ＧＰＩＦworking paper

Japanese Private Real Estate Models and Portfolio Selection

February 10, 2023
Koichi Miyazaki
Kazuhiro Shimada

EXECUTIVE SUMMARY➢

Private real estate is attracting attention as one of the alternative investment assets from theviewpoint of high income gain and its low correlation with traditional assets due to low liquidity.
The prices of privately placed real estate are mainly based on “appraisal prices” by real estateappraisers. The return of private real estate has autocorrelation due to the smoothing effects ofappraisals, and therefore, appropriate de-smoothing of the return is necessary for practical use.

➢

However, even though it is de-smoothed, the return is still based only on appraisal prices, so itcannot be said to be based on prices actually traded in the market (hereinafter referred to as“transaction-based prices”).

➢

To solve the issue, we propose a modeling of mean and variance of the transaction-based returnfor privately placed real estate, using both the de-smoothed returns of privately placed real estate
and the J-REIT returns obtained from the listed market.

（Note）This paper is a compilation of research results by GPIF staff, and the contents and opinions expressed in the text do not represent
the official views of the GPIF.

１．Introduction
Although global interest rates are in a phase of rising around the first half of 2022, the bond yields in Japanremains extremely low, and under such circumstances, the risk-return characteristics of J-REITs, which are
Japanese version of the listed real estate investment trust, are attracting attention. In addition, privately
placed REITs, which are less liquid than J-REIT, are expected to generate even higher returns with an added
liquidity risk premium. Against this backdrop, the Japanese real estate securitization market has grown to amarket size of 40 trillion yen as of 2022.
Tanabe (2022) evaluates Japanese real estate securitization as one of the few innovations, reviews thehistory of the real estate securitization market to date, and summarizes the future growth direction of themarket with the keywords Arbitrage, Borderless, and Concentration. In particular, with regard to “Arbitrage,”
the report states that the market for securitized real estate will further promote the integration of real estateand finance, as well as arbitrage not only within the real estate market, but also with other financial products.
The securitization market is expected to serve as a platform for connecting the real estate market and thep. 1

financial market. Among the assets traded in the securitized real estate market, J-REITs, which are listed
REITs, have some liquidity, and their prices are marked to market in a daily basis as traditional assets, so it
is relatively easy to analyze their relative value to traditional assets. Arbitrage between REITs and real estate
has already been pointed out by Kawaguchi (2004), it stated that the ratio of a REIT’s stock price (P) to its
corresponding NAV provides investors with extremely important information. Kawaguchi (2004) explains thatthe indicator can be used for arbitrage, citing a study by Gentry et al. (2003) on U.S. REITs. Also, Shimada,
Miyazaki, and Oishi (2021) discuss whether alternative assets are attractive or not, in comparison withtraditional assets in terms of risk-return characteristics and the effect of including then in a portfolio
consisted with traditional assets. As a first step to examine these issues, Shimada, Miyazaki, and Oishi (2021)
considers J-REITs as alternative assets and discuss the necessity of managing J-REIT separately as anindependent asset class in a portfolio.
Tokushima (2022) points out that among alternative investments, investments in low-liquidity assets such asreal estate should be treated separately from traditional assets classes and considered as a new asset classwhen the investment amount expands, because their risk-return characteristics differ from those of
traditional assets. It goes on to say, “Due to the highly idiosyncratic nature of the investments, it is probably
inappropriate to use the historical standard deviation and correlation coefficients as they are in the
optimization for asset allocation.”
In order to conduct arbitrage trade within real estate market, as well as between real estate market andfinancial markets, it is necessary that risk-return characteristics can be compared with a high degree of
accuracy. As mentioned above, the price for private REITs and real estate is based on the appraisal value,
and is smoothed so that it does not dynamically change from the past appraisal value. In other words, withregard to privately placed REITs and real estate, it cannot be assumed that daily returns are independently,
identically, normally distributed as traditional assets and the simple mean-variance method is inappropriate.
In this study, we first remove the autocorrelation inherent in appraisal price returns for real estate by desmoothing, and transform the returns so that they are independently, identically, normally distributed. We callthem “de-smoothed returns.” There are two main methods of de-smoothing according to Spencer, Andrei,
and Andrea (2020). One is the Geltner’s (1991, 1993) approach that relies on AR(H)-type time series models,
mainly for real estate returns, and the other is Getmansky, Lo, and Makarov (2004) that relies on MA(H)type time series models, mainly for hedge fund and PE returns. In this study, we adopt Geltner’s (1991, 1993)
approach because the subject is the appraised price return of real estate.
Next, regarding the modeling of mean and variance of real estate return, preceding researches have simply
obtained average and volatility of the de-smoothed returns. The problem in it is that the price assumed byreal estate appraisers does not necessarily coincide with the transaction price. Therefore, the de-smoothed
return itself cannot be regarded as the return based on the transaction price of privately placed real estate.
Therefore, focusing on the fact that the J-REIT price is a transaction price with transfer of real estate
ownership, we propose a model of mean and variance on a transaction-based return for privately placed realestate by utilizing both the de-smoothed return and the J-REIT return. An empirical analysis will also be
conducted to understand the characteristics of the model. The model will make it possible to compare riskreturn characteristics among real estate and financial market with a high degree of accuracy.

p. 2

In the empirical analysis of portfolio selection, the optimal portfolio selection will be examined from theperspective of Sharpe ratio maximization and risk-return by adding low-liquidity private real estate and JREIT, to the four traditional assets. In doing so, we will utilize data from the Association for Real Estate
Securitization (ARES) for privately placed real estate, and will analyze the risk-return of privately placed realestate in three ways: “based on the original ARES data (hereafter, ARES)”, “based on the de-smoothed ARES
data (hereafter, de-smoothed ARES),” and “based on our proposed model (hereafter, the model).” The
usefulness of the model for portfolio selection will be examined.
The paper is organized as follows. Section 2 compactly introduce de-smoothing method that relies on an
AR(H)-type time series model for real estate return. Section 3 proposes the model to derive mean andvariance of transaction-based return for privately placed real estate and identifies the characteristics of themodel through empirical analysis. Section 4 presents an empirical analysis of portfolio selection by adding two
assets (privately placed real estate and J-REIT) to four traditional assets from the perspective of Sharpe
ratio maximization. The final section is accompanied by a summary and future issues.

