Journal of Real Estate Finance and Economics, 4:273-281 (1991) �9 1991 Kluwer Academic Publishers

Modeling the Behavior of Real Asset Prices

TAEWON KIM Department of Finance and Law, School of Business and Economics, California State University, Los Angeles, 5151 State University Drive, Los Angeles, CA 90032

Abstract

Recently there has been much research treating housing and other real assets as financial claims, primarily in order to value their derivative assets, such as mortgages and mortgage-backed securities. Real asset prices are then typically modeled as a lognormal process, in the same manner that has traditionally been applied to firm value. The service flow or implicit value of a house is thus considered, in analogy with stock dividends, to be a fixed proportion of the fluctuating house price. We consider the appropriateness of this formulation and draw some distinctions between real assets, such as a house, and investment enterprises, such as a firm. We then pro- pose an alternative method of formulating the service flow and the price of real assets which seems more appro- pilate to the economic characteristic of such assets.

Keywords: Equilibrium asset pricing models, consumption-based CAPM, interest-dependent rates of service flow, contingent claim analysis

1. Introduction

There has been much recent research treating housing and other real assets as f inancial claims, pr imar i ly in order to value their derivative assets (mortgages, mortgage-backed

securities, etc.). Bui lding prices are typically then modeled as a log-normal process, in precisely the manne r that has tradit ionally been applied to f i rm equity. The service flow or implici t rental value of a house is thus considered, in analogy with stock dividends,

to be a fixed proport ion of the f luctuating house price. We consider the appropriateness of this formulat ion and, in so doing, draw some per t inent dist inctions be tween fixed real assets, such as a house, and entire investment enterprises, such as a firm. As a conse- quence of these distinctions, we propose an alternative method of formulat ing the service flow and price of a given real asset which seems more appropriate to the actual economic

character of such assets.

2. Real assets

2.1. Lognormal form

Real assets, such as bui ldings and other physical structures, give off a flow of benefits, which are either explictly valued, in case the bui ld ing is rented out, or implici t ly valued,

274 TAEWON KIM

in case the building is owner-occupied. Invariably, movements in price X(t) of such a real asset are described, in the fashion of a stock with dividends, by

d X _ (a - s)dt + adz. (1) X

The stochastic differential equation is composed of both a deterministic part and a ran- dom one. The second, random part is driven by the white noise process dz, where each z(t) is a standard normal random variable (see Ingersoll, 1987). When the parameters of the total rate of return, or, the rate of service flow, s, and the instantaneous standard devia- tion, o, are treated as constant, as usually they are, then the distribution of X(t) becomes log-normal. When doing contingent claim analysis, though, there is no actual need to assume a constant, since if X(t) represents a traded asset, then by well-known arbitrage reason- ing, the return o~ will not appear as part of the valuation of derivative claims on X. 1 Our attention will focus instead on the appropriateness of treating the service flow as a fixed proportion of the asset, in analogy to the practice with stock dividends. In contrast to the gross return a , the service flow of the real asset underlying a contingent claim has a con- siderable effect on the value of the derivative security, since this "dividend" represents the portion of the underlying real asset's return that is not currently captured by the con- tingent claim. Further, it is the future flow of dividends that is the ultimate source of value for the real asset, and thereby for all its derivative securities.

2.2. Dividends and service flows

Presumably, when such economic variables as prices are assumed stochastic, the source of uncertainty lies in the primitives of some equilibrium model generating these prices. When modeling anything less than a full equilibrium setup, however, one has considerable freedom in where to introduce the stochasticity. Nonetheless, it is important that the assumed stochastic forms be compatible with basic economic reasoning. An example of such a basic economic principle is the idea that the value of an asset is the expected discounted value of its flow of returns. It is the value of these services that is determined by underlying economic forces, and it is from this that should come the value of the asset itself. It is not the case at all that the value of the asset instead determines the service flow, as sug- gested by the fixed proportionate relationship of the log-normal asset form (equation (1)). Of course, a simple proportional relationship does allow causality to go either way, so it might simply turn out that the capitalized value of the asset is always in some fixed propor- tion to the current level of the dividend flow that generates the asset's value. As we shall see, however, this is unlikely to be the case in a stochastic environment.

