# Valuation of a firm with a tax loss carryover.

ABSTRACT: This paper examines the effects of a tax loss carryover on the market and book values of a firm's assets. The loss carryover has a direct effect on market value by sheltering future income from tax, and a direct effect on book value due to the recognition of a deferred tax asset. The failure to discount the deferred tax asset to its present value causes the market-to-book ratio of the deferred tax asset to be less than 1. However, positive skewness in the distribution of future taxable income can cause the market-to-book ratio to exceed 1 because the market value depends on the mean level of future tax benefits, while the book value is based on the median level of future tax benefits. The loss carryover also has an indirect effect on firm value in that it induces the firm to exercise its real option to invest early. This reduces firm value before investment takes place and decreases the market-to-book ratio of physical assets after investment takes place.Keywords: deferred taxes; net operating loss carryovers; valuation; real options.

JEL Classification: H25.

INTRODUCTION

A tax loss carryover is a valuable asset for a firm because it shelters some portion of the firm's future income from tax. The financial accounting system reflects a tax loss carryover as a deferred tax asset, perhaps offset by a valuation allowance. This paper characterizes the direct and indirect effects of a firm's tax loss carryover on the market value of its stock and the book value of its assets.

We first examine the direct effect of the loss carryover on firm value and characterize the market-to-book ratio of the deferred tax asset. This ratio can diverge from 1 for four reasons. First, neither the deferred tax asset nor the valuation allowance is discounted to its present value, which decreases the market-to-book ratio. Second, a valuation allowance is not established under generally accepted accounting principles (GAAP) if the probability that some of the loss carryover will expire is less than 50 percent. This also decreases the ratio for those firms without a valuation allowance, but with some positive probability of having a tax loss carryover expire. Third, the market value of the tax loss carryover reflects the mean level of future tax savings associated with the carryover, whereas the net book value reflects the median level of future tax savings. Positive (negative) skewness in the probability distribution of future income increases (decreases) the ratio. Fourth, a positive probability of negative returns before expiration of the loss carryover increases the market-to-book ratio, even for symmetric income distributions. The option to abandon the project in the case of negative returns induces positive skewness into the distribution of realized returns. We also show that the effect of the size of the tax loss carryover on the market-to-book ratio of the deferred tax asset depends on whether the firm has a valuation allowance. The ratio is decreasing in the size of the loss for a firm without a valuation allowance; the ratio is increasing for a firm with a valuation allowance.

Next we examine the indirect effects of a loss carryover on firm value. We consider a setting in which the expenditure that creates the loss carryover also provides the firm with an opportunity to make an investment in physical capacity, so the initial expenditure provides the firm with a real option (Dixit and Pindyck 1994; Amram and Kulatilaka 1999). We show that the greater the loss carryover, the faster the firm makes the investment, which weakly decreases the value of the firm's real option. This decrease is an implicit tax associated with the tax loss carryover (Scholes et al. 2002). This implicit tax manifests itself as a decrease in the market value of the tax loss carryover prior to when the investment in capacity occurs. It also manifests itself as a decrease in the market-to-book ratio of a firm's physical assets after the investment in capacity takes place.

Our study relates to Amir et al. (2001), which argues that the deferred taxes recorded under U.S. GAAP are overstated because they are not discounted to their present value. While we agree that failure to discount causes the book value of the deferred tax asset to exceed the market value of the tax loss carryover, we identify other factors that can cause the book value to be less than the market value. Specifically, we show that when the mean tax benefits exceed the median tax benefits, the market-to-book ratio can exceed 1 by a substantial margin.

Amir et al. (1997), Ayers (1998), and Amir and Sougiannis (1999) empirically investigate the relations between market and book values using linear regression models in which stock price is explained by accounting variables. The regression coefficient on the deferred tax asset from a tax loss carryover represents the market-to-book ratio of the deferred tax asset. Amir et al. (1997), Ayers (1998), and Amir and Sougiannis (1999) each report regression coefficients on either the deferred tax asset or the valuation allowance that exceed 1. If the only difference between book value and market value relates to present value discounting, one would expect the regression coefficient to be between 0 and 1. The results of our model provide possible explanations for why regression coefficients could exceed 1.

Miller and Skinner (1998) and Schrand and Wong (2000) examine the use of the deferred tax asset valuation allowance to manage earnings and do not find evidence of earnings management; therefore, we do not consider earnings management in this study. Instead, we examine a benchmark case in which both the deferred tax asset and valuation allowance comply with the literal requirements of GAAP.

Our study also relates to the literature on the effect of tax loss carryovers on the timing and magnitude of investment decisions. Wielhouwer et al. (2000) examine the effects of a net operating loss (NOL) arising from accelerated tax depreciation on optimal investment decisions. In this paper, we focus on the effects of a NOL created by an initial tax deductible investment (e.g., R&D) on the timing of a subsequent investment in physical capacity.

The second section derives the direct effects of a tax loss carryover on firm value and on the market-to-book ratio of the deferred tax asset. The third section extends our analysis to cases following a merger in which the use of the acquired corporation's loss carryover is limited under Internal Revenue Code (IRC) [section] 382. The fourth section derives the effects of the tax loss carryover on the value of a real option and on the cost of investment once the real option is exercised. The fifth section summarizes the empirical implications of our results, and the last section concludes the paper.

DIRECT EFFECTS OF THE TAX LOSS CARRYOVER

In this section, we derive the market value of the firm's tax loss carryover and the book value of the associated deferred tax asset. We then characterize the market-to-book ratio of the deferred tax asset, net of any valuation allowance.

Model

The firm owns nondepreciable physical assets that will generate income in perpetuity. The stochastic rate of income is denoted Y, and a possible realization is denoted y. We assume that Y is a random variable with a probability density function f(.) and a cumulative density function F(.). We let p denote the probability that Y > 0, so that Y [less than or equal to] 0 with probability 1-p. In the latter case, the firm decides to abandon the project, so that realized income is equal to zero. The fact that the firm has the option to abandon the project if Y [less than or equal to] 0 implies that the market value of the firm is determined by [Y.sup.+] = max{0,Y}. Note that Y is uncertain as of the date that the investment occurs, but is constant in the sense that once Y is realized, it does not vary over time. The assumption of constant future cash flows enhances the tractability of the model without qualitatively changing the results.

The firm faces a tax rate [tau] on its taxable income. The firm also has a tax net operating loss (NOL) carryover L. This loss carryover could arise due to a previous investment (e.g., in R&D) that, along with the physical assets, makes it possible to generate income. In the fourth section, we examine in detail a two-stage investment model. In this section, we simply assume the existence of a NOL and examine its market value and book value. Given a realization y of Y, taxable income is y per unit of time if there is no NOL, and zero otherwise; in the latter case, the NOL decreases at the rate of y per unit of time until it is either fully used, or until it expires on date w.

All after-tax cash flows are distributed to the shareholders as dividends as they are generated. The stock price V is equal to the present value of all future after-tax cash flows, discounted at the interest rate r. In the absence of a loss carryover (L = 0) the present value of future after-tax cash flows for a given y is equal to:

(1) [[integral].sup.[infinity].sub.0]] (1 - t)y[e.sup.-rt]dt = y(1 - t)/r.

Therefore, the firm's stock price [V.sub.0] is:

(2) [V.sub.0] = [[integral].sup.[infinity].sub.0]] y(1 - [tau])/r f(y)dy = E[[Y.sup.+]](1 - [tau])/r.