２ ． De-smoothing for real estate returns (Geltner’s (1991, 1993)
approach)
2.1 De-smoothed ARES Returns
The observable return for any given period (ARES return) 𝑅𝑡𝑜 shall be formed as in equation (1) using thede-smoothed return for this period (the return that appraiser assumes without smoothing) 𝑅𝑡 and observable𝑜𝑜𝑜returns prior to this period (𝑅𝑡−1, 𝑅𝑡−2, ⋯ , 𝑅𝑡−𝐻).
(ℎ)
𝑜
𝑅𝑡𝑜 = 𝜃 (0) ∙ 𝑅𝑡 + ∑𝐻
∙ 𝑅𝑡−ℎℎ=1 𝜃(1)where each θ is a parameter expressing the degree of obsolescence in the observable returns (the older
the data, the less it should be influenced) and satisfy the following conditions.
(ℎ)
∑𝐻
=1
ℎ=0 𝜃

(2)

In addition, the de-smoothed return at the period 𝑡 𝑅𝑡 follows the Brownian motion with drift in equation (3).
𝑅𝑡 = 𝜇 + 𝜂𝑡

, E[η𝑡 ] = 0 , η𝑡 ~𝑖. 𝑖. 𝑑(3)

Substituting equation (3) into equation (1) and rearranging, we obtain equation (4) as follows.
(ℎ)
𝑜𝑅𝑡𝑜 = 𝜃 (0) ∙ (𝜇 + 𝜂𝑡 ) + ∑𝐻
∙ 𝑅𝑡−ℎℎ=1 𝜃(ℎ)
𝑜= 𝜃 (0) ∙ 𝜇 + ∑𝐻
∙ 𝑅𝑡−ℎ+ 𝜃 (0) ∙ 𝜂𝑡
ℎ=1 𝜃
(ℎ)
(ℎ)
𝑜
= (1 − ∑𝐻) ∙ 𝜇 + ∑𝐻
∙ 𝑅𝑡−ℎ+ 𝜃 (0) ∙ 𝜂𝑡
ℎ=1 𝜃
ℎ=1 𝜃
(ℎ)
𝑜
= 𝜇 + ∑𝐻
∙ (𝑅𝑡−ℎ− 𝜇) + 𝜃 (0) ∙ 𝜂𝑡
ℎ=1 𝜃

(4)

2.2 Parameter estimation for de-smoothed returns
The observable mean-deducted return 𝑋𝑡 is defined as equation (5) by deducting the mean from theobservable return 𝑅𝑡𝑜 . The term 𝜃 (0) ∙ 𝜂𝑡 in equation (4) being taken as the error term 𝜀𝑡 = 𝜃 (0) ∙ 𝜂𝑡 , it canbe quantified from the regression equation (6) regarding 𝑋𝑡 .
𝑋𝑡 = 𝑅𝑡𝑜 − 𝜇(5)𝑋𝑡 = 𝜃 (1) 𝑋𝑡−1 + 𝜃 (2) 𝑋𝑡−2 ⋯ + 𝜃 (𝐻) 𝑋𝑡−𝐻 + 𝜀𝑡

(6)

p. 3

Once the parameters of the multiple regression equation (6), 𝜃 (1) , 𝜃 (2) , ⋯ , 𝜃 (𝐻) and the error term 𝜀𝑡𝜀are obtained, 𝜂𝑡 = 1−∑𝐻 𝑡 𝜃(ℎ) is also computed. Substituting it into equation (3), we are able to estimate theℎ=1

de-smoothed return 𝑅𝑡 as equation (7).
𝜀𝑅𝑡 = 𝜇 + 1−∑𝐻 𝑡 𝜃(ℎ)

(7)

ℎ=1

３．Model and parameter estimation
3.1 Model
In section 2, we first follow preceding research to model the de-smoothed return 𝑅𝑡 as equation (3). In
constructing our model, we were conscious of the following question: “Since the price assumed by real estateappraisers does not necessarily coincide with the transaction price, can we not regard the de-smoothed
return as a return that represents the actual investment return?” More specifically, the situation is explainedby using the two NAVs shown in Figure 1. The left side of the balance sheet (the Assets) of J-REIT has a
lot of properties which are based on the appraised price real estate appraisers assign. We call the part of
the Assets minus the liabilities (bonds, loans, etc.) “appraisal NAV”. The price of the investment units (which
is based on the appraised price) corresponds to the original ARES price data. Although the appraisal value ofindividual properties is basically updated twice a year, this does not mean that the properties can be tradedat the price. In contrast, if the investment units are listed and traded in the capital market on a daily basis,the investment units (hereinafter, referred to as “listed NAV”) are marked to market on a daily basis. Even
if there is a divergence between the two, they should ultimately eventually converge because they are thesame real estate.
【Figure 1: Appraisal NAV and Listed NAV】

（Source: Prepared by the authors based on data from Japan Real Estate Institute）
For this reason, in order to estimate the transaction-based return of real estate traded as private placements,
de-smoothed return 𝑅𝑡 alone is not sufficient and the J-REIT return should also be utilized. Therefore, in ourp. 4model, we consider the situation where the transaction price deviates from the price assumed by the realestate appraiser, due in part to the effects of transaction costs (equivalent to the Bid-Offer Spread in liquid
assets) and other factors. The mean（𝜇）and variance（𝜎 2 ）of the transaction-based return（𝑅𝑡𝑃 ）areestimated using both de-smoothed return 𝑅𝑡 and the J-REIT return. Utilizing them, the transaction-based
return（𝑅𝑡𝑃 ）is modeled by transforming the de-smoothed return 𝑅𝑡 .
Where, the de-smoothed return 𝑅𝑡 is newly denoted 𝑅𝑡𝐴 . Equation (7) becomes equation (8) in the newnotation.
Where, the de-smoothed return 𝑅𝑡 is newly denoted 𝑅𝑡𝐴 . Equation (7) becomes equation (8) in the newnotation.
𝑅𝑡𝐴 = 𝜇𝐴 + 𝜀𝑡𝐴 , 𝜀𝑡𝐴 ~𝑖. 𝑖. 𝑑. 𝑁 (0, (𝜎 𝐴 )2)(8)