The principle whereby flow value determines stock value must be true of all assets, not just housing stock. One may ask, then, how it is that the log-normal form (1) should be used so widely in modeling firm equity [e.g., Merton, 19"/3b), and yet we could find it inadequte for describing real assets? The basic point we seek to establish is that a real asset, such as a standing house, is a fixed structure whose physical flow of services, if not their value, is predetermined. Rational owners will always accept these services, since

MODELING THE BEHAVIOR OF REAL ASSET PRICES 275

if they are not used they are lost. Capital gains aside, the return and the "dividend" on a real asset are one and the same thing. Furthermore, the only source of expected capital gains is the anticipation that the given dividend flow will increase in value in the future.

In contrast, a firm is an entity able to engage in an ongoing sequence of investment pro- jects. It is not necessarily the case that the physical flow of return be emitted as dividends, since it may instead be reinvested in new or expanded projects, resulting in further capital gains to the firm. In fact, the traditional Modigliani-Miller reasoning, applicable in com- plete markets, is that the value of a finn is independent of dividend policy, and so dividends can be chosen at will, without affecting anything of real consequence. In particular, the policy could well be to set the dividend flow equal to some fixed proportion of current firm value, as indicated by equation (1).

Unlike a firm, in the case of a single fixed asset, one cannot independently establish the principle that the service flow be a given proportion on asset value; the economic struc- ture will have to allow this to be so. The general form for the evolution of an asset's value (Cox, Ingersoll, and Ross, 1985) is

dX = (t~X - 6)dt + ~dz (2)

where 6 (t) is the instantaneous flow of dividends from the asset. We seek the circumstances under which

6(0 = sX(t) , (3)

so that the general form (2) can reduce to (1). The first part of our investigation explores conditions under which the service proportion s is independent of 6, though it may still depend on other stochastic factors. These are the minimal circumstances in which the pro- portionate relation (3) has any meaning. We then consider the stricter conditions necessary for s to be in fact constant. Our general conclusion will be that the service proportion s is unlikely to be strictly constant and must be allowed to depend on such factors as the spot interest rate r(t). We therefore discuss alternative methods for modeling dividend flows that reflect this dependence.

3. Constant rates of service flow

3.1. Rates independent o f the f low

Our argument that it is dividend flow that determines asset value and not vice versa leads us to the conclusion that we should model dividend flow as an exogenous stochastic factor with asset value as an induced stochastic variable. Nonetheless, there is nothing incorrect about employing asset X( t ) as an instrumental variable to proxy for dividend flow, when valuing financial claims on the asset ? We are in fact assuming that this standard practice will be continued. What we are requiring, though, is that the specification of the asset process recognize the source of the asset's value in terms of some dividend flow.

2 7 6 TAEWON KIM

The solution for the value of an asset moving according to the general process (2) is (Cox, Ingersoll, and Ross, 1985):

X ( t ) = E e ~, 6 ( r ) d r . (4)

Here, again, 6 is the dividend flow, whereas o~ (t) represents the required rate of return on the asset X ( t ) at t ime t. This form makes evident the proposition that asset value is the appropriately discounted expected value of the stream of dividends. Note, though, that besides its dependence on the dividend flow 6, the asset X will be affected by any other stochastic variables that influence the required return a . Making use of (4), we can re- express our desired condition (3) as

1 1 1 E e ~, b ( r ) d r . (5) s 6( t )

I f s is to be independent of 6 (t), it must be possible to factor that term out of the expecta- tion operator. There are two cases of interest where this is possible.

