CRES: 2004-003
COVARIANCE, COSKEWNESS AND COKURTOSIS IN GLOBAL REAL ESTATE SECURITIES
Kim Hiang LIOW and Lanz C.W.J. CHAN, Department of Real Estate, National University of Singapore
Contact Author
Dr Kim Hiang LIOW
Associate Professor and Deputy Head (Academic)
Department of Real Estate
School of Design and Environment
National University of Singapore
4 Architecture Drive
Singapore 117566
Tel: (65)68743420
Fax: (65)67748684
E-mail: rstlkh@nus.edu.sg
28 February 2004
COVARIANCE, COSKEWNESS AND COKURTOSIS IN GLOBAL REAL ESTATE SECURITIES
Abstract
This research explores whether there is a significant relationship between expected return, covariance, co-skewness and
co-kurtosis of global real estate securities and the resulting impact on risk premia estimation. Using a monthly dataset
comprising 19 international real estate securities from January 1994 to January 2004 and 4 real estate funds of various
periods from Datastream, we develop higher-moment Capital Asset Pricing Models (CAPMs) and test them using Hansen’s
(1982) Generalized Method of Moments (GMM). We employ data generating processes (DGPs) that mimics the Linear,
Quadratic and Cubic Market Models. Our results reveal that expected return and covariance, co-skewness and co-kurtosis
are significantly related and the number of significant higher-order systematic risks supports the modeling of higher
moments and risk premia estimation in global real estate securities. However, the choice of an appropriate market portfolio
is important in estimating an appropriate higher-moment CAPM and the resulting risk premium. Our results also reveal that
co-kurtosis has more explanatory power than co-skewness in pricing global real estate securities. The findings of this paper
are expected to provide alternative risk-return perspectives regarding the pricing of international real estate securities and
portfolio design.
Keywords: CAPM, higher moments, co-skewness, co-kurtosis, real estate securities, risk premia
1.
INTRODUCTION
The returns from any asset are usually described in terms of four moments. These are the mean (M1), variance
(M2), skewness (M3) and Kurtosis (M4). For example, if two assets have the same expected return and variance, investors
would view these as equivalent unless they have some knowledge about the skewness and / or kurtosis of returns. With this
additional information, investors would prefer the asset (say) with the greatest positive skewness. The logic behind this is
that investors are much happier with above-target returns but less happy with below-target returns. Nevertheless, the
majority of prior finance and real estate studies employ only M1 and M2 to characterize stock and real estate returns.
Additionally, a common assumption underlying these studies is that returns are normally distributed. While this assumption
is often made for convenience in theoretical models, it might be acceptable for returns over medium to long horizons, such
as quarterly or annual returns. However, it is less appropriate for more frequently observed data (daily, weekly or monthly).
Many studies of equity performance have found that a normal distribution does not adequately describe individual
stock returns. Instead the distribution of returns often have “fat tails” and more peaked than would be expected with a normal
distribution (Brown and Matysiak, 2000). For example, Simkowitz and Beedles (1980), Singleton and Wingender (1986) and
Badrinath and Chatterjee (1988) find evidence of skewness in individual stock returns as well as market indices in US stock
markets. In real estate markets, as in other financial markets, some evidence in favour of skewness has been presented.
For example, Young and Graff (1995) investigate the return distribution of individual properties in the Russell-NCREIF
database. They find evidence of time-varying heteroscedasticity and skewness over the period of the study. In particular, for
most years in the sample period (1980-1992), the returns on individual properties are negatively skewed. Bond and Patel
(2003) find that a large portion of property company returns in the UK does exhibit skewness in the conditional return
distribution.
With increase allocation of US pension funds to global investments and an expansion in global market
capitalization represented by Asian markets as well as the rise of China as a new economic giant, considerable attention
has been given to various aspects of property company performance in Asia and internationally. As property company
returns are likely to have different risk-return profiles from the underlying stock and property markets, it is important to
assess the fuller investment characteristics of property stocks with respect to their risk measures and expected return
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determination. An example in point is that since property stock return volatility (measured by M2) is generally higher than the
market, could additional risk factors such as higher-moment skewness and kurtosis better explain the distribution of property
stock returns? Additionally, Liow and Sim (2004) find that the majority of Asian property stock index returns are not normally
distributed and that the main source of non-normality is kurtosis rather than skewness.
The presence of skewness and kurtosis in property stock return distribution is reasonably documented. However,
little is known about the presence of co-skewness and co-kurtosis and, if any, their relevance in modeling asset pricing. The
main objective of this research is thus to investigate whether expected returns of global real estate securities and their
covariance, co-skewness and co-kurtonis are significantly related and the resulting impact on risk premia estimation. Using
monthly return data of 23 global real estate securitiesand real estate funds, our study reveals that there is a statistically
significant relationship between expected return and covariance, co-skewness and co-kurtosis and the number of significant
higher-order systematic risks supports the modeling of higher moments and risk premia estimation in global real estate
security markets. However, the choice of an appropriate market portfolio is important in estimating an appropriate highermoment CAPM and the resulting risk premium. Our results also reveal that co-kurtosis has more explanatory power than coskewness in the pricing global real estate securities. In all, the findings of this paper provide alternative risk-return
perspectives regarding the pricing of international real estate securities and portfolio design
The remainder of this paper is organized as follow. Section 2 provides a brief review of relevant finance and real
estate literature. In section 3, the theoretical framework of higher order CAPM is discussed. This is followed by an
explanation of three empirical CAPM models (linear, quadratic and cubic market models) to mimic the data generating
process in Section 4. The three models are estimated and results are reported and compared in Section 5. Section 6
continues with the risk premia estimation and discussion. The final section concludes the study.
2.
LITERATURE REVIEW
A large body of finance literature has documented that stock returns are affected by skewness and/ or kurtosis.
Skewness characterizes the degree of asymmetry of a distribution around its mean. Positive (negative) skewness indicates
a distribution with an asymmetric tail extending toward more positive (negative) values. Kurtosis characterizes the relative
peakness or flatness of a distribution compared with the normal distribution. Kurtosis higher (lower) than 3 indicates a
distribution more peaked (flatter) than a normal distribution. For example, Simkowitz and Beedles (1980), Singleton and
Wingender (1986) and Badrinath and Chatterjee (1988) find evidence of skewness in individual stock returns as well as
market indices in US stock markets. Bekaert et al. (2001) find that the majority of emerging country stock returns are not
normally distributed. The combination of skewness and kurtosis will contribute to different volatilities for different classes of
investment (Brown and Matysiak, 2000).
Similarly, prior real estate research has shown that the returns on individual properties and listed property
securities are skewed. For example, Young and Graff (1995) investigate the return distribution of individual properties in the
Russell-NCREIF database. They find evidence of time-varying heteroscedasticity and skewness over the period of the
study. Lizieri and Ward (2001) also strongly reject the assumption of normality for IPD commercial property returns in the
UK. Bond and Patel (2003) find that a large portion of property company returns in the UK does exhibit skewness in the
conditional return distribution.
From an asset pricing perspective, skewness and kurtosis of a given asset are also jointly analyzed with the
skewness and kurtosis of the reference market. Similar to the so-called systematic risk or beta, some authors examine if
there exists a systematic skewness and systematic kurtosis and, if any, whether they are priced in asset prices. Systematic
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skewness and kurtosis are also called co-skewness and co-kurtosis (Christie-David and Chaudry, 2001). Provided that the
market has a positive skewness of returns, investors will prefer an asset with positive co-skewness. Co-kurtosis measures
the likelihood that extreme returns will jointly occur in a given asset and in the market. One common characteristic of the
models accounting for co-skewness and co-kurtosis is to incorporate higher moments into the classical two-moment CAPM
model. In the literature, two main approaches have been investigated, namely the three-moment and four-moment CAPMs.
Kraus and Litzenberger (1976) and Sears and Wei (1985) extend the classical CAPM to incorporate the effect of
skewness on portfolio evaluation and provide mixed results. Barone-Adesi (1985) proposes a Quadratic Model to test a
three-moment CAPM. Homaifar and Graddy (1988) derive a higher moment CAPM and test it using principal component
analysis, latent root regression and OLS. Fang and Lai (1997), on the other hand, examine the effect of co-kurtosis on asset
prices using a four-moment CAPM. Their results show that expected excess rate of return is related to systematic variance,
systematic skewness and systematic kurtosis. Investors are generally compensated for taking high risk as measured by high
systematic variance and systematic kurtosis. Investors also forgo the expected returns for taking the benefit of increasing
the skewness. Additionally skewness and kurtosis cannot be diversified away by increasing the size of portfolios. Thus nondiversified skewness and kurtosis play an important role in determining security valuations. Harvey and Siddique (2000)
examine an extended CAPM that includes systematic skewness (co-skewness). They find that the higher moment is priced
and suggest a model incorporating skewness helps explain the cross-sectional variations of stock returns.
Other finance researchers like Hwang and Satchell (1999), Christie-David and Chaudry (2001), Berenyi (2002),
Jurczenko and Maillet (2002), and Galagedera, Henry and Silvapulle (2002) propose the use of the Cubic Model as a test for
coskewness and cokurtosis. Berenyi (2002) applies the four-moment CAPM to mutual fund and hedge fund data. He shows
that volatility is an insufficient measure for the risk of hedge funds and for medium risk adverse agents. Christie-David and
Chaudry (2001) employ the four-moment CAPM on the future markets. They show that systematic skewness and systematic
kurtosis increase the explanatory power of the return generating process of future markets. Hwang and Satchell (1999)
investigate co-skewness and co-kurtosis in emerging markets. Using a GMM approach, they show that systematic kurtosis
explains the emerging market returns better than systematic skewness. Finally, Dittmar (2002) analyzes skewness and
kurtosis across industry indices.
In real estate literature, Liu, Hartzell and Grissom (1992) consider the presence of skewness (relative to other
assets) and any pricing implications for real estate assets. Using the three-moment model of Kraus and Litzenberger (K-L)
(1976), they suggest that investors are willing to accept a lower expected return on real estate assets (relative to other risky
assets) because of the lower negative co-skewness. Using K-:L model, Vines, Hsieh and Hatem (1994) examine the role of
systematic covariance and co-skewness in the pricing of equity real estate investment trusts. Their findings are that systemic
risk impacts return as predicted. However, they find no evidence that co-skewness is a determinant of EREIT return.
So far, no real estate research has explored the joint pricing implications of covariance, co-skewness and cokurtosis in global real estate securities. Following broadly from Hwang and Satchell (1999), we extend the traditional twomoment CAPM to the three-moment and four-moment CAPM i.e. less restrictive forms of the traditional CAPM that
accommodate systematic volatility (i.e. beta or covariance), systematic skewness and systematic kurtosis. We further
employ higher moment data generating processes (DGPs) to test whether coskewness and cokurtosis are priced in real
estate securities. Finally, we estimate preference risk premia values for the 23 real estate securities analyzed and present
the simulated forecast risk premia estimates. Such analyzes (not found in Hwang and Satchell, 1999) will allow us make
some interesting observations and infer the general applicability of higher-moment models in asset pricing and portfolio
design of global real estate securities.