Value of the Tax Loss Carryover

In the presence of a loss carryover (L > 0), the present value of future after-tax cash flows depends on whether part of the loss carryover will expire unused. We distinguish between two different cases. In the first case, L [less than or equal to] wy, which implies that the NOL is fully used before it expires. The stock price then consists of two parts. The first part is the present value of pretax cash flows earned between dates zero and L/y, at which point the NOL is fully used. The second part is the present value of future after-tax cash flows earned after date L/y. This yields for a given y:

(3) [V.sub.1](y) = [[integral].sup.L/y.sub.0] y[e.sup.-rt]dt + [[integral].sup.[infinity].sub.L/y] y(1 - [tau])[e.sup.- rt]dt

= y(1 - [tau])/r + [tau]y(1 - [e.sup.-rL/y)/r.

In the second case, L > wy, which implies that some of the loss L expires on date w. The stock price in this case also consists of two parts. The first part is the present value of pretax cash flows earned between date zero and w, at which point the NOL expires. The second part is the present value of future after-tax cash flows earned after date w. This yields for a given y:

(4) [V.sub.2](y) = [[integral].sup.w.sub.0]] y[e.sup.-rt]dt + [[integral].sup.[infinity].sub.w]] y(1 - [tau])[e.sup.- rt]dt

= y(1 - [tau])/r + [tau]y(1 - [e.sup.-rw)]/r

The value of the loss carryover on date zero reflects the possibility that some of the NOL carryover will expire on date w (y < L/w), and the possibility that all of the NOL carryover will be used (y [greater than or equal to] L/w). Therefore, the value reflects an average of [V.sub.2], the value when y < L/w, and [V.sub.1], the value when y [less than or equal to] L/w. Because the market value of the firm is equal to zero when y [less than or equal to] 0, we have:

(5) V = [[integral].sup.L/w.sub.0]] [V.sub.2](y)f(y)dy + [[integral].sup.[infinity].sub.L/w]] [V.sub.1](y)f(y)dy.

Substituting in the values of [V.sub.1] and [V.sub.2] from Equations (3) and (4) into Equation (5) and subtracting the value of [V.sub.0] from Equation (2) (the value of the firm when L = 0) yields the value of a firm's tax loss carryover, denoted VCF.

(6) VCF = [tau]/r [[integral].sup.L/w.sub.0]] y(1 - [e.sup.-rw])f(y)dy + [[integral].sup.[infinity].sub.L/w]] y(1 - [e.sup.- rL/y])f(y)dy.

Deferred Tax Asset and Valuation Allowance

We now consider how L is reflected on the firm's balance sheet. The firm pays zero tax when the loss is incurred and, assuming the loss cannot be carried back, records a deferred tax asset (DTA) equal to [tau]L, less any valuation allowance under SFAS No. 109 (FASB 1992). Because NOLs can only be carried forward a limited number of years (IRC [section] 172(b)(1)), SFAS No. 109 requires that a valuation allowance be established under certain circumstances. Paragraph 96 reads as follows:

The Board believes that the criterion required for measurement of a deferred tax asset should be one that produces accounting results that come closest to the expected outcome, that is, realization or nonrealization of the deferred tax asset in future years. For that reason, the Board selected more likely than not as the criterion for measurement of a deferred tax asset. Based on that criterion, (a) recognition of a deferred tax asset that is expected to be realized is required, and (b) recognition of a deferred tax asset that is not expected to be realized is prohibited.

Paragraph 97 reads in part:

The Board intends more likely than not to mean a level of likelihood that is more than 50 percent.

Paragraph 98 reads in part:

The Board acknowledges that future realization of a tax benefit sometimes will be expected for a portion but not all of a deferred tax asset, and that the dividing line between the two portions may be unclear. In those circumstances, application of judgment based on a careful assessment of all available evidence is required to determine the portion of a deferred tax asset for which it is more likely than not a tax benefit will not be realized.

The median [y.sup.*] of the stochastic rate of income Y is defined to be the solution to:

(7) F([y.sup.*]) = 1/2.

Because a valuation allowance is required if there is a greater than 50 percent probability that some of the loss L will not yield a future tax benefit, a valuation allowance must be established if L > w[y.sup.*], and cannot be established otherwise. In case [y.sup*] [less than or equal to] 0, the book value of the NOL is equal to zero because the valuation allowance is equal to the deferred tax asset. Therefore, we focus on the case where [y.sup.*] > 0, or equivalently, p > 1/2. The valuation allowance is:

(8) VA = max{0,[tau] (L - w[y.sup.*])}.

Market-to-Book Ratio

Next, we derive the ratio of the market value of the net operating loss carryover and the book value of the NOL. We consider two types of firms that differ in the size of their NOL and in the time it takes before the NOL expires. A type-A firm is one that has not recognized a valuation allowance because [L.sub.A] [less than or equal to] [w.sub.A][y.sup.*], so VA = 0. A type-B firm has recognized a valuation allowance because [L.sub.B] > [w.sub.B][y.sup.*], so VA > 0.

Let [[beta].sub.A] denote the ratio of the market value of the loss carryover, VCF, to its book value for a type A firm.

(9) [[beta].sub.A] VCF/DTA.

For a type B-firm, the market-to-book ratio equals:

(10) [[beta].sub.B] = VCF/DTA - VA

These ratios are determined by two main effects: discounting and uncertainty. In order to isolate the discounting effect, we first consider the situation in which Y is a known constant. The market-to-book ratio then only reflects time value of money considerations.

Proposition 1: When Y is known with certainty to be y, then;

(11) 0 < [[beta].sub.A] = y(1 - [e.sup.-rLA/y])/r[L.sub.A] < 1, and

(12) 0 < [[beta].sub.B] = (1 - [e.sup.-rwB])/r[w.sub.B] < 1

Furthermore, for firms of type A (y[w.sub.A] [greater than or equal to] [L.sub.A]) and B (y[w.sub.B] < [L.sub.B]) with identical pretax cash flows y:

[[beta].sub.A] > [[beta].sub.B] iff [L.sub.A]/y < [w.sub.B].

All proofs are in the appendix.

Under certainty, the future tax savings associated with L is equal to the book value of the deferred tax asset. Therefore, the term [[beta].sub.A] diverges from 1 only because of time value of money considerations. The factors that cause [[beta].sub.A] to diverge from 1 are the length of time it takes to realize the tax benefits (L/y) and the opportunity cost to the firm of delaying the realization of the tax benefits (r). As either the length of time it takes to realize the benefits or the interest rate approaches zero, [[beta].sub.A] approaches 1. As was the case of [[beta].sub.A], [[beta].sub.B] is between zero and 1 because the coefficient only reflects time value of money considerations. In this case, the length of time it takes to use the tax loss L reflects the remaining carryover period w instead of L/y. Proposition 1 shows that the market-to-book ratio of a firm's NOL carryover depends on the length of time that the loss carryover shelters the firm's income from tax. A firm that fully uses its loss carryover does so by date L/y, while a firm that loses part of its loss carryover uses losses until date w. The longer it takes a firm to use the loss carryover, the lower the market-to-book ratio of that loss carryover.

