Accelerated Depreciation and Income Growth William Beranek and Edward B. Selby, Jr. * This paper challenges the belief that accelerated depreciation methods are always superior to the straight-line method-especially for low-tax bracket owners of highly leveraged investments who have prospects for income growth. The root of the problem is our highly progressive income tax structure and the nature of loan amortization schedules which increase a debtor's taxable income while decreasing his net cash flows. When the entire personal and corporate tax schedules are used to test alternative depreciation methods, our simulation results demonstrate that the much maligned straight-line method is optimal for suitably low discount rates. INTRODUCTION Many developments have occurred recently in real estate investment analysis, a summary of which appears in Jaffe [1979]. While tax law changes have increased the complexity of the analysis of tax-depreciation methods (Sirmans [1980]), there is one important simplifying assumption the literature has not abandoned: a constant marginal income-tax rate for the investor. If the investor's marginal tax rate is constant over time, numerous studies have shown how to derive the optimal depredation method (DM) from a set of federally permitted methods (e.g.. Levy and Samat [1978], Bierman and Smidt [1980]). Since the present value of depreciation charges should be maximized, College of Business Administration, The University of Georgia, Athens, Georgia 30602. **The authors are indebted to Woon Youl Choi and Vincent B. Crawford for computational assistance. 67 68 AREUE A JOURNAL [ VoL 9 subject to the obvious constraint that over the life of the asset they can not exceed its initial depreciable value, any rearrangement of these charges which can shift larger charges to an assets' early years will be favored over the straightline (SL) method of depreciation. This can easily lead to the misinterpretation that any accelerated method is always favored over the SL method, forgetting the fact that the proposition only applies to investors in constant marginal tax brackets. Since many investor-taxpayers (both corporations and individuals) do not expect to be in a constant marginal tax bracket over the holding period of the depreciable asset, the choice of an optimal DM requires much more analysis than the familiar rule of thumb that the SL method is always inferior, and that the optimal accelerated method can be obtained, for a zero salvage value, by consulting a set of tables (see Levy and Sarnat or Bierman and Smidt). The reason for this conclusion is easy to explain. If a low-to-medium tax bracket investor can expect growth in taxable income over the duration of the depreciable asset's life, then the investor's marginal tax bracket will increase with income growth. A varying marginal tax rate, however, renders invalid evaluation procedures based on a constant tax rate. Similarly for a low-bracket corporation, its managers may expect income growth leading to an upward climb in marginal tax rates. It is necessary to evaluate each DM available to the taxpayer by calculating the present value of the expected after-tax cash flows of the proposed investment. A present-value model must be constructed which captures the essential ingredients of the analysis, including financing, tailored to fit each taxpayer's circumstances. The purpose of this paper is to illustrate how this may be done in a simplified set of circumstances, to develop examples illustrating the application of the approach, and to demonstrate that an accelerated DM is not always optimal. THE PRESENT-VALUE MODEL To suggest a general model for the individual real estate investor let us define: K = original depreciable value of the asset, L = amount of loan to finance acquisition of asset, r = rate of interest on L, T, = total taxes in year t, Aj = amortization of loan in year t, p = rate of discount appropriate to investor, R = annual rental received by investor from leases on asset, a constant, P = annual payment on loan, Yf = other income of investor-taxpayer, n = useful life of asset, g = annual rate of growth in Yj. 1981 ] Accelerated Depreciation 69 For the sake of simplicity, we assume no salvage value, K = L, R = P, and the life ofthe loan equals the depreciable, useful life of asset, n. The after-tax cash fiow for year t, C^, must be. t-1 (11.1) Ct = (Yt + R)-Tt-r( L T=l t - 1 because R = P, Aj = P r(L SA^) , and there are no operating expenses. T = l The present value V of these cash fiows to the investor is n (11.2) V=*S t = i because the investor's equity in this venture at time 0 is zero. This remarkedly simple expression stems from our simplifying assumptions, assumptions which imply that the impact of the accelerated DM on V is transmitted solely through Tt. Since the series Yj is given, to maximize V in equation (11.2) implies minimizing t=i (1+py As noted before, because the