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3.
THEORETICAL FRAMEWORK
The theoretical derpinning this research is the four-moment CAPM. It considers a pricing model for the covariance,
coskewness, and cokurtosis of the real estate securities. Let i denote a generic asset and m the reference market and ri and
rm denote their respective returns. The investment problem for an investor is to maximize the expected utility at the end of
the period. The investor’s expected utility can be represented as a Taylor expansion of order n:
E[U (ri ,t )] = U [ E (ri ,t )] +
1 ''
1
U [ E (ri ,t )]E[ri ,t − E (ri ,t )]2 + U ''' [ E (ri ,t )]E[ri ,t − E (ri ,t )]3
2!
3!
∞
1
+ ∑ U n [ E (ri ,t )]E[ri ,t − E (ri ,t )]n
n =5 n!
(1)
OR
E[U (ri ,t )] = U [ E (ri ,t )] +
1 ''
1
1
U [ E (ri ,t )]σ 2 (ri ,t ) + U ''' [ E (ri ,t )]S 3 + U '''' [ E (ri ,t )]K 4 + ε i ,t
2!
3!
4!
(2)
with
σ = [ E (ri ,t − r i ,t ) 2 ]1 / 2
S = [ E (ri ,t − r i ,t ) 3 ]1 / 3
K = [ E (ri ,t − r i ,t ) 4 ]1 / 4
(3)
where ri,t is the return of the asset i at time t,
r i ,t is the expected return of the asset i at time t, σ is the volatility, S is the
Un
is the nth derivative of the utility function U. In this paper the terms S and K
third moment, K is the fourth moment and
stand for third and fourth moments respectively and not for skewness and kurtosis. In statistics, skewness and kurtosis are
the third and fourth moments standardized respectively by the cube of volatility and volatility to the power of four.
The four-moment CAPM, which is the solution of the maximization of equation (1), is given by
E (ri ,t ) − r f ,t = α 1 β i ,m + α 2 S i ,m + α 3 K i ,m
(4)
with
systematic beta:
β i ,m =
E[(ri ,t − r i ,t )(rm ,t − r m,t )]
E[(rm,t − r m,t ) 2 ]
systematic skewness: S i ,m =
systematic kurtosis: K i ,m =
E[(ri ,t − r i ,t )(rm ,t − r m ,t ) 2 ]
E[(rm ,t − r m ,t ) 3 ]
E[(ri ,t − r i ,t )(rm,t − r m ,t ) 3 ]
E[(rm,t − r m ,t ) 4 ]
(5)
where rf,t is the return of the risk-free asset at time t. The three terms above in equations (5) are respectively the standard
beta from the standard CAPM model, the coskewness divided by the skewness (or third moment) and the cokurtosis divided
by the kurtosis (or fourth moment). Like in the two-moment CAPM where systematic risk is priced, the assumption in this
four-moment CAPM is that the systematic skewness and systematic kurtosis are also priced. We expect a positive risk
premium for positive beta since investors require higher return for a higher beta. We expect a negative risk premium for
positive systematic skewness since, in equilibrium, investors require a lower return for less downside risk. We expect a
positive risk premium for positive systematic kurtosis since investors require a higher return for assets with higher probability
of extreme price variations.
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In equation (4), the three alphas are respectively the market price, or risk premium, for an increase in beta, a
decrease in systematic skewness, and an increase in systematic kurtosis. These three alphas are given by
α1 =
dE (ri ,t )
dσ (ri ,t )
2
σ 2 (rm ,t )
α2 =
dE (ri ,t )
3
dS (ri ,t )
α3 =
S 3 (rm ,t )
dE (ri ,t )
dK 3 (ri ,t )
K 4 (rm ,t )
(6)
The four-moment CAPM in equation (4) is the combination of the systematic beta, systematic skewness, and
systematic kurtosis with the respective market price alphas. If the investor prices the co-moments β i ,m , S i ,m and K i ,m ,
α1 ,α 2 and α 3 should be significantly different from zero. Thus α1 ,α 2 and α 3 are the risk premia to
bear respectively positive beta β i,m , negative systematic skewness S i ,m , and positive systematic kurtosis K i ,m . α 1 can
the alpha values
be seen as the marginal investor risk aversion to variance multiplied with the portfolio variance;
preference for skewness multiplied with the portfolio skewness; and
α3
α2
is the investor marginal
is the investor aversion for kurtosis multiplied with
the portfolio kurtosis.
4.
EMPIRICAL MODELS
Based on the general four-moment CAPM discussed above, as in Huang and Satchell ((1999), we then specify
three DGPs (proxy for empirical models) to test the role of co-skewness and co-kurtosis in asset pricing. The three DGPs
mimic the linear market model, the quadratic market model and the cubic market model respectively. They are briefly
described below.
The Linear Market Model
The market model (7) is an empirical version of the classical CAPM model. In addition, it is used as the benchmark
model to compare with the quadratic and cubic models. The classical market model is a linear equation that relates the
equilibrium expected return on each asset to a single identifiable risk measure. That is, the asset return is linked to the
market risk premium with its beta. We appeal to Generalized Method of Moments (GMM) to estimate the regression.
ri ,t − r f ,t = α 0,t + α 1,t (rm ,t − r f ,t ) + ε i ,t ………………………………………………………………..(7)
The Quadratic Market Model
The Quadratic Model (8) extends the pricing relation to the third moment. This approach assumes that investors
take into consideration the skewness of return distributions. The Quadratic Model states that the relation between an asset
and the market portfolio is quadratic.
ri ,t − r f ,t = α 0,t + α 1,t ( rm ,t − r f ,t ) + α 2,t [rm ,t − E ( rm ,t )]2 + ε i ,t ………………………(8)
The Cubic Market Model
The Cubic Model (9) is the four-moment specification of the CAPM model. It extends the Market Model by
including squared and cubic unexpected market returns as additional factors. This extension allows us to test the role of coskewness and co-kurtosis in asset pricing.
ri ,t − r f ,t = α 0,t + α 1,t ( rm ,t − r f ,t ) + α 2,t [rm ,t − E ( rm ,t )]2 + α 3,t [ rm ,t − E (rm ,t )]3 + ε i ,t
(9)
Here, we assume that the asset returns are a function of a polynomial expansion of the market return. In this
Cubic Model, the aim is to test whether the alphas are significantly different from zero. ri,t is the security return at time t,
α0
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is the asset intercept,
α1 , α 2 and α 3 are respectively the sensitivity of asset i to excess returns of the market portfolio
(proxy for beta), to the market portfolio’s unexpected returns squared (proxy for co-skewness), and to the market portfolio’s
unexpected returns cubed (proxy for co-kurtosis). We test equation (9) in an unconditional framework. Additionally, the
systematic risks of the four-moment CAPM (4) can be expressed as:
systematic beta:
β i , m = α 2 ,i +
α 3,i S m3 + α 4,i K m4
σ m2
α 3,i ( K m4 − σ m4 ) + α 4,i (θ m5 − S m3 σ m2 )
systematic skewness: S i ,m = α 2 ,i +
systematic kurtosis: K i ,m = α 2,i +
(10)
(11)
S m3
α 3,i (θ m5 − σ m2 S m3 ) + α 4,i (δ m6 − S m6 )
(12)
K m4
with
fifth moment: θ i ,m = E[(rm − r m ) ]
5 1/ 5
sixth moment:
(13)
δ i ,m = E[(rm − r m ) 6 ]1 / 6
The expressions (10), (11) and (12) show how the systematic risks (i.e.
(14)
β i ,m , S i ,m andK i ,m ) are related to the
alphas of equation (9). If the asset return distribution can be adequately described by a quadratic model (i.e.
α 3,t [rm ,t − E (rm ,t )]3 = 0
This is because
in equation (9)], then a four-moment CAPM (i.e. cubic model) specification is meaningless.
α 3,t would have no additional explanatory value. Thus, a four-moment CAPM could only be employed if the
data generating process [i.e. equation (6)] is at least cubic. If not, there will be collinearity in the systematic risk of the fourmoment CAPM [i.e. collinearities between equation (10), (11) and (12)] (Hwang and Satchell, 1999).
5.
EMPIRICAL RESULTS
Descriptive Statistics of Monthly returns
The data used are monthly returns of 19 international securities from Datastream (DS) from January 1994 to
January 2004. They cover G7 countries, namely US, UK, Canada, Germany, France, Italy and Japan and also securitized
real estate markets in Hongkong, Singapore, and Asia-Pacific countries. Additionally, four international real estate funds are
included (Alpha International Real Estate Equity Fund, Alpha US Real Estate Equity Fund, Goldman Sachs International
Real Estate Securities Fund and Henderson Horizon Pan European Property Equities Fund). As a proxy of the market
portfolio, Morgan Stanley Capital International (MSCI) world total returns (to represent the world equity market) and the DS
world real estate returns (to represent the world real estate securities market) are used. All empirical tests are presented
from an American investor’s point of view. Returns of all the indices are represented in US dollars. The risk-free rate chosen
is the US 1-month Certificate of Deposit (CD) (the shortest U.S. savings rate available).
Table 1 provides the usual descriptive statistics of monthly returns. Over the full period, the best performing region
is Germany with an annualized return of 23.66%, outperforming other broader market portfolio as represeted by DS world
real estate, MSCI world real estate, DS world market and MSCI world returns; while returns in the South East Asian (SEA)
region posted negative returns. The wide range of the standard deviation (S.D.) measures denotes a great difference among
the various regional real estate return performance. In particular, the standard deviation of China is the highest (14.09%)
and Europe-ex UK is the lowest (3.62%). Skewness in the return distribution is negative for 4 out of the 19 assets, while both
Alpine funds also display negative skewness over the period analyzed. This suggests that extreme positive price increases
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are more likely than extreme price decreases for most of the assets analyzed. We also observe a high probability of extreme
price variations in Asia, Asia-ex Jap, SEA and Singapore where the kurtosis is high (13 of 19 regional assets and 3 of 4
funds display excess kurtosis of above 3). Hence, as in Liow and Sim (2004), the main source of non-stationarity in global
real estate securities might be kurtosis rather than skewness.
(Table 1 here)
The Market Model
Table 2a displays the regression results of the Linear Market Model using MSCI world as the market portfolio.
