3.1 Long-term development of accounting measures and firm value
First, we use the definitions in the previous section to develop inductively closed-form solutions for the business units’ accounting measures. Lemma
1 illustrates how the value drivers of both units jointly affect the accounting measures if we consider cross-unit investments.
19
All investments in business unit A are funded from its NOPLAT according to the net investment rate
\(n^A\) and yield a return of
\(ROIC^A\). Consequently, the invested capital, NOPLAT, and net investments of this unit grow at the nominal rate
\(g^A\). By contrast, the expected accounting measures of unit B depend on both growth rates
\(g^A\) and
\(g^B\), which is an intuitive result. Periodically, a constant share
\(\overline{n}^{AB}\) of the NOPLAT in business unit A is invested into unit B. The rate
\(g^A\) determines the growth of this NOPLAT and therefore, the magnitude of the cross-unit investments. Once the firm has diverted the budget from business unit A into unit B, all subsequent investments yield
\(ROIC^B\) and are characterized by the growth rate
\(g^B\). Lemma
1 (b) shows that the invested capital, NOPLAT, and net investments in unit B do
not grow at a constant rate.
To understand the long-term development of a firm, we analyze the growth rates of invested capital, net investments, NOPLAT, and free cash flow at the firm level:
$$\begin{aligned} g_t^x = \frac{E[\tilde{x}_t] - E[\tilde{x}_{t - 1}]}{E[\tilde{x}_{t - 1}]} \ \ \ \text {with}\ x \in \left\{ {IC,NI,NOPLAT,FCF} \right\} . \end{aligned}$$
(7)
We further study the firm’s aggregate return
\(ROIC_t\) and total payout ratio
\(q_t\) at time
t:
Proposition
1 shows the effects of the planning premises on the long-term development of these measures. Based on Lemma
1 , we determine the growth rates, ROIC, and payout ratio according to equations (
7,
8) and study their limit values.
The growth rates of the accounting measures converge to the maximum of \(g^A\) and \(g^B\). Thus, we find that long-term forecasts of growth rates should not be interpreted as the average growth of a firm’s business units, but reflect the business that yields the maximum growth. This unit has a dominant effect on firm financials in the long-run. Interestingly, this result is independent of the cross-unit investment rate \(n^{AB}\). Even if the investments in unit B are primarily funded by cross-unit investments, the long-term growth of the firm’s aggregate free cash flow solely depends on the growth established within both units.
The development of the firm’s ROIC depends on the relative magnitude of both growth rates. For
\(g^B \ge g^A\), the total ROIC converges to the return in business unit B. By contrast, the limit of the total ROIC is a weighted average of the returns
\(ROIC^A\) and
\(ROIC^B\) of both units for
\(g^B < g^A\). This result reflects the fact that the growth rate
\(g^B\) exclusively influences financial surpluses of unit B, while the growth rate
\(g^A\) has an impact on both units. For high values of
\(g^A\), both units continue to be important for the firm, and the long-term ROIC depends on the specific investment rates. A similar result is obtained with regard to the firm’s total payout ratio
\(q_t\). It approaches the payout ratio
\(1-n^B\) of unit B if this unit’s growth rate exceeds the growth in unit A. For
\(g^B < g^A\), the long-term payout ratio is a weighted average of the effective payout ratios
\((1-n^{AB}) \cdot (1-n^A)\) and
\(1-n^B\) of both units.
20
In Corollary
1, we focus on the case wherein business unit A is characterized by a higher growth rate than unit B. We study how the value drivers affect the weights
\(\lambda ^{ROIC}\) and
\(\lambda ^{q}\) of unit A’s ROIC and the payout ratio in the determination of
\(ROIC^\infty\) and
\(q^\infty\).
Business unit A is more important for the long-term development of the firm if this unit yields a higher return and if a larger part of its NOPLAT is reinvested within unit A. By contrast, the firm’s long-term ROIC and payout ratio are rather aligned with unit B if this business unit is characterized by a high ROIC and high net investment rate. If the firm chooses a higher cross-unit investment rate, this implies additional investments into unit B. Thus, the total ROIC and the payout ratio approach \(ROIC^B\) and \(1-n^B\).
Our model provides a deeper understanding of the development of accounting measures in a steady-state period. Although our assumptions do not adhere to the common idea of constant and uniform growth of all relevant metrics, we provide a closed-form valuation formula that offers a tractable way to determine terminal value in a free cash flow calculus. The process of firm valuation can be divided into three steps. First, we determine the stand-alone values
\(E[\tilde{V}_0^A]\) and
\(E[\tilde{V}_0^B]\) of both units according to benchmark model (a). Second, we identify the net contribution
\(E[\tilde{V}_0^{AB}]\) of cross-unit investments to firm value. Cross-unit investments reduce the free cash flow in early periods, but increase the cash flow in later periods.
