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The valuation relevance of reversing deferred tax liabilities.

ABSTRACT: This paper compares two attributes of a deferred tax liability (DTL) that arise from differences in book and tax depreciation methods. The first attribute is the effect of the DTL on the market value of the firm. The second is the length of time between when the asset is placed into service and when the DTL associated with that asset begins to reverse. The paper shows that a decrease in the time it takes for the DTL to begin to reverse is neither necessary nor sufficient for the value of the DTL to increase. It also shows that the value of the DTL is not equal to the present value of the future deferred tax expense. The effect of one dollar of DTL on firm value depends only on the tax depreciation rate and the discount rate.

Keywords: deferred income taxes; depreciation; valuation.

I. INTRODUCTION

This paper extends Sansing (1998) and Guenther and Sansing (2000) (hereafter GS) by investigating the relation between the effect of a firm's deferred tax liability (DTL) on firm value and how quickly that DTL begins to reverse. To illustrate our research question, consider two depreciable assets owned by different firms with different book and tax depreciation rates. Assume that at some point in time the book values of the DTLs related to these two assets are equal, but one of the DTLs will begin to reverse sooner than the other. Our question is: Will the effect of the DTLs on firm value be different because one DTL reverses more quickly than the other?

Sansing (1998) shows that the value of one dollar of depreciation-related DTL is equal to [delta]/([delta] + [rho]) dollars, where [delta] is the rate of tax depreciation and [rho] is the discount rate. Because the valuation factor [delta]/([delta] + [rho]) does not depend on when of whether the DTL reverses, GS concludes that the time of reversal of a depreciation-related DTL is not valuation relevant. (1) In a later paper using a different analytical model, Amir et al. (2001) (hereafter AKW) also find that the valuation factor [delta]/([delta] + [rho]) is appropriate. AKW also agree that reinvestment that delays or prevents the reversal of the DTL at the firm level does not affect firm value, as demonstrated in GS.

However, AKW argue that the value relevance of DTL reversals should be investigated at the asset level rather than at the firm level, stating "the reversal rate, or rate at which the deferred tax liability on each asset reverses, plays a critical role in the value relevance of deferred taxes" (AKW, 277). (2) AKW also suggest that their finding supports the "partial allocation" method of recording DTLs previously used in the U.K.

Resolving the question of the value relevance of the timing of DTL reversals has important implications for both academic accounting research and financial reporting practice. First, prior empirical studies of the valuation of DTLs (Givoly and Hayn 1992; Amir et al. 1997) document valuation coefficients that are less than 1 on depreciation-related deferred taxes, and infer that the expected time to reversal affects the value of DTL. Our results in this paper call that inference into question.

Second, some financial statement analysis texts argue that deferred future taxes represented by the DTL should be discounted to their present value, where present value is based on the time that the DTL is expected to reverse. (3) We show in this paper that the value of the DTL is not equal to the present value of future deferred tax expenses. Third, the U.K. Accounting Standards Board has in the past argued that the partial allocation method of recording DTLs should be used, a method under which only those DTLs that are expected to reverse quickly would be recorded, while DTLs expected to reverse slowly would not be recorded. (4) A proper understanding of the value relevance of DTL reversals has implications for standard setters.

Our research question in this paper is to examine the relation between (1) the effect of the DTL on firm value, and (2) the length of time until the DTL begins to reverse (which is equivalent to the reversal rate). In other words, given two equal DTL balances, where one balance will begin to reverse sooner than the other, will there be a difference in the effects these two equal DTLs have on firm value? Our results show that there is no consistent relation between the rate of DTL reversal and the value of the DTL, even when reversal is defined at the asset level, as proposed by AKW. We also demonstrate that the value of the DTL is not equal to the present value of future deferred tax expenses because the value of the DTL only depends on cash flows associated with tax depreciation, whereas the changes in the DTL depend on both tax and book depreciation. Finally, we show that these results do not depend on whether the benefits of rapid tax depreciation (relative to economic depreciation) are capitalized into asset prices.

