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Implementing discounted cash flow valuation models: what is the correct discount rate?

The rate at which to discount expected future cash flows is a fundamental issue in the valuation of income-producing real estate. Until quite recently, professional appraisers have been primarily concerned with the appropriate derivation or construction of the "cap rate"; that is, the denominator in the simple income capitalization formula for estimating market value. As stated in The Appraisal of Real Estate, ninth edition, simple income capitalization is applicable only to properties "with stable or stabilized income streams and properties with uneven income streams that are expected to change according to the J- or K-factor pattern." (1) Multiperiod discounted cash flow (DCF) analysis is not subject to these constraints; on the contrary, it is appropriate for any pattern of future income. As appraisers of income property increase their reliance on DCF techniques, the issue of the appropriate discount rate to use in these multiperiod valuation models has received more attention.

The Appraisal of Real Estate contains an example that employs two DCF approaches to value an income-producing property. (2) In the first approach, annual, before-debt cash flows from operations (i.e., net operating income [NOI]) and from the sale of the property are discounted at a rate that reflects potential investors' weighted average cost of debt and equity capital. This weighted average cost of capital (WACC) approach to discounting multiperiod cash flows is similar in concept to construction of capitalization rates using mortgage-equity analysis.

In the second DCF approach to valuing the example property, all after-debt cash flows received by an equity investor are discounted by the investor's required after-debt return on equity. The present value of the equity cash flows is then added to the loan proceeds (net of financing costs) received from the lender at closing. This "equity valuation" approach explicitly values the equity cash flows separately from the mortgage debt cash flows. A separate valuation of the two rate-of-return components is necessary because the subject property in The Appraisal of Real Estate example must be valued as an all-cash ("free and clear") deal and valued subject to its existing, and assumable, financing.

In his recent article, "The Weighted Average Cost of Capital--The Correct Discount Rate," Stephen C. Kincheloe correctly points out that both the WACC approach and the equity valuation approach, if acceptable, should produce the same estimate of market value. Kincheloe then provides an example in which the estimate of market value derived through the WACC technique is significantly higher than the market value estimate obtained by separately valuing the equity cash flows. He concludes that the WACC approach is the one correct approach, and that the equity valuation technique is "theoretically indefensible and can cause investors to make less than optimal investment decisions." (3)

Kincheole is partially correct--as it is typically used, the equity valuation technique results in a lower value estimate than the use of WACC. This is because the standard application of the equity valuation model incorrectly specifies the cost of equity financing after the first year of operations. The purpose of this article is to demonstrate that separately discounting each source of capital by its respective cost is theoretically correct and, if correctly done, produces the same market value estimate as the WACC approach. Market analysts need not remove the equity valuation technique from their appraisal tool kit; rather, they simply need to avoid the most common mistake associated with its application.

REVIEW OF EXAMPLE

PROPERTY AND

CONSTRUCTION OF

DISCOUNT RATES

The income property DCF analysis example presented by Kincheloe is summarized in Table 1 and is used as a starting point for this discussion. Potential gross income (PGI) in the year of acquisition is expected to be $1,000,000 and to grow at an annual rate of 5%. Vacancy and collection losses are estimated at 8% of PGI and operating expenses will consume 16.3% of effective gross income. Typical loan terms include a 70% loan-to-value ratio, a 10% interest rate, and a 30-year amortization. The market value of the property at the end of any year is set equal to NOI in the subsequent year capitalized at 8%. Selling expenses are assumed to be 4%.

If a valuation is to be performed on a before-debt basis, the projected stream of NOI, including the selling price net of expenses, must be discounted or converted into an estimate of current market value. The correct rate at which to discount

[TABULAR DATA OMITTED]

the before-debt cash flows in year t is the [WACC.sub.t] or

[WACC.sub.t] = [m.sub.t][y.sub.T] + (1 - [m.sub.t])[y.sub.et] (1)

where

[m.sub.t] = Outstanding loan-to-market-value ratio at the beginning of year t [y.sub.T] = Constant (assumed) before-tax yield on a Treasury security of a comparable maturity [y.sub.et] = Rate of return required in year t on the equity capital invested in the property

Why is the use of the WACC theoretically correct? The NOI generated by the property in each year is essentially purchased by the investor with a combination of debt and equity financing. Thus, discounting operating income by the weighted average cost of the purchase in each year is an appropriate method for valuing the income stream.

