Capital investments and non-linear residual income dynamics: the cost of equity capital bias

Capital investments and non-linear residual income

dynamics: the cost of equity capital bias

Ludo Wurfbain S1864858 Thesis MSc. Finance June 2015 Supervisor: L. Dam ABSTRACT

This study answers the call of Beaver (1999) to incorporate varying costs of equity capital into the residual income dynamic. Following the linear information dynamics (LID) proposed by Ohlson (1995) and the “capital follows profitability” dynamic by Biddle et al. (2001), this study tests for the most appropriate functional form of the residual income dynamic. Additionally, the test are performed for both industry specific and standardized cost of equity capital with supportive results of a bias created by assuming constancy in the cost of equity capital. These findings suggest that results from previous research is biased and therefore inaccurate. From a practical perspective, such a bias can deceptively influence the managers capital investment decision.

Contents

1. Introduction ... 2

2. Literature Review ... 4

2.1 Residual income dynamics... 4

2.2 Economic driver of the investment dynamic ... 5

3. Methodology ... 6

3.1 Model review ... 6

3.2 Model development ... 9

4. Hypotheses ... 10

5. Data & Descriptive Statistics ... 11

6. Results ... 14

6.1 Capital investments follow profitability ... 14

6.2. Cost of equity bias on the relation between period-ahead and current residual income ... 15

6.3 Robustness Checks ... 21

6.3.1. Firm size ... 21

6.3.2. Investment vs. Divestment ... 22

6.3.3. Variable definition for capital investment growth ... 22

7. Conclusion ... 22

Appendix A ... 25

Appendix B ... 29

Appendix C ... 33

1. Introduction

The development of the residual income model can be perceived as a revelation in the domain of valuation, as it presents a bridge between equity values and accounting variables, and functions as an interchangeable model for the dividend discount model (Bernard, 1995). The model combines current book value of equity and future residual income1_{, which is unobservable, to}

derive the current market value of a firm. In order to form a link between current observables and unobservable future residual income, The Linear Information Dynamic (LID) is proposed by Ohlson (1995). He argues that the relationship between current residual income and future expected residual income follows a linear stochastic process. This proposition was empirically tested by Dechow, Hutton and Sloan (1999), who report mean-reversion of excess returns but also conclude that simply capitalizing period-ahead earnings in explaining equity values provides a similar performance. Biddle, Chen and Zhang (2001) argue that this is due to the exclusion of capital investment effects. They introduce the “capital follows profitability” dynamic and empirically test for a convex relationship between current and future residual income. A S-shaped relationship is found, rather than convexity, but it does conclude that the linear stochastic process of the LID is insufficient to provide accurate measures of earnings persistence in all situations. The literature has been reaching out to find the most accurate functional form of the persistence parameter of residual income but has been blind to a noteworthy bias that is the result of one of the major assumption that is embedded in all models; a standardized cost of equity capital.

This study answers the call of William Beaver (1999) to incorporate the effects of varying costs of equity capital in defining excess returns and the persistence thereof over time. The costs of equity capital is a major factor that can explain the difference between normal earnings and excess returns. The above mentioned authors that empirically tested residual income models assumed a constant cost of equity capital (at a 12% rate). The fact that the effect of variation in the cost of equity capital is ignored in these studies has the clear potential to impose a bias on the presented results. Taking variation of the cost of equity capital into consideration is therefore likely to provide a more accurate indicator of the persistence of excess returns over time and should increase explanatory power.

In order to allow for industry specific costs of equity capital, this study builds on the industry specific equity premium database provided by Damodaran (2014). By cross-matching the industry classifications proposed by Damodaran (2014) and the SIC-code industry

classifications, a dataset of 61129 firm-year observations over the period 1998-2013 is composed. Empirical tests, regarding the LID model of Ohlson (1995), the “capital follows profitability” dynamic, and the convex relationship of current residual income and future expected residual income proposed by Biddle et al. (2001), are conducted for both industry specific and standardized cost of equity capital. Considering both the adjusted and non-adjusted sample allows for proper inference on the bias imposed by a standardized cost of equity capital. Multiple findings emerge from the three empirical tests, the first one being that “capital follows profitability”, meaning that current residual income leads to future capital investments. Although, it must be noted that the results show that only a small share of current profitability is reinvested2 _{and explanatory power is lacking. This indicates that the results should be}

interpreted with caution. The second result coming forward is that standardized cost of equity capital impose only a minor downward bias on the persistence parameter found by the LID model. However, the piece-wise regression of future on current residual income displays signs of convexity through increasing slope parameters, which signifies that the third result conflicts with the linear relationship proposed by the LID model. Though, since the highest values of residual income do not comply with convexity, a S-shape relationship is observed between future and current residual income rather than convexity. The fourth, and most pronounced result is that when the more appropriate functional form is considered, accounting for capital investments, using standardized cost of equity capital does impose a significant bias on the persistence parameter. This result is robust for firm size and investment/divestment variations. The findings of this study hold two implications, the first being that a S-shaped functional form appears to be more appropriate when aiming at capturing the persistence of excess returns while accounting for capital investments that follow from current profitability. This might imply that a competitive advantage can be strengthened by managers through new capital investments, rather than see it deteriorate over time due to competition effects. The second implication is that studies that have assumed a constant cost of equity capital are bound to provide significantly biased results through this assumption. With differences ranging from 3-7%3_{, managers might deceptively be made to believe that part of their residual income is}

persistent (or transitory), while this is not the case.

To present an overview of previous findings and to motivate the economic relevancy of this paper, Section 2 provides a literary review. Section 3 provides a review of the LID model proposed by Ohlson (1995) and continues by elaborating on the model proposed by Biddle et al. (2001) including the “capital follows profitability” assumption. Section 4 describes the drafted hypothesis and Section 5 follows this by describing the dataset used to empirically test these

2 _{Only 5% is reinvested compared to the 96% found by Biddle et al. (2001)}

hypothesis. Section 6 precisely describes the performed empirical tests and explains the results that come forward. Section 7 concludes and provides suggestions for future research.

2. Literature Review

2.1 Residual income dynamics

The last decade of the previous century can be characterized by the introduction of the residual income substitute of the dividend discount model, when considering the domain of valuation (Bernard, 1995). This residual income dynamic, as proposed by Ohlson (1995), provides the essential link between equity values and accounting variables, such as earnings and book value of equity. The model shows that book value of equity combined with the present value of future residual income, which is unobservable, is equal to the intrinsic value of a firm (Myers, 1999). To deal with the fact that future residual income is unobservable, Ohlson (1995) and Feltham and Ohlson (1996) have proposed the “linear information dynamics” (LID)4_{. These LID state that}

future residual income relates to current residual income via a linear stochastic process. Residual income is defined as the difference between accounting earnings and the cost of equity capital, which symbolizes the normal expected return (Myers, 1999). Empirical tests of LID have resulted in varying findings. Frankel and Lee (1998) conclude that a more complete valuation level might provide superior return prediction, whereas Dechow et al. (1999) do find evidence of mean reverting behavior of residual income but are also troubled with the finding that LID does not outperform the simple capitalization of period-ahead earnings in explaining equity values. Myers (1999) complements this by empirically testing four forms of the LID model. He finds that none of these forms outperform book equity as a standalone factor. This suggests that the LID model might be “incomplete” or that residual income does not follow a stationary process. So, the assumption that excess returns follow a stationary process, merely forms a starting point for empirical research (Lee, 1999; Lo and Lys, 2000). Hayn (1995) and Collines, Pincus and Xie (1999) find that asymmetry exists when comparing positive and negative earnings, again suggesting non-linearity. Considering the limited support for the linear relation suggested by the LID, several authors included “capital investment dynamics” including G. Zhang (2000), Chen and Zhang (2003) and Yee (2000). However these previously mentioned studies do not consider the residual income dynamics. These two features, residual income and capital investments, are combined by Biddle et al. (2001), who find partial evidence that the relationship of future residual income and current residual income is convex. Despite being