These findings contribute to accounting research by clearly distinguishing between the value of the DTL (a levels effect) and the rate at which the DTL reverses (a changes effect). A rigorous analysis of how tax and book depreciation rates are linked to both firm value and the rate of DTL reversal helps contribute to our understanding of the relation between financial accounting numbers and firm value.

We illustrate the relation between tax and book depreciation rates, the level of the DTL, and how quickly the DTL reverses with numerical examples in Section II. In Section III we derive the valuation factor [delta]/([delta] + [rho]) and demonstrate the relations among the value of the DTL, tax and book depreciation rates, and the discount rate. In Section IV we derive a measure of DTL reversal and describe the properties of this measure. In Section V we examine the relation between the value of the DTL and the present value of future deferred tax expenses. Section VI extends our analysis to a setting with positive net present value (NPV) investments. Section VII concludes. All proofs are in the Appendix.

II. EVOLUTION OF THE DEFERRED TAX LIABILITY

To motivate our analytical work and to provide intuition for our results, we present a graphical analysis of how the book value of the DTL evolves over time. The DTL initially increases (assuming tax depreciation exceeds book depreciation), but later begins to decrease. The time at which the DTL balance begins to decrease is the reversal date, and the sooner the DTL balance begins to reverse, the higher is the rate of reversal. Our analysis also shows how the DTL balance and the rate of reversal are affected by changes in the rate of tax or book depreciation. Consistent with the argument in AKW, these examples focus on the DTL related to a single asset rather than a firm's total DTL balance. Although our later analytical model is based on an infinite horizon, continuous time model, in these examples we use discrete time and focus on the first ten years of the asset's life.

A firm invests $1,000 in a single depreciable asset that is never replaced. The tax rate is 40 percent. Depreciation is based on a declining balance method, with different rates for tax and book purposes. Under U.S. generally accepted accounting principles, the book value of the DTL at the end of any year is:

(1) DTL = tax rate x (book value of asset - tax basis of asset).

In the first two examples we show how the book value of the DTL and the date at which the DTL begins to reverse behave for different combinations of tax and book depreciation.

Example 1

In the first example we hold the book depreciation rate constant at 10 percent and allow the tax depreciation rate to vary from 20 percent to 50 percent. The book values of the DTL for this example are presented graphically in Figure 1. As this example illustrates, increasing the rate of tax depreciation while holding the book depreciation rate fixed has two effects. First, the book value of the DTL increases. Second, the DTL begins to reverse sooner (i.e., the length of time until the DTL begins to reverse decreases). For example, the DTL of the asset with the 50 percent depreciation rate begins to reverse in year four, whereas the DTL of the asset with the 30 percent depreciation rate does not begin to reverse until year six. In this case, an increase in the level of the DTL is associated with a more rapid reversal of the DTL.

[FIGURE 1 OMITTED]

Example 2

In the second example we hold the tax depreciation rate constant at 25 percent and allow the book depreciation rate to vary from 5 percent to 20 percent. The book values of the DTL for this example are presented graphically in Figure 2. Increasing the rate of book depreciation while holding the tax depreciation rate fixed decreases the book value of the DTL and causes the DTL to reverse sooner. In this case, a decrease in the level of the DTL is associated with a more rapid reversal of the DTL. Therefore, increasing either the tax depreciation rate or the book depreciation rate will increase the rate of reversal of the DTL. However, changing the two depreciation rates has opposite effects on the level of the DTL--increasing the tax depreciation rate increases the DTL, while increasing the book depreciation rate decreases the DTL.

[FIGURE 2 OMITTED]

Taken together these examples indicate that there is no consistent relation between the factors that determine the level of the DTL and how quickly the DTL reverses. An increase in the tax depreciation rate increases the level of the DTL and causes the DTL to reverse more quickly; an increase in the book depreciation rate decreases the level of the DTL and causes the DTL to reverse more quickly.

Our next two examples compare assets with the same DTL to illustrate the relation between the effect of a DTL on firm value and how quickly the DTL begins to reverse.