This formulation of the WACC highlights two important points. First, the current cost of typical debt financing does affect the estimated market value of the property, even if an analyst only estimates cash flow down to the level of NOI. Although the analyst may not explicitly include debt financing flows in the valuation model, debt financing effects are thus implicitly captured via the WACC. Second, subscripting the variables by t emphasizes that the loan-to-market value ratio, the equity discount rate, and perhaps the WACC change over the investment holding period.

How is [y.sub.et] selected? As a practical matter, the equity discount rate used in the valuation process is generally based on what equity investors appear to require at the time of the appraisal. A financial economist, however, would likely specify the components of [e.sub.t] as

[y.sub.et] = [y.sub.T] + [(RP/(1 - [m.sub.t])] (2)

where RP = Risk premium required on an investment of similar risk that is financed entirely with equity (i.e., cash)

RP reflects the anticipated amount of variation in NOI and is sometimes referred to as the "business risk" premium. RP is not a function of [m.sub.t] because operating cash flows do not vary with the amount of debt financing. By not subscripting RP with t, it is assumed that the riskness of NOI will not vary over time.

Default risk is the risk that borrowers will cease to make timely payments of principle and interest (as per the terms of the promissory note). Such behavior could lead to lender foreclosure on the property if not cured by the borrower. Pre-payment (i.e., "call") risk is the risk that the borrower will prepay the remaining loan balance (if contractually allowed to do so) before the end of the loan term. Early repayments on fixed-rate loans are more likely during periods of declining interest rates and thus usually occur to the detriment of the lender. It should be noted that the before-tax yield on Treasury securities ([y.sub.T]) is free from the risk of borrower default and prepayment, but is not free from interest-rate risk. That is, real yields earned by owners of Treasury securities are affected by unanticipated changes in general inflation and interest rates.

THE EFFECTS OF DEBT

FINANCING ON REQUIRED

EQUITY RETURNS

Although they do not affect NOI, the method and amount of debt financing do affect the amount and the variation (from expectations) of after-debt cash flows that are available for distribution to an equity investor. More specifically, the expected variability of the after-debt return to equity investors decreases over time along with the amount of financial leverage, even if default risk is not altered. This is because as the property increases in value and the loan balance is amortized, larger amounts of equity capital are, barring early repayment, invested in the property. As a result, a given amount of variation in NOI will have increasingly smaller effects on the equity return.

That decreasing amounts of financial leverage reduce the variation or riskiness of equity returns can be demonstrated by analysis of the equity dividend rate, which is defined as the equity cash flow in year t divided by the equity investment at the beginning of year t. In this example, the equity investor is assumed to borrow 70% of the $9,815,603 purchase price, implying an original equity investment of $2,944,681 ($9,815,604 minus $6,870,922). If NOI in year 1 equals $770,000, the equity or residual cash flow in year 1 equals $41,138 and the equity dividend rate equals 1.4% ($41,138/ $2,944,681).

The selling price at the end of year 1 equals $10,106,250 ($808,500/0.08) and the remaining loan balance equals $6,829,152. This implies that [m. sub.12] = 0.675 ($6,829,152/$10,106,250) and that $3,277,098 of equity capital is invested in the property at the beggining of year 2. This increase in equity capital increases the denominator in the year 2 equity dividend rate. A variation in NOI in year 2 thus will have a relatively smaller impact on the equity dividend rate in year 2 than would the same variation in NOI in year 1. As the amount of invested equity increases over the holding period, the after-debt rate of return to an equity investor becomes even less sensitive to a given percentage variationm in net operating income.

