Valuing lease contracts A real-options approach

JOURNAL,OF hclal ELSlWlER

Journal of Financial Economics 38 (1995) 297-331

ECONOMICS

Valuing lease contracts A real-options approach Steven R. Grenadier Graduate School of Business, Stanford University, Stanford, CA 94305, USA

(Received July 1994;final version received November 1994)

Abstract Using a real-options approach to endogenously derive the entire term structure of lease rates, I develop a unified framework for pricing a wide variety of leasing contracts. The structure of the model is analogous to traditional models of the term structure of interest rates. I show how the model is flexible enough to determine equilibrium lease rates for leases of any term and practically any structure, including forward leases, leases with options to renew or cancel, lease insurance contracts, adjustable-rate leases, and leases with payments contingent on asset usage. Key words: Leasing; Options JEL classz$cation: G13; G12; E43

1. Introduction

Historical evidence suggeststhat leasing was in widespread use by the ancient Sumerians several thousand years B.C. and has continued through almost all advanced civilizations since. Until the 1950’s,however, modern-day leasing was predominantly confined to the- rental of real estate. Since then, leasing has burgeoned to include practically any assetimaginable, from copiers to satellites, trucks to airplanes, and power plants to zoo animals.

I am grateful for the comments of Geert Bekaert, Michael Harrison, Ayman Hindy, Francis Longstaff, Jeff Zwiebel, and seminar participants at the 1994Western Finance Association Meetings. Comments and suggestions made by Clifford Smith (the editor) and James Schallheim (the referee) led to substantial improvements in this paper. 0304-405X/95/$09.50 0 1995 Elsevier Science S.A. All rights reserved SSDI 0304405X9400820

Q

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Leasing plays a critical but often-overlooked role in the U.S. economy. It is clearly one of the most important sources of corporate financing. The rental payments made by corporations are substantial, even when compared with the interest expense incurred on all forms of debt. In 1993, the dollar amount of rental expenseswas more than 40% of the dollar amount of interest expensesfor all active companies reported in COMPUSTAT. In 1991, one-third of new business equipment was leased (World Leasing Yearbook, 1993), and the volume of equipment leasing almost tripled from 1982 to 1991 to a level of $124 billion. Automobile leasing alone has risen to unprecedented levels. In 1993, one-fourth of all cars and trucks sold - $43 billion worth of vehicles, a fourfold increase in just ten years - went out the door under lease. In the luxury car market, the percentage of leasing is more than 50%.’ Finally, while difficult to quantify, the largest sector of leasing is the rental of real estate. The commercial property market, which is dominated by office and retail properties, is valued at over $2 trillion (IREM Foundation and Arthur Andersen, 1991), and is prei dominantly leased under long-term rental agreements. This paper provides a unified framework for determining equilibrium lease rates. I present a model which determines the entire term structure of leaserates. Moreover, I apply the model to ‘real-world’ leasing contracts. Consider the following common leasing arrangements (from the real estate arena) which the model specifically addresses: 1. You are building an office tower which will be completed in one year. You want to sign a tenant to a ten-year lease,where occupancy begins in one year. The lease rate, however, must be set today. What is the equilibrium rent on such a lease? 2. A standard leasing arrangement for retail tenants in shopping malls is as follows. Every month, a fixed base rent is paid. However, if store sales rise above a given threshold level, a percentage of sales is paid to the landlord. What are the equilibrium combinations of base rent, sales threshold, and percentage rent? 3. An office tenant would like to lease one floor of your office building for ten years. However, the tenant suspects that it may be necessary to relocate in five years. Thus, you agree to a five-year lease, with the. tenant holding an option to renew for an additional five years. How much should you charge for such an option? I show that the model is flexible enough to determine equilibrium lease rates for leases of any term and with practically any assortment of embedded options, contingent fees,and structure. ‘In fact, the Jaguar XJ models holds the distinction

that over 90% are leased; Woodruff et al. (1994).

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The underlying approach of this paper is in the tradition of Miller and Upton (1976) with the focus on the economic aspects of leasing. The concept is very simple: leasing an asset is simply a purchase of the use of the asset over a specified period of time. Thus, leasing provides a mechanism for the separation of ownership from use, with the lesseereceiving the benefits of use and the lessor receiving the value of the leasepayments plus the residual value of the asset.This intuitive and simple framework allows the use of an option-pricing approach to lease valuation. For example, the value of leasing an asset for T years is economically equivalent to a portfolio which simultaneously purchases the underlying asset and writes a European call option on the assetwith expiration date T and zero exercise price. This characterization of leasing is explicitly derived by Smith (1979) using an option-pricing approach to valuing corporate liabilities. While this approach is intuitive and facilitates tackling some of the more complicated aspectsof real-world leasing, it also leaves out some very important leasing considerations. In particular, there is a vast literature on the importance of taxes in the lease-versus-buy decision (e.g.,see Schall, 1974; Myers, Dill, and Bautista, 1976; Brealey and Young, 1980; Lewis and Schallheim, 1992). My intention is not, in any way, to minimize the importance of this literature. Rather, I hope to provide a framework which focuses on the nontax aspects of leasing and applies to the majority of actual lease structures. Incorporating tax considerations into the present model would vastly increase its complexity. The structure of the model is analogous to traditional models of the term structure of interest rates. Term structure models basically have three steps. First, the short-rate processis determined. In models such as Vasicek (1977),the short-rate process is exogenous, whereas in models such as Cox, Ingersoll, and Ross (1985b), the short-rate process is endogenously determined in an equilibrium framework. Second, using the short-rate process, an equilibrium term structure of interest rates is determined, capable of pricing zero-coupon bonds of any maturity. Finally, the models are used to price contracts representing contingent claims on term structure variables, as in options, collars, floaters, and others. Following this analogy, I begin by deriving the short-term lease rate. The concept of a short-term leaserate is really a ‘shadow rate’. For example, the term structure would be equivalent if the fundamental lease rate were that for a one-year lease. The derivation of the short-term lease rate falls somewhere between the exogenous approach of Vasicek and the general equilibrium approach of Cox, Ingersoll, and Ross.I derive the short-term rent in the context of an intertemporal, rational-expectations competitive equilibrium, as in Lucas and Prescott (1971). The demand for the use of the asset (the service flow) is prone to stochastic shocks. The supply of the underlying asset depends on competitive firms which may enter the industry at any time by paying the cost of constructing the asset, also subject to stochastic shocks. The interaction of

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competitive, value-maximizing suppliers and the derived demand for the use of the underlying asset results in an equilibrium determination of the short-rent process. Since this is a dynamic, rational-expectations model, the entire shortrent process is determined. Given an asset-pricing model (taken as exogenous in the industry equilibrium framework), the value of the underlying asset is also determined. All in all, the industry equilibrium provides an endogenous determination of the short-rent process, the supply process, and the asset value process, which are then used in the following stages of the model. Continuing the term structure of interest rates analogy, I then determine (in closed-form) the equilibrium term structure of lease rates, i.e., an equilibrium rental rate for leasesof any term T > 0. Brennan and Kraus (1982) also develop a term structure of lease rates, but in a framework with an exogenous, lognormal short-term rent process. My industry equilibrium model, which begins with economic fundamentals such as supply and demand shocks (rather than an exogenous rent process), fosters the intuition to analyze the term structure results. For example, the shape of the term structure of lease rates can be interpreted as rational responsesto anticipations of future supply increases due to supply or demand shocks. Finally, I use the term structure of lease rates to analyze specific cases of real-world leasing contracts. The first application is the case of forward leasing, or pre-leasing. In virtually all realistic leasing situations, the signing of the lease predates the initiation of the lease payment period. In some leasing markets, such as the car rental business, the lag may be only a few moments. In other markets, however, the lag may be substantial. For example, it is quite common in the real estate industry for large development projects to be preleased; lease contracts are signed between the developer and tenants before or during the construction process. In fact, lenders often insist on the pre-signing of one or more ‘anchor tenants’ before construction funds are disbursed. A second application of the model addresses options to renew or cancel a lease. Leases often contain options (for either the lesseeor lessor) to alter the future course of the leasing contract by choosing between alternatives at a future date. The model provides an equilibrium rent on leaseswith options, in which the lease-option holder is assumed to choose the optimal exercise strategy. McConnell and Schallheim (1983) provide an especially thorough analysis of a wide variety of lease options. Their derivation of lease-option pricing differs from the present case only in that they take the underlying asset price as exogenous. The third application of the model is the equilibrium determination of lease insurance premiums. I analyze lease-cancellation insurance and residual-value insurance, two of the most common forms of asset leasing insurance. Leasecancellation insurance can be purchased by lessors who enter into a lease agreement which provides the lesseewith a cancellation option. In the event of cancellation, the insurance company guarantees lease payments until the