Example 3

A firm with a 40 percent tax rate acquires an asset for $1,000. The tax depreciation rate is 25 percent and the book depreciation rate is 10 percent. (5) At the end of four years the asset's tax basis is 316, its book value is 656, and the DTL is equal to 0.4 x (656 - 316) = 136. A second firm (also with a 40 percent tax rate and a 25 percent tax depreciation rate) acquires an asset for $1,000, but this firm has a book depreciation rate of 5 percent. At the end of two years the asset's tax basis is 563, its book value is 903, and the DTL is equal to 0.4 x (903 - 563) = 136. Figure 3 shows how the 136 book value of the DTL for these two assets will change over a ten-year period of time. As Figure 3 demonstrates, the DTL for the firm with the 10 percent book depreciation rate begins to reverse much sooner than the DTL for the firm with the 5 percent rate.

[FIGURE 3 OMITTED]

At the point in time when the DTL for the each firm is equal to 136, the effect of the DTL on firm value is also the same. Assume that the discount rate for both firms is 6 percent. Using these assumptions, the value of the DTL, as demonstrated by Sansing (1998) and AKW, is:

Value of DTL = 0.25/0.25 + 0.6 x 136 = 110

for both firms. Therefore, despite the fact that the DTL balance reverses sooner for the firm with the 10 percent book depreciation rate, the value of the DTL for both firms is the same.

This result may at first appear counterintuitive. Because the DTL for the 10 percent rate firm will begin to reverse in year 4 (see Figure 3), while the DTL for the 5 percent rate firm will not begin to reverse until year 8 (see Figure 3), it may appear that the present value of the 10 percent firm's DTL should be higher, since it will reverse sooner. Under this (incorrect) view, the value of the DTL would be equal to the present value of future deferred tax expense represented by the DTL. However, as we demonstrate in Section V, the value of the DTL is not equal to the present value of future deferred tax expense.

The reason the value of the DTL is the same for the two firms is as follows. Income tax expense affects firm value only to the extent it represents cash flows, and tax-related cash flows for depreciation are based solely on the asset's tax depreciation rate. (6) Since in this example the book-tax difference for each firm is the same (equal to 340), and since both firms have the same tax depreciation rate (25 percent), the future tax-related cash flows associated with the 340 book-tax difference will be realized in the same year by each firm. This means that the present value of future tax-related cash flows arising from the 340 book-tax difference will be the same for each firm. The time at which future deferred tax expense is recognized has no effect on the value of the DTL.

Here is an alternative way to think about the value of the DTL in the above example. Each example firm has an asset recorded on its balance sheet--one firm at a book value of 656, and one firm at a book value of 903. If book and economic depreciation are the same, a new buyer would be willing to pay 656 for the first firm's asset and 903 for the second firm's asset. However, the present values of the after-tax cash flows to the current owner of the assets are 546 and 793, respectively. These amounts are the sum of the present value of the after-tax cash flows before depreciation (A) and the present value of the tax savings from depreciation (B). The amount (A) is the same for both the new buyer and the current owner, so the differences are attributable to differences in (B). In the first example, the present value of tax savings from depreciation is .4(.25)656/1.06 in year one, .4(.25)(.75)656/1.06 (2) in year two, and .4(.25)[(.75).sup.n-1]656/[1.06.sup.n] in year n. Therefore, the present value of the depreciation tax shield to the new buyer is:

(2) [[infinity].summation over (n=0)] 656(.25)(.4)/1.06 [[.75/1.06].sup.n].

Using the fact that [[SIGMA].sup.[infinity].sub.n = 0] [x.sup.n]= 1/(1 - x), Equation (2) equals 212. The tax basis to the current owner is 316, so the present value of the remaining depreciation tax shield to the current owner is:

(3) [[infinity].summation over (n=0)] 316(.25)(.4)/1.06 [[.75/1.06].sup.n] = 102.

