# Effects of depreciation and corporate taxes on asset life under debt-equity financing.

This paper examines the effects of depreciation and corporate taxes
on the optimal life of an asset financed by a combination of debt and
equity. In a single-cycle problem, the optimal life literature purports
to determine how long an asset should be usefully employed before
termination. In a replacement problem, the focus is on determining how
long the asset should be held before being replaced.

In the existing literature, Robichek and Van Horne [20] indicate that an asset will be terminated in the earliest year its salvage value exceeds the net present value, NPV, of its future cash flows. Dyl and Long [5] show that this does not recognize that later termination could be more valuable than that in the earliest year. Hence, they propose that asset termination should take place in the year which provides the highest NPV. Gaumnitz and Emery [7] demonstrate that the optimal replacement time in a like-for-like replacement problem will generally differ from the optimal asset life in a single-cycle problem.(1) Howe and McCabe [10] also analyze the finite- and infinite-cycle replacement problems and indicate that if future asset replications have a zero NPV, the replacement problem analysis reduces to that of the single-cycle problem.

As is typical in asset life literature, the above studies share two assumptions. First, they assume that at time zero the firm precommits itself to holding the asset for the period of time that maximizes its NPV. Second, these studies assume that the asset is all-equity financed.(2)

Under the first assumption, the firm views future cash flows and salvage values as certain. Certainty equivalents or expected values are used if future salvage values and cash flows are uncertain. Recently, Myers and Majd [18] relax this assumption. They indicate that future cash flows and salvage values can deviate from their expected values and the firm has the option to postpone the abandonment decision to a later date. Using numerical approximation, they evaluate the option to abandon as an American put option on a stock with varying dividend yield and uncertain exercise price. In doing so, they assume that asset value follows a longnormal diffusion process with known expected value and variance for its rate of change. They suggest that an asset will be abandoned when its salvage value exceeds the sum of its value without abandonment plus the value of its option to abandon. Nevertheless, they retain the all-equity financing assumption.

The present paper, on the other hand, relaxes the all-equity financing assumption and allows for debt-equity financing, but assumes that the firm decides irrevocably about asset duration at time zero; i.e., it assumes that the value of the abandonment option is zero. As a result, it is demonstrated that optimal asset life increases with the use of riskless debt financing. It is also shown that higher allowable depreciation does not affect optimal life when the asset is totally depreciated before its termination; otherwise, optimal life increases with depreciation. Further, if the tax life of the asset is shorter than its economic life, optimal life increases with the corporate tax rate. However, if the asset is not fully depreciated before its termination, an increase in the corporate tax rate reduces its optimal life if salvage value is less than asset book value; otherwise, it has an ambiguous effect.

The Tax Reform Act of 1986 (TRA) reduces corporate tax rates and, in general, increases the depreciation life of assets (i.e., reduces the depreciation allowance in each ensuing period). As a result, it is shown that for an asset not fully depreciated before disposal, TRA has an ambiguous effect on its optimal life; otherwise, TRA reduces optimal life.

The paper's results are derived in Section I with all major proofs presented in the Appendix. Section II concludes the paper.

I. Optimal Asset Life Under Debt-Equity Financing

The firm invests in an asset and selects [T*, the asset's holding period, that maximizes V(T), the net present value of its after-tax cash flows over the holding period. Upon termination of the asset, the firm is taxed on the difference between salvage value and book value, the latter being equal to the asset's initial cost minus accumulated depreciation. Depreciation is exogenous and is based on the asset's historical cost. There is no investment tax credit and the anticipated rate of inflation is assumed to be zero.(3) The asset's required rate of return, k, is used to discount the project's expected cash flows.

The firm issues riskless debt to partially finance the asset; the remaining balance is financed with equity. Miles and Ezzell [13] and Ezzell and Miles [6], henceforth ME, demonstrate that if the discounting rate for unlevered cash flows r, the corporate tax rate [tau], the cost of debt [r.sub.D] (r > [r.sub.D]), and the market value leverage ratio L are constant over the asset's life, asset value is determined by discounting its unlevered cash flows at the single rate k = r - [[tau]r.sub.D] L [(1 + r)/ (1 + [r.sub.D])]. This rate is independent of the asset's duration and cash flow pattern and equals the textbook formula or weighted average cost of capital, WACC.(4) Defining L in terms of realized market values, implies that the amount of outstanding debt will adjust to follow fluctuations in future asset value unless the investment is riskless. Hence, the investment decision affects both the size and riskiness of future tax shields associated with debt financing.(5) The ME expression for k, however, applies to discrete time discounting only. When adjusted for continuous time, this discounting rate becomes:(6) k=r-[[tau]r.sub.D]L. (1)

Equation (1) assumes corporate taxes only.(7) Under both corporate and personal taxes, Miller [14] shows that the net tax advantage to corporate debt is [1 - (1 - [[tau].sub.pE])(1 - t)/(1 - [[tau].sub.p])], where [[tau].sub.p] and [[tau].sub.pE] are the tax rates investors pay on debt and equity income respectively. If, as in ME, only the first debt tax shield is certain and if, for simplicity, it is assumed that all investors are subject to the same tax rates, then Taggart [23] shows that an asset's value is determined by discounting its expected unlevered cash flows at a single rate, the continuous time equivalent of which is:(8)

[Mathematical Expression Omitted]

where [r.sub.fE] is the cost of risk-free equity. Equation (2) is the generalization of ME's cost of capital in Equation (1) enhanced to include both personal and corporate taxes. Note that Equation (2) utilizes the risk-free equity rate instead of the riskless debt rate used in Equation (1). Also, the tax advantage of corporate debt differs in these two scenarios since only corporate taxes exist in Equation (1), as opposed to both corporate and personal taxes in Equation (2).

Assuming continous discounting, the net present value of an asset held for T periods is given by:

[Mathematical Expression Omitted]

where

X(t) = The expected unlevered operating cash flow occurring continuously in period t of the asset's life;

D(t) = The depreciation allowance occurring continuously in period t of the asset's life;

S(T) = The asset's expected salvage value at time T;

B(T) = The asset's book value at time T; and

I = The asset's initial cost.