In addition, the J-REIT return 𝑅𝑡𝐽 follows equation (9) and equation (10).
𝑅𝑡𝐽 = 𝜇𝐽 + 𝜀𝑡𝐽 , 𝜀𝑡𝐽 ~𝑖. 𝑖. 𝑑. 𝑁 (0, (𝜎 𝐽 )2 )𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛(𝑅𝑡𝐽 , 𝑅𝑡𝐴 ) = 𝜌𝐽,𝐴

(9)
(10)

The transaction-based return on real estate traded as a private placement 𝑅𝑡𝑃
𝑅𝑡𝑃 −𝜇𝜎:=

𝑅𝑡𝐴 −𝜇𝐴𝜎𝐴.

is modeled as follows.
(11)

Rewriting equation (11),𝑅𝑡𝑃 : = (𝜇 −𝜎𝜎𝐴𝜇𝐴 ) +

𝜎
𝜎𝐴

𝑅𝑡𝐴 .

(12)

Where,1(13)

𝜇 = (𝜇𝐴 + 𝜇𝐽 )
2

1

1

2

4

𝜎2 = ((𝜎𝐴 )2 + (𝜎𝐽 )2 ) + (𝜇𝐴 − 𝜇𝐽 )
2

1

2

(14)

1

1

2

ρ = [𝜌𝐽,𝐴 𝜎 𝐴 𝜎 𝐽 − 4 (𝜇𝐴 − 𝜇𝐽 ) ]⁄[2 ((𝜎 𝐴 )2 + (𝜎 𝐽 )2 ) + 4 (𝜇𝐴 − 𝜇𝐽 ) ](15)

With the background given in this section, we now provide the way of estimating the parameters of the modelin equation (13) through (15). 𝑅𝑡𝐴 and 𝑅𝑡𝐽 are assumed to be given by equation (8) through equation (10)follow the two-dimensional normal distributions in equations (16-1) and (16-2).
𝑓(𝑅𝑡𝐴 , 𝑅𝑡𝐽 , 𝜽) =1
2𝜋𝜎 𝐴 𝜎 𝐽 √1−(𝜌(𝑅𝑡𝐴 −𝜇𝐴 )(𝜎 𝐴 )2

𝑒𝑥𝑝 {−𝐽,𝐴 )1
2

2(1−(𝜌𝐽,𝐴 ) )𝐽

2

𝑄(𝑥𝑖 , 𝑦𝑖 ) =2− 2𝜌𝐽,𝐴(𝑅𝑡𝐴 −𝜇𝐴 )(𝑅𝑡 −𝜇𝐽 )𝜎𝐴 𝜎𝐽

𝑄(𝑅𝑡𝐴 , 𝑅𝑡𝐽 )}𝐽+(𝑅𝑡 −𝜇𝐽 )(𝜎 𝐽 )2(16-1)

2(16-2)

As noted earlier, total capitalization units and market capitalization units should ultimately converge on thesame risk-return characteristics, even if there is a temporary divergence between the two, since they areboth valuations of the same property in a private placement and in the capital markets. Therefore, the average
𝜇 and the variance 𝜎 2 of the transaction price return in the private placement 𝑅𝑡𝑃 are modeled with therestriction of parameters as 𝜇𝐴 = 𝜇𝐽 = 𝜇, 𝜎 𝐴 = 𝜎 𝐽 = 𝜎, and 𝜌𝐽,𝐴 = 𝜌 in equations (16-1) and (16-2) andestimated by way of maximum likelihood method. See the Appendix for details on the derivation of equation(13) through (15).

p. 5

(Remark)
This section presents a method for estimating the transaction-based return in a private placement for the
total equity units of the entire real estate in which the J-REIT invests. To estimate the transaction-based
return in the case of a private placement of individual properties, it is necessary to construct data on thelisted NAV (for J-REIT return) customized to match the region and use of the property, in addition to theappraisal NAV (for ARES return) data for the property in question.
3.2 Parameter estimation
3.2.1 Data
The data in the parameter estimation is as follows.
(1) Time series of the de-smoothed return 𝑅𝑡𝐴 obtained by de-smoothing the return of the ARES Japan
Property Index (hereinafter referred to as the ARES return) for the equity portion of the individual
properties held by J-REITs.
(2) Time series data for J-REIT return 𝑅𝑡𝐽 .
3.2.2 Estimation procedure
The procedure for parameter estimation for the model is as follows.
Step1：
Computing the mean and the variance of data (1), we obtain the mean 𝜇𝐴 and the variance (𝜎 𝐴 )2 ,respectively. As the same manner, computing the mean and variance of data (2), we obtain the mean 𝜇𝐽 andthe variance (𝜎 𝐽 )2, respectively. In addition, computing the correlation coefficient for data (1) and data (2),
we obtain the correlation coefficient 𝜌𝐽,𝐴 .
Step2：The estimated parameters in Step 1 are substituted into equation (13) through (15) to obtain the parameterof the model 𝜇, 𝜎, and 𝜌.
3.3 Empirical analysis to capture the characteristics of the model
3.3.1 Parameter estimation for de-smoothed returns
First, the autoregressive parameters obtained by applying the AR(H) model to the monthly ARES returns
are estimated by the maximum likelihood method; the number of lags H in the AR model is selected basedon the AIC criterion. The return obtained by substituting the parameter values estimated here into equation(7) is the de-smoothed ARES return. Next, to confirm that these de-smoothed returns are notautocorrelated, the AR(H) model is applied again to the de-smoothed returns to check the significance of theestimated autoregressive parameters, and if the null hypothesis that the autoregressive parameters are zerois rejected, the returns are considered to be autocorrelated. The estimated parameters obtained using monthly
ARES returns from April 2003 to December 2021 are shown in Table 1. The results obtained by applying theAR (10) model to the monthly ARES returns show that the value of the autoregressive coefficient for lag 1
is positive and its t-value of 18.62 is significant, suggesting the existence of a strong positive autocorrelation.