Equation (2) incorporates this clear dependence of the required equity return on the amount of financial leverage. [m. sub.t] increases as [m. sub.t] decreases over time; thus the unlevered business risk premium (RP) is scaled up by successively smaller amounts. To clarify these relationships, the loan- to-market-value ratios, equity discount rates, and weighted average costs of capital for the example property over 20-year investment horizon are reported in Table 2. These calculations assume that RP is a constant equal to 0.024. With the 70% initial loan-to-value ratio and a 10% yield available on Treasury securities, the required equity rate in the first year of operations (from Equation [2] is 18% (0.10 + [0.024/[1 - 0.70]]). This in turn implies from Equation [1]) that [WACC sub.1] = 0.1240 (0.70 x 0.10] + [1 - 0.70] x 0.18). With 5% nominal price appreciation and normal loan amortization, the loan- to-market-value ratio at the beginning of the year 2 is 67.5%, which implies that [y sub.e2] = 0.174. Substitution of [m sub.2] = 0.675 and [y sub.e2] = 0.174 into Equation (1) produces [WACC sub.2] = 0.1240. The time-dependent relationship between [m sub.t] and [y sub.et] is depicted graphically in Figure 1.

Kincheloe states that the advantage of discounting before-debt cash flows using a fixed WACC is that "The costs of debt and equity capital are adjusted automatically for a varying capital structure." (4) This is an oversimplification; the opportunity cost of equity capital is not automatically adjusted when the WACC technique is used. Rather, use of a constant WACC implicity assumes that the cost of equity financing is being adjusted over time as the loan-to-market-value ratio

TABLE 2 Leverage Rates, Equity Discount Rates, and Implied Weighted Average Costs of Capital
 [m.sub.t] [u.sub.et]
 Loan-to-Market-Value Equity Discount
Year Ratio * Rate ** [WACC.sub.t] ***
 1 70.0% 18.0% 12.4%
 2 67.5% 17.4% 12.4%
 3 63.9% 16.6% 12.4%
 4 60.4% 16.0% 12.4%
 5 57.0% 15.5% 12.4%
 6 53.8% 15.2% 12.4%
 7 50.7% 14.8% 12.4%
 8 47.8% 14.6% 12.4%
 9 44.9% 14.3% 12.4%
10 42.2% 14.1% 12.4%
11 39.5% 13.9% 12.4%
12 37.0% 13.8% 12.4%
13 34.5% 13.6% 12.4%
14 32.2% 13.5% 12.4%
15 29.9% 13.4% 12.4%
16 27.7% 13.3% 12.4%
17 25.5% 13.2% 12.4%
18 23.4% 13.1% 12.4%
19 21.4% 13.0% 12.4%
20 19.4% 12.9% 12.4%
 (*) Loan-to-market-value ratio ([m.sub.t]) is the
outstanding loan balance divided by the market value of the
property at the beginning of the year. Market value is equal
to year-end net operating income
capitalized at 8%. Gross income is expected to increase 5%
per year. The assumed loan terms are:
10% Treasury yield ([y.sub.t]), 30-year term; annual
amortization; and an initial 70% loan-to-value ratio
([m.sub.t])
 (**) Equity discount rate = [y.sub.et] = [y.sub.t] +
[RP/(1 - [m.sub.t])], where RP is the required risk premium on a
project financed entirely with equity
 (***) [WACC.sub.t] = [m.sub.t][y.sub.t] + (1 -
[m.sub.t])[y.sub.et]


declines. the question that Kincheloe does not address is whether the cost of equity over the expected holding period is being adjusted in a way that is consistent with economic and appraisal theory.

A more correct statement is that using a fixed WACC implicitly assumes the relationship between [m.sub.t] and [y.sub.et] that is depicted by Equation (2). That is, use of WACC is based on the assumption that the appropriate risk premium on a leveraged investment is equal to RP/(1 - [m.sub.t]). This can be verified by noting that the annual equity discount rates calculated from Equation (2) and displayed in Table 2 are the only discount rates that imply a constant annual WACC equal to 12.4%. Further, setting the risk premium on a leveraged investment equal to RP/(1 - [m.sub.t]), while reasonable, is by no means the only assumption that could be made concerning risk and required equity returns.

COMPARNG VALUE

ESTIMATES FOR THE

EXAMPLE PROPERTY

Kincheloe sets the equity discount rate in year 1 equal to 18%. This implies a WACC in year 1 of 12.4%, as has been previously shown. By assuming that the WACC does not change over time, however, Kincheloe also implicitly assumes that the required equity return in any year is determined from Equation (2). The present value of the all-equity cash flows--that is, the stream of NOI, discounted at 12.4%--is equal to $9,815,603.