unable to entirely conform convexity of the relationship, they do find that in most instances the LID is inapplicable, or at least provides an inaccurate measure of excess return persistency. With the study of Biddle et al. (2001) the call of Beaver (1999) for further inference of the LID is partially answered. However, a considerable research opportunity that Beaver (1999) proposes is still left untouched. He states that by assuming a constant cost of equity across all firms and over time, one potential difference between returns and excessive returns is not accounted for. This bold assumption of constant cost of equity capital both over time and between firms is highly likely to impose a bias on the results following from the residual income model. Authors using a constant discount rate (a.o. Dechow et al., 1999; and Biddle et al., 2001) file as an argument that previous research has relied on the same assumption, which can be considered a fallacy since justification is lacking in the literature. Another argument that is brought forward states that results are robust for various levels of cost of equity capital. However, rather than allowing the constant rate to shift, the bias imposed by this constancy could be disposed by allowing for varying cost of equity capital along industries or even firms (Beaver, 1999). Lee (1999) supports this view and points to this rather restrictive assumption as a major concern for future research. However, he also states that the use of historical data when calculating the cost of capital suffers from estimation errors. This is confirmed by Claus and Thomas (1999) who show that the equity premium, when calculated according to the residual income model (referred to as the “implied method” by Damodaran (2014)), is much lower than the equity premium computed via the traditional way. This study follows the equity premium calculation method proposed by Damodaran (2014) so that industry varying cost of equity capital can be computed and the initial research by Miller and Modigliani (1966) on variance in equity capital costs can be fused into the residual income dynamic. Given that the mean of the varying cost of equity capital5 _{(varying between 8% and 11%) is substantially lower than 12%, it is reasonable}

to expect that the bias created by standardized cost of equity capital will be downward. When cost of equity capital are relatively lower, excess returns increase. Provided that Biddle et al. (2001) propose that 96% of excess returns will be reinvested, it can be expected (assuming zero or positive NPV projects) that the extra acquired residual income will cause higher persistence of returns. Additionally, the explanatory power is expected to increase if variation in the cost of equity capital between industries is accounted for.

2.2 Economic driver of the investment dynamic

Although the functional form of the relationship is still questionable, the persistence of current residual income to expected future residual income is clearly observable. Persistence of earnings is of particular interest since this specific form of earnings is valued over transitory earnings

since persistence assures quality earnings in the future (Barth, Elliott and Finn, 1999). However, what economic factors are able to explain this persistence and what determines the magnitude of the persistence? To start with, residual income seemingly persists due to the development of a competitive advantage. Multiple streams of literature have developed over the past few decades in the field of strategy, of which the Resource Based View (RBV) is the most renowned (e.g. Barney, 1991; Barney, 2001; Porter, 1985). This RBV states that a competitive advantage is developed by resources that are rare, valuable, inimitable and sustainable. To ensure continuity of this competitive advantage, the literature generally agrees that capital investments are essential (Baginski, Lorek, Willinger, and Branson, 1999; Porter and Millar, 1985; Sirmon, Hitt, Ireland, and Gilbert, 2011). In particular, R&D investments are positively related to the market value of a firm (Chan, Lankonishok, and Sougiannis, 2001; Hand, 2001). However, the RBV only analysis and aims at explaining internal factors to the company, while the persistence of residual returns is expected to be dependable on external factors as well (Zhang, 2013). These factors include fierceness of competition (Dechow et al. 1999) and industry entry barriers (Cheng, 2005). When excess returns are realized within an industry, new entrants are attracted. The higher the entry barriers, the higher the excess returns have to be to make entrance feasible. However, once competitors have entered, prices are likely to be driven down, resulting in diminished excess returns. Especially since inimitability of a product is often temporarily, the entrance of new competitors appears to be only a matter of time.

Another influential factor when examining the persistence of excess returns is conservative accounting (Chambers, Jennings and Thompson II, 2002; Hand, 2001; Zhang, 2000). Returns on capital investments are highly uncertain and therefore need to be expensed. Expensing these capital investments has a negative influence on the residual income of period t, but the capital investment is expected to have a positive effect on future residual income. Overall the benefits, in the form of persistent earnings, of capital investments outweigh the associated risks (Asthana and Zhang, 2006). Therefore, including the capital investment dynamic should result in a positive impulse to the persistence parameter of residual returns.

3. Methodology

3.1 Model review

This section describes the pure form of the residual income valuation model developed by Ohlson (1995), also referred to as the Linear Information Dynamic approach (LID). From this approach, the derivations made by Biddle et al. (2001) and Zhang (2013) are discussed.

assumptions. The first assumption is that price is equal to the present value of expected dividends: 𝑃_𝑡 = ∑ 𝐸𝑡[𝑑𝑡+𝜏] (1+𝑟)𝜏 ∞ 𝜏=1 , (1)

where Pt is the value of the firm’s equity at time t, dt is net dividends paid at time t, r is the

discount rate (which is assumed constant by Ohlson (1995)) and Et[ ] is the expectation operator

conditioned on time t information. The second assumption entails the clean surplus accounting relation:

𝑏𝑡 = 𝑏𝑡−1+ 𝑥𝑡− 𝑑𝑡, (2)

where bt is the book value of equity at time t, and xt represents earnings for the period from

(t-1) to t. The clean surplus relation in equation 2 and the dividend discount model in equation 1 can now be combined since the previously mentioned assumptions allow future dividends to be expressed in terms of future earnings and book values. The combination of equation 1 and 2 yields:

𝑃𝑡 = ∑𝜏=1∞ 𝐸𝑡[𝑏𝑡+𝜏−1_(1+𝑟)+ 𝑥𝑡+𝜏 𝜏 −𝑏𝑡+𝜏]. (3)

This can be rewritten, following algebraic manipulation as

𝑃_𝑡 = ∑ 𝐸𝑡[𝑥𝑡+𝜏 − 𝑟.𝑏𝑡+𝜏−1]

(1+𝑟)𝜏

∞

𝜏=1 , (4)

and ‘abnormal earnings’ can be defined as

𝑥_𝑡𝑎_{= 𝑥}

𝑡− 𝑟. 𝑏𝑡−1.

So that the price of equity can be expressed as the sum of the current book value and the present value of future abnormal earnings’:

𝑃𝑡 = 𝑏𝑡+ ∑ 𝐸𝑡[ 𝑥𝑡+𝜏

𝑎 _]

(1+𝑟)𝜏

∞

𝜏=1 . (5)

The third assumption grasps the development of profitability over time. Profitability is assumed to persist from one period to the next. This final assumptions leads to the basic residual income valuation model adopted by Ohlson (1995):

𝑋_𝑡+1𝑎 _{= 𝜔𝑋}

where 𝑋_𝑡𝑎 _≡X

t - rBt-1 refers to the abnormal earnings in period t, 0 ≤ ω ≤ 1 represents a

persistence parameter of the abnormal returns over two consecutive time periods. εt+1 is a zero

mean disturbance term that is not predictable at time t. However, this basic residual income model suggests that any differences between current book value and book value from the previous period plus the abnormal returns should be explained by a zero mean error term (Zhang, 2013). Since this is unlikely to be the case, other variables should explain this variance. In order to deal with this, Ohlson (1995) extends its basic version of the LID approach by providing an additional parameter concerning the variable ‘other information’. The inclusion of the parameter ‘other information’ forms the foundation for the third assumption of Ohlson, which entails that abnormal earnings satisfy the following modified autoregressive process:

𝑋_𝑡+1𝑎 = 𝜔𝑋𝑡𝑎 + 𝜐𝑡 + 𝜀𝑡+1, (7a)

𝜐_𝑡+1= 𝛾𝜐_𝑡 + 𝜀_𝑡+1, (7b)

The parameter υt refers to non-accounting information that influences the one period ahead

abnormal earnings. Given the above stated equations, the ‘other information’ variable affects abnormal earnings in two ways. Firstly, once the effect is absorbed in the abnormal earnings, it will gradually decay at the persistence rate of abnormal earnings ω. Secondly, 𝜐𝑡 follows an

autoregressive process in itself at a persistence rate of γ. This shows the influence of current non-accounting information on future non-accounting information, υt+1, υt+2… υt+n. These separate

values for 𝜐𝑡 will be absorbed in the abnormal earnings metric and will, as stated before, persist

at rate ω. This model by Ohlson (1995) is empirically tested by Dechow et al (1999). In line with equation 2a and 2b, Dechow et al (1999) propose that future abnormal earnings depend on the two persistence parameters related to the current abnormal earnings and non-accounting variables. In order to make the the non-observable future residual earnings measurable, Dechow et al. (1999) propose multiple factors that influence the future period abnormal earnings including the magnitude of special items, operating assets, dividend pay-out policy and historical persistence of abnormal earnings for firms in the same industry. Considering the ‘other information’ variable 𝜐_𝑡, Dechow et al. (1999) do not propose specific factors that capture the effect of non-accounting information. In fact, it can be considered as any event or news outside the financial report that is relevant to valuation. Examples that can be thought of include, changes in macroeconomic conditions, industry regulations and product inventions (Zhang, 2013). Rather than specifying possible factors, Dechow et al. and several other studies (including Begley and Feltham (2002), Liu and Ohlson (2000) and Ohlson (2001)) back 𝜐_𝑡 out from analyst earnings forecasts. 𝜐𝑡 is inferred from 𝑋𝑡+1𝑎 based on equation 7a as 𝜐𝑡 = 𝑋𝑡+1𝑎 − 𝜔𝑋𝑡𝑎. Instead of

returns. The force considered in this paper, which is expected to influence excess return persistence, is the (re)investment of excess returns.

