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A modified equity valuation model that yields the true maximum value.

Investment value is a maximum price to be paid by a particular buyer or a minimum price to a particular seller. According to The Appraisal of Real Estate:

Each voluntary purchase or sale of real estate is based on an investment value decision made by the parties to the transaction. Thus, the market is made up of transactions in which willing participants make investment decisions...Sellers are normally expected to accept a price that equals or exceeds their investment value, while buyers will pay a price that does not exceed their investment value.(1)

In another leading real estate textbook, authors Jaffe and Sirmans discuss the equity valuation model, a discounted cash flow technique that calculates the investment value of equity. This amount "represents the greatest amount that the specific investor is justified in paying to acquire the property rights of the investment."(2) This sentiment is echoed by Greer and Farrell, who wrote, "Investment value is the greatest amount an investor is justified in paying for an asset, given the anticipated after-tax cash flows the asset will generate and the investor's minimum acceptable rate of return."(3)

However, a problem that arises in attempting to find the maximum investment value is that depreciation (and therefore after-tax cash flow including after-tax equity reversion) is a function of the price or total value, but the price or value itself is a function of depreciation. This is a simultaneity problem.(4) As a result of it, the traditional investment value of equity model does not provide the correct maximum price an investor can pay and still earn the required (minimum) return since the model does not correctly account for the value of the depreciation tax savings.

This simultaneity problem can be addressed using a modified discounted cash flow model, which provides the true maximum value that a specific investor can invest and still earn the required return on equity.

TRADITIONAL EQUITY VALUATION MODEL

The equity valuation model is traditionally used to measure the investment value of equity (E), the net present value of equity ([NPV.sub.e]), and/or the internal rate of return ([IRR.sub.e]) on equity. In addition, by adding the amount of mortgage debt (MD) to the investment value of equity, the model can also be used to measure the investment value of the total capital or total investment value ([V.sub.O]). That is:

[V.sub.O] = E + MD (1)

This total investment value represents the maximum purchase price and is

therefore of particular importance in this paper. Formally, the investment value of equity (E) is defined as:(5)

E = [summation of] [ATCF.sub.t]/[(1 + [k.sub.e]).sup.t] + [ATER.sub.n]/[(1 + [k.sub.e]).sup.n] where t = 1 to n (2)

where,

[ATCF.sub.t] = After-tax cash flow in period t

[ATER.sub.n] = After-tax equity reversion in period n

n = Expected holding period

[k.sub.e] = Required rate of return on equity

By substituting equation 2 into equation 1, the following is obtained:

[V.sub.O] = [summation of] [ATCF.sub.t]/[(1 + [k.sub.e]).sup.t] + [ATER.sub.n]/[(1 + [k.sub.e]).sup.n] + MD where t = 1 to n (3)

The net present value of equity, however, is defined as the difference between the investment value of equity and the required equity outlay. Equation 4 formalizes this relationship:

[NPV.sub.t] = E - (MV - MD)

= E + MD-MV

= [V.sub.O] - MV (4)

where,

MV = Current market value of the property

The wealth-maximizing investor should invest in a property if NPV is positive,(6) that is, if E + MD [greater than] MV or [V.sub.O] [greater than] MV or E [greater than] MV - MD. E would normally represent the greatest amount that the specific investor is justified in paying (in equity) to acquire the property rights of the investment. E plus the given MD would then represent the maximum price for the property. However, E (and therefore [V.sub.O] = E + MD) as calculated does not represent the maximum value because it is based on the investor purchasing the property at the asking price (MV). It does not reflect the simultaneity between depreciation and value.

SIMULTANEITY OF CASH FLOW AND VALUE

The traditional model values two major components of equity value: the periodic after-tax cash flows and the one-time after-tax equity reversion. After-tax cash flow (ATCF) is defined as net operating income less debt service and tax payments, as shown in equation 5:

[ATCF.sub.t] = [NOI.sub.t] - [DS.sub.t] - [T.sub.t] (5)

where,

[ATCF.sub.t] = After-tax cash flow in period t

[NOI.sub.t] = Net operating income in period t

[DS.sub.t] = Debt service in period t

[T.sub.t] = Income tax liability in period t

Debt service constitutes interest expense and principal amortization:

[DS.sub.t] = [I.sub.t] + [A.sub.t] (6)

where,

[I.sub.t] = Interest expense in period t

[A.sub.t] = Principal amortization in period t

Income taxes are calculated by multiplying the investor's marginal tax rate times the net operating income plus the replacement reserve less interest expense and depreciation as follows:

[T.sub.t] = T([NOI.sub.t] + [RR.sub.t] - [I.sub.t] - [D.sub.t]) (7)

where,

[T.sub.t] = Total income taxes in period t

T = Investor's marginal tax rate

[RR.sub.t] = Reserve for replacement in period t

[D.sub.t] = Depreciation allowance in period t

By substituting equation 6 for [DS.sub.t] and equation 7 for [T.sub.t] into equation 5 for [ATCF.sub.t] obtains:

[ATCF.sub.t] = [NOI.sub.t] - ([I.sub.t] + [A.sub.t]) - T([NOI.sub.t] + [RR.sub.t] - [I.sub.t] - [D.sub.t]) (8)

Gathering all like terms, the above equation then becomes:

[ATCF.sub.t] = [NOI.sub.t] (1 - T) - [I.sub.t] (1 - T) - [A.sub.t] - [TRR.sub.t] + [TD.sub.t] (9)

In this particular form of the equation, the contribution of depreciation to cash flow (i.e., the tax shield or tax savings as a result of depreciation expense) becomes more apparent in the last term.

Depreciation expense is a function of the price paid by the investor. Since the Tax Reform Act of 1986,(7) real estate depreciation has been determined on a straight-line basis of 27.5 years for residential property and 31.5 years for nonresidential property (now 39 years for nonresidential property since the Revenue Reconciliation Act of 1993).(8) Therefore, the amount of depreciation (and thus the amount of the tax shield) in each year is an equal fraction of the price paid less the value attributable to land. The half-month convention is ignored due to its negligible effect on value. In addition, the depreciable basis is generally the original cost ([V.sub.O]) plus the cost of any capital improvements less the cost of land. The allocation of the total investment is based on relative market values at the time of acquisition. This particular model assumes a given (depreciable) building to total value ratio similar to that obtained by tax assessor estimates and contract allocations. The model is easily modified to permit a separate appraisal and input of land value.

Let N equal the depreciable life in years (i.e., 27.5 or 39), and let b represent the proportion of value that is depreciable (i.e., not land). The amount of depreciation in each year is then expressed by:

D = 1/Nb[V.sub.O]

The after-tax cash flow in each period becomes:

[ATCF.sub.t] = [NOI.sub.t] (1 - T) - [I.sub.t] (1 - T) - [A.sub.t] - [TRR.sub.t] + T(1/N)(b[V.sub.O]) (10)

Note that the first four terms in the equation 10 are independent of [V.sub.O] (MD is a given, and therefore so are [I.sub.t] and [A.sub.t]). Letting [ATCF.sup.*] represent those first four terms in equation 10 except the depreciation tax shield, the following is obtained:

[ATCF.sub.t] = [[ATCF.sup.*].sub.t] + T (1/N)(b[V.sub.O]) (11)

Therefore, after-tax cash flow in each period, the major component contributing value to [V.sub.O], is clearly dependent on [V.sub.O]. But [V.sub.O] is dependent on ATCF by equation 3. Thus, there is a simultaneous relationship between value and the periodic after-tax cash flow used to find value.

The after-tax equity reversion (ATE[R.sub.n]) is a one-time cash flow to the real estate investor at the (expected) time of sale designated period n.

The equation for ATER is:

[ATER.sub.n] = [SP.sub.n] - [SE.sub.n] - [UM.sub.n] - [GT.sub.n] (12)

where,

[ATER.sub.n] = After-tax equity reversion at the end of period n

[Sp.sub.n] = Estimated gross selling price at the end of period n

[Se.sub.n] = Selling expenses at the end of period n

[UM.sub.n] = Unpaid mortgage balance at the end of period n

[GT.sub.n] = Taxes on total gain at the end of the period n

The components of this equation are independent of [V.sub.O] with the exception of [GT.sub.n]. The taxes on the gain of the sale are related to [V.sub.O] through the adjusted basis used to calculate the gain. Specifically:

[GT.sub.n] = T([SP.sub.n] - [SE.sub.n] - adjusted basis)

The adjusted basis is equal to the price paid by the investor ([V.sub.O]) less the accumulated depreciation. The accumulated depreciation is determined by n/N times the depreciable basis (b[V.sub.O]). The adjusted basis is then:

Adjusted basis = [V.sub.O] - (n/N)b[V.sub.O]

= [V.sub.O] [1 - (bn/N)] (13)

The taxes on the total gain can be expressed as:

[GT.sub.n] = T [[SP.sub.n] - [SE.sub.n] - [V.sub.O] [1 - (bn/N)]] (14)