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maturity date of the lease. A comprehensive treatment of lease-cancellation insurance is provided by Schallheim and McConnell (1985). The value of this insurance is a simple extension of the results provided in the lease-option section of the paper. Residual-value insurance provides protection to the lessor in the event of a decline in assetvalue at the end of the leaseterm by guaranteeing that the residual value cannot fall below a prespecified floor. Once again, the equilibrium value of this insurance is a simple extension of the treatment on lease options. The fourth application of the model addressesvariable-rate leases.In one type of variable-rate lease,the current rent is periodically adjusted to equal the then-prevailing equilibrium lease rate. At each adjustment date, the rent is changed to ensure indifference between the newly adjusted rent and the signing of a new lease.In a second type of variable lease,the rent is adjusted according to the growth of an exogenous index such as the CPI. The final application of the model addressesleasesin which the rent level is contingent on the intensity of the asset’suse. For example, copy machine leases are frequently contingent on the number of copies, and computer lease rates may be contingent on CPU cycles. While virtually any form of contingent lease can be priced in this model, I focus on one of the most common forms of contingent leasing: the percentage rent under retail leases. I find that the equilibrium rent on such a leaseis simply a fixed payment plus a time-integral of call options on the sales of the firm. The paper proceeds as follows. Section 2 develops the competitive equilibrium in the spot rental market. Section 3 characterizes the term structure of lease rates. Section 4 derives forward lease rates. Section 5 analyzes lease options. Section 6 analyzes lease insurance. Section 7 values variable rate leases. Section 8 analyzes leasesin which rent is contingent on the intensity of use, and Section 9 concludes. 2. Competitive equilibrium in the spot rental market This section presents the derivation of the competitive equilibrium processes for short-term rents, industry supply, and assetvalues. The general framework of the spot market equilibrium is the real-options approach to investment under uncertainty. [The reader familiar with the real-options approach, or who prefers to jump ahead to the applications, may do so by simply noting the asset value solution in Eq. (18) and proceeding to the next section.] Pindyck (1991) does an excellent survey of the real-options literature. Dixit (1989a,b) uses an optionpricing framework to analyze the optimal entry decision of firms under the presence of demand uncertainty and lump-sum costs of adjustment. In the majority of financial as well as real-options models, the starting point is an exogenous process for underlying asset values (e.g., the stock price in the

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Black-Scholes framework) or cash flows (e.g., Brennan and Schwartz, 1985; McDonald and Siegel, 1986).However, since the goal is to derive an equilibrium distribution of cash flows (rent levels), the analysis begins with more fundamental underlying uncertainties. Here, I consider an industry subject to two sources of uncertainty: demand shocks and construction-cost shocks. I will then determine how the interaction of competitive firms, whose value-maximizing entry decisions are formed with rational expectations, leads to an equilibrium determination of rent, supply, and asset values. Consider a competitive leasing market in which a large number of firms lease an underlying asset. For easeof presentation, assume that each firm owns one unit of the underlying asset. Thus, the terms ‘firm’ and ‘asset’ can be used interchangeably. New supply may enter the market through new firms constructing additional units of the asset.Assume that the units are small enough and the number of firms large enough so that current supply may be represented as a continuum whose mass at time t is Q(t). The essential determinant of value from owning or leasing the asset is its underlying service flow. In this frictionless model, the economic benefits from the service flow are realized by the user of the asset,while the owner of the asset retains the right to sell this service flow to potential lessees.At each point in time, the price of the flow of services from the asset,or the instantaneous lease rate P(t), evolves in order to clear the market. Assume that the market inverse demand function is of a constant-elasticity form: P(t) = X(t).Q(t)-I”‘, where y > 0. Such a market is characterized by evolving uncertainty in the state of demand for the service flow from using the asset.At each point in time, even demand at the next instant is uncertain. X(t) represents a multiplicative demand shock, and evolves as a geometric Brownian motion: dX = a,Xdt + o,Xdz, ,

(2)

where a, is the instantaneous conditional expected percentage change in X per unit time, 0, is the instantaneous conditional standard deviation per unit time, and dz, is the increment of a standard Wiener process. Consider potential examples of representations for the shock term, X. If the market were for the leasing of office space,then a stochastic factor which could affect demand for the use of space might be corporate profits. If the space is industrial, then a potential shock term might be changes in industrial production. If the space is residential, then perhaps demographic variables such as household formation would be suitable. Firms may add to the supply of assetsby incurring a one-time construction cost of C(t) proportional to the quantity of units supplied. The new units are irreversibly committed thereafter. Construction costs are also assumed to

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evolve according to a geometric Brownian motion, correlated with the demand shock: dC = a,Cdt + o,Cdz, , where a, is the instantaneous conditional expected percentage change in C per unit time, cc is the instantaneous conditional standard deviation per unit time, and dz, is the increment of a standard Wiener process. Let p denote the instantaneous correlation coefficient between the Wiener processesdz, and dz,. An assumption of the model is that the construction process is instantaneous. For a majority of leased assets,however, the construction process is prone to substantial lags. Grenadier (1994)solves for a continuous-time rational-expectations equilibrium with construction lags. The additional complexity introduced by construction lags is substantial. Consider the relevant state variables in such an economy. Asset values will depend not only on the current levels of construction costs, demand, and asset supply, but also on the entry times of all assets currently under construction. Thus, the state space may become of infinite dimension. Nonetheless, the substantive results of this paper continue to apply. In fact, all of the applications in the following sections of the paper continue to hold, conditional on substituting the asset valuation function for an economy with lags for the present valuation. By definition, there are two types of firms: firms currently leasing the assetand idle firms waiting for the optimal moment to enter the industry. Let W denote the value of an existing firm leasing the underlying asset and let v denote the value of an idle firm (a firm which may, in the future, undertake construction of a new asset).These values are interrelated and must be solved simultaneously. Using Ito’s Lemma and the demand function specified in (l), the evolution of the equilibrium rent process can be written as2 dP = a,Pdt + o,Pdz, - +Q dQ. Over the range in which supply does not change, rents follow a geometric Brownian motion process, and the instantaneous mean and variance of the equilibrium rent process are independent of the level of supply. Changes in supply will, however, shift the level of the rent process downward. Consider the value of an existing firm, W. Its value will depend on P and C, the former because the rent level is its cash inflow and the latter because the equilibrium supply response of new entrants will prevent W from rising beyond the cost of construction. Thus C will serve to bound the value function from

‘There are no second-order terms in Q because the supply process is variation-finite. the case, the economy would be incurring infinite costs of adjustment.

