Structural properties of the price-to-earnings and price-to-book ratios


We examine the structural properties of a firm’s price-to-earnings (P/E) and price-to-book (P/B) ratios and the relation between these two ratios. A benchmark result is obtained under the hypothesis that firms use replacement cost accounting to value their operating assets, so that the P/B ratio coincides with Tobin’s q. The firm’s P/E ratio can then be expressed as a convex combination of the P/E ratios suggested respectively by the permanent earnings model and the Gordon growth model, with the relative weight to be placed on these two endpoints determined entirely by Tobin’s q. Under current financial reporting rules, the accounting for operating assets is likely to be more conservative than replacement cost accounting. Our findings characterize how the magnitude and behavior of the P/E and P/B ratios are jointly shaped by several key variables, including both past and anticipated future growth, economic profitability, and accounting conservatism

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Fig. 1


  1. 1.

    See, for example, Tobin (1969), Hayashi (1982), and Abel and Eberly (2011).

  2. 2.

    For instance, Lindenberg and Ross (1981) submit that q “exceeds one by the capitalized value of the Ricardian and monopoly rents which the firm enjoys.”

  3. 3.

    See Basu (1977), Jaffe et al. (1989), and Lakonishok et al. (1994). At the aggregate level, the Fed model states that the stock market earnings yield (the inverse of the P/E ratio) should equal the 10-year nominal Treasury yield; see Asness (2003) and Bekaert and Engstrom (2010).

  4. 4.

    The empirical relation between the P/E and P/B ratios is studied by Penman (1996).

  5. 5.

    Alternative benchmarks for the P/E ratio are discussed by Feltham and Ohlson (1996), Ohlson and Juettner-Nauroth (2005), Penman (1996, 2013), and Zhang (2000).

  6. 6.

    This benchmark can be justified in a setting where firm sales grow at a constant rate and the firm has only variable cash operating expenses. Accounting earnings are presumed equal to cash flow, and, as a consequence, firm value can be expressed as a multiple of forward earnings. See, for example, Beaver and Morse (1978), Zarowin (1990), and Damodaran (2006, p. 245).

  7. 7.

    Our model framework of capacity investments and replacement cost accounting builds on that of Rogerson (2008). This framework has been used in a number of recent studies spanning managerial performance evaluation (Rogerson 2008; Dutta and Reichelstein 2010), monopoly regulation (Rogerson 2011; Nezlobin et al. 2012), and financial statement analysis (Nezlobin 2012; McNichols et al. 2014).

  8. 8.

    Consistent with much of the investment literature in finance and economics, the firm’s market price in our model is equal to the replacement cost of assets in place plus the discounted sum of future economic profits (e.g. Thomadakis 1976; Lindenberg and Ross 1981; Fisher and McGowan 1983; Salinger 1984; Abel and Eberly 2011). This result is obtained under the assumption that the firm’s price is equal to the present value of future cash flows under the optimal investment policy. The price of a firm’s stock can, of course, deviate from its fundamental value due to market inefficiencies or agency problems, two issues that are ignored in our analysis.

  9. 9.

    Our analysis builds on the work of McNichols et al. (2014), who seek to obtain a measure of Tobin’s q by applying a “conservatism correction” factor to the P/B ratio. The empirical part of their analysis shows that this measure of Tobin’s q has better predictive power for future investments than the P/B ratio. Like McNichols et al. (2014), our focus is on unconditional conservatism, as contrasted with the conditional conservatism studies, e.g., Basu (1977) or Beaver and Ryan (2005).

  10. 10.

    This finding is conceptually related to a “quadrant result” obtained in connection with the Accounting Rate-of-Return: see, for instance, Salamon (1985), Fisher and McGowan (1983), and Rajan et al. (2007).

  11. 11.

    Rogerson (2008) has shown that the present model can be extended to settings where the cost of new assets changes over time.

  12. 12.

    In the geometric scenario with \(T=\infty \), all inequalities in (1) are satisfied as equalities. If the productive capacity declines geometrically over time but the useful life of assets is finite, then inequality (1) is strict for \(\tau =T\).

  13. 13.

    Extensions of the base model to stochastic environments are discussed in Sect. 3 below.

  14. 14.

    Since we do not impose any assumptions on the composition of the firm’s asset base in period 0, there is no loss of generality in evaluating the price-to-book and price-to-earnings ratios at date T. Our results hold for the financial ratios calculated at any date t, if the corresponding relevant investment history is understood to be \(\left( I_{t},\ldots,I_{t-T+1}\right) \).

  15. 15.

    Replacement cost accounting for operating assets, like plant, property and equipment, was permissible under U.S. Generally Accepted Accounting Principles in the 1970s. Lindenberg and Ross (1981) base their estimates of Tobin’s q on companies that adopted this asset valuation rule. Subsequent literature suggested several methods for estimating the replacement cost of assets based on the information available in the published accounting reports; see, for instance, Salinger and Summers (1983), Perfect and Wiles (1994), and Lewellen and Badrinath (1997). Erickson and Whited (2006) evaluate the accuracy of different methods for computing Tobin’s q.

  16. 16.

    Arrow (1964) provided a general expression for the user cost of capital in terms of a certain series of recursively defined functions. The simple expression for c in Eq. (5) is due to Rogerson (2008).

