# The Gordon Growth Model and the income approach to value.

The idea that property has a value directly proportional to its earning ability has been around since at least the sixteenth century.(1) At that time, though, a distinct and informed property market was not readily apparent, and the factors used to convert or discount the earnings stream were determined or explained largely in terms of the prevailing rate of interest on money, or alternatively, increases in the money supply.(2)Later, in the eighteenth century, the increasing separation and sophistication of the property market in general (especially in the commercial, retail, and industrial sectors) resulted in the income approach (to market value) becoming increasingly prominent. A critical factor in this prominence was the increasing separation of property ownership from property function or use.(3) As greater numbers of people came to own property but not specifically use it, the change in ownership perspective saw property move from simply an additional factor (or cost) in the overall production process to more of an independent, income-earning commodity in itself, with no necessary ties to its existing use or current fluctuation.

In fact, at this time the whole theoretical concept of economic or exchange value was changing. The traditional "objective" (i.e., cost-oriented) theories argued by Marx and Ricardo (and upon which the cost approach itself is still based) were being challenged by a new "subjective" (i.e., consumer-oriented) theory, championed first by Auguste Comte and culminating in H. H. Gossen's theory of marginal utility that underlies virtually all modern economic value thinking today.

The perceived market or exchange value of property was thus becoming less tied to its current function or existing use, and more tied to its potential utility or "highest and best" use, as determined in the marketplace. The factor used to discount a current rental stream increasingly became analyzed within that marketplace itself, reflecting a property's potential worth, compared with other similar properties, as a market-sale-price-to-earnings ratio (or its reciprocal). This factor has subsequently become the cornerstone of the traditional income approach to market value and is usually expressed as a percentage via the expression

(Income/Sale price) x 100

The income approach uses this market-based percentage to discount in perpetuity the existing (and assumed constant) income derivable from a particular property. While simple to use, the ensuing problem was relating this unique property return or market-based capitalization rate to rates of return on other available investments, including other forms of property investments.

Many instruments explaining property return in relation to rates on other investments have, over the years, been put forward, including:

* The built-up capitalization rate

* The component-weighted capitalization rate

* Ellwood capitalization

* Dual-rate and tax-adjusted annuity capitalization

* Finance-weighted capitalization

All of these attempt to model the basic rate by breaking it down into various factors directly affecting the decision of the average investor. These factors include the timing of cash flows; individual rates of return on the land, buildings, and equity; mortgage interest rates; taxation rates; and so forth. While these techniques have increased our understanding of the capitalization rate, they have also been more complicated than is practical for many appraisal purposes and thus not commonly used. A further technique was popularized in the 1960s by Myron J. Gordon for the analysis and valuation of financial securities.(4) As with the traditional income approach, it is another derivation of discounted cash flow (DCF) analysis now commonly called the Gordon (constant) Growth Model. While not often used now for the valuation of real estate, it can be used to relate an investment return to other available investments by incorporating an expected constant growth component. In addition, it is relatively simple to use and can be analyzed within the relevant market.

THE TRADITIONAL INCOME APPROACH AND THE GORDON GROWTH MODEL

Both approaches have as their origins the simple and logical cash flow timeline. Because the future value of an income stream is that income stream plus the rate of return able to be earned on that income, the present value (PV) of an income stream is the reciprocal, that is

PV = Income in first period discounted for one period + Income in second period discounted for two periods ... + Income in the nth period discounted for n periods

The traditional income approach is not only empirically based but is mathematically derivable, assuming the income in each period is a constant with zero growth and existing in perpetuity. This assumption of a perpetual income stream is important as it is in line with the perpetual nature of rights attaching to property in general. In addition, a constant reversion value can be directly incorporated into present value as part of the perpetual income stream.

The derivation of the traditional income approach from the basic DCF formula is as follows

P|V.sub.0~ = (|Inc.sub.1~/|(1 + r).sup.1~) + (|Inc.sub.2~/|(1 + r).sup.2~) + ... (|Inc.sub.n~/|(1 + r).sup.n~) (1)

where

P|V.sub.0~ = Present value

|Inc.sub.n~ = Income in the nth period

r = Discount rate applicable to that period

First, because income is assumed constant (with zero growth), it may be isolated out

P|V.sub.0~ = Inc x (1/|(1 + r).sup.1~ + 1/|(1 + r).sup.2~ + ... 1/|(1 + r).sup.n~) (2)

Now the equation may be simplified by multiplying both sides by (1 + r):

P|V.sub.0~ (1 + r) = Inc x (1 + 1/|(1 + r).sup.1~ + 1/|(1 + r).sup.2~ + ... 1/|(1 + r).sup.n-1~) (3)

and second, subtracting equation (2) from equation (3) to give the result

P|V.sub.0~ x (1 + r - 1) = Inc (1 - 1/|(1 + r).sup.n~) (4)

The resulting formula may be further simplified by remembering that the second assumption is of a perpetual income stream, thus as n tends to infinity,

1/|(1 + r).sup.n~ |approaches~ 0

Therefore, if we remove the above component, we arrive at

P|V.sub.0~ x (r) = Inc

or, finally, rearranged as the traditional income approach formula with resulting basic capitalization rate (r) incorporating a perpetual income component with zero growth:

P|V.sub.0~ = (Inc/r)

While the resulting formula is simple and easily applied, it unfortunately also rests upon a fundamental assumption not found in the "real world"--a constant, never-changing income stream.