The difference in the values in Equations (2) and (3) is 110, the value of the deferred tax liability.

In the second example, replacing 656 with 903 in Equation (2) yields a value of the depreciation tax shield of 292; replacing 316 with 563 in Equation (3) yields a value of 182. In each case the difference between the amount a new buyer would pay and the value of the current owner's depreciation tax shield diverges by 110, the value of the DTL.

Example 4

A firm with a 40 percent tax rate acquires an asset for $1,000. The asset has a 20 percent tax depreciation rate and a 10 percent book depreciation rate. At the end of five years the asset's tax basis is 328, its book value is 590, and the DTL is equal to 0.4 x (590 - 328) = 105. As Figure 4 shows, the DTL begins to reverse in year 7. At the end of year 8 the asset's tax basis is 168, its book value is 430, and the DTL is equal to 0.4 x (430 - 168) = 105. In this example the same asset has a DTL book value equal to 105 at two different points in time, once before the DTL begins to reverse, and once after the DTL has already reversed.

The value of the DTL in both of these cases is exactly the same, equal to:

Value of DTL = 0.20/0.20 + 0.06 x 105 = 81

even though the first DTL balance will not begin to reverse for another two years, and the second DTL balance has already begun to reverse. As was the case with Example 3, this 81 represents the present value of the cash flow effect of future tax depreciation deductions related to the 262 book-tax difference. The tax-related cash flow effect will be realized at the tax depreciation rate of 20 percent, regardless of the timing of the reversal of the DTL balance.

The examples in this section are meant to provide some intuition for the analytical results which follow. In the remainder of the paper we use a continuous time model to demonstrate that the relations suggested by our examples still hold when we move beyond the parameters in the examples.

III. THE EFFECT OF THE DEFERRED TAX LIABILITY ON FIRM VALUE

In this section we consider the relation between the firm's stock price and the level of the DTL. We begin by restating and explaining the result from both Sansing (1998) and AKW: the valuation factor associated with book-tax differences from depreciation is [delta]/([delta] + [rho]), where [delta] is the tax depreciation rate and [rho] is the discount rate.

The firm owns depreciable assets with a book value (net of accumulated depreciation) of [B.sub.0] and an adjusted tax basis of [V.sub.0], [B.sub.0] [greater than or equal to] [V.sub.0]. The assets have a replacement cost [K.sub.0], [K.sub.0] [greater than or equal to] [V.sub.0]. We assume that the rate of tax depreciation ([delta]) is weakly greater than the rate of book depreciation ([beta]) and economic depreciation ([lambda]) and that reinvestment occurs at the rate [alpha] [greater than or equal to] 0. The net book value of the assets on an arbitrary date u is [B.sub.0][e.sup.([alpha]-[beta])u], denoted B(u). The adjusted tax basis on date u is [B.sub.0][e.sup.([alpha]-[beta])u], denoted V(u), and the replacement cost on date u is [K.sub.0][e.sup.([alpha]-[lambda])u], denoted K(u).

In the case of zero net present value investments, the stock price P(u) on date u is that derived in Sansing (1998):

(4) P(u) = K(u) - (t[delta]/[delta] + [rho]) [K(u) - V(u)].

Equation (4) shows that stock price equals the replacement cost of the assets minus a tax term, where t is the tax rate. Partway through the useful life of an asset, the difference between the rate of tax depreciation ([delta]) and economic depreciation ([lambda]) creates a difference between the tax basis of the assets (V(u)) and their replacement cost (K(u)). A portion of the replacement cost of the asset represents the present value of the future tax savings related to tax depreciation deductions that a new buyer of the asset at its replacement cost would receive. This amount is equal to:

(5) MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

However, the current owner of the asset has already depreciated the asset down to an adjusted tax basis of [V.sub.0]. The present value of the remaining future tax savings related to tax depreciation deductions that the current owner of the asset will receive is t[delta][V.sub.0]/([delta] + [rho]). Therefore, the value of the asset to its current owner is equal to the asset's replacement cost of [K.sub.0] reduced by the difference in the present values of the future depreciation tax savings between what a new buyer would receive (based on replacement cost) and what the current owner will receive (based on tax basis). This difference in the present values of future depreciation tax savings is:

(6) (t[delta]/[delta] + [rho]) (K(u) - V(u)].