If this were not

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above. However, the value will not depend on the level of supply, Q. When supply is not changing, the equilibrium rent process does not depend on Q becausethe rent diffusion process in (4) is independent of the level of Q at all points at which supply is unchanged (dQ = 0). In addition, even at the point at which supply is changing, free-entry will ensure that W = C, which is independent of Q. The functional dependencies of W are denoted by W(P, C). In the traditional option-pricing literature, contingent claims are priced through arbitrage arguments. However, such an approach requires assumptions about the liquidity of the underlying asset.In the caseof leasing markets, where the underlying assetssuch as office buildings are subject to substantial transactions costs, indivisibility, and the inability to be short sold, such arbitrage arguments are particularly questionable. An equilibrium approach relaxes the tradability assumptions needed for arbitrage pricing, although an appropriate equilibrium model must be chosen. Clearly, the appropriate assetpricing model for such markets remains an open question and must be confronted by empirical evidence. I use the continuous-time version of the capital assetpricing model of Merton (1973b). While the actual valuations vary according to the choice of asset pricing model, the modeling procedure remains intact under alternative models such as that of Cox, Ingersoll, and Ross (1985a),with the more general demand and cost factor risk premiums. Consider the instantaneous return on W(P, C) over a region in which supply is unchanging. By It& Lemma, the instantaneous change in W is dW = [;a:P2Wp,

+ po,o,PCW,,

+$~J,~C~W~~+ a,PW, + a,CW,]dt

+ o,PW,dz, + o,CW,dz,.

(5)

In addition to the capital gain earned on the asset,d W, the asset also yields a cash inflow due to rents of P. Therefore, the total expected return on W per unit time, CI,, is c1 =E w

[

dw+P W

1

1

‘Z

+ a,PW, + a,cw,

+ PI +.

P-5)

Setting CI, according to the CAPM equilibrium return and simplifying yields the following equilibrium partial differential equation: 0 = ;o;P2wp,

+ po,o,PCW*,

+ :0,2c2wcc

+ 4PWp + P;,cw, - rw + P)

(7)

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where

and where r is the (instantaneous) riskless rate of interest, CI, is the expected return on the market portfolio, a,,, is the standard deviation of the market’s return, and px,,, and pc,,, denote the correlations of dX/X and dC/C with the return on the market, respectively. The terms 2, and 2, can be interpreted ,as risk-neutral growth rates for X and C, respectively. Thus, in the manner of Cox and Ross (1976), assetsmay be priced as if the economy were risk-neutral by discounting expected cash flows at the riskless rate of interest, after substituting (4, &-) for (G, 4). Similarly, the value of an idle firm V(P, C) will derive its return solely from capital gains. Over the range of no output changes, V(P, C) must satisfy the following partial differential equation in equilibrium: 0 = $a:P2VPP+ pa,a,PCI/,, + iafC2KEc + L?,PV, f di,CK - rl/.

(8)

The values of idle and existing firms are connected by individual firm behavior and equilibrium conditions. I focus first on individual firm behavior. An idle firm will choose to enter and pay the cost of construction C(t) when it is individually optimal, which is precisely the problem of investment under uncertainty with the option to wait described in Brennan and Schwartz (1985) and McDonald and Siegel (1986). Solving this problem necessitates two types of conditions: value-matching and smooth-pasting conditions. These can be written as v(P*, c*) = w(P*, c*) - c* ) v,(P*, c*) = w,(P*,

c*) ,

(9)

v,(P*, c*) = w,(P*, c*) - 1.

Here, P* and C* are the values of P(t) and C(t) which trigger entry. The first boundary condition is the value-matching condition, which simply states that at any point of entry, the value of an idle firm equals the value of an existing firm after paying the cost of construction. The second and third conditions are smooth-pasting or high-contact conditions. [See Merton (1973a) for a derivation of the high-contact condition.] Essentially, these are conditions which ensure that the idle firm choosesto enter at the most advantageous point. These optimality conditions determine the trigger values that maximize the value of an idle firm, and are thus individually rational.

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Up until this point the model has been in the tradition of the real-options literature with individual optimality. The present model now deviates from the standard, single-firm framework. In a competitive equilibrium in which individual firms compete to undertake positive net present value projects, competitive forces will ensure that new entry will eat away all excessprofits. Thus, the free-entry condition of competitive equilibrium is imposed: w(P*, c*) = c*.

(10)

This condition ensures that, at any point of entry, competition leads to a zero net present value of investment. That is, since all firms are solving the same optimization problem and will each choose the same optimal policy, there can be no excessprofits accruing to entry. This condition ensures that V(P, C) G 0; the value of an idle firm is always zero. The individual firm optimality conditions are now combined with the freeentry condition [V(P, C) E 0] to solve for the equilibrium value of an existing asset.W(P, C) must satisfy the following partial differential equation in equilibrium: 0 = :cJ;P2w,,

+ pa,o,PCW,,

+ :o,2c2 WC,

+ di,PW, + &CW, - rw + P)

(11)

subject to w(P*,

c*) = c* )

w,(P*,

c*) = 0 )

w,(P*, c*) = 1 )

(12)

W(O,C)=O, W(P, 00) = P/(r - a,),

where the first three conditions are simply those of (9), with the free-entry condition (10) imposed. The final two equations are regularity conditions. The fourth condition reflects the fact that if prices ever fell to zero, they would remain there forever since X has an absorbing barrier at zero. The fifth condition ensures that, for very large construction costs, the probability of new entry approaches zero. Therefore, the value of an existing asset no longer depends on construction costs and equals the perpetuity value of rent flow. To solve partial differential equation (11) subject to boundary conditions (12), the following change in variables is applied:

Id,

F(K) = ; W(P, C) .

(13)

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Applying the above substitution yields the ordinary differential equation 0 = go; - 2po,o, + o,2)K2F” + (& - a*,)KF’ - (r - &,)F + K )

(14)

which is subject to the following boundary conditions: F(0) = 0,

F(K*) = 1, F’(K*) = 0 )

(15)

where K* = P*/C*. For convergence, it is assumed that Y > B,. The solution to this differential equation is

P11BKB +(r-Ka,) F(K) =-/j-1[P(). -4,) ' 1

(16)

with

K* = P(r - &A p-1 * Finally, using the definition that W(P, C) = C. F(K), the value of an existing asset can be written as 1 W(P,C)=

-p-1

[

b-1

P(r-o;x)

p

1

c’“~P~+r-;x.

P

(18)

This equilibrium asset value function is critical to the determination of generalized leasing. While W(P, C) represents the value of using the asset forever, the use of the asset over arbitrary periods of time may be valued by pricing option contracts on the asset. From Eq. (17), the entry of new supply is triggered when the ratio of rent-tocost reaches the upper barrier K*, implying that the equilibrium ratio of rentto-cost follows a geometric Brownian motion regulated at the upper barrier K*. Whenever rents rise relative to costs so as to hit the upper barrier, new supply enters the market in order to keep the ratio from breaking through. Using Eqs. (1) and (17), new supply is triggered when the ratio of X(t) to C(t) rises to a function of existing supply Q(t). Define the ratio of X(t) to C(t) as e(t) E X(t)/C(t) .

(19)

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S.R. GrenadierjJournal

Thus, new supply is triggered when the demand-to-cost ratio reaches the following function of existing supply: e*(Q)

=

P(r

-

‘,).

p-1

Ql/Y *

Using the supply trigger equation, Eq. (20), equilibrium supply Q*(t)and rent P*(t) can be expressed as functions of the exogenous state variables:

P*(t) = X(t).Q*(t)-l".