\(E[\tilde{V}_0^{AB}]\) represents the present value of these changes. The firm’s terminal value is determined in a third step by adding up the three components. Proposition
2 shows the free cash flow calculus.
If we neglect cross-unit investments, the business units would become independent of each other. The value \(E[\tilde{V}_0^i]\) of unit \(i \in \{ A,B\}\) can be determined according to the value driver model with growth rate \(g^i\) and cost of capital k. Cross-unit investments change free cash flows to the shareholders and debtors in two ways. First, in each period, only a share \(1 - n^{AB}\) of business unit A’s surplus is distributed, which reduces the firm’s free cash flow. Second, the retained amount is used to fund investments in unit B and results in incremental free cash flow in all future periods. The net value contribution \(E[\tilde{V}_0^{AB}]\) aggregates the value-reducing effect from higher retention of the surpluses in unit A and the value-enhancing effect due to the increasing future free cash flow generated in unit B.
Note that the net value contribution of cross-unit investments \(E[\tilde{V}_0^{AB}]\) depends on the growth rates of both business units. If the growth rate \(g^A\) of unit A increases, this implies higher periodic NOPLAT in this unit and higher subsidies to unit B. Each investment in unit B is followed by an infinite sequence of reinvestments according to the net investment rate \(n^B\) and thus, yields financial surpluses that grow at the nominal rate \(g^B\).
We assume that the rentability of business unit B exceeds the cost of capital. Thus, cross-unit investments into unit B always generate a positive value
\(E[\tilde{V}_0^{AB}]\): It is beneficial to retain and invest cash flows into unit B rather than distribute them. Therefore, it is not surprising that a higher rentability
\(ROIC^B\) and net investment rate
\(n^B\) increase
\(E[\tilde{V}_0^{AB}]\). Moreover, a higher cross-unit investment rate
\(n^{AB}\) and a higher return
\(ROIC^A\) in unit A increase the magnitude of the value-enhancing cross-unit investments and the component
\(E[\tilde{V}_0^{AB}]\). The effect of the net investment rate in unit A is less obvious. Increasing
\(n^A\) reduces the share
\(\overline{n}^{AB}\) of the NOPLAT of unit A that is invested into unit B. At the same time, the additional net investments in unit A raise future surpluses of this unit and allow for higher cross-unit investments in all subsequent periods. Corollary
2 shows that the latter effect is dominant. Not surprisingly, a higher cost of capital reduces the benefits of cross-unit investments.
The benchmark models mainly differ in the value contribution of the cross-unit investments
\(E[\tilde{V}_0^{AB}]\). The application of the value driver model at the business unit level according to the benchmark model (a) neglects this component completely. Benchmark model (b) presumes a constant net investment rate at the firm level and does not reflect the dynamic development of the investments in both business units. Models (c) and (d) impose assumptions on the level of the cross-unit investment rate, either equalizing
\(n^{AB}\) with the net investment rate of unit B (Koller et al.
2015) or assuming that the firm’s free cash flow grows at a constant rate (Meitner
2013). Moreover, model (c) does not consider the nominal growth within unit A and the complementary effect of both units’ growth rates on
\(E[\tilde{V}_0^{AB}]\). Ignoring the growth within unit A may lead to severe forecasting errors. Proposition 1 shows that the accurate depiction of the relative magnitude of
\(g^A\) and
\(g^B\) crucially affects the forecasts of accounting measures.
3.2 Business unit strategy and residual income measures
An obvious application of our results concerns value-based management. Firms may translate alternative business unit strategies into value drivers and assess their implications for the development of accounting measures and firm value. The literature on value-based management emphasizes the role of adequate performance metrics in strategy implementation and particularly highlights potential benefits of residual income: This measure is closely related to value creation and can help prevent myopic behavior of a firm’s management (Reichelstein
1997; Young and O’Byrne
2001). Furthermore, empirical work shows that valuation methods based on residual income are frequently used in corporate practice (Plenborg
2002; Damodaran
2005; Imam et al.
2008). Therefore, we highlight the link between business unit strategy and residual income valuation.
