数理解析研究所講究録
第 1983 巻 2016 年 22-26

22A Note on an Extension of Asset Pricing Models*Hiroshi ISHIJIMA
Graduate School of Intemational Accounting, Chuo University
Akira MAEDA
Graduate School ofArts and Sciences, University ofTokyo

1.

Introduction

In this note, we review the recent advances of asset pricing models by Ishijima and Maeda (2015) anddiscuss the direction of extending these models.
As in Ishijima and Maeda (2015), we consider an economy that comprises three types of exchange

market; i.e. financial market, real estate market and its leasing market. We then start our discussion fromthe conventional dynamic portfolio choice problem with a representative agent. Besides the utility ofnondurable-goods consumption, we focus on the utility that stems from the bundle of real estate
attributes which benefit the agent activities. Examples of those characteristics include square footage,

year built, walking distance from the nearest subway/railway station and so on. Given an occupancy
rate with plausible market clearing conditions at any point in time, we endogenously provide acompetitive pricing system for financial assets, real estate and its rent. We then we discuss the directionof extending the pricing system. The rest of this note is organized as follows: Section 2 reviews anddiscusses the asset pricing models of Ishijima and Maeda (2015) and in Section 3, we conclude.

Review of asset pricing models for real estate and financial assets
In addition to financial security markets, we assume that there are real estate property markets and real2.

estate lease markets within the economy. We then introduce the following notation for real estate $i=$$1,$$N^{H}$and financial asset $j=1$ ,$t=0$ to$\infty$:$P_{t}=(P_{j,t})_{1\leq j\leq N^{P}}$,$N^{P}.$Discrete timing of market trades.

: Financial security price vector at time

$H_{t}=(H_{i,t})_{1\leq i\leq N^{H}}$: Real estate price vector at time

$t.$

$t.$

This paper is presented at RIMS Workshop on Financial Modeling and Analysis 2015. We are grateful toProfessor Toshikazu Kimura (organizer) and the participants for their comments. We remark that we areplanning to submit the extended version of this paper in which some parts of this paper might be included.
$*$

23$D_{t}^{P}=(D_{j,t}^{P})_{1\leq j\leq N^{P}}$: Vector of dividends yielded by financial securities at time

$D_{\tau}^{H}=(D_{i,t}^{H})_{1\leq i\leq N^{H}}$: Vector of rents paid by lessees to lessors at time$\theta_{t}=(\theta_{j,t})_{1\leq j\leq N^{P}}$$t.$: Portfolio vector of financial security holdings at time$\varphi_{t}=(\varphi_{i,t})_{1\leq i\leq N^{H}}$$\phi_{t}=(\phi_{i,t})_{1\leq i\leq N^{H}}$

$t.$

$t.$

: Portfolio vector of real estate property right holdings at time
: Portfolio vector of real estate leasing at time , i.e., the portfolio of real estate
$t.$

$t$

properties currently under lease contracts at time
: Diagonal matrix of occupancy rates of real estate at time$t.$

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

Representative agent’s income at time$t.$

Representative agent’s consumption at time

:

$t.$

$t.$

Representative agent’s portfolio value before portfolio rebalancing at time:

Representative agent’s portfolio value after portfolio rebalancing at time

$t.$

$t.$

A self-financing portfolio strategy of the representative agent is described as follows:(1.1)$V_{t}=V_{t}^{-}+Y_{t}-C_{t}-\phi_{t}’D_{t}^{H}+\varphi_{t}’L_{\acute{t}}D_{t}^{H}$$V_{t}$

andare represented as follows:$V_{\mathfrak{t}}$$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, the representative agent’s consumption at is given by:
$t$

(1.4)

$C. =\theta_{t-1}’(P_{t}+D_{t}^{P})+\varphi_{t-}’{}_{1}H_{t}+Y_{t}-\phi\’{i} D_{t}^{H}+\varphi_{t}’L_{t}’D_{t}^{H}-\theta_{t}’P_{\mathfrak{l}}-\varphi_{t}’H_{t}$It is worthwhile to note that the timing of cash-in and cash-out expressed as Eqs. $(1.1)-(1.4)$ are twofolds; the timing of rent payout is diﬀerent from that of financial securities’ dividend payment$\theta_{t}’D_{t+1}^{P}.$It is because lease (or rent) contracts are usually cash-in-advance contracts: the lessee must pay the rentat the beginning of each period, which brings the rent of“dividends”$\phi_{t}’D_{t}^{H}$$\varphi_{t}’L_{t}’D_{t}^{H}$to the owner ofthe property. This fact contrasts to financial security investment, in which dividends are brought to thesecurity holder at the end of each period. Also we remark that the representative agent has a choice
between consumption goods C. and housing goods$\phi_{t}’D_{t}^{H}.$As a source of benefits from real estate, each piece of real estate is a representation of a bundle ofattributes. That is, with the following notation:$b_{ik,t}$:Unit content of attribute$1,$$Z_{k,t}$:$N^{H})$$t,$that is included in real estate$i$

at time$(k=1,$$\ldots,$$K,$$i=$. Lancaster $(1966, 1971)$ referred this variable as consumption technology,

Amount of attribute

time

$k$

$k$

$(=1, K)$ that is included in the entire real estate portfolio$\varphi_{t}$at24
the bundle of attributes can be represented as:$Z_{k,t}= \sum_{i=1}^{N^{H}}b_{tk,t}\varphi_{i,t}$