p. 6

Furthermore, the autoregressive parameters for lags 5 and 10 are negative and their absolute t-values are
significant above 3, suggesting the necessity of taking the number of lags up to 10 to eliminate
autocorrelation. In contrast, the results obtained by applying the AR (10) model to the de-smoothed returns
show that the t-values of all the autoregressive parameters are less than about 1, confirming that the desmoothed returns were indeed obtained by properly removing the autocorrelation inherent in the monthly
ARES returns.
【Table１：Autoregressive coefficients and t-values】
ARES returns
μ

(1)

(2)

( )

(4)

( )

( )

( )

( )

( )

(10)

coefficients

0.00

1.25

0.16

-0.15

-0.07

-0.43

0.06

0.02

0.14

0.21

-0.22

t-values

2.17

18.62

1.48

-1.35

-0.67

-4.03

0.57

0.23

1.37

2.02

-3.29

(1)

(2)

( )

(4)

( )

( )

( )

( )

( )

(10)

de-smoothed returns
μcoefficients

0.01

-0.03

0.03

0.04

0.02

-0.01

-0.06

-0.03

0.02

0.05

0.09

t-values

2.36

-0.40

0.37

0.54

0.30

-0.15

-0.84

-0.45

0.34

0.67

1.23

（Source：Authors’ compilation from ARES data）
3.3.2 Volatility and Correlation Coefficients
To get an idea of the level of volatility generated by our model, the volatility of the model, the volatility of
the ARES returns, the volatility of the de-smoothed returns, and the volatility of the J-REIT returns, four intotal, are obtained based on monthly returns for the 10-year and three-year periods and compared on arolling basis. With respect to the correlation coefficients, because the model revamps only the volatility ofthe de-smoothed return to make it more realistic by taking market return into account, the correlation
coefficient between the transaction-based return traded in private placement and other assets’ returns isstill assumed to be equal to that between the de-smoothed return and other assets returns. We compare
correlation coefficients computed based on monthly returns for two periods, the past 10 years and the past
three years on a monthly basis among the correlation coefficient between the ARES return and the composite
benchmark return (representing the return of a portfolio with one-quarter each of domestic bonds, domestic
stocks, foreign bonds, and foreign stock; the same applies hereafter), the correlation coefficient between thede-smoothed return and the composite benchmark return, and the correlation coefficient between the J-REIT
return and the composite benchmark return. These correlation coefficients will be an important factor when
discussing the Sharpe ratio for a portfolio that incorporates real estate into a portfolio of four traditional
assets.
Long term volatilities (obtained from rolling over the past 10 years) are shown in Figure 2. The four longterm volatilities have generally the same time-series trends but differ significantly in level. A closer lookshows that for “J-REIT” and “the model,” volatility, which was high in the 2014-2017 period, declined rapidly
in the period from 2017 to 2019, rose again in 2020, and has remained stable since then. This is due toseveral factors such as economic shock in the past 10 years, for example, the GFC affecting the volatility
from 2014 to 2017 and the COVID-19 Shock influencing on the volatility around 2020.

Interestingly, withp. 7

respect to the “de-smoothed”, the volatility, which was high during the 2014 period, has decreased prior tothat of “J-REIT” and “the model” and was not affected by the COVID-19 shock. As for “ARES,” theautocorrelation of monthly return is high, so it can be noticed that the time when the impact of the GFC on
the volatility is clearly slipping away is about one to two years later than that of “J-REIT” and “the model,”and it has not been affected by the COVID-19 Shock.
【Figure２：Long-term volatility (120-month rolling, annualized)】（Source: Author’s calculations based on FactSet and ARES data）
Short-term volatility (obtained on a rolling basis based on the last three years of data) is shown in Figure 3.
As expected, it can be seen that the change in short-term volatility is much larger than that in long-term
volatility. It is interesting to note that the increase in short-term volatility is also confirmed in “de-smoothed
ARES” and “ARES” due to the impact of the GFC, and in particular, for “de-smoothed ARES”, it is also
affected by the COVID-19 shock to some extent. In addition, the level of short-term volatilities of “J-REIT,”“the model,” and “de-smoothed ARES” were almost the same when they were not affected by the economic
shocks such as the GFC in 2008 and the COVID-19 Shock in 2020.

p. 8

【Figure３：Shier-term volatility（36-month rolling, annualized）】（Source：Authors’ calculations based on FactSet and ARES data）
The long-term correlation coefficients (obtained by rolling basis on the last 10 years of data) are shown in
Figure 4. Since the correlation coefficient is the one between the return on real estate and the composite
benchmark return, the level varies greatly depending on whether the return on real estate is a market return(J-REIT return) or an appraisal return (ARES return). For “J-REITs,” the coefficient is around 0.6 when datain the GFC period is included, and it remains at around 0.5 even when the data in the period is dropped, whilefor “de-smoothed ARES” and “ARES,” it is around 0.2 when data in the GFC period is included, and as thedata in the period is dropped out, it drops and remains stable around 0. Therefore, when quantifying the riskof portfolio by its variance, real estate that is traded in private placement will provide greater diversification
benefits than J-REIT into a portfolio over the long term.
【Figure４：Correlation coefficient with composite BM（120-month rolling）】（Source：Authors’ calculations based on FactSet and ARES data）