First, all the regression intercepts are statistically insignificant. Hence the hypothesis that the intercepts are significantly
different from zero is rejected. Second, the values of the regression slopes are very diverse across the sample. Specifically
the respective beta coefficients show that the covariances of the return with the market portfolio of the 19 assets are
extremely different. In particular, all except the Goldman Sachs fund have significant positive betas, while both the China
securitized real estate and the Henderson property fund have insignificant negative betas. Third, the German real estate
security returns have a relatively low beta (0.8703) but the highest performance (23.66% on an annualized basis), while the
Singapore returns have the highest beta (1.7428) and only an annualized return of 3.22%. Hence the standard two-moment
CAPM might not adequately price risk-return tradeoff of these securities. Finally, the linear market model results of Table 2b
using DS world real estate as the market portfolio are qualitatively similar
(Tables 2a and 2b here)
The Quadratic Model
Table 3a contains the results. It further compares the adjusted R2 values from the Linear Market Model (from
Table 2a) and the Quadratic Market Model (both with MSCI world as the market portfolio). We observe that using the
Quadratic Model the adjusted R2 increases for 16 of the 23 real estate assets. Nevertheless, the beta estimates are
statistically significant for all except China, Goldman Sachs and Henderson funds. Only 5 quadratic results (Japan,
European Union (EU), Europe, North America and US) show that co-skewness (represented by Alpha2) is significant. This
means that the excess returns of Japan, EU, Europe, North America and US have a non-linear relationship with the market
portfolio, which implies that these assets will significantly increase or decrease market skewness if added to the market
portfolio. Indeed, 14 of the 23 assets derive negative co-skewness (Alpha2), which have concave payoffs with respect to
their market portfolio. On the contrary, assets with positive co-skewness coefficients have convex payoffs.
(Table 3a here)
Using DS world real estate as the market portfolio, Table 3b shows the the adjusted R2 values for the Quadratic
Market Model increases for 14 of 23 real estate assets. In addition, only 4 significant quadratic results (Asia-ex Japan,
Australia, Hong Kong and Italy) show that co-skewness (represented by Alpha2) is priced. On the other hand, the market
betas for all assets except the Henderson fund are statistically significant.
(Table 3b here)
In all, our results here suggest that the addition of co-skewness as a risk measure increases explanatory power of
the three-moment CAPM (i.e. quadratic market model) for some real estate security markets. In particular, our results
broadly suggest that Asian-Pacific countries can be better explained with co-skewness when the DS world real estate is
considered as the market portfolio.
The Cubic Market Model
Table 4a reports the regression results of the Cubic Market Model. A general observation is that co-kurtosis
(represented by the coefficient Alpha3) appears to be an appropriate risk measure in explaining global real estate security
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returns. Specifically, the co-kurtosis coefficient is significant for 8 securities, namely China, SEA, Singapore, EU, Europe,
Germany UK and the Alpine international fund. A positive co-kurtosis coefficient means that the asset is adding kurtosis to
the market portfolio. China, EU, Europe and UK real estate markets possess negative co-kurtosis, while Germany, SEA,
Singapore and Alpine fund possess positive co-kurtosis. Adding these positive kurtotic assets into the market portfolio will
increase the market’s kurtosis. In contrast, an asset with a negative cokurtosis coefficient asset will decrease the market
portfolio’s kurtosis.
We further compare the adjusted R2 values from the three DGPs (i.e. linear, quadratic and cubic). We note that
the Cubic Model provides better explanatory power for 14 of the 23 investments analyzed. On the other hand, the
corresponding numbers are 4 and 5 respectively for the quadratic model and linear model respectively.
(Table 4a here)
Table 4b reports the regression results of the Cubic Model with the DS world real estate as the market portfolio.
Notably, the co-kurtosis coefficient is statistically significant for 10 of the 23 assets, namely: Asia, Asia-ex Jap, Australia, EU,
Europe, Hong Kong, UK, Alpine US fund, Goldman Sachs (GS) fund and the Henderson fund. Based on the comparative
adjusted R2 values of the three models, again the Cubic Model provides slightly better explanatory power for 17 of the 23
real estate securities analyzed
(Table 4b here)
In all, our results suggest the addition of co-kurtosis as a risk measure increases explanatory power of the Cubic
Market Model for over 50% of the international real estate security markets and provides support for the modeling of higher
moments for securitized real estate markets. It further appears that co-kurtosis rather than co-skewness might be a more
appropriate additional risk measure for global real estate securities.
Additional comments
When the Cubic Model (with DS world real estate market portfolio) is tested, all the 3 co-moments (covariance, coskewness and co-kurtosis) are statistically significant for the Hong Kong and Asia-ex Japan markets. We propose a new
terminology and say that these assets exhibit a ‘tri-moments’ condition (or state) for a particular period of time, where all
the 3 co-moments are priced or concomitant in the Cubic Model. Another situation happens when say, for Henderson fund
(with DS world real estate market portfolio), both the co-skewness and co-kurtosis coefficients are statistically significant in
the Cubic Model. Additionally, another common and interesting observation of the results is when co-kurtosis is statistically
significant, the co-skewness estimate is not significant (7 of the 23 assets display significant co-kurtosis and none display
significant co-skewness with the DS world real estate market portfolio), except for the abovementioned 2 “tri-moments”
securities which is more of a rarity or phenomenon than a common outcome. Hence there might be a tendency for coskewness and co-kurtosis not to co-exist (or they ‘counter-exist’) for a given period of time (see for example Asia, EU and
UK in Table 4b). We label this condition the ‘anti-moment’ state of the Cubic Model - where either co-skewness or
cokurtosis is priced, but not both. Similar “anti-moment” observations can be made when the MSCI world index is used to
represent the market (Table 4a), 12 of the 23 assets display the tendency for either co-skewness or co-kurtosis to be priced
(8 with significant co-kurtosis). However, none of the 23 assets display the “tri-moments” condition.
6.
ESTIMATION OF RISK PREMIA
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One of the key concerns of institutional investors is risk premium estimation (and hence required rate of return)
that should ideally commensurate with the expected returns for more accurate return forecasts. The required rate of return is
defined as the investor’s compensation for bearing the risk associated with the investment. The extent of risk remuneration
depends on the relationship between the equilibrium expected return on each real estate security and the identifiable risk
measures. We will therefore expect that the required rates of return for the real estate securities deriving from the linear,
quadratic and cubic models to be different.
In particular, our expectation can be described as follow. A rational investor dislikes (prefers) negative (positive)
co-skewness. Thus, by comparing the linear and quadratic models, we will expect that the required of return increases
(decreases) for those real estate securities with significantly negative (positive) significant co-skewness coefficients. In
addition, a rational investor dislikes (prefers) positive (negative) co-kurtosis. Thus, by comparing the linear and cubic model,
we will expect that the required rate of return increases (decreases) for those real estate securities with positive (negative)
co-kurtosis coefficients.
To calculate the required rate of return, we use the estimated coefficients from Tables 2a, 2b, 3a, 3b, 4a and 4b.
The expected market and risk-free returns are represented by the historical mean returns from Table I, namely 6.76% (MSCI
world), 3.26% (DS world real estate) and 4.14% (US 1-month CD) per annum.
The appropriate pricing model that is chosen to compute the required rate of return are selected based on the
significance tests of the coefficients of covariance, co-skewness and co-kurtosis, together with the improvement in the
adjusted R2 statistics. For example, we will choose the Cubic Model as the most appropriate asset pricing model only when
the GMM estimates produce a significant coefficient for co- kurtosis and an increase in the adjusted R2 statistic. Conversely,
we will not select the Cubic Model as the most appropriate model when the GMM estimates failed to produce a significant
coefficient for cokurtosis even if the adjusted R2 increases, since most of the R2 values only improve marginally with the
cubic model estimation.
Table 5 produces the empirical estimation of the required rate of return for both proxies of market portfolio (MSCI
world and DS world real estate) for the 23 real estate securities. The results presented are highly interesting. First, the
market model is the most appropriate asset-pricing model for 12 of the 23 real estate securities when the MSCI world is
used as the market portfolio (8 assets use cubic model while 3 use quadratic model). On the contrary, when the DS world
real estate is taken as the market portfolio, the number of assets based on the linear model, quadratic model and cubic
model estimations are 12, 1 and 10 respectively. Second, 5 of the 23 markets (Canada, Europe, Europe-ex UK, France and
UK) use the same model for risk premia estimation regardless of which proxy is used to represent the market portfolio.
(Table 5 here)
Appendix 2 provides simulated forecast comparisons for all the securities. Together with the Theil inequality
coefficients, we find that the best forecast of risk premia for the 23 securities are: cubic model (market portfolio: DS world
real estate): 10; linear model (market portfolio: DS world real estate): 9; linear model (market portfolio: MSCI world): 2; cubic
model (market portfolio: MSCI world): 1 and quadratic model (market portfolio: MSCI world): 1. Our results thus imply that
the choice of a higher-moment model together with an appropriate market portfolio can influence the forecasting of risk
premia.
The hypothesis that rational investors dislike negative co-skewness and prefer positive co-skewness is further
supported. In Table 5 we observe a higher required rate of return each for those assets with negative co-skewness in the
EU, Europe, North America and US markets. These real estate securities display negative co-skewness and hence require a
positive risk premium each (with MSCI world as the market portfolio). The respective positive risk premia are EU(2.74%),
10
Europe (2.87%), North America (5.25%) and US (5.37%). With the DS world real estate market portfolio, the positive risk
premia are for Australia (2.99%), Europe (0.87%), UK (1.67%), Alpine US fund (0.16%) and the GS fund (4.62%)
respectively.
On the co-kurtosis dimension, Table 5 reveals that with the MSCI world market portfolio, the risk premium
estimation for Germany (7.13%) and Alpine international fund (2.08%) again provides support to the hypothesis that rational
investors dislike positive co-kurtosis. Another interesting observation is that UK securitized real estate (2.20%) displays a
negative co-skewness and co-kurtosis coefficients each. While negative co-skewness increases the risk premium, negative
co-kurtosis decreases the required risk premium. Hence the estimated risk premium of 2.20% for the UK market would seem
to account for the excess negative co-skewness although the cubic model does not reveal a significant co-skewness
coefficient. With the DS world market portfolio, the risk premium estimation for GS RealEstate Fund again supports the
hypothesis that rational investors dislike positive co-kurtosis, while the risk premia results for Australia, Europe, UK and
Alpine US fund (with both negative co-skewness and co-kurtosis) appear to reflect the excess negative co-skewness albeit
all the 4 securities do not have significant co-kewness coefficients from the Cubic Model regressions, while all except
Australia have insignificant co-skewness coefficients from the Quadratic Model.
Finally, when the MSCI world is used as the market portfolio, the risk premia results for China, Japan, SEA and
Singapore are negative. These results imply the required rate of return is smaller than what the linear market model would
estimate. As an illustration, the cubic model (MSCI world market portfolio) for the China market is selected and estimates an
annual required return of 5.45% and a risk premium of –7.56% or a “risk discount” of 7.56%. This means that by using the
cubic model to price the China securities, the risk level is reduced compared to what the market model estimates and hence
the required rate of return is less than that of the market model’s expected return (5.45% vs 13.01%). One possible
explanation is that the cubic model has taken into account either the positive co-skewness or negative co-kurtosis (or both)
in security pricing and hence the overall effect is to reduce the systematic risk of the asset. From Table 4a, we further note
that for China, it is the significant and high negative co-kurtosis coefficient that contributed to the overall risk reduction and
hence displays a “risk discount”. Similarly, when the DS world real estate represents the market, securitized real estate
markets in Asia, Asia-ex Japan, Europe-ex EU, Hong Kong, Italy and the Henderson fund display a “risk discount” each,
i.e. the standard two-moment CAPM has overestimated the overall risk level and the use of higher-moment CAPMs helps
ascertain the appropriate and lower risk level.
