CRES: 2005-002
Regime Switching and Real Estate Asset Allocation: Some
Evidence from International Markets
K. H. LIOW and H. ZHU, Department of Real Estate, National University of Singapore
Corresponding authors
Dr Kim Hiang LIOW
Associate Professor
Department of Real Estate
National University of Singapore
4 Architecture Drive
Singapore 117566
Republic of Singapore
Tel: (65)68743420
Fax: (65)67748684
Email: rstlkh@nus.edu.sg
Web-page: http://courses.nus.edu.sg/course/rstlkh/
February 15, 2005
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Regime Switching and Real Estate Asset Allocation: Some Evidence from
International Markets
ABSTRACT
The presence of asymmetric correlations in real estate markets has cast doubt on the benefits of international
diversification in real estate. This paper explores a regime switching asset allocation model that includes six major
real estate markets (USA, UK, Japan, Australia, Hong Kong and Singapore) and focuses on how the presence of
regimes affects portfolio composition. We find strong evidence of regimes in these real estate markets. Importantly,
the correlations between the various real estate markets’ returns are higher in the bear market regime than in the
bull market regime. Consequently the optimal real estate portfolio in the high volatility–low return (bear market)
regime is very different from the optimal real estate portfolio in the low volatility-high return (bull market) regime.
Additional out-of-sample tests suggest that the regime-switching strategy outperforms the non-regime dependent
strategy, world real estate portfolio and equally-weighted portfolio from risk-adjusted performance perspective.
Therefore, real estate investors are able to improve their portfolio performance under a bear market and high
correlation regime.
Keywords: regime switching, asymmetric correlations, volatility, asset allocation, risk-adjusted portfolio
performance, international real estate.
1.
INTRODUCTION
Real estate is the world’s biggest business accounting for approximately 15% of gross domestic product with
assets of US$50 trillion compared with US$30 trillion in equities (Bloomberg, 2004). Recently, several studies have
highlighted that listed real estate companies make a significant contribution to the market capitalization of Asian
stock markets (Liow, 2001 and Steinert and Crowe, 2001). Similarly, securitized real estate has become an
increasingly important real estate investment vehicle in Asia and internationally (Steinert and Crowe, 2001),
particularly through the success of REITs in the USA, LPT in Australia and the long-established track record of
listed real estate companies in Asia and Europe.
In the same vein, a number of studies have also highlighted the benefits of international real estate
diversification. These research studies have typically developed correlation matrices in constructing efficient
portfolios. In this process, real estate return, risk and correlation coefficients between the assets are assumed to
be constant over the sample period. Consequently, this approach ignores the salient features of regime switching
in international real estate return data that may affect asset allocation (Liow et al. 2005). Given that global real
estate over the coming decade is expected to become an increasingly important aspect of portfolio construction, it
is thus important to incorporate the presence of regimes in active real estate asset allocation.
The issue of regime changes and their implications for asset allocation has also been examined in the stock
market literature. There is strong evidence of regimes in international stock market returns (Ang and Bekaert,
2002). Consequently, international stock market returns are more highly correlated with each other during bear
markets than during bull markets and in normal times. This asymmetric correlation phenomenon is statistically
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significant (Longin and Solnik, 2001). Furthermore, the presence of regime switching has been incorporated into
investors’ portfolio selection and asset allocation process (Ang and Bekaert, 2002, Guidolin and Timmermann,
2002).
Similarly, securitized real estate markets perform differently in different economic environments (e.g. bull and
bear markets) and this change in behavior results in discrete changes in the time-series risk-return characteristics
of securitized real estate indexes (Liow et al. 2005). Consequently, failure to consider changing behaviors and
time-varying correlations of the markets due to regime shifts might result in sub-optimal asset allocation and
inaccurate portfolio performance measurement.
The resulting key investment issue is whether the presence of regimes does affect the benefits of
international real estate diversification. This issue has not been formally examined in the real estate literature. In
this study we consider mean-variance optimal real estate asset allocation of a regime switching model. Our work
builds on the framework developed in Ang and Bekaert (2003) and focuses on six major real estate markets of the
US, UK, Singapore, Hong Kong, Japan and Australia over 1992-2004. The specific objectives of this research are:
(a) To investigate regime-dependent returns, variances and correlations across the six real estate markets
(b) To consider a regime switching asset allocation model that includes the six real estate markets.