Example 1

Suppose Y = y = 2, [L.sub.A] = [L.sub.B] = 30, [w.sub.A] = 20, and [w.sub.B] = 10. In this case, DTA = 30[tau], firm A has no valuation allowance, and VA = 10[tau] for firm B. [[beta].sub.A] < [[beta].sub.B] because firm B uses its asset faster than does firm A (in 10 years for B as opposed to 15 years for A). Alternatively, suppose Y = y = 2, w = 20, [L.sub.A] = 30, and [L.sub.B] = 50. Then [[beta].sub.A] > [[beta].sub.B] because firm A uses its loss carryover for 15 years, whereas firm B uses its loss carryover for 20 years.

Therefore, if firms A and B have the same L but some of firm B's loss expires unused because it has a shorter carryover period w over which the NOL can be used, the market-to-book ratio of firm B is higher than that of firm A. In contrast, if firms A and B have the same w, but some of the NOL of firm B expires because it has a greater loss carryover, then the market-to-book ratio of firm A is higher. Therefore, the effect of the expiration of some of the NOL on the market-to-book ratio is theoretically ambiguous in the certainty case.

We now consider the effect of uncertainty. Substituting DTA = [tau]L and VCF from Equation (6) yields:

(13) [[beta].sub.A] = [[integral].sup.L/w.sub.0]] y(1 - [e.sup.-rw])f(y)dy + [[integral].sup.[infinity].sub.L/w]] y (1 - [e.sup.-rL/y]) f (y)dy/rL

The difference compared to [[beta].sub.A] from Proposition 1 is that Y is a random variable instead of a constant. Whereas [[beta].sub.A] only reflected time value of money considerations in the certainty case, [[beta].sub.A] in Equation (13) reflects both time value considerations and the possibility that part of the loss L expires unused. To quantify the effect of an expiring loss on [[beta].sub.A], we derive upper and lower bounds on [[beta].sub.A] in absence of time value considerations, i.e., when the interest rate r equals zero.

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The lower and upper bounds of [[beta].sub.A] reflect the valuation allowance rules of SFAS No. 109. The probability of losing a portion of a firm's loss carryover can be as low as zero percent or as high as 50 percent without recognizing a valuation allowance. When L is sufficiently small, the probability that the tax benefit associated with L is fully used is close to p, and so [[beta].sub.A] is close to p when L is close to zero. As L increases, [[beta].sub.A] falls because the probability that some of the loss L will expire unused grows. This probability can be as high as 50 percent without recognizing a valuation allowance. This result shows that even in the absence of discounting, the book value of the deferred tax asset can exceed its market value.

Next, we consider the market-to-book ratio for a type-B firm, for which VA = [tau](L - w[y.sup.*]). Because VA > 0 for a type-B firm, [[beta].sub.B] = VCF/(DTA - VA). Substituting DTA = [tau]L, VA = [tau](L - w[y.sup.*]) and VCF from Equation (6) yields:

(14) [[beta].sub.B] = [[integral].sup.L/w.sub.0]] y(1 - [e.sup.-rw] dy + [[integral].sup.[infinity].sub.L/w] y(1 - [e.sup.-rL/y])f(y)dy/rw[y.sup.*]

As was the case with [[beta].sub.A], [[beta].sub.B] reflects both time value of money considerations and the difference between the expected future tax savings and the book value of the tax loss carryover. To quantify this difference, we again determine the upper and lower bounds of [[beta].sub.B] when r = 0.

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As L grows sufficiently large, the probability that some of the loss will expire converges to 1. As that happens, the market-to-book ratio [[beta].sub.B] converges to E[[Y.sup.+]]/[y.sup.*], which is the ratio of the expected level of future tax benefits ([tau]w [[integral].sup.[infinity].sub.0]][yf(y)dy = [tau]wE[[Y.sup.+]]) to the amount of future tax benefits that are reflected on the balance sheet (DTA - VA = [tau]w[y.sup.*]). The VA may overstate the expected unused portion of the loss because VA reflects the median unused loss while the stock price reflects the mean unused loss. Therefore, [[beta].sub.B] may exceed 1. The fact that the coefficient can become greater than 1 is illustrated in the following example, where f(y) has positive skew and there is zero probability on negative income.

Example 2

Let Y be lognormally distributed with a location parameter [mu] and dispersion parameter [[sigma].sup.2]] Then E[Y] = [e.sup.[mu]+[sigma]2/2], [y.sup.*] = [e.sup.[mu]], and E[Y]/[y.sup.*] = [e.sup.[sigma]2/2]. Because an increase in [[sigma].sup.2] increases E[Y] but not [y.sup.*], E[Y]/[y.sup.*] could exceed 1 by a substantial margin.

The following example shows that the market-to-book ratio can exceed 1 even when the underlying income distribution is symmetric. This is due to the fact that the market value reflects the expected income if positive, which is different from the expected income without accounting for the option to abandon the project.

Example 3

Let Y ~ N([mu],[sigma].sup.2]). Then [y.sup.*] = [mu] and E[[Y.sup.+]] > E[Y] = [mu]. In particular, E[[Y.sup.+]] converges to 1/[square root of (term)]2[pi] as [mu] converges to zero. In that case, the market-to-book ratio can become arbitrarily large.

Finally, the following example illustrates the market-to-book ratio for a discrete probability distribution. (1)

Example 4

A firm has a tax loss carryover L with a remaining life of 5 years (w = 5). Its future annual income prospects are 10 with probability 49 percent, 50 with probability 2 percent and 1,000 with probability 49 percent. These assumptions imply that [y.sup.*] = 50 and E[Y] = 496. A valuation allowance is recognized when L > 250 and the market-to-book ratio becomes:

[[beta].sub.B] = 0.49 x 50 + 0.02 x 250 + 0.49 x L /250 = 29.5 + 0.49 x L/250.

For low values of L (L [less than or equal to] 450), the expected use is lower than the median use; the opposite holds for higher values of L. This implies that depending on L, [[beta].sub.B] can vary between 0.608 and 9.918.

Note that because the ratio E[[Y.sup.+]]/[y.sup.*] can become substantially larger than 1, the market-to-book ratio can also exceed 1 for positive r. When E[[Y.sup.+]] > [y.sup.*] and r > 0, the market-to-book ratio could be either greater than or less than 1, depending on the relative sizes of the two effects.

Next, we examine the effects of the parameter L on the market-to-book ratios [[beta].sub.A] and [[beta].sub.B]. As before, we focus on the special case in which r = 0 so as to distinguish between the effects of present value discounting from the differences between the book value of the deferred tax asset and the expected future tax savings associated with that asset.

Proposition 4: When r = 0:

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Proposition 4 shows that the relation between the NOL carryover L and the market-to-book ratio [beta] is not monotone. When L is close to zero, the probability that it will yield a tax benefit of [tau]L is close to p, so the market-to-book ratio is close to p. As L increases, both the market and book values increase; however, the market value grows more slowly because the probability that some of the loss will expire unused increases with L; this causes the market-to-book ratio to decline when 0 < L < w[y.sup.*]. When L [greater than or equal to] w[y.sup.*], the market value continues to increase with L, while the book value remains at [tau]w[y.sup.*]; this causes the market-to-book ratio to increase as L increases. As L becomes arbitrarily large, the market-to-book ratio converges to the ratio of the mean future tax savings to the median future tax savings.

EFFECTS OF THE [section]382 LIMITATION

Section 382 of the Internal Revenue Code limits the use of the tax loss carryover of a corporation that is acquired in a merger or stock purchase. The annual limitation is the product of the value of the acquired corporation and the long-term tax-exempt interest rate (IRC [section]382(b)(1)). In this section, we examine the effects of the [section]382 limitation on the market-to-book ratio of the NOL carryover.