7.
CONCLUSION
In this paper, we explore the question of whether global real estate securities could be better explained with
additional systematic risks such as co-skewness and co-kurtosis and the resulting impact on risk premium estimation. Based
on a generalized four-moment CAPM, we examine three candidate DGPs; these are the linear, quadratic and cubic models;
and test them using generalized method of moment (GMM).
Overall, our comparative analysis of the linear (i.e. two-moment CAPM) and quadratic and cubic models (i.e.
higher-moment CAPMs) suggests that beta is a significant measure of risk for most of the real estate securities analyzed.
However some securities display significant co-skewness and/or co-kurtosis. The lack of consideration of higher moments in
pricing securitized real estate in such cases can lead to an inadequate estimation and compensation for the additional risk
involved. This is because investing in these assets requires a higher risk premium each for bearing negative co-skewness or
positive co-kurtosis. On the other hand, the standard two-moment CAPM could overestimate the market risk level for some
11
of the securities in the form of a ‘risk discount”. Additionally, we have also observed other interesting conditions displayed
for some of the securities analyzed; these are the “tri-moments”, “bi-moments” and “anti-moment” effects.
The different responses from the three pricing models have wide implications for risk premia modeling and returns
forecasting especially in asset management applications; for example, a natural corollary is that institutional investors or
fund managers would like to know whether the combination of assets display the conditions of “tri-moments” or “antimoment”; i.e. “anti-moment” imply either co-skewness or co-kurtosis is priced in the asset and would entail an increase or
decrease in the estimated risk premium, but since both are not statistically significant at the same time, it would not serve as
counteractive to each other. On the other hand, under tri-moments” situation, all the 3 co-moments are statistically
significant at the same time. Investor would then have to assess the combined moment effect on the resulting risk premium.
Finally It should be noted that these higher-moment models and conditions are applicable to assets for a particular time
frame under study and does not take into account time-varying parameters for higher moments.
In conclusion, our study reveals that there is a significant relationship between expected return and covariance,
co-skewness and co-kurtosis and the number of significant higher-order systematic risks supports the modeling of higher
moments and risk premia estimation in global real estate security market. However, the choice of an appropriate market
portfolio is important in estimating an appropriate higher-moment CAPM and the resulting risk premia. Our results also
reveal that co-kurtosis has more explanatory power than co-skewness in pricing global real estate securities. The findings of
this paper are indeed encouraging and provide alternative risk-return perspectives regarding the pricing of global real estate
securities and portfolio design.
.
12
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14
Table 1 Descriptive Statistics of Monthly Returns
This table shows the descriptive statistics of monthly returns for all 19 real estate assets. The sample period is from Jan-94 to Jan-04. In addition we present 2
Alpine funds (both international and US real estate equity funds), the Goldman Sachs international real estate securities fund, the Henderson property fund, 2
proxies to the world real estate securities market portfolio (MSCI world real estate and DS world real estate), 2 proxies to the world equity market (MSCI world
and DS world market) and finally, the risk-free rate (US 1-month CD). The normality test is based on the Jarque-Bera (JB) statistic at 95% confidence level. 13
of 19 securitized real estate returns and the Alpine international real estate equity fund returns display non-normality. This is in line with both MSCI world real
estate and DS world real estate returns. It should be noted that the sample periods of the funds presented are limited to the availability of data since 3 of the
funds began post Jan-94, while MSCI world real estate index only started in Jan-95.
No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Region / Asset
Asia
Asia ex-Jap
Australia
Canada
China
EU
Europe
Europe ex-EU
Europe ex-UK
France
Germany
Hong Kong
Italy
Japan
North America
SEA
Singapore
UK
US
Alpine Intl RE EQ Fund
Alpine US RE EQ Fund
GS RE Sec Fund Class A
Henderson Prop Fund
DS World RE
MSCI World RE
DS World Market
MSCI World
US CD 1 month
Asset Code
REAS
REAJ
REAU
RECN
RECH
REEU
REER
RENE
REEX
REFR
REGE
REHK
REIT
REJP
RENA
RESE
RESG
REUK
REUS
ALILREFD
ALUSREFD
GSRESFDA
HEPYFD
REWD
MSWDRE
MKWD
MSWD
USCD1M
Period
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 – Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Mar-95 – Jan-04
Feb-99 - Jan-04
Sep-97 - Jan-04
Jan-94 - Jan-04
Jan-95 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Jan-94 - Jan-04
Mean
0.325%
0.347%
0.718%
0.891%
1.429%
0.478%
0.521%
1.639%
0.520%
0.435%
1.972%
0.573%
0.792%
0.247%
0.672%
-0.098%
0.268%
0.541%
0.665%
0.442%
1.298%
0.793%
0.633%
0.272%
0.294%
0.573%
0.563%
0.345%
S.D.
9.272%
11.239%
4.592%
5.518%
14.085%
3.644%
3.683%
10.743%
3.624%
4.645%
10.744%
12.012%
8.046%
9.800%
4.331%
11.668%
13.379%
4.987%
4.391%
4.646%
6.475%
4.146%
2.192%
5.050%
5.739%
4.317%
4.250%
0.134%
Skewness
2.025
1.602
-0.106
0.336
1.321
-0.031
0.046
0.991
0.072
0.215
1.320
1.241
1.015
1.084
0.005
2.393
2.367
-0.108
-0.034
-0.013
-0.257
0.192
0.376
0.170
0.257
-0.174
-0.123
-0.740
Kurtosis
15.241
10.849
3.046
5.119
5.357
2.900
3.022
5.496
2.933
2.561
6.456
8.010
6.618
6.615
4.732
18.898
18.289
3.208
4.451
4.409
3.651
3.599
2.935
5.129
6.546
3.543
3.386
2.260
JB Prob
0.00%
0.00%
88.81%
0.00%
0.00%
96.58%
97.78%
0.00%
93.84%
38.55%
0.00%
0.00%
0.00%
0.00%
0.05%
0.00%
0.00%
79.75%
0.49%
0.67%
21.86%
53.68%
40.63%
0.00%
0.00%
35.02%
58.98%
0.10%
Normality
No
No
Yes
No
No
Yes
Yes
No
Yes
Yes
No
No
No
No
No
No
No
Yes
No
No
Yes
Yes
Yes
No
No
Yes
Yes
No
Note: Values in bold font represent the highest and lowest figures in both the regional and fund samples. See Appendix for details of the funds presented herein.
Descriptive statistics for both MSCI world real estate and DS world real estate are similar. These indices are presented here to serve as a check against one
another and also for estimating the risk premium for a global securitized real estate asset (proxied by MSCI / DS world real estate) with the total world market
portfolio (proxied by MSCI world).
15
Table 2a
Model:
GMM estimates for the Two-Moment CAPM Market Model (MSCI world as market portfolio)
ri ,t − r f ,t = α 0,t + α1,t (rm ,t − r f ,t ) + ε i ,t
This table shows the regression coefficients from the Market Model presented for the 19 securitized real estate assets, 4 real estate funds and also for the 2 world
real estate indices. The t-stat shows the significance of the coefficients (t-stats above 1.96 indicate 5% significance level). The risk-free rate used is the US 1month CD and the proxy to market portfolio is the MSCI world returns.
Region / Asset
Alpha0
T-stat
Alpha1
T-stat
Adj R2 Mkt
Asia
-0.0029
-0.362
1.2423
0.318
4.539**
Asia ex-Jap
-0.0030
-0.310
1.3996
0.274
5.185**
Australia
0.0024
0.826
0.6259
6.362**
0.328
Canada
0.0044
0.723
0.5033
0.143
4.357**
-0.484
China
0.0111
0.761
-0.1191
-0.007
EU
0.0005
0.139
0.4027
0.212
4.934**
0.215
Europe
0.0009
0.254
0.4101
5.061**
0.059
Europe ex-EU
0.0115
1.013
0.6546
2.940**
0.268
Europe ex-UK
0.0008
0.226
0.4500
7.977**
0.078
France
0.0002
0.043
0.3220
3.375**
1.501
0.8703
0.111
Germany
0.0144
4.976**
0.236
Hong Kong
-0.0007
-0.073
1.3909
5.678**
0.095
Italy
0.0032
0.496
0.6056
3.735**
0.125
Japan
-0.0028
-0.398
0.8387
2.639**
0.145
North America
0.0024
0.591
0.3976
3.733**
-0.741
1.5305
0.305
SEA
-0.0078
3.837**
0.301
Singapore
-0.0046
-0.400
1.7428
3.957**
0.089
UK
0.0012
0.251
0.3644
2.534*
0.135
US
0.0023
0.592
0.3897
3.601**
Alpine Intl RE EQ Fund
-0.0006
-0.127
0.7099
7.771**
0.414
Alpine US RE EQ Fund
0.0077
1.136
0.266
0.7692
7.184**
GS RE Sec Fund Class A
0.0059
1.179
0.2170
1.879
0.045
-0.789
Henderson Prop Fund
0.0031
1.180
-0.0473
-0.003
DS World RE
-0.0024
-0.551
0.7558
0.398
7.987**
MSCI World RE
-0.0024
-0.485
0.8015
0.366
6.550**
* Denotes significance at 5% level. ** Denotes significance at 1% level. Values in bold font
indicate highest and lowest values for each column and significant t-stat values.
Table 2b GMM estimates for the Two-Moment CAPM Market Model (DS world real estate as market portfolio)
Model:
ri ,t − r f ,t = α 0,t + α1,t (rm ,t − r f ,t ) + ε i ,t
Region / Asset
Asia
Asia ex-Jap
Australia
Canada
China
EU
Europe
Europe ex-EU
Europe ex-UK
France
Germany
Hong Kong
Italy
Japan
North America
SEA
Singapore
UK
US
Alpine Intl RE EQ Fund
Alpine US RE EQ Fund
GS RE Sec Fund Class A
Henderson Prop Fund
Alpha0
0.0010
0.0015
0.0041
0.0059
0.0114
0.0016
0.0020
0.0134
0.0020
0.0011
0.0166
0.0038
0.0047
-0.0004
0.0036
-0.0031
0.0007
0.0023
0.0036
0.0015
0.0097
0.0039
0.0030
T-stat
0.248
0.282
1.546
1.141
0.845
0.560
0.679
1.166
0.613
0.252
1.517
0.699
0.661
-0.063
1.041
-0.409
0.077
0.557
1.033
0.406
1.711
1.031
1.153
Alpha1
1.6542
1.9884
0.5392
0.6075
0.7562
0.4053
0.4110
0.5924
0.3536
0.2845
0.4006
2.0782
0.3228
0.7967
0.5227
1.7673
1.9771
0.4691
0.5132
0.6871
0.7932
0.6316
-0.0293
T-stat
10.107**
12.316**
7.442**
7.517**
2.696**
5.318**
5.325**
3.469**
6.132**
3.510**
2.001*
15.668**
2.221*
3.373**
7.013**
5.374**
5.026**
3.422**
6.717**
8.398**
7.604**
6.942**
-0.472
Adj R2 Mkt
0.814
0.801
0.346
0.305
0.066
0.308
0.310
0.070
0.234
0.088
0.028
0.766
0.033
0.162
0.368
0.584
0.556
0.220
0.345
0.553
0.340
0.364
-0.009
* Denotes significance at 5% level. ** Denotes significance at 1% level. Values in bold font indicate
highest and lowest values for each column and significant t-stat values.