As valuation is primarily concerned with anticipated or future benefits and returns, the constant (zero-growth) income stream assumption, even though in perpetuity, seems an important contradiction. What is required to remove this contradiction is some method of allowing for any anticipated growth (or decline) in both the perpetual income stream and the anticipated reversionary value. One method of allowing for such growth may be shown by expanding the original cash flow timeline as

P|V.sub.0~ = |Inc.sub.0~|(1 + g).sup.1~/|(1 + |r.sub.m~).sup.1~ + |Inc.sub.0~|(1 + g).sup.2~/|(1 + |r.sub.m~).sup.2~ + ... |Inc.sub.0~|(1 + g).sup.n~/|(1 + |r.sub.m~).sup.n~ (5)

where

|Inc.sub.0~ = Existing or current rental

|r.sub.m~ = Discount rate that equates to the overall market rate of return, including an income growth component

g = Expected growth component

By adding this (1 + g) component to the basic cash flow timeline and considering it again in perpetuity, an expected growth component is now allowed for in both the income stream and subsequently the value of the asset over time. This not only removes the constant (zero-growth) income assumption but also results in an overall market-based return that may be directly compared with other investment opportunities on the basis of expected growth potential.

While the above equation is seen as the theoretical framework for all traditional cash flow analysis, it may also be simplified by following the same steps as before; that is, first removing the existing income as a constant within the equation (while retaining the growth component),(5)

P|V.sub.0~ = |Inc.sub.0~ x (|(1 + g).sup.1~/|(1 + |r.sub.m~).sup.1~ + |(1 + g).sup.2~/|(1 + |r.sub.m~).sup.2~ + ... |(1 + g).sup.n~/|(1 + |r.sub.m~).sup.n~) (6)

and second, multiplying both sides by

(1 + |r.sub.m~)/(1 + g)

P|V.sub.0~ ((1 + |r.sub.m~)/(1 + g)) = |Inc.sub.0~ (1 + |(1 + g).sup.1~/|(1 + |r.sub.m~).sup.1~ + |(1 + g).sup.2~/|(1 + |r.sub.m~).sup.2~ + ... |(1 + g).sup.n - 1~/|(1 + |r.sub.m~).sup.n - 1~) (7)

and third, subtracting equation (6) from equation (7)

((1 + |r.sub.m~)/(1 + g) - 1) x P|V.sub.0~ = |Inc.sub.0~ x (1 - |(1 + g).sup.n~/|(1 + |r.sub.m~).sup.n~)) (8)

which is rearranged as

((1 + |r.sub.m~) - (1 + g)/(1 + g)) x P|V.sub.0~ = |Inc.sub.0~ x (1 - |(1 + g).sup.n~/|(1 + |r.sub.m~).sup.n~)) (9)

Finally, assuming the overall market return (|r.sub.m~) will always be greater than the expected growth component (g), the perpetual income stream again means that n will tend toward infinity, consequently

|1 - (|(1 + g).sup.n~)/(|(1 + |r.sub.m~).sup.n)~ |approaches to~ 1

Thus, as the above component tends toward 1 the equation is further simplified as

((1 + |r.sub.m~) - (1 + g)/(1 + g)) x P|V.sub.0~ = |Inc.sub.0~ (10)

or

(|r.sub.m~ - g) x |PV.sub.0~ = |Inc.sub.0~ x (1 + g) (11)

and finally, rearranged as the Gordon Growth Model

P|V.sub.0~ = |Inc.sub.1~/(|r.sub.m~ - g) (12)

where

|Inc.sub.1~ = Expected income in one year, that is |Inc.sub.0~ x (1 + g)

Mathematical gymnastics aside for a moment, the resulting formula shows that the Gordon Growth Model (equation 12) is based on an important assumption found also in the property market. That is, the overall rate of return (|r.sub.m~) on any investment is a composite, first of initial cash flow generation and second of the anticipated growth in that cash flow over time. The capitalization rate used on the expected income in year 1 is therefore explained as a residual function of:

1. The overall rate of return required by the average investor in a particular sector of the investment market, as compared with other investment markets such as shares, bonds, interest rates, and property (|r.sub.m~)

2. The expected (and assumed constant) growth rate in investment return (g)

The constant growth equation therefore models a property's growth potential by breaking it down into its initial income and expected growth components. For example, while the commercial property market generally has an equivalent overall rate of return as compared with other available investments, each sector (and property) within that market also has a unique initial return (or capitalization rate), dependent upon the level of expected growth. The central business district, with higher expected asset growth rates, generally has lower income capitalization rates, and, in comparison, the entire suburban sectors with lower expected asset growth rates will experience higher income capitalization rates.