We let the parameter [pi] denote the maximum amount of loss carryover that can be used per unit of time under [section]382. This implies that the amount of the loss that can be used per unit of time equals:

(15) Z = min{[tau], Y}.

Valuation

We first consider the value of the loss carryover. There are two cases to consider. First, when [tau]w < L, some part of the loss will expire unused at date w, because the maximum amount of loss that can be used equals min{[tau],Y}w [less than or equal to] [tau]w < L Equation (6) and the fact that the amount of the loss used per unit of time equals [pi] when y > [pi] implies that:

(16) VCF =[tau]/r [[integral].sup.[tau].sub.0]] y(1 - [e.sup.-rw])f(y)dy + [[integral].sup.[infinity].sub.[tau] [pi](1 - [e.sup.-rw]) f(y)dy.

Second, when [tau]w [greater than or equal to] L, the level of income y will determine whether some of the loss will expire unused. When y < L/w, part of the loss will expire unused at date w. When L/w < y < [pi], all the loss will be used by date L/y. In both cases the amount of loss that is used per unit of time equals y. Finally, when y > [pi], all the loss will be used; but this will only happen at date L/[pi] > L/y because the amount that can be used per unit of time equals [pi] < y due to the [section]382 limitation. Therefore, as in Equations (6) and (16), we find the following expression for the market value of the NOL carryover:

(17) VCF = [tau]/r [[integral].sup.L/w.sub.0]] y(1 - [e.sup.-rw]) f(y)dy + [tau/r [[integral].sup.[pi].sub.L/w]] y(1 - [e.sup.-rL/y])f(y)dy + [tau]/r [[integral].sup.[infinity].sub.[pi]] [pi](1 - [e.sup.-rL/[pi]])f(y)dy.

Valuation Allowance

Because the amount of the loss that can be used per unit of time is the stochastic variable Z = min{[pi] Y}, SFAS No. 109 implies that the valuation allowance equals:

(18) VA = max{0, [tau](L - w[z.sup.*])},

where [z.sup.*] denotes the median of Z. This in turn implies [z.sup.*] = min{[pi], [y.sup.*]}. Therefore, it follows that:

VA = max{0, [tau](L - w[y.sup.*])} if [y.sup.*] < [pi] = max{0, [tau](L - w[pi]} if [y.sup.*] [greater than or equal to] [pi],

so that the valuation allowance is not affected by [section]382 as long as either [pi] [greater than or equal to] [y.sup.*] or w[pi] [greater than or equal to] L. If [pi] < [y.sup.*] and w[pi] < L, then the [section]382 limitation changes the book value of the deferred tax asset by increasing the valuation allowance.

Market-to-Book Ratio

The effect of a [section]382 limitation on the market-to-book ratio depends on whether the limitation changes the valuation allowance. If it does not, then the limitation decreases the market value of the carryover without decreasing its book value, which causes the market-to-book ratio to decrease.

Proposition 5: If either [pi] [greater than or equal to] [y.sup.*] or [pi] [greater than or equal to] L/w, the [section]382 limitation decreases the market-to-book ratio.

Next, we consider the case in which the [section]382 limitation affects both the market value and the book value of the loss carryover, which occurs when [pi] < [y.sup.*] and [pi] < L/w. In that case, the net book value of the loss carryover is [tau]w[pi], and the market-to-book ratio ([[beta].sub.c]) is:

(19) [[beta].sub.C] = [[integral] .sup.[pi].sub.0]] y(1 - [e.sup.-rw])f(y)dy + [[integral].sup.[infinity].sub.[pi]] [pi](1 - [e.sup.-rw]) f(y)dy/rw[pi]

Recall from Equations (9) and (10) that [[beta].sub.A] and [[beta].sub.B] denote the market-to-book ratios of type A and B firms, respectively, in the absence of the [section]382 limitation. The following proposition deals with the market-to-book ratio when the [section]382 limitation reduces the net book value of the loss carryover by increasing the valuation allowance VA.

Proposition 6: If [pi] < L/w and [pi] < [y.sup.*], so that the [section]382 limitation affects both the market value and the valuation allowance, then there exists a [pi] [less than or equal to] min{L/w, [y.sup.*]} such that:

(a) The [section]382 limitation increases the market-to-book ratio if:

[[beta].sub.A]([[beta].sub.B]) < 1 - [e.sup.-rw]/rw p and [pi]<[pi]

(b) The [section]382 limitation decreases the market-to-book ratio if:

[[beta].sub.A]([[beta].sub.B]) [greater than or equal to] 1 - [e.sup.-rw]/rw p and [pi]<[pi].

Proposition 6 implies that the [section]382 limitation can either increase or decrease the market-to-book ratio of the NOL when it affects the valuation allowance. When without the [section]382 limitation the market-to-book ratio is high (>((1 - [e.sup.-rw]/rw)p), the reduction in market value due to the [section]382 limitation dominates the reduction in book value. When without the [section]382 limitation the market-to-book ratio is moderate (<((1 - [e.sup.-rw])/rw)p < 1), the [section]382 limitation increases the market-to-book ratio for low values of [pi] and decreases the market-to-book ratio for higher values of [pi].

Our analysis shows the effects of both discounting and uncertainty. When eliminating discounting by setting r = 0, we find that when the limitation creates a valuation allowance for a type A firm, then the [section]382 limitation always weakly increases the market-to-book ratio. This is a consequence of the fact that if r = 0, then [[[beta].sub.A] < p and [pi] = min{L/w, [y.sup.*]}. In contrast, when r > 0, [pi] < min{L/w, [y.sup.*]}, so that a decrease occurs for sufficiently high values of [pi].

INDIRECT EFFECTS OF A TAX LOSS CARRYOVER

In this section, we examine the indirect effects of the tax loss carryover on market and book values. The carryover affects the timing of the firm's investment decision, which in turn affects the cost of the investment when it is actually made.

Model

A firm invests an amount L on date zero. This investment provides the firm with an opportunity to invest K[e.sup.-[alpha][Tau]] on a subsequent date [T.sup.2]. The second investment generates income in perpetuity. The first expenditure can be thought of as an investment in intellectual property that provides the opportunity for the second expenditure, which can be thought of as an investment in capacity. One example of a two-stage investment is an R&D expenditure in the first stage that yields a patent, followed by the construction of physical plant to produce and sell a product using the patent. A second example is an investment in mineral exploration in the first stage, followed by mineral extraction in the second stage? The parameter [alpha] [greater than or equal to 0 represents the rate at which the cost of the investment in capacity is decreasing over time due to improvements in production technology. Because the cost of physical capacity is weakly decreasing over time, the optimal way to exploit the intellectual property acquired in stage one may be to wait before making the second stage investment.

Effect of a Tax Loss Carryover on the Value of a Real Option

We consider first the optimal investment date [T.sup.*.sub.0] for a firm without a NOL carryover. The firm chooses the investment date T that maximizes the value [U.sub.0] of the real option on date zero.

(20) [U.sub.o] = [e.sup.-rT]{[[integral].sup.[infinity].sub.0]] [[integral].sup.[infinity].sub.T]] y(1 - [tau]) [e.sup.-r(t -T]) f(y)dydt - K[e.sup.-[alpha]T]}

[e.sup.-rT]{[[integral].sup.[infinity].sub.T] E[Y.sup.+]](1 - [tau]) [e.sup.r](t - T) dt - K[e.sup.-[alpha]T}.