p. 9

The short-term correlation coefficients (obtained on a rolling basis based on the last three years of data)
are shown in Figure 5. The short-term correlation coefficients look quite different from the long-termcorrelation coefficients. Even for the correlation coefficient between the J-REIT return, which is a market
return and the composite benchmark return fluctuates widely from 0 to 0.8 in the short term. Therefore, if
rebalancing in the short term is assumed, a diversification effect can be expected even for J-REITs dependingon timing. It is interesting that while the movement of the correlation coefficient between “de-smoothed
ARES” return and the composite benchmark return across 0, which is the level of the long-term correlation
coefficient, is similar to that between “J-REIT” return and the composite benchmark return, the movementof the correlation coefficient between “ARES” return and the composite benchmark return is significantly
different from that between “J-REIT” return and the composite benchmark return. This is due to the factthat the de-smoothed return is a return obtained by de-smoothing the ARES return and is similar to the JREIT return in that there is generally no time-series correlation.
【Figure 5: Correlation coefficient with composite BM（36 month rolling）】（Source：Authors’ calculations based on FactSet and ARES data）
3.3.3 Real estate returns and relative value analysis
Figure 6 shows rolling averages of annualized monthly returns over the past 10 years for the J-REIT return,the return of the model, and the de-smoothed return. The average returns fluctuate, in order of magnitude,“J-REIT”（3%～15%）, “the model”（5%～12%）, and “de-smoothed ARES”（6%～11%）. Equation (13), which
represents the return of the model, is consistent with the results in Figure 6, since it implies the average of
the J-REIT return and the de-smoothed return. The rolling 10 years average difference, which is the desmoothed return minus the return of the model, is shown in Figure 7 as a relative value analysis. The valuationof the model for real estate is in between the valuation in the private placement and the valuation in thecapital market and is a fair value for real estate. Positive relative value in Figure 7 means that the desmoothed return (the return on real estate when traded in private placement) over the past 10 years hasoutperformed the fair return on real estate and is overvalued within the real estate market. Figure 7 showsp. 10that the degree of overvaluation and undervaluation are all generally within about 2% per annum, suggestingthat over the long period, the valuation of real estate in private placement and that in the market converge.
【Figure 6: Monthly returns（120-month rolling, annualized）】（Source：Authors’ calculations based on FactSet and ARES data）【Figure 7: Relative Value Comparison（De-smoothed Returns-Returns for the Model, 120-monthrolling）】（Source：Authors’ calculations based on FactSet and ARES data）
Similarly, for the J-REIT return, the return of the model, and the de-smoothed return, Figure 8 shows therolling average of the annualized monthly returns over the past three years. The rolling averages fluctuate,
in order of magnitude, for the J-REIT（−25%～35%）, the model（−14%～24%）, and the de-smoothed ARES（−5%～15%）. The ranking of the magnitude of fluctuations is the same as that of the past 10 years period
and the consistency with equation (13) is also maintained, but the level of volatility is significantly different.

p. 11

Reflecting this, the relative value analysis shown in Figure 9 shows that the extent of overvaluation orundervaluation ranges from about 11% per year in any case. In other words, it confirms that over a shortperiod of time (the past three years), the valuation of a real estate in the private placement and its valuationin the capital market can diverge significantly.
【Figure 8: Monthly returns（36-month rolling, annualized）】（Source：Authors’ calculations based on FactSet and ARES data）【Figure 9: Relative Value Comparison（De-smoothed Returns-Returns for the Model, 36-month rolling）】（Source：Authors’ calculations based on FactSet and ARES data）

４．Empirical analysis of portfolio selection
4.1 Data, setup and methods for analysis
The data for real estate used in the empirical analysis of portfolio selection with Sharpe ratio maximization isas described in Section 3.2.1 Data. For the four traditional assets, we use the monthly returns of the following

p. 12

indices: Nomura BPI for domestic bonds, TOPIX for domestic stocks, WGBI for foreign bonds, and ACWI ex
Japan for foreign stocks.
The definition of the Sharpe ratio is originally “the value attained by subtracting the risk-free interest ratefrom the portfolio return and divide it with the standard deviation of the portfolio return.” However, since theperiod under analysis includes long periods of negative short-term interest rates, here the value obtained byassuming that the risk-free interest rate is 0% is called the Sharpe ratio. The portfolio to be analyzed are(1) a portfolio consisting of four traditional assets (domestic bonds, domestic stocks, foreign bonds, andforeign stocks) plus only J-REITs, (2) a portfolio consisting of four traditional assets plus J-REITs and private
real estate assuming risk-return characteristics based on ARES returns, and (3) a portfolio consisting of four
traditional assets plus J-REITs and private real estate assuming risk-return characteristics based on desmoothed ARES returns, and (4) a portfolio consisting of four traditional assets plus J-REITs and private realestate assuming risk-return characteristics based on the model.
The reason for adopting these four different portfolios in the analysis is that when optimizing a portfolio
including real estate in the four traditional assets, if the risk of real estate is simply calculated from ARES
returns, the risk is underestimated due to the time series correlation inherent in the ARES returns and realestate is excessively included in the optimal portfolio and it causes practical problem. First, we clarify these
problems based on the optimal portfolio in portfolio (2). Next, preceding researches solved the problem tosome extent by considering the standard deviation of the de-smoothed real estate return as the real estate
risk, and this point is discussed based on the optimal portfolio in portfolio (3).
With respect to real estate traded in private placement, this study proposes a method for deriving the riskof transaction-based return that also takes the J-REIT returns into account in the de-smoothed returns. We
discuss the impact of the risk derived from the model on the weight of real estate in the optimal portfolio,
based on the optimal portfolio in portfolio (4). Since portfolio (1) is a portfolio that includes only J-REITs in
addition to the four traditional assets, by comparing it with portfolio (4), we can grasp the impact ofdifferences in risk characteristics between J-REIT returns and returns of the model, especially differences incorrelation coefficients with the four traditional assets, on the weight of real estate in the optimal portfolio.
4.2 Empirical Results and Discussion
In portfolio (1) through (4), we attempted to maximize the Sharpe ratio based on risk-return characteristics
computed from monthly returns over past three years on a monthly basis. The weight of each asset in the
optimal portfolio (“optimal weight”) on a monthly basis are shown in Figure 10 through 13 for portfolios (1)through (4), in that order. Figure 11, which shows the optimal weight of portfolio (2), reflects the previously
mentioned problem, as it means that almost all weights are allocated to real estate, except for the period
from 2009 to 2013.

