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Deferred taxation under default risk


In this article, we have used a continuous EBIT-based model to study deferred tax liabilities under default risk. Quite surprisingly, default risk has been disregarded in research on deferred taxation. In order to underline its importance, we first calculated the probability of default, over a given time period, together with the contingent value of tax deferral. We then applied our theoretical model to a sample of 27,749 OECD companies. We showed that, when accounting for both firms with a negative EBIT and firms with a probability of default higher than 50% (over a 10-year period), a relevant percentage of firms were close enough to default. Hence, the expected present value of deferred taxes is much lower than that obtained in a deterministic context. From the Government’s point of view, deferred tax liabilities are a risk-free loan. Since only a portion are subsequently repaid, the Government should account for future losses due to companies’ default. So far, these estimates have been missing, although techniques do exist and are quite practical.

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  1. 1.

    Positive temporary differences generate deferred tax liabilities, i.e., taxes to be paid in the future, which increase the total tax liability of a corporation; negative temporary differences generate deferred tax assets, i.e., credits against current taxes, thus reducing the total tax liability of a corporation.

  2. 2.

    See, e.g., Mills and Plesko (2003), Hanlon and Shevlin (2005) as well as Poterba et al. (2011). See also Spengel et al. (2018) who analyzed the recent US Tax Cuts and Jobs Act, which allows the immediate expensing of fixed assets: such a reform is expected to increase the value of deferred tax liabilities.

  3. 3.

    See also Edgerton (2012), who compared the impact of investment tax credits and accelerated tax depreciation. He showed that the former relief has more impact on investment choices than the latter. For further details see also Burnham and Ozanne (2006) and Hafkenscheid and Janssen (2009).

  4. 4.

    For further details see Graham (2005), Hanlon and Heitzman (2010), Graham et al. (2012).

  5. 5.

    Tax uncertainty combines three aspects: an investment decision, tax compliance, and the financial reporting matters. Investment decision, tax effects and financial reporting incentives can potentially interact in ways that affect investment decisions (Hanlon and Heitzman 2010). First of all, value, timing and uncertainty of tax payments affect the present value of a project and therefore invest choices. Secondly, through deprecation or expensing, the investment decision will affect pre-tax accounting earnings. Lastly, as pointed out by Edgerton (2012), tax policies that do not affect accounting profit (e.g., accelerated depreciation) are less effective in stimulating investment than those that increase accounting profit (e.g., investment tax credit). In conclusion, tax uncertainty may be harmful for investment (Edmiston 2004; Scholes et al. 2015; Diller et al. 2017; De Simone et al. 2013; Mills et al. 2010).

  6. 6.

    See, e.g., Sansing (1998), Guenther and Sansing (2000), Sikes and Verrecchia (2012), Diller et al. (2017), and Langenmayr and Lester (2018).

  7. 7.

    For simplicity, we assumed that this firm cannot postpone its investment decision. For a discussion of this choice, see Panteghini (2012), and Panteghini and Vergalli (2016). For simplicity, we also assumed that economic and book depreciation coincide. This allowed us to focus only on the tax effects.

  8. 8.

    For further details on the EBIT-based models see also Goldstein et al. (2001), and Panteghini (2006, 2007, 2012).

  9. 9.

    Notice that the difference \( r - \alpha \) is the well-known convenience yield (see, e.g., McDonald and Siegel 1985).

  10. 10.

    Given this coupon and the risk-free interest rate r, the market value of debt can be calculated in the absence of arbitrage (see Leland 1994).

  11. 11.

    The absence of debt renegotiation simplifies our analysis, although it does not affect the qualitative properties of the model. For a detailed analysis of financial decisions, with costly debt renegotiation, see e.g., Goldstein et al. (2001), and Hennessy and Whited (2005).

  12. 12.

    For instance, according to the International GAAP 2009, (Generally Accepted Accounting Practice under International Financial Reporting Standards, vol. 2, Wiley, West Sussex): Deferred tax is an accounting model based on the premise that, for financial reporting purposes, the tax effects on transactions should be recognized in the same period as the transactions themselves". Moreover, "[t]he tax authorities cannot demand payment of an entity's deferred tax liability until it forms part of the legal tax liability for a future period; equally, an entity cannot recover its deferred tax assets from the tax authorities until they form a deduction in arriving at the legal tax liability for a future period". This means that, in the event of default, deferred tax liabilities simply vanish.