Differentiation yields the following investment strategy [T.sup.*.sup.0]:

(21) [T.sup.*.sub.0] = max 0, 1/[alpha] log [K([alpha] + r)/E[y.sup.+] (1 - [tau])]}.

Next, we consider the investment strategy of a firm with the same investment opportunity, but with a tax loss carryover of L that can only be used to offset the income generated by the physical investment. Firm value then consists of two components, the real option to invest and the tax loss carryover. Whether part of the loss will expire unused depends on the firm's choice of the investment date T and the realized income y. If the physical investment occurs on date T < w, whether the loss expires depends on the realization of y. If income y is such that T + L/y [less than or equal to] w, then the firm will be able to use all of its loss, and the value of the firm on date zero is given by:

(22) [U.sub.1](y) = [e.sup.-rT] {[[integral].sup.T+L/y.sub.T]] y[e.sup.-r(t-T]) dt + [[integral].sup.[infinity].sub.T + L/y]] y (1 - [tau]) [e.sup.-r(t - T)] dt - K[e.sup.-aT]]}.

If income y is such that T + L/y > w, part of the firm's loss carryover will expire on date w, then its value is given by:

(23) [U.sub.2](y) = [e.sup.-rT] {[[integral].sup.w.sub.T]] y[e.sup.-r(t-T]) dt + [[integral].sup.[infinity].sub.w]] y (1 - [tau]) [e.sup.-r(t - T)] dt - K[e.sup.-[alpha]T]]}.

The value of the firm reflects a weighted average of [U.sub.1] and [U.sub.2]:

(24) U = [integral].sup.L/(w - T).sub.0] [U.sub.2](y)f(y)dy + [[integral].sup.[infinity].sub.L/(w - T) [U.sub.1](y)f(y)dy.

If the firm chooses an investment date T [greater than or equal to] w, then all of the loss will expire at date w, so the value of the firm equals [U.sub.0], i.e., the value of the real option for a firm without an NOL carryover. The firm chooses the optimal investment date [T.sup.*] in order to maximize its market value. It can be seen easily that K[e.sup.-[alpha]w] < (E[Y.sup.+] (1 - [tau]))/([alpha] + r) implies that [U.sub.0] is decreasing in T for T [greater than or equal to] w, which implies that [T.sup.*] < w. Therefore, we assume that K[e.sup.-[alpha]w] < (E[Y.sup.+](1 - [tau]))/([alpha] + r). The resulting optimal investment date affects the probability that the firm will lose part of its loss carryover.

In the following proposition we show how the investment date is affected by the NOL carryover.

Proposition 7: The optimal investment time [T.sup.*] is decreasing in L.

The intuition behind Proposition 7 is that the cost of waiting is increasing in L. All firms incur the opportunity cost of postponing the benefits from the investment by waiting, which is offset by the lower investment cost associated with waiting. In addition, however, firms with L > 0 face an additional cost of waiting because waiting postpones the use of the tax loss carryover. Therefore, firms with loss carryovers exercise their real options earlier because they have higher waiting costs, which in turn implies that the value of the real option from Equation (20) (i.e., the value of the real option excluding the value of the NOL carryover) is lower for a firm with an NOL carryover.

Effect of a Tax Loss Carryover on Investment Costs

A second indirect effect of the NOL carryover is on the market-to-book ratio of the firm's physical assets once the investment in capacity takes place on date [T.sup.*]. The book value of the assets is K[e.sup.-[alpha]T.sup.*], and so the market-to-book ratio of the physical asset on the investment date [T.sup.*] is:

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The market-to-book ratio of the physical asset is an increasing function of [T.sup.*], and so Proposition 7 implies that the market-to-book ratio of the physical assets will be highest for firms without a tax loss carryover and lower for firms with a tax loss carryover. Therefore, there is a tax-induced difference in the relation between the market and book values of the firm's physical assets depending on whether a firm had a tax loss carryover when the investment was made.

EMPIRICAL IMPLICATIONS

In this section, we summarize the empirical implications of our model. We consider first the implications that relate to the deferred tax asset and valuation allowance, then we address the indirect effects of the tax loss carryover.

Deferred Tax Asset and Valuation Allowance

The market-to-book ratio of the net deferred tax asset corresponds to the coefficient on the deferred tax asset that one would expect in a regression that associates stock price with accounting balance sheet items. We show that the properties of the market-to-book ratio depend on whether a valuation allowance is associated with the deferred tax asset. If there is no valuation allowance, then the ratio is less than 1 because (1) the deferred tax asset is not discounted to its present value and (2) there is a possibility that part of the loss carryover will expire unused. In addition, the ratio is decreasing in the size of the loss carryover.

When there is a valuation allowance, none of these relations necessarily hold. First, the market-to-book ratio could exceed 1 because the market value reflects the mean future tax savings while the book value reflects the median future tax savings. Positive skewness in the distribution of future taxable income can cause the regression coefficient on the deferred tax asset to be greater than 1. Note that even if the underlying distribution is symmetric, the distribution of future taxable income is positively skewed because the firm will exercise its option to abandon the project when y < 0. Second, an increase in the loss carryover increases the market-to-book ratio because an increase in the loss increases both the gross deferred tax asset and the valuation allowance. A coefficient in excess of 1 is likely for start-up firms, because these more often have a highly skewed distribution.

A [section]382 limitation on the use of its tax loss carryover decreases the market value of the carryover and may decrease the book value by increasing the valuation allowance. This implies an ambiguous theoretical relation between the market-to-book ratio and the presence of a [section]382 limitation. If either (1) the limitation does not change the valuation allowance, (2) the market-to-book ratio was already high, or (3) the maximum amount of loss carryover that can be used per unit of time under [section]382 is high, then the limitation decreases the market-to-book ratio. If the limitation increases the valuation allowance, then the market-to-book ratio was already low, and the maximum amount that can be used under the limitation is low, then the limitation increases the market-to-book ratio.

Indirect Effects

The fourth section of this paper showed that the real option to invest held by a firm with a loss carryover is worth less than the real option of a firm without a loss carryover because the loss carryover induces the firm to exercise its option early. This implies that the valuation effects of a tax loss carryover are not limited to the reduction in future taxes; to the extent the tax loss carryover distorts the firm's investment decisions, there is a countervailing implicit tax effect that reduces firm value.

The fourth section also showed how a tax loss carryover affects the market-to-book ratio of a firm's physical assets. Because a firm with a loss carryover will exercise its real option earlier than a firm without a loss carryover, we expect the loss firm to have lower market-to-book ratios with respect to its assets (other than deferred tax assets) than a similar firm without a loss carryover.

CONCLUSIONS

This paper examines the direct and indirect effects of a tax loss carryover on a firm's value. It also examines how these effects are reflected in the market-to-book ratios of the firm's assets.

A tax loss carryover has a direct effect on firm value by sheltering some portion of the firm's future income from tax. The carryover is recognized by the financial accounting system as a deferred tax asset, offset by a valuation allowance if the probability that part of the loss will go unused exceeds 50 percent. If the firm does not recognize a valuation allowance, then the market-to-book ratio of the deferred tax asset is less than 1. However, this ratio can exceed 1 when a valuation allowance is present because the market value of the loss depends on the mean future tax savings, whereas the book value of the deferred tax asset depends on the median future tax savings. This provides a possible explanation for why regression coefficients on deferred tax assets associated with tax loss carryovers sometimes exceed 1.