(21)

In the Appendix, the equilibrium rent and supply processesare fully characterized through a closed-form representation of their conditional density functions. 3. The term structure of lease rates

Having now established the competitive equilibrium evolution of short-term rents, P*(r), the equilibrium rent on leasesof any term is derived. For all T > 0, define a T-year fixed leaseas a contract in which the lesseeobtains the use of the assetfor T years beginning at time 0 and, in return, the lessor receivesthe flow of rental payments of R( T) until time T, with the first payment made immediately upon the signing of the lease.The goal is to derive the equilibrium characterization of R as a function of T. In this section, and in the applications which follow, an important assumption is that the fundamental determinant of value is the service flow of the asset,and that the contractual ownership is not relevant. This assumption implies that any two methods of obtaining the use of the assetover the samelength of time must have the same equilibrium valuation. However, this assumption abstracts from some of the important incentive effectsin actual leasing arrangements. As Smith and Wakeman (1985) point out, if the right to use the asset is unbundled from the right to the ownership of the residual assetvalue, then the incentives for the use and maintenance of the asset can change. For assets whose values are sensitive to these decisions, this assumption becomes less realistic. The simplest manner of deriving the term structure is from the following economic characterization of the leasing process. A lease of term T gives the lesseethe use of the assetfor T years and nothing thereafter. The same service flow can be achieved by forming a portfolio which involves purchasing the underlying asset and writing a call option on the underlying asset with expiration date T and an exercise price of zero. This portfolio also provides the economic benefits of using the asset over the term of the lease. To avoid dominance, the value of the leasemust equal the value of the portfolio. Thus, the

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value of a T-year lease is equal to W(P, C) less the value of a European call option on W with an exercise price of zero and expiration T. The value of W(P, C) was obtained in the previous section. What remains is the valuation of the call option. The call option evaluation, however, is quite complicated. Its terminal value depends not only on the final realizations of the random variables X(T), C(T), and Q*(T) but also on the entire path of X(t) and C(t) from time 0 to time T, becausethe equilibrium level of Q*(T) depends on realizations of their ratio at all previous dates. This dependenceis apparent from the equilibrium characterization of Q*(T) in Eq. (21). Thus, this option is of a variety known as ‘path-dependent’ options, a class for which few closedform solutions have been found. Let L(P, C, T) denote the value of the T-year call option on W when the current levels P(0) and C(0) equal P and C, respectively. The solution is derived in the Appendix, where the closed-form valuation is IQ,

C, T) = C.a,(T)

j-(P, C, T, 1) - (’ ,,)“‘i(P,

C, T, p)

1

+ B.g(p, C, T, 1) - df’, C, T, P) ,

f,(p, C, T, v) =f,(f’, C, T, v) -Mp, fdp, C, T, v) = @

f3(p,

c,

T,

v)

=

ofi Wv);W 1

g(P, CTT, v) = exp

C, T, v),

M(P, C) - u2(v). T

exp

Ic

P-4

1 1 [ OJT 1 ’

0

. @

-

@(P?

si(f’, C, T, v)

p-1 :

sl(p, C, T, v) = gn(P, C, T, v) + ss(p, C, T, v),

Cl

-

%(V).~

)

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s2(P, c, T,v) = [+(v&y2].exp[-

1

a@ a2(0)*7[ az(v)

- jQ(P, Cl cifi ’

g3(P, c T, v) = * exp %(V) [

-@ [

v.u3(V)‘Tl

24v)4qP, u2

Q(V). T - Al(P, C)

c-$6

[C

=

ay2v

’

2f?~(fi - r) - oP(pa, - GA2- Pl?f7c(P@x- 0,) + 202

al(T) = exp

a2(v)

1 1

C)

2

- 1)

PI . T

) I>

+ oi, - oi, ,

(qv) = 02.CV - l) + a^X- & , 2

with

and @(.) denotes the cumulative standard normal distribution function. Since the value of a T-year lease is equal to that of a portfolio which is long one unit of the asset and short one call, its value is equal to W(P, C) - L(P, C, T).

Finally, it is simple to solve for the equilibrium long-term lease payment which provides an annuity value equal to the equilibrium leasevalue. The risk of default for the stream of lease payments is ignored in this case;for lesseeswith poor creditworthiness this simplification may be unrealistic, but such a complication would not alter the basic structure of the model. Thus, the equilibrium term structure of lease rates can be expressed as R(P, C, T) =

Y

1 -exp(-

r-T)

C) -W’, C, 1.[WV’,

T)l ,

(23)

where R(P, C, T) is the equilibrium rent on a T-year lease when the current levels of rent and construction costs are P and C, respectively.

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Having derived a closed-form representation for the term structure of lease rates, I compare the results with those from standard term structure of interest rate models. First, as in the Cox, Ingersoll, and Ross or the Vasicek models of the term structure of interest rates, the term structure of leaserates converges to a well-defined perpetual lease rate. Note that a perpetual lease is economically equivalent to ownership of the asset.Thus, the equilibrium value of the underlying asset, W(P, C), must equal the discounted value of the equilibrium perpetual lease rate, R(P, C, co): R(P, c, co) = Y’ W(P, C)

(24)

Differentiating R(P, C, co), it is seenthat the perpetual leaserate is increasing in both state variables:

aqP,c, 00yap > 0,

aqP,c, ~yac > 0.

(25) Second,as with the Cox, Ingersoll, and Ross or the Vasicek models of the term structure of interest rates, the yield curve for rents may take on three possible shapes: downward-sloping, upward-sloping, and single-humped. Each of these casesis depicted in Fig. 1. The model’s structure permits someintuition as to why the curve takes on one of the potential shapes.First, define a ‘hot’ market as one in which the current ratio of the spot rent to construction cost is high, or near the upper bound at which new supply is triggered. Conversely, a ‘cold’ market displays a low rent-to-cost ratio. The shape of the term structure turns out to hinge on the ‘temperature’ of the market: hot markets are downward-sloping, cold markets are upward-sloping, and ‘lukewarm’ markets are single-humped. Why does a hot market lead to a downward-sloping term structure of lease rates?Given that the ratio is near the level at which new supply is triggered, the market rationally anticipates new supply in the near future, along with the concomitant decreasein future short-term lease rates. If the term structure did not adjust to such expectations, lesseeswould prefer to roll over a series of short-term leasesrather than accept a long-term lease.However, in equilibrium lessors and lesseesmust be indifferent to the form of financing of the use of the asset. Thus, the term structure adjusts to allow long-term rents to fall. This situation is depicted in the upper graph of Fig. 1, where the current spot rent of 20 is very near 22.54, the level at which new supply would be triggered. In a cold market, on the other hand, the ratio of rent-to-cost is far from the trigger level, so the market does not anticipate new asset supply in the near future. Thus, the market expects short-term rents to rise with overall demand over time. In this case,if the term structure did not adjust, lessors would prefer rolling over a series of short-term leases to accepting a single long-term lease. Once again, to ensure indifference in equilibrium, the term structure adjusts to an upward-sloping shape. This situation is depicted in the bottom graph of Fig. 1, where the current spot rent of 5 is very far from 22.54, the level at which new supply would be triggered.

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RentCT) 20

19 16 17 16 15

14 13

12 0

! 5

: I ,015

: : 20 25

: : : : ! ! I : I 30 35 40 45 So 55 60 65 70

; 75

: : : : , 80 69 so 95100

T Rent(T)

T

10.5

9.6 I. 0

RentflY)

5

10

l5

20

6-

5.5

--

R(m)

25

3035

40 45 50 $5 60 65

70

75

W 6590

95109

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Economics 38 (1995) 297-331

313

For intermediate cases,the term structure takes on a single-humped shape. While supply is not expected to be triggered in the very short run, it is anticipated further down the road. Thus, short-term rents are expected to rise for a few years and then come down once the new supply kicks in. As a result, the term structure takes on an upward-sloping shape in the short run, and then falls back down and converges to the perpetual lease rate. This situation is depicted in the middle graph of Fig. 1, where the current spot rent of 10 is at an intermediate distance from 22.54, the level at which new supply would be triggered.