From Equation (3), the asset's net present value equals the present value of the unlevered after-tax future cash flows (including depreciation tax shields), plus the present value of the after-tax salvage value, minus the asset's initial cost. Also, Equation (3) assumes that operating cash flows, future salvage value, depreciation tax savings, and tax on the difference between salvage and book value are all being discounted at the same rate; i.e., their perceived riskiness is the same.(9) It should be noted that discounting takes place at the required rate of return adjusted for the effect of debt financing as determined by Equations (1) and (2) under the different debt scenarios. Finally, as indicated in the introduction, Equation (3) assumes that the value of the implicit abandonment option is zero.

The effect of a marginally longer holding period on the asset's net present value is determined by differentiating(10) Equation (3) with respect to T. Since D(T*) = - B'(T*), this gives:

[Mathematical Expression Omitted]

where primes denote partial derivatives with respect to time. The optimal asset life, T*, that maximizes V(T) must satisfy:(11)

Z(T*) = O. (5)

The intuition behind Equation (5) is straightforward. In present value terms, asset life is increased until the instantaneous after-tax cash flows from operations and from the decline in salvage value become equal to the instantaneous return the firm can obtain by terminating the asset and investing the resulting after-tax cash flow at the rate k.

Solving Equation (5) for k, provides an alternative condition for T*, i.e.,

k= (1 - [tau])[X(T*) + S' (T*)]/(1 - [tau])S(T*) + [tau]B(T*). (6)

Intuitively, asset duration is increased until the after-tax instantaneous rate of return the firm obtains by holding the asset equals the asset's after-tax required rate of return.(12)

The effect of a marginal increase in leverage on the asset's optimal life is determined by totally differentiating Equation (5) with respect to L to obtain:

dT*/dL = - dZ(T*)/dL/Z'(T*), (7)

where Z'(T*) < 0 (see footnote 11) and, from Equations (4) and (5),(13)

dZ(T*)/dL = - dk/dL/[(1 -[tau])S(T*) + [tau]B(T*)[e.sup.-kT*]. (8)

Because the sign of dT* /dL in Equation (7) is the same as that of dZ(T*)/dL, it follows from Equation (8) that higher leverage could affect optimal asset life through its effect on the required rate of return. Since k and L are inversely related,(14) higher leverage reduces the instantaneous return the firm obtains by terminating the asset and investing the resulting after-tax cash flows at the rate k. As a result, the loss associated with asset continuation is reduced and optimal asset life increases.(15) Hence, ceteris paribus, firms in industries with higher leverage ratios will be expected to optimally invest in assets with longer lives. Utilities provide an example of such companies.

From Equation (5), Z(T*) depends on the depreciation allowance through the asset's book value and it also depends on the corporate tax rate. Hence, optimal asset life will change in response to exogenous changes in either parameter. The effect of an exogenous increase in allowable depreciation on T* is determined by totally differentiating Equation (5) with respect to D(t) to obtain:

dT*/dD(t) = - dZ(T*)/dD(t)/Z'(T*), (9)

where Z'(T*) < 0 and, since the asset's book value declines with an increase in allowable depreciation,(16)

dZ(T*)/dD(t) = -kt/dB(T*)e-kT* >0/dD(t). (10)

Because the signs of dT*/dD(t) and dZ(T*)/dD(t) are the same in Equation (9), an increase in allowable depreciation results in longer asset life. Essentially, an increase in depreciation reduces the asset's book value at T* and the loss from delaying investment of the book value tax shields, [tau]B(T*), at the rate k is reduced. Hence, asset life increases.

From Equations (9) and (10), dT* /dD(t) depends on leverage. To determine the impact of financial leverage on the increment of optimal asset life caused by higher depreciation, Equation (9) can be differentiated with respect to L to obtain:

[Mathematical Expression Omitted]

It is shown in the Appendix that: (i) [d.sup.2]Z(T*)/dLdD(t) is positive (negative) for "long" ("short") assets, i.e., assets for which T* is greater (less) than 1/k, and (ii) [d.sup.2]Z(T*)/dLdT < 0. It therefore follows from Equation (11) that the increase in an asset's optimal life associated with higher depreciation declines as leverage increases for "short" assets, while it could either increase or decline for "long" assets. Thus, ceteris paribus, lower depreciation reduces the optimal life of "short" assets with lower debt ratios by more than the life of "short" assets with higher debt ratios. For "long" assets, lower depreciation will always reduce optimal asset life, but the impact of a higher debt ratio on this reduction is ambiguous. For example, as depreciation increases, a utility with higher leverage which invests in a "long" asset may or may not postpone its termination by more than a utility with lower leverage.

Next, the effect of an exogenous increase in the corporate tax rate on T* is determined by totally differentiating Equation (5) with respect to [tau] to obtain:

dT*/d[tau] = -dZ(T*)/d[tau]/,Z'(T*), (12) where, as shown in the Appendix:

[Mathematical Expression Omitted]

Since Z'(T*) < 0, it follows from Equation (12) that dT*/d[tau] has the same sign as dZ(T*)/d[tau].

Equation (13) indicates that an exogenous increase in the tax rate has two effects on optimal asset life. The first term is the direct effect [tau] has on T* regardless of the form of financing and it is negative. Intuitively, continuing the asset provides the firm with a pre-tax cash flow of X(T*) + S'(T*) from operations and from the reduction in salvage value. This cash flow exceeds the pre-tax return kS(T*) the firm could realize by terminating the asset and investing the salvage value at a rate k (see Equation (A1) in the Appendix). Hence, a higher [tau] increases the firm's tax liability if asset termination is postponed. Furthermore, the higher [tau] increases the return the firm realizes at T* by investing the tax shields associated with its book value by an amount equal to kB(T*); thus, delaying asset liquidation provides the firm with a loss. Combining these two components implies that the asset is terminated earlier as the corporate tax rate increases.

The second term of Equation (13) is the indirect effect of [tau] on T* and it exists because, in the presence of debt financing, a higher [tau] lowers [k;.sup.17] hence reducing the loss the firm suffers by delaying asset termination, i.e., by postponing the investment of the after-tax cash flows resulting from the termination of the asset at the rate k. This indirect effect of [tau] on T* is, therefore, positive. As shown in the Appendix, when combining the direct and indirect effects, optimal asset life declines as the corporate tax rate increases if salvage value is less than asset book value upon termination. If salvage value exceeds book value, however, optimal asset life could either increase or decline.