investor's marginal tax rate is not constant, T^ is not constant but will grow with Y^. Outside income, Y^, need not literally be restricted to "outside income," i.e., income from sources other than the depreciable asset. Any rental R in excess of P, the annual loan payment, is included in Y^, so that growth in rents is refiected in the analysis. By the same token, the amount by which P exceeds R in any year must be treated as a negative quantity in arriving at Y^. SOME SIMULATION RESULTS To maximize V or equation (11.2) the investor must choose among several DMs: straight-line (SL), sum-of-years-digits (SYD), and, among the declining balance (DB) methods he can choose 125%, 150%, or 200%. Consider the following case for an individual investor: 70 AREUE A JOURNAL [ Vol. 9 n = K = r = Yo = s = 10 years L = $100,000 12% $25,000 10% The 1979 personal income tax rates were used in the computations. Switching from a declining-balance method to a SL method was done according to the formula n t* = n + 1 , X where t* denotes the optimal year to switch and X the declining-balance factor (1.25, 1.50 or 2.0). Exhibits 1 and 2 contain the present values of after-tax cash flows for the individual and corporate cases, respectively. A glance at Exhibit 1 quickly shows that for discount rates below .11, the SL method is optimal. For rates .11 to .50, the SYD method is optimal. It is notable that not one of the DB methods is best for the range of discount rates studied. For the corporation, we assume the same data as above except that YQ = $40,000 instead of $25,000. Of course, the corporate-tax structure is different from the individual structure and this impacts, as expected, on the analysis. Exhibit 2 presents the results of simulations for discount rates from .01 to .50. Again the SL method becomes optimal for discount rates below .10, 1.25 DB is best for a /a of .10, 2.0 DB is optimal for p = .11, and for rates in excess of .12, SYD is most appropriate, the same method that is favored for these rates in the individual investor case. It is notable that for any discount rate the variation in present values among the DMs is small, typically less than 1%. This is in contrast to the constant marginal tax-rate case where the double DB and SYD methods, when evaluated by computing present values of depreciation charges, can diverge by 9%. It is also significant that only rarely is a DB method a winner. If the taxpayers' marginal tax rate is constant over time, it is well established that for assets of zero-salvage value the double DB method is preferable (i.e., maximizes asset value of depreciation charges) to the SYD method for discount rates below 14% and assets with useful lives of less than 6 years, while for discount rates at or above 14% and useful lives of 7 or more years the order of preference is reversed.' However, our results show that even with an outside income growth rate of 10% it is not optimal to be guided by rules derived from this case. For a discount rate of less than .11, our individual taxpayer would opt for the SL ' Levy and Sainat, pp. 345 to 348. 1981 ] Accelerated Depreciation 71 method, while the use of tables currently found in the literature would lead one to choose the double DB method. Looking at Exhibit 1, the difference in present values between these two methods for, say, a discount rate of .05, would be $1,909 ($219,566 - $217,657), a small quantity to be sure. However, we do not know whether such a small difference is true in general or if it depends upon the specific parameters of the project. Based upon our current knowledge, the investor that is faced with growth in income is best advised to calculate present values of altemative DMs. CONCLUSIONS Choosing a DM so as to maximize the present value of depreciation charges is not always optimal. This is particularly true when the investor's marginal tax rate is expected to increase over the assets' holding period, stemming from growth in the taxpayer's taxable income. When faced with such a prospect, the investor is advised to choose an optimal DM by the process of enumerating the present values of the asset's cash flows, rather than observing rules based on a constant marginal tax rate. Our simulation results demonstrate that an accelerated method is not always optimal; the much maligned SL method was optimal for suitably low discount rates for both the individual and the corporation. At higher rates the SYD method almost always dominated the other methods, as it does in the constant marginal tax-rate case. In our analysis, property income in excess of P, the annual loan payment, was included in "other income" and the resulting Yj was assumed to grow at a constant rate. The reader is entitled to know how sensitive these conclusions are to changes in the assumed values of the parameters chosen, but these answers must be developed by another study.

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