Residual income accounts for the fact that revenues should cover operating expenses as well as the firm’s capital cost. Thus, the residual income of unit
\(i \in \{ A,B\}\) at time
t is
According to the conservation property of residual income, the terminal value of a firm equals the sum of the initially invested capital and the market value added (MVA):
$$\begin{aligned} E[\tilde{V}_0] = E[\widetilde{IC}_0] + E[\widetilde{MVA}_0] \ \ \ \mathrm{with}\ E[\widetilde{MVA}_0] = \sum \limits _{t = 1}^\infty \frac{E[\widetilde{RI}_t]}{(1 + k)^t}. \end{aligned}$$
(10)
The MVA
\(E[\widetilde{MVA}_0]\) can be interpreted as the value contribution of investments in the steady-state period. In Corollary
3, we illustrate the development of the business units’ residual income and state a valuation formula based on a residual income approach.
Unit A does not receive any cross-unit investments, and the unit’s residual income grows at the rate \(g^A\). However, the residual income of unit B is affected by the growth established by both units and does not increase at a constant rate. The residual income of unit \(i \in \{ A,B\}\) is proportional to the excess return \(ROIC^i - k\) that is attainable in this unit.
The formula in Corollary
3 (b) has a similar structure as the discounted cash flow calculus in Proposition
2. If we neglect cross-unit investments, both NOPLAT and invested capital of unit
\(i \in \{ A,B\}\) grow at the rate
\(g^i\). Thus, the residual income grows at the same rate, and we obtain
\(E[\widetilde{MVA}_0^i]\). The cross-unit investments at time
t are given by
\(\overline{n}^{AB} \cdot ROIC^A \cdot E[\widetilde{IC}_{t - 1}^A]\), which implies an incremental residual income of
\(\overline{n}^{AB} \cdot ROIC^A \cdot E[\widetilde{IC}_0^A] \cdot (ROIC^B - k)\) at time
\(t+1\). This residual income grows at the rate
\(g^B\). If we consider the growth
\(g^A\) of unit A’s NOPLAT, we obtain
\(E[\widetilde{MVA}_0^{AB}]\) according to Corollary
3. The terminal value is the sum of the initially invested capital and the MVAs.
3.3 Learning value drivers from business unit data
The results of the previous sections may be useful for a firm’s management to assess business unit strategies with cross-unit investments. External parties, such as investors or financial analysts, typically do not have access to firms’ strategic plans and cannot directly apply our findings. Yet, our results offer guidance with regard to estimating a firm’s value drivers if external parties have basic financial information on the firm’s business units.
For instance, the IASB requires public companies to disclose such information in their segment reporting. According to IFRS 8, affected firms are supposed to report information on total assets, profit, interest expense, depreciation, and income tax expense for sufficiently large operating segments. Identification of the relevant segments and the presented accounting numbers must be consistent with internal management information. Firms are allowed to aggregate segments only if they share similar economic characteristics. Arguably, external parties do not make significant mistakes if they interpret these segments as the firm’s strategic business units and use segment data to infer information about business unit strategy.
21
To illustrate a simple estimation that is consistent with our model framework, assume that there is financial information on a firm with two business units,
\(i \in \{ A,B\}\), for three years,
\(t \in \{ 0,1,2\}\). If we assume that the three years represent the beginning of the steady-state period and that the realized accounting variables are close to their expected values (i.e.,
\(x_t^i = E[\tilde{x}_t^i]\) for
\(x \in \{NOPLAT,IC\}\)), it is sufficient to know the units’ invested capital and NOPLAT. Based on this information, the net investments are given by
\(NI_t^i = IC_t^i - IC_{t - 1}^i\). The units’ returns can be estimated using the formulas in Lemma
1:
$$\begin{aligned} ROIC^A = \frac{NOPLAT_1^A}{IC_0^A}, \ ROIC^B = \frac{NOPLAT_1^B}{IC_0^B}. \end{aligned}$$
(11)
Given these returns, we can use the data to infer the net investment rate of unit A,
$$\begin{aligned} n^A = \frac{NI_1^A}{NOPLAT_1^A}, \end{aligned}$$
(12)
the net investment rate
\(n^B\), and the cross-unit investment rate
\(\overline{n}^{AB}\) related to the NOPLAT as follows:
$$\begin{aligned} \begin{aligned}&n^B = \frac{NOPLAT_1^A \cdot NI_2^B - NOPLAT_2^A \cdot NI_1^B}{NOPLAT_1^A \cdot NOPLAT_2^B - NOPLAT_2^A \cdot NOPLAT_1^B},\\&\overline{n}^{AB} = \frac{{NOPLAT_1^B \cdot NI_2^B - NOPLAT_2^B \cdot NI_1^B}}{{NOPLAT_2^A \cdot NOPLAT_1^B - NOPLAT_1^A \cdot NOPLAT_2^B}}. \end{aligned} \end{aligned}$$
(13)
These estimates may be used to forecast the business units’ accounting measures (Lemma
1 and Proposition
1) or to determine the firm’s terminal value (Proposition
2). Note that this approach may lead to considerable estimation errors if the accounting variables are very volatile and realized values deviate from the expected values. In this case, it may be inevitable to consider data for more than three periods.
3.4 Analysis of special cases
Empirical research on financial strategy choices shows that firms frequently pursue strategies that reduce the volatility of accounting variables over time. For instance, the survey by Graham et al. (
2005) indicates that firms pursue a smooth development of accounting earnings because “smooth earnings make it easier for analysts and investors to predict future earnings”, and such dynamics are associated with less risk.
22 In line with this observation, Stone (
1972) argues that firms engage in cash management to smooth cash flows. Brav et al. (
2005) find evidence regarding firms’ endeavors to smooth dividends and highlight the prevalence of target payout ratios. We conclude that strategy formulation typically relies on the target levels of value drivers, such as growth rates and payout ratios.
Due to their high practical relevance, we analyze the conditions for a constant payout or constant growth strategy within our model setting. Specifically, we assume that firms try to maintain either a time-invariant payout ratio or constant growth of free cash flow (\(q_t = q\) or \(g_t^{FCF} = g^{FCF}\) for \(t \ge 1\)), for instance, as a signal to potential investors. Both objectives are consistent with the assumptions of the value driver model according to benchmark case (b). A constant net investment rate n at the firm level implies a constant payout ratio \(q=1-n\) and (for a given ROIC) a constant growth rate \(g^{FCF} = n \cdot ROIC\). The previous analysis documents that strategy formulation at the business unit level does not necessarily yield such results, even if intra- and cross-unit investment rates are constant.
Proposition
3 shows under which conditions our model can be aligned with the objectives of a constant payout ratio or constant growth of the aggregate free cash flow.
23
The proposition shows that constant payout and constant growth strategies represent special cases of our model. The conditions in parts (a) and (b) ensure that the payout ratio according to (
8) and the growth rate of the free cash flow according to (
7) are time invariant. Both conditions have a very intuitive interpretation. A constant payout strategy can be achieved if both units effectively apply the same payout ratio.
24 In this case, the limit of the total ROIC according to Proposition
1 (b) can be simplified as follows
$$\begin{aligned} ROIC^\infty = \left\{ \begin{array}{ll} ROIC^B &{} \text {if}\ g^B \ge g^A\\ ROIC^A \cdot \frac{n^A}{n^B} &{}\mathrm{if} \ g^B < g^A \end{array} \right. . \end{aligned}$$
(14)
Furthermore, the valuation formula can be restated in the following way:
The special case in Proposition
3 (b) is useful for understanding valuation models that assume constant growth of the free cash flow. These models do not necessarily assume economic homogeneity of a firm’s business units; instead, constant growth of the free cash flow may be the result of a complex disaggregate budgeting process with multiple business units that potentially grow at different rates. Our analysis highlights the assumptions that nevertheless yield constant growth of the firm’s free cash flow.
25 The firm’s free cash flow grows at the rate
$$\begin{aligned} g^{FCF} = (1 - n^{AB}) \cdot n^A \cdot ROIC^A + n^{AB} \cdot ROIC^B = g^B. \end{aligned}$$
(16)
It is important to note that a constant growth strategy neither implies constant growth rates of total NOPLAT, net investments, and invested capital nor is accompanied by a constant ROIC at the firm level. Thus, the development of the firm’s accounting measures cannot be depicted by the value driver model. The valuation formula can be simplified to
The simplicity of (
17) comes at the cost of the cross-unit investment rate having to meet the condition in Proposition
3 (b). If a firm’s strategy violates this condition, the formula may cause significant errors. For a constant growth strategy, the firm’s long-term ROIC and payout ratio approximate the ROIC and payout ratio of unit B:
$$\begin{aligned} ROIC^\infty = ROIC^B, \ \ \ q^\infty = 1 - n^B. \end{aligned}$$
(18)
Corollary
4 studies the applicability of the value driver model in our setting with two business units.
Corollary
4 highlights the role of intra-firm heterogeneity in our valuation approach. The application of the value driver model according to benchmark case (b) requires constant payout ratios
and constant growth of the firm’s free cash flows. To obtain this result, the two special cases must coincide. However, this cannot occur unless the units generate identical ROIC. For all other cases, firm value must be determined with the valuation formula presented in Section
3.1.