A Note on an Extention of Asset Pricing Models (Financial Modeling and Analysis)

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A

Note

on an

Extension

of

Asset Pricing

Models*

Hiroshi ISHIJIMA

Graduate School of Intemational Accounting, ChuoUniversity

Akira MAEDA

GraduateSchool ofArtsandSciences,UniversityofTokyo

1. Introduction

In thisnote,

we

reviewthe recent advances of asset pricing models by Ishijimaand Maeda(2015) and discuss the direction of extending these models.

Asin Ishijima and Maeda(2015),

we

consider

an

economythat comprises three types ofexchange

market;

i.e.

financialmarket,realestate market and

its

leasingmarket. We then start

our

discussion from

the conventional dynamic portfolio choice problem with

a

representative agent. Besides the utility of

nondurable-goods consumption,

we

focus

on

the utility that stems from the bundle of real estate

attributes which benefitthe agent activities. Examples of thosecharacteristics include

square

footage,

yearbuilt,walking distance from the nearestsubway/railway station and

so

on.

Given

an

occupancy

rate with plausible market clearing conditions at any point in time, we endogenously provide a

competitive pricingsystemforfinancialassets,real estateand itsrent. We thenwediscussthedirection

of extending the pricing system. The rest ofthis note isorganized

as

follows: Section 2 reviews and

discusses theasset

pricing

modelsof Ishijima and Maeda(2015) and in Section3,

we

conclude.

2.

Review

of

asset pricing

models for

real estate

and financial

assets

In additiontofinancial securitymarkets,

we

assume

that there

are

real estate propertymarkets andreal

estatelease markets within the economy. We then introduce the following notationforreal estate $i=$

$1,$ $N^{H}$ and financial asset _{$j=1$ ,} _,$N^{P}.$

$t=0$to$\infty$: Discrete timing of market trades.

$P_{t}=(P_{j,t})_{1\leq j\leq N^{P}}$

:

Financial securitypricevector attime $t.$

$H_{t}=(H_{i,t})_{1\leq i\leq N^{H}}$

:

Realestate price vectorattime $t.$

$*$

Thispaper ispresentedatRIMS WorkshoponFinancial Modeling and Analysis2015.Wearegrateful to

ProfessorToshikazu Kimura(organizer)and theparticipants for theircomments. Weremarkthatwe

are

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$D_{t}^{P}=(D_{j,t}^{P})_{1\leq j\leq N^{P}}$: Vector of dividends yielded by financialsecuritiesattime $t.$

$D_{\tau}^{H}=(D_{i,t}^{H})_{1\leq i\leq N^{H}}$: Vector ofrentspaid by lesseesto lessorsat time $t.$

$\theta_{t}=(\theta_{j,t})_{1\leq j\leq N^{P}}$

:

Portfoliovectorof financial security holdings attime $t.$

$\varphi_{t}=(\varphi_{i,t})_{1\leq i\leq N^{H}}$: Portfolio vector ofreal estate propertyright holdings attime $t.$

$\phi_{t}=(\phi_{i,t})_{1\leq i\leq N^{H}}$: Portfoliovectorof real estate leasing attime $t$,i.e., the portfolio of real estate

properties currently under lease contracts attime $t.$

$L_{t}=diag(L_{i_{\mathfrak{e}}t})_{1\leq i\leq N^{H}}$: Diagonalmatrixof

occupancy

ratesofreal estate attime $t.$

$Y_{t}$: Representative agent’s incomeattime $t.$

$C_{t}$: Representativeagent’s consumptionattime $t.$

$V_{t}^{-}$: Representative agent’sportfoliovaluebefore portfolio rebalancing attime $t.$

$V_{t}$

:

Representative agent’s portfolio value after portfolio rebalancing attime $t.$

Aself-financingportfolio strategyof therepresentativeagentisdescribed

as

follows:

$V_{t}=V_{t}^{-}+Y_{t}-C_{t}-\phi_{t}’D_{t}^{H}+\varphi_{t}’L_{\acute{t}}D_{t}^{H}$ (1.1)

$V_{t}$ and $V_{\mathfrak{t}}$

are

represented

as

follows:

$V_{t}=\theta_{t}’P_{t}+\varphi_{t}’H_{t}$ (1.2)

$V_{t+1}^{-}=\theta_{t}’(P_{t+1}+D_{t+1}^{P})+\varphi_{t}’H_{t+1}$ (1.3) Thus,therepresentativeagent’sconsumption at$t$is given by:

$C. =\theta_{t-1}’(P_{t}+D_{t}^{P})+\varphi_{t-}’{}_{1}H_{t}+Y_{t}-\phi\’{i} D_{t}^{H}+\varphi_{t}’L_{t}’D_{t}^{H}-\theta_{t}’P_{\mathfrak{l}}-\varphi_{t}’H_{t}$ (1.4)