Substituting equation 14 into equation 12 and simplifying to isolate the [V.sub.O] terms, the following is obtained:

[ATER.sub.n] = [SP.sub.n] (1 - T) - [SE.sub.n] (1 - T) - [UM.sub.n] + [TV.sub.O] [1 - (bn/N)] (15)

Letting [ATER.sup.*] represent all components of ATER that are independent of value would include all but the last term above in equation 15 which captures the effect on cash flow of the adjusted basis at the time of sale. Specifically:

[ATER.sub.n] = [[ATER.sub.n].sup.*] + [TV.sub.O] [1 - (bn/N)] (16)

As was the case with the periodic ATCF, the one-time after-tax cash flow from equity reversion is clearly a function of [V.sub.O]; yet [V.sub.O] is also based on the after-tax equity reversion by equation 3.

The interdependence of investment value with the two major components of cash flow in the traditional equity valuation model has thus been established. If an investor or appraiser is interested in the maximum investment value to a specific investor, the traditional model must be modified to reflect this interdependency.

MODIFIED MODEL

This section develops a model to determine the true maximum price that a specific investor is justified in paying for an investment property. The model results in the maximum price for a given set of estimated cash flows (other than the unknown depreciation tax shields) and the required return on equity. Like the traditional model. the mortgage loan (MD) is assumed to be based on market value (the value to a typical or marginal buyer) and is therefore independent of investment value (the value to a specific buyer).

As previously defined, total investment value ([V.sub.O]) is the sum of the investment value of equity and mortgage debt:

[V.sub.O] = E + MD (1)

and

[V.sub.O] = [summation of] [ATCF.sub.t]/[(1 + [k.sub.e]).sup.t] + [ATER.sub.n]/[(1 + [k.sub.e]).sup.n] + MD where t = 1 to n (3)

where,

[ATCF.sub.t] = [[ATCF.sub.t].sup.*] + T (1/N)(b[V.sub.O]) (11)

and

[ATER.sub.n] = [[ATER.sub.n].sup.*] +. [TV.sub.O] [1 - (bn/N)] (16)

Substitution of equations 11 and 16 into equation 3 and isolation of the terms containing [V.sub.O] results in:

[V.sub.O] = [summation of] [[ATCF.sub.t].sup.*]/[(1 + [k.sub.e]).sup.t] + [[ATER.sub.n].sup.*]/[(1 + [k.sub.e]).sup.n] where t = 1 to n + [summation of] T(1/N)(b[V.sub.O])/[(1 + [k.sub.e]).sup.t] + [TV.sub.O][1 - (1 - bn/N)]/[(1 + [k.sub.e]).sup.n] + MD where t = 1 to n

For simplification, let,

[PV.sup.*] = [summation of] [[ATCF.sub.t].sup.*]/[(1 + [k.sub.e]).sup.t] + [[ATER.sub.n].sup.*]/[(1 + [k.sub.e]).sup.n] + MD where t = 1 to n (17)

Substituting [PV.sup.*] into equation 17 and using the annuity form of the tax shields because of straight-line depreciation(9) results in the following:

[V.sub.o] = P[V.sup.*] + MD + T(1/N)(b[V.sub.o])PVIF[[[A.sub.k].sub.e[prime]].sup.n] + T[V.sup.o] [1 - (bn/N)] PVI[[[F.sub.k].sub.e[prime]].sup.n] (18)

which further simplifies to:

[V.sub.o] = P[V.sup.*] + MD/[1 - (Tb/N)] PVIF[[[A.sub.k].sub.e[prime]].sup.n] - T[1 - (bn/N)]PVIF[[[A.sub.k].sub.e[prime]].sup.n]] (19)

PVIFA and PVIF are the present value interest factor of an annuity and single payment, respectively. Also note that P[V.sup.*] represents the present value of all cash flows except the periodic tax shield of depreciation and the adjusted basis at the time of sale. In other words, the appraiser can determine the value of all other cash flows while ignoring the depreciation problem. The resulting value is then "grossed up" with the adjustment factor shown in the denominator of equation 19.

APPLICATION OF THE MODEL

To illustrate the application of this model, the example from Jaffe and Sirmans (J/S hereafter) is used.(10) This example assumes an asking price (MV) of $75,000 that is fully depreciable (i.e., because no value is attributed to land, b = 1). The J/S inputs are reproduced in modified form in tables 1 and 2.