3.2 Model development

The model exercised in this paper follows Biddle et al. (2001) and Zhang (2013) who attempt to enhance the LID model by proposing a fourth assumption namely that capital investment follows profitability. This assumption describes the guiding effect of profitability signals and the constraining effect of available investment opportunity on investment decisions. The guiding role of profitability signals should lead to well informed investment decisions that form the basis of value creation. To capture this effect, the LID model of Ohlson (1995) is extended with a capital investment term. Following the profitability persistence assumption, an investment at date t is expected to result into a future residual income stream 𝜔𝑋_𝑡𝑎_{, 𝜔}2_𝑋

𝑡𝑎,… The NPV of this

investment, derived by discounting the stream of future residual income, yields 𝜔𝑋𝑡𝑎

(1+𝑟)+ 𝜔2_𝑋 𝑡𝑎 (1+𝑟)2+ ⋯ = 𝜔𝑋𝑡𝑎 (1+𝑟−𝜔)= 𝜓𝑋𝑡𝑎, where 𝜓 ≡ 𝜔

(1+𝑟−𝜔) . It follows from this persistence of investment yields

term that

𝐼_𝑡= 𝜋𝐵_𝑡−1[𝜓𝑋_𝑡𝑎_{], (8)}

Where 𝐼𝑡 is the amount of capital from the operation that is invested (or divested in the case of a

negative value) and 𝜋 is the parameter that displays the investment opportunities that arise to a firm. So, the influence of capital investments on future excess returns depends on the reinvestment of current excess returns (indicated by 𝜓) and the investment opportunities available to the firm.

Besides this additional assumption, Biddle et al. (2001) and Zhang (2013) also use a slightly altered notation of equation 6 to describe the persistence of profitability

𝑝𝑡+1 = 𝜔𝑝𝑡+ 𝜀𝑡+1, (9)

Where 𝑝𝑡 ≡ 𝑋𝑡𝑎/𝐵𝑡−1= 𝑋𝑡/𝐵𝑡−1− 𝑟, which measures the profitability in period t, referring to

the return on equity minus the cost of equity. The persistence parameter 0<ω≤1 is similar to Ohlson’s (1995) and 𝜀_𝑡+1 is a zero mean disturbance term. This altered notation of the profitability persistence assumption can be combined with equation 8 in order to follow Biddle et al. (2001) and Zhang (2013) to derive the income in period t+1

𝑋_𝑡+1 𝑎 _{≡ 𝑝}

𝑡+1𝐵𝑡 = (𝑝𝑡+ 𝜀𝑡+1)(𝐵𝑡−1+ 𝐼𝑡) = 𝜔𝑝𝑡𝐵𝑡−1+ 𝜋1𝐵𝑡−1𝜔𝜓𝑝𝑡2+ 𝑒𝑡+1

= 𝜔𝑋_𝑡𝑎_{+ 𝜋}

This equation aims at accounting for capital investments when estimating the persistence of excess returns. The quadratic term in equation 10 suggests a convex relation between current residual income and future residual income, rather than a linear one. This suggested convexity will be empirically tested in Section 6. Besides confirming convexity of excess return persistence, this study aims at designating the potential bias that the use of standardized cost of capital incurs. This implies that residual income will be derived for both varying and standard cost of equity capital, Section 6 will elaborate on the empirical alterations that follow from accounting for variation in the cost of equity capital.

4. Hypotheses

As stated before, this study aims at describing the righteous functional form of excess return persistence. This functional form depends on the persistence parameter 𝜔 but, as shown in the methodology section, this study aims at providing evidence that it additionally depends on the quadratic capital investment term. In order for excess returns to be influenced by capital investments, this study assumes that excess returns are followed by capital investments. Following from this assumption the first hypothesis can be drawn. Besides the functional form, this study looks to describe the influence of varying cost of equity capital on the persistence of excess returns. Hence, the first hypothesis is altered to include variation in the cost of equity capital, resulting in hypothesis 1.1.

1. Capital follows profitability which should result in a positive relation between excess returns and year-ahead capital investment.

1.1 Standardized cost of equity imposes a downward bias on the positive relationship between excess returns and year-ahead capital investment

In order to fully capture the effect that varying cost of equity capital has on the persistence of excess returns, the initial model by Ohlson (1995) will be considered first. Since the cost of capital overall ,when allowing for variation, is lower than 12%, the persistence parameter is expected to become larger since the additional funds that become available can be fully invested. When fully invested, this could slightly push the persistence parameter to 1. Additionally, since variation in the cost of equity capital should more accurately explain the persistence of excess returns, the explanatory power is expected to increase. These expectations result in the second hypothesis.

2.1 Accounting for variation in cost of equity capital provides a more accurate prediction of excess return persistence.

As specified before, the functional form of the persistence of excess return remains a source of debate, therefore the third hypothesis concerns this matter. Due to the inclusion of capital investments, the functional form is expected to become convex rather than linear. Here too, the assumption of a standard cost of equity capital is expected to impose a bias on the persistence parameter. Following a similar reasoning as for hypothesis 2, an overall lower cost of equity capital is expected to provide higher persistence results. Allowing for industry varying cost of equity capital is also expected to increase the explanatory power when considering a convex relationship between future and current residual returns.

3. Period ahead residual income is a convex function of current residual income

3.1 Standardized cost of equity imposes a downward bias on the persistence of residual return.

3.2 Accounting for variation in cost of equity capital provides a more accurate prediction of excess return persistence when considering a convex relationship between future and current residual return.

The empirical tests related to these hypothesis will be elaborated on in Section 6 where the results following from these tests will be discussed.

Results by Biddle et al. (2001) provide partial evidence of convexity. However, since their dataset covers the years 1981-1998, their findings might be outdated or even irrelevant for current economic conditions. Therefore it is relevant to re-perform the empirical tests with a dataset covering the years 1998-2013. The dataset will be elaborated on in the subsequent section.

5. Data & Descriptive Statistics

provide one-year ahead estimates of capital growth. By scaling residual income in year t and t+1 by operating assets in year t-1 (OAt-1), cross-sectional variation in firm size is controlled for. The

last variable that is measured is the investment opportunity. This is measured ex post using the percentage change in operating assets over a one year time horizon (from year t to t+1, it+1). In

section 6.3, this percentage change in operating assets is also measured over a three year time horizon (from year t to t+3, it+3), which functions as a robustness check. Table 1 provides an

overview of the input variables, their abbreviations and, if applicable, the method of calculation. Next to using Compustat for the extraction of general accounting data, the online Damodaran database was consulted for the collection of industry betas and the cost of equity.6 _{The indicated}

industries by Damodaran were manually cross-referenced with the SIC industry classification indicated by Compustat. In this way industry varying cost of equity was extracted for the years 1998 to 2013. Missing observations were given the value of the closest value that is observable. An overview of the industries and the respective cost of equity capital are provided in Appendix A. In order to be able to infer the possible bias that is imposed by disallowing variance in the cost of equity capital, the same sample of excess returns is also computed for a standard cost of equity of 12%.

The sample is adjusted in concordance with Bidle et al. (2001). The first adjustment entails the deletion of all negative values for book value or operating assets. This adjustment reduces the sample to 68941 observations. In order to deal with outliers, the sample is further reduced by

Table 1. Data descriptions

Variable Notation Definition and method of calculation

Book Equity Bt Book value of common equity at the end of year t.

Operating Assets OAt Total assets minus short-term investments at the end of year t, denoted as the book value of operating assets.

Net Income Xt Net income, adjusted for extraordinary items and

discontinued operations in year t.

Cost of Equity per Industry ri Implied cost of equity per industry as provided by Damodaran, matched with Standard Industrial Classification (SIC) codes.