Itis worthwhileto notethatthetimingof cash-in andcash-out expressed

as

Eqs. $(1.1)-(1.4)$

_are

two

folds;thetimingof rentpayoutisdiﬀerent from that of financialsecurities’dividend payment $\theta_{t}’D_{t+1}^{P}.$

Itis because lease(or rent)contractsareusuallycash-in-advancecontracts: thelessee mustpaytherent

$\phi_{t}’D_{t}^{H}$ atthebeginning of each period, which brings the rentof“dividends” $\varphi_{t}’L_{t}’D_{t}^{H}$ tothe

owner

of

the property. This fact contraststofinancial securityinvestment,in which dividends

are

broughttothe

security holder atthe end of eachperiod. Also

we

remark that the representative agent has

a

choice

betweenconsumption goods

_C.

and housing goods $\phi_{t}’D_{t}^{H}.$

Asa sourceofbenefitsfrom realestate, eachpiece of real estate is arepresentation ofabundle of

attributes. Thatis, with thefollowing notation:

$b_{ik,t}$: Unit content of attribute $k$ that is included in real estate $i$ at time $(k=1,$

$\ldots,$$K,$ $i=$

$1,$ $N^{H})$

.

Lancaster$(1966, 1971)$referredthis variable asconsumptiontechnology,

$Z_{k,t}$: Amountof attribute $k$$(=1, K)$ thatisincluded in theentirereal estate portfolio $\varphi_{t}$ at

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thebundle of attributes

can

berepresented

as:

$Z_{k,t}= \sum_{i=1}^{N^{H}}b_{tk,t}\varphi_{i,t}$

$or$ $Z_{t}=B_{t}’\varphi_{t}$ (1.5)

The representative agentistomaximize the

sum

of the instantaneous utility derived duringeach

period fromthe presentintothe infinite future. As theinstantaneousutility attime $t$ is time-additive

and assumed to be

a

function ofconsumptionatthetimeand the bundle ofattributes, theobjective to

bemaximizedis definedasfollows:

$U( \{C_{\mathfrak{t}},Z_{t}\}, \{C_{t+\tau/}Z_{t+\tau}\}, =E_{t}[\sum_{\tau=0}^{\infty}\delta^{\tau}u(C_{t+\tau/}Z_{t+\tau})]$ (1.6)

Theagent’sproblemisdescribedasfollows:

$\max_{\mathfrak{t}\prime}imize\varphi_{C} E_{t}[\sum_{\tau=0}^{\infty}\delta^{\tau}u(C_{t+\tau},Z_{t+\tau})]$ subjectto $C_{t+\tau}=\theta_{t+\tau-1}’(P_{t+\tau}+D_{t+\tau}^{P})+\varphi_{t+\tau-}’{}_{1}H_{t+\tau}+Y_{t+\tau}-\phi_{t+\tau}’D_{t+\tau}^{H}$ (1.7) $+\varphi_{t+\tau}’L_{t+\tau}D_{t+\tau}^{H}-\theta’{}_{t+\tau}P_{t+\tau}-\varphi’{}_{t+\tau}H_{t+\tau}$ $Z_{t+\tau}=B_{t+\tau}’\phi_{t+\tau}$ $\tau=0,1,$

It

can

clearlybe observedthat the problemismerely

an

extension

oftypicaldynamic portfolio selection

problems. Thefirst-order

necessary

conditions forEq.(1.7)and theplausible market clearing conditions

expressedconstitute

a

competitive equilibriumasstatedinProposition 1.

Proposition 1 (PHDEquations)

Lettheoccupancyrates $L_{t}(\forall t)$ anddividendsyielded by

financial

securities $D_{t}^{P}(\forall t)$ be exogenous.

Within the

_framework

and accordingtotheassumptionsdescribedabove,

_financial

securityprices,real

estateprices, andrealestate rentsare determined by thefollowingequations:

$P$: Financial_assetequilibriumprices($P$-equation)

$P.$ $=E_{t}[(D_{t+1}^{P}+P_{t+1})M_{t:t+1}^{C}]\Leftrightarrow$ (1.8)

$P_{j,t}=E_{t}[(D_{j,\mathfrak{t}+1}^{P}+P_{j,t+1})M_{t:t+1}^{C}](i=1, N^{P})$ _(1.9)

$H$: Real_estateequilibrium_prices ₍$H$_-equation)

$H_{t}=L_{t}D_{t}^{H}+E_{t}[H_{t+1}M_{t:t+1}^{C}]=L_{t}B_{t}M_{t:t}^{z}+E_{t}[H_{t+1}M_{\mathfrak{c}:t+1}^{C}]\Leftrightarrow$ (1.10)

$H_{i_{\mathbb{R}}t}=L_{\mathfrak{l},t}D_{i_{\fbox{Error::0x0000}}t}^{H}+E_{t}[H_{i,t+1}M_{t:t+1}^{C}]=L_{i,t}b_{i,t}M_{t:t}^{Z}+E_{t}[H_{i,t+1}M_{ \tau:t+1}^{C}]$

