° ^, ° ^,

(22) holds under the assumption that ke₁^κ,s is suited for the assessment of the value effect.

Plugging (13) in (22) and solving for [°ET^κ,1] yields:

Equation (23) shows that the additional retention of earnings leads to a value increase in previ-ous periods, even if the respective additional investments are invested in value neutral projects.

This occurs whenever dividends are taxed differently than changes in market values, which is a common assumption in corporate valuation practice. Only if the personal tax rates s_d and s_g are identical and additional retained earnings are invested in value neutral projects, value neu-tral investments do not have a value effect in previous periods. Beyond the scope of (23), the assumption of an active debt management induces additional financing effects. We will explic-itly address this issue in the next section.