16
Table 3a GMM estimates for the Three-Moment CAPM Quadratic Model (MSCI world as market portfolio)
Model:
ri ,t − r f ,t = α 0,t + α 1,t (rm ,t − r f ,t ) + α 2,t [ rm ,t − E ( rm ,t )]2 + ε i ,t
Region / Asset
Asia
Asia ex-Jap
Australia
Canada
China
EU
Europe
Europe ex-EU
Europe ex-UK
France
Germany
Hong Kong
Italy
Japan
North America
SEA
Singapore
UK
US
Alpine Intl RE EQ Fund
Alpine US RE EQ Fund
GS RE Sec Fund Class A
Henderson Prop Fund
DS World RE
MSCI World RE
Alpha0
-0.0187
-0.0176
0.0040
0.0082
0.0089
0.0040
0.0045
0.0202
0.0033
0.0026
0.0167
-0.0143
0.0024
-0.0229
0.0068
-0.0245
-0.0245
0.0060
0.0068
-0.0006
0.0127
0.0124
0.0041
-0.0038
-0.0079
T-stat
-1.543
-1.291
1.079
1.199
0.633
1.039
1.130
1.849
0.800
0.483
1.409
-1.011
0.344
-2.214*
1.642
-1.716
-1.671
1.089
1.700
-0.117
1.874
2.179*
1.286
-0.703
-1.183
Alpha1
1.2878
1.4414
0.6211
0.4922
-0.1128
0.3926
0.3997
0.6296
0.4427
0.3150
0.8635
1.4301
0.6077
0.8967
0.3850
1.5788
1.8002
0.3506
0.3768
0.7100
0.7551
0.1607
-0.0502
0.7598
0.8176
T-stat
5.442**
5.972**
6.281**
4.059**
-0.462
5.144**
5.286**
2.833**
7.321**
3.287**
4.714**
6.404**
3.721**
3.480**
3.515**
4.387**
4.596**
2.621**
3.400**
7.717**
7.260**
1.532
-0.879
8.061**
7.212**
Alpha2
8.7528
8.0593
-0.9265
-2.1211
1.1975
-1.9403
-2.0047
-4.8112
-1.4132
-1.3346
-1.3026
7.5463
0.4064
11.1733
-2.4290
9.3126
11.0498
-2.6556
-2.4868
0.0245
-2.5405
-3.0467
-0.4097
0.7773
2.9064
T-stat
1.630
1.278
-0.927
-1.559
0.214
-2.655**
-2.675**
-1.378
-1.715
-1.309
-0.346
1.202
0.280
3.661**
-2.384*
1.260
1.355
-1.746
-2.445*
0.022
-1.660
-1.728
-0.564
0.529
1.279
Adj R2 Mkt
0.318
0.274
0.328
0.143
-0.007
0.212
0.215
0.059
0.268
0.078
0.111
0.236
0.095
0.125
0.145
0.305
0.301
0.089
0.135
0.414
0.266
0.045
-0.003
0.398
0.366
Adj R2 Quad
0.382
0.308
0.325
0.147
-0.015
0.227
0.231
0.067
0.274
0.077
0.105
0.260
0.087
0.219
0.162
0.348
0.348
0.103
0.153
0.409
0.272
0.063
-0.013
0.394
0.382
* Denotes significance at 5% level. ** Denotes significance at 1% level. Values in bold font indicate highest and lowest values for each column,
significant t-stat values and the larger of the two adjusted R2 values (i.e. from market model or quadratic model)
Table 3b GMM estimates for the Three-Moment CAPM Quadratic Model (DS world real estate as market portfolio)
Model:
ri ,t − r f ,t = α 0,t + α 1,t (rm ,t − r f ,t ) + α 2,t [ rm ,t − E ( rm ,t )]2 + ε i ,t
Adj R2 Quad
Region / Asset
Alpha0
T-stat
Alpha1
T-stat
Alpha2
T-stat
Adj R2 Mkt
Asia
-0.0073
-1.638
1.6268
3.2786
1.609
0.814
14.050**
0.846
Asia ex-Jap
-0.0089
-1.927
1.9542
4.0748
0.801
20.489**
2.720**
0.834
Australia
0.0079
0.5515
-1.4759
0.346
2.326*
9.962**
-2.934**
0.368
Canada
0.0085
1.594
0.6162
-1.0437
-1.064
0.305
7.517**
0.308
China
0.0129
0.898
0.7611
-0.5865
-0.151
0.059
2.985**
0.066
-1.0769
-1.245
0.308
EU
0.0044
1.512
0.4143
5.864**
0.325
Europe
0.0048
1.603
0.4202
-1.1006
-1.231
0.310
5.863**
0.328
Europe ex-EU
0.0153
1.336
0.5989
-0.7787
-0.282
0.064
3.487**
0.070
Europe ex-UK
0.0026
0.735
0.3556
-0.2326
-0.634
0.229
5.936**
0.234
France
0.0018
0.360
0.2867
-0.2646
-0.476
0.081
3.403**
0.088
1.340
0.3971
0.4191
0.193
0.020
Germany
0.0155
2.049*
0.028
3.8681
0.766
Hong Kong
-0.0060
-1.181
2.0458
27.373**
3.165**
0.792
Italy
-0.0009
-0.121
0.3043
2.2061
0.033
2.375*
3.750**
0.045
Japan
-0.0065
-0.680
0.7765
2.4065
0.745
0.162
3.854**
0.171
North America
0.0075
0.5355
-1.5303
-1.806
0.368
2.144*
7.558**
0.396
SEA
1.7266
4.8580
1.568
0.584
-0.0155
-2.039*
6.822**
0.627
Singapore
-0.0129
-1.610
1.9324
1.436
0.556
6.237**
5.3444
0.595
UK
0.0075
1.873
0.4862
-1.332
0.220
4.142**
-2.0400
0.258
US
0.0076
0.5265
-1.5835
-1.862
0.345
2.175*
7.280**
0.374
Alpine Intl RE EQ Fund
0.0004
0.119
0.6837
0.4110
0.539
0.551
8.479**
0.553
Alpine US RE EQ Fund
0.0120
0.7935
-0.9791
-0.574
0.339
2.188*
7.855**
0.340
GS RE Sec Fund Class A
0.0052
0.933
0.6326
-0.8467
-0.334
0.355
6.982**
0.364
Henderson Prop Fund
0.0022
0.726
-0.470
0.3070
0.465
-0.017
-0.0300
-0.009
* Denotes significance at 5% level. ** Denotes significance at 1% level. Values in bold font signify highest and lowest values for each column,
significant t-stat values and the larger of the two adjusted R2 values (i.e. from market model or quadratic model)
17
Table 4a GMM estimates for the Four-Moment CAPM Cubic Model (MSCI world as market portfolio)
Model:
ri ,t − r f ,t = α 0,t + α 1,t ( rm ,t − r f ,t ) + α 2,t [rm ,t − E ( rm ,t )]2 + α 3,t [ rm ,t − E (rm ,t )]3 + ε i ,t
Region / Asset
Asia
Asia ex-Jap
Australia
Canada
China
EU
Europe
Europe ex-EU
Europe ex-UK
France
Germany
Hong Kong
Italy
Japan
North America
SEA
Singapore
UK
US
Alpine Intl RE EQ Fund
Alpine US RE EQ Fund
GS RE Sec Fund Class A
Henderson Prop Fund
DS World RE
MSCI World RE
Alpha0
-0.0134
-0.0120
0.0037
0.0098
0.0037
0.0025
0.0030
0.0194
0.0038
0.0032
0.0213
-0.0101
0.0053
-0.0195
0.0082
-0.0134
-0.0109
0.0023
0.0082
0.0016
0.0136
0.0131
0.0029
-0.0032
-0.0067
T-stat
-1.357
-1.053
0.946
1.438
0.262
0.598
0.697
1.792
0.901
0.580
1.961*
-0.811
0.749
-2.123*
2.030*
-1.196
-0.965
0.391
2.098*
0.305
2.001*
2.212*
0.911
-0.610
-1.049
Alpha1
0.7878
0.9171
0.6526
0.3384
0.3860
0.5339
0.5450
0.7017
0.3934
0.2587
0.4319
1.0266
0.3358
0.5726
0.2515
0.5173
0.5117
0.7015
0.2446
0.4965
0.6725
0.0226
0.0403
0.7021
0.7029
T-stat
2.965**
2.850**
4.093**
2.115*
1.172
4.013**
4.095**
1.925
3.548**
1.385
1.617
3.055**
1.073
1.806
1.589
1.430
1.111
3.597**
1.493
4.583**
3.968**
0.112
0.488
4.722**
3.953**
Alpha2
6.8446
6.0582
-0.8061
-2.7084
3.1015
-1.4010
-1.4504
-4.5359
-1.6014
-1.5496
-2.9501
6.0066
-0.6312
9.9365
-2.9385
5.2610
6.1320
-1.3163
-2.9915
-0.7905
-2.8554
-2.9228
-0.0312
0.5571
2.4726
T-stat
2.097*
1.523
-0.779
-1.688
0.591
-1.688
-1.782
-1.177
-1.884
-1.373
-1.065
1.367
-0.422
4.621**
-2.630**
1.386
1.429
-1.075
-2.726**
-0.782
-1.623
-1.869
-0.036
0.545
1.479
Alpha3
81.0540
84.9975
-5.1143
24.9461
-80.8724
-22.9084
-23.5434
-11.6943
7.9912
9.1302
69.9801
65.4013
44.0759
52.5345
21.6431
172.0977
208.8905
-56.8876
21.4395
34.6163
12.8199
26.7361
-12.9069
9.3507
17.8403
T-stat
1.863
1.549
-0.327
1.161
-2.125*
-1.966*
-2.022**
-0.324
0.754
0.545
2.490*
1.214
1.549
1.355
1.169
2.638**
2.737**
-3.036**
1.153
2.633**
0.829
1.086
-1.265
0.441
0.766
Adj R2 Mkt
0.318
0.274
0.328
0.143
-0.007
0.212
0.215
0.059
0.268
0.078
0.111
0.236
0.095
0.125
0.145
0.305
0.301
0.089
0.135
0.414
0.266
0.045
-0.003
0.398
0.366
Adj R2 Quad
0.382
0.308
0.325
0.147
-0.015
0.227
0.231
0.067
0.274
0.077
0.105
0.260
0.087
0.219
0.162
0.348
0.348
0.103
0.153
0.409
0.272
0.063
-0.013
0.394
0.382
Adj R2 Cubic
0.411
0.328
0.320
0.149
-0.009
0.238
0.243
0.060
0.270
0.071
0.117
0.267
0.093
0.225
0.166
0.441
0.453
0.154
0.156
0.429
0.267
0.057
-0.006
0.391
0.380
* Denotes significance at 5% level. ** Denotes significance at 1% level. Values in bold font signify highest and lowest values for each column, significant t-stat values and
the largest of the three adjusted R2 values (i.e. from market model, quadratic model or cubic model)
Table 4b GMM estimates for the Four-Moment CAPM Cubic Model (DS world real estate as market portfolio)
Model:
ri ,t − r f ,t = α 0,t + α 1,t ( rm ,t − r f ,t ) + α 2,t [rm ,t − E ( rm ,t )]2 + α 3,t [ rm ,t − E (rm ,t )]3 + ε i ,t
Region / Asset
Asia
Asia ex-Jap
Australia
Canada
China
EU
Europe
Europe ex-EU
Europe ex-UK
France
Germany
Hong Kong
Italy
Japan
North America
SEA
Singapore
UK
US
Alpine Intl RE EQ Fund
Alpine US RE EQ Fund
GS RE Sec Fund Class A
Henderson Prop Fund
Alpha0
-0.0050
-0.0063
0.0067
0.0074
0.0125
0.0025
0.0029
0.0123
0.0021
0.0011
0.0167
-0.0041
0.0004
-0.0052
0.0068
-0.0114
-0.0086
0.0040
0.0069
-0.0006
0.0100
0.0075
0.0013
T-stat
-1.374
-1.507
1.893
1.466
0.899
0.938
1.069
1.087
0.612
0.221
1.458
-0.836
0.060
-0.659
1.978*
-1.530
-1.168
1.229
2.006*
-0.177
1.959
1.542
0.498
Alpha1
1.3534
1.6604
0.6874
0.7451
0.8016
0.6341
0.6456
0.9571
0.4093
0.3692
0.2541
1.8191
0.1495
0.6293
0.6211
1.2556
1.4317
0.8868
0.6093
0.8044
1.0272
0.2873
0.1454
T-stat
13.681**
15.719**
8.967**