(1.11) $(i=1, \prime N^{H})$

$D$:Realestateequilibriumrent($D$-equation)

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$D_{i,t}^{H}=b_{i_{\fbox{Error::0x0000}}t}M_{t:t}^{Z}= \sum_{k=1}^{K}b_{ik_{\fbox{Error::0x0000}}t}M_{k,t:t}^{Z}(i=1, , N^{H})$ (1.13)

where

$M_{t:t+1}^{C}= \delta\cdot\frac{\partial u(C_{t+1\prime}Z_{t+1})/\partial C_{t+1}}{\partial u(C_{t},Z_{t})/\partial C_{t}}$ (1.14)

$M_{t:t}^{z}= \frac{\partial u(C_{t\prime}Z_{t})/\partial Z_{t}}{\partial u(C_{t/}Z_{t})/\partial C_{t}}$ (1.15)

$C_{t}=1’D_{t}^{P}+Y_{t}$ (1.16) $Z_{t}=B_{t}’L_{t}1$ (1.17)

Tointerpretthepricingsystem,the financialassetprice isgiven asastochastically discounted value

oﬀuture dividends

as

showninthefinancialeconomics literature sinceMerton(1969)andLucas(1978).

Similarly, the real estate priceisgivenas astochasticallydiscounted value of future rents which

can

be

regardedas dividends offinancialassets. Moreover, the$fi_{J}$ture rentsof realestate

can

be representedas

alinear combination ofattributeprices for each of real estateas quoted in theliteratures of real estate

economics

or consumer

choicesinceLancaster$(1966, 1971)$, Rosen(1974),andEkelandetal.(2004).

Weremark that these attribute prices arethe product of two components thatcanbeinterpreted as

the cash-flow pricing kernel and hedonic pricing kernel, respectively. The first component is a

cash-flowpricingkernel (orstochastic discountfactor) whichis

a

marginal rate ofsubstitutionbetween the

present and future nondurable-goods consumptions along time horizon. The second component is

a

hedonic pricing kernel which

a

substitution between the nondurable-goods consumption and the

real-estate attributes benefit atany point in time inthe future. Inthis regard,our pricingkernel could be an

extensiontocombine two existing pricing kernels. We might also extend the discussion along

discrete-timehorizon to provide

a

stochasticprocessofreal estateprices.Onthe basisofthesetheoreticalpricing

systems,

we

might provide

some

statistical models that

are

ready to implement empirical analyses to

explore the determinants of real estate prices. That is,

our

statistical pricing model allows

us

to

incorporatenot only the hedonic variables of real estateattributes but also the exogenous variablesas

thecash-flowpricingkernel. These model specifications would help understand the pricing mechanism

ofreal estatein detail.

3. Conclusion

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then

we

discussthedirectionofextending thepricingsystemin orderto explore thedeterminants ofreal

estateprices inconjunctionwith financialassets,

References

Ekeland, I., Heckman, J.J.

&

Nesheim, L. (2004). Identification and estimation of hedonic models.

Journal of PoliticalEconomy, 112(1),

60-109.

Ishijima, H. andMaeda, A. (2015).Real Estate Pricing Models: Theory,Evidence,and Implementation.

Asia-PacificFinancialMarkets, 22(4),369-396

Lancaster,K. (1966).A

new

approach to

consumer

theory. Journal ofPoliticalEconomy,74, 132-157.

Lancaster, K. (1971). Consumer demand: A

new

approach. (New York and London: Columbia

UniversityPress)

Lucas,R. E. Jr. (1978).Assetprices in

an

exchangeeconomy,Econometrica,46, 1429-45.

Merton,R.C. (1969).Lifetime portfolio selection under uncertainty: The

continuous-time case.

Review

ofEconomics andStatistics, 51(3),247-57

Rosen, S. (1974). Hedonic prices and implicit markets: Product diﬀerentiation in pure competition.

Journal of Political Economy, 82, 34-35.

Hiroshi ISHIJIMA

GraduateSchoolofInternationalAccounting

Chuo University

Tokyo

162-8478

JAPAN

$E$-mail address: $ishii\dot{:}ma@$tamacc.$chuoarrow u.ac.iP$

$\iota F\#\star^{\frac{\backslash }{\neq}}$

.

$\star\neq\mapsto^{\backslash }\beta_{JD}^{g}$ $/\Psi_{\Gamma\backslash =p}^{\angle\sim\ni+\Re aE}$

$E\Leftrightarrow ffl$

AkiraMAEDA

GraduateSchool ofArtsandSciences

University ofTokyo

Tokyo 153-8902JAPAN

$E$-mail address: [email protected]

$\ovalbox{\tt\small REJECT}\overline{R}\lambda^{\star^{\backslash }}\neq^{\wedge}\cdot\star^{\mapsto\backslash }\neq P_{Jt}^{\Leftrightarrow R_{\omega ロ}^{A}X4bffl\mathfrak{f}a\ovalbox{\tt\small REJECT}^{\backslash }\}\backslash }/\backslash$