From Equation (12), dT*/d[tau] depends on financial leverage through k. To determine the impact of L on the change of an asset's optimal life caused by a higher tax rate, Equation (12) can be differentiated with respect to L to obtain:

[Mathematical Expression Omitted]

As shown in the Appendix, [d.sup.2]Z(T*)/dLdT < 0 and [d.sup.2]Z(T*)/dLd[tau] is ambiguous in sign; thus from Equation (14), the impact of leverage on the effect of a higher corporate tax rate on optimal asset life is ambiguous.

In a recent paper, Dammon and Senbet [3] analyze the effects of taxes and depreciation on the firm's investment and debt financing decisions under uncertainty. They demonstrate that an increase in allowable investment-related tax shields due to changes in the corporate tax code could lead either to an increase or a decline in the optimal amount of debt depending on whether the DeAngelo-Masulis [4] substitution effect is less than or greater than the income effect resulting from an increase in investment. The present paper, however, abstracts from these effects by assuming that investment and debt are exogenously chosen; thus it abstracts from the interactions between investment and debt.

The Tax Reform Act of 1986 (TRA), in general, lengthens the depreciation period of assets (equivalently, lowers allowable depreciation in each of the ensuing periods) and reduces the corporate tax rates.[18] The combined effect of a lower corporate tax rate and lower allowable depreciation on T* is given by the total differential:

[Mathematical Expression Omitted]

Using the results obtained in Equations (9) and (12), it follows that if T* increases with [tau], TRA reduces optimal asset life. In any other case, the lower [tau] associated with TRA increases T* while the lower D(t) associated with TRA reduces T* and no one effect will always dominate; hence TRA has an ambiguous effect on optimal asset life.

The results obtained in Equations (9), (12), and (15) depend on the implicit assumption that, upon termination, the book value B(T*) is positive. Nevertheless, if, as in Howe [8], the asset's tax life is less than its economic life (equivalently, if the asset is fully depreciated for tax purposes before disposal), then B(T*) = 0. Therefore, from Equations (10) and (13):

dZ(T*)/dD(t) = 0 (16)

and

dZ(T*)/d[tau] = -(1 - [tau])S(T*)dk/e-kT*/d[tau], (17)

respectively. Combining Equations (9) and (16) indicates that, if the asset is completely written off for tax purposes before termination, an increase in allowable depreciation does not affect its optimal life. On the other hand, combining Equations (12) and (17) gives the effect of [tau] on T*. Intuitively, the direct effect of [tau] on T* vanishes and the indirect effect reduces to the impact [tau] has via k on the loss the firm suffers by postponing the investment of the after-tax salvage value at the rate of return k. Since k and [tau] are inversely related (see footnote 17), an increase in the corporate tax rate optimally postpones asset termination. Further, since asset life increases with [tau] and is independent of changes in the allowable depreciation when B(T*) = 0, it follows from Equation (15) that TRA will reduce optimal asset life.

If the asset is all-equity financed, k = r. Then, using Equations (3) through (17), one can verify that depreciation and taxes have no effect on the optimal life of an asset fully depreciated upon termination; otherwise, higher depreciation (corporate taxes) increases (reduce(19)) optimal life and TRA has an ambiguous effect.(20)

II. Conclusions

This paper examined the effects of debt financing, allowable depreciation, and corporate tax rate on optimal asset life. The following conclusions were reached:

(i) Optimal asset life increases with the level of debt financing.

(ii) If the asset has not been fully depreciated upon disposal, its optimal life increases with allowable depreciation; declines with higher corporate tax rate if book value exceeds salvage value while it could increase if book value is less than salvage value; and could either increase or decline as a result of the changes introduced by the TRA.

(iii) If the asset has been fully depreciated upon disposal, its optimal life is unaffected by changes in allowable depreciation; increases with the corporate tax rate; and declines as a result of the TRA changes.

For capital budgeting practices, the above analysis implies that the optimal life of a given asset should differ across companies depending on their debt ratio, marginal tax rate, and the depreciation schedule used for tax purposes. Alternatively, changes in any of these three parameters will generally induce the companies to recalculate the optimal life of the asset.

(1)The two are equal when the internal rate of return is used as the discounting rate.

(2)On the other hand, most studies dealing with the effect of inflation on optimal asset life appear to assume only debt financing. In Nelson [19], higher inflation defers asset replacement. Brenner and Venezia [2] show that, in the replacement case, higher inflation increases the duration of assets that were fully depreciated before termination while it tends to increase (decrease) the duration of "long" ("short") assets in the "no-reinvestment" case. In Howe and Lapan [9], the effect of higher inflation on optimal asset life is indetermine in most cases under either the Fisher or Darby effect, while in Howe [8], inflation has no effect on optimal life if the asset's tax life is shorter than its economic life.

(3)Under a positive, neutral, and fully anticipated rate of inflation, the paper's qualitative results do not change under either the Fisher or Darby effect. A proof is available to the interested reader upon request.

(4)WACC is the weighted sum of the firm's after-tax costs of the debt and equity.

(5)Defining L as the target ratio of debt to present market value, Modigliani and Miller [16], henceforth MM, show that the value of an asset which has level perpetual cash flows and supports permanent debt is determined by discounting its unlevered after-tax cash flows at the single rate: k = r(1 - [tau]L). Myers [17] indicates that using the MM scenario to value assets with finite lives or irregular cash flows (i) does not adequately capture the valuation effects of the underlying borrowing plan, and (ii) results in a two to six percent errror. However, the paper's qualitative results hold even under the MM scenario.

(6)Modigliani [15] was the first to recast the ME formula in continuous time. For details on this adjustment, see, for example, Sick [22].

(7)Lewellen and Emery [12] interpret [tau] in the MM and ME scenarios as the generic debt-advantage "multiplier" which can take on a range of non-negative values which do not depend exclusively on tax considerations. Hence, depending on the value of [tau], these two k-specifications can encompass a variety of capital structure theories like DeAngelo and Masulis [4], Jensen and Meckling [11], and Ross [21].

(8)The discrete time discounting rate as given in Taggart [23] is:

[Mathematical Expression Omitted]

(9)Using a single rate is justified because: (i) S(t) can be thought of as the present value of the remaining operating cash flows, thus being perfectly correlated with them; (ii) as a result of (i) and given that B(t) is determined by a fixed depreciation schedule, the difference S(t) - B(t) should also be perfectly correlated with operating cash flows; and (iii) the value of depreciation tax savings is contingent upon and highly correlated with the level of the operating cash flows.