5.309**
2.331*
6.667**
6.657**
3.821**
3.943**
2.601*
0.832
14.534**
0.675
2.650**
7.099**
5.975**
5.889**
7.167**
7.046**
8.739**
8.166**
2.143*
1.792
Alpha2
2.0835
2.7903
-0.8818
-0.4802
-0.4094
-0.1158
-0.1150
0.7874
0.0024
0.0960
-0.2060
2.8769
1.5290
1.7630
-1.1564
2.7987
3.1554
-0.2886
-1.2214
0.9388
-0.0931
-2.1028
0.9140
T-stat
1.870
3.964**
-1.508
-0.461
-0.146
-0.226
-0.217
0.348
0.006
0.153
-0.132
3.812**
1.654
0.821
-1.345
1.253
1.236
-0.398
-1.427
1.576
-0.097
-1.6008
4.2800**
Alpha3
21.9413
23.5835
-10.9073
-10.3456
-3.2508
-17.6448
-18.0950
-28.7548
-4.3149
-6.6204
11.4765
18.1977
12.4318
11.8142
-6.8656
37.8081
40.1893
-32.1542
-6.6493
-9.6905
-19.4285
71.0498
-12.8542
T-stat
2.182*
3.652**
-3.120**
-1.160
-0.117
-3.628**
-3.608**
-1.734
-1.007
-1.142
0.710
2.802**
1.257
0.554
-1.121
1.810
1.741
-4.690**
-1.108
-1.724
-3.095**
3.7094**
-4.2583**
Adj R2 Mkt
0.814
0.801
0.346
0.305
0.066
0.308
0.310
0.070
0.234
0.088
0.028
0.766
0.033
0.162
0.368
0.584
0.556
0.220
0.345
0.553
0.340
0.364
-0.009
Adj R2 Quad
0.846
0.834
0.368
0.308
0.059
0.325
0.328
0.064
0.229
0.081
0.020
0.792
0.045
0.171
0.396
0.627
0.595
0.258
0.374
0.551
0.339
0.355
-0.017
Adj R2 Cubic
0.861
0.846
0.379
0.313
0.051
0.389
0.393
0.077
0.226
0.079
0.015
0.797
0.044
0.169
0.398
0.655
0.619
0.375
0.376
0.560
0.360
0.433
0.099
* Denotes significance at 5% level. ** Denotes significance at 1% level. Values in bold font signify highest and lowest values for each column, significant t-stat values, and
the largest of the three R2 values (i.e. from market model, quadratic model or cubic model)
18
Table 5 Required Rates of Return for International Securitized Real Estate Assets
Premium1 is the difference between the pricing model stated in the third column and the Market Model (with MSCI world as market portfolio), while Premium2
is the difference between the pricing model stated in the sixth column and the Market Model (with DS world real estate as market portfolio). A positive
premium is the additional return the investor should require in order taking on significant negative coskewness and/or significant positive excess kurtosis. RRR1
are the required rates of return of the assets with a market historical annual return of 6.76% (MSCI world) and a risk free rate of 4.14% (US 1-month CD
annualized). RRR2 are the required rates of return of the assets with a market historical annual return of 3.26% (DS world real estate) and a risk free rate of
4.14%.
Region / Asset
Asia
Asia ex-Jap
Australia
Canada
China
EU
Europe
Europe ex-EU
Europe ex-UK
France
Germany
Hong Kong
Italy
Japan
North America
SEA
Singapore
UK
US
Alpine Intl RE EQ Fund
Alpine US RE EQ Fund
GS RE Sec Fund Class A
Henderson Prop Fund
Normality
No
No
Yes
No
No
Yes
Yes
No
Yes
Yes
No
No
No
No
No
No
No
Yes
No
No
Yes
Yes
Yes
Model
(MSCI world)
Market
Market
Market
Market
Cubic
Cubic
Cubic
Market
Market
Market
Cubic
Market
Market
Quadratic
Quadratic
Cubic
Cubic
Cubic
Quadratic
Cubic
Market
Market
Market
RRR1
Premium1
-0.23%
0.06%
4.52%
6.60%
5.45%
4.40%
5.03%
15.51%
2.14%
1.08%
26.69%
2.80%
5.42%
-25.13%
9.17%
-14.73%
-11.74%
4.60%
9.15%
3.22%
11.25%
7.65%
3.60%
0.00%
0.00%
0.00%
0.00%
-7.56%
2.74%
2.87%
0.00%
0.00%
0.00%
7.13%
0.00%
0.00%
-23.97%
5.25%
-9.37%
-10.78%
2.20%
5.37%
2.08%
0.00%
0.00%
0.00%
Model
(DS world RE)
Cubic
Cubic
Cubic
Market
Market
Market
Cubic
Cubic
Market
Market
Market
Cubic
Quadratic
Market
Market
Market
Market
Cubic
Market
Market
Cubic
Cubic
Cubic
RRR2
Premium2
-7.19%
-9.01%
7.44%
6.55%
13.02%
1.56%
2.91%
13.92%
2.09%
1.07%
19.57%
-6.51%
-1.35%
-1.18%
3.86%
-5.27%
-0.89%
4.02%
3.87%
1.20%
11.10%
8.75%
1.43%
-6.94%
-9.07%
2.99%
0.00%
0.00%
0.00%
0.87%
-1.64%
0.00%
0.00%
0.00%
-9.25%
-6.70%
0.00%
0.00%
0.00%
0.00%
1.67%
0.00%
0.00%
0.16%
4.62%
-2.19%
Note: Values in bold font signify highest and lowest values in both the region and fund samples, and also to highlight the similarity in pricing
models for some of the assets. The assets in bold font show that although normality exists (does not exist), the selected model based on the GMM
estimates may not be the market model (quadratic or cubic models), which occurs in 3 cases: Canada (not normal but market models are chosen),
Europe and UK (in both assets normality exists but cubic models are chosen).
19
Appendix 1
Description of Funds
Alpine International Real Estate Equity Fund: The Fund seeks long-term capital growth with a secondary emphasis on
income. Holdings include owners, operators and developers as well as other companies, which derive a majority of their
income or value from real estate.
Alpine U.S. Real Estate Equity Fund: The Fund seeks long-term capital growth with a secondary emphasis on income. The
Fund’s objective is to provide diversified exposure to the U.S. Real Estate securities market, including REITs, operating
companies, homebuilders and other companies, which derive a majority of their income or value from real estate.
Goldman Sachs International Real Estate Securities Fund: The total return is comprised of long-term growth of capital and
dividend income, which offers portfolio diversification from participation in the real estate market, which tends to move
independently of the stock bond market.
Henderson Horizon Pan European Property Equities Fund or known as the Henderson Property Fund: The objective of the
Fund is to achieve long-term capital growth, by investing in the quoted equity securities of companies involved in the
ownership, management and/or development of real estate in Europe. The investment approach is a combination of top
down analysis with bottom up stock selection.
Generalized Method of Moments (GMM)
The GMM estimator belongs to a class of estimators known as M-estimators that are defined by minimizing some
criterion function. GMM is a robust estimator in that it does not require information of the exact distribution of the
disturbances. GSS estimation is based upon the assumption that the disturbances in the equations are uncorrelated with a
set of instrumental variables. The GMM estimator selects parameter estimates so that the correlations between the
instruments and disturbances are as close to zero as possible, as defined by a criterion function. By choosing the weighting
matrix in the criterion function appropriately, GMM can be made robust to heteroskedasticity and/or autocorrelation of
unknown form.
All the procedures are standardized to the conditions where it is set that each of the right-hand side variables is
uncorrelated with the residual i.e. the residuals are uncorrelated with the regression intercept, the excess market returns or
market risk premia, the squared as well as the cube of this excess market returns. Hence the estimates are chosen to
minimize the weighted distance between the theoretical and actual values. GMM is a robust estimator in that, unlike
maximum likelihood estimation, it does not require information of the exact distribution of the disturbances.
The theoretical relation that the parameters should satisfy is usually orthogonality conditions between some
(possibly nonlinear) function of the parameters and a set of instrumental variables. The GMM estimator selects parameter
estimates so that the sample correlations between the instruments and the function are as close to zero as possible, as
defined by the criterion function.
The heteroskedasticity-autocorrelation consistent (HAC) or robust weighting matrix is used which results in GMM
estimates that are robust to heteroskedasticity and autocorrelation of unknown form. The kernel option used is the Bartlett
functional form to weight the autocovariances in computing the weighting matrix, while the Newey and West's fixed
bandwidth selection criterion is used, which determines how the weights given by the kernel change with the lags of the
autocovariances in the computation of the weighting matrix. Prewhitening is performed which runs a preliminary VAR(1)
prior to estimation to "soak up" the correlation in the moment conditions.