(c)
To assess whether the regime switching dynamic strategy dominates the conventional (i.e. non-regime
dependent) asset allocation strategy by out-of-sample tests.
To establish a background for this study, Section 2 provides a brief review of relevant literature. The
underlying regime switching asset allocation methodology is discussed in section 3. The data requirements are
outlined in Section 4. Section 5 discusses the results. Section 6 contains some conclusions.
2.
LITERATURE REVIEW
Markowitz (1952) proposes a conventional approach for optimal asset allocation, using estimates of risk and
expected return for each component asset class and the covariance between them over some specified
investment horizon. However, one major shortcoming of the conventional approach is that it ignores the dynamic
movement of international stock returns that are characterized by two regimes: a normal regime and a bear market
regime (Ang and Bekaert, 2002). More importantly, the correlations between various returns are higher in the bear
market regime than in the normal one. Schaller and Van Norden (1987) find strong evidence of switching behavior
in the US stock market excess returns. Nishiyama (1998) finds that international stock markets display distinct
regimes in volatility, but not in expected mean return. The persistence of the regimes and the frequency of regime
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swifts are significantly different among the markets. Moreover, the inter-market correlations of regimes are
significantly higher in the post 1987 crash period. Clark and de Silva (1998) demonstrate how the existence of two
regimes affects the mean-variance asset allocation, but their study is silent on the characteristics of the return
series in the two regimes and appropriate portfolio construction strategies with regime switching. Finally, Das and
Uppal (2001) model jumps in equity return correlations using a continuous-jump model and investigate the
implications for asset allocation.
In the real estate literature, Addae-Dapaah and Kion (1996) investigate international diversification of
property stocks from the perspective of a Singaporean investor. Eichholtz (1997) examines real estate stock
diversification by property type and regions. Pierzak (2001) explores the potential benefits of holding international
real estate stocks from a US investor’s perspective. They also analyze the regional risk-return tradeoff, currency
outlook and efficient frontiers in real estate. Worzala and Sirmans (2003) caution that the use of the
mean-variance framework in international real estate diversification is questionable, given the typical reliance on
the historical data that are not necessarily stable over time and therefore not a good predictor of the future.
Furthermore, there is some evidence that real estate stock returns are nonlinear, time varying and unstable over
time (Ambrose et al, 1992 and Newell et al, 2004). Liow et al (2005) formally explore the presence of regimes in
real estate return and volatility using a set of international exchange-based real estate index data from the USA,
the UK, Singapore, Hong Kong, Japan and Australia markets. They find that regime changes in securitized real
estate markets result in different states of the markets with different patterns of risk-return behavior and state
interactions.
This study extends Liow et al. (2005)‘ s work to model the impact of regime switching on international real
estate asset allocation. Finally, although evidence of regime swifts using Markov switching model has been
documented in stock market, its application to real estate markets is relatively new and is definitely of great
importance to this complex issue of international real estate diversification. With increased significance of
international securitized property as a real estate investment vehicle for institutional investors to gain worldwide
real estate exposure, this study is significant to help investors understand better optimal asset allocation when
returns follow a regime switching process.
3.
RESEARCH METHODOLOGY
The principal task in this research is to explore a regime-switching model for international real estate. As an
extension of work in Ang and Bakaert (2003) and Liow et.al (2005), the main components are described below.
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3.1 Theoretical support
Equation (1) is a standard version of the world CAPM for real estate returns:
Rit − R ft = α i + β iw [ Rwt − R ft ] + ε it
(1)
where Rit and R ft denote real estate return and risk-free return respectively, Rwt − R ft is the world
market excess return,
β
is the systematic risk for each real estate market and
ε it
is real estate market
idiosyncratic risk and cannot be diversified. Assume further that the world market excess return is drawn from a
single Gaussian distribution with expected return
u w and variance σ wε tw , then:
y tw = u w + σ wε tw
where y
w
(2)
= Rwt − R ft
Next, suppose that the world market expected excess return and conditional volatility can be characterized
by two different regimes as contraction and expansion (Liow et al.2005). The regime specification of the world
market excess return is given by Equation (3) where st defines an unobservable state or regime; which is
denoted by 0 (contraction) or 1(expansion). The transition between the states is governed by a first-order Markov
process as shown in Equations (4.1- 4.4).