p. 13

【Figure 10: Optimal weights (traditional assets + J-REITs, 36-month rolling)】【Figure 11: Optimal weights (traditional assets + J-REITs + ARES, 36-month rolling)】【Figure 12: Optimal weights (traditional assets+ J-REITs + de-smoothed ARES, 36-month rolling)】

p. 14

【Figure 13: Optimal weights (traditional assets + J-REITs+ the model, 36-month rolling)）】（Source: Figure 10 through 13, all prepared from FactSet and ARES data by authors）
Figure 12, which shows the optimal portfolio in portfolio (3) is closely examined. The optimal portfolio for theperiod from November 2008 to May 2011, was to hold almost all weights in domestic bonds. Since June
2011, due to the effect of BOJ’s zero interest rate policy, the less the domestic bond yield, the less thedomestic bond weight in the optimal portfolio and it finally diminished around 0% at December 2020.
During the period from June 2011 to December 2019, assets such as domestic stocks, foreign bonds, andforeign stocks were often chosen to replace domestic bonds in the optimal portfolio, but sum of their weights
are limited at most to several or around 10 % of total weight. During this period, the consistently increasing
weight of private real estate with risk-return characteristic based on de-smoothed ARES returns amounted
to about 40% of the optimal portfolio in 2018 when combined with J-REITs, which had a negligible weight,
and decreased in weight from 2020. The largest increase in weight since 2020 has been in foreign bonds,
which have benefited from the FED’s quantitative easing policy. It can also be noticed that the weight offoreign bonds was relatively large in 2007-2008, which corresponds to the period of the BNP Paribas Shockand the GFC. The comparison between Figure 11 and Figure 12 shows that the use of risk-return
characteristics based on the de-smoothed ARES return instead of the ARES return is reasonably effective
when optimizing a portfolio that includes privately placed real estate.
To examine the impact of risk-return characteristics based on the model on the optimal portfolio, we compare
Figure 13, which shows the optimal weights for portfolio (4), with Figure 12. The optimal weights shown in
Figure 13 are generally similar to those in Figure 12. A more detailed comparison shows that in Figure 13,
private real estate has increased its weight from June 2011 to December 2019, but it is included in the
optimal portfolio with 10% to 15% and is slightly less compared to the weight in Figure 12. One possible
reason for this is that the risk of privately placed real estate by the model is usually higher than the risk
based on de-smoothed ARES return because it reflects the J-REIT returns, which are evaluated by daily
market transactions.
We now examine the extent to which adding J-REITs and private real estate to the four traditional assets
improves the Sharpe ratio of the optimal portfolio. In Figure 14, we attempt to maximize the Sharpe ratio forp. 15

three different portfolios, portfolio (1), portfolio (4), and a portfolio consisting of only four traditional assets(5) on a monthly basis, and show the Sharpe ratio of the optimal portfolio in time series. Figure 14 showsthat the Sharpe ratio of portfolios (1) and (5) generally overlap, except for some periods in 2007 and 2019.
For the period up to July 2015, the Sharpe ratio of portfolio (4) is also generally the same as the other two,with the exception of 2007. However, the Sharpe ratio of portfolio (4) has been higher than the other two
since August 2015, with Sharpe ratios ranging from 0.5 to1.5. Thus, adding J-REITs to the four traditional
assets would hardly improve the Sharpe ratio, but adding private real estate could improve the Sharpe ratio
in some periods.
【Figure 14: Sharpe Ratio (36-month rolling)】（Source: Authors’ calculations based on FactSet and ARES data）
This may be due to the low correlation coefficient between the returns of private real estate and the returnsof the four traditional assets and the fact that the risk of these assets is usually smaller than that of JREITs. Here, the optimal portfolio of only the four traditional assets is shown in Figure 15. A closer look at
Figure 15 shows that the J-REIT weights in portfolio (1) in Figure 10 are almost replaced by foreign stock
for some periods in 2007 and 2019. In other words, in terms of Sharpe ratio maximization, the risk-return
characteristics of J-REITs are similar to those of foreign stocks, but they do not exceed the performanceof foreign stock for so many periods.

p. 16

【Figure 15: Optimal weights (traditional assets only, 36-month rolling)】（Source: Authors’ calculations based on FactSet and ARES data）

５．Summary and future researches
In this study, as a research question, we stated, “The prices assumed by real estate appraisers do notnecessarily coincide with the prices at which properties are actually traded. Therefore, we cannot regard thede-smoothed ARES return itself as the transaction-based return of privately placed real estate, can we?”To solve this question, we focused on the fact that the J-REIT price is the price at which the real estate is
actually traded with the transfer of ownership, and proposed the model to derive mean and standard deviationof transaction-based return by using both the de-smoothed ARES return and the J-REIT return. We also
conducted an empirical analysis to understand the characteristics of the model and an empirical analysis of
portfolio selection based on Sharpe ratio maximization for a portfolio that includes J-REITs and privately
placed real estate in the four traditional assets. Based on the results of the empirical analysis, it was foundthat the use of the model enables the construction of portfolios that do not excessively include private realestate even if private real estate is included in the four traditional assets, and that in Sharpe ratio maximization,
private real estate is included in the optimal portfolio for a larger period than J-REITs, and thus contributes
more to building an efficient portfolio.
There are three main issues to be addressed in the future.
The first is to confirm to what extent the model is effective in optimizing the pension fund under the conditionof securing a 1.7% real return on pension fund investments (the return on pension fund investments minus thenominal wage growth rate) over the long term with minimal risk, which is a requirement in practice.
The second issue concerns the usage of the model. In the empirical analysis in Section 4, the fair return
proposed in the model was used as the return on privately placed real estate in order to eliminate arbitrariness.
However, as confirmed in Section 3.3.3, in the short term, the de-smoothed ARES return and the J-REIT
return can diverge significantly, and at the same time, both returns can deviate significantly from the fairlevel. The issue is how to quantify the period until both returns converge to fair levels and the excess returnor losses obtained in the process.
Third issue, related to the second issue, is how to connect the model to the Black-Litterman model. If de-p. 17

smoothed ARES returns are used for privately traded real estate returns as in preceding researches, theBlack-Litterman model can be used directly because the short-term optimization is equivalent to the longterm optimization due to the i.i.d. de-smoothed ARES returns. However, when using the model, it will benecessary to specify parameters that represent the investment horizon, and the appropriate setting of suchparameters will be a challenge.