20
Appendix 2
Simulated Risk Premia Forecast Charts of Securitized Real Estate Assets
[The blue middle line in each chart depicts the simulated risk premium forecast (additional required rate of return above the
risk-free rate: USCD1M), while the red lines above and below the middle line represent the 95% forecast confidence interval.
If the Theil inequality coefficient is close to 0, covariance proportion is close to one and bias proportion is very small, it
means that there is a “perfect fit” between actual and simulated data i.e. the forecasting performance of the model is
relatively good.]
Asia (MSCI world)
Market Model
Asia (DS world real estate)
Cubic Model
(Note: REASF is the simulated Asian real estate risk premium forecast)
.4
.8
Forecast: REASF
Actual: REAS-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.3
.2
.1
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
-.2
-.3
0.075993
0.054218
244.5257
0.523871
0.000000
0.274443
0.725557
-.4
Forecast: REASF
Actual: REAS-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.6
.4
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.2
.0
-.2
0.033995
0.022388
130.4730
0.190565
0.000000
0.036315
0.963685
-.4
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
REASF
99
00
01
02
03
REASF
The model for Asia (DS world real estate) has a Theil inequality coefficient (0.19) that is the lower of the 2 models estimated
and also with a larger covariance proportion value (0.96). We conclude that the Cubic Model utilizing DS world real estate as
the market portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for the Asian
asset.
Asia-ex Jap (MSCI world)
Market Model
Asia (DS world real estate)
Cubic Model
(Note: REAJF is the simulated Asian-ex Japan real estate risk premium forecast)
.5
.8
Forecast: REAJF
Actual: REAJ-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.4
.3
.2
.1
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
-.2
-.3
-.4
0.095025
0.069383
164.6081
0.554771
0.000000
0.307771
0.692229
Forecast: REAJF
Actual: REAJ-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.6
.4
.2
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.2
-.4
0.043378
0.034039
113.3730
0.201511
0.000000
0.040607
0.959393
-.6
94
95
96
97
98
99
00
01
02
03
94
95
96
97
REAJF
98
99
00
01
02
03
REAJF
The model for Asia-ex Japan (DS world real estate) has a Theil inequality coefficient (0.20) that is the lower of the 2 models
estimated and also with a larger covariance proportion value (0.96). We conclude that the Cubic Model utilizing DS world
real estate as the market portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for
the Asian-ex Japan asset.
Australia (MSCI world)
Market Model
Australia (DS world real estate)
Cubic Model
(Note: REAUF is the simulated Australian real estate risk premium forecast)
21
.20
.15
Forecast: REAUF
Actual: REAU-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.15
.10
.05
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.05
-.10
0.037492
0.030209
215.7775
0.514709
0.000000
0.267950
0.732050
-.15
Forecast: REAUF
Actual: REAU-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.10
.05
.00
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
-.05
-.10
-.15
0.035732
0.027563
274.2192
0.475477
0.000000
0.228452
0.771548
-.20
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
REAUF
99
00
01
02
03
REAUF
The model for Australia (DS world real estate) has a Theil inequality coefficient (0.48) that is the lower of the 2 models
estimated and also with a larger covariance proportion value (0.77). We conclude that the Cubic Model utilizing DS world
real estate as the market portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for
the Australian asset.
Canada (MSCI world)
Market Model
Canada (DS world real estate)
Market Model
(Note: RECNF is the simulated Canadian real estate risk premium forecast)
.20
.3
Forecast: RECNF
Actual: RECN-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.16
.12
.08
Forecast: RECNF
Actual: RECN-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.2
.1
.04
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.04
-.08
-.12
0.050732
0.038293
173.9334
0.656278
0.000000
0.441557
0.558443
-.16
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
0.045700
0.034029
147.1950
0.528737
0.000000
0.284485
0.715515
-.2
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
RECNF
99
00
01
02
03
RECNF
The model for Canada (DS world real estate) has a Theil inequality coefficient (0.53) that is the lower of the 2 models
estimated and also with a larger covariance proportion value (0.72). We conclude that the Market Model utilizing DS world
real estate as the market portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for
the Canadian asset.
China (MSCI world)
Cubic Model
China (DS world real estate)
Market Model
(Note: RECHF is the simulated Chinese real estate risk premium forecast)
.6
.3
Forecast: RECHF
Actual: RECH-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.4
.2
Forecast: RECNF
Actual: RECN-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.2
.1
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.2
-.4
-.6
0.139036
0.101410
118.1432
0.860265
0.000000
0.772473
0.227527
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
0.045700
0.034029
147.1950
0.528737
0.000000
0.284485
0.715515
-.2
94
95
96
97
98
99
00
01
02
03
94
95
RECHF
96
97
98
99
00
01
02
03
RECNF
The model for China (DS world real estate) has a Theil inequality coefficient (0.53) that is the lower of the 2 models
estimated and also with a larger covariance proportion value (0.72). We conclude that the Market Model utilizing DS world
real estate as the market portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for
the Chinese asset.
EU (MSCI world)
Cubic Model
EU (DS world real estate)
Market Model
(Note: REEUF is the simulated EU real estate risk premium forecast)
22
.12
.15
Forecast: REEUF
Actual: REEU-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.08
.04
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.04
-.08
0.031462
0.025223
365.4457
0.571117
0.000000
0.327027
0.672973
-.12
Forecast: REEUF
Actual: REEU-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.10
.05
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.05
-.10
0.030237
0.023357
280.1387
0.530280
0.000000
0.281862
0.718138
-.15
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
REEUF
99
00
01
02
03
REEUF
The model for European Union (DS world real estate) has a Theil inequality coefficient (0.53) that is the lower of the 2
models estimated and also with a larger covariance proportion value (0.72). We conclude that the Market Model utilizing DS
world real estate as the market portfolio is a better forecast and hence a more appropriate model for estimating the risk
premia for the EU asset.
Europe (MSCI world)
Cubic Model
Europe (DS world real estate)
Cubic Model
(Note: REERF is the simulated European real estate risk premium forecast)
.12
.12
Forecast: REERF
Actual: REER-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.08
.04
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.04
-.08
0.031709
0.025183
356.3099
0.567047
0.000000
0.322965
0.677035
-.12
Forecast: REERF
Actual: REER-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.08
.04
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.04
-.08
0.028384
0.022148
298.5213
0.468340
0.000000
0.220120
0.779880
-.12
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
REERF
99
00
01
02
03
REERF
The model for Europe (DS world real estate) has a Theil inequality coefficient (0.47) that is the lower of the 2 models
estimated and also with a larger covariance proportion value (0.78). We conclude that the Cubic Model utilizing DS world
real estate as the market portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for
the European asset.
Europe-ex EU (MSCI world)
Market Model
Europe-ex EU (DS world real estate)
Cubic Model
(Note: RENEF is the simulated European-ex EU real estate risk premium forecast)
.4
.4
Forecast: RENEF
Actual: RENE-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.3
.2
.1
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
-.2
-.3
0.103275
0.072586
140.7257
0.746700
0.000000
0.588194
0.411806
Forecast: RENEF
Actual: RENE-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.3
.2
.1
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
-.2
0.101419
0.072762
162.6043
0.704355
0.000000
0.518668
0.481332
-.3
94
95
96
97
98
99
00
01
RENEF
02
03
94
95
96
97
98
99
00
01
02
03
RENEF
The model for Europe-ex EU (DS world real estate) has a Theil inequality coefficient (0.70) that is the lower of the 2 models
estimated and also with a larger covariance proportion value (0.48). We conclude that the Cubic Model utilizing DS world
real estate as the market portfolio is a better forecast (although not a very good model) and hence a more appropriate model
for estimating the risk premia for the Europe-ex EU asset.
Europe-ex UK (MSCI world)
Market Model
Europe-ex UK (DS world real estate)
Market Model
(Note: REEXF is the simulated European-ex UK real estate risk premium forecast)
23
.15
.15
Forecast: REEXF
Actual: REEX-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.10
.05
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.05
-.10
0.031002
0.025405
445.9026
0.557809
0.000000
0.312528
0.687472
-.15
Forecast: REEXF
Actual: REEX-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.10
.05
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.05
-.10
0.031720
0.025728
546.0946
0.583460
0.000000
0.342035
0.657965
-.15
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
REEXF
99
00
01
02
03
REEXF
The model for Europe-ex UK (MSCI world) has a Theil inequality coefficient (0.56) that is the lower of the 2 models
estimated and also with a larger covariance proportion value (0.69). We conclude that the Market Model utilizing MSCI world
as the market portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for the
European-ex UK asset.
France (MSCI world)
Market Model
France (DS world real estate)
Market Model
(Note: REFRF is the simulated French real estate risk premium forecast)
.15
.16
Forecast: REFRF
Actual: REFR-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.10
.05
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.05
-.10
0.044456
0.036637
148.0461
0.738752
0.000000
0.546444
0.453556
-.15
Forecast: REFRF
Actual: REFR-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.12
.08
.04
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.04
-.08
-.12
0.044229
0.035856
128.2337
0.726324
0.000000
0.528180
0.471820
-.16
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
REFRF
99
00
01
02
03
REFRF
The model for France (DS world real estate) has a Theil inequality coefficient (0.73) that is just marginally lower of the 2
models estimated and also with a slightly larger covariance proportion value (0.47). We conclude that the Market Model
utilizing DS world real estate as the market portfolio is a better forecast (although not a good model) and hence a more
appropriate model for estimating the risk premia for the French asset.
Germany (MSCI world)
Cubic Model
Germany (DS world real estate)
Market Model
(Note: REGEF is the simulated German real estate risk premium forecast)
.5
.4
Forecast: REGEF
Actual: REGE-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.4
.3
.2
.1
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
-.2
-.3
-.4
0.099227
0.070238
187.4783
0.656312
0.000000
0.457096
0.542904
Forecast: REGEF
Actual: REGE-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.3
.2
.1
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
-.2
0.104998
0.074113
158.9505
0.782975
0.000000
0.682023
0.317977
-.3
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
REGEF
99
00
01
02
03
REGEF
The model for Germany (MSCI world) has a Theil inequality coefficient (0.66) that is the lower of the 2 models estimated and
also with a larger covariance proportion value (0.54). We conclude that the Cubic Model utilizing MSCI world as the market
portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for the German asset.
Hong Kong (MSCI world)
Market Model
Hong Kong (DS world real estate)
Cubic Model
(Note: REHKF is the simulated Hong Kong real estate risk premium forecast)
24
.5
.8
Forecast: REHKF
Actual: REHK-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.4
.3
.2
.1
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
-.2
-.3
0.104166
0.077710
202.4011
0.583057
0.000000
0.340206
0.659794
-.4
Forecast: REHKF
Actual: REHK-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.6
.4
.2
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.2
-.4
0.053261
0.042447
143.4496
0.234746
0.000000
0.055128
0.944872
-.6
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
REHKF
99
00
01
02
03
REHKF
The model for HK (DS world real estate) has a Theil inequality coefficient (0.23) that is the lower of the 2 models estimated
and also with a larger covariance proportion value (0.94). We conclude that the Cubic Model utilizing DS world real estate as
the market portfolio is a much better forecast and hence a more appropriate model for estimating the risk premia for the
French asset.