y tw = u0w (1 − st ) + u1w st + [σ 0w (1 − st ) + σ 1w st ]
(3)
P[ st = 0 | st −1 = 0] = p
P[ st = 1 | st −1 = 0] = 1 − p
P[ st = 1 | st −1 = 1] = q
(4.1-4.4)
P[ st = 0 | st −1 = 1] = 1 − q
The regime switching mechanisms are further illustrated in Exhibit 1. As noted in Equations (4.1-4.4), the
regime variable follows a Markov process with constant transition probabilities p and q. For example, if investors
are currently (at time t) in regime 1, the probability of remaining in that regime is q and hence the probability of
w
transitioning in the other regime (i.e. regime 0) is (1-q). The expected return ( u1 ) specification is shown in
Equation (5). Next the expected return for next period (t+1) depends on the investor’s expectations for the regime
realization
u w at time t+1with relevant probability, and if the world market at time t+1 is in regime 0, the expected
w
return is u 0 with transition probability P (Equation 6).
(Exhibit 1 here)
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u1w = (1 − q)u w ( st +1 = 0) + qu w ( st +1 = 1)
u 0w = pu w ( st +1 = 0) + (1 − p )u w ( st +1 = 1)
st = 1
(5)
st = 0
(6)
The expected variance for the world market excess return also depends on the regimes. Its specification
has two components. The first component is the weighted average of conditional variance in two regimes; the
second component is a jump component that originates because the condition mean is different across the
regimes. The specifications for regimes 1 and 0 are shown in Equations 7 and 8 respectively.
(σ 1 ) 2 = (1 − q)(σ w (st +1 = 0))2 + q(σ w (st +1 = 1))2 + q(1 − q)[u w (st +1 = 1) − u w (st +1 = 1)]2 ..(7)
w
(σ 0 ) 2 = p(σ w (st +1 = 0))2 + (1 − p)(σ w (st +1 = 1))2 + p(1 − p)[u w (st +1 = 1) − u w (st +1 = 1)]2 ..(8)
w
3.2 Expected returns and volatilities for individual real estate markets
First, expected returns differ across the individual real estate markets through their different betas (i.e.
systematic risks) relative to the world market. Since the mean of the world market excess return switches between
regimes, the expected excess return of each market is given by Equation 9:
u j = α i + β e wj (j denotes two different regimes: 0 and 1)………..(9)
Next, the expected variance-covariance matrix has three components. First, there is an idiosyncratic
volatility term (unrelated to its beta exposure). Second, when the world market excess return switches between
regimes, the market’s conditional variance also depends on the regime prevailing at time t. Hence there are two
possible variance matrices for the unexpected returns next period ( Ω ), given by Equation (10).Third, the variance
of an individual asset depends on both the realization of the current regime and a jump component. Consequently,
the conditional variance of individual markets can be written as shown in Equation 11 (regime 0) and Equation 12
(regime 1):
Ω i = ( ββ ' )(σ w ( st +1 = i )) 2 + V
(10)
(σ 0 ) 2 = pΩ 0 + (1 − p )Ω1 + p(1 − p)(e1 − e0 )(e1 − e0 ) '
(11)
(σ 1 ) 2 = (1 − q )Ω 0 + qΩ1 + q (1 − q )(e1 − e0 )(e1 − e0 ) '
(12)
where V captures the idiosyncratic volatility term and is a matrix of 0 with (σ i ) along the diagonal.
2
3.3 Asset allocation under regime-switching
We use mean-variance optimization under regime switching with monthly rebalancing for the portfolio,
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consistent with our data frequency. The standard optimal mean-variance portfolio specification is given in Equation
(13), where
γ
is real estate investor’s risk aversion,
∑ ( j)
is the covariance matrix with regime j and
e( j ) is the vector of conditional means for regime j. In addition, we specify 3-month T-bill rate as the portfolio
risk-free rate.
w( j ) =
1
γ
∑ ( j)
−1
e( j )
(13)
In the proposed asset allocation model, there will be two optimal tangency portfolios the investor would
choose, one for each regime. Finally, we will show how mean-variance asset allocation with regime switching
performs in an out-of-sample exercise.