First, condition (5) will obtain if every possible trajectory of dividend value is homogenous of degree one in the current dividend level. In terms of the dividend process itself, this requires

d5 _ t3dt + ~dz, (6) 6

where 13 and ~ are independent of 6. Let O (6 (r)16 (t)) be the density function of b(r) for ~- _ t, given 6 ( 0 . We then have 4~(6(r)16(t)) = 6(t)qb(6(r)]l) , and so

Is, 1 1 X ( t ) = E e J, 6(r)16(t) = 6 ( t ) E Jt e , 6( r ) l l __ _1 6(t), (7) t S

as desired. The other case giving the desired form occurs when the average effect E[6(r)16 (t)] is

separable from the discount term. Thus, if the stochastic disturbances determining 6 are independent of those affecting o~, one has

i f E E e - f t ~(u)6(r)dr = E [ 6 ( r ) ] E e , dr , (8)

and so, if E[6 (7)16 (t)] = ~ ( t ) f ( t , r ) , where f ( t , r ) is independent of 6 (t), then one can again factor 6(t) out of the expectation operator. This last condition will hold when

d6 _ fldt + ~ (6 )dz , (9)

MODELING THE BEHAVIOR OF REAL ASSET PRICES 277

where now only/3 need be independent of S, since E[ f ~(5)Sdz = 0, by properties of the Brownian motion process z. A case of particular interest is when B -- 0, so that ~ is a martingale, that is, E[5(r)l~(t)] -- 5(t). This means that the service flow is not anticipated to trend either upward o r downward. Such a situation results in X(t) = E l f e-Y ~(u) 5(r)l~(t)] = 5( t )E[ f e-Y ~(u)] __ 1/s ~(t), where not only is s independent of the cur- rent value ~ (t), it is independent of the dividend process altogether.

The scenarios described in the previous paragraphs would lead to ~ (t) = sX(t), where s is independent of 6. On the other hand, they would still not result in s being a fixed constant, since s would still depend on those stochastic factors that influence or, of which the interest rate r(t) is most conspicuous. This is scarcely surprising and agrees with the intuition one brings from the simplest possible examples. For instance, if an asset emits a dollar's worth of benefit in a deterministic steady state described by interest rate r, then the asset value X will simply be 1/r and the dividend/value ratio will, of course, be s = r. Obvious as this may be, the intuition that s depends (positively) on r is violated when working with processes in the form of equation (1).

3.2. Absolutely constant rates of f low

In order to understand their limitations, we investigate those exact conditions necessary in order that the proportion s be truly constant. The horizon will need to be infinite or else the value X(t) = E [ f f e-f , ~(u)du 6( r )d r ] will depend not only on the current value 6(0 but on the time to termination. More important, it is also clear that the only source of stochasticity of X(t) will have to be through 6 (t), since unanticipated movements of other factors would destroy any fixed relation between X and &3

Having required dividends to be the only source of asset variability, the differential form (2) for asset price movement can then be written more explicitly as

dX(t) = (otX(t) - ~3)dt + X~(t)~dz, (10)

where 8 is the instantaneous variance of the dividend process. We desire that 6 (t) = sX(t) for all t, which in turn implies dt ( t ) = sdX(t). Substituting (10) into this condition gives

d6 = s(o~X - 6 ) d t + s X ~ S d z (11)

= ~(o~ - s ) d t + 8dz , (12)

where we again make use of the condition 6(t) = sX(t), and thereby the condition sX~= 1. Finally, we need to take ~ = 6a if we want X(t) to satisfy equation (1). Thus, the dividend process must be of precisely the same form

d8 _ (at - s )d t + adz, (13) ~5

as the asset process itself if the log-normal form (1) is to hold?

278 T A E W O N K I M

3. 3. Asset value without service flow

We are now in a good position to see an explicit example of the need to found asset value on a dividend process. Consider what happens if one simply evaluated (1) directly. The solution is easily seen to be

T

X(t) = E[e -ft (~-s)du X(T)], (14)

where T is some later time. This solution, however, begs the question of valuing X(T). It is far from satisfactory to consider X(T) as some terminal value of the asset, since this value is likely to be zero, in which case one is forced to conclude that the value X(t) is zero over all time? One could alternatively write down the solution

t t

X(t) = e-f-'o (~-s)au + fro dz X(to) (15)

based on some earlier value X(to) and the actual value z(t) - z(t0), but besides raising the question of how the earlier value X(to) came to be, this approach violates the princi- ple that X(t) should only depend on future occurrences, not past ones. Of themselves, solu- tions to equation (1) are very much like bubbles, since any level X(t) can support itself through time. On the other hand, when equation (1) arises through the dividend process (13), so that

X(t) = 6(t)E e-f, sdu dr _ 6(0, t S

then the value of X(t) is pinned down by 6(0, which in turn depends on real economic forces in the here and now.