  17. 17.

    To be sure, our model does not assume the existence of such a rental market, yet the construct is useful in defining the user cost of capacity and the replacement cost of assets in place.

  18. 18.

    We recall that at date T the investment decision \(I_{T}\) has been made, and therefore the capacity level for period \(T+1\) has already been decided. We further assume that at date T (when the P/E and P/B ratios are evaluated) the firm is already on the optimal investment path, i.e., the investment \(I_{T}\) was chosen so as to maximize \(\pi _{T+1}.\)

  19. 19.

    Specifically, the consistency condition will be met if \(\mu _{t}\ge -\min _{1\le \tau \le T}\frac{(x_{\tau }-x_{\tau +1})}{x_{\tau }}\) for all t.

  20. 20.

    See the proof of Proposition 1 for details. This result generalizes similar findings of Lindenberg and Ross (1981) and Salinger (1984) to settings with a general vintage composition of assets.

  21. 21.

    This relation holds because for any history of investments,

    $$\begin{aligned} D_{t}^{*}+r\cdot BV_{t-1}^{*}&=(d_{1}^{*}+r\cdot bv_{0}^{*})\cdot I_{t-1}+\cdots +(d_{T}^{*}+r\cdot bv_{T-1})\cdot I_{t-T}\\&=c\cdot \left( x_{1}\cdot I_{t-1}+\cdots +x_{T}\cdot I_{t-T}\right) =c\cdot K_{t}. \end{aligned}$$
  22. 22.

    The first \(\mu _{t}\) that matters in capitalizing future economic profits is \(\mu _{T+2}\) because the baseline value for capitalizing future economic profits is \(\pi _{T+1}^{o}\). We note that firm value, \(P_{T}\), is well defined, provided the sequence \(\varvec{\mu }\) is such that the denominator on the right-hand side of (12) is positive.

  23. 23.

    In particular, \(ROE=r\) under replacement cost accounting whenever the firm operates in a competitive environment, resulting again in a market-to-book ratio equal to one.

  24. 24.

    The resulting sequence of book values will grow at the rate \(\mu \), irrespective of the accounting rules, in the special case of a constant growth rate for all investments, both past and future.

  25. 25.

    To illustrate this point, assume that \(T=2,\) \(x_{1}=x_{2}=1\) and \(r=10\%\). Assume further that \(K_{3}^{o}=100\) and the firm expects its sales to remain constant after period \(T+1,\) i.e., \(s(\varvec{\mu })=0\), \(K_{4}^{o}=100,\) \(K_{5}^{o}=100,\) and so on. Consider the following investment history that leads to \(K_{3}^{o}=100\): \(\left( I_{2}=0,I_{1}=100\right) \). To implement the optimal capacity levels going forward, the firm will need to make a replacement investment of 100 in years 3, 5, 7... Therefore, the firm’s net cash flows will alternate between the values of \(100\cdot p^{o}-100\) and \(100\cdot p^{o}\). It can be verified that, for this investment history, \(P_{T}=1,000\cdot p^{o}-1,000\cdot 1.1/2.1.\) Under the straight-line depreciation rule, \(\left( d_{1}=0.5,d_{2}=0.5\right) \), \(BV_{T}=50\) and \(E_{T+1}=100\cdot p^{o}-50\), \(ROE_{T+1}=2p^{o}-1.\) It is straightforward to check that Eq. (14) does not hold under the straight-line rule if \(g=0\) (the demand growth rate). It will, however, hold under replacement cost accounting (annuity depreciation) where \(d_{1}=\frac{1}{2.1}\), \(d_{2}=\frac{1.1}{2.1}\).

  26. 26.

    Ohlson and Juettner-Nauroth (2005) derive the fundamental result that firm value can be expressed as capitalized forward earnings plus the capitalized value of future abnormal earnings growth. Their result is obtained irrespective of the accounting rules, provided the first difference of the residual income series grows or declines geometrically over time. This specification will be met in our model only in special cases. For instance, residual income grows at the same rate as market demand for the firm’s product, given replacement cost accounting. For other accounting rules, though, the residual income series will no longer correspond to a geometric series, even if the future growth rates do. See Nezlobin (2012) for numerical examples illustrating this point.

  27. 27.

    The contribution margin ratio \(\left( p^{o}-c\right) /c\) is a monotone transformation of the Lerner index of monopoly power, \(L\equiv \left( p^{o}-c\right) /p^{o}\) (Martin 2002). In particular for a demand curve exhibiting constant price elasticity of demand, say \(\epsilon \), one obtains \(\frac{p^{o}-c}{c}=\frac{1}{\epsilon -1}\).

  28. 28.

    To have equal capacity in period \(T+1\), the firms then must have different investments in the first period, \(I_{1}\). Yet \(I_{1}\) cancels out from the calculation of Tobin’s q in Eq. (18).

  29. 29.

    See, for example, Dixit and Pindyck (1994, p. 374), Feltham and Ohlson (1996), and Biglaiser and Riordan (2000).

  30. 30.

    Consistent with our characterization, Carlton and Perloff (2005) refer to \(c=r+\alpha \) as the marginal cost of capital.

  31. 31.