Standard Cost of Equity rs Assumed to be 12% (for comparability purposes with Biddle et al., 2001)

Scaled Residual Income in Year t with Industry Specific Cost of Equity 𝑋𝑡

𝑎 _{= (X}

t – riBt-1)/OAt-1. Scaled Residual Income in Year t with

Standard Cost of Equity 𝑋𝑡

𝑏 _{= (X}

t – rsBt-1)/OA t-Scaled Residual Income in Year t+1

with Industry Specific Cost of Equity 𝑋𝑡+1

𝑎 _{= (X}

t+1 – riBt)/OAt-1. Scaled Residual Income in Year t+1

with Standard Cost of Equity 𝑋𝑡+1

𝑏 _{= (X}

t+1 – rsBt)/OAt-1. Scaled Capital Investment in Year t+1 𝑖𝑡+1 =(OAt+1 – OAt)/OAt.

Table 2. Descriptive Statistics

Panel a: Descriptive statistics for the pooled samplea

Variable Notation Mean Median Std. Dev. Min. 1

st

Quartile 3

th

Quartile Max.

Scaled Residual Income in Year t with Industry Specific Cost of Equity

𝑋𝑡𝑎 -0.06 0.00 0.21 -1.41 -0.06 0.03 0.31

Scaled Residual Income in Year t with Standard Cost of Equity 𝑋𝑡

𝑏 _{-0.07 -0.01 0.21 -1.43 -0.07 0.02 0.30}

Scaled Residual Income in Year t+1 with Industry Specific Cost of Equity

𝑋𝑡+1𝑎 -0.05 -0.00 0.24 -1.77 -0.06 0.03 0.67

Scaled Residual Income in Year

t+1 with Standard Cost of Equity 𝑋𝑡+1

𝑏 _{-0.07 -0.01 0.25 -1.80 -0.07 0.02 0.65}

Capital Investment in Year t+1 𝑖𝑡+1 0.13 0.06 0.34 -0.48 -0.05 0.20 2.31

a _{The sample consists of 61129 firm-year observations for the period 1998-2013}

Panel b: Annual descriptive statistics

𝑋𝑡𝑎 𝑋𝑡+1𝑎 𝑖𝑡+1

Year t Observa

tions Mean Median Std. Dev Mean Median Std. Dev Mean Median Std. Dev

98 2747 -0.06 0.00 0.22 -0.04 0.00 0.24 0.19 0.09 0.39 99 2932 -0.05 0.00 0.22 -0.06 0.00 0.28 0.18 0.07 0.40 00 3149 -0.07 0.00 0.24 -0.09 -0.01 0.29 0.08 0.02 0.32 01 3340 -0.09 -0.00 0.23 -0.08 -0.01 0.24 0.09 0.05 0.31 02 3489 -0.08 -0.01 0.23 -0.04 0.00 0.21 0.14 0.08 0.31 03 3606 -0.05 0.00 0.20 -0.04 0.01 0.24 0.16 0.09 0.33 04 3724 -0.04 0.01 0.21 -0.03 0.01 0.24 0.16 0.08 0.35 05 3849 -0.04 0.00 0.21 -0.04 0.01 0.26 0.19 0.10 0.36 06 3983 -0.04 0.00 0.21 -0.05 0.00 0.27 0.18 0.09 0.36 07 4014 -0.06 0.00 0.22 -0.09 -0.01 0.27 0.07 0.03 0.32 08 4158 -0.08 -0.01 0.22 -0.06 -0.01 0.21 0.06 0.02 0.28 09 4306 -0.06 -0.01 0.20 -0.04 -0.01 0.21 0.13 0.05 0.31 10 4373 -0.04 -0.00 0.19 -0.04 -0.00 0.22 0.12 0.05 0.31 11 4514 -0.05 -0.00 0.20 -0.05 -0.00 0.23 0.10 0.05 0.30 12 4698 -0.06 -0.00 0.20 -0.05 -0.00 0.23 0.12 0.04 0.33 13 4247 -0.06 0.00 0.21 -0.05 -0.00 0.24 0.13 0.05 0.33 𝑋𝑡𝑏 𝑋𝑡+1𝑏

Year t Observations Mean Median Std. Dev Mean Median Std. Dev

deleting the highest and lowest 2% of residual income. The same reduction is done for the capital expenditures (measured as yearly change in operating assets), in concordance with the sample treatment of Biddle et al (2001). These adjustments mentioned, allow for comparability but also assure a sound and representative dataset and result in a final sample size of 61129.

6. Results

6.1 Capital investments follow profitability

The first empirical test that is executed aims at describing the “capital follows profitability” assumption and to test hypothesis 1. In concurrence with Biddle et al (2001) the assumption is tested by regressing year t+1 investment growth (𝑖_𝑡+1), which is measured ex post on year t residual income (𝑋_𝑡𝑎_):

𝑖_𝑡+1= 𝛽₀+ 𝛽₁𝑋_𝑡𝑎_{+ 𝜇}

𝑡+1, (11)

where 𝛽₀ and 𝛽₁ are estimated intercept and slope parameters respectively; 𝜇_𝑡+1is a zero mean random disturbance term. The results of this empirical test are displayed in Panel A of Table 3 where β0 can be interpreted as the yearly growth rate of operating assets and β1 as the

percentage of prior residual income that is invested in operating assets. The year to year results for the reinvested percentage of prior residual income in operating assets show a fluctuating pattern. Small positive percentages are exchanged for small negative percentages but given the low statistical significance and explanatory power of the year to year results, inferences from these data are hard to justify. The pooled sample however, provides a slightly positive result of 5% of prior residual income and 12% of year ahead operating assets growth. This is consistent with hypothesis 1 that capital follows profitability, however, it must be noted that the 5% reported in this study is considerably lower than the 96% found by Biddle et al. (2001) and intuitively seems rather low given that excess returns are the least expensive form of funding capital expenditures (Titman and Wessels, 1988). Another issue that arises, as indicated before, is the lack of explanatory power, even for the pooled sample. Due to the difference in result in comparison to Biddle et al. (2001) additional research is desirable.

Rather than reporting the difference between this study and the results of Biddle et al. (2001), it is of interest to grasp any bias that might be imposed by standardized cost of equity. Therefore, the subsequent equation is adjusted to test if a standardized cost of capital imposes a bias (Hypothesis 1.1) on the results that follow from equation 11 (𝑋_𝑡𝑎 _{is replaced by 𝑋}

𝑡𝑏), leading to:

𝑖_𝑡+1 = 𝛽₀+ 𝛽₁𝑋_𝑡𝑏_{+ 𝜇}

where β₀ and β₁, again, are estimated intercept and slope parameters respectively and μ_t+1is a zero mean random disturbance term. The regressions following equation 11 and 12 are effectively measuring the percentage of operating assets growth over a one-year time span on time t operating residual income. This use of operating residual income rather than residual income is consistent with Feltham and Ohlson (1995), who assume that financial assets and liabilities have a zero net present value. By making this assumption, they argue that residual income is regarded as being similar to operating residual income. The intercept parameter is exactly similar (12%) to the one reported in Panel A, and the slope parameter drops a single percent (4%). This clearly indicates that only a minor bias is imposed by using standardized cost of capital and support for Hypothesis 1.1 is limited. Yet, it should be noted that the explanatory power is nihil and care should be taken when inferring these results.

6.2. Cost of equity bias on the relation between period-ahead and current residual

income

To test if the assumption of a standardized cost of equity of 12% imposes a considerable bias on the results of the LID model proposed by Ohlson (1995), the original linear regression is first performed with varying cost of equity:

𝑋_𝑡+1𝑎 _{= 𝛼 + 𝜔𝑋}

𝑡𝑎+ 𝜀𝑡+1 (13)

Where 𝛼 and 𝜔 are intercept and slope parameters, respectively; 𝜀_𝑡+1is a random, zero mean disturbance term. Results are provided in Panel A of Table 4. In order to enable a proper comparison between the effect of variable and standardized cost of equity on the persistence of excess returns, the same linear regression is performed, replacing 𝑋_𝑡𝑎 _{and 𝑋}

𝑡+1𝑎 , with 𝑋𝑡𝑏and

𝑋_𝑡+1𝑏 _:

𝑋_𝑡+1𝑏 _{= 𝛼 + 𝜔𝑋}

𝑡𝑏+ 𝜀𝑡+1 (14)

Where 𝛼 and 𝜔, similarly to the previous equation, are intercept and slope parameters and 𝜀_𝑡+1is a random, zero mean disturbance term. Results are provided in Panel B of Table 4. When comparing the results for regressions 13 and 14, it becomes evident that differences are small for year to year observations and even non-existent for the pooled sample, which both provide a value for 𝜔 of 0.76. This value for the persistence of residual returns is comparable to the findings of Biddle et al. (2001), who found a persistence rate of 0.71.