4. Interest-dependent rates of service flow

4.1. Alternatives to the log-normal form

We have just argued that the log-normal process (1) needs to be founded on an explicit dividend process, and we have seen what conditions must hold in order that this be so. We do not, however, regard those conditions as very plausible. Uppermost among our ob- jections is the requirement that the risk-adjusted rate of return must depend only on 6, when in fact economic intuition would require that it depend on other factors as well, such as the interest rate r(t). A fixed proportion relation between X and 6 would also force other relationships that do not necessarily agree with one's economic sense. One expects the value X(t) of the asset to depend negatively on r, due to the heavier discounting of the dividend flow that would presumably occur when interest rates rise. On the other hand, to the extent that one has an intuition as to how the interest rate affects the dividend flow, one expects that the dividend flow will rise in value, in concert with r. These two positions,

MODELING THE BEHAVIOR OF REAL ASSET PRICES 279

however, are made incompatible by the fixed proportion dividend rule, which requires that both the dividend and asset value move in proportion to any change in an exogenous variable.

I f the fixed proportion dividend relation is not to be used, then another must be found. The most satisfactory method is, of course, to directly modei the dividend flow. Given the intimate relation between real asset dividends and the rental values, this is by no means an impossible task. However, for the very reason that the problem seems straightforward conceptually, we have little more to say concerning it, and turn instead to the problem of inferring the dividend flow indirectly through movements in the asset prices themselves.

4.2. Consumption-based capital asset pricing models

The basic relation to be employed is

E[dX(t ) l = ~X( t ) - 6, (16)

stating that the required rate of return c~ consists of dividends, together with capital gains, that is, c~ = E[dX/X] + s. There is again no difficulty in principle in estimating expected capital gains, using the same past information that rational agents would use to form their anticipations of the capital gain process. Without any loss in generality, then, we will con- sider the case where the entire expected return consists of dividends. This is the benchmark case where the asset's value is a strict martingale, that is there are no reasons to anticipate either capital gains or losses in the future. Unless one has specific knowledge of a future different from the present, this is the case on which one should concentrate.

The problem we face, then, reduces to estimating the required rate of return on the asset, since now s = ~. This is, however, a well-understood problem that lends itself naturally to empirical estimation. It is known from consumption-based asset pricing models that the anticipated excess return of an asset depends on that of the market portfolio as well as those of certain other portfolios hedging against shifts in investment opporttmities. If, following Merton (1973a), we suppose that these latter shifts can be captured by fluctua- tions in the interest rate, then we have

- r = ~ M ( O I M - - r ) q- ~ l ( O l l - - r ) . (17)

Here, aM is the anticipated return on the market portfolio M, whereas at is the expected return on some default-free bond, whose movements, given the assumption of a single- state term structure, would be perfectly correlated with interest rate fluctuations. The coef- ficients/3 M and ~l are, of course, ordinary multiple regression coefficients of the given asset with the market portfolio and default-free bond, respectively. Estimating these coef- ficients is a standard problem in finance, and thus it is possible by entirely orthodox methods to recover an expression for the service flow of the given asset.