4. The valuation of forward leases

In many real-world applications of leasing, there is a lag between the signing of the lease and the initial rent payment. In a literal sense,virtually all leasesare technically forward leases,since the payment of rent does not begin until after the rent is contractually determined. The development of an equilibrium term structure of lease rates provides the necessary tools for valuing leases signed anywhere from seconds to decadesin advance. A forward lease contract (more commonly known as ‘pre-leasing’) is defined as follows. At the current date (time 0) the lessor conveys to the lesseethe right to use the assetfor z years, beginning at time T1. In return, the lesseepromises to make r years of leasepayments, beginning at time T1 and ending at time T1 + z. The lease payments are, however, agreed on today. Thus, this is a I’,-year forward contract on a z-year lease. Given the analysis of the equilibrium term structure of lease rates, the determination of forward lease rates is straightforward. The T1-year forward contract on a z-year lease provides the lesseewith the use of the asset over the period from time T1 to time T1 + z. However, the sameeconomic benefits arise from purchasing a European call option on the asset with expiration date T1 and zero exercise price and simultaneously writing a European call option

Fig. 1. Term structure of lease rates. Each graph in the figure shows the equilibrium rental rate as a function of the term of the lease, T. The shape can take on one of only three potential shapes:upward-sloping, downward-sloping, and single-humped. The shape is determined by the distance of the initial spot rent, Rent(O),from the level at which new supply is triggered. Under the assumed parameter values, new supply would be triggered if the initial rent equaled or exceeded$22.54.The top graph has an initial spot lease rate of $20 and is downward-sloping. The middle graph has initial spot lease rate of $10 and has a single hump. The bottom graph has an initial spot lease rate of $5 and is upward-sloping. The default parameter values are B, = 0.02, Li, = 0, a, = 0.25, oC= 0.15, p = 0.5, r = 0.07, y = 1.2, and C(0) = $200.

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with an expiration date of T1 + r and a zero exerciseprice. In both the portfolio and the forward lease,the service flow of the assetaccrues over the period T1 to T1 + z. Therefore, the equilibrium value of the forward contract is up, c, T,) - up, c, T1 + z), where L(P, C, T) is the value of the call option with expiration date T derived in Eq. (22). Finally, let the equilibrium z-year forward leaserate, beginning in year T1 and agreed upon at time 0, be denoted by RF(P, C, T1, 7). The equilibrium forward leasepayment which provides an annuity value equal to the equilibrium forward contract value is RF(P, C, T1, z) =

r

exp(- reTI) - exp(- r.(T1 + 7))

*CUP,C, T,) - UP, C, TI + 711.

1 (26)

Note that for T1 = 0, the equilibrium forward rate is simply the equilibrium rate on a r-year lease, evaluated in Section 3. As the term of the underlying lease (z) and the time until payment (T,) vary, the effectsare similar to those obtained for the underlying term structure derived in the previous section. Once again, there is an analogous result in the Cox, Ingersoll, and Ross model of the term structure of interest rates; the yield and forward rate curves share the same shape. If the underlying term structure of lease rates is downward-sloping, then the forward leaserates will be downwardsloping in both arguments. If the underlying term structure is upward-sloping, then forward rents will be upward-sloping in both arguments. For singlehumped term structures, the forward rent results will be indeterminate. Fig. 2 plots equilibrium forward leaserates as functions of both the term of the forward leaseas well as the time until the first leasepayment is made. The upper graph of Fig. 2 depicts an industry in which the current spot rent is 20 and which therefore has a downward-sloping term structure. The equilibrium forward rents on 5, lo-, and 15-year leases(r’s effect) are plotted as functions of the time at which the leasestake effect (T,‘s effect).The longer the term of the underlying lease,the lower is the forward rent, in keeping with the downward slope of the underlying yield curve. The intuition is similar to that for the shape of the yield curve. Short-term rents are high, and thus entry is likely in the near future. As new entry occurs, rents in the future are more likely to be lower. Therefore, lesseeswill only be willing to lock themselvesinto a long-term leasein the future if the forward rate is lower. Otherwise, they will prefer to gamble on rolling over future short-term rental contracts. In addition, for leasesof any term, the longer the period until the lease takes effect, the lower is the rent. The intuition is the same.To entice lesseesinto locking themselves into leasesfar out in the future, the forward rent must be low enough to steer them away from the alternative of

S.R. Grenadier/Journal of Financial Economics 38 (1995) 297-331

Equilibrium Forward Rent 18T

Equilibrium Forward Rent 10.7

315

Initial Spot Rent = 20

Initial Spot Rent = 10

T

10.6 10.5

10.4 ./ / 10.3 .10.2 .10.1 .10.9.9 .-

Equilibrium Forward Rent 6.4 T

Initial Spot Rent = 5

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signing a series of short-term leases.Thus, forward lease rates are decreasing in the time until payment. The bottom graph of Fig. 2 depicts an industry in which the current spot rent is 5, and which therefore has an upward-sloping term structure. Now, forward leaserates are increasing in the term of the underlying leaseas well as in the time until payment. Since short-term rents are far from the point which would trigger new entry, short-term rents in the future are now more likely to be higher. Forward rents will be higher to deter lessorsfrom gambling on signing lesseesto a series of future short-term leases. Finally, the center graph of Fig. 2 depicts an intermediate case in which the spot rent is 10 and the yield curve is single-humped. Forward leases are no longer monotonic in either the term of the underlying lease or in the time until payment. The underlying term structure dictates mixed comparative-static results.

5. The valuation of options to renew or cancel a lease Lease agreements may contain a wide variety of options for both the lessor and lessee.Two of the most common options are the options to renew and to cancel. This section provides a valuation of both options and derives the equilibrium lease rate consistent with the pricing of the options. [McConnell and Schallheim (1983) provide an in-depth analysis of lease options.] The characterization of the renewal option is as follows. The lessor and lessee agree to a T1-year leasewith fixed payment Ry,y{. At time T1, the lesseehas the option to renew the leaseuntil time T2 at the leaserate of Ry,y$:.3 The leaserate is contractually set at time 0. 31nsome cases,there may be a different rent after the option is exercised. Such a contract would be priced in a manner virtually identical to the present case.In addition, there are casesin which the lessor holds the option to renew (or cancel).

Fig. 2. Equilibrium forward lease rates. A forward lease is an agreement to lease the asset for a given term r, but where the lease does not begin until a fixed future date, Ti . Each graph in the figure shows the equilibrium forward lease rate as a function of both T1 and r. The shape of the forward rent curve with respect to both ‘fr and r will be of the same form as the underlying term structure of lease rates. The top graph has a spot rental rate of $20. As in Fig. 1, the underlying term structure is downward-sloping. The forward rent curve is also downward-sloping in both arguments. The middle graph has a spot rental rate of $10. As in Fig. 1, the underlying term structure is single-humped.,The forward rent curve is also single-humped in both arguments. The bottom graph has a spot rental rate of $5. As in Fig. 1, the underlying term structure is upward-sloping. The forward rent curve is also upward-sloping in both arguments. The default parameter values are 6, = 0.02, B, = 0, cX = 0.25, eC= 0.15, p = 0.5, r = 0.07, y = 1.2, and C(0) = $200.

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317

The characterization of the cancellation option is as follows. The lessor and lesseeagree to a &-year lease with fixed payment RyI”‘i. At time T1, the lessee has the option to cancel the lease. The lease rate is contractually set at time 0. It is immediately apparent that there exists a direct correspondence between lease-renewal and lease-cancellation options. A T1-year lease with an option to renew until time Tz is economically equivalent to a T,-year leasewith an option to cancel at year T1. Therefore, in equilibrium Ry,:>: = Ry,Ty:. This common option lease rate is denoted R$,,T2. Consider a lease with term T1 and an option to renew until time Tz (or equivalently, a leasewith term Tz and an option to cancel at time T,). To value the option to renew, consider the following two ways in which the lesseecould obtain the service flow from an asset for T, years: 1: Sign a T,-year lease with no options.