The above model expresses stock price in terms of the replacement cost of an asset. To investigate what happens when replacement cost differs from book value, we substitute for the replacement cost K(u) the term B(u) + [K(u) - B(u)] where B(u) is the book value of the asset at time u. The book value is based on book depreciation at rate [beta]. We also express the DTL in terms of book values: DTL = t[B(u) - V(u)]. This results in a stock price of:

(7) P(u) = B(u) - [delta]/[delta] + [rho] DTL + [K(u) - B(u)] [1 - t[delta]/[delta] + [rho]]].

In Proposition 1 we establish the relations among the parameters of the model and the following measures: (1) the DTL, (2) the valuation factor [delta]/([delta] + [rho]), and (3) the value of the DTL.

Proposition 1:

(a) An increase in the tax depreciation rate [delta] increases the DTL, increases the valuation factor, and increases the value of the DTL.

(b) An increase in the book depreciation rate [beta] decreases the DTL, has no effect on the valuation factor, and decreases the value of the DTL.

(c) An increase in the discount rate [rho] has no effect on the DTL, decreases the valuation factor, and decreases the value of the DTL.

Table 1 summarizes the effects of the depreciation parameters and discount rate on the book value of the DTL (row 1), the valuation factor (row 2), and the effect of the DTL on the market value of the firm (row 3).

IV. THE REVERSAL OF THE DEFERRED TAX LIABILITY

In this section we examine the reversal of the DTL. We begin by defining the rate at which the DTL changes. The instantaneous change in the DTL on date u is:

(8) [delta]DTL/[delta]u = t[([alpha] - [beta])B(u) - ([alpha] - [delta]V(u)],

and so the rate at which the DTL changes is the change from Equation (8) divided by the DTL:

(9) [delta]DTL/[delta]u/DTL = t[([alpha] - [beta])B(u) - ([alpha] - [delta]V(u)]/t[B(u) - V(u)] = - B + ([delta] - [beta])V(u)/B(u)- V(u).

Equation (9) implies that if the reinvestment rate [alpha] is at least as large as the book depreciation rate [beta], then the rate of the change is always positive and thus the DTL never reverses.

AKW define deferred tax reversal at the asset level rather then the firm level, which can be though of as a special case of Equation (9) in which [alpha] = 0. (7) When [alpha] = 0, Equation (9) becomes:

(10) t[([alpha] - [beta])B(u) - ([alpha] - [delta]V(u)]/t(B(u) - V(u)] = [delta]V(u) - [beta]B(u)/B(u) - V(u).

Equation (10) makes it clear that at the asset level, the DTL begins to reverse once book depreciation expense [beta]B(u) exceeds tax depreciation expense [delta]V(u). This occurs because the change in the DTL, [delta]V(u)--[beta]B(u), can be decomposed into the difference between two distinct effects, ([delta]--[beta])V(u) and [beta][B(u)--V(u)]. The former is the increase in the DTL because the rate of tax depreciation exceeds the rate of book depreciation. The latter is the decrease in the DTL because the net book value of the asset exceeds the tax basis of the asset. When the second effect exceeds the first, the DTL begins to reverse.

The above analysis helps to clarify what factors affect the rate of DTL reversal. However, when financial analysts (and accounting researchers) talk about DTLs reversing quickly or slowly, or reversing sooner or later, they are generally focusing on the time at which the DTL balance begins to decrease. Therefore, the measure of DTL reversal we use in this analysis is the length of time from when the asset is placed into service until the DTL begins to decrease. (8)

In the absence of reinvestment the DTL begins reversing on date u = 1n([delta]/[beta])/([delta] - [beta]), which is found by setting Equation (10) equal to 0 and solving for u. Proposition 2 sets out the properties of the reversal date.