Since the regression coefficient/31 undoubtedly should be positive, one is confident that the real asset's risk premium X --- a - r will also be positive. It is this term that now appears in the fundamental valuation equation for a derivative asset F, contingent on X. Indeed, by standard reasoning, one obtains

280 TAEWON K1M

1/2var(r)Fr~ + cov(r , X )Fr x + 1 / 2 v a r ( X ) F x x +

(IZr + ~kl)F r - (or(r) - r ) F x - rF + Ft = O. (18)

Here/z r is the drift term of the spot interest rate process and kl = O/l - - r is the interest rate risk premium. This valuation equation is of the same form as used in the literature for pricing mortgage instruments, except that the rate of return a (r) has replaced the fixed parameter s that arises when a process of form (1) is used. The solution of this problem is formally identical to the one pertaining to a risk-neutral individual who faces an asset

X that depreciates at expected rate k = ot - r. This declining value of the risk-neutralized

process for X increases the value of put-like securities, such as mortgages, over the value they would have when s < r.

Regressing o~ in the fashion we have described will yield it as a function of stochastic influences other than X, which we assume are being adequately captured in the term struc-

ture r(t) . In general, the dividend proportion s --- ~/Xmay differ from o~ by expected capital

gains. We have already discussed the conditions of lognormality under which E[dX/X]

is independent of X. In the remaining cases, the expression s = o~ - E[dX/X] will then depend on both X and r.

5. Conclus ion

We have argued that the constant dividend rate assumption of the log-normal form is inap- propriate when pricing real assets. Dividends then take the form of a service flow, whose value will surely depend on such economic considerations as the interest rate. We have

instead proposed decomposing the service flow into the difference between the required rate of return and expected capital gains. The latter can be easily estimated on the basis of past experience, while the former can be obtained in typical CAPM fashion from a regres- sion on both the market portfolio and some representative default-free bond. If contingent claims such as mortgages are to be accurately priced, it is essential that the service flows

of the underlying real assets be correctly modeled.

Notes

1. Examples of papers using the lognormal form to model the value of buildings include Cunningham and Hender- shott (1984), Kau and Associates (1987, 1990a, 1990b), and Titman and Torous (1989).

2. For instance, the dividend process is obviously an instrumental variable itself, reflecting changes in underly- ing tastes and technology of the equilibrium model in which asset pricing must be embedded.

3. Actually, anticipated movements are also likely to destroy any fixed relation, since at the very least they will make s time dependent.

4. Notice that when ~5 = sX, then E[dX] = 0 exactly when E[d~] = 0, both of which require that a = s. If we have a > s, both 6 and X grow unboundedly large, whereas if a < s, both ~5 and X exponentially decline toward zero. We instead regard the steady state case, where c~ = s, as the benchmark in the discussion to follow. ]an this case, both ~ and X are martingales.

5. Note that in order for equation (1) to correctly hold, it was required that there be no terminal date.

MODELING THE BEHAVIOR OF REAL ASSET PRICES 281

References

Cox, J.C., Ingersol, J.E. Jr., and Ross, S.A. "An Intertemporal General Equilibrium Model of Asset Prices?' Econometrica 53 (1985), 363-384.

Cunningham, R.E, and Hendershott, P.H. "Pricing FHA Mortgage Default Insurance?' Housing Finance Review 3 (1984), 373-403.

Ingersoll, J.E. Theory of Financial Decision Making. Rowman and Littlefield, 1987. Kau, James B., Keenan, Donald C., Muller, Walter J., 1ii, and Epperson, James F. "The Valuation and Securit-

zation of Commercial and Multifamily Mortgages?' Journal of Banking and Finance 11 (1987), 526-546. Kau, James B., Keenan, Donald C., Muller, Walter J., III, and Epperson, James F. "Floating-Rate Securities

with Default and Prepayment?' Working paper, University of Georgia, 1990a. Kau, James B., Keenan, Donald C., Muller, Walter J., III, and Epperson, James E "Pricing Fixed Rate Mort-

gages with Default and Prepayment?' Working paper, University of Georgia, 1990b. Merton, Robert, ' ~n Intertemporal Capital Asset Pricing Model?' Econometrica 41 (1973a), 867-887. Merton, Robert, "Theory of Rational Option Pricing" Bell Journal of Economics and Management Science 4

(1973b), 141-183. Titman, S., and Torous, W. "Valuing Commercial Mortgages: An Emperical Investigation of the Contingent Claims

Approach to Risky Debt" Journal of Finance 44 (1989), 392-412.