Italy (MSCI world)
Market Model
Italy (DS world real estate)
Quadratic Model
(Note: REITF is the simulated Italian real estate risk premium forecast)
.3
.4
Forecast: REITF
Actual: REIT-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.2
.1
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
-.2
0.076074
0.057412
7862.311
0.714668
0.000000
0.515690
0.484310
-.3
Forecast: REITF
Actual: REIT-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.3
.2
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.1
.0
-.1
0.077797
0.056688
2900.699
0.772333
0.000000
0.603936
0.396064
-.2
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
REITF
99
00
01
02
03
REITF
The model for Italy (MSCI world) has a Theil inequality coefficient (0.71) that is lower of the 2 models estimated and also
with a larger covariance proportion value (0.48). We conclude that the Market Model utilizing MSCI world as the market
portfolio is a better forecast (although not a good model) and hence a more appropriate model for estimating the risk premia
for the Italian asset.
Japan (MSCI world)
Quadratic Model
Japan (DS world real estate)
Market Model
(Note: REJPF is the simulated Japanese real estate risk premium forecast)
.6
.4
Forecast: REJPF
Actual: REJP-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.5
.4
.3
.2
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.1
.0
-.1
-.2
-.3
0.085626
0.064232
641.0690
0.591617
0.000000
0.350085
0.649915
Forecast: REJPF
Actual: REJP-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.3
.2
.1
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
-.2
-.3
0.089036
0.065888
676.6465
0.645630
0.000000
0.416941
0.583059
-.4
94
95
96
97
98
99
00
01
02
REJPF
03
94
95
96
97
98
99
00
01
02
03
REJPF
The model for Japan (MSCI world) has a Theil inequality coefficient (0.59) that is just marginally lower of the 2 models
estimated and also with a larger covariance proportion value (0.65). We conclude that the Quadratic Model utilizing MSCI
world as the market portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for the
Japanese asset.
North America (MSCI world)
Quadratic Model
North America (DS world real estate)
Market Model
(Note: RENAF is the simulated North American real estate risk premium forecast)
25
.12
.20
Forecast: RENAF
Actual: RENA-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.08
.04
.00
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
-.04
-.08
-.12
0.039186
0.030061
161.2651
0.634929
0.000000
0.408609
0.591391
-.16
Forecast: RENAF
Actual: RENA-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.16
.12
.08
.04
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.04
-.08
-.12
0.034183
0.025190
152.1317
0.489203
0.000000
0.241557
0.758443
-.16
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
RENAF
99
00
01
02
03
RENAF
The model for North American (DS world real estate) has a Theil inequality coefficient (0.49) that is lower of the 2 models
estimated and also with a larger covariance proportion value (0.76). We conclude that the Market Model utilizing DS world
real estate as the market portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for
the North American asset.
SEA (MSCI world)
Cubic Model
SEA (DS world real estate)
Market Model
(Note: RESEF is the simulated South East Asian real estate risk premium forecast)
1.0
.6
Forecast: RESEF
Actual: RESE-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
0.8
0.6
0.4
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
0.2
0.0
-0.2
-0.4
0.085862
0.067001
203.8752
0.440160
0.000000
0.194157
0.805843
-0.6
Forecast: RESEF
Actual: RESE-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.4
.2
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.2
-.4
0.074731
0.051778
226.2867
0.363274
0.000000
0.132218
0.867782
-.6
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
RESEF
99
00
01
02
03
RESEF
The model for SEA (DS world real estate) has a Theil inequality coefficient (0.36) that is lower of the 2 models estimated and
also with a larger covariance proportion value (0.87). We conclude that the Market Model utilizing DS world real estate as
the market portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for the SEA
asset.
Singapore (MSCI world)
Cubic Model
Singapore (DS world real estate)
Market Model
(Note: RESGF is the simulated Singaporean real estate risk premium forecast)
1.2
.8
Forecast: RESGF
Actual: RESG-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
0.8
0.4
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
0.0
-0.4
-0.8
0.097364
0.075520
209.2357
0.433846
0.000000
0.188231
0.811769
Forecast: RESGF
Actual: RESG-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.6
.4
.2
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.2
-.4
0.088452
0.062338
349.3165
0.379462
0.000000
0.143998
0.856002
-.6
94
95
96
97
98
99
00
01
02
03
94
95
RESGF
96
97
98
99
00
01
02
03
RESGF
The model for Singapore (DS world real estate) has a Theil inequality coefficient (0.38) that is the lower of the 2 models
estimated and also with a larger covariance proportion value (0.85). We conclude that the Market Model utilizing DS world
real estate as the market portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for
the Singaporean asset.
UK (MSCI world)
Cubic Model
UK (DS world real estate)
Cubic Model
(Note: REUKF is the simulated UK real estate risk premium forecast)
26
.15
.15
Forecast: REUKF
Actual: REUK-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.10
.05
.00
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
-.05
-.10
-.15
0.045208
0.036589
203.8831
0.639018
0.000000
0.409859
0.590141
-.20
Forecast: REUKF
Actual: REUK-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.10
.05
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.05
-.10
0.038848
0.031268
208.6756
0.479588
0.000000
0.230577
0.769423
-.15
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
REUKF
99
00
01
02
03
REUKF
The model for UK (DS world real estate) has a Theil inequality coefficient (0.48) that is the lower of the 2 models estimated
and also with a larger covariance proportion value (0.77). We conclude that the Cubic Model utilizing DS world real estate as
the market portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for the UK asset.
US (MSCI world)
Quadratic Model
US (DS world real estate)
Market Model
(Note: REUSF is the simulated US real estate risk premium forecast
.12
.20
Forecast: REUSF
Actual: REUS-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.08
.04
.00
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
-.04
-.08
-.12
0.039948
0.030659
151.9482
0.643885
0.000000
0.419988
0.580012
-.16
Forecast: REUSF
Actual: REUS-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.16
.12
.08
.04
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.04
-.08
-.12
0.035282
0.026081
153.8579
0.504219
0.000000
0.256528
0.743472
-.16
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
REUSF
99
00
01
02
03
REUSF
The model for US (DS world real estate) has a Theil inequality coefficient (0.50) that is the lower of the 2 models estimated
and also with a larger covariance proportion value (0.74). We conclude that the Market Model utilizing DS world real estate
as the market portfolio is a better forecast and hence a more appropriate model for estimating the risk premia for the US
asset.
Alpine Intl (MSCI world)
Cubic Model
Alpine Intl (DS world real estate)
Market Model
(Note: ALILREFDF is the simulated Alpine international real estate fund risk premium forecast)
.3
.3
Forecast: ALILREFDF
Actual: ALILREFD-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.2
.1
Forecast: ALILREFDF
Actual: ALILREFD-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 121
.2
.1
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
-.2
0.034697
0.027764
136.1252
0.448032
0.000000
0.200864
0.799136
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
0.030963
0.025079
133.0700
0.381427
0.000000
0.145572
0.854428
-.2
94
95
96
97
98
99
00
ALILREFDF
01
02
03
94
95
96
97
98
99
00
01
02
03
ALILREFDF
The model for Alpine international real estate fund (DS world real estate) has a Theil inequality coefficient (0.38) that is the
lower of the 2 models estimated and also with a larger covariance proportion value (0.85). We conclude that the Market
Model utilizing DS world real estate as the market portfolio is a better forecast and hence a more appropriate model for
estimating the risk premia for the Alpine Intl asset.
Alpine US (MSCI world)
Market Model
Alpine US (DS world real estate)
Cubic Model
(Note: ALUSREFDF is the simulated Alpine US real estate fund risk premium forecast)
27
.3
.3
Forecast: ALUSREFDF
Actual: ALUSREFD-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 106
.2
Forecast: ALUSREFDF
Actual: ALUSREFD-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 106
.2
.1
.1
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
0.055099
0.043802
232.2915
0.548822
0.000000
0.313636
0.686364
-.2
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.1
-.2
0.050948
0.041574
243.4967
0.479768
0.000000
0.238264
0.761736
-.3
94
95
96
97
98
99
00
01
02
03
94
95
96
97
98
ALUSREFDF
99
00
01
02
03
ALUSREFDF
The model for Alpine US real estate fund (DS world real estate) has a Theil inequality coefficient (0.48) that is the lower of
the 2 models estimated and also with a larger covariance proportion value (0.76). We conclude that the Cubic Model utilizing
DS world real estate as the market portfolio is a better forecast and hence a more appropriate model for estimating the risk
premia for the Alpine US asset.
GS RE (MSCI world)
Market Model
GS RE (DS world real estate)
Cubic Model
[Note: GSRESFDAF is the simulated Goldman Sachs real estate securities fund (class A) risk premium forecast]
.16
.8
Forecast: GSRESFDAF
Actual: GSRESFDA-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 59
.12
.08
.04
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.04
-.08
0.039852
0.032139
165.7532
0.754803
0.000000
0.603105
0.396895
-.12
Forecast: GSRESFDAF
Actual: GSRESFDA-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 59
.6
.4
.2
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.0
-.2
-.4
0.030154
0.024032
191.3173
0.431629
0.000000
0.190399
0.809601
-.6
94
95
96
97
98
99
00
01
02
03
94
95
96
97
GSRESFDAF
98
99
00
01
02
03
GSRESFDAF
The model for Goldman Sachs real estate securities fund (DS world real estate) has a Theil inequality coefficient (0.43) that
is the lower of the 2 models estimated and also with a larger covariance proportion value (0.80). We conclude that the Cubic
Model utilizing DS world real estate as the market portfolio is a better forecast and hence a more appropriate model for
estimating the risk premia for the GS asset.
Henderson Pty (MSCI world)
Market Model
Henderson Pty (DS world real estate)
Cubic Model
(Note: HEPYFDF is the simulated Henderson property fund risk premium forecast)
.06
.12
Forecast: HEPYFDF
Actual: HEPYFD-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 76
.04
.02
Forecast: HEPYFDF
Actual: HEPYFD-USCD1M
Forecast sample: 1993:12 2004:01
Adjusted sample: 1994:01 2004:01
Included observations: 76
.08
.04
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.02
-.04
-.06
0.022031
0.017545
173.2515
0.839057
0.000000
0.811615
0.188385
Root Mean Squared Error
Mean Absolute Error
Mean Abs. Percent Error
Theil Inequality Coefficient
Bias Proportion
Variance Proportion
Covariance Proportion
.00
-.04
0.020602
0.016046
300.9433
0.662719
0.000000
0.462736
0.537264
-.08
94
95
96
97
98
99
00
HEPYFDF
01
02
03
94
95
96
97
98
99
00
01
02
03
HEPYFDF
The model for Henderson property fund (DS world real estate) has a Theil inequality coefficient (0.66) that is the lower of the
2 models estimated and also with a larger covariance proportion value (0.54). We conclude that the Cubic Model utilizing DS
world real estate as the market portfolio is a better forecast and hence a more appropriate model for estimating the risk
premia for the Henderson asset.
28
29