Proposition 2: When [alpha] = 0 (i.e., there is no reinvestment), the date at which the DTL begins to reverse is:

(a) decreasing in the tax depreciation rate [delta];

(b) decreasing in the book depreciation rate [beta];

(c) independent of the discount rate [rho].

Table 1 summarizes the effects of the depreciation parameters and discount rate on the DTL's reversal date (row 4). A comparison of rows 3 and 4 of Table 1 shows that a more rapid reversal of the DTL is neither necessary nor sufficient for the value of the DTL to increase. An increase in the tax depreciation rate [delta] increases the value and accelerates the reversal; an increase in the book depreciation rate [beta] decreases the value and accelerates the reversal; and an increase in the discount rate decreases the value but has no effect on the reversal date.

V. PRESENT VALUE OF FUTURE DEFERRED TAX EXPENSE REVERSAL

In this section we address the intuitive notion that the value of the DTL should be equal to the present value of future deferred tax expense. This notion seems likely to be the reason for the commonly held assumption that the timing of the DTL reversal affects the value of the DTL. Under GAAP, future deferred tax expense will be equal to future changes in the DTL. We therefore begin our analysis by deriving the present value of future changes in the DTL.

The present value of future changes in the DTL in the absence of reinvestment ([alpha] = 0) is:

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In contrast, the value of the DTL itself, as derived by Sansing (1998) and AKW, is:

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Subtracting Equation (11) from Equation (12) yields:

(13) t[[BETA].sub.0]([rho]([delta] - [beta])/[delta] + [rho])([beta] + [rho]) > 0.

Equation (13) indicates that the present value of the future changes in the DTL is less than the effect of the DTL on firm value. The reason for this difference is as follows. The value of the DTL reflects the difference between the present value of future tax savings that the current owner will receive (based on the current tax basis, [V.sub.0]) and the present value of future tax savings that a new owner would receive if it acquired the asset at its book value, [B.sub.0]. In contrast, the present value of future changes in the DTL reflects the difference between the present value of future tax savings from depreciation to the current owner and the present value of future book depreciation expense multiplied by the tax rate. Because book depreciation has no cash flow consequences, the present value of future changes in the DTL is not value relevant.

An alternative way to think about the issue of the value relevance of deferred tax expenses is based on an analysis of the firm's cash flows. Let [ye.sup.-[lambda]u] denote the pretax cash flow from the asset on date u. Then the after-tax cash flow from the asset as of date u is:

(14) [ye.sup.-[lambda]u](1 - t) + t[delta][Ve.sup.-[delta]u].

The accounting earnings on date u is:

(15) ([ye.sup.-[lambda]u] - [beta][BETA][e.sup.-[beta]u])(1 - t).

The cash flows in Equation (14) can be obtained from Equation (15) by adding back book depreciation and the change in deferred tax expense.

(16) ([ye.sup.-[lambda]u] - [beta][BETA][e.sup.-[beta]u])(1 - t) + [beta][BETA][e.sup.-[beta]u] + t([delta][Ve.sup.-[delta]u] - [beta][BETA][e.sup.-[beta]u]).

The change in deferred tax expense on date u reflects the difference between tax and book depreciation expense on that date, multiplied by the tax rate. If an analyst (of accounting researcher) considered only the change in the DTL on date u, t([delta][Ve.sup.-[delta]u] - [beta][BETA][e.sup.-[beta]u]), then the change appears to be valuation relevant because it appears in Equation (16). We regard this result as illusory because all of the terms in Equation (16) involving book depreciation ([beta][BETA][e.sup.-[beta]u]) cancel out, leaving only the terms in Equation (14).

VI. THE POSITIVE NPV CASE

The relation between book value, replacement cost, the DTL, and stock price from Equation (7) is derived under the assumption that investments in depreciable assets are zero net present value investments, so that any tax benefits from rapid tax depreciation are reflected in the replacement cost of the asset. We now relax the zero net present value assumption and allow the replacement cost of the asset to differ from the present value of future cash flows generated by the asset.