(10)Liebnitz's rule is used to differentiate integrals.

(11)Furthermore, from Equation (4)

[Mathematical Expression Omitted]

where double primes denote second order derivatives with respect to time. The second order condition sufficient for V(T) to reach its maximum at T* requires that Z'(T*) < 0. It is assumed that this condition is satisfied. If Z(T) > 0 for every T, the asset is held until the end of its physical life. If Z(T) < 0 for every T, the asset is not purchased.

(12)Equation (5) or (6) determines a local optimum. In the presence of multiple optima, the global optimum is determined by substituting the salvage values in Equation (3) to determine which one provides the highest NPV. To assure a unique global optimum, some restrictions must be placed on X(t), S(t), and B(t). As in Howe and McCabe [10], a sufficient condition is that the rate of return the firm obtains by postponing liquidation of the asset, as determined by the right hand side of Equation (6), declines with asset life over the relevant time horizon.

(13)dZ(T*)/dL is the derivative of Z(T) with respect to L evaluated at T*. To calculate it, one should differentiate Z(T) in Equation (4) with respect to L and then substitute T* or T in the resulting expression. All the derivatives of Z(T*) with respect to any of the problem's parameters should be interpreted similarly in the remainder of the paper.

(14)From Equations (1) and (2), it follows that

dk/dL = -[[tau]r.sub.D] < 0

and

[Mathematical Expression Omitted]

respectively.

(15)In terms of Myers' [17] APV, the project's value equals its unlevered value plus the extra value provided by debt. Hence, the value of continuing the project increases with L. This alternative justification for the positive relationship between T* and L does not ensue from the analysis in the present paper which, instead of using the additivity principle implicit in APV, discounts the project's unlevered cash flows at a single leverage-adjusted rate.

(16)By definition,

[Mathematical Expression Omitted]

Hence, dB(T*)/dD(t) = -T* < 0.

(17)From Equations (1) and (2), dk/d[tau] equal [-r.sub.D] and [-r.sub.fE][(1 - [[tau].sub.pE])/(1 - [[tau].sub.p])], respectively. Here, it is implicitly assumed that only the tax rate of the firm under question is increased.

(18)For the effect of TRA on the internal rate of return and the net present value of capital investment, see Angel [1].

(19)Optimal life and the corporate tax rate are inversely related in Brenner and Venezia [2] also. Unlike the present paper, however, where cash inflows accrue continuously, they examine a growth asset case where the only cash inflow takes place at the end of the asset's life.

(20)The conclusions of Section I can also be reached for an infinite-cycle replacement problem if perfect competition eliminates economic rents past the first cycle, reducing the NPV of the chain of replacements to that of the first cycle.

References

[1.] R. Angel, "The Effect of the Tax Reform Act on Capital Investment Decisions," Financial Management (Winter 1988), pp. 82-86.

[2.] M. Brenner and I. Venezia, "The Effects of Inflation and Taxes on Growth Investment and Replacement Policies," Journal of Finance (December 1983), pp. 1519-1528.

[3.] R. Dammon and L. Senbet, "The Effect of Taxes and Depreciation on Corporate Investment and Financial Leverage," Journal of Finance (June 1988), pp. 357-373.

[4.] H. DeAngelo and R. Masulis, "Optimal Capital Structure Under Corporate and Personal Taxation," Journal of Financial Economics (March 1980), pp. 3-29.

[5.] E. Dyl and H. Long, "Abandonment Value and Capital Budgeting," Journal of Finance (March 1969), pp. 88-95.

[6.] J. Ezzell and J. Miles, "Capital Project Analysis and the Debt Transaction Plan," Journal of Financial Research (Spring 1983), pp. 25-31.

[7.] J. Gaumnitz and D. Emery, "Asset Growth, Abandonment Value and the Replacement Decision of Like-for-Like Capital Assets," Journal of Financial and Quantitative Analysis (June 1980), pp. 407-419.

[8.] K. Howe, "Does Inflationary Change Affect Capital Asset Life?," Financial Management (Summer 1987), pp. 63-67.

[9.] K. Howe and H. Lapan, "Inflation and Asset Life: The Darby vs. the Fisher Effect," Journal of Financial and Quantitative Analysis (June 1987), pp. 249-258.

[10.] K. Howe and G. McCabe, "On Optimal Asset Abandonment and Replacement," Journal of Financial and Quantitative Analysis (September 1983), pp. 295-305.

[11.] M. Jensen and W. Meckling, "Theory of the Firm: Managerial Behavior, Agency Costs, and Ownership Structure," Journal of Financial Economics (October 1976), pp. 305-360.

[12.] W. Lewellen and D. Emery, "Corporate Debt Management and the Value of the Firm," Journal of Financial and Quantitative Analysis (December 1986), pp. 415-426.

[13.] J. Miles and J. Ezzell, "The Weighted Average Cost of Capital, Perfect Capital Markets, and Project Life: A Clarification," Journal of Financial and Quantitative Analysis (September 1980), pp. 719-730.

[14.] M. Miller, "Debt and Taxes," Journal of Finance (May 1977), pp. 261-275.

[15.] F. Modigliani, "Debt, Dividend Policy, Taxes, Inflation and Market Valuation," Journal of Finance (May 1982), pp. 255-273.

[16.] F. Modigliani and M. Miller, "Corporate Income Taxes and the Cost of Capital: A Correction," American Economic Review (June 1963), pp. 433-443.

[17.] S. Myers, "Interactions in Corporate Financing and Investment Decisions--Implications for Capital Budgeting," Journal of Finance (March 1974), pp. 1-25.

[18.] S. Myers and S. Majd, "Abandonment Value and Project Life," Advances in Futures and Options Research (1990), pp. 1-21.

[19.] C. Nelson, "Inflation and Capital Budgeting," Journal of Finance (June 1976), pp. 923-931. [20.] A. Robichek and J. Van Horne, "Abandonment Value and Capital Budgeting," Journal of Finance (December 1967), pp. 577-589.

[21.] S. Ross, "The Determination of Financial Structure: The Incentive-Signalling Approach," Bell Journal of Economics and Management Science (Spring 1977), pp. 23-40.

[22.] G. Sick, "Tax-Adjusted Discount Rates," Management Science (December 1990), pp. 23-40.

[23.] R. Taggart, "Consistent Valuation and Cost of Capital Expressions with Corporate and Personal Taxes," Financial Management (Autumn 1991), pp. 8-20.