Table 3. Relation between current residual income and future capital investment

of excess returns have a transitory nature but an ample three quarters consist of persistent, or quality earnings. The differences that do come forward from the linear regressions are inconsistent with Hypothesis 2 that standardized cost of capital imposes a downward bias on the persistence parameter. Year 2001 and 2010 do show a slight increase in persistence of excess returns but 2006, 2012 and 2013 show the opposite effect with a comparable magnitude.

Panel a: Linear regression of capital investments in year (t+1) on residual income of year t with industry specific cost of equity a_.

Year t Observations β0 tβ0 β1 tβ1 Adj. R2

98 2721 0.17 24.08 0.02 0.54 0.00 99 2900 0.16 23.03 -0.08 -2.33** 0.00 00 3114 0.08 14.47 0.16 6.22*** 0.01 01 3312 0.09 16.83 0.16 6.23*** 0.01 02 3456 0.13 24.63 0.03 1.38 0.00 03 3566 0.15 27.56 -0.01 -0.44 0.00 04 3687 0.15 27.10 0.03 1.22 0.00 05 3812 0.16 30.92 -0.03 -1.01 0.00 06 3952 0.17 31.56 0.10 3.47*** 0.00 07 3980 0.06 13.91 0.09 4.22*** 0.00 08 4123 0.05 11.40 0.03 1.49 0.00 09 4271 0.12 25.60 0.11 4.30 0.00 10 4336 0.11 24.70 0.00 0.16 0.00 11 4480 0.09 22.15 -0.02 -0.74 0.00 12 4661 0.09 21.68 -0.06 -2.60*** 0.00 13 4216 0.12 24.60 0.05 2.04** 0.00 98-13 pooled 60587 0.12 91.37 0.05 7.02*** 0.00

Panel b. Linear regression of capital investments in year (t+1) on residual income of year t with standard cost of equity (12%)b_.

Year t Observations β0 tβ0 β1 tβ1 Adj. R2

98 2747 0.17 23.95 0.01 0.32 0.00 99 2932 0.16 22.79 -0.10 -2.95 0.00 00 3149 0.08 14.79 0.15 6.41*** 0.01 01 3340 0.09 17.25 0.18 7.01*** 0.01 02 3489 0.13 24.45 0.04 1.78* 0.00 03 3606 0.14 27.10 -0.01 -0.49 0.00 04 3724 0.15 26.91 0.04 1.59 0.00 05 3849 0.17 30.51 -0.04 -1.25 0.00 06 3983 0.17 31.55 0.10 3.56*** 0.00 07 4014 0.06 13.79 0.09 3.77*** 0.00 08 4158 0.05 11.09 0.02 1.03 0.00 09 4306 0.12 25.27 0.10 3.92*** 0.00 10 4373 0.10 24.27 -0.01 -0.24 0.00 11 4514 0.09 21.77 -0.02 -0.86 0.00 12 4698 0.09 21.17 -0.05 -2.20** 0.00 13 4247 0.12 24.32 0.05 2.13 0.00 98-13 pooled 61129 0.12 90.61 0.04 6.73*** 0.00

a _{This panel displays the results of the linear regression: 𝑖}

𝑡+1= 𝛽0+ 𝛽1𝑋𝑡𝑎+ 𝜇𝑡+1,

b _{This panel displays the results of the linear regression: 𝑖}

𝑡+1= 𝛽0+ 𝛽1𝑋𝑡𝑏+ 𝜇𝑡+1,

Additionally, no evidence is found for Hypothesis 2.1, since the explanatory power of both pooled samples is similar.

However, since the linear form might not be able to accurately predict the relation between one period ahead residual income and current residual income, a piece-wise linear regression is performed to test for convexity (Hypothesis 3). This piece-wise linear regression of period ahead residual income adjusted for industry

specific cost of equity 𝑋

_{on current residual}

income 𝑋

_{follows Burgstahler and Dichev (1997) and Biddle et al. (2001):}

𝑋

_{= 𝛼}

0

+ 𝛼

𝑀 + 𝛼

𝐻 + 𝜔

X

+ 𝜔

MX

+ 𝜔

𝐻X

+ 𝜀

(15)

Where estimated intercepts are denoted by αj, j = {0,1,2}, and slope parameters by 𝜔𝑗, j =

{0,1,2}. Dummy variables M and H are indicators for values of residual income in the middle and high thirds, respectively. 𝜀_𝑡+1functions as a random, zero mean disturbance term. The results are displayed in Panel A of Table 5 for the low, middle (M) and high (H) thirds of residual income X_t𝑎_{. Again, to enable proper examination of the bias effect of varying cost of capital, the same}

regression analysis is performed for standardized cost of capital

𝑋_𝑡+1𝑏 _{= 𝛼}

0+ 𝛼1𝑀 + 𝛼2𝐻 + 𝜔0Xt𝑏+ 𝜔1MXt𝑏+ 𝜔2𝐻Xt𝑏+ 𝜀𝑡+1 (16)

Where αj denotes the estimated intercepts, j = {0,1,2}, and 𝜔𝑗 the slope parameters, j = {0,1,2}.

Variables M and H indicate the middle and high thirds of values for residual income, respectively, and 𝜀𝑡+1 is a random, zero mean disturbance term. The results are reported in

Panel B of Table 5 for the low, middle (M) and high (H) of residual income X_t𝑏_{. In order to}

provide a clear overview of the differences between the regression 15 and 16, Table 6 reports the findings of the slope parameters. It comes forward from Table 5 and 6 that the slope parameter ω0 +ω1 is considerably higher than ω0, indicating an increasing slope between the

low and middle thirds of cost of equity adjusted residual income Xt𝑎 and standard cost of equity

residual income X_t𝑏_{. This increase in slope is statistically significant for both adjusted and}

unadjusted residual income for both the yearly samples except for 2003 and 2005, and the pooled sample which is consistent with Hypothesis 3 that convexity can be observed as the functional form. Though, an increase in slope does not hold when the middle (M) and high (H) thirds of both form of residual income are compared. When looking at ω0 +ω1 and ω1 +ω2 a clear

Ryan (2005), and Burgstahler and Dichev (1997). A possible explanation for this S-shaped curve is the predomination of transitory earnings in the highest excess return segments rather than quality and persistent earnings. Another reason why the persistence parameter is lower for the extreme ranges of excess return might be the entry of competitors in the market due to the luring profitability. The higher the realized excess returns, the more attention the market is likely to attract and the more insignificant the barriers to market entry become.

Next to the convexity Hypothesis, Table 6 also provides an insight on the bias that is created by the usage of standardized cost of equity. Whereas the low third does not show a considerable difference when comparing the persistence of industry cost of equity adjusted residual income and standardized cost of equity residual income, there is a clear difference observable for the middle and high third. The pooled sample for both the middle and the high third of the residual income show a downward bias when using standardized cost of capital. The middle third differs 3% and the high third has a somewhat larger difference of 7%. When considering the yearly sample, it stands out that the persistence parameter for the high third is higher for all years, except for 1999, when industry specific costs of equity are used in the regression analysis. These results are consistent with Hypothesis 3.1 stating that the lower cost of equity results in higher persistence of excess returns. However, despite the supportive finding of the pooled sample of the middle range, the yearly samples show significantly varying results. The yearly samples that show a downward bias are only slightly more frequent than years that propose an upward bias when using standardized cost of equity. These findings indicate that when considering the bias effect over time its effect is downward, but individual years are almost as likely to show an upward bias for the middle range of residual income. This variation in both upward and downward biases might be explained by the distribution of the cost of equity capital for the middle section. To be more precise, when the distribution includes more observations with a higher cost of capital than the mean value, the persistence parameter is expected to drop and vice versa. In addition to differences between the adjusted and the non-adjusted persistence parameter, the changes in explanatory power are also of interest (Hypothesis 3.2). Although minor fluctuations in both upward and downward direction are observable for the year-to-year samples, the explanatory power for the pooled sample is similar for both regressions.