In the traditional model, the value of E from table 1 is simply the present value (PV) of the periodic after-tax cash flow (ATCF) [TABULAR DATA FOR TABLE 1 OMITTED] plus the present value of the after-tax equity reversion (ATER): $24,449.79 + $43,237.70 = $67,687.49. With a mortgage balance of $50,000, the "maximum" investment value is then $117,687.50. Since E + MD [greater than] $75,000, NPV is positive ($42,687.49), and the decision is made to accept the investment.

According to these numbers, this is a very attractive investment opportunity. The investor can purchase a property for $75,000 that has a value to him or her of $117,687.50. However, a positive NPV indicates that this is not the maximum the investor can pay and still earn his or her required return. If the investor or appraiser is interested in the maximum investment value (i.e., the value at which NP[V.sub.e] = 0 and IR[R.sub.e] = [k.sub.e].) the model must base depreciation on that maximum price. The modified model developed in this paper determines this true maximum price because it correctly accounts for the relationship between value (price) and cash flow resulting from the tax writeoff on the depreciation. Recall that the J/S example assumes a land [TABULAR DATA FOR TABLE 2 OMITTED] value of zero and therefore b = 1. Table 2 provides the value of P[V.sup.*] (equal to PV(ATC[F.sup.*]) + PV(ATE[R.sup.*])) in the modified model. Substitution of the remaining inputs into equation 19 yields the true maximum value of:

[V.sub.o] = $55,185 + $50,000/1 - ((0.28)(1)/27.5) (3.6048) - (0.28)(1 - (1)(5)/27.5) (0.5674) = $105,185/0.8333 = $126,226.81

Assuming the investor purchases the property at this maximum price of $126,226.81, the traditional model provides a "check" on the modified version. Table 2 provides the ATCF for each period and the ATER based on this assumption. Their present values are $26,330.21 and $49,896.60, respectively. Adding the mortgage debt of $50,000 to their sum results in $126,226.81 as before.

Further, the equity investment of $76,226.81 (PV(ATCF) + PV(ATER)) provides future cash flows with a present value equal to the investment. The NP[V.sub.e] = 0, and the IR[R.sub.e] = 12%, the investor's required return. The maximum investment value is therefore established.

Note that the traditional model found a maximum value of $117,687.50, but the modified model determined the true maximum value to be $126,226.81. The difference of $8,539.31 is reconciled when the marginal tax savings from depreciation and the adjusted basis are computed on a present-value basis. Specifically, from tables 1 and 2, the depreciation amounts for the traditional and modified models are $2,727 and $4,590, respectively, for a difference of $1,863. This difference multiplied by the tax rate of 28% produces an annual tax savings of $521.64 which, on a present-value basis, computes to $1,881.53. The adjusted basis at the time of reversion for the traditional model is $75,000 - (5)($2,727) = $61,365, which saves taxes of $17,182.20. For the modified model, the adjusted basis is $126,227 - (5)($4,590) = $103,277, which saves taxes of $28,917.56. The difference in tax savings at the end of period 5 is $11,735.36, which has a present value of $6,658.96. This amount added to the $1,881.53 explains the difference in value of $8,540 (a slight difference due to rounding errors).

Although the maximum value found is 6.7% lower than the true maximum, this difference will vary according to the divergence between the asking price and the true investment value. Using the traditional model, the relationship between the asking price (investment value) and the NPV is linear, as shown in figure 1. As the market value (asking price) increases, the maximum investment value also increases, but the NPV decreases. The true maximum is at the point that the market value and investment value lines intersect and simultaneously where NPV equals zero. Figure 1 shows that the traditional model determines the true maximum only if the asking price is coincidentally equal to the true maximum. If the asking price is below the true maximum, the traditional model finds a maximum value below the true maximum. If the asking price is above the true maximum value, the traditional model determines a maximum above the true maximum. Such a false maximum, would, however, be rejected on a net present value basis. In short, the modified model does not depend on a market value or asking price as an input to its estimate of investment value. Further, the modified model generates only the one true (maximum) value.

Keeping in mind from the example above that the true investment value is $126,226.81, one could obtain greater or lesser errors directly related to the magnitude of divergence from the true investment value. For example, if the asking price is only $60,000 (versus the $75,000 in the previous example), the maximum investment value (according to the traditional model) would be $115,187, resulting in a difference of $11,039 or 8.7% of the true maximum price (from the modified model). On the other hand, if the asking price is as high as $140,000 (a lesser divergence from the true investment value), then the maximum investment value is $128,522, resulting in a difference of $2,296 or 1.8% of the true price. (But note that the investor would reject the investment due to the negative NPV.) In summary, the error that occurs, whether in a dollar or percentage figure, depends on the divergence between the asking price and the true investment value.