    The firm then has enough information in period t to implement the optimal capacity level in period \(t+1\), \(K_{t+1}^{o}\).

  32. 32.

    This assumption is frequently made in the investment literature, beginning with Arrow (1964). See also Abel and Eberly (2011).

  33. 33.

    Assume that the secondary market satisfies the following “no-arbitrage” condition: for any two streams of asset purchases that result in the same capacity levels in all periods, the total discounted cost of the purchases must be the same. It can be verified that this condition implies that an asset of age \(\tau \) will be priced at \(bv_{\tau }^{*}\) in this market.

  34. 34.

    Proportional depreciation accords with the IAS 16 requirement that “... the depreciation method used shall reflect the pattern in which the asset’s future economic benefits are expected to be consumed by the entity.” In our model, the revenues generated by an asset are proportional to current productive capacity. The proportional depreciation rule allocates the cost of investment according to the (nominal) cash flows generated by the asset, ignoring the time value of those cash flows.

  35. 35.

    See, for example, Penman (2013, p. 580).

  36. 36.

    The dashed line in Fig. 1 depicts the P/E ratio as a function of growth under the proportional depreciation rule, which is more conservative than replacement cost accounting. We will formally show that \(PE_{T}\left( \varvec{\lambda },\varvec{d}^{p}\right) \) is increasing in past growth in Proposition 4 below. Earlier accounting literature has considered “liberal” as opposed to conservative accounting; see, for example, Rajan et al. (2007) or Li (2013). We note that the inequality in Proposition 3 would be reversed for a depreciation schedule that is more liberal than replacement cost accounting.

  37. 37.

    See, for instance, Salamon (1985) or Fisher and McGowan (1983).

  38. 38.

    These results have been obtained in a “representative project” model where the firm effectively invests in the same representative project, with exogenously determined growth rates. This framework is equivalent to our capacity model in the special case of zero economic profits, that is, \(p^{o}=c\).

  39. 39.

    If this assumption is not satisfied, the firm’s accounting earnings can be negative for certain investment histories. Nonetheless, it can still be shown that the earnings yield, or the forward E/P ratio, is monotonic in each \(\lambda _{t}\).

  40. 40.

    The logic of this argument is related to the so-called old plant trap usually associated with biases in the Accounting Rate-of-Return (see, for instance Lundholm and Sloan 2013). The common feature is that differences in the age of incumbent assets may not be properly reflected in earnings, thus causing an accounting-induced bias in the respective financial ratios.

  41. 41.

    McNichols et al. (2014) refer to this ratio as the conservatism correction factor, since Tobin’s q is obtained by dividing the price-to-book ratio by the conservatism correction factor. In their sample, the median value of the correction factor, \(\frac{BV_{T}^{*}\left( \varvec{\lambda },\varvec{d}\right) }{BV_{T}\left( \varvec{\lambda },\varvec{d}\right) }\), was 1.37.

  42. 42.

    It follows from Eq. (18) that Tobin’s q is strictly increasing in \(p^{o}\). Proposition 1 then implies that the P/E ratio under replacement cost accounting is also strictly increasing in \(p^{o}\), unless there is no anticipation of growth in the product market (i.e., unless \(s\left( \varvec{\mu }\right) =0\)).

  43. 43.

    In his empirical investigation, Penman (1996) also cites continuity considerations for studying the E/P rather than the P/E ratio.

  44. 44.

    A related argument, which relies on a weaker notion of accounting conservatism, is provided in the proof of Proposition 2 in McNichols et al. (2014).

  45. 45.

    Recall that \(x_{T+1}=0\).

  46. 46.

    If one interprets \(\lambda \) as the interest rate, this equivalence relation relies on the identity between the initial investment and the present value of future depreciation- and imputed interest charges (Preinreich 1935).


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We are grateful to Tim Baldenius, Nicole Johnson, Stephen Penman, and seminar participants at UC Berkeley, UCLA, and the 2014 Colorado Summer Accounting Research Conference for their helpful comments on this paper. Particular thanks to James Ohlson, Russell Lundholm (editor), and an anonymous reviewer for their many constructive and detailed suggestions.

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Correspondence to Madhav V. Rajan.



Proof of Proposition 1

The residual earnings valuation model provides the identity:

$$\begin{aligned} P_{T}=BV_{T}+\sum _{t=1}^{\infty }RI_{T+t}\cdot \gamma ^{t}, \end{aligned}$$

for any accounting rules, provided the condition of comprehensive income measurement is met; see Preinreich (1935) and Feltham and Ohlson (1995, 1996). Since income is measured comprehensively in our model, we can apply the residual income formula to replacement cost accounting in particular. Thus:

$$\begin{aligned} P_{T}=BV_{T}^{*}+\sum _{t=1}^{\infty }RI_{T+t}^{*}\cdot \gamma ^{t}=BV_{T}^{*}+\sum _{t=1}^{\infty }\pi _{T+t}^{o}\cdot \gamma ^{t}. \end{aligned}$$