First, we consider the stock price (P) on the date a new asset is purchased, where stock price is equal to the present value of future after-tax cash flows generated by the asset, discounted at rate p. The asset generates pre-tax cash flows at rate y. Because the asset is new, both the book value ([B.sub.0]) and tax basis ([V.sub.0]) are equal to the replacement cost ([K.sub.0]).

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We define the excess, if any, of P over [K.sub.0] on the acquisition date as X, so that X = P - [K.sub.0], or:

(18) X = y(1 - t)/[lambda] + [rho] + [delta][K.sub.0]t/[delta] + [rho] - [K.sub.0].

In the case of a zero net present value investment, P and K are equal, so X = 0. In cases where the asset's price (K) does not reflect all of the future cash flow benefits of rapid tax depreciation, X will be positive.

On a future date u*, the stock price P(u*) is:

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using the definition of X from Equation (18) and the facts that DTL = t[B(u) - V(u)], [K.sub.0][e.sup.-[beta]u] = [BETA](u), and [V.sub.0][e.sup.-[delta]u] = V(u) allows Equation (19) to be rewritten as:

(20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus the value of the DTL from Equation (20) is consistent with the DTL value from both Sansing (1998) and AKW, even after relaxing the zero net present value assumption. Equation (20) demonstrates that whether the tax benefits from rapid depreciation are capitalized into asset prices has no bearing on the valuation relevance of the DTL. Regardless of whether some future cash flow is reflected in the original purchase price ([K.sub.0]) or excess value (X), it decays in value at the rate of economic depreciation X. The relation between DTL and firm value does not depend on how the original stock price Pis divided between K and X.

VII. CONCLUSION

In this paper, we investigate the relation between two attributes of the DTL that arises from book-tax differences in depreciation--its effect on the value of the firm and how quickly it begins to reverse. We find that the DTL reverses more quickly when either the tax depreciation rate or the book depreciation rate increases. However, while increasing the tax depreciation rate increases the value of the DTL itself, increasing the book depreciation rate has the opposite effect. In addition, an increase in the discount rate decreases the value of the DTL, but has no effect on the reversal rate. We conclude that an increase in the rate of reversal is neither necessary nor sufficient for the value of the DTL to increase. We also find that the value of the DTL is not equal to the present value of future changes in the DTL because the value of the DTL only depends on cash flows associated with tax depreciation, whereas the changes in the DTL depend on both tax and book depreciation. These results do not depend on whether the asset purchase is a zero or positive NPV investment.

Relating our results to financial statement analysis, our results demonstrate that when two firms have identical book values of depreciation-related DTLs, the values of the DTLs depend only on the tax depreciation rate and the discount rate, and are not related to the time at which the DTLs reverse. The paper thus helps clarify an important financial accounting issue and should be useful to academics teaching financial statement analysis or other valuation courses, as well as to standard setters considering the partial allocation method of recording deferred taxes.

APPENDIX

Proof of Proposition 1

Normalizing the original cost of the asset to one implies that V(u) = [e.sup.-[delta]u] and [BETA](u) = [e.sup.-[beta]u] which implies:

DTL = t[[e.sup.-[beta]u] - [e.sup.-[delta]u].

In addition, let [theta] = [delta]/[delta] + [rho]).

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof of Proposition 2

Let u* = ln([delta]/[beta])/([delta] - [beta]).

(A1) (a) [delta]u*/[delta][delta] = [delta] - [beta] - [delta]ln([delta]/[beta])/[delta][([delta] - [beta]).sup.2]

Let x = [delta]/[beta]. The expression (A1) has the same sign as x - 1 - xln(x), which is negative for all x > 1.

(A2) (b) [delta]u*/[delta][beta] = [beta] - [delta] + [beta]ln([delta]/[beta]/[beta][([delta] - [beta]).sup.2]

Let x = [delta]/[beta]. The expression (A2) has the same sign as 1 - x + ln(x), which is negative for all x > 1.