In the existing literature, Robichek and Van Horne [20] indicate that an asset will be terminated in the earliest year its salvage value exceeds the net present value, NPV, of its future cash flows. Dyl and Long [5] show that this does not recognize that later termination could be more valuable than that in the earliest year. Hence, they propose that asset termination should take place in the year which provides the highest NPV. Gaumnitz and Emery [7] demonstrate that the optimal replacement time in a like-for-like replacement problem will generally differ from the optimal asset life in a single-cycle problem.(1) Howe and McCabe [10] also analyze the finite- and infinite-cycle replacement problems and indicate that if future asset replications have a zero NPV, the replacement problem analysis reduces to that of the single-cycle problem.

As is typical in asset life literature, the above studies share two assumptions. First, they assume that at time zero the firm precommits itself to holding the asset for the period of time that maximizes its NPV. Second, these studies assume that the asset is all-equity financed.(2)

Under the first assumption, the firm views future cash flows and salvage values as certain. Certainty equivalents or expected values are used if future salvage values and cash flows are uncertain. Recently, Myers and Majd [18] relax this assumption. They indicate that future cash flows and salvage values can deviate from their expected values and the firm has the option to postpone the abandonment decision to a later date. Using numerical approximation, they evaluate the option to abandon as an American put option on a stock with varying dividend yield and uncertain exercise price. In doing so, they assume that asset value follows a longnormal diffusion process with known expected value and variance for its rate of change. They suggest that an asset will be abandoned when its salvage value exceeds the sum of its value without abandonment plus the value of its option to abandon. Nevertheless, they retain the all-equity financing assumption.

The present paper, on the other hand, relaxes the all-equity financing assumption and allows for debt-equity financing, but assumes that the firm decides irrevocably about asset duration at time zero; i.e., it assumes that the value of the abandonment option is zero. As a result, it is demonstrated that optimal asset life increases with the use of riskless debt financing. It is also shown that higher allowable depreciation does not affect optimal life when the asset is totally depreciated before its termination; otherwise, optimal life increases with depreciation. Further, if the tax life of the asset is shorter than its economic life, optimal life increases with the corporate tax rate. However, if the asset is not fully depreciated before its termination, an increase in the corporate tax rate reduces its optimal life if salvage value is less than asset book value; otherwise, it has an ambiguous effect.

The Tax Reform Act of 1986 (TRA) reduces corporate tax rates and, in general, increases the depreciation life of assets (i.e., reduces the depreciation allowance in each ensuing period). As a result, it is shown that for an asset not fully depreciated before disposal, TRA has an ambiguous effect on its optimal life; otherwise, TRA reduces optimal life.

The paper's results are derived in Section I with all major proofs presented in the Appendix. Section II concludes the paper.

I. Optimal Asset Life Under Debt-Equity Financing

The firm invests in an asset and selects [T*, the asset's holding period, that maximizes V(T), the net present value of its after-tax cash flows over the holding period. Upon termination of the asset, the firm is taxed on the difference between salvage value and book value, the latter being equal to the asset's initial cost minus accumulated depreciation. Depreciation is exogenous and is based on the asset's historical cost. There is no investment tax credit and the anticipated rate of inflation is assumed to be zero.(3) The asset's required rate of return, k, is used to discount the project's expected cash flows.

The firm issues riskless debt to partially finance the asset; the remaining balance is financed with equity. Miles and Ezzell [13] and Ezzell and Miles [6], henceforth ME, demonstrate that if the discounting rate for unlevered cash flows r, the corporate tax rate [tau], the cost of debt [r.sub.D] (r > [r.sub.D]), and the market value leverage ratio L are constant over the asset's life, asset value is determined by discounting its unlevered cash flows at the single rate k = r - [[tau]r.sub.D] L [(1 + r)/ (1 + [r.sub.D])]. This rate is independent of the asset's duration and cash flow pattern and equals the textbook formula or weighted average cost of capital, WACC.(4) Defining L in terms of realized market values, implies that the amount of outstanding debt will adjust to follow fluctuations in future asset value unless the investment is riskless. Hence, the investment decision affects both the size and riskiness of future tax shields associated with debt financing.(5) The ME expression for k, however, applies to discrete time discounting only. When adjusted for continuous time, this discounting rate becomes:(6) k=r-[[tau]r.sub.D]L. (1)

Equation (1) assumes corporate taxes only.(7) Under both corporate and personal taxes, Miller [14] shows that the net tax advantage to corporate debt is [1 - (1 - [[tau].sub.pE])(1 - t)/(1 - [[tau].sub.p])], where [[tau].sub.p] and [[tau].sub.pE] are the tax rates investors pay on debt and equity income respectively. If, as in ME, only the first debt tax shield is certain and if, for simplicity, it is assumed that all investors are subject to the same tax rates, then Taggart [23] shows that an asset's value is determined by discounting its expected unlevered cash flows at a single rate, the continuous time equivalent of which is:(8)

[Mathematical Expression Omitted]

where [r.sub.fE] is the cost of risk-free equity. Equation (2) is the generalization of ME's cost of capital in Equation (1) enhanced to include both personal and corporate taxes. Note that Equation (2) utilizes the risk-free equity rate instead of the riskless debt rate used in Equation (1). Also, the tax advantage of corporate debt differs in these two scenarios since only corporate taxes exist in Equation (1), as opposed to both corporate and personal taxes in Equation (2).

Assuming continous discounting, the net present value of an asset held for T periods is given by:

[Mathematical Expression Omitted]

where

X(t) = The expected unlevered operating cash flow occurring continuously in period t of the asset's life;

D(t) = The depreciation allowance occurring continuously in period t of the asset's life;

S(T) = The asset's expected salvage value at time T;

B(T) = The asset's book value at time T; and

I = The asset's initial cost.

From Equation (3), the asset's net present value equals the present value of the unlevered after-tax future cash flows (including depreciation tax shields), plus the present value of the after-tax salvage value, minus the asset's initial cost. Also, Equation (3) assumes that operating cash flows, future salvage value, depreciation tax savings, and tax on the difference between salvage and book value are all being discounted at the same rate; i.e., their perceived riskiness is the same.(9) It should be noted that discounting takes place at the required rate of return adjusted for the effect of debt financing as determined by Equations (1) and (2) under the different debt scenarios. Finally, as indicated in the introduction, Equation (3) assumes that the value of the implicit abandonment option is zero.