As argued in the main text, the proportionate growth assumption in (A1) implies that \(\pi _{t+1}^{o}=(1+\mu _{t+1})\cdot \pi _{t}^{o}\). We then obtain:

$$\begin{aligned} P_{T}=BV_{T}^{*}+\frac{RI_{T+1}^{*}}{r-s(\varvec{\mu })}=BV_{T}^{*}+\frac{\pi _{T+1}^{o}}{r-s(\varvec{\mu })}, \end{aligned}$$


$$\begin{aligned} \frac{1}{r-s(\varvec{\mu })}\equiv \gamma +\gamma ^{2}\cdot \left( 1+\mu _{T+2}\right) +\gamma ^{3}\cdot \left( 1+\mu _{T+2}\right) \cdot \left( 1+\mu _{T+3}\right) +\cdots \end{aligned}$$

Dividing both sides in (30) by \(P_{T}\) yields

$$\begin{aligned} 1=\frac{1}{r-s(\varvec{\mu })}\frac{E_{T+1}^{*}}{P_{T}} -\frac{s(\varvec{\mu })/q_{T}}{r-s(\varvec{\mu })}, \end{aligned}$$

which, in turn, implies

$$\begin{aligned} \frac{P_{T}}{E_{T+1}^{*}}=\frac{1}{r-s(\varvec{\mu })\cdot \frac{q_{T}-1}{q_{T}}}. \end{aligned}$$

\(\square \)

Some of the proofs below will rely on the following auxiliary lemma.


For any numbers \(a_{1},\ldots,a_{n}\), positive numbers \(b_{1},\ldots,b_{n}\), and growth rates \(\xi _{2},\ldots,\xi _{n}\ge -1\), the function

$$\begin{aligned} f\left( \xi _{1},\ldots,\xi _{n}\right) =\frac{a_{n}+\left( 1+\xi _{2}\right) a_{n-1}+\cdots +\left( 1+\xi _{2}\right) \ldots\left( 1+\xi _{n}\right) a_{1}}{b_{n}+\left( 1+\xi _{2}\right) b_{n-1}+\cdots +\left( 1+\xi _{2}\right) \ldots\left( 1+\xi _{n}\right) b_{1}} \end{aligned}$$

is everywhere increasing (decreasing) in each \(\xi _{i}\) for \(2\le i\le n\), if the sequence \(\frac{a_{i}}{b_{i}}\) is decreasing (increasing) in i.

Proof of Lemma

The claim obviously holds for \(n=2\). For \(n>2\), the function \(f\left( \xi _{1},\ldots,\xi _{n}\right) \) can be written as:

$$\begin{aligned} f\left( \xi _{1},\ldots,\xi _{n}\right) =\frac{A_{2}+\left( 1+\xi _{i}\right) A_{1}}{B_{2}+\left( 1+\xi _{i}\right) B_{1}}, \end{aligned}$$


$$\begin{aligned} A_{2}&=a_{n}+\left( 1+\xi _{2}\right) a_{n-1}+\cdots +\left( 1+\xi _{2}\right) \ldots\left( 1+\xi _{i-1}\right) a_{n-i+2},\\ B_{2}&=b_{n}+\left( 1+\xi _{2}\right) b_{n-1}+\cdots +\left( 1+\xi _{2}\right) \ldots\left( 1+\xi _{i-1}\right) b_{n-i+2},\\ A_{1}&=\frac{1}{\left( 1+\xi _{i}\right) }\left\{ \left( 1+\xi _{2}\right) \ldots\left( 1+\xi _{i}\right) a_{n-i+1}+\cdots +\left( 1+\xi _{2}\right) \ldots\left( 1+\xi _{n}\right) a_{1}\right\} ,\\ B_{1}&=\frac{1}{\left( 1+\xi _{i}\right) }\left\{ \left( 1+\xi _{2}\right) \ldots\left( 1+\xi _{i}\right) b_{n-i+1}+\cdots +\left( 1+\xi _{2}\right) \ldots\left( 1+\xi _{n}\right) b_{1}\right\} . \end{aligned}$$

The fact that \(\frac{a_{i}}{b_{i}}\) is decreasing in i implies that

$$\begin{aligned} \frac{A_{2}}{B_{2}}\le \frac{a_{n-i+2}}{b_{n-i+2}}\le \frac{a_{n-i+1}}{b_{n-i+1}}\le \frac{A_{1}}{B_{1}}. \end{aligned}$$

The representation in (31) and the inequality above reduce the problem for a general n to the special case of \(n=2\). \(\square \)

Proof of Proposition 2

From Proposition 1, we know that:

$$\begin{aligned} \frac{P_{T}}{E_{T+1}^{*}}=\frac{1}{r-s(\varvec{\mu })\cdot \frac{q_{T}-1}{q_{T}}}. \end{aligned}$$

It thus suffices to show that Tobin’s q is decreasing in \(\lambda _{t}\). Tobin’s q can be rewritten as

$$\begin{aligned} q_{T}= & {} \frac{P_{T}}{BV_{T}^{*}}=\frac{E_{T+1}^{*}-s(\varvec{\mu })\cdot BV_{T}^{*}}{\left( r-s(\varvec{\mu })\right) \cdot BV_{T}^{*}}\\= & {} \frac{1}{\left( r-s(\varvec{\mu })\right) }\cdot \frac{\left( x_{1}p^{o}-d_{1}^{*}-s(\varvec{\mu })\cdot bv_{0}^{*}\right) I_{T}+\cdots +\left( x_{T}p^{o}-d_{T}^{*}-s(\varvec{\mu })\cdot bv_{T-1}^{*}\right) I_{1}}{bv_{0}^{*}I_{T}+\cdots +bv_{T-1}^{*}I_{1}}. \end{aligned}$$

