Abstract
The paper develops the valuation implications of the Ohlson/Juettner-Nauroth (Rev Account Stud 10(2–3):349–365, 2005, OJ) model having two growth parameters in the residual income dynamic—one for the long-term (g) and one for the short-term (g h ). The central result shows that the model can be transformed into \(V_{0} = BV_{0} + (RE_{ 1} /(r - g)) \cdot Scalar,\) where Scalar = 1 + (g h − g)/r. As a benchmark, the scalar equals one if and only if g = g h . The Ohlson (Contemp Account Res 11(2):661–687, 1995) residual income valuation (RIV) model with a single-growth parameter then holds. The OJ model thus generalizes and enriches the set of residual income growth patterns when g ≠ g h . A scalar greater than one admits residual income that (i) grows at a relatively large short-term rate fading downward to a long-term rate or (ii) declines over the long-term. A scalar less than one admits residual income that (iii) grows at a rate fading upward to a long-term rate or (iv) starts out negative and turns positive. Under these scenarios, the RIV model cannot hold in any meaningful sense, even as an approximation. The transformed OJ model also provides a unique angle in explaining market-to-book ratios as a function of profitability and short- and long-term growth, generating empirical implications beyond the constrained RIV model.
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Notes
BV is the book value of equity; x is the comprehensive earnings; and d represents dividends (Ohlson 1995).
For simplicity, the expectation operator is suppressed.
\(\mathop {\lim }\limits_{t \to \infty } (RE_{t + 1} /RE_{t} - 1) = g\quad if \, g > 0\) and \(\mathop {\lim }\limits_{t \to \infty } (RE_{t + 1} /RE_{t} - 1) = 0\quad if \, g \le 0\). The residual income growth rate geometrically reverts to the long-term rate (Penman 2005).
The assumptions of the growth patterns are indicated in the figures.
In the Scalar > 1 and g < 0 and Scalar < 1 and g > 0 groups, two growth patterns are possible. The zero residual income case (case 1 in Penman 2012) is also included for completeness. This yields a total of 12 cases.
\(\frac{{RE_{t + 1} }}{{RE_{t} }} - 1 = g\left( {1 - \frac{{(g - g_{h} )}}{{(g - g_{h} ) + g_{h} (1 + g)^{t - 1} }}} \right) \, if \, g \ne 0\) and \(\frac{{RE_{t + 1} }}{{RE_{t} }} - 1 = \frac{{g_{h} }}{{1 + (t - 1)g_{h} }} \, if \, g = 0\). When g h = g, the item in the brackets of the first equation becomes one. In any period t, residual income grows at the rate of g, which is also the growth rate of residual income growth. Similarly, when g h and g are both zero in the second equation, residual income and residual income growth are constant over time.
When t is very large, the residual income growth under negative g would be so close to zero that it would be economically insignificant. However, the growth rate would remain constant at g.
When ∆RE 2 > 0 and g > 0, residual income growth increases exponentially without limit at the rate of g, and residual income grows by an increasing amount over time. When ∆RE 2 > 0 and g = 0, residual income growth is constant, and residual income grows by a constant amount over time. When ∆RE 2 > 0 and g < 0, residual income growth declines towards zero at the rate of g, and residual income increases by a decreasing amount over time. The reverse is true in the case of ∆RE 2 < 0. Depending on whether g is positive, zero, or negative, negative residual income growth declines exponentially, stays constant, or declines towards zero, and residual income declines by an increasing, constant, or decreasing amount over time. Formally,
if g ≠ 0, \(RE_{t + 1} = \left( {1 + ((1 + g)^{t} - 1)(g_{h} /g)} \right)RE_{1}\) and \(\, \partial RE_{t + 1} /\partial t = RE_{1} \left( {\left( {1 + g} \right)^{t} \cdot \log \left( {1 + g} \right) \cdot \left( {g_{h} /g} \right)} \right)\). If g = 0, \(RE_{t + 1} = \left( {1 + t \cdot g_{h} } \right)RE_{1}\) and \(\partial RE_{t + 1} /\partial t =\Delta RE\).
When t is very large, the (1 + g)t−1 in the denominator dominates the first formula in footnote 6, and the term in the big brackets approaches one.
Under negative g and large t, the (1 + g)t−1 in the denominator of the first formula in footnote 6 becomes negligible, and the term in the big brackets approaches zero. Accounting numbers measure residual income growth in the nominal term. If the long-term growth rate is zero, residual income is not growing in the nominal term and decreases by the rate of inflation in the real term.
Negative g h cases due to positive RE 1 and negative ∆RE 2 are ruled out because they would result in negative long-term residual income. See the first paragraph of Sect. 3.
As residual income in a particular period equals RE 1 plus accumulated residual income growth, the effect of the latter eventually dominates. With the power of compounding, firms with negative RE 1 may outgrow firms with positive RE 1 if g is high enough.
There are missing data points at the beginning of the series because the graph only depicts sensible growth rates with a positive denominator.
One sensible growth rate is the long-term rate of inflation or GDP growth, capturing the notion that accounting value added is growing in the nominal term but maintained over time in real terms. A growth rate greater than the inflation rate implies residual income is growing in real terms.
These scenarios are ruled out in the previous analysis. See the first paragraph of Sect. 3.
A sufficiently positive ∆RE 2 (the numerator of g h ) would result in a highly negative g h .
If ROE > r, the −(r − g)*[ROE 1 /(ROE 1 –r)] < g h < −(r − g) condition ensures that 0 < MTB < 1.
The reverse is true when ROE < r, and the linear relation turns negative and is less negative with a higher ROE, a lower r, or both.
Page 29 of Ohlson and Gao (2006) makes similar points.
Under the clean surplus relation, ∆RE is the same as abnormal earnings growth in the OJ model but on a total dollar rather than per share basis.
The clean surplus relation admits the role of accounting in valuation, addressing a problem in the OJ model where the EPS is primitive and arbitrarily chosen (Penman 2005). The book value and residual income in model (2) are both unambiguously relevant. As clean surplus accounting holds for both biased and neutral accounting principles, expressions (2) and (3) are versatile to any accounting policy.
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Acknowledgments
I owe a debt of gratitude to Jim Ohlson (editor) for generously sharing his time and advice. I thank the anonymous reviewers of this journal for their constructive comments. I also wish to thank Stephen Penman for his helpful guidance on an early version of this paper.
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Appendix: Derivation of the clean surplus OJ model
Appendix: Derivation of the clean surplus OJ model
Combining PVED, the clean surplus relation (∆BV t = x t − d t , where BV is the book value of equity, x is the comprehensive earnings, and d represents dividends), and assumption (1) yields the Ohlson and Juettner-Nauroth (2005) model under clean surplus accounting:
Rearranging yields:
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Lai, C. Growth in residual income, short and long term, in the OJ model. Rev Account Stud 20, 1287–1296 (2015). https://doi.org/10.1007/s11142-015-9320-4
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DOI: https://doi.org/10.1007/s11142-015-9320-4