Appendix
Derivation of equation (13) through (15)
To simplify the notation, time-series data of the de-smoothed return 𝑅𝑡 and the J-REIT return 𝑅𝑡𝐽 are
expressed by 𝒙 = (𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 ) and 𝒚 = (𝑦1 , 𝑦2 , ⋯ , 𝑦𝑛 ), respectively and define the triad of the parametersas 𝜽 = (𝜇, 𝜎 2 , 𝜌). The density function of transaction-based return 𝑅𝑡𝑃 for privately traded real estate using
the data set (𝑥𝑖 , 𝑦𝑖 ) is𝑓(𝑥𝑖 , 𝑦𝑖 , 𝜽) =𝑄(𝑥𝑖 , 𝑦𝑖 ) =1
2𝜋𝜎 2 √1−𝜌2(𝑥𝑖 −𝜇)2𝜎2− 2𝜌

𝑒𝑥𝑝 {−12(1−𝜌2 )(𝑥𝑖 −𝜇)(𝑦𝑖 −𝜇)𝜎2+(A-1-1)

𝑄(𝑥𝑖 , 𝑦𝑖 )},(𝑦𝑖 −𝜇)2(A-1-2)𝜎2So, likelihood function 𝐿(𝒙, 𝒚, 𝜽) and the log-likelihood function 𝑙(𝒙, 𝒚, 𝜽) for the time-series data set 𝒙 and𝒚 are𝐿(𝒙, 𝒚, 𝜽) = ∏𝑛𝑖=1 𝑓(𝑥𝑖 , 𝑦𝑖 , 𝜽)(A-3)

𝑙(𝒙, 𝒚, 𝜽) = 𝑙𝑜𝑔𝐿(𝒙, 𝒚, 𝜽)1122(1−𝜌2 )𝜎 2

= −𝑛𝑙𝑜𝑔2𝜋 − 𝑛𝑙𝑜𝑔(𝜎 2 ) − 𝑛𝑙𝑜𝑔(1 − 𝜌2 ) −{∑𝑛𝑖=1(𝑥𝑖 − 𝜇)2 −2𝜌 ∑𝑛𝑖=1(𝑥𝑖 − 𝜇)(𝑦𝑖 − 𝜇) + ∑𝑛𝑖=1(𝑦𝑖 − 𝜇)2 }(A-4)

Differentiating the log-likelihood function 𝑙(𝒙, 𝒚, 𝜽) with respect to each of the three parameters 𝜇, 𝜎 2 , 𝜌and setting them equal to 0, we obtain three equations to estimate the parameters.
𝑛

𝑛

𝑛

𝑖=1

𝑖=1

𝑖=1

𝜕𝑙
1
=−
{− ∑ 2(𝑥𝑖 − 𝜇) − 2𝜌 ∑{−(𝑦𝑖 − 𝜇) − (𝑥𝑖 − 𝜇)} − ∑ 2(𝑦𝑖 − 𝜇)}𝜕𝜇
2(1 − 𝜌2 )𝜎 2
1

= (1−𝜌2 )𝜎2 {∑𝑛𝑖=1(𝑥𝑖 − 𝜇) − 𝜌 ∑𝑛𝑖=1{𝑥𝑖 + 𝑦𝑖 − 2𝜇} + ∑𝑛𝑖=1(𝑦𝑖 − 𝜇)} = 0

(A-5)

𝑛

𝑛

𝑛

𝑖=1

𝑖=1

𝑖=1

𝜕𝑙
1
1
= −𝑛 2 +{∑(𝑥𝑖 − 𝜇)2 − 2𝜌 ∑(𝑥𝑖 − 𝜇)(𝑦𝑖 − 𝜇) + ∑(𝑦𝑖 − 𝜇)2 } = 0(𝜎 ) 2(1 − 𝜌2 )(𝜎 2 )2𝜕(𝜎 2 )
(A-6)
𝑛

𝑛

𝑛

𝑖=1
𝑛

𝑖=1

𝑖=1

𝜕𝑙
1
−2𝜌
−(−2𝜌)=− 𝑛− 2{∑(𝑥𝑖 − 𝜇)2 − 2𝜌 ∑(𝑥𝑖 − 𝜇)(𝑦𝑖 − 𝜇) + ∑(𝑦𝑖 − 𝜇)2 }2𝜕𝜌
2 (1 − 𝜌 ) 2𝜎 (1 − 𝜌2 )2
1−{− 2𝜌 ∑(𝑥𝑖 − 𝜇)(𝑦𝑖 − 𝜇)}2(1 − 𝜌2 )𝜎 2
𝑖=1

𝑛𝜌
𝜌
= (1−𝜌2 ) − 𝜎2 (1−𝜌2)2 {∑𝑛𝑖=1(𝑥𝑖 − 𝜇)2 − 2𝜌 ∑𝑛𝑖=1(𝑥𝑖 − 𝜇)(𝑦𝑖 − 𝜇) + ∑𝑛𝑖=1(𝑦𝑖 − 𝜇)2 } −

p. 18

1
2(1−𝜌2 )𝜎 2{− 2 ∑𝑛𝑖=1(𝑥𝑖 − 𝜇)(𝑦𝑖 − 𝜇)} = 0(A-7)

Multiplying equation (A-5) by (1 − 𝜌2 )𝜎 2 and dividing 𝑛, we obtain1𝑛{∑𝑛𝑖=1(𝑥𝑖 − 𝜇) − 𝜌 ∑𝑛𝑖=1{𝑥𝑖 + 𝑦𝑖 − 2𝜇} + ∑𝑛𝑖=1(𝑦𝑖 − 𝜇)} = 0
1

1

Putting 𝑥̅ = 𝑛 ∑𝑛𝑖=1 𝑥𝑖 and 𝑦̅ = 𝑛 ∑𝑛𝑖=1 𝑦𝑖 , we are able to transform above equation to𝑥̅ − 𝜇 − 𝜌(𝑥̅ + 𝑦̅ − 2𝜇) + 𝑦̅ − 𝜇 = 0
and we finally obtain
1

(A-8)

