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Conservatism correction for the market-to-book ratio and Tobin’s q


We decompose the market-to-book ratio into two additive components: a conservatism correction factor and a future-to-book ratio. The conservatism correction factor exceeds the benchmark value of one whenever the accounting for past transactions has been subject to an (unconditional) conservatism bias. The observed history of a firm’s past investments allows us to calculate the magnitude of its conservatism correction factor, resulting in an average value that is about two-thirds of the overall market-to-book ratio. We demonstrate that our measure of Tobin’s q, obtained as the market-to-book ratio divided by the conservatism correction factor, has greater explanatory power in predicting future investments than the market-to-book ratio by itself. Our model analysis derives a number of structural properties of the conservatism correction factor, including its sensitivity to growth in past investments, the percentage of investments in intangibles, and the firm’s cost of capital. We provide empirical support for these hypothesized structural properties.

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  1. Without attempting to summarize the extensive literature on accounting conservatism, we note that parts of the theoretical literature on unconditional conservatism take a market-to-book ratio greater than one as a manifestation of conservative accounting; see, for example, Feltham and Ohlson (1995, 1996), Zhang (2000), and Ohlson and Gao (2006).

  2. This calibration is consistent with Ross et al. (2005, p. 41) who state: “Firms with high q ratios tend to be those firms with attractive investment opportunities, or a significant competitive advantage.” See also Lindenberg and Ross (1981), Landsman and Shapiro (1995) and Roll and Weston (2008).

  3. Our model framework builds on the notion that firms undertake a sequence of overlapping investments in productive capacity. That feature is also central to the models in Arrow (1964), Rogerson (2008), Rajan and Reichelstein (2009), and Nezlobin (2012).

  4. For the market-to-book ratio, the predicted impact is ambiguous since both the numerator and the denominator of this ratio are increasing in higher past growth.

  5. This approach is broadly consistent with the valuation model formulated in Nezlobin (2012), where the capitalization of current economic profits reflects both the discount rate and the rate of growth in the firm’s sales revenues.

  6. We note that our present model formulation is not suited to address issues of conditional conservatism, as considered, for instance, in Basu (1997) and Watts (2003).

  7. The addition of accounting information is, of course, the general motivation for studies like those in Piotroski (2000), Mohanram (2005), and Penman and Reggiani (2013). By including firm-specific scores derived from financial statement analysis, these authors are able to refine the association between market-to-book ratios and stock returns by partitioning firms with similar market-to-book ratios into different subgroups.

  8. In connection with solar power panels, it is commonly assumed that electricity yield is subject to “systems degradation,” which is modeled as a pattern of geometrically declining capacity levels (Campbell 2008).

  9. Conceptually, it would not be difficult to extend our model formulation so as to include uncertainty and investors’ expectations. Such an extension would, however, not serve any particular purpose for either our theoretical or our empirical analysis.

  10. Without reference to a hypothetical rental market, Arrow (1964) and Rogerson (2008) derive the same unit cost of capacity in an infinite horizon setting with new investments in each period.

  11. Our notion of replacement cost accounting differs from the concept of unbiased accounting in Feltham and Ohlson (1995, 1996), Zhang (2000), and Ohlson and Gao (2006). Their notion of unbiased accounting is that the market-to-book ratio approaches a value of 1 asymptotically. In the literature on ROI, the concept of unbiased accounting is operationalized by the criterion that for an individual project the accounting rate of return should be equal to the project’s internal rate of return; see, for instance, Beaver and Dukes (1974), Rajan et al. (2007), and Staehle and Lampenius (2010). To satisfy this criterion, the accruals must generally reflect the intrinsic profitability of the project. In the special case where all projects have zero NPV, this criterion does coincide with our notion of unbiased accounting. In contrast, our notion of replacement accounting is consistent with the accounting treatment recommended in the managerial performance evaluation literature; see, for instance, Rogerson (1997), Reichelstein (1997), and Dutta and Reichelstein (2005)

  12. When assets are not in productive use during the first L periods, they become more valuable over time. Therefore the depreciation charges in the first L − 1 periods are negative with \(d^{*}_{t} = - r \cdot (1+r)^{t-1}\) for 1 ≤ t ≤ L − 1. This is exactly the accounting treatment that Ehrbar (1998) recommends for so-called “strategic investments,” which are characterized by a long time lag between investments and subsequent cash inflows.

  13. See also Proposition 2 in Staehle and Lampenius (2010).

  14. The AICPA’s (2007, p. 399) Accounting Trends & Techniques survey of 600 Fortune 1,000 firms reports that 592 of the sample firms applied straight-line accounting in reporting the value of their operating assets.

  15. It is readily verified that, if d o is uniformly more accelerated than d * = (0, d *1 d *2 , ..., d * T ), then so is d o t .

  16. The condition on the x t ’s in the statement of Proposition 3 is sufficient but not necessary. This condition is also not very restrictive. For instance, it is satisfied by any \({\varvec{x}}\) vector that decreases over time in either a linear or geometric fashion. The one-hoss shay scenario, where all x t  = 1, is one particular admissible case.

  17. Informally, this inequality follows from the following two observations. (1) On the interval [0, L − 1], it is clearly true that bv * t  > bv o t ; (2) on the interval [L,T − 1] it must also be true that bv * t  > bv o t , because bv * T  = bv o T  = 0 and bv * t is decreasing and concave on [LT], while bv o t is a linear function of time.

  18. For general L > 1, it can be shown that at least half of the drop in CC T occurs in the range of negative growth rates, provided productivity conforms to the one-hoss shay scenario.

  19. This finding can be extended to general values of β and L. The limit values are available from the authors upon request. We note that \( \lim_{\lambda \rightarrow -1} CC_T = \frac{bv^{*}_{T-1}}{bv^{o}_{T-1}}\) and \(\lim_{\lambda \rightarrow \infty} CC_T = \frac{bv^{*}_{1}}{bv^{o}_{1}}\). Here, bv o t bv t (d o).

  20. We shall from hereon use the more compact notation BV o T instead of BV T (I T d o). Similarly, we use the shorter BV * T (or OA * T ) instead of BV T (I T d *) (or OA T (I T d *)).

  21. This representation is, of course, consistent with the studies in Feltham and Ohlson (1995) and Penman et al. (2007), which presume that financial assets are carried at their fair market values on the balance sheet.

  22. See, for example, the discussion by Erickson and Whited (2000), p. 1029.

  23. Lindenberg and Ross did not test whether this improved their measure of q, and the SEC subsequently abandoned the requirement to disclose replacement cost of property and plant.

  24. It goes without saying that our approach to forecasting future value is somewhat ad hoc. There appear to be many promising avenues for refining the approach taken here in future studies.

  25. Our approach of incorporating income taxes avoids the issues of estimating the firm’s actual tax rate or taxes to be paid in future periods.

  26. Our capitalization of current economic profit is broadly consistent with the valuation model developed in Nezlobin (2012). We use the average growth rate over the past three years as a proxy for anticipated future growth in the firm’s product markets.

  27. Throughout our empirical analysis, we set the lag factor L equal to 1. It seems plausible that there are significant variations in L across industries, an aspect we do not pursue in this paper.

  28. See, for instance, Fazzari et al. (1988, 2000), Kaplan and Zingales (1997), Erickson and Whited (2000), Baker et al. (2003), Rauh (2006), and McNichols and Stubben (2008).

  29. It should be recalled at this stage that our framework allows for only a single category of operating assets and correspondingly growth in one dimension. Zhang (1998) considers the impact of differential growth rates for PPE and intangible assets.

  30. A caveat to this interpretation is that measurement error in our estimate of \(\widehat{FB}_T\) is not highly correlated with past growth. To the extent such a correlation arises, it could induce a negative correlation between \(\widehat{CC}_T\) and past growth. We do not expect this effect to be driving our results as the correlation between \(\widehat{CC}_T\) and past growth is largely comparable to the correlation between CC T and past growth.

  31. A proof of this assertion can be found in Claim 2 in the proof of Proposition 3 in Rajan and Reichelstein (2009).


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We thank William Beaver, James Ohlson, Alexander Nezlobin, Stephen Penman (editor), Stephen Ryan, two anonymous reviewers, and workshop participants at Berkeley, Copenhagen (Interdisciplinary Workshop), Harvard, ISB (Hyderabad), Michigan, Muenchen (LMU), Northwestern, NYU, WHU, and Stanford for their valuable comments. We also acknowledge the excellent research assistance of Maria Correia, Moritz Hiemann, Julia Reichelstein, Eric So, Yanruo Wang, and Anastasia Zakolyukina.

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Correspondence to Maureen McNichols.


Appendix 1: Description of variables


= Market value of equity at end of fiscal year T (CSHO * PRCC_ F)

MB un T

= Market value of equity at time T divided by book value of equity for fiscal year T


= Net financial assets at end of fiscal year t, measured as total assets minus net plant and intangibles minus liabilities (AT − PPENT − INTAN − LT)


= \(\frac{\displaystyle{MV_T - FA_T}}{\displaystyle{OA^{o}_T}} = \frac{\displaystyle{MV_T - FA_T}}{\displaystyle{BV^{o}_T - FA_T}}\). Adjusted market-to-book ratio

OA o t

= Net plant + intangibles, at end of fiscal year T, (PPENT + INTAN)

Total Investment

= Advertising expense plus R & D expense plus Capital expenditures for period T − 1 to T (XAD + XRD + CAPXV)


= Operating income after depreciation, amortization



α t

= Conservatism in fiscal year t, measured as (XAD + XRD)/(XAD + XRD + CAPXV), where XAD is advertising expense; XRD is Research and Development expense; and CAPXV is capital expenditures

α a T

= Growth weighted average of directly expensed investments

\(\frac{\displaystyle{\alpha_1 (1 + \lambda_1) + \cdots + \alpha_{T} \prod\nolimits^{T}_{i=1}(1 + \lambda_i)}}{\displaystyle{ (1 + \lambda_1) + \cdots+\prod\nolimits^{T}_{i=1} (1 + \lambda_i)}}\)


= Useful life of plant and intangibles, (average gross\ plant+intangibles)/ (depreciation+amortization of intangibles) measured as (average PPEGT + INTAN)/dep

λ t

= \(\frac{\displaystyle{XAD_{t}+XRD_{t}+CAPXV_{t}}}{\displaystyle{XAD_{t-1}+XRD_{t-1}+CAPXV_{t-1}}} -1\)

λ a T

= Geometric mean of growth over T periods

λ a3

= Geometric mean of growth rates (λ T λ T−1λ T−2)

r e

= Equity cost of capital for firm i and year T, estimated with coefficients from the Fama and French (1992) two-factor model:

\(R_i - R_f = \delta_0 + \delta_1 (R_m - R_f) + \delta_2 (SMB) + \epsilon\)


using CRSP monthly returns from five preceding years and Ken French’s data on market and size factors (SMB)

τ T

= Statutory income tax rate in year T

r d

= Cost of Debt: \((1 - \tau_T) \cdot\) Interest Expense divided by the average of beginning and ending balance of interest-bearing debt


= Weighted average cost of capital: \(\frac{\displaystyle{BV^{o}_T}}{\displaystyle{AT^{o}_T}}\cdot r_e + \frac{\displaystyle{AT^{o}_T - BV^{o}_T}}{\displaystyle{AT^{o}_T}}\cdot r_d \cdot (1-\tau_T) \)


\(\equiv \gamma + \gamma^2 + \dots + \gamma^n\)


= \(\Upgamma^1 + \Upgamma^2 (1 + \lambda_2) + \dots + \Upgamma^T \prod\limits^{T}_{i=2} (1 + \lambda_i)\)


= \((1 - \alpha_1) \left( 1 - \frac{\displaystyle{T-1}}{\displaystyle{T}}\right) + (1-\alpha_2) \left( 1 - \frac{\displaystyle{T-2}}{\displaystyle{T}}\right) (1 + \lambda_2) + \dots\)

\((1 - \alpha_{T-1}) \prod\limits^{T-1}_{i=2} (1 + \lambda_i) \cdot \left( 1 - \frac{\displaystyle{1}}{\displaystyle{T}}\right) + (1 - \alpha_{T}) \prod\limits^{T}_{i=2} (1 + \lambda_i)\)


= \(\frac{\displaystyle{1}}{\displaystyle{1+r}}\) + \(\frac{\displaystyle{1}}{\displaystyle{{(1+r)}^2}} +\cdots + \) \(\frac{\displaystyle{1}}{\displaystyle{{(1+r)}^T}} \)


= \(\frac{\displaystyle{N_T}}{\displaystyle{D_T}} \cdot \frac{\displaystyle{1}}{\displaystyle{\Upgamma^T}}\) with Tr, γ, and λ as defined above

CC λ T

= Same as CC T except that λ t  = λ a T for all t


= MB T  − CC T


= Estimated Future-to-Book Value, defined in equation (23)


= \(MB_T- \widehat{FB}_T\)

EconCost T

= Expenses T  − dep T + \(\frac{\displaystyle{1}}{\displaystyle{\Updelta_T}}(dep_T + r\cdot OA_{T-1})\)


= \(\Upgamma^T\cdot \frac{\displaystyle{u_0 + u_1 (1 + \lambda_1) + \cdots + u_{T-1} \prod\nolimits^{T-1}_{i=1} (1 + \lambda_i)+ \alpha_T \cdot \prod\nolimits^{T}_{i=1} (1 + \lambda_i)}}{\displaystyle{1 + (1 + \lambda_1) + \cdots + \prod\nolimits^{T-1}_{i=1} (1 + \lambda_i)}}\)

u t

= \((1-\alpha_t)\left[\frac{\displaystyle{1}}{\displaystyle{T}} + r \cdot\left(1- \frac{\displaystyle{T-1-t}}{\displaystyle{T}}\right) \right]\) for 0 ≤ t ≤ T − 1


= x if x ≥ 0 and \(\mathcal{I}\{x\}=0\) if x ≤ 0


= Capitalization factor, given by \(\sum_{i=1}^{5} \left(\frac{\displaystyle{1+\lambda^a_3}}{\displaystyle{1+r}}\right)^i\)

Appendix 2: Proofs

Proof of Proposition 1 We can set α to 0 without loss of generality for this proof. From (8), CC T equals (after dividing through by the common term \(I_0\cdot (1+\lambda_{1})\)):

$$ CC_T = \frac{BV_{T}({\bf I}_T, {\bf d}^*)}{BV_{T}({\bf I}_T, {\bf d}^o)} = \frac{bv^{*}_{T-1}+ bv^{*}_{T-2} \cdot(1+\lambda_{2})+\cdots+ bv^{*}_{0}\cdot\prod^{T}_{i=2}(1+\lambda_{i})}{bv^{o}_{T-1}+ bv^{o}_{T-2} \cdot(1+\lambda_{2})+\cdots+ bv^{o}_{0}\cdot\prod^{T}_{i=2}(1+\lambda_{i})}. $$

This ratio is decreasing in λ t if and only if the sequence \(\frac{bv^*_t}{bv^{o}_t}\) is an increasing function of t. Footnote 31 Given the replacement cost accounting rule, d *, we know from (6) and (11) that z * t  = x t  = 0 for 1 ≤ t ≤ L − 1 and \(z^{*}_{t}= c \cdot x_{t}\), for L ≤ t ≤ T. It follows that for all \(t, z^{*}_{t}=c\cdot x_t\). We recall the following identity linking book values to future “residual income charges.”

$$ bv_t({\bf d}) = \sum_{i=t+1}^{T} z_{i}({\bf d}) \cdot\gamma^{i-t}. $$

Denoting bv o t bv t (d o), we have:

$$ \frac{bv^*_t}{bv^o_t } = \frac{\displaystyle \sum\nolimits_{i=t+1}^{T} z_{i}({\bf d}^*) \cdot \gamma^{i-t}}{\displaystyle \sum\nolimits_{i=t+1}^{T} z_{i}({\bf d}^o) \cdot\gamma^{i-t}} = \frac{\displaystyle c \cdot \sum\nolimits_{i=t+1}^{T} x_{i} \cdot\gamma^{i}}{\displaystyle \sum\nolimits_{i=t+1}^{T} z_{i}({\bf d}^o) \cdot\gamma^{i}}. $$

Analogously, we have:

$$ \frac{bv^*_{t-1}}{bv^o_{t-1}} = \frac{\displaystyle c \cdot \sum\nolimits_{i=t}^{T} x_{i} \cdot\gamma^{i}}{\displaystyle \sum\nolimits_{i=t}^{T} z_{i}({\bf d}^o) \cdot\gamma^{i}} = \frac{\displaystyle c \cdot \left[x_t \cdot \gamma^t + \sum\nolimits_{i=t+1}^{T} x_{i} \cdot\gamma^{i}\right]}{\displaystyle z_{t}({\bf d}^o) \cdot \gamma^t + \sum\nolimits_{i=t+1}^{T} z_{i}({\bf d}^o) \cdot\gamma^{i}}. $$

To establish that (27) ≥ (28), we note that, for 1 ≤ t ≤ L − 1, the inequality follows immediately since x t  = 0 and z t (d o) ≥ 0 (see Definition 2). For t ≥ L, the result holds if and only if

$$ \frac{z_{t}({\bf d}^o)}{x_t} \ge \frac{\displaystyle \sum\nolimits_{i=t+1}^{T} z_{i}({\bf d}^o)\cdot\gamma^{i}}{\displaystyle \sum\nolimits_{i=t+1}^{T} x_{i} \cdot\gamma^{i}}. $$

We demonstrate (29) by induction. For t = T − 1, (29) requires:

$$ \frac{z_{T-1}({\bf d}^o)}{x_{T-1}} \ge \frac{z_{T}({\bf d^o})}{x_{T}}, $$

which is true as \(\frac{z_t({\bf d}^o)}{x_t}\) decreases in t. Now suppose that (29) holds for t = k. Then, for t=k−1,

$$ \frac{z_{k-1}({\bf d}^o)}{x_{k-1}} \ge \frac{z_k({\bf d}^o)}{x_k} = \frac{z_k({\bf d}^o) \cdot \gamma^k}{x_k \cdot \gamma^k} \ge \frac{\displaystyle z_k({\bf d}^o) \cdot \gamma^k+\sum\nolimits_{i=k+1}^{T} z_{i}({\bf d}^o) \cdot\gamma^{i}}{\displaystyle x_k \cdot \gamma^k+\sum\nolimits_{i=k+1}^{T} x_{i} \cdot\gamma^{i}} = \frac{\displaystyle \sum\nolimits_{i=k}^{T} z_{i}({\bf d}^o) \cdot\gamma^{i}}{\displaystyle \sum\nolimits_{i=k}^{T} x_{i} \cdot\gamma^{i}}, $$

where the second inequality arises from the induction hypothesis. We have thus shown that (29) holds. To conclude, note that we have stated the proof in terms of weak inequalities. However, if either z t (d o) > 0 for some t ≤ L − 1 or \(\frac{z_t({\bf d^o})}{x_t}\) strictly decreases in t for t ≥ L, it follows that \(\frac{bv^*_t}{bv^o_t}\) strictly increases for some subset of values of t and therefore that CC T is monotone decreasing in each λ t .

Proof of Proposition 2

Consider CC T as represented in equation (26). The denominator, BV T , is determined by the depreciation scheme under consideration and is independent of the cost of capital, r. So it is sufficient to show that the numerator, BV * T , increases in r, or, equivalently, that it decreases in γ. We use the following formulation of BV T (I T d *):

$$ BV_{T}({\bf I}_T, {\bf d}^*) = bv^{*}_{T-1}\cdot I_1 + bv^{*}_{T-2} \cdot I_2 +\cdots+ bv^{*}_{0}\cdot I_T. $$

As in the proof of Proposition 1, we set \(bv^*_t = c \cdot \sum_{i=t+1}^{T} x_{i} \cdot\gamma^{i-t}\). BV T (I T d *) therefore equals:

$$ c \cdot \left[I_1\cdot \sum_{i=T}^{T} x_i \cdot \gamma^{i-(T-1)} + I_2 \cdot \sum_{i=T-1}^{T} x_i \cdot \gamma^{i-(T-2)} + \ldots+ I_T\cdot \sum_{i=1}^{T} x_i \cdot \gamma^{i}\right]. $$

As \(c=\frac{1}{\sum_{i=L}^{T} x_{i} \cdot\gamma^{i}}\), we need to show that the following expression decreases in γ:

$$ \frac{\displaystyle I_1\cdot x_{T} \cdot \gamma + I_2\cdot \sum\nolimits_{i=T-1}^{T} x_i \cdot \gamma^{i-(T-2)} + \ldots+ I_T\cdot \sum\nolimits_{i=1}^{T} x_i \cdot \gamma^{i}}{\sum\nolimits_{i=L}^{T} x_{i} \cdot\gamma^{i}}. $$

We do so one term at a time. Ignoring the positive constant I t , an arbitrary term in (30) is of the form:

$$ \frac{\displaystyle{\sum_{i=k}^{T} x_i \cdot \gamma^{i-k+1}}}{\displaystyle{\sum_{i=L}^{T} x_i \cdot \gamma^{i}}}, \quad k\in\{1,2,\ldots, T\}. $$

Consider k ≤ L. As \(x_1=\cdots=x_{L-1}=0\), (31) is equivalent to:

$$ \frac{\displaystyle{\sum_{i=L}^{T} x_i \cdot \gamma^{i-k+1}}}{\displaystyle{\sum_{i=L}^{T} x_i \cdot \gamma^{i}}} = \gamma^{-(k-1)}, $$

which is decreasing in γ as k ≥ 1.

For k > L, (31) decreases in γ if and only if

$$ \left(\sum_{i=L}^{T} x_i \cdot \gamma^i\right)\cdot \left[\sum_{i=k}^{T} x_i \cdot (i-k+1)\cdot \gamma^{i-k}\right]\leq \left[\sum_{i=k}^{T} x_i \cdot \gamma^{i-k+1}\right] \cdot \left(\sum_{i=L}^{T} x_i\cdot i\cdot \gamma^{i-1}\right),\quad \hbox {or} \frac{\displaystyle{\sum_{i=L}^{T} x_i \cdot \gamma^{i-1}}}{\displaystyle{\sum_{i=L}^{T} x_i\cdot i\cdot \gamma^{i-1}}} \leq \frac{\displaystyle{ \sum\nolimits_{i=k}^{T} x_i \cdot \gamma^{i-k}}}{\displaystyle{\sum\nolimits_{i=k}^{T} x_i \cdot (i-k+1)\cdot \gamma^{i-k}}}. $$

With regard to the left-hand side of (33), note that:

$$ \frac{\displaystyle \sum\nolimits_{i=L}^{T} x_i \cdot \gamma^{i-1}}{\displaystyle \sum\nolimits_{i=L}^{T} x_i\cdot i\cdot \gamma^{i-1}} < \frac{\displaystyle \sum\nolimits_{i=L}^{L+T-k} x_i \cdot \gamma^{i-1}}{\displaystyle \sum\nolimits_{i=L}^{L+T-k} x_i\cdot i\cdot \gamma^{i-1}}. $$

since each additional term in the former has a numerator-to-denominator ratio of less than 1/(L + T − k). So it is sufficient to demonstrate that

$$ \begin{aligned} \frac{\displaystyle \sum\nolimits_{i=L}^{L+T-k} x_i \cdot \gamma^{i-1}}{\displaystyle \sum\nolimits_{i=L}^{L+T-k} x_i \cdot i\cdot \gamma^{i-1}} \leq& \frac{\displaystyle \sum\nolimits_{i=k}^{T} x_i \cdot \gamma^{i-k}}{\displaystyle \sum\nolimits_{i=k}^{T} x_i \cdot (i-k+1) \cdot \gamma^{i-k}}\\ \Leftrightarrow \quad \frac{\displaystyle \sum\nolimits_{i=L}^{L+T-k} x_i \cdot \gamma^{i-L}}{\displaystyle \sum\nolimits_{i=k}^{T} x_i \cdot \gamma^{i-k}} \leq& \frac{\displaystyle \sum\nolimits_{i=L}^{L+T-k} x_i \cdot i \cdot \gamma^{i-L}}{\displaystyle \sum\nolimits_{i=k}^{T} x_i \cdot (i-k+1)\cdot \gamma^{i-k}}. \end{aligned} $$

Since x i /x i+1 increases in i, we know that \(\frac{x_L}{x_k}\leq \frac{x_{L+1}}{x_{k+1}} \cdots\leq \frac{x_{L+T-k}}{x_T}. \) The left-hand side of (34) places equal weight on these ratios, which implies that

$$ \frac{\displaystyle \sum\nolimits_{i=L}^{L+T-k} x_i \cdot \gamma^{i-L}}{\displaystyle \sum\nolimits_{i=k}^{T} x_i \cdot \gamma^{i-k}} < \frac{\displaystyle \sum\nolimits_{i=L}^{L+T-k} x_i \cdot (i-L+1) \cdot \gamma^{i-L}}{\displaystyle \sum\nolimits_{i=k}^{T} x_i \cdot (i-k+1)\cdot \gamma^{i-k}}, $$

since the expression on the right-hand side of (35) places increasingly higher weights on the higher ratios. Finally, L ≥ 1 implies that the right-hand side of (34) exceeds the right-hand side of (35). Hence, the inequality in (34) holds, and we have shown that BV T (I T d *) (and hence CC T ) increases in r.

Proof of Proposition 3

For L = 1, we have \(x_t=1-\beta\cdot (t-1)\). The capital charges in (16) simplify to:

$$ z_t = \frac{1-\alpha}{T} \cdot \left[1+r \cdot (T-t+1)\right]. $$

Using these expressions, as well as the definition of c, we can rewrite CC T in (15) as:

$$ CC_T = \frac{T}{\displaystyle \sum\nolimits_{i=1}^{T} [1-\beta\cdot (i-1)] \cdot \gamma^i}\cdot \frac{\displaystyle \sum\nolimits_{i=1}^{T} [1-\beta\cdot (i-1)] \cdot (\gamma^i-\mu^i)}{\displaystyle \sum\nolimits_{i=1}^{T} [1+r \cdot (T-i+1)] \cdot (\gamma^i-\mu^i)}\cdot \frac{1}{1-\alpha}. $$

Expanding this expression, it can then be shown that the limit values of the CC T function are as follows:

$$ \begin{aligned} &\lim_{\lambda \rightarrow -1} CC_T(\cdot) \thinspace= \left(\frac{1}{1-\alpha}\right)\cdot \frac{T \cdot r^2 \cdot (1+r)^{T-1} \cdot [1-\beta \cdot(T-1)]}{(r-\beta)\cdot [(1+r)^T-1]+\beta \cdot r \cdot T} \\ &\lim_{\lambda \rightarrow 0} CC_T(\cdot) \enspace = \left(\frac{1}{1-\alpha}\right) \cdot \left[\frac{T \cdot [2+\beta \cdot (1-T)] \cdot r^2 \cdot (1+r)^{T}}{(r-\beta) \cdot [(1+r)^T-1]+\beta \cdot r \cdot T}-2\right] \cdot \frac{1}{r \cdot [T+1]}\\ &\lim_{\lambda \rightarrow \infty} CC_T(\cdot) \thinspace= \frac{1}{1-\alpha}. \end{aligned} $$

The limit results for the β = 0 case follow directly from these expressions.

We next prove the claim regarding the bounds on the ratios of the CC T variables. Note that the term 1/(1 − α) enters in a multiplicative fashion in each of the CC T expressions in (37) and, as such, can be ignored. Also, when T = 2, direct computations on (37) reveal that the ratio in question always equals \(\frac{2}{3}\). We therefore restrict attention to values of T > 2. We first show the upper bound result that

$$ \frac{CC_T(\lambda=-1)-CC_T(\lambda=0)}{CC_T(\lambda=-1)-CC_T(\lambda=\infty)} \leq \frac{T}{T+1}. $$

To do so, we will demonstrate the equivalent result that

$$ \frac{CC_T(\lambda=0)-CC_T(\lambda=\infty)}{CC_T(\lambda=-1)-CC_T(\lambda=\infty)}\geq \frac{1}{T+1}. $$

Using the limits in (37), (38) reduces to the following inequality:

$$ \begin{aligned} &(T+1)\cdot \left[\frac{1}{r(1+T)}\cdot \frac{T r^2 (1+r)^T(2+\beta-\beta T)}{\beta r T +(r-\beta)[(1+r)^T-1]}-\frac{2}{r(1+T)}-1\right] \\ &\quad \geq \frac{T r^2 (1+r)^{T-1}[1+\beta-\beta T]}{\beta r T+(r-\beta)[(1+r)^T-1]}-1. \end{aligned} $$
$$ \begin{aligned} &\Leftrightarrow (T+1)T r^2 (1+r)^T (2+\beta-\beta T)-(T+1)\left[2 + r (1+T)\right] \cdot \left[\beta r T+ (r-\beta) \cdot \left[(1+r)^T -1\right]\right]\\ &\quad \geq T r^2 (1+r)^{T-1}[1+\beta-\beta T]r(1+T)-r(1+T)\big[\beta r T+(r-\beta)[(1+r)^T-1]\big]\\ &\Leftrightarrow T r^2 (1+r)^{T-1}(1+\beta-\beta T)+T r^2 (1+r)^T-\big[\beta r T+(r-\beta)[(1+r)^T-1]\big](2+rT) \geq 0\\ & \Leftrightarrow T r^2(1+r)^{T-1}(2+\beta-\beta T+r)-(2+rT)\big[\beta r T+(r-\beta)[(1+r)^T-1]\big] \geq 0. \end{aligned} $$

But (39) is a linear function of β. At \(\beta=\frac{r}{1+rT}\), (39) equals

$$ \frac{T r^2 (1+r)^T(2+rT)}{(1+rT)}-\frac{T r^2 (1+r)^T (2+rT)}{(1+rT)}=0. $$

At β = 0, the expression in (39) reduces to (after dividing through by r):

$$ T\cdot r \cdot (1+r)^{T-1}(2+r)-(2+rT)[-1+(1+r)^T] \geq 0. $$

Letting s = (1 + r) ≥ 1, this inequality holds if and only if

$$ T\cdot (s-1)\cdot s^{T-1}\cdot (s+1)-2(s^T-1)-(s^T-1)\cdot T \cdot (s-1) \geq 0, $$


$$ T\cdot (s-1)(s^{T-1}+1)-2(s^T-1) \geq 0. $$

But this function and its first derivative equal 0 at s = 1, while the second derivative is

$$ (s-1)\cdot T\cdot (T-1)s^{T-3}\cdot (T-2) \geq 0, $$

for all s ≥ 1. So the function is convex and positive everywhere. Thus (39) ≥ 0 for all \(\beta \in [0, \frac{r}{1+rT}]\). We have therefore shown that (38) holds.

We next demonstrate the lower bound inequality:

$$ \frac{CC_T(\lambda=-1)-CC_T(\lambda=0)}{CC_T(\lambda=-1)-CC_T(\lambda=\infty)}\geq \frac{2}{3}. $$

Expanding these expressions using (37), we seek to show that

$$ \begin{aligned} &\frac{Tr^2(1+r)^{T-1}(1+\beta-\beta T)}{(r-\beta)[(1+r)^T-1]+\beta r T}- \frac{6 T r^2(1+r)^T(2+\beta-\beta T)}{2r(1+T)\big[\beta r T+(r-\beta)[(1+r)^T-1]\big]}+\frac{6}{r(1+T)}+2 \geq 0,\\ &\Leftrightarrow T r^3 (1+r)^{T-1}(1+T)(1+\beta-\beta T)-3T r^2(1+r)^T(2+\beta-\beta T)+6\big[\beta r T+(r-\beta)[(1+r)^T-1]\big] +2r(1+T)\big[\beta r T+(r-\beta)[(1+r)^T-1]\big] \geq 0,\\ &\Leftrightarrow T r^2 (1+r)^{T-1}(1+\beta-\beta T)[r+rT-3(1+r)]-3T r^2 (1+r)^T+2[3+r+rT]\big[\beta r T+(r-\beta)[(1+r)^T-1]\big]\geq 0. \end{aligned} $$

Again, this is a linear function of β, so it suffices to show that (40) holds at its end points. At \(\beta=\frac{r}{1+rT}\), the expression reduces to

$$ \begin{aligned} &T r^2 (1+r)^{T-1} \frac{(1+r)}{(1+rT)}[r(1+T)-3(1+r)]-3Tr^2(1+r)^T+2[3+r+rT]\cdot \frac{r^2(1+r)^T T}{(1+rT)} \\ =&\frac{T r^2 (1+r)^T}{(1+rT)}[r+rT-3-3r+6+2r+2rT]-3Tr^2(1+r)^T \\ =&\frac{T r^2 (1+r)^T}{(1+rT)}[3(1+rT)]-3Tr^2(1+r)^T=0. \end{aligned} $$

At β = 0, we need to show that

$$ \begin{aligned} & Tr^3(1+r)^{T-1}(1+T)-6Tr^2(1+r)^T+6r((1+r)^T-1)+2r^2(1+T) ((1+r)^T-1)\geq 0\\ & \Leftrightarrow T(1+T)s^{T-1}(s^2-2s+1)-6T(s-1)s^T+6(s^T-1)+2(1+T)(s-1)(s^T-1)\geq 0\\ & \Leftrightarrow T(1+T)s^{T+1}-2T(1+T)s^T+T(1+T)s^{T-1}-6Ts^{T+1}+6T s^T+6s^T-6 +2(1+T)s^{T+1}-2(1+T)s^T-2s(1+T)+2(1+T)\geq 0\\ & \Leftrightarrow (T-2)(T-1)s^{T+1}-2(T-2)(T+1)s^T+T(1+T)s^{T-1}-2(1+T)s+2T-4\geq 0. \end{aligned} $$

Again, this equals 0 at s = 1. In addition, its derivative is

$$ \begin{aligned} (T-2)(T-1)(T+1)s^T-2(T-2)(T+1)Ts^{T-1}+T(1+T)(T-1)s^{T-2}-2(1+T)\\ \propto (T-2)(T-1)s^T-2(T-2)Ts^{T-1}+T(T-1)s^{T-2}-2. \end{aligned} $$

This equals 0 at s = 1. Its derivative in turn is

$$ \begin{aligned} &T(T-2)(T-1)s^{T-1}-2T(T-1)(T-2)s^{T-2}+T(T-1)(T-2)s^{T-3} \\ =&T(T-1)(T-2)s^{T-3}(s-1)^2 > 0, \end{aligned} $$

for all T > 2 and all s > 1. We have thus established that (40) is strictly positive for values of β between 0 and \(\frac{r}{1+rT}\). We conclude that for any level of decay in that range, the ratio bounds of \(\frac{2}{3}\) and \(\frac{T}{T+1}\) hold.

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McNichols, M., Rajan, M.V. & Reichelstein, S. Conservatism correction for the market-to-book ratio and Tobin’s q . Rev Account Stud 19, 1393–1435 (2014).

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