If we show that

$$\begin{aligned} \frac{x_{i+1}p^{o}-d_{i+1}^{*}-s(\varvec{\mu })\cdot bv_{i}^{*}}{bv_{i}^{*}} \end{aligned}$$

increases in i, then the monotonicity of Tobin’s q will follow from Lemma A by setting \(n=T\), \(a_{i}=x_{i}p^{o}-d_{i}^{*}-s(\varvec{\mu })\cdot bv_{i-1}^{*}\), \(b_{i}=bv_{i-1}^{*}\), and \(\xi _{i}=\lambda _{i}\).

Observe that

$$\begin{aligned}&\frac{x_{i+1}p^{o}-d_{i+1}^{*}-s(\varvec{\mu })\cdot bv_{i}^{*}}{bv_{i}^{*}}=\frac{x_{i+1}p^{o}-d_{i+1}^{*}-r\cdot bv_{i}^{*}+\left( r-s(\varvec{\mu })\right) bv_{i}^{*}}{bv_{i}^{*}}\\&\quad =\frac{p^{o}-c}{\left( bv_{i}^{*}/x_{i+1}\right) }+r-s(\varvec{\mu }). \end{aligned}$$

It remains to show that \(bv_{i}^{*}/x_{i+1}\) is decreasing in i. To that end, we note that

$$\begin{aligned} \frac{bv_{i-1}^{*}}{bv_{i}^{*}}=\frac{\gamma x_{i}+\cdots +\gamma ^{T-i+1}x_{T}}{\gamma x_{i+1}+\cdots +\gamma ^{T-i+1}x_{T+1}}, \end{aligned}$$

where \(x_{T+1}=0\).

We can apply Lemma A to the sequences defined by the following equations:

$$\begin{aligned}&\left( a_{1},\ldots,a_{n}\right) =\left( x_{T},\ldots,x_{i}\right) ,\\&\left( b_{1},\ldots,b_{n}\right) =\left( x_{T+1},\ldots,x_{i+1}\right) ,\\&\left( 1+\xi _{2},\ldots,1+\xi _{n}\right) =\left( \gamma ,\ldots,\gamma \right) ,\\&\left( 1+\xi '_{2},\ldots,1+\xi '_{n}\right) =\left( 0,\ldots,0\right) . \end{aligned}$$

Since the productivity pattern satisfies the condition:

$$\begin{aligned} \frac{x_{t}-x_{t+1}}{x_{t}}\ge \frac{x_{t-1}-x_{t}}{x_{t-1}}, \end{aligned}$$

it follows that:

$$\begin{aligned} \frac{a_{t-1}}{b_{t-1}}\ge \frac{a_{t}}{b_{t}}. \end{aligned}$$

Therefore the function f from Lemma A will be increasing in each \(\xi _{i}\), and its value at \(\left( \xi _{2},\ldots,\xi _{n}\right) \) will be greater than its value at \(\left( \xi '_{2},\ldots,\xi '_{n}\right) \). Hence,

$$\begin{aligned} \frac{bv_{i-1}^{*}}{bv_{i}^{*}}\ge \frac{\gamma \cdot x_{i}+0\cdot x_{i+1}+\cdots +0\cdot x_{T}}{\gamma \cdot x_{i+1}+0\cdot x_{i+2}+\cdots +0\cdot x_{T+1}}=\frac{x_{i}}{x_{i+1}}, \end{aligned}$$

and it follows that the sequence \(\frac{bv_{i-1}^{*}}{x_{i}}\) is decreasing in i. \(\square \)

Proof of Observation 2

Assume that \(\varvec{d}\) is more conservative than \(\varvec{d}^{\prime }\). Since

$$\begin{aligned} \sum d_{\tau }=1, \end{aligned}$$

we can rewrite \(d_{1}\) as:

$$\begin{aligned} d_{1}=\frac{1}{1+\frac{d_{2}}{d_{1}}+\cdots +\frac{d_{2}}{d_{1}}\frac{d_{3}}{d_{2}}\ldots\frac{d_{T}}{d_{T-1}}}. \end{aligned}$$

Since \(\varvec{d}\) is more conservative than \(\varvec{d}^{\prime }\),

$$\begin{aligned} \frac{d_{\tau +1}}{d_{\tau }}\le \frac{d_{\tau +1}^{^{\prime }}}{d_{\tau }^{^{\prime }}}. \end{aligned}$$

Equation (33) then implies that \(d_{1}\ge d_{1}^{^{\prime }}\). Note that if \(d_{\tau }\le d_{\tau }^{^{\prime }}\) for some \(\tau \), then \(d_{\tau +1}\le d_{\tau }\frac{d_{\tau +1}^{^{\prime }}}{d_{\tau }^{^{\prime }}}\le d_{\tau +1}^{^{\prime }}\). Applying the same argument iteratively, one can verify that \(d_{\tau }\le d_{\tau }^{^{\prime }}\) implies \(d_{\tau +i}\le d_{\tau +i}^{\prime }\) for any i.