(c) [delta]u* / [delta]p = 0
TABLE 1 Effects of an Increase in a Parameter on the Value of
Different Measures

                        Parameter

                       Rate of Tax    Rate of Book
                       Depreciation   Depreciation   Discount
                       [delta]        [beta]         Rate
   Measures            (Figure 1)     (Figure 2)     [rho]

1. Book value of DTL   increases      decreases      no effect
2. Valuation factor    increases      no effect      decreases
3. Value of DTL (a)    increases      decreases      decreases
4. Reversal date       decreases      decreases      no effect

(a) The value of the DTL in row 3 is the product of the book value from
row 1 and the valuation factor in row 2.


We thank Mike Kirschenheiter, Linda Krull, workshop participants at the 2003 American Accounting Association Annual Meeting, and two anonymous reviewers for helpful comments.

Editor's note: This paper was accepted by Terry Shevlin, Senior Editor.

(1) This result applies only to deferred taxes (such as those related to depreciation) for which the firm does not receive an income tax deduction at the time the deferred tax balance sheet amount reverses. For example, the reversal of deferred tax assets associated with restructuring charges or employee post-retirement benefits occurs at the time the firm makes a cash payment, resulting in a tax deduction. Therefore, the timing of reversals of deferred tax assets associated with restructuring charges or employee post-retirement benefits affects the present value of future cash flows for income taxes, Contrast this with the case of depreciation, where income tax deductions are based on statutory depreciation rates that are unaffected by the time of reversal. This distinction between depreciation-related deferred taxes and other types of deferred taxes is explained more fully in Guenther and Sansing (2000).

(2) AKW refer to the valuation factor [delta]/([delta] + [rho]) as the net present value (NPV) factor. We deliberately use the term "valuation factor" because we consider the term NPV factor misleading for reasons we discuss in Section V.

(3) A similar suggestion made by some financial statement analysis texts is to ignore DTL balances that are expected to grow indefinitely, in effect discounting them to zero.

(4) Gordon and Joos (2004) examine the reporting behavior of U.K. managers under the partial allocation method.

(5) Assume for purposes of this example that the rate of economic depreciation is also equal to the book depreciation rate.

(6) We emphasize that this result holds only for depreciation-related DTLs. Deferred taxes associated with restructuring charges or employee post-retirement benefits will reverse in the year the firm makes cash payments, and the cash payments will result in income tax deductions. For these types of deferred tax assets the timing of the reversal does change the timing of when taxes are paid. This issue is explored in more detail in Guenther and Sansing (2000).

(7) When [alpha] = 0, the firm invests in a single asset at time zero and never reinvests in new assets. Therefore, Equation (10) represents the rate at which the DTL reverses for this single asset.

(8) The reversal rate from Equation (10) has the same properties as the reversal date.

REFERENCES

Amir, E., M. Kirschenheiter, and K. Willard. 1997. The valuation of deferred taxes. Contemporary Accounting Research 14: 597-622.

--, --, and --. 2001. The aggregation and valuation of deferred taxes. Review of Accounting Studies 6: 275-297.

Givoly, D., and C. Hayn. 1992. The valuation of the deferred tax liability: Evidence from the stock market. The Accounting Review 67: 394-410.

Gordon, E., and P. Joos. 2004. Unrecognized deferred taxes: Evidence from the U.K. The Accounting Review 79: 97-124.

Guenther, D., and R. Sansing. 2000. Valuation of the firm in the presence of temporary book-tax differences: The role of deferred tax assets and liabilities. The Accounting Review 75: 1-12.

Sansing, R. 1998. Valuing the deferred tax liability. Journal of Accounting Research 36: 357-364.

David A. Guenther

University of Colorado at Boulder

Richard C. Sansing

Dartmouth College

Tilburg University

Submitted July 2002

Accepted November 2003
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Author:Guenther, David A.; Sansing, Richard C.
Publication:Accounting Review
Date:Apr 1, 2004
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