Table 4. Linear regression of residual income; industry specific cost of equity versus standard cost of

equity

Panel a: Linear regression of residual income with industry specific cost of equity a

Year Observations α tα ω tω Adj. R2

98 2795 -0.00 -1.77 0.69 45.39 0.42 99 2968 -0.01 -3.51 0.90 52.39 0.48 00 3228 -0.04 -10.74 0.76 46.47 0.40 01 3475 -0.02 -5.54 0.64 50.86 0.43 02 3547 0.01 2.85 0.58 51.56 0.43 03 3658 0.01 3.09 0.83 61.94 0.51 04 3787 -0.00 -1.31 0.83 65.09 0.53 05 3907 -0.00 -0.40 0.97 79.61 0.62 06 4069 -0.01 -4.32 0.88 64.20 0.50 07 4181 -0.04 -13.76 0.77 55.54 0.42 08 4253 -0.01 -5.48 0.52 49.66 0.37 09 4397 0.01 3.99 0.74 65.00 0.49 10 4492 -0.01 -3.73 0.78 62.72 0.47 11 4652 -0.02 -7.73 0.84 70.21 0.51 12 4848 -0.01 -4.29 0.82 73.91 0.53 13 5035 -0.01 -3.53 0.80 65.91 0.46 98-13 63292 -0.01 -13.97 0.76 234.23 0.46

Panel b: Linear regression of residual income with standard cost of equity b

Year Observations α tα ω tω Adj. R2

98 2822 -0.01 -2.83 0.69 45.52 0.42 99 2998 -0.01 -3.94 0.90 52.12 0.48 00 3262 -0.04 -11.47 0.76 46.37 0.40 01 3505 -0.02 -6.67 0.63 60.68 0.42 02 3581 -0.00 -0.25 0.58 52.79 0.44 03 3699 0.00 1.08 0.83 61.68 0.51 04 3827 -0.01 -3.58 0.83 64.67 0.52 05 3948 -0.00 -1.62 0.97 78.85 0.61 06 4101 -0.02 -5.58 0.89 64.47 0.50 07 4215 -0.04 -14.55 0.77 55.60 0.42 08 4289 -0.02 -7.83 0.52 48.42 0.35 09 4432 0.00 1.90 0.74 65.59 0.49 10 4526 -0.01 -6.14 0.76 61.59 0.46 11 4687 -0.02 -9.10 0.84 70.11 0.51 12 4890 -0.01 -5.75 0.83 74.73 0.53 13 5068 -0.01 -5.82 0.82 66.79 0.47 98-13 63850 -0.01 -20.52 0.76 234.11 0.46

a_{This panel displays the results of the linear regression: 𝑋}

𝑡+1𝑎 = 𝛼 + 𝜔𝑋𝑡𝑎+ 𝜀𝑡+1

b _{This panel displays the results of the linear regression: 𝑋}

Table 5. Piece-wise regression of residual income; industry specific cost of equity versus standard cost of

equity

Panel a: Piece-wise regression of residual income of year t+1 on residual income of year t with industry specific cost of equity Year (t+1) α0 tα0 α0+ α1 tα1 α0+ α2 tα2 ω0 tω0 ω0+ ω1 tω1 ω0+ ω2 tω2 Adj. R2 98 -0.02 -1.60 -0.01 -3.47 0.00 0.28 0.67 23.09 1.11 3.62*** 0.68 8.93*** 0.37 99 -0.02 -1.46 -0.01 -2.81 0.01 1.84 0.91 28.92 1.55 6.61*** 0.53 6.39*** 0.45 00 -0.05 -4.44 -0.01 -5.10 0.00 0.73 0.77 24.87 1.13 4.09*** -0.01 -0.13 0.36 01 -0.05 -6.04 -0.00 -1.64 -0.01 -0.87 0.58 25.90 1.24 4.98*** 0.68 9.81*** 0.31 02 -0.00 -0.64 -0.00 -0.12 -0.00 -0.63 0.54 26.40 0.64 3.03*** 0.92 13.26*** 0.34 03 0.01 1.44 0.01 3.92 0.02 4.52 0.84 31.44 0.48 2.64*** 0.60 9.73*** 0.45 04 -0.02 -1.96 -0.00 -1.68 0.01 1.81 0.80 28.19 1.45 7.87*** 0.73 14.37*** 0.44 05 0.00 0.25 -0.00 -0.15 0.02 4.13 0.98 37.61 0.76 3.68*** 0.72 16.14*** 0.55 06 -0.03 -2.75 -0.01 -3.09 -0.01 -1.43 0.85 29.81 1.08 4.99*** 0,85 13.94*** 0.44 07 -0.07 -7.43 -0.02 -9.77 -0.01 -1.52 0.72 27.40 1.80 5.38*** 0.43 6.84*** 0.36 08 -0.02 -3.32 -0.01 -7.30 0.00 0.17 0.51 27.23 0.88 4.43*** 0.42 7.70*** 0.29 09 0.02 2.99 0.00 0.24 0.01 2.10 0.75 38.95 0.98 4.80*** 0.74 14.28*** 0.45 10 -0.02 -2.61 -0.00 -1.00 0.00 0.35 0.76 31.85 1.18 7.07*** 0.66 13.39*** 0.40 11 -0.02 -2.99 -0.01 -3.26 0.00 0.23 0.85 37.66 1.35 6.62*** 0.53 11.12*** 0.47 12 -0.01 -1.31 -0.00 -2.82 0.00 0.95 0.83 39.81 1.12 5.92*** 0.58 12.49*** 0.48 13 -0.01 -0.67 -0.00 -1.10 0.01 1.39 0.82 34.36 1.07 6.05*** 0.49 9,14*** 0.42 98-13 -0.02 -8.18 -0.01 -11.57 0.00 3.09 0.75 120.62 1.14 21.11*** 0.60 40.42*** 0.40

Panel b: Piece-wise regression of residual income of year t+1 on residual income of year t with standard cost of equity Year (t+1) α0 tα0 α0+ α1 tα1 α0+ α2 tα2 ω0 tω0 ω0+ ω1 tω1 ω0+ ω2 tω2 Adj. R2 98 -0.02 -1.68 -0.01 -2.25 0.00 0.29 0.67 22.24 1.26 4.28*** 0.62 8.54*** 0.36 99 -0.02 -1.29 -0.01 -1.40 0.01 1.08 0.91 26.47 1.53 5.73*** 0.55 6.97*** 0.43 00 -0.05 -4.40 -0.01 -2.19 -0.00 -0.57 0.77 23.87 1.71 6.84*** -0.05 -0.66 0.35 01 -0.05 -6.24 -0.01 -3.09 -0.01 -1.62 0.57 25.04 1.20 7.66*** 0.68 9.23*** 0.31 02 -0.01 -1.60 -0.01 -2.46 -0.00 -0.01 0.55 27.97 0.79 4.52*** 0.84 11.08*** 0.35 03 0.01 0.99 0.00 0.88 0.02 3.03 0.85 32.80 0.57 4.01*** 0.54 7.23*** 0.46 04 -0.03 -2.47 -0.00 -0.46 0.00 0.76 0.80 29.20 1.30 8.34*** 0.66 11.82*** 0.44 05 0.00 0.02 -0.00 -0.43 0.01 2.76 0.99 37.75 0.90 5.54*** 0.67 13.54*** 0.55 06 -0.03 -2.93 -0.01 -3.20 -0.00 -0.83 0.86 29.83 1.06 4.76*** 0.75 12.30*** 0.44 07 -0.08 -7.46 -0.03 -7.67 -0.01 -2.00 0.72 26.28 1.63 6.08*** 0.40 6.33*** 0.35 08 -0.03 -4.32 -0.01 -4.99 -0.00 -0.72 0.50 26.25 0.95 5.72*** 0.35 5.72*** 0.28 09 0.01 2.52 -0.00 -0.27 0.01 1.04 0.76 39.30 0.85 5.78*** 0.69 11.70*** 0.45 10 -0.03 -3.85 -0.00 -1.61 -0.00 -0.54 0.74 31.12 0.98 8.17*** 0.58 10.27*** 0.38 11 -0.02 -3.57 -0.01 -2.28 -0.00 -0.48 0.85 37.69 1.19 7.30*** 0.44 8.31*** 0.47 12 -0.01 -2.01 -0.00 -1.09 -0.00 -0.97 0.84 40.60 1.18 10.41*** 0.55 9,94*** 0.47 13 -0.00 -1.30 -0.01 -2.54 0.01 1.65 0.85 37.05 0.97 5.52*** 0.28 4.46*** 0.43 98-13 -0.02 -9,93 -0.01 -10.05 0.00 0.69 0.75 121.16 1.11 24.73*** 0.53 32.61*** 0.40

a _{This panel displays the results from the following piece-wise linear regression: 𝑋}