$or$

(1.5)

$Z_{t}=B_{t}’\varphi_{t}$The representative agent is to maximize the sum of the instantaneous utility derived during eachperiod from the present into the infinite future. As the instantaneous utility at time$t$

is time-additive

and assumed to be a function of consumption at the time and the bundle of attributes, the objective to
be maximized is defined as follows:(1.6)$U( \{C_{\mathfrak{t}},Z_{t}\}, \{C_{t+\tau/}Z_{t+\tau}\}, =E_{t}[\sum_{\tau=0}^{\infty}\delta^{\tau}u(C_{t+\tau/}Z_{t+\tau})]$The agent’s problem is described as follows:$\max_{\mathfrak{t}\prime}imize\varphi_{C} E_{t}[\sum_{\tau=0}^{\infty}\delta^{\tau}u(C_{t+\tau},Z_{t+\tau})]$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 clearly be observed that the problem is merely an extension oftypical dynamic portfolio selectionproblems. The first-order necessary conditions for Eq. (1.7) and the plausible market clearing conditions
expressed constitute a competitive equilibrium as stated in Proposition 1.

Proposition 1 (PHD Equations)be exogenous.
Within the framework and according to the assumptions described above, financial security prices, realestate prices, and real estate rents are determined by thefollowing equations:: Financial asset equilibrium prices ( -equation)

Let the occupancy rates$L_{t}(\forall t)$and dividends yielded by financial securities$P$$D_{t}^{P}(\forall t)$$P$$P.$ $=E_{t}[(D_{t+1}^{P}+P_{t+1})M_{t:t+1}^{C}]\Leftrightarrow$(1.8)$P_{j,t}=E_{t}[(D_{j,\mathfrak{t}+1}^{P}+P_{j,t+1})M_{t:t+1}^{C}](i=1, N^{P})$(1.9): Real estate equilibrium prices ($H$-equation)$H$$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})$$D$: Real estate equilibrium rent ( -equation)$D$$D_{t}^{H}=B_{t}M_{t:t}^{z}\Leftrightarrow$(1.12)

25$D_{i,t}^{H}=b_{i_{\fbox{Error::0x0000}}t}M_{t:t}^{Z}= \sum_{k=1}^{K}b_{ik_{\fbox{Error::0x0000}}t}M_{k,t:t}^{Z}(i=1, , N^{H})$(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)

To interpret the pricing system, the financial asset price is given as a stochastically discounted value

oﬀuture dividends as shown in the financial economics literature since Merton (1969) and Lucas (1978).
Similarly, the real estate price is given as a stochastically discounted value of future rents which can beregarded as dividends of financial assets. Moreover, the ture rents of real estate can be represented asa linear combination of attribute prices for each of real estate as quoted in the literatures of real estate$fi_{J}$economics or consumer choice since Lancaster $(1966, 1971)$ , Rosen (1974), and Ekeland et al. (2004).
We remark that these attribute prices are the product of two components that can be interpreted as
the cash-flow pricing kernel and hedonic pricing kernel, respectively. The first component is a cash-

flow pricing kernel (or stochastic discount factor) which is a marginal rate of substitution between thepresent and future nondurable-goods consumptions along time horizon. The second component is ahedonic pricing kernel which a substitution between the nondurable-goods consumption and the realestate attributes benefit at any point in time in the future. In this regard, our pricing kernel could be anextension to combine two existing pricing kernels. We might also extend the discussion along discretetime horizon to provide a stochastic process ofreal estate prices. On the basis ofthese theoretical pricing
systems, we might provide some statistical models that are ready to implement empirical analyses toexplore the determinants of real estate prices. That is, our statistical pricing model allows us toincorporate not only the hedonic variables of real estate attributes but also the exogenous variables as
the cash-flow pricing kernel. These model specifications would help understand the pricing mechanismof real estate in detail.

3.

Conclusion

In this note, we review the recent advances of asset pricing models by Ishijima and Maeda (2015). We

26then we discuss the direction ofextending the pricing system in order to explore the determinants ofreal
estate prices in conjunction with financial assets,

References
Ekeland, I., Heckman, J.J. & Nesheim, L. (2004). Identification and estimation of hedonic models.

Journal of Political Economy, 112(1), 60-109.
Ishijima, H. and Maeda, A. (2015). Real Estate Pricing Models: Theory, Evidence, and Implementation.
Asia-Pacific Financial Markets, 22(4), 369-396
Lancaster, K. (1966). A new approach to consumer theory. Journal of Political Economy, 74, 132-157.
Lancaster, K. (1971). Consumer demand: A new approach. (New York and London: Columbia

University Press)
Lucas, R. E. Jr. (1978). Asset prices in an exchange economy, Econometrica, 46, 1429-45.
Merton, R.C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. ReviewofEconomics and Statistics, 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

Graduate School of International Accounting
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}$

$E\Leftrightarrow ffl$

Akira MAEDA

Graduate School ofArts and Sciences
University of Tokyo

Tokyo 153-8902 JAPAN
$E$-mail address:

maeda@global.c.u-tokyo.ac.jp$\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$$\vec{B/\rfloor}ffl\Rightarrow$