The effect of a marginally longer holding period on the asset's net present value is determined by differentiating(10) Equation (3) with respect to T. Since D(T*) = - B'(T*), this gives:

[Mathematical Expression Omitted]

where primes denote partial derivatives with respect to time. The optimal asset life, T*, that maximizes V(T) must satisfy:(11)

Z(T*) = O. (5)

The intuition behind Equation (5) is straightforward. In present value terms, asset life is increased until the instantaneous after-tax cash flows from operations and from the decline in salvage value become equal to the instantaneous return the firm can obtain by terminating the asset and investing the resulting after-tax cash flow at the rate k.

Solving Equation (5) for k, provides an alternative condition for T*, i.e.,

k= (1 - [tau])[X(T*) + S' (T*)]/(1 - [tau])S(T*) + [tau]B(T*). (6)

Intuitively, asset duration is increased until the after-tax instantaneous rate of return the firm obtains by holding the asset equals the asset's after-tax required rate of return.(12)

The effect of a marginal increase in leverage on the asset's optimal life is determined by totally differentiating Equation (5) with respect to L to obtain:

dT*/dL = - dZ(T*)/dL/Z'(T*), (7)

where Z'(T*) < 0 (see footnote 11) and, from Equations (4) and (5),(13)

dZ(T*)/dL = - dk/dL/[(1 -[tau])S(T*) + [tau]B(T*)[e.sup.-kT*]. (8)

Because the sign of dT* /dL in Equation (7) is the same as that of dZ(T*)/dL, it follows from Equation (8) that higher leverage could affect optimal asset life through its effect on the required rate of return. Since k and L are inversely related,(14) higher leverage reduces the instantaneous return the firm obtains by terminating the asset and investing the resulting after-tax cash flows at the rate k. As a result, the loss associated with asset continuation is reduced and optimal asset life increases.(15) Hence, ceteris paribus, firms in industries with higher leverage ratios will be expected to optimally invest in assets with longer lives. Utilities provide an example of such companies.

From Equation (5), Z(T*) depends on the depreciation allowance through the asset's book value and it also depends on the corporate tax rate. Hence, optimal asset life will change in response to exogenous changes in either parameter. The effect of an exogenous increase in allowable depreciation on T* is determined by totally differentiating Equation (5) with respect to D(t) to obtain:

dT*/dD(t) = - dZ(T*)/dD(t)/Z'(T*), (9)

where Z'(T*) < 0 and, since the asset's book value declines with an increase in allowable depreciation,(16)

dZ(T*)/dD(t) = -kt/dB(T*)e-kT* >0/dD(t). (10)

Because the signs of dT*/dD(t) and dZ(T*)/dD(t) are the same in Equation (9), an increase in allowable depreciation results in longer asset life. Essentially, an increase in depreciation reduces the asset's book value at T* and the loss from delaying investment of the book value tax shields, [tau]B(T*), at the rate k is reduced. Hence, asset life increases.

From Equations (9) and (10), dT* /dD(t) depends on leverage. To determine the impact of financial leverage on the increment of optimal asset life caused by higher depreciation, Equation (9) can be differentiated with respect to L to obtain:

[Mathematical Expression Omitted]

It is shown in the Appendix that: (i) [d.sup.2]Z(T*)/dLdD(t) is positive (negative) for "long" ("short") assets, i.e., assets for which T* is greater (less) than 1/k, and (ii) [d.sup.2]Z(T*)/dLdT < 0. It therefore follows from Equation (11) that the increase in an asset's optimal life associated with higher depreciation declines as leverage increases for "short" assets, while it could either increase or decline for "long" assets. Thus, ceteris paribus, lower depreciation reduces the optimal life of "short" assets with lower debt ratios by more than the life of "short" assets with higher debt ratios. For "long" assets, lower depreciation will always reduce optimal asset life, but the impact of a higher debt ratio on this reduction is ambiguous. For example, as depreciation increases, a utility with higher leverage which invests in a "long" asset may or may not postpone its termination by more than a utility with lower leverage.

Next, the effect of an exogenous increase in the corporate tax rate on T* is determined by totally differentiating Equation (5) with respect to [tau] to obtain:

dT*/d[tau] = -dZ(T*)/d[tau]/,Z'(T*), (12) where, as shown in the Appendix:

[Mathematical Expression Omitted]

Since Z'(T*) < 0, it follows from Equation (12) that dT*/d[tau] has the same sign as dZ(T*)/d[tau].

Equation (13) indicates that an exogenous increase in the tax rate has two effects on optimal asset life. The first term is the direct effect [tau] has on T* regardless of the form of financing and it is negative. Intuitively, continuing the asset provides the firm with a pre-tax cash flow of X(T*) + S'(T*) from operations and from the reduction in salvage value. This cash flow exceeds the pre-tax return kS(T*) the firm could realize by terminating the asset and investing the salvage value at a rate k (see Equation (A1) in the Appendix). Hence, a higher [tau] increases the firm's tax liability if asset termination is postponed. Furthermore, the higher [tau] increases the return the firm realizes at T* by investing the tax shields associated with its book value by an amount equal to kB(T*); thus, delaying asset liquidation provides the firm with a loss. Combining these two components implies that the asset is terminated earlier as the corporate tax rate increases.

The second term of Equation (13) is the indirect effect of [tau] on T* and it exists because, in the presence of debt financing, a higher [tau] lowers [k;.sup.17] hence reducing the loss the firm suffers by delaying asset termination, i.e., by postponing the investment of the after-tax cash flows resulting from the termination of the asset at the rate k. This indirect effect of [tau] on T* is, therefore, positive. As shown in the Appendix, when combining the direct and indirect effects, optimal asset life declines as the corporate tax rate increases if salvage value is less than asset book value upon termination. If salvage value exceeds book value, however, optimal asset life could either increase or decline.

From Equation (12), dT*/d[tau] depends on financial leverage through k. To determine the impact of L on the change of an asset's optimal life caused by a higher tax rate, Equation (12) can be differentiated with respect to L to obtain:

[Mathematical Expression Omitted]

As shown in the Appendix, [d.sup.2]Z(T*)/dLdT < 0 and [d.sup.2]Z(T*)/dLd[tau] is ambiguous in sign; thus from Equation (14), the impact of leverage on the effect of a higher corporate tax rate on optimal asset life is ambiguous.