Let \(\delta \left( i\right) =bv_{i}-bv_{i}^{^{\prime }}\). We have the following observations:

  1. 1.

    \( \delta \left( 1\right) =bv_{1}-bv_{1}^{^{\prime }}\le 0.\)

  2. 2.

    \({\text {If }}\,\delta ( \tau ) -\delta ( \tau +1) \le 0\,{\text{ for}}\,{\text{ some }}\,\tau,\,{\text { then }}\,\delta ( \tau +i) -\delta \left( \tau +i+1\right) \le 0\,{\text { for}}\, {\text{any }}\,i\ge 0.\)

  3. 3.

    \(\delta \left( T\right) =0. \)

The function \(\delta \) is negative at one, and once it becomes increasing, it continues to increase up to the end of the useful life. Therefore \(\delta \left( i\right) \) can only cross zero once and this happens at \(i=T\). The three observations hence imply that \(\delta \left( i\right) \le 0\) and \(bv_{i}\le bv_{i}^{^{\prime }}\) for all i. \(\square \)

Proof of Observation 3

Assume that

$$\begin{aligned} d_{1}'=\left( 1-\eta \right) d_{1}+\eta , \end{aligned}$$


$$\begin{aligned} d'_{\tau }=\left( 1-\eta \right) d_{\tau } \end{aligned}$$

for \(\tau >1\) and some \(\eta >0\).

Note that

$$\begin{aligned} \frac{d_{\tau }}{d_{\tau +1}}=\frac{d'_{\tau }}{d'_{\tau +1}} \end{aligned}$$

for \(\tau >1\). For \(\tau =1\), we have

$$\begin{aligned} \frac{d'_{1}}{d'_{2}}=\frac{\left( 1-\eta \right) d_{1}+\eta }{\left( 1-\eta \right) d_{2}}>\frac{d_{1}}{d_{2}}. \end{aligned}$$

Therefore \(\varvec{d}'\) is more conservative than \(\varvec{d}\). \(\square \)

Proof of Proposition 3

Let \(D_{T+1}\) and \(D'_{T+1}\) denote the aggregate depreciation expenses under rules \(\varvec{d}\) and \(\varvec{d}'\), respectively, where \(\varvec{d}\) is more conservative than \(\varvec{d}'.\) We will show that \(D_{T+1}\ge D_{T+1}^{\prime }\) (\(D_{T+1}\le D_{T+1}^{\prime }\)) if investments \(I_{1},\ldots,I_{T}\) are monotonically increasing (decreasing). From this it will follow that if \(\varvec{d}\) is more conservative than replacement cost accounting and the firm is growing, then

$$\begin{aligned} PE_{T}\left( \varvec{\lambda },\varvec{d}\right) =\frac{P{}_{T}\left( \varvec{\lambda }\right) }{E_{T+1}\left( \varvec{\lambda },\varvec{d}\right) }\ge \frac{P{}_{T}\left( \varvec{\lambda }\right) }{E_{T+1}^{*}}=\frac{1}{r-s\left( \varvec{\mu }\right) \cdot \frac{q_{T}\left( \varvec{\lambda }\right) -1}{q_{T}\left( \varvec{\lambda }\right) }}. \end{aligned}$$

Observe that

$$\begin{aligned} D_{T+1}= & {} I_{1}\cdot d_{T}+I_{2}\cdot d_{T-1}+\cdots +I_{T}\cdot d_{1}\\= & {} I_{1}+\left( I_{2}-I_{1}\right) \cdot \left( 1-bv_{T-1}\right) +\cdots +\left( I_{T}-I_{T-1}\right) \cdot \left( 1-bv_{1}\right) . \end{aligned}$$


$$\begin{aligned} D_{T+1}-D_{T+1}^{\prime }=\left( I_{2}-I_{1}\right) \cdot \left( bv'_{T-1}-bv_{T-1}\right) +\cdots +\left( I_{T}-I_{T-1}\right) \cdot \left( bv'_{1}-bv_{1}\right) \ge 0, \end{aligned}$$

where the last inequality holds by Observation 2 if investments are increasing and \(\varvec{d}\) is more conservative than \(\varvec{d}'\). \(\square \)

Proof of Observation 4

In the proof of Proposition 2, we have verified that \(\frac{bv_{i-1}^{*}}{x_{i}}\) is decreasing in i. Recall that

$$\begin{aligned} \frac{d_{i}^{*}+r\cdot bv_{i-1}^{*}}{x_{i}}=c. \end{aligned}$$

Since \(\frac{bv_{i-1}^{*}}{x_{i}}\) is decreasing in i, the sequence \(\frac{d_{i}^{*}}{x_{i}}\) must increase in i. Therefore,

$$\begin{aligned} \frac{d_{i}^{*}}{d_{i+1}^{*}}\le \frac{x_{i}}{x_{i+1}}=\frac{d_{i}^{p}}{d_{i+1}^{p}}. \end{aligned}$$

\(\square \)