Alternative

Alternative 2: Sign a T,-year lease with an option to renew for an additional (Tz - T,) years. At the end of T1 years, either renew the leaseat the rate R‘$,, T2, or instead sign a simple (Tz - T,)-year lease at the then-prevailing market lease

rate. Whether or not to renew will be determined optimally by the lesseeby choosing the cheaper of the renewal rate or the equilibrium ( Tz - T1)-year lease rate prevailing at T1. In equilibrium, the value (or the cost) of these two methods must be equal since each provides the same service flow from the underlying asset. The value of the first alternative (obtaining the asset’s service flow for T2 years) was determined in Section 3. The present value of this first alternative is Pl(T,)

=

1 - exp(- Y. Tz) Y

>

. R(P, C, T,) .

(27)

The value of the second alternative can be broken down into two parts: the payment flow up to time T, and the payment flow from T1 to Ta. The initial payments up to time T1 are simply the rent set under the lease option, ROT,,T2. The remaining cash flows over the period T1 to T2 will be the minimum of the lease renewal rent, R$,,TI, and the market rent for a (TZ - T1)-year lease prevailing at time T1, R[P(T,), C(T,), Tz - T,]. The present value of the second alternative, P2(R”,,, T2, T1, T2), is

1-expl-r-7,)].R~~,,*+[l-exp(-:(T,-Tl))]

= [

.-PC- Y.T~).E*C~~~(ROT,,T~,RCP(T~), C(Tl), T2 - T1l}l,

(28)

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where E* denotes expectations under risk-neutral probabilities (i.e., with CI,and CI,replaced by B, and a,). Finally, the equilibrium option rent, Rt,,T2, is determined by equating the values of the first and second alternatives. Thus, the equilibrium renewal-option rent must satisfy PI ( Tz) = P2(R",, , TP,T1, T,):

.exp(- r.T1)

.E*Cmin{ROTI,T2, RCP(T,)>C(T,)>T2 - T11)l.

(29)

Note that the equilibrium option lease rate will be a function of P, C, T1, and T2 as well as the parameters of the processesfor X(t) and C(t). While no closed-form solution to the above equation has been derived, solutions can be found using Monte Carlo simulation or numerical integration. Fig. 3 plots the equilibrium rent on a five-year lease with a renewal option as the option period varies from 5 to 25 years.4 Notice that the equilibrium leaseoption rent is not monotonic in the option expiration date. While the underlying term structure (given the assumed spot rent of 20) is downward-sloping, the option rent has both upward- and downward-sloping segments. Fig. 3 also demonstrates the premium the market places on the embedded lease-option. Consider the difference between a five-year lease with an option to renew for another ten years and a simple lease to rent for 15 years. While both leases provide the opportunity to lease for 15 years, the first lease provides the lessee with the option not to renew if future rents turn out to be lower than the market anticipates. This ‘option premium’ will, of course, always be nonnegative.

6. The equilibrium determination of lease insurance premiums Lessors can obtain insurance against adverse future contingencies through the purchase of leasing insurance. This section focuses on two of the most

4The results are obtained through Monte Carlo simulation. The risk-neutral processesfor dX and dC are discretized as

where the initial five-year path is divided into 500 subintervals of length At = 0.01. s,,j and EC,)are random samples from a bivariate standard normal distribution, with a correlation coefficient of p. 1,000simulation runs were conducted.

S.R. Grenadier/Journal of Financial Economics 38 (1995) 297-331

Equilibrium Rent 16.5

319

5-Year lease with an option to renew until time T,

T

18 -17.5

-\

17 --

\ '1

Option Premium

'1 16.5 --

\ \

16 T

,t

\ \

15.5

. 15

T,-Year lease with no option . .-----

14.5 I 14 -I 5

/ 7

9

11

13

15

17

19

21

23

25

T* Fig. 3. Equilibrium rent on a lease with an option to renew. The upper curve in the figure shows the equilibrium rent on a five-year lease which contains an option to renew until time Tz. As Tz increases, the equilibrium rent on the five-year lease with a renewal option first decreases,then increases.The bottom curve shows the equilibrium rent on a lease of term Tz with no renewal option. The difference between these two curves is the renewal option premium, which is increasing in Ts. The default parameter values are B, = 0.02, B, = 0, 6, = 0.25, o, = 0.15, p = 0.5, r = 0.07, y = 1.2, C(0) = $200, and P(0) = $20.

common forms of leasing insurance: lease-cancellation insurance and residualvalue insurance. In return for making contingent payouts to the lessor, the insurance company charges a premium. Using the underlying equilibrium of the present model, an equilibrium premium is derived in terms of the lease-option values from Section 5. Lease-cancellation insurance was originated by Lloyd’s of London in 1974. The insurance covered lessors who leased computer equipment under leases with cancellation options. Since then, lease-cancellation insurance has evolved to cover a wide variety of leased assetsunder leaseswith cancellation clauses. Lease-cancellation insurance offers protection to the lessor from early cancellation by guaranteeing the leasepayments until the maturity date of the lease.For a comprehensive treatment of lease-cancellation insurance, see Schallheim and McConnell (1985). Consider the terms of a standard lease-cancellation insurance policy. A lessor and lesseehave agreed to a T,-year lease,and the lesseehas the option to cancel

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at time T1. As determined in the previous section, the equilibrium rent under such a lease is ROT,,T2. If the lesseechooses to cancel at time T1, the lessor must lease the asset over the remainder of the lease (time T1 to T,) at the thenprevailing market rent of R[P(T,), C(T,), T2 - T,]. The lessor is concerned about this potential realization, since the lesseewill cancel when the market rent is lessthan the contract rent. In the event of cancellation, the insurance company guarantees the lessor the rental stream of R $,, T2over the remainder of the lease. Thus, the insurance company pays the lessor the cash flow stream of ROTI, TI - R[P(T,), C(T,), T2 - T,] over the period T1 to T2 if the lease is cancelled. If the lease is not cancelled, there is no payment by the insurance company. In return for the insurance, the lessor pays the insurer an up-front premium, LCP(T1, Tz). To derive the equilibrium lease-cancellation insurance premium, consider the following two alternative methods of leasing the asset for T2 years: Alternative I: Sign a standard T,-year lease. This lease provides a fixed rental flow of R(P, C, TJ. The value of this cash flow stream is simply (1 - exp(- r.T,))/r.R(P, C, Tz).

2: Sign a T,-year lease with an option to cancel at time T1. Simultaneously, purchase lease-cancellation insurance. This combination provides a fixed rental flow of ROT,,T2,with an up-front cost of LCP( T1, T2). The value of this cash flow stream is simply (1 - exp(- r. T2))/r. ROT,,T2- LCP(T1, Tz). Alternative

Since each alternative represents a sale of the asset’suse over T2 years, their values must be equal in equilibrium. The equilibrium lease-cancellation insurance premium is then LCP(T1, TX) =

1 -exp(-

r.T2) Y

* CROT,,T,- W’, k 7’211.

Thus, the equilibrium premium is a function of R$,,Tz, determined in Eq. (29), and R(P, C, T,), determined in Eq. (23), and is the present value of the difference between two constant cash flows: the equilibrium rent on a leasewith a cancellation option and the equilibrium rent on a lease with no cancellation option. Residual-value insurance provides protection to the lessor in the event of a decline in asset value at the end of the lease term. With residual-value insurance, the lessor is guaranteed that the residual value cannot fall below a prespecified floor value at the maturity date of the lease. The World Leasing Yearbook (1991) gives a general overview of residual-value insurance. Such insurance is increasing in popularity, and has been provided for such assetsas automobiles, aircraft, construction equipment, ships, and commercial real estate.