The loss carryover also has an indirect effect on firm value when the firm has a real option to invest. A firm with a loss carryover has a higher opportunity cost of waiting to invest and thus exercises its real option sooner than a firm without a loss carryover. Prior to the date of the investment, this indirect effect will manifest itself as a decrease in the value of the firm's real options. Following the investment, it will manifest itself as a lower market-to-book ratio of the firm's physical assets.

APPENDIX

Proof of Proposition 1

When Y is known to be y, a type-A firm is one for which y [greater than or equal to] L/w and a type-B firm is one for which y < L/w. Equations (6), (8), and (9) imply that [[beta].sub.A] = y(1 - [e.sup.-rL/y])/rL; Equations (6), (8), and (10) imply that 138 = (1 - [e.sup.-rw])/rw. Furthermore, the definitions of type-A and type-B firms imply that ([L.sub.A]/[w.sub.A]) [less than or equal to] y < ([L.sub.B]/[w.sub.B]). The ratios [[beta].sub.A]a and [[beta].sub.B] are both of the form (1 - [e.sup.-z])/z, where z = r[L.sub.A]/y for firm A and z = r[w.sub.B] for firm B. The expression (1 - [e.sup.-z])/z is decreasing in z, which implies that [[beta].sub.A] > [[beta].sub.B] if and only if [L.sub.A]/y <[w.sub.B].

Finally, we need to prove that the coefficients [[beta].sub.A] and [[beta].sub.B] are between 0 and 1. The function (1 - [e.sup.-z]/z is positive for positive values of z, and less than 1. The latter follows from the fact that [e.sup.-z] > 1 - z.

Proof of Proposition 2

Applying L'Hopital's rule to Equation (13) and evaluating it at r = 0 yields:

(A.1) [[beta].sub.A] = w/L [[integral].sup.L/w.sub.0]] yf(y)dy + [[integral].sup.[infinity].sub.L/w] f(y)dy.

Differentiating [[beta].sub.A] with respect to L indicates that when r = 0, [[beta].sub.A] is decreasing in L, since:

[[diff][beta].sub.A]]/[diff]L = w [[integral].sup.L/w.sub.0] yf(y)dy/[L.sup.2] < 0.

For any type-A firm, 0 [less than or equal to] L [less than or equal to] w[y.sup.*]. Once again applying L'Hopital's rule shows that, when r = 0, [[beta].sub.A] approaches [[integral].sup.[infinity].sub.0] f(y)dy = p as L approaches 0, which yields the upper bound of [[beta].sub.A] = p. Substituting L = w[y.sup.*] into Equation (A.1) yields:

[[beta].sub.A] = [[integral].sup.y*.sub.0]] y/[y.sup.*] f(y)dy + [[integral].sup.[infinity].sub.y*]] f(y)dy.

The first term could be arbitrarily close to 0; the second term equals 1 - F([y.sup.*]) = 1/2, and thus the lower bound of [[beta].sub.A] is 1/2.

Proof of Proposition 3

Applying L'Hopital's rule to Equation (14) and evaluating it at r = 0 yields:

(A.3) [[beta].sub.B] = w[[integral].sup.L/w.sub.0]] yf(y)dy + L [[integral].sup.[infinity].sub.L/w]] f(y)dy/w[y.sup.*]

Differentiating [[beta].sub.B] with respect to L yields:

(A.4) [diff][[beta].sub.B]/[diff]L = [[integral].sup.[infinity].sub.L/w]] f(y)dy/w[y.sup.*] > 0,

so that when r = 0, [[beta].sub.B] is increasing in L. For any type-B firm, L > w[y.sup.*]. Substituting L = w[y.sup.*] into Equation (A.3) yields:

[[beta].sub.B] = [[integral].sup.y*.sub.0]] y/[y.sup.*] f(y)dy + [[integral].sup.[infinity].sub.y*]] f(y)dy.

The first term could be arbitrarily close to 0; the second term equals 1 - F([y.sup.*]) = 1/2, and thus the lower hound of [[beta].sub.B] is 1/2. Applying L'Hopital's rule shows that when r = 0, [[beta].sub.B] converges to [[integral].sup.[infinity].sub.0] y/[y.sup.*] f(y)dy as L approaches infinity, so the upper bound of [[beta].sub.B] is E[[Y.sup.+]]/[y.sup.*].

Proof of Proposition 4

When 0 < L [less than or equal to] w[y.sup.*], VA = 0 and thus the market-to-book ratio is [[beta].sub.A]. When L > w[y.sup.*], VA > 0 and thus the market-to-book ratio is [[beta].sub.B]. The statements (i)-(v) immediately follow from the proofs of Propositions 2 and 3.

Proof of Proposition 5

There are two cases to consider. If [pi] < L/w, then the effect of the [section]382 limitation on VCF is equal to the difference between Equations (6) and (16). Differentiating Equation (16) with respect to [pi] yields:

(A.5) [diff]VCF/[diff][pi] = [tau](1 - [e.sup.-rw]) [[integral].sup.[infinity].sub.[pi]] f(y)dy/r > 0.

Therefore, we need only show that Equation (6) exceeds Equation (16) when [pi] = L/w. Subtracting Equation (16) from Equation (6) and setting [pi] = L/w yields:

(A.6) [[integral].sup.[infinity].sub.L/w][y(1 - [e.sup.-rL/y])/rL - (1 - [e.sup.-rw])/rw]f(y)dy>0.

If [pi] [greater than or equal to] L/w, then the effect of the [section]382 limitation on VCF is equal to the difference between Equations (6) and (17). Differentiating Equation (17) with respect to [pi] yields:

(A.7) [diff]VCF/[diff][pi] = [tau](1 - [e.sup.-rL/[pi] - rL[e.sup.-rL/[pi]]/[pi])[[integral].sup.[infinity].sub.[pi]] f(y)dy/r > 0.

Furthermore, Equation (17) converges to Equation (6) as [pi] approaches infinity, and thus Equation (6) exceeds Equation (17) for all finite values of [pi].

Proof of Proposition 6

First, notice that [[beta].sub.c] is decreasing in [pi] on the interval [0,min{L/w, [y.sup.*]}], since:

(A.8) [diff][[beta].sub.c]/[diff][pi] = - 1/rw[[pi].sup.2] [[integral].sup.[pi].sub.0]] y(1 - [e.sup.-rw])f(y)dy<0.

When [pi] approaches 0, one finds by applying L'Hopital's rule:

(A.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we show that when L/w < [y.sup.*], it holds that [lim.sub.[pi][right arrow]L/w] [[beta].sub.c] < [[beta].sub.A]. For [pi] = L/w one has:

(A.10) [[beta].sub.C] = [[integral].sup.L/w.sub.0]] y/rL (1 - [e.sup.-rw]) f(y)dy + [[integral].sup.[infinity].sub.L/w 1/rw (1 - [e.sup.-rw])f(y)dy,

so that:

(A.11) [[beta].sub.C] - [[beta].sub.A] = [[integral].sup.[infinity].sub.L/w]] (1 - [e.sup.-rw]) f(y)dy - [[integral].sup.[infinity].sub.L/w]] y/rL (1 - [e.sup.-rL/y])f(y)dy[less than or equal to] 0

with a strict inequality for r > 0. Similarly one can show that when [pi] = [y.sup.*] < L/w it holds that [[beta].sub.c] [less than or equal to] [[beta].sub.B], with a strict inequality for r > 0. Therefore, the [section]382 limitation causes the value of [[beta].sub.c] to decrease from an upper bound of ((1 - [e.sup.-rw])/rw)p (when [pi] is close to 0) to a level strictly less than (for r > 0) or equal to (for r = 0) [[beta].sub.A] ([[beta].sub.B]) as [pi] approaches min{L/w, [y.sup.*]}.