𝜇 = 2 (𝑥̅ + 𝑦̅)
Equation (A-8) is the same as equation (13).
Next, we arrange {･} of equation (A-6).
∑𝑛𝑖=1(𝑥𝑖 − 𝜇)2 − 2𝜌 ∑𝑛𝑖=1(𝑥𝑖 − 𝜇)(𝑦𝑖 − 𝜇) + ∑𝑛𝑖=1(𝑦𝑖 − 𝜇)2= ∑𝑛𝑖=1{(𝑥𝑖 − 𝑥̅ ) + (𝑥̅ − 𝜇)}2 − 2𝜌 ∑𝑛𝑖=1{(𝑥𝑖 − 𝑥̅ ) + (𝑥̅ − 𝜇)}{(𝑦𝑖 − 𝑦̅) + (𝑦̅ − 𝜇)} + ∑𝑛𝑖=1{(𝑦𝑖 − 𝑦̅) + (𝑦̅ −𝜇)}2= ∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅ )2 + 2(𝑥̅ − 𝜇) ∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅ ) + 𝑛(𝑥̅ − 𝜇)2 − 2ρ ∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅ )(𝑦𝑖 − 𝑦̅) − 2𝜌 ∑𝑛𝑖=1(𝑥𝑖 −𝑥̅ )(𝑦̅ − 𝜇) − 2𝜌 ∑𝑛𝑖=1(𝑥̅ − 𝜇)(𝑦𝑖 − 𝑦̅) − 2𝜌𝑛(𝑥̅ − 𝜇)(𝑦̅ − 𝜇) + ∑𝑛𝑖=1(𝑦𝑖 − 𝑦̅)2 + 2(𝑦̅ − 𝜇) ∑𝑛𝑖=1(𝑦𝑖 −𝑦̅) + 𝑛(𝑦̅ − 𝜇)2
1

Dividing {∙} in equation (A-6) by 𝑛, we obtain 𝑛 {∙} and arrange it using equation (A-8),1𝑛{∑𝑛𝑖=1(𝑥𝑖 − 𝜇)2 − 2𝜌 ∑𝑛𝑖=1(𝑥𝑖 − 𝜇)(𝑦𝑖 − 𝜇) + ∑𝑛𝑖=1(𝑦𝑖 − 𝜇)2 }= 𝜎𝑥2 + (𝑥̅ − 𝜇)2 − 2𝜌𝜌𝑥,𝑦 𝜎𝑥 𝜎𝑦 − 2𝜌(𝑥̅ − 𝜇)(𝑦̅ − 𝜇) + 𝜎𝑦2 + (𝑦̅ − 𝜇)2𝑥̅ −𝑦̅ 2𝑥̅ −𝑦̅𝑦̅−𝑥̅𝑦̅−𝑥̅ 2

= 𝜎𝑥2 + ( 2 ) − 2𝜌𝜌𝑥,𝑦 𝜎𝑥 𝜎𝑦 − 2𝜌 ( 2 ) ( 2 ) + 𝜎𝑦2 + ( 2 )1= 𝜎𝑥2 − 2𝜌𝜌𝑥,𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 + 2 (1 + 𝜌)(𝑥̅ − 𝑦̅)2(A-9)

In the process of deriving equation (A-9) above,1𝑛1∑𝑛𝑖=1(𝑥𝑖 − 𝜇)(𝑦𝑖 − 𝜇) = 𝜌𝑥,𝑦 𝜎𝑥 𝜎𝑦 − (𝑥̅ − 𝑦̅)2

(A-10)

4can also be obtained.
𝜎2Multiplying both sides of equation (A-6) by 𝑛 , and substituting equation (A-9), we obtain
1

1−1 + 2(1−𝜌2)𝜎2 {𝜎𝑥2 − 2𝜌𝜌𝑥,𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 + 2 (1 + 𝜌)(𝑥̅ − 𝑦̅)2} = 0
Then arranging this, we obtain11

𝜎 2 = 2(1−𝜌2) {𝜎𝑥2 − 2𝜌𝜌𝑥,𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 + 2 (1 + 𝜌)(𝑥̅ − 𝑦̅)2}Also, by multiplying both sides of equation (A-7) by(1−𝜌2 )𝑛𝜌

(A-11)and substituting equation (A-9) and equation (A-10), we obtain
1

1

1

1

1 − (1−𝜌2)𝜎2 {𝜎𝑥2 − 2𝜌𝜌𝑥,𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 + 2 (1 + 𝜌)(𝑥̅ − 𝑦̅)2 } + 𝜌𝜎2 {𝜌𝑥,𝑦 𝜎𝑥 𝜎𝑦 − 4 (𝑥̅ − 𝑦̅)2 } = 0
Arranging above equation with the fact that the second term on the left-hand side is equal to 2 using equation(A-11), we obtain

p. 19

1

𝜌𝜎 2 = 𝜌𝑥,𝑦 𝜎𝑥 𝜎𝑦 − (𝑥̅ − 𝑦̅)2

(A-12)

4

Now, multiplying both sides of equation (A-11) by 2(1 − 𝜌2 ) and substituting equation (A-12) after
arranging the right-hand side, we obtain1

1

2(1 − 𝜌2 )𝜎 2 = 𝜎𝑥2 − 2𝜌𝜌𝑥,𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 + 2 (1 + 𝜌)(𝑥̅ − 𝑦̅)2 = 𝜎𝑥2 + 𝜎𝑦2 + 2 (𝑥̅ − 𝑦̅)2 −
1

1

4

2

2ρ (𝜌𝑥,𝑦 𝜎𝑥 𝜎𝑦 − (𝑥̅ − 𝑦̅)2 ) = 𝜎𝑥2 + 𝜎𝑦2 + (𝑥̅ − 𝑦̅)2 − 2𝜌2 𝜎 2and this is simplified to11

𝜎 2 = 2 (𝜎𝑥2 + 𝜎𝑦2 )+4 (𝑥̅ − 𝑦̅)2

(A-13)

Equation (A-13) is the same as equation (14). Substituting this into equation (A-12) and rearranging it, we
obtain14122(𝜎 +𝜎𝑦 )+ (𝑥̅ −𝑦̅)22 𝑥
4

𝜌𝑥,𝑦 𝜎𝑥 𝜎𝑦 − (𝑥̅ −𝑦̅)2

𝜌=1(A-14)

Equation (A-14) is the same as equation (15).

References
Geltner, David (1993) “Estimating Market Values from Appraised Values without Assuming
an Efficient Market”. Journal of Real Estate Research 8 (3) pp.325–345.
Geltner, David (1991) “Smoothing in Appraisal-Based Returns”. Journal of Real Estate Finance andEconomics 4:327–345.
Gentry, W, M., Jones, C. M. and C. J. Mayer (2003) “Deviations Between Stock Price and Fundamental Valuefor Real Estate Investment Trusts,” Discussion Paper, Columbia Business School.
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