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321

For example, at the end of a standard T-year lease the lesseereturns to the lessor an asset with value W [P(T), C(T)], a value which is uncertain at the initiation of the lease. The lessor can purchase residual-value insurance which guarantees that the residual value cannot fall below a certain floor, FLR. In the event that W [P(T), C(T)] < FLR, the insurer pays out FLR - W[P(T), C(T)]. Otherwise, no payout is made. In return for making such contingent payouts, the insurer charges an up-front insurance premium. It is immediately clear that residual-value insurance is equivalent to a put option on the underlying asset,with exercise price equal to FLR and expiration date T. The equilibrium insurance premium, RI/P(T) , must equal the value of the put option. The value of the put option can be derived in terms of the function P2(RtI, TZ,T1, Tz) in Eq. (28). Simple algebraic manipulation allows the equilibrium residual-value insurance premium to be written as RI/P(T)

7. Variable-rate

= FLR - P2(v.FLR,

T, co).

(31)

leases

While the basic leasing model in this paper focuses on a fixed level of rent throughout the lease term, it can be modified to account for rents which are adjusted during the lease term. Adjustable leases may provide the lessor with a hedge against such factors as unexpected inflation or cost fluctuations. One common form of lease escalation is to adjust the rent each year to the then-prevailing market rate on one-year leases.That is, instead of signing an N-year lease with fixed rent R(P, C, N), the parties commit to a variable-rate N-year lease in which market-determined one-year lease rates are sequentially rolled over. Therefore, the market-adjusted rent is simply a payment flow of R[P(r), C(z), 1.01 in each year r. Another common form of variable-rate lease structure is to adjust leases according to an exogenous index, such as a price or operating-cost index. Let I(t) denote the value of the index at time t. Suppose I(t) evolves as a geometric Brownian motion: dl = a,Idt + alIdzl ,

(32)

where a1 is the instantaneous conditional expected percentage change in I per unit time, (rl is the instantaneous conditional standard deviation per unit time, and dzl is the increment of a standard Wiener process. Let B1= CQ- P1(a, - Y) denote the risk-adjusted growth rate of I(t), as in Eq. (7), where PI = pl,,,dl/d,. According to such an escalation index, the lease rate is adjusted for any percentage change in the level of the index. Let R’(t, T) denote the indexed rent at time t on a lease which expires at time T. Let R’(0, T) be initially set. Therefore, R’(t, T) equals (R’(0, T) . I(t))/l(O). Notice that the rental payment

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flow at time t is equivalent to the payoff of R’(O, T)/Z(O) call options on Z(t), with an exercise price of 0 and expiration date oft. The value of such a call option is Z(O)*exp[- (r - &,).t] .

(33)

Once again, any two methods of achieving the service flow from the assetfrom time 0 to time T must have the same market value. The simplest alternative is to sign a fixed-rent lease of term T. From Section 3, this value is [(l - exp( - Y* T))/r] *R(P, C, T). Alternatively, the value of the variable-rate lease is simply the sum of the call options represented in Eq. (33), setting up the equilibrium condition on R’(0, T): l-exp(-r*T) r

T

1

-R(P,C,T)=w. =%.[I

s0 I

Z(O)exp[- (Y - &,)*t]dt -exp[-(r-$).T]].

(34) Solving for the equilibrium R'(0, T) yields

RI(O,T) = (r - OC1 - ew(- r’T)l . RtP c TJ r.[l-exp[-(r-&).T]]

’ ’

’

(35)

The equilibrium level of R'(0, T) will be a function of P, C, and T as well as the parameters of the processesfor X(t), C(t), and Z(t). Note that the case&I = 0 is equivalent to setting R'(0, T) equal to the equilibrium fixed lease rate of Section 3, R(P, C, T). The equilibrium initial variable rate rent R'(0, T) depends on the choice of index only through the parameter BI. Differentiation easily reveals that aR'/SI < 0. This is independent of the slope of the underlying term structure of lease rates. 8. Analysis of leaseswith payments contingent on the intensity of the asset’suse Another application of the equilibrium valuation of rents is the analysis of leasesin which rent is tied to some measure of the intensity of the asset’suse, sometimesreferred to as ‘metering’. Examples of metering include car-lease rates linked to mileage, copy-machine rents linked to the number of copies, and computer leaseslinked to CPU cycles. Smith and Wakeman (1985) list four potential purposes of such metering clauses. First, metering may permit the lessor to price-discriminate by charging higher prices to those with more inelastic demands. Second, by charging for the intensity of the asset’suse, the lessor is more easily able to charge for the actual wear and tear of the asset, and can better control the moral-hazard and

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323

adverse-selection problems associated with charging the samerent to all lessees. Third, by linking lessor revenues to asset usage, the lessor is bonding his obligation to provide services to maintain the asset’sservice flow, particularly when the lessor provides the servicing most essential to the asset’sperformance. Finally, metering may provide risk reduction to the lessee.When business is good and the assetis put to considerable use, the rent is higher; when businessis poor, the rent is lower. Thus, the rent payments will serve to dampen the swings in overall cash flow. While almost any form of contractual metering can be analyzed using the equilibrium model, this section focuses on a form of metering prevalent in shopping center retail leases:the percentage lease. [See Benjamin, Boyle, and Sirmans (1990) for a discussion and empirical analysis of shopping center leases.] A typical percentage lease specifies the payment of a base rent which is assured to the landlord plus a percentage (termed ‘overage’) of sales above a stated threshold. Let RB denote the baserent, S the threshold level of sales,and p the percentage of sales paid to the landlord. Let sales at time t be denoted as S(t). Therefore, the lease-payment flow at time z, Rperc(~), is RPerc(r) = RB + max[O, p.(S(z) - S)] ,

=RB+p.max[O,S(r)-$1.

(36)

The payment on a percentage lease is equal to a fixed payment RB plus the payoff of p call options on S with expiration date r and exercise price S. Denote the value of a call option on S with expiration z and exerciseprice S as C’(S, z, S) . Once again, the value of a T-year percentage lease must equal the value of a T-year fixed lease in equilibrium, since each provides the service flow of the underlying asset for T years. This equality provides a locus of combinations of (RB, p, S) which satisfy the equilibrium condition. Therefore, equilibrium combinations of (RB, p, S) for a T-year percentage leaseare solutions to the following equality: R(P, C, T) = RB +

Y

l-exp(-r.T)

t,

CS(S(t), S)dt .

(37)

Note that, in equilibrium, RB, p, and swill all be functions of P, C, and T as well as the parameters of the relevant stochastic processes. 9. Conclusion Rather than relying on an exogenous spot rental rate to build a term structure of leaserates,the model presentedhere usesa foundation of fundamental economic uncertainty and competitive interaction of individual value-maximizing firms to

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construct endogenous processesfor rent, supply, and asset values. While complicated, this structure facilitates economic intuition for a wide variety of leasing phenomena. For example, the analysis implies a downward-sloping term structure for leaserates in markets in which supply has recently been added; similarly, the model predicts an upward-sloping term structure for markets in which the economic fortunes of active firms are severely depressed. The development of an equilibrium term structure of lease rates permits analysis of a number of realistic leasing structures. First, the model provides a structure for leases whose rents are determined prior to the actual payment period. Second, the model determines equilibrium rents on leaseswhich contain options to renew or cancel. Third, the equilibrium value of lease insurance is derived. Fourth, equilibrium variable-rate rents are constructed to account for several varieties of real-word adjustable leasestructures. Finally, rental rates for leaseswhich make rent levels contingent upon the intensity of the asset’suse are determined. In general, virtually any contingent claim structure applied to leases can be valued in the model’s framework. Several extensions of the model would prove interesting. First, while the competitive paradigm may be a reasonable approximation in some leasing markets, a strategic, game-theoretic approach may be more meaningful in other markets. Second, the underlying assumption that the value of the service flow from the asset is independent of the ownership structure could be relaxed. For some assetswhose value is sensitive to the intensity of both maintenance and use, separating use rights from the property rights to the salvage value can have a substantial impact on assetvalue. Third, in many actual leasing markets (real estate in particular), the construction process will be subject to substantial lags. Thus;not only will demand and cost shocks be relevant but so too will the quantity and vintage of all units in the construction pipeline. Finally, the model should be subjected to empirical testing. In particular, the term structure and forward lease structure implications should be compared with actual market rental rates. Appendix This appendix presents the derivations of the distributions of future rent and supply conditional on current state variables as well as the value of a call option on the leased asset. While the derivations are quite involved, the basic calculations are sketched out. To facilitate the derivations, it is useful to define the following random variables: y(t) = ln[$$/.$/]