Therefore, there exists a [pi] [less than or equal to] min{L/w, [y.sup.*]}, such that [[beta].sub.c] > [[beta].sub.A] ([[beta].sub.B]) for all [pi] < [pi] and [[beta].sub.c] < [[beta].sub.A] ([[beta].sub.B]) for all [pi] > [pi]. For r = 0, it holds that [pi] = min{L/w, [y.sup.*]}.

Proof of Proposition 7

Setting the derivative of U with respect to T equal to 0 yields the following equation:

(A.12) E[Y.sup.+] - [tau] [[integral].sup.[infinity].sub.L/(w-T)]] y[e.sup.-rL/y] f(y)dy = ([alpha] + r) K[e.sup.- [alpha]T].

Let us denote [t.sup.*] for the solution of (A.12). Now since U is strictly concave in T on the interval [0,w], and since K[e.sup.[alpha]w] < (E[[Y.sup.+]](1 - [tau]))/([alpha] + r) implies that [t.sup.*] [less than or equal to] w, it follows that [T.sup.*] = max{0,[t.sup.*]}.

The left-hand side of (A. 12) is increasing in L and increasing in T, and the right-hand side is decreasing in T. Therefore it follows immediately that [t.sup.*] is decreasing in L. Consequently, [T.sup.*] is also decreasing in L.

We thank Shelley Rhoades-Catanach and the 2003 JATA Conference participants for useful comments.

REFERENCES

Amir, E., M. Kirschenheiter, and K. Willard. 1997. The valuation of deferred taxes. Contemporary

Accounting Research 14 (Winter): 597-622.

--, and T. Sougiannis. 1999. Analysts' interpretation of investors' valuation of tax carryovers. Contemporary Accounting Research 16 (Spring): 1-33.

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

Amram, M., and N. Kulatilaka. 1999. Real Options: Managing Strategic Investment in an Uncertain World. Boston, MA: Harvard Business School Press.

Ayers, B. 1998. Deferred tax accounting under SFAS No. 109: An empirical investigation of its incremental value-relevance relative to APB No. 11. The Accounting Review 73 (April): 195-212.

Dixit, A., and R. Pindyck. 1994. Investment under Uncertainty. Princeton, NJ: Princeton University Press.

Financial Accounting Standards Board (FASB). 1992. Accounting for Income Taxes. Statement of Financial Accounting Standards No. 109. Norwalk, CT: FASB.

Miller, G., and D. Skinner. 1998. Determinants of the valuation allowance for deferred tax assets under SFAS No. 109. The Accounting Review 73 (April): 213-233.

Scholes, M., M. Wolfson, M. Erickson, E. Maydew, and T. Shevlin. 2002. Taxes and Business Strategy: A Planning Approach. 2nd edition. Upper Saddle River, NJ: Prentice Hall.

Schrand, C., and M. Wong. 2000. Earnings management and its pricing implications: Evidence from banks' adjustments to the valuation allowance for deferred tax assets under SFAS No. 109. Working paper, University of Pennsylvania.

Wielhouwer, J. L., A. De Waegenaere, and E M. Kort. 2000. Optimal dynamic investment policy for different tax depreciation rates and economic depreciation rates. Journal of Optimization Theory and Applications 106 (l): 23-48.

Anja De Waegenaere is an Associate Professor and Jacco L. Wielhouwer is an Assistant Professor, both at Tilburg University, and Richard C. Sansing is an Associate Professor at Dartmouth College and is a Visiting Professor at Tilburg University.

(1) This example is based on the example presented by Shelley Rhoades-Catanach when discussing our paper at the 2003 JATA Conference.

(2) Alternatively, this setting applies to firms with a real investment option and an existing loss carryover of size L.

(3) There is an extensive literature on real options to invest (Dixit and Pindyck 1994; Amram and Kulatilaka 1999).

DISCUSSION OF Valuation of a Firm with a Tax Loss Carryover

Shelly C. Rhoades-Catanach

De Waegenaere et al. (2003) examine the properties of the market-to-book ratio of deferred tax assets arising from tax loss carryovers. The numerator of this ratio reflects the net present value of future cash flows attributable to the deferred tax asset, while the denominator reflects an undiscounted book value. Lack of discounting in the denominator suggests that the market-to-book ratio would be between 0 and 1. However, prior empirical results report market-to-book ratios greater than 1. This paper attempts to provide a theoretical explanation of conditions under which market-to-book ratios might be greater than 1.

The paper has two important strengths that broaden its audience appeal. First, its consideration of deferred tax assets focuses on the interface between tax and financial accounting. Second, it attempts to provide theory that explains puzzling empirical results and motivates future empirical testing to validate its predictions. However, the link between any theoretical model and empirical observation focuses attention on the underlying assumptions of the model. This discussion examines several assumptions of this model that may render it inappropriate to the empirical setting it attempts to explain.

DEFINING THE VALUATION ALLOWANCE

As defined in the paper, the market-to-book ratio of the tax loss carryforward is:

VCF / DTA - VA

where VCF is the market value of the loss carryforward, DTA is the gross book deferred tax asset, and VA is any valuation allowance required to reduce the book value of the deferred tax asset. When a valuation allowance is needed, VA = [tau](L - wy*), where [tau] is the tax rate, L is the tax loss carryforward, w is the number of years until the loss carryforward expires, and y* is the median of the distribution of possible annual income realizations. The paper focuses on the median in defining VA because SFAS No. 109 requires a valuation allowance if it is "more likely than not" that the deferred tax asset will not be realized. Defining the valuation allowance in this way is critical to the paper, since most of the paper's results depend on the difference between the median value, y*, and E([Y.sup.+]), the mean of the positive values of the income distribution. For skewed distributions or distributions with negative values of y, these two measures can be quite different.

To illustrate the importance of using the median in determining the valuation allowance, as well as the importance of the difference between the mean and median, consider a simple example. Suppose a firm has a tax loss carryforward L = $275 with a remaining life of 5 years. Its future annual income prospects are: y = $10 with 49 percent probability, y = $50 with 2 percent probability, and y = $1,000 with 49 percent probability. Given this income distribution, y* = $50 and E(Y) = $496. The modeling approach used in this paper suggests that a valuation allowance of $25[tau] is needed. In practice, however, do we believe such a valuation allowance will be recorded? If not, why not?

To answer these questions, let's return the sections of SFAS No. 109 cited in the paper. SFAS No. 109, paragraph 96, states in part that "the criterion required for measurement of a deferred tax asset should be one that produces accounting results that come closest to the expected outcome." In determining the amount of any valuation allowance required, SFAS No. 109, paragraph 98, states that "application of judgment based on a careful assessment of all available evidence is required." Note that SFAS No. 109 refers both to expected outcomes and the application of judgment. Yet the model in this paper does not allow for judgment, and focuses on circumstances in which the "expected outcome," i.e., the expected value or the mean, is very different from the median. I would argue that using the median of the income distribution to define the valuation allowance is a theoretically correct application of statistical theory, but has little relation to the practical real-world application of SFAS No. 109. If so, then the model's predictions may be of little use in explaining observed empirical results.