Proof of Proposition 4

We expand the P/E ratio as:

$$\begin{aligned}&\frac{P_{T}}{E_{T+1}}=\frac{1}{\left( r-s\left( \varvec{\mu }\right) \right) }\cdot \frac{E_{T+1}^{*}-s\left( \varvec{\mu }\right) \cdot BV_{T}^{*}}{E_{T+1}}\\&\quad =\frac{1}{\left( r-s\left( \varvec{\mu }\right) \right) }\cdot \frac{I_{T}\left( x_{1}p^{o}-d_{1}^{*}-s\left( \varvec{\mu }\right) \cdot bv_{0}^{*}\right) +\cdots +I_{1}\left( x_{T}p^{o}-d_{T}^{*}-s\left( \varvec{\mu }\right) \cdot bv_{T-1}^{*}\right) }{I_{T}\left( x_{1}p^{o}-d_{1}\right) +\cdots +I_{1}\left( x_{T}p^{o}-d_{T}\right) }. \end{aligned}$$

To apply Lemma A, we need to show that

$$\begin{aligned} \frac{x_{i}p^{o}-d_{i}^{*}-s\left( \varvec{\mu }\right) \cdot bv_{i-1}^{*}}{x_{i}p^{o}-d_{i}} \end{aligned}$$

is decreasing in i. Note that

$$\begin{aligned}&\frac{x_{i}p^{o}-d_{i}^{*}-s\left( \varvec{\mu }\right) \cdot bv_{i-1}^{*}}{x_{i}p^{o}-d_{i}}=\frac{x_{i}p^{o}-d_{i}^{*}-r\cdot bv_{i-1}^{*}+\left( r-s\left( \varvec{\mu }\right) \right) bv_{i-1}^{*}}{x_{i}\left( p^{o}-\frac{d_{i}}{x_{i}}\right) }\nonumber \\&\quad =\frac{p^{o}-c+\left( r-s\left( \varvec{\mu }\right) \right) \frac{bv_{i-1}^{*}}{x_{i}}}{p^{o}-\frac{d_{i}}{x_{i}}}. \end{aligned}$$

Assumption (1) implies that \(\frac{bv_{i-1}^{*}}{x_{i}}\) is decreasing in i (see the proof of Proposition 2), and since \(r>s\left( \varvec{\mu }\right) \), the numerator is decreasing in i. If \(\varvec{d}\) corresponds to proportional depreciation, then the denominator does not depend on i. If \(\varvec{d}\) is more conservative than the proportional depreciation rule, then \(\frac{d_{i}}{x_{i}}\) is decreasing in i, and the denominator is increasing in i. Therefore the ratio (34) is decreasing in i.

To show that the price-to-book ratio is decreasing in each \(\lambda _{t}\), we recall that the firm’s market value is given by:

$$\begin{aligned} P_{T}=BV_{T}^{*}+\frac{\left( p^{o}-c\right) K_{T+1}^{o}}{r-s \left( \varvec{\mu }\right) }. \end{aligned}$$

\(PB_{T}\) is therefore equal to:

$$\begin{aligned} PB_{T}=\frac{BV_{T}^{*}}{BV_{T}} +\frac{\left( p^{o}-c\right) }{r-s\left( \varvec{\mu }\right) }\cdot \frac{K_{T +1}^{o}}{BV_{T}}. \end{aligned}$$

It remains to show that both \(\frac{BV_{T}^{*}}{BV_{T}}\) and \(\frac{K_{T+1}^{o}}{BV_{T}}\) decrease in each \(\lambda _{t}.\) Footnote 44 Note that

$$\begin{aligned} \frac{BV_{T}^{*}}{BV_{T}}=\frac{bv_{0}^{*}I_{T}+\cdots +bv_{T-1}^{*}I_{1}}{bv_{0}I_{T} +\cdots +bv_{T-1}I_{1}}. \end{aligned}$$

By Lemma A, to show that \(BV_{T}^{*}/BV_{T}\) is declining in \(\lambda _{t}\), it suffices to check that \(\frac{bv_{i}^{*}}{bv_{i}}\) is increasing in i. Recall that

$$\begin{aligned} bv_{i}^{*}=\gamma cx_{i+1}+\cdots +\gamma ^{T-i}cx_{T} \end{aligned}$$


$$\begin{aligned} bv_{i}=d_{i+1}+\cdots +d_{T}. \end{aligned}$$

Using the assumption that \(\frac{x_{\tau }}{x_{\tau +1}}\) increases in \(\tau \) and Lemma A, we obtain:Footnote 45

$$\begin{aligned} \frac{bv_{i}^{*}}{bv_{i+1}^{*}}=\frac{\gamma cx_{i+1}+\cdots +\gamma ^{T-i}cx_{T}}{\gamma cx_{i+2}+\cdots +\gamma ^{T-i-1}cx_{T}}\le \frac{x_{i+1}+\cdots +x_{T}}{x_{i+2}+\cdots +x_{T}}. \end{aligned}$$

If we now show that

$$\begin{aligned} \frac{bv_{i}}{bv_{i+1}}\ge \frac{x_{i+1}+\cdots +x_{T}}{x_{i+2}+\cdots +x_{T}}, \end{aligned}$$

it will follow that \(\frac{bv_{i}^{*}}{bv_{i}}\) increases in i.