The value of any property is thus assessed as the expected income in year 1, capitalized, using its general market sector rate of return, net of the expected growth in the property itself.

Equation 12 can also be taken one step further, showing the initial capitalization rate (|Inc.sub.0~/P|V.sub.0~) in terms of the following

function:

y = |r.sub.m~ - g/1 + g

or more simply as

y = 1 + |r.sub.m~/1 + g - 1

where

g = Implied growth rate (pa) in income and asset value

|r.sub.m~ = Overall market sector rate of return, including the growth component.

A sale in the marketplace may then be analyzed for both the implied growth rate and implied overall return using the following derived formulas:

g = 1 + |r.sub.m~/1 + y -1

|r.sub.m~ = (1 + y) x (1 + g) - 1

For example, consider an industrial property with a current annual net rental of $8,000 pa, which has just sold for $100,000. If the overall sector rate of return (|r.sub.m~) is, say, 16% pa, the implied growth rate is

g = 1.16/1.08 - 1

= 7.407%

This market-derived growth rate can then be used to develop a capitalization rate for similar industrial property, that is

y = .16 - .07407/1.07407

= 8.00%

A comparable industrial property with an income stream of $7,000 pa can then be valued in two ways, first through the basic capitalization formula

P|V.sub.0~ = 7000/.08

= $87,500

and second through the original Gordon constant growth equation

P|V.sub.0~ = 7000 x (1.07407)/.16 - .07407

= $87,500 (allowing for rounding errors)

Notice that if each and every property had equivalent growth expectations both the traditional capitalization technique and the constant growth approach would result in the same figure, that is

(7000/87,500) = (.16 - .07407/1.07407)

= 8.00%

This situation, unfortunately, is seldom found in the "real world," and while the traditional approach allows little flexibility to objectively incorporate differing growth levels, the Gordon Growth Model may be used not only to analyze similar properties with varying growth expectations but also to incorporate these differences into a subsequent valuation.

The link between the two approaches is clear when we reapply our traditional constant, zero-growth assumption to the Gordon Growth Model. If the estimated growth in rental is assumed zero, then |Inc.sub.0~ = |Inc.sub.1~ and thus

P|V.sub.0~ = |Inc.sub.0~/(|r.sub.m~ - 0)

which simplifies to

P|V.sub.0~ = |Inc.sub.0~/|r.sub.m~

where |r.sub.m~ equates to the basic, or traditional, capitalization rate, again net of any growth component (since we have assumed it to be zero).

This approach is also useful in the calculation of long-term partial interests. Here, long-term lease provisions may allow for a constant expected growth in the rental stream (e.g., Consumer Price Index escalation clauses), with or without a reversionary input at the end of the lease term. Unlike a lengthy cash flow analysis, the Gordon Growth Model allows any expected growth to be incorporated (as a constant) into an assumed perpetual income stream. The approach is not only reasonably simple, but incorporates an inflated reversionary value as part of the perpetually growing income stream. The overall rate of return, including the income growth component, allows the appraiser to compare the capitalization rate directly with available returns on other investments and may be calculated in several ways:

1. As a built-up calculation

2. With the capital asset pricing model (or other multiple-factor models)

3. Analyzed within the market using the above formula where growth rates are empirically determined or known

The approach therefore offers a realistic compromise between the simplistic traditional approach and the lengthy versions of DCF analysis. In addition, it removes the sometimes nagging assumption of a constant income stream and offers a new insight into capitalization rates in a way far simpler and more practical than other, earlier explanations.

1. Fernand Braudel, Civilization and Capitalization--15th to 18th Century: The Wheels of Commerce (New York: Collins, 1982), 51.

2. P. H. Kelly, ed., Locke on Money, Clarendon edition (Oxford/New York: Clarendon Press, 1991), 50-52.

3. R. H. Tawney, "Property and Creative Work," in C. B. Macpherson, ed., Property: Mainstream and Critical Positions (Oxford: B. Blackwell, 1978), 140.

4. See part two, chapters four through six of Eugene F. Brigham and Louis C. Gapenski, Financial Management--Theory and Practice (New York: Dryden Press, 1988).

5. See chapter 15 and Appendix A of Thomas E. Copeland and J. Fred Weston, Financial Theory and Corporate Policy (Redding, Massachusetts: Addison-Wesley Publishing, 1988).

Marcus Jackson is a fully qualified urban valuer with Valuation New Zealand in Dunedin, New Zealand. He is working toward his registration and is completing a bachelor of commerce degree in finance and quantitative analysis at the University of Otago.

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Author: | Jackson, Marcus |
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Publication: | Appraisal Journal |

Date: | Jan 1, 1994 |

Words: | 2654 |

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