CONCLUSION

The traditional investment value of equity model is touted for determining the maximum price that an investor should pay for a particular real estate investment. In fact, the traditional model provides the maximum price only in the unlikely event that the asking price is the true maximum. If the asking price is below the maximum, the traditional model finds a maximum value below the true maximum. For an asking price above the maximum, the traditional model finds a maximum investment value above the maximum, but one that would be rejected on an NP V basis.

A modified form of the traditional investment value of equity model provides the true maximum price. The model accounts for the simultaneity between investment value and the cash flows used in estimating that value. Further, this modified model does not require the input of an asking price since the valuation herein is independent of the asking price.

The primary benefit of the modified model is that it provides the true maximum price or investment value that a specific investor can pay and still earn the required return. Estimating the maximum purchase price to a specific investor is important for intelligent negotiating between the buyer and seller.

Another benefit of the modified model is that it explicitly separates the tax shields from the operating cash flows, letting the appraiser focus clearly on the sources of value in the analysis. In addition, such separation accommodates the application of appropriate growth rates to income, expenses, and selling price (all in P[V.sup.*]) while isolating the depreciation tax savings that are solely dependent on present value. Further, the appraiser may want to justify a higher discount rate for the more risky P[V.sup.*] cash flows compared with the more certain tax shields. The modified model accommodates this approach to reflecting differential risk in the analysis.

Finally, the modified model makes clear the nature of the simultaneity and provides a workable formula to incorporate it into models using discounted cash flow techniques. Even appraisers who may have encountered this problem and "solved" it with computerized iterative techniques will now have available an understanding of the problem as well as a more efficient and convenient formula to solve it. All an appraiser would need is a cell formula devoted to the adjustment factor in the numerator as a function of the inputs T, N, n, [k.sub.e[prime]] and b.

1. Appraisal Institute, The Appraisal of Real Estate, 11th ed. (Chicago, Illinois: Appraisal Institute, 1996), 639.

2. Austin J. Jaffe and C.F. Sirmans, Fundamentals of Real Estate Investment, 2nd ed. (Englewood Cliffs, New Jersey: Prentice-Hall, 1989), 279.

3. Gaylon E. Greer and Michael D. Farrell, Investment Analysis for Real Estate Decisions, 2nd ed. (Chicago, Illinois: Longman Financial Services Publishing, 1988), 304.

4. A general model of a simultaneity is presented along with an application in Richard Bums and Joe Walker, "Small Business Valuation: Simultaneity of Value and Non-Cash Expenses," Journal of Small Business Management (January 1991): 10-14.

5. Although the [V.sub.O] term is original to the authors of this paper, the rest of the notation that follows is taken largely from Jaffe and Sirmans's Fundamentals of Real Estate, chapter 12, due to its clarity and familiarity in the literature. The example in that chapter will also be used here for the same reasons.

6. Strictly speaking, the strong inequalities ought to be weak inequalities since the maximum value is that which provides a zero net present value.

7. Tax Reform Act of 1986, PL 99-514.

8. Reconciliation Act of 1993 (Omnibus Budget), PL 103-66.

9. Any other depreciation schedule beside that of straight-line would require each of the b[V.sub.O] terms to have a different fractional multiplier and be discounted separately and then summed, complicating the notation but not to the application of these concepts since a spreadsheet could easily accommodate such changes.

10. Jaffe and Sirmans, 286-289. Other notable examples among many that reveal the same discrepancy include Greer and Farrell, 298-303, and Stephen A. Pyhrr and J. R. Cooper, Real Estate Investment: Strategy, Analysis, Decision (Boston, Massachusetts: Warren, Gorham & Lamont, 1982), 297-309.

Richard Burns, PhD, is associate professor of finance at the University of Alabama at Birmingham. He holds a PhD in finance from the University of Georgia, Athens. He specializes in valuation, small firm finance, and capital budgeting. Contact: (205) 934-8832. Fax (205) 975-4428. Email Rbums@UAB.edu.

Joe Walker. PhD, is associate professor of finance at the University of Alabama at Birmingham. He holds a PhD in economics from Texas A & M University, College Station. His areas of concentration are capital budgeting, small firm finance, and investments. Contact: (205) 934-8867. Fax (205) 975-4428. Email jwalker@UAB.edu.
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Author:Burns, Richard; Walker, Joe
Publication:Appraisal Journal
Date:Jan 1, 1998
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