Inequality (36) is equivalent to:

$$\begin{aligned} \frac{d_{i+1}+\cdots +d_{T}}{x_{i+1}+\cdots +x_{T}}\ge \frac{d_{i+2}+\cdots +d_{T}}{x_{i+2}+\cdots +x_{T}}. \end{aligned}$$

The inequality above holds by Lemma A, since \(\frac{d_{\tau }}{x_{\tau }}\) decreases in \(\tau \). This concludes the proof that \(\frac{BV_{T}^{*}}{BV_{T}}\) declines in each \(\lambda _{t}\).

To verify that \(\frac{K_{T+1}^{o}}{BV_{T}}\) also decreases in \(\lambda _{t}\), we will check that \(\frac{K_{T+1}^{o}}{BV_{T}^{*}}\) declines in \(\lambda _{t}\). Note that

$$\begin{aligned} \frac{K_{T+1}^{o}}{BV_{T}^{*}}=\frac{x_{1}I_{T}+\cdots +x_{T}I_{1}}{bv_{0}^{*}I_{T}+\cdots +bv_{T-1}^{*}I_{1}}. \end{aligned}$$

We have shown earlier that \(\frac{bv_{\tau -1}^{*}}{x_{\tau }}\) is decreasing in \(\tau \), and therefore \(\frac{x_{\tau }}{bv_{\tau -1}^{*}}\) is increasing in \(\tau \). Then, by Lemma A, the right-hand side of (37) decreases in each \(\lambda _{t}\). \(\square \)

Proof of Observation 5

\(\frac{\partial PE_{T}}{\partial p^{o}}\) has the same sign as \(\frac{\partial \ln (PE_{T})}{\partial p^{o}}\).

$$\begin{aligned}&\frac{\partial \ln (PE_{T})}{\partial p^{o}}=\frac{\partial \ln P_{T}}{\partial p^{o}}-\frac{\partial \ln (E_{T+1})}{\partial p^{o}}=\frac{1}{P_{T}}\left( \frac{\partial P_{T}}{\partial p^{o}}\right) -\frac{1}{E_{T+1}}\left( \frac{\partial E_{T+1}}{\partial p^{o}}\right) ,\\&\frac{\partial P_{T}}{\partial p^{o}}=\frac{K_{T+1}^{o}}{r-s\left( \varvec{\mu }\right) },\\&\frac{\partial E_{T+1}}{\partial p^{o}}=K_{T+1}^{o}. \end{aligned}$$

Therefore \(\frac{\partial PE_{T}}{\partial p^{o}}\) has the same sign as

$$\begin{aligned} \frac{K_{T+1}^{o}}{\left( r-s\left( \varvec{\mu }\right) \right) P_{T}} -\frac{K_{T+1}^{o}}{E_{T+1}}=\left( \frac{1}{r-s\left( \varvec{\mu }\right) } -PE_{T}\right) \frac{K_{T+1}^{o}}{P_{T}}. \end{aligned}$$

\(\square \)

Proof of Proposition 5

We show that if investments have grown at the constant rate \(\lambda \), the difference

$$\begin{aligned} E_{T+1}-\lambda \cdot BV_{T} \end{aligned}$$

is invariant to the accounting rules. To that end, it suffices to demonstrate that

$$\begin{aligned} D_{T+1}+\lambda \cdot BV_{T}=(1+\lambda )^{T}I_{1}, \end{aligned}$$

for any depreciation schedule, \(\varvec{d}\). By definition:

$$\begin{aligned} D_{T+1}&= d_{1}\cdot I_{T}+d_{2}\cdot I_{T-1}+\cdots +d_{t}\cdot I_{1}\\&= [d_{1}\cdot (1+\lambda )^{T-1}+d_{2}\cdot (1+\lambda )^{T-2}+\cdots +d_{T}]\cdot I_{1}. \end{aligned}$$


$$\begin{aligned} BV_{T}&= bv_{0}\cdot I_{T}+bv_{1}\cdot I_{t-1}+\cdots +bv_{T-1}\cdot I_{1}\\&= [bv_{0}\cdot (1+\lambda )^{T-1}+bv_{1}\cdot (1+\lambda )^{T-2}+\cdots +bv_{T-1}]\cdot I_{1} \end{aligned}$$


$$\begin{aligned} D_{T+1}+\lambda \cdot BV_{T}=(1+\lambda )^{T}\cdot \left[ \sum _{t=1}^{T}(d_{t}+\lambda \cdot bv_{t-1})\cdot (1+\lambda )^{-t}\right] I_{1} \end{aligned}$$

For any depreciation schedule and any \(\lambda >{-}1\), the expression in brackets in the right-hand side of (39) is equal to one.Footnote 46 We may evaluate (38) by supposing replacement cost accounting. Proposition 1 then yields \(E_{T+1}^{*}=r\cdot P_{T}\) and \(BV_{T}^{*}=P_{T}\). Thus,

$$\begin{aligned} E_{T+1}-\lambda \cdot BV_{T}=(r-\lambda )\cdot P_{T}, \end{aligned}$$

and the claim follows immediately. \(\square \)

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Nezlobin, A., Rajan, M.V. & Reichelstein, S. Structural properties of the price-to-earnings and price-to-book ratios. Rev Account Stud 21, 438–472 (2016).

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  • Price-to-earnings
  • Price-to-book
  • Tobin’s q
  • Growth
  • Profitability
  • Conservatism

JEL Classification

  • M41
  • L25
  • G11