In a recent paper, Dammon and Senbet [3] analyze the effects of taxes and depreciation on the firm's investment and debt financing decisions under uncertainty. They demonstrate that an increase in allowable investment-related tax shields due to changes in the corporate tax code could lead either to an increase or a decline in the optimal amount of debt depending on whether the DeAngelo-Masulis [4] substitution effect is less than or greater than the income effect resulting from an increase in investment. The present paper, however, abstracts from these effects by assuming that investment and debt are exogenously chosen; thus it abstracts from the interactions between investment and debt.

The Tax Reform Act of 1986 (TRA), in general, lengthens the depreciation period of assets (equivalently, lowers allowable depreciation in each of the ensuing periods) and reduces the corporate tax rates.[18] The combined effect of a lower corporate tax rate and lower allowable depreciation on T* is given by the total differential:

[Mathematical Expression Omitted]

Using the results obtained in Equations (9) and (12), it follows that if T* increases with [tau], TRA reduces optimal asset life. In any other case, the lower [tau] associated with TRA increases T* while the lower D(t) associated with TRA reduces T* and no one effect will always dominate; hence TRA has an ambiguous effect on optimal asset life.

The results obtained in Equations (9), (12), and (15) depend on the implicit assumption that, upon termination, the book value B(T*) is positive. Nevertheless, if, as in Howe [8], the asset's tax life is less than its economic life (equivalently, if the asset is fully depreciated for tax purposes before disposal), then B(T*) = 0. Therefore, from Equations (10) and (13):

dZ(T*)/dD(t) = 0 (16)

and

dZ(T*)/d[tau] = -(1 - [tau])S(T*)dk/e-kT*/d[tau], (17)

respectively. Combining Equations (9) and (16) indicates that, if the asset is completely written off for tax purposes before termination, an increase in allowable depreciation does not affect its optimal life. On the other hand, combining Equations (12) and (17) gives the effect of [tau] on T*. Intuitively, the direct effect of [tau] on T* vanishes and the indirect effect reduces to the impact [tau] has via k on the loss the firm suffers by postponing the investment of the after-tax salvage value at the rate of return k. Since k and [tau] are inversely related (see footnote 17), an increase in the corporate tax rate optimally postpones asset termination. Further, since asset life increases with [tau] and is independent of changes in the allowable depreciation when B(T*) = 0, it follows from Equation (15) that TRA will reduce optimal asset life.

If the asset is all-equity financed, k = r. Then, using Equations (3) through (17), one can verify that depreciation and taxes have no effect on the optimal life of an asset fully depreciated upon termination; otherwise, higher depreciation (corporate taxes) increases (reduce(19)) optimal life and TRA has an ambiguous effect.(20)

II. Conclusions

This paper examined the effects of debt financing, allowable depreciation, and corporate tax rate on optimal asset life. The following conclusions were reached:

(i) Optimal asset life increases with the level of debt financing.

(ii) If the asset has not been fully depreciated upon disposal, its optimal life increases with allowable depreciation; declines with higher corporate tax rate if book value exceeds salvage value while it could increase if book value is less than salvage value; and could either increase or decline as a result of the changes introduced by the TRA.

(iii) If the asset has been fully depreciated upon disposal, its optimal life is unaffected by changes in allowable depreciation; increases with the corporate tax rate; and declines as a result of the TRA changes.

For capital budgeting practices, the above analysis implies that the optimal life of a given asset should differ across companies depending on their debt ratio, marginal tax rate, and the depreciation schedule used for tax purposes. Alternatively, changes in any of these three parameters will generally induce the companies to recalculate the optimal life of the asset.

(1)The two are equal when the internal rate of return is used as the discounting rate.

(2)On the other hand, most studies dealing with the effect of inflation on optimal asset life appear to assume only debt financing. In Nelson [19], higher inflation defers asset replacement. Brenner and Venezia [2] show that, in the replacement case, higher inflation increases the duration of assets that were fully depreciated before termination while it tends to increase (decrease) the duration of "long" ("short") assets in the "no-reinvestment" case. In Howe and Lapan [9], the effect of higher inflation on optimal asset life is indetermine in most cases under either the Fisher or Darby effect, while in Howe [8], inflation has no effect on optimal life if the asset's tax life is shorter than its economic life.

(3)Under a positive, neutral, and fully anticipated rate of inflation, the paper's qualitative results do not change under either the Fisher or Darby effect. A proof is available to the interested reader upon request.

(4)WACC is the weighted sum of the firm's after-tax costs of the debt and equity.

(5)Defining L as the target ratio of debt to present market value, Modigliani and Miller [16], henceforth MM, show that the value of an asset which has level perpetual cash flows and supports permanent debt is determined by discounting its unlevered after-tax cash flows at the single rate: k = r(1 - [tau]L). Myers [17] indicates that using the MM scenario to value assets with finite lives or irregular cash flows (i) does not adequately capture the valuation effects of the underlying borrowing plan, and (ii) results in a two to six percent errror. However, the paper's qualitative results hold even under the MM scenario.

(6)Modigliani [15] was the first to recast the ME formula in continuous time. For details on this adjustment, see, for example, Sick [22].

(7)Lewellen and Emery [12] interpret [tau] in the MM and ME scenarios as the generic debt-advantage "multiplier" which can take on a range of non-negative values which do not depend exclusively on tax considerations. Hence, depending on the value of [tau], these two k-specifications can encompass a variety of capital structure theories like DeAngelo and Masulis [4], Jensen and Meckling [11], and Ross [21].

(8)The discrete time discounting rate as given in Taggart [23] is:

[Mathematical Expression Omitted]

(9)Using a single rate is justified because: (i) S(t) can be thought of as the present value of the remaining operating cash flows, thus being perfectly correlated with them; (ii) as a result of (i) and given that B(t) is determined by a fixed depreciation schedule, the difference S(t) - B(t) should also be perfectly correlated with operating cash flows; and (iii) the value of depreciation tax savings is contingent upon and highly correlated with the level of the operating cash flows.

(10)Liebnitz's rule is used to differentiate integrals.

(11)Furthermore, from Equation (4)

[Mathematical Expression Omitted]

where double primes denote second order derivatives with respect to time. The second order condition sufficient for V(T) to reach its maximum at T* requires that Z'(T*) < 0. It is assumed that this condition is satisfied. If Z(T) > 0 for every T, the asset is held until the end of its physical life. If Z(T) < 0 for every T, the asset is not purchased.