  13. 13.

    Of course, this simplifying assumption could be relaxed and we intend to leave these extensions for future research.

  14. 14.

    In our framework we focused on deferred taxation and therefore, we let the inequality \( \lambda_{F} \ge \lambda \) hold. In doing so, we disregarded the effects of reversal, which occurs when tax depreciation is less than economic depreciation. It is worth noting that we used a straight-line depreciation method. Of course, the qualitative properties of our results would not change if we used the declining balance method for tax depreciation allowances. To show this, let us compare our framework with Guenther and Sansing (2004). In their model, these authors define the tax depreciation rate and the economic depreciation rate under the declining balance method as δ and b, respectively. The present value of the tax benefit due to tax and economic depreciation will then be \( \tfrac{\delta }{r + \delta } \) and \( \tfrac{b}{r + b}, \) respectively. According to our model we have thus assumed that \( \tfrac{\delta }{r + \delta } \ge \tfrac{b}{r + b}, \) or equivalently \( \delta \ge b. \)

  15. 15.

    As pointed out by Polito (2009), if tax is deferred, there is some undistributed cash, which could be re-invested in a risk-free bond. For simplicity, we did not account for this extra-provision and considered this cash as a collateral that reduces the risk of default. As long as the level of risk-free interest rates is low enough, this simplifying assumption does not affect the quality of results.

  16. 16.

    For simplicity, we assumed that the cutoff level of \( \Pi , \) below which default takes place, was exogenously given. It is worth noting that this default assumption causes no loss of generality. If, for instance, we assumed that shareholders can decide when to default, the threshold level would be lower. However, the new threshold would be similar to \( \underline{\Pi }^{D} . \) For a detailed analysis of default conditions see, e.g., Leland (1994) and Panteghini (2006, 2007), who show that the quality of results does not change when an endogenously given threshold point is found.

  17. 17.

    Using the notation of footnote 14, under a declining balance method, the deterministic value of the benefit arising from deferred taxation would be \( \tau \left( {\tfrac{\delta }{r + \delta } - \tfrac{b}{r + b}} \right)\,I \ge 0. \) Of course, the quality of results would not change.

  18. 18.

    For sake of brevity, we did not add the results obtained with T = 15. Data are available upon request. In any case, according to our theoretical framework, the longer the arrival time T the higher the probability of default and the smaller the contingent value of deferred taxation is.

  19. 19.

    Of course, the choice of the threshold probability of 50% is fully arbitrary. However, it is useful to stress the relevance of default risk even if EBIT is positive.


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We thank Vito Polito for his stimulating discussions on deferred taxation. Many thanks also go to Alberto Quagli, Amedeo Pugliese and Christoph Spengel for their helpful comments. Usual disclaimers hold.

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Correspondence to Sergio Vergalli.

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Appendix A

The calculation of probability

Let us focus on a stochastic process that follows a geometric Brownian motion. If at time 0 we have \( \Pi_{0} > 0 \) then at time \( t > 0 \) we will have \( \Pi_{t} = \Pi_{0} \exp \left[ {\left( {\alpha - \tfrac{{\sigma^{2} }}{2}} \right)\,t + \sigma W_{t} } \right]. \) Let us next introduce a lower bound \( \underline{\Pi }^{D} < \Pi_{0} \) and calculate the probability to hit \( \underline{\Pi }^{D} \) within time \( T \). To do so we must define the hitting time as follows: \( \left\{ {\tau_{{\underline{\Pi }^{D} }} \le T} \right\} = \left\{ {\Pi^{ * } \le \underline{\Pi }^{D} } \right\} \) where \( \Pi^{ * } = \left( {\Pi_{t}^{ * } } \right)_{t \ge 0} \) and \( \Pi_{t}^{ * } = \min_{t\varepsilon [0,\,T]} . \) Given this notation the probability that \( \Pi_{t} \) hits \( \underline{\Pi }^{D} \) in the \( [0,\,T] \) period is:

$$ P\left[ {\tau_{{\underline{\Pi }^{D} }} \le T} \right] = P\left[ {\Pi^{ * } \le \underline{\Pi }^{D} } \right] = P\left[ {\xi_{T}^{\left( \theta \right)} \le \ln \left( {\frac{{\underline{\Pi }^{D} }}{{\Pi_{0} }}} \right)} \right], $$


$$ \xi_{t}^{\left( \theta \right)} = \mathop {\hbox{min} }\limits_{s \in [0,\,t]} \left\{ {\bar{\alpha }s + \sigma W_{s} } \right\}. $$

Using the definitions \( \bar{\alpha } \equiv \left( {\alpha - \tfrac{{\sigma^{2} }}{2}} \right) \) and \( \theta \equiv \tfrac{{\bar{\alpha }}}{{\sigma^{2} }}, \) we can rewrite (7) as follows:

$$ \xi_{t}^{\left( \theta \right)} = \mathop {\hbox{min} }\limits_{s \in [0,\,t]} \left\{ {\bar{\alpha }s + \sigma W_{s} } \right\} = \mathop {\hbox{min} }\limits_{s \in [0,\,t]} \left\{ {\theta s + W_{s} } \right\}. $$

Let us next use (6) and (8). Following Harrison (1985, 11–14), Sarkar (2000, 222–223) and Cappuccio and Moretto (2001, 8–9) gives:

$$ \begin{aligned} P\left[ {\tau_{{\underline{\Pi }^{D} }} \le T} \right] = & P\left[ {X_{T}^{ * } \le B} \right] = P\left[ {\xi_{T}^{\left( \theta \right)} \le \ln \left( {\frac{{\underline{\Pi }^{D} }}{{\Pi_{0} }}} \right)} \right] \\ = & e^{{2\theta \ln \left( {\tfrac{{\underline{\Pi }^{D} }}{{\Pi_{0} }}} \right)}} \Phi \left( {\frac{{\ln \left( {\tfrac{{\underline{\Pi }^{D} }}{{\Pi_{0} }}} \right) + \sigma^{2} \theta T}}{\sigma \sqrt T }} \right) + \Phi \left( {\frac{{\ln \left( {\tfrac{{\underline{\Pi }^{D} }}{{\Pi_{0} }}} \right) - \sigma^{2} \theta T}}{\sigma \sqrt T }} \right) \\ = & \left( {\frac{{\underline{\Pi }^{D} }}{{\Pi_{0} }}} \right)^{2\theta } \Phi \left( {\frac{{\ln \left( {\tfrac{{\underline{\Pi }^{D} }}{{\Pi_{0} }}} \right) + \sigma^{2} \theta T}}{\sigma \sqrt T }} \right) + \Phi \left( {\frac{{\ln \left( {\tfrac{{\underline{\Pi }^{D} }}{{\Pi_{0} }}} \right) - \sigma^{2} \theta T}}{\sigma \sqrt T }} \right). \\ \end{aligned} $$

Formula (4) is thus obtained.

Appendix B

The derivation of (5)

The value of deferred taxation can be written as:

$$ DT\left( {\Pi ;\,C} \right) = \left\{ {\begin{array}{*{20}l} 0 \hfill & {\Pi = \underline{\Pi }^{D} ,} \hfill \\ {\left[ {\tau \left( {\lambda_{F} - \lambda } \right)\,I} \right]\,dt + e^{ - rdt} \left[ {DT\left( {\Pi + d\Pi ;\,C} \right)} \right]} \hfill & {\Pi > \underline{\Pi }^{D} .} \hfill \\ \end{array} } \right. $$

\( DT\left( {\Pi ;\,C} \right) \) is given by the solution of the following system of Second Order Differential Equation, and the threshold level \( \underline{\Pi }^{D} \) is given by (dividend threshold). Differentiating (equity) and applying Itô’s Lemma gives the following non-arbitrage condition:

$$ \frac{1}{2}\sigma^{2} \Pi^{2} DT_{\Pi } \left( {\Pi ;\,C} \right) + \alpha \Pi DT_{\Pi } \left( {\Pi ;\,C} \right) - rDT\left( {\Pi ;\,C} \right) = - \tau \left( {\lambda_{F} - \lambda } \right)\,I. $$