The equity valuation technique explicitly incorporates the cash flows associated with the debt financing; the amount disbursed by the lender at loan origination; the periodic payments of interest and principal; and the repayment of the remaining loan balance at the end of the investment holding period. In this example, the investor promises to make annual mortgage payments of $728,862 in exchange for a lump sum amount equal to $6,870,922 (0.70 x $9,815,604) at loan origination. Debt financing thus lowers both the amount of equity capital that is invested in the property at any time and the amount of residual cash flow that can be paid out to the equity investor after servicing the debt. The remaining loan balance at the end of the 10-year holding period will be $6,205,215, leaving a residual cash flow from sale to the equity investor of $9,311,432.

The present value of the equity cash flows (the last line in Table 1) equals $2,495,954 when discounted at a constant 18% rate. Thus the total indicated value of the property (equity + debt) is $9,366,876. This is $450,464 (or about 5%) less than the value of the property derived from the WACC technique.

WHICH INDICATED

MARKET VALUE IS

CORRECT?

Kincheloe concludes that the estimate of $9,815,603 derived through the WACC is the correct estimate of market value and that the equity valuation technique should never be employed because it is "theoretically indefensible." He is half correct. The amount $9,815,603 derived from the WACC approach is the correct estimate of value in light of the appropriateness of the other variable assumptions. However, the equity valuation technique will in fact produce the same $9,815,603 estimate of value if it is properly employed.

More specifically, as the absolute and relative amounts of equity financing increase over time, the expected variability of equity returns decreases, and the required equity return must accordingly decrease. Setting the equity rate equal to a constant 18% is thus incorrect because it involves an internal inconsistency. If the equity flows in each year are discounted using the equity discount rates in Table 2 (i.e., the specification of [y.sub.et] in Equation [2]), the equity valuation technique produces an estimate of market value equal to that obtained using the WACC approach. That is, both approaches produce the same estimate of market value as long as the equity discount rates explicitly assumed when using the equity valuation model are equal to the equity discount rates that are implicitly assumed in the WACC approach.

CONCLUSION

According to economic theory as well as appraisal theory, the variability of the after-debt returns to equity investors decreases over time as the portion of the ongoing investment that is financed with equity increases. A major problem with the typical application of the discounted cash flow equity valuation model is that the equity discount rate is assumed not to vary over time. As Kincheloe correctly points out, this common mistake produces value estimates from the equity valuation model that are lower than those obtained using an approach that discounts before-debt cash flows (i.e., NOI) by the weighted average cost of the debt and equity capital.

This article explores the subtle relationship between the WACC and equity valuation techniques, demonstrating that the two approaches will produce consistent estimates of value if analysts correctly specify the equity discount rate when using the equity valuation technique. More specifically, this article demonstrates that, when using the WACC technique, an appraiser is, in fact, implicitly assuming that equity discount rates decline over time as the loan-to-market-value decreases. If these implicitly assumed equity discount rates are used when applying the equity valuation technique, the two approaches will yield consistent estimates of value. Thus, the equity valuation model need not be scrapped, as recommended by Kincheloe. Rather, analysts simply need to avoid the most common mistake associated with its application--equity discount rates that are assumed not to decrease over time.

(1) American Inst. of Real Estate Appraisers, The Appraisal of Real Estate, 9th ed. (Chicago: American Inst. of Real Estate Appraisers, 1987), 537-539.

(2) Ibid., 541-548.

(3) Stephen C. Kincheloe, "The Weighted Average Cost of Capital--The Correct Discount Rate," The Appraisal Journal (January 1990): 88-95.

(4) Ibid., 93.

David C. Ling, PhD, is a professor at the graduate school of business administration in the University of Florida. He received an MBA in finance and a PhD in finance and real estate economics from the Ohio State University.
COPYRIGHT 1992 The Appraisal Institute
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 1992 Gale, Cengage Learning. All rights reserved.

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Author:Ling, David C.
Publication:Appraisal Journal
Date:Apr 1, 1992
Words:3187
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