(12)Equation (5) or (6) determines a local optimum. In the presence of multiple optima, the global optimum is determined by substituting the salvage values in Equation (3) to determine which one provides the highest NPV. To assure a unique global optimum, some restrictions must be placed on X(t), S(t), and B(t). As in Howe and McCabe [10], a sufficient condition is that the rate of return the firm obtains by postponing liquidation of the asset, as determined by the right hand side of Equation (6), declines with asset life over the relevant time horizon.

(13)dZ(T*)/dL is the derivative of Z(T) with respect to L evaluated at T*. To calculate it, one should differentiate Z(T) in Equation (4) with respect to L and then substitute T* or T in the resulting expression. All the derivatives of Z(T*) with respect to any of the problem's parameters should be interpreted similarly in the remainder of the paper.

(14)From Equations (1) and (2), it follows that

dk/dL = -[[tau]r.sub.D] < 0

and

[Mathematical Expression Omitted]

respectively.

(15)In terms of Myers' [17] APV, the project's value equals its unlevered value plus the extra value provided by debt. Hence, the value of continuing the project increases with L. This alternative justification for the positive relationship between T* and L does not ensue from the analysis in the present paper which, instead of using the additivity principle implicit in APV, discounts the project's unlevered cash flows at a single leverage-adjusted rate.

(16)By definition,

[Mathematical Expression Omitted]

Hence, dB(T*)/dD(t) = -T* < 0.

(17)From Equations (1) and (2), dk/d[tau] equal [-r.sub.D] and [-r.sub.fE][(1 - [[tau].sub.pE])/(1 - [[tau].sub.p])], respectively. Here, it is implicitly assumed that only the tax rate of the firm under question is increased.

(18)For the effect of TRA on the internal rate of return and the net present value of capital investment, see Angel [1].

(19)Optimal life and the corporate tax rate are inversely related in Brenner and Venezia [2] also. Unlike the present paper, however, where cash inflows accrue continuously, they examine a growth asset case where the only cash inflow takes place at the end of the asset's life.

(20)The conclusions of Section I can also be reached for an infinite-cycle replacement problem if perfect competition eliminates economic rents past the first cycle, reducing the NPV of the chain of replacements to that of the first cycle.

References

[1.] R. Angel, "The Effect of the Tax Reform Act on Capital Investment Decisions," Financial Management (Winter 1988), pp. 82-86.

[2.] M. Brenner and I. Venezia, "The Effects of Inflation and Taxes on Growth Investment and Replacement Policies," Journal of Finance (December 1983), pp. 1519-1528.

[3.] R. Dammon and L. Senbet, "The Effect of Taxes and Depreciation on Corporate Investment and Financial Leverage," Journal of Finance (June 1988), pp. 357-373.

[4.] H. DeAngelo and R. Masulis, "Optimal Capital Structure Under Corporate and Personal Taxation," Journal of Financial Economics (March 1980), pp. 3-29.

[5.] E. Dyl and H. Long, "Abandonment Value and Capital Budgeting," Journal of Finance (March 1969), pp. 88-95.

[6.] J. Ezzell and J. Miles, "Capital Project Analysis and the Debt Transaction Plan," Journal of Financial Research (Spring 1983), pp. 25-31.

[7.] J. Gaumnitz and D. Emery, "Asset Growth, Abandonment Value and the Replacement Decision of Like-for-Like Capital Assets," Journal of Financial and Quantitative Analysis (June 1980), pp. 407-419.

[8.] K. Howe, "Does Inflationary Change Affect Capital Asset Life?," Financial Management (Summer 1987), pp. 63-67.

[9.] K. Howe and H. Lapan, "Inflation and Asset Life: The Darby vs. the Fisher Effect," Journal of Financial and Quantitative Analysis (June 1987), pp. 249-258.

[10.] K. Howe and G. McCabe, "On Optimal Asset Abandonment and Replacement," Journal of Financial and Quantitative Analysis (September 1983), pp. 295-305.

[11.] M. Jensen and W. Meckling, "Theory of the Firm: Managerial Behavior, Agency Costs, and Ownership Structure," Journal of Financial Economics (October 1976), pp. 305-360.

[12.] W. Lewellen and D. Emery, "Corporate Debt Management and the Value of the Firm," Journal of Financial and Quantitative Analysis (December 1986), pp. 415-426.

[13.] J. Miles and J. Ezzell, "The Weighted Average Cost of Capital, Perfect Capital Markets, and Project Life: A Clarification," Journal of Financial and Quantitative Analysis (September 1980), pp. 719-730.

[14.] M. Miller, "Debt and Taxes," Journal of Finance (May 1977), pp. 261-275.

[15.] F. Modigliani, "Debt, Dividend Policy, Taxes, Inflation and Market Valuation," Journal of Finance (May 1982), pp. 255-273.

[16.] F. Modigliani and M. Miller, "Corporate Income Taxes and the Cost of Capital: A Correction," American Economic Review (June 1963), pp. 433-443.

[17.] S. Myers, "Interactions in Corporate Financing and Investment Decisions--Implications for Capital Budgeting," Journal of Finance (March 1974), pp. 1-25.

[18.] S. Myers and S. Majd, "Abandonment Value and Project Life," Advances in Futures and Options Research (1990), pp. 1-21.

[19.] C. Nelson, "Inflation and Capital Budgeting," Journal of Finance (June 1976), pp. 923-931. [20.] A. Robichek and J. Van Horne, "Abandonment Value and Capital Budgeting," Journal of Finance (December 1967), pp. 577-589.

[21.] S. Ross, "The Determination of Financial Structure: The Incentive-Signalling Approach," Bell Journal of Economics and Management Science (Spring 1977), pp. 23-40.

[22.] G. Sick, "Tax-Adjusted Discount Rates," Management Science (December 1990), pp. 23-40.

[23.] R. Taggart, "Consistent Valuation and Cost of Capital Expressions with Corporate and Personal Taxes," Financial Management (Autumn 1991), pp. 8-20.

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Title Annotation: | Issues in Corporate Investments |
---|---|

Author: | Prezas, Alexandros P. |

Publication: | Financial Management |

Date: | Jun 22, 1992 |

Words: | 4682 |

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