Unfortunately, SFAS No. 109 provides little direct guidance on how to apply the "more likely than not" criteria for recording a valuation allowance. In practice, how do accountants and auditors make these decisions? Since I am not an auditor or financial accountant, I sought such guidance from a Big 4 firm. PricewaterhouseCoopers' (2002) Accounting and Reporting Manual, Income Taxes (ARM 5790, paragraph 34) addresses the determination of a valuation allowance as follows:

Realization Test ... The need for a valuation allowance is a subjective judgment made by management ... The final amount recorded should be based on management's judgment of what is more likely than not based on all available information, quantitative and qualitative.

Again, note the focus on judgment (management's judgment) and on both quantitative and qualitative information.

SFAS No. 109 does provide some guidance on when a tax asset is realizable. Paragraph 21 of SFAS No. 109 states in part:

Future realization of the tax benefit of an existing deductible temporary difference or carryforward ultimately depends on the existence of sufficient taxable income of the appropriate character (for example, ordinary income or capital gain) within the carryback, carryforward period available under the tax law. The following four possible sources of taxable income may be available under the tax law to realize a tax benefit for deductible temporary differences and carryforwards: a. Future reversals of existing taxable temporary differences b. Future taxable income exclusive of reversing temporary differences and carry forwards c. Taxable income in prior carryback year(s) if carryback is permitted under the tax law d. Tax-planning strategies (paragraph 22) that would, if necessary, be implemented to, for example: (1) Accelerate taxable amounts to utilize expiring carryforwards, (2) Change the character of taxable or deductible amounts from ordinary income or loss to capital gain or loss, (3) Switch from tax-exempt to taxable investments.

The model in this paper focuses on part b, future taxable income other than reversing temporary differences. PricewaterhouseCoopers provides guidance on how to schedule out these sources of income, and on their estimation. ARM 5790, paragraph 45, states:

Future Taxable Income Other Than Reversals--Since detailed calculations will usually be undertaken only in connection with assessing the valuation allowance, management must be comfortable that the future taxable income is reasonable in order to conclude that it is not more likely than not that the deferred tax assets will not be realized. Once the schedule reflects the temporary difference reversals in the years they are expected to take place and, if appropriate, future taxable income other than reversals, the amounts for each year should be added up, leaving either net taxable income or a net taxable loss for each future year or possibly a breakeven. The example at ARM 5790.471 depicts this process; the middle portion of the schedule reflects expected future taxable income other than reversals.

The example referred to above creates a schedule, for several future periods, of income from the first two sources referred to in paragraph 21 of SFAS No. 109 (reversing temporary differences and future taxable income other than reversals). To the extent these income sources are sufficient to absorb a loss carryforward prior to its expiration, the ARM concludes that a valuation allowance is not required. In particular, note the final sentence of ARM 5790, paragraph 45, which refers to "expected future taxable income," another focus on expected value. In my earlier example, this approach would indicate that a valuation allowance is not needed.

I am not advocating any particular methodology for determining a valuation allowance or arguing that the model should define the valuation allowance using the mean rather than the median. My concern is simply with the link between the paper's theory and the observed empirical evidence. There are several critical questions that we must consider in order to address this concern:

* Do we believe that, in practice, the valuation allowance is based on the median income prediction, its expected value, or some other amount (i.e., a judgmental value)? * As the difference between the mean and median increases, does the exercise of judgment become more or less important in determining the valuation allowance? * If in practice the valuation allowance is not based on the median, then what explains the observed empirical results?

Clearly, additional research is needed to answer these questions.

OTHER ISSUES AFFECTING THE MODEL'S RELATION TO OBSERVED RESULTS

The assumption that the valuation allowance is based on the median is not, unfortunately, the only weakness I see in relating this model to observed empirical results. The model assumes income is constant over time, once the initial realization occurs. Thus, no uncertainty exists about the need for a valuation allowance after year 0. The model also assumes that income is either positive or the project is abandoned. This setting would seem to apply only in the start-up phase of a new business, yet the empirical results the model is attempting to explain are not limited to such firms. For post-start-up firms, prior year income is considered a strong predictor of future income (see SFAS No. 109, paragraph 24). However, the modeling approach does not consider such a predictor.

OTHER QUESTIONS

In this discussion, I have focused primarily on the model in the second section of the paper. The fourth section considers the impact of a tax loss carryover on firm investment decisions and concludes that firms accelerate investments, reducing their value in the presence of a loss carryover. This model considers the strategic timing of exercise of an investment option, yet ignores the strategic timing of incurring the initial loss carryover (which critically determines w). Shouldn't both decisions be made strategically, so as to maximize overall firm value before the loss is incurred? This approach offers a possible extension of the paper that might yield interesting insights.

Finally, I would like to relay a comment made by members of the 2003 JATA Conference Selection Committee. "The model seems to imply that the market-to-book ratio can be just about anything. What do we really learn from this result? Does it improve decision making? Does it improve understanding of some other observed phenomenon?" These are the types of questions theoretical researchers often get and must consider in order to motivate interest in our work. (While we may feel that analytic research is interesting in and of itself, most non-analytics require some convincing.) I think the paper's attempt to explain empirical results serves to motivate additional, more detailed empirical testing. In particular, the paper describes specific circumstances in which market-to-book ratios will be greater than 1 versus less than 1. These results should permit empirical testing that relates market-to-book ratios to specific firm characteristics. If further empirical analysis produces results consistent with the predictions of the model, then such validation would address many concerns regarding the model's underlying assumptions.

Valuation of a Firm with a Tax Loss Carryover

Anja De Waegenaere, Richard c. Sansing, and Kacco L Wielhouwer

This paper examines the effects of a tax loss carryover on the market and book values of a firm's assets. A firm with a tax loss carryover records a deferred tax asset for financial accounting purposes. The asset is discounted if it is more likely than not that some part of the asset will expire unused. We show that the book value of the deferred tax asset can be either higher than or lower than the effect of the carryover on the market value of the firm. This occurs because the book value (net of the valuation allowance) reflects the undiscounted median future tax savings, whereas the market value reflects the present value of the mean future tax savings.

We also examine the indirect effects of a loss carryover on firm value. We consider a setting in which the expenditure that creates the loss carryover also provides the firm with an opportunity to make an investment in physical capacity, so the initial expenditure provides the firm with a real option. We show that the greater the loss carryover, the faster the firm makes the investment, which weakly decreases the value of the firm's real option. This decrease manifests itself as a decrease in the market value of the tax loss carryover prior to when the investment in capacity occurs. It also manifests itself as a decrease in the market-to-book ratio of a firm's physical assets after the investment in capacity takes place.

REFERENCES

De Waegenaere, A., R. C. Sansing, and J. L. Wielhouwer. 2003. Valuation of a firm with a tax loss carryover. The Journal of the American Taxation Association (Supplement): 65-82.

Financial Accounting Standards Board (FASB). 1992. Accounting for Income Taxes. Statement of Financial Accounting Standards No. 109. Norwalk, CT: FASB.

PricewaterhouseCoopers. 2002. Accounting and Reporting Manual 5790, Income Taxes. September 10, 2002 update. New York, NY: PWC.

Shelley C. Rhoades-Catanach is an Associate Professor at Villanova University.

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Author: | De Waegenaere, Anja; Sansing, Richard C.; Weilhouwer, Jacco L. |
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Publication: | Journal of the American Taxation Association |

Date: | Sep 22, 2003 |

Words: | 11298 |

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