4.
RESEARCH DATA
We focus on six major real estate markets for a US-based investor. Apart from the US and UK, the
remaining four are Asian-Pacific markets of Singapore, Hong Kong, Japan and Australia. Japan is a significantly
developed economy in Asia and has a long history of listed real estate. Other countries like Australia, Hong Kong
and Singapore have track records of listed real estate companies that play a relatively important role in general
stock market indexes. In particular, Australian securitized real estate sector is a leading player in global real estate.
The UK real estate market plays a key role in the European property market. Moreover, the six markets represent
about 91 percent of the global securitized real estate market and have the world’s most significant listed real
estate markets in the respective regions (UBS Investment Bank, 2003). All data are Dow Jones real estate stock
indexes1 and the sample period is from January 1992 till August 2004. We take the US 3-month T-bill rate to be
the risk-free rate.
Monthly real estate stock returns (R) are obtained by taking the logarithmic difference of the stock index
(P) times 100. That is, R t = 100 * (log P t – log P
t – 1).
In addition, the Dow Jones World Real Estate Index
The Dow Jones Global Indexes (DJGI) provides comprehensive world indexes to help international investors in
portfolio management and benchmarking. DJGI calculates indexes on 80% of the investable market capitalization
in 34 countries including both developed and developing markets. In addition, the DJGI family includes indexes for
each of the 10 economic sectors, 18 market sectors, 51 industry groups and 89 subgroups defined by the Dow
Jones Global Classification Standard. Real estate index (code: 8730) is one of the industry sector indexes and
comprises two sub-sector indexes: (a) code 8733: real estate holding and development; and (b) code 8737: real
estate investment trusts (http://djindexes.com). Datastream only has the real estate sector price indexes but does
not have the index data for the two real estate sub-sectors.
1
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(DJWRE) is chosen as a proxy for the world market portfolio. All data are expressed in US dollars. Exhibit 2
displays the index movement over the study period.
(Exhibit 2 here)
Exhibit 3 reports several descriptive statistics for the monthly return series of the six real estate
markets and world real estate. These include the mean, standard deviation, maximum and minimum of returns,
and the measures of skewness and kurtosis. Over the full sample period, the UK real estate market reports the
highest average monthly returns (0.56%) and the US market presents the lowest standard deviation (3.47%).
Moreover, Hong Kong, Japan and Singapore are more volatile (standard deviations are 11.74%, 10.61% and
9.62% respectively) than the other three markets. The skewness statistic shows that 50% of the country returns
series are positively skewed although the respective skewness statistics are not large (between -0.9 and 0.012).
The kurtosis measure is more than 3 in all the return series (between 3.14 and 6.62). This evidence suggests that
for all real estate markets, the distribution of returns has fat tails compared with the normal distribution. Over the
study period, the world real estate index (DJWRE) has a mean monthly return of 0.35% and standard deviation of
4.49% respectively. In addition, it is negatively skewed (-0.26) and exhibits moderate kurtosis (5.23)
(Exhibit 3 here)
5.
RESULTS
5.1 Parameter Estimates
Exhibit 4 contains the estimation results for the regime-switching (RS) real estate model. As the figures
indicate, the world real estate returns (DJWRE) are characterized by switching means and volatility across two
regimes. The first regime (State 0) is characterized by a low excess return of -1.46% per month, with high volatility
of 8.11% per month. However there is also a low volatility regime (State 1) with standard deviation of 4.12% per
month and a higher mean excess return of 0.44% per month. The two states are both persistent with high
transitional probabilities, of 0.8921(state 0) and 0.9722 (state 1) respectively.
(Exhibits 4 here)
The estimates of real estate market beta β i (systematic risk) are also reported in Exhibit 4. All the beta
estimates are statistically significant at the 1% level. The ranges of betas are between 0.49 (Australia) and 1.82
(Hong Kong), with only Hong Kong and Singapore (beta=1.68) being more volatile than the world real estate, an
evidence which is reasonably expected. The CAPM estimates also report each market’s Jensen abnormal return
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index ( α i ) which indicates whether the real estate market has outperformed the world benchmark on a
risk-adjusted basis or not. In particular, the Jensen
α s for the US, UK and Australia real estate markets are