Eln[g]y

M(t) = sup[ Y(s): 0, $ s < t] )

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X(t) [X(O) 1’

G(t) E In -

325

(38)

1

@[P(O), C(O)] E In P(r - 4) C(O) ‘p(0) ’ [ o-1

where I shall use the shorthand expression ii? for M[P(O), C(O)]. Denote the equilibrium supply and rent processesas Q*(t) and P*(t), respectively. Therefore, Q*(t) can be written as

and P*(t) as X(t).Q(O)-“’

if M(t) < A,

[~~(l~~~~o~].X(t)e-I(‘)

if M(t)>li;l.

p*i”=i

A. 1. Distribution

(40)

of equilibrium rent

Define the distribution function:

J,(P) = PV*(t)

G PI,

(41)

where ‘Pr’ represents probabilities conditional on all current information. By the law of total probability, J,(p) can be broken up into two parts: J,(p) = Pr[P*(t) < p, M(t) < M] + Pr[P*(t) d p, M(t) > M] .

(42)

Through algebraic manipulation, the above expression can be rewritten as

(43)

M(t),

Letting v,(g, m) denote the joint (conditional) density function of G(t) and J,(p) can be expressed as M WP/P(O)) J,(P) =

ss0

--oo

vdg, m)dgdm +

30 ssM

m + ln(p/P(O))

-cc

- M

v,b, m)dgdm. (44)

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of Financial Economics 38 (1995) 297-331

The probability density function of equilibrium rent can be obtained by differentiating J,(p). Letting j,(p) denote the p.d.f. of P*(t) dP) = (i).{{O‘v,(ln(&),m)dm

(45) Deriving a closed-form solution toj,(p) requires deriving the joint p.d.f. of G(t) and M(t), ut(g, m). First, the joint distribution of Y(t) and M(t) is obtained using Harrison (1985, Sec.1.8).Second,define the random variable N(t) as a Brownian motion, independent of Y(t), with mean a,$ and variance &t. Becauseof N(t)% independence with Y(t) (and therefore with M(t) also), the joint density of Y(t), M(t), and N(t) can be derived by merely multiplying the joint density of. Y(t) and M(t) with the marginal density of N(t). Third, using a well-known property of normal random variables, the random variable G(t) can always be expressed as G(t) = AY(t) + N(t).

(46)

One simply chooses 1, CI,, and 0,’ to allow G(t) to have the correct mean, variance, and correlation with Y(t). Using the change of variables procedure, the distribution of Y(t), M(t), and G(t) is obtained. Finally, integrating with respect to Y(t) over the region - co to m results in the distribution u,(g, m). Using the distribution vt(g, m) and then performing the integration in Eq. (45) provides a closed-form solution to the density function of P*(t). The solution is quite complicated, and can be written as .a)

where

= 4P).

jil

d(P)

+ MJ)

. $ j=4

gj(p),

(47)

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of Financial Economics 38 (1995) 297-331

2 dpt ).exP[(2(2i!$2JC:(p)]i bt(p) = ( ( 1 - 211 d(P) =

- Ac?(l - 2&/i ox\1 -21) (

(1 - 1)fJc,Ji g:(p) = (1 - 2A)o;d2

>

~43N~

@C&P) + 44P)l?

~[~].{l-O[d2.U:(8)+Ct(~~dl]j,

gp(P)=(~).{m[~]-N,[~,d(P),~]j,

,,.(,,t-ln(&))-p”:l c:(P) =

AcT2a,J;

3

P A4 + Apt + pzt - In po 2&a ( 1 C3P) = + (22 - 1)202 ’ 1 - 21 -A?-pL,t--&t(U-1)

cB(p)=402(2A--1).

c,‘(p,=(~).[(~)-c:(P)].

c:(P) = d = 1

c: + A(1 - A)02 -

&,2(1

- 21)2’

1

+,ut(40z+a2),

321

328

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of Financial Economics 38 (1995) 297-331

and n( .) denotes the standard normal density function, @(*) denotes the cumulative standard normal distribution function, and N2(x1, x2, p) denotes the cumulative standard bivariate normal distribution function. That is, N2(xI, x2, p) is the area under a standard bivariate normal distribution function covering the portion from - co to x1 and - cc to x2, where the two random variables have correlation p. A.2. Distribution

of equilibrium supply

Define the distribution function f&(q) = WQ*(t) < 4,

(48)

where ‘Pr’ represents probabilities conditional on all current information. From the definition of equilibrium supply in Eq. (39), the distribution of Q*(t) is truncated below at Q(0). The density function will be continuous for all q > Q(O),but with a probability mass at the point Q(0). First, let q > Q(0). H,(q) can be expressed as Wd

= Pr(Q*(t) d d

= Pr

1I

(P - l).wJ expW(t)) ’ G 4 i[ P(r - 4)

=Pr{M(t)
(49)

=m[hl(~~p-t]-exp(2P~~(q))..[ where jqr - a^,) ql’y B- 1 .e(0)

h(q)=ln [

1 ,

-hJyq

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of Financial Economics 38 (1995) 297-331

329

The derivation in the fourth line comes from Harrison (1985, Ch. 1, Sec. 8). For the case q = Q(O),H,(q) can be written as

K(Q(W = Pr(Q*(t) 6 Q(O)) = Pr[M(t)

d NJ

(50)

The density function for Q*(t), denoted as h,(q), can now be derived by differentiating the distribution function H,(q), while allowing for a discrete probability mass at the point Q(0):

A notion that may be of some interest is the likelihood that supply in the industry will grow to any level Q in the future. Two characterizations of this notion are provided: the probability that Q will ever be reached, and the expected time until Q is reached. Suppose the current levels of 0(O)and Q*(O) are 8 and Q, respectively, where Q d Q. Let P(Q; 13,Q) denote the probability that Q will ever be reached, conditional on the current values of 8 and Q. P(Q; 0, Q) can be expressed as

p(Q; 4Q) =

(52)

if ~20. Define T(Q; 8, Q) as the expected time until & is first reached. T(Q; 8, Q) can be expressed as 00

if j.
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of Financial Economics 38 (1995) 297-331

A.3. The value of a call option on the asset

In Section 3, an important input into the determination of the term structure is the valuation of a European call option on the asset W (P, C) with expiration date T and zero exercise price. Let the value of this call be denoted as up, c, 0 Since the option has a zero exerciseprice, it will always be exercised.Thus, it is economically equivalent to owning the assetfrom year T onwards, and its value is simply the expected discounted value of rents from time T to infinity, where expectations are taken under the risk-neutral probabilities, as in Cox and Ross (1976). This is accomplished by substituting 2, and B, for a, and ~1,and then discounting expected values at the risk-free rate. Section A.1 of this appendix derives the probability density function for the short-term rental rate P(t) conditional upon the current levels of rent and costs, P and C. The risk-neutral distribution is obtained by simply substituting B, and B, for CI, and ,a,. Denote the risk-neutral distribution of P(t) conditional on P and C as f(p,; P, C). The value of the call option is the solution to the following integral: Up, C, T) =

m m eerfpt .f(pr; P, C)dpt dt .

ssT

0

(54)

Following a grueling integration, the solution in Eq. (22) is obtained.

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