𝑡+1𝑎 = 𝛼0+ 𝛼1𝑀 + 𝛼2𝐻 +

𝜔0Xta+ 𝜔1MXat + 𝜔2𝐻Xta+ 𝜀𝑡+1

b _{This panel displays the results from the following piece-wise linear regression: 𝑋}

𝑡+1𝑏 = 𝛼0+ 𝛼1𝑀 + 𝛼2𝐻 +

𝜔0Xt𝑏+ 𝜔1MXt𝑏+ 𝜔2𝐻Xt𝑏+ 𝜀𝑡+1

Table 6. Piece-wise linear regression results comparison for industry specific and standard cost of equity

Varying cost of equity ri Standardized cost of equity rs

Year (t+1) _{ω0 ω0+ ω1 ω0+ ω2 ω0 ω0+ ω1 ω0+ ω2} 98 0.67 1.11*** 0.68*** 0.67 1.26*** 0.62*** 99 0.91 1.55*** 0.53*** 0.91 1.53*** 0.55*** 00 0.77 1.13*** -0.01 0.77 1.71*** -0.05 01 0.58 1.24*** 0.68*** 0.57 1.20*** 0.68*** 02 0.54 0.64*** 0.92*** 0.55 0.79*** 0.84*** 03 0.84 0.48*** 0.60*** 0.85 0.57*** 0.54*** 04 0.80 1.45*** 0.73*** 0.80 1.30*** 0.66*** 05 0.98 0.76*** 0.72*** 0.99 0.90*** 0.67*** 06 0.85 1.08*** 0,85*** 0.86 1.06*** 0.75*** 07 0.72 1.80*** 0.43*** 0.72 1.63*** 0.40*** 08 0.51 0.88*** 0.42*** 0.50 0.95*** 0.35*** 09 0.75 0.98*** 0.74*** 0.76 0.85*** 0.69*** 10 0.76 1.18*** 0.66*** 0.74 0.98*** 0.58*** 11 0.85 1.35*** 0.53*** 0.85 1.19*** 0.44*** 12 0.83 1.12*** 0.58*** 0.84 1.18*** 0.55*** 13 0.82 1.07*** 0.49*** 0.85 0.97*** 0.28*** 98-13 0.75 1.14*** 0.60*** 0.75 1.11*** 0.53***

*** indicates significance at a 1% level

6.3 Robustness Checks

6.3.1. Firm size

In order to account for the possibility that the findings on convexity are due to the pooling of firms with different sizes, the sample is split up in quantiles. First, observations for positive capital investments are parted from observations for negative capital investments (divestments). These two sub-groups are then both split into four quantiles (Q1 representing the lowest investment/divestment quantile and Q4 the highest). The same steps are repeated for the standardized cost of equity capital sample. Results for both the simple linear regression and the piece-wise linear regression of current residual income on future expected residual income are provided in Tables 9-12 in Appendix B. It comes forward from Panel B of the afore mentioned tables that variation in the persistence parameter of the simple linear regression is limited. This confirms the findings in Table 4 that standardized cost of equity capital imposes only a minor bias on excess return persistence when considering the LID model.

results again show supportive evidence of convexity in the middle range. The results on the highest residual income however are in conflict with convexity, again reaffirming the S-Shape that comes forward from Table 5. Though, since the sample size for negative capital investments is rather small and almost half of the findings insignificant, these results might not be representative.

6.3.2. Investment vs. Divestment

The method to account for large fluctuations in size of observed firms, also accounts for large fluctuations in capital investments. By splitting up investments and divestments and then dividing each in four quantiles, these fluctuations are accounted for. The case of positive investments provides additional evidence on hypothesis 1 since the persistence parameter moves monotonically with capital investments (i+1) (by reading vertically down the columns of the persistence parameters ω0, ω0+ ω1 andω0+ ω2) when industry specific cost of equity capital

are considered. Divestments do not provide evidence for the “capital follows profitability” hypothesis. Where one would expect the persistence parameter to decrease monotonically with capital divestment, an increase is shown for most quantiles. It must be noted however, that observations for the divestment quantiles were limited and that this result might not be representative.

6.3.3. Variable definition for capital investment growth

This robustness checks aims at controlling random variation in the one-year ahead growth measure. For both the positive and negative investment firms the regression (provided in Tables 9-12) was repeated using the three year ahead growth measure (i+3). Similar results were found on the “capital follows profitability” and the convexity hypothesis. Considering the bias that is imposed by disregarding industry specific cost of equity capital, the results are even larger when using the three year ahead growth measure (i+3). Variation between adjusted and un-adjusted persistence parameters range from 4% to 32% (Results are reported in Tables 13-16 in Apendix C).

7. Conclusion

relationship between current residual income and future expected residual income since the latter is also related to capital investments that follow from current residual income. This inclusion of capital investments in the residual income model oppose the LID model defined by Ohlson (1995).

Three empirical tests are performed, one considering the “capital follows profitability” assumption and the others testing the proposed models of Ohlson (1995) and Biddle et al. (2001) respectively. All these tests are performed for both varying and constant cost of equity capital, to infer whether a bias is imposed by the latter. The results from the empirical tests are partially supportive of the bias that this study proposes. The first finding entails that the bias on the “capital follows profitability” assumption is only minor, varying cost of capital only realizes an one percent increase in the excess earnings that is reinvested. The more notable result that is observed regarding capital investments is the very low percentage of excess returns being reinvested. These results conflict with Biddle et al. (2001) who find a reinvestment parameter of 0.96 for the pooled sample. These contradictory results suggest that excess returns might not be reinvested in their entirety but might be used for other purposes inlcuding debt repayment or improving cash balances. However, it must be noted that explanatory power is lacking for this empirical test and therefore caution should be taken when interpreting the results.

account, especially with large differences in the persistence parameters, this is not clearly observable. Overall, the explanatory power tends to increase but by less than one percent. The inclusion of capital investments suggests non-linearity of the relationship between current residual income and future expected residual income. The empirical results in this study confirm non-linearity and reaffirm the findings of Biddle et al. (2001) that the relationship can be depicted by a S-shaped curve. This rejection of the linear form is in line with the findings in this study that accounting for variation in the cost of equity capital does not point out a bias for the linear functional form but does so for the S-shaped relationship.

It can be concluded that the findings in this study hold two implications. To start with, the linear relationship between residual income and future expected residual income is found to be an inappropriate functional form and evidence for a S-shaped relation is provided. But more importantly, a clear and significant bias in results ignoring variation in the cost of equity capital is identified. This suggests that the long preserved assumption that cost of equity capital can be assumed to be 12%, without significantly distorting the results, is an invalid one. In order to provide accurate results in the field of valuation, variation in the cost of capital should be taken into account. From a practical perspective, the finding that a lower cost of capital results in higher persistence in and quality of earnings may assist managers in deciding on capital investments.