To solve Eq. (10), let us apply the following general function:

$$ DT\left( {\Pi ;\,C} \right) = \left\{ {\begin{array}{*{20}l} 0 \hfill & {\Pi = \underline{\Pi }^{D} ,} \hfill \\ {\Phi + A_{1} \Pi^{{\gamma_{1} }} + A_{2} \Pi^{{\gamma_{2} }} } \hfill & {\Pi > \underline{\Pi }^{D} ,} \hfill \\ \end{array} } \right. $$

where \( \Phi \equiv \tfrac{{\tau \left( {\lambda_{F} - \lambda } \right)\,I}}{r} \) is the present value of the deferred tax liability, in a deterministic context. Parameters \( \gamma_{1} > 1 \) and \( \gamma_{2} < 0 \) are the positive and negative roots of the auxiliary quadratic equation \( \Phi (z) = \tfrac{1}{2}\sigma^{2} z(z - 1) + \alpha z - r = 0, \) respectively.

To solve (10) we used (11) and applied the following boundary condition (see Dixit and Pindyck 1994):

$$ DT\left( {\underline{\Pi }^{D} ;\,C} \right) = 0. $$

Moreover, we assumed the absence of financial bubbles. This means that the equality \( A_{1} = 0 \) must hold. Setting \( A_{1} = 0 \), and using (12) gives:

$$ \Phi + A_{2} \underline{\Pi }^{{D^{{\gamma_{2} }} }} = 0, $$

and thus \( A_{2} = - \,\Phi \underline{\Pi }^{{D^{{ - \gamma_{2} }} }} . \) Substituting these results into (11) gives function (5).

Appendix C

Comparative statics

Given \( \underline{\Pi }^{D} = C - \tfrac{\tau }{1 - \tau }\lambda_{F} I \) and \( \gamma_{2} < 0 \) it is easy to show that:

$$ \begin{aligned} \frac{{\partial \underline{\Pi }^{D} }}{\partial \tau } = & - \frac{1}{1 - \tau }\lambda_{F} I - \frac{\tau }{{\left( {1 - \tau } \right)^{2} }}\lambda_{F} I = \left[ { - \frac{1 - \tau }{{\left( {1 - \tau } \right)^{2} }} - \frac{\tau }{{\left( {1 - \tau } \right)^{2} }}} \right]\,\lambda_{F} I \\ = & - \frac{{\lambda_{F} I}}{{\left( {1 - \tau } \right)^{2} }} < 0, \\ \frac{{\partial \underline{\Pi }^{D} }}{{\partial \lambda_{F} }} = & - \frac{\tau }{1 - \tau }I < 0, \\ \end{aligned} $$


$$ \begin{aligned} \frac{{d\left[ {\left( {\tfrac{\Pi }{{\underline{\Pi }^{D} }}} \right)^{{\gamma_{2} }} } \right]}}{d\tau } = & \left[ {\left( { - \,\gamma_{2} } \right)\,\Pi^{{\gamma_{2} }} \underline{\Pi }^{{D^{{ - \gamma_{2} - 1}} }} } \right]\frac{{\partial \underline{\Pi }^{D} }}{\partial \tau } < 0, \\ \frac{{d\left[ {\left( {\tfrac{\Pi }{{\underline{\Pi }^{D} }}} \right)^{{\gamma_{2} }} } \right]}}{{d\lambda_{F} }} = & \left[ { - \,\gamma_{2} \Pi^{{\gamma_{2} }} \underline{\Pi }^{{D^{{ - \gamma_{2} - 1}} }} } \right]\frac{{\partial \underline{\Pi }^{D} }}{{\partial \lambda_{F} }} < 0. \\ \end{aligned} $$

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Carini, C., Moretto, M., Panteghini, P.M. et al. Deferred taxation under default risk. J Econ 129, 33–48 (2020).

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  • Capital structure
  • Contingent claims
  • Corporate taxation
  • Tax depreciation allowances

JEL Classification

  • H25
  • M41