Appendix A

Table 7. Industry specific cost of equity for 1998-2006

Industry 1998 1999 2000 2001 2002 2003 2004 2005 2006 Agricultural Production-Crops 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% Agricultural Prod-Livestock 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% Agricultural Services 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% Forestry 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01%

Fishing, Hunting and

Trapping 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01% 7,01%

Metal Mining 8,62% 8,23% 8,28% 6,98% 5,60% 6,21% 6,21% 7,60% 9,11%

Bituminous Coal & Lignite

Mining 10,14% 9,55% 10,02% 10,29% 8,57% 8,99% 8,99% 9,33% 9,81%

Oil and Gas Extraction 11,62% 11,23% 11,26% 10,23% 8,41% 8,59% 8,98% 9,30% 9,86%

Mining & Quarrying of

Nonmetallic Minerals 10,14% 9,55% 10,02% 10,29% 8,57% 8,99% 8,99% 9,33% 9,81%

General Bldg Contractors -

Residential Bldgs 10,71% 10,10% 9,53% 9,47% 7,67% 8,09% 8,33% 8,82% 9,53%

Heavy Construction 10,82% 10,82% 10,82% 10,82% 10,82% 10,82% 10,82% 10,82% 10,82%

Construction Special Trade 11,12% 11,12% 11,12% 11,12% 11,12% 11,12% 11,12% 11,12% 11,12%

Food and Kindred Products 9,66% 9,27% 8,75% 8,46% 6,83% 7,33% 7,05% 7,31% 8,26%

Tobacco Products 9,77% 9,43% 8,25% 8,30% 7,04% 7,47% 7,09% 7,58% 8,58%

Textile Mill Products 10,73% 10,22% 9,50% 9,71% 7,97% 8,50% 7,91% 8,69% 9,26%

Apparel 10,73% 10,22% 9,50% 9,71% 7,97% 8,50% 7,91% 8,69% 9,26%

Lumber & Wood Products 10,46% 9,72% 9,65% 9,56% 7,66% 8,11% 7,79% 9,00% 9,50%

Furniture and Fixtures 10,85% 10,27% 9,64% 9,47% 7,79% 7,93% 8,18% 8,80% 8,97%

Papers & Allied Products 10,46% 9,89% 9,30% 9,57% 7,67% 8,29% 8,40% 8,34% 8,83% Printing, Publishing, and

Allied Industries 10,68% 10,22% 9,35% 9,69% 8,44% 8,57% 7,81% 7,95% 9,09%

Chemicals & Allied Products 10,54% 9,74% 10,33% 9,74% 7,78% 8,51% 8,62% 9,31% 9,49% Petroleum Refining and

Related Industries 10,27% 9,69% 8,93% 9,06% 7,40% 7,75% 7,24% 7,61% 9,02%

Rubber and Miscellaneous

Plastics Products 10,67% 10,40% 9,79% 9,92% 8,41% 8,83% 9,16% 10,11% 9,42%

Leather & Leather Products 10,61% 9,81% 9,73% 9,65% 8,03% 8,22% 8,95% 9,27% 9,99% Stone, Clay, Glass, and

Concrete Products 10,37% 9,60% 9,10% 8,92% 7,37% 8,00% 8,02% 7,78% 9,69%

Primary Metal Industries 10,70% 10,23% 9,47% 9,54% 8,00% 7,90% 8,11% 8,62% 9,68%

Fabricated Metal Products 10,70% 10,23% 9,47% 9,54% 8,00% 7,90% 8,11% 8,62% 9,68%

Computer Equipment 10,58% 10,05% 9,28% 9,29% 7,47% 8,07% 7,96% 8,39% 9,64%

Electronic Equipment 11,77% 11,06% 11,06% 12,03% 10,54% 11,03% 11,25% 12,31% 12,04%

Transportation Equipment 10,58% 9,99% 9,71% 9,63% 7,75% 8,27% 8,00% 8,60% 9,53%

Measuring, Analyzing, and

Controlling Instruments; 11,31% 10,51% 9,57% 9,72% 7,83% 8,28% 8,34% 9,39% 10,14%

Miscellaneous

Manufacturing Industries 11,51% 10,93% 11,48% 10,55% 8,40% 8,77% 9,11% 9,55% 9,52%

Railroad Transportation 11,14% 10,06% 9,25% 9,17% 7,57% 8,26% 7,47% 7,90% 9,40%

Hwy Passenger Trans 10,58% 9,99% 9,71% 9,63% 7,75% 8,27% 8,00% 8,60% 9,53%

Industry 1998 1999 2000 2001 2002 2003 2004 2005 2006

Water Transportation 10,75% 10,16% 9,52% 9,20% 7,61% 8,12% 7,48% 7,87% 8,95%

Transportation by Air 12,24% 11,41% 10,71% 11,10% 10,25% 10,60% 10,69% 11,12% 11,48%

Pipelines, except Natural Gas 7,34% 7,34% 7,34% 7,34% 7,34% 7,34% 7,34% 7,34% 7,34%

Transportation Services 10,58% 9,99% 9,71% 9,63% 7,75% 8,27% 8,00% 8,60% 9,53%

Communications 12,02% 11,41% 12,96% 13,74% 11,17% 10,77% 10,61% 12,48% 11,71%

Electric, Gas & Sanitary

Services 8,94% 8,30% 8,16% 8,11% 7,11% 8,06% 7,89% 8,29% 9,29%

Wholesale-Durable Goods 9,70% 9,67% 8,84% 8,81% 6,96% 7,49% 7,26% 7,27% 8,23%

Retail-Building Materials,

Hardware, Garden Supply 11,24% 10,43% 9,95% 9,74% 8,12% 9,14% 8,50% 8,68% 9,36%

Retail-Food Stores 11,65% 11,09% 10,69% 10,07% 8,20% 8,63% 8,93% 9,16% 9,30%

Retail-Auto Dealers &

Gasoline Stations 11,02% 10,07% 9,92% 10,21% 8,63% 9,00% 9,44% 10,30% 11,04%

Retail-Apparel & Accessory

Stores 10,73% 10,22% 9,50% 9,71% 7,97% 8,50% 7,91% 8,69% 9,26%

Retail-Home Furniture 10,85% 10,27% 9,64% 9,47% 7,79% 7,93% 8,18% 8,80% 8,97%

Retail-Eating & Drinking

Places 10,84% 10,03% 9,13% 9,22% 7,35% 7,83% 7,54% 7,64% 8,63%

Retail-Miscellaneous Retail 11,51% 10,93% 11,48% 10,55% 8,40% 8,77% 9,11% 9,55% 9,52%

Depository Institutions 10,18% 9,29% 9,31% 8,70% 6,90% 7,24% 6,76% 7,05% 7,62%

Non-Depository Credit

Institutions 12,12% 11,20% 10,33% 10,25% 8,09% 8,59% 7,98% 7,98% 9,09%

Security & Commodity 12,28% 12,33% 11,62% 12,00% 10,23% 10,55% 10,62% 10,90% 11,05%

Insurance Carriers 10,69% 10,35% 9,78% 9,69% 7,89% 8,51% 7,84% 7,90% 9,28%

Insurance Agents, Brokers

and Service 9,68% 9,68% 9,68% 9,68% 9,68% 9,68% 9,68% 9,68% 9,68%

Real Estate 9,76% 9,36% 8,79% 8,57% 6,94% 7,34% 7,28% 7,63% 8,50%

Holding and other

Investment Offices 9,76% 9,36% 8,79% 8,57% 6,94% 7,34% 7,28% 7,63% 8,50%

Hotels, Rooming Houses 11,20% 10,44% 9,63% 9,95% 7,95% 8,31% 7,81% 8,34% 8,47%

Personal Services 11,23% 11,23% 11,23% 11,23% 8,84% 9,27% 8,75% 8,54% 9,69% Business Services 11,23% 11,23% 11,23% 11,23% 8,84% 9,27% 8,75% 8,54% 9,69% Automotive Repair 9,97% 9,72% 8,70% 9,77% 7,96% 8,58% 9,37% 10,25% 9,56% Miscellaneous Repair Services 11,32% 10,72% 10,07% 9,53% 7,70% 8,08% 8,32% 9,04% 9,64% Motion Picture 11,70% 11,21% 12,22% 14,45% 10,12% 10,01% 10,18% 11,89% 9,55%

Amusement & Recreation

Services 10,71% 10,15% 9,72% 9,93% 7,92% 8,46% 8,74% 9,59% 10,19% Health Services 11,83% 10,51% 9,77% 10,30% 8,41% 9,09% 9,35% 11,02% 10,69% Legal Services 11,23% 11,23% 11,23% 11,23% 8,84% 9,27% 8,75% 8,54% 9,69% Educational Services 11,54% 10,26% 9,62% 9,95% 9,08% 9,23% 9,55% 9,64% 10,04% Social Services 11,23% 11,23% 11,23% 11,23% 8,84% 9,27% 8,75% 8,54% 9,69% Membership organizations 11,23% 11,23% 11,23% 11,23% 8,84% 9,27% 8,75% 8,54% 9,69% Engineering 11,23% 11,23% 11,23% 11,23% 8,84% 9,27% 8,75% 8,54% 9,69% Services, NEC 11,23% 11,23% 11,23% 11,23% 8,84% 9,27% 8,75% 8,54% 9,69% Administration of Environmental Quality 10,58% 9,75% 8,80% 9,11% 7,38% 7,88% 7,55% 8,02% 8,97%

National Security and

International Affairs 10,86% 10,39% 9,52% 9,31% 7,46% 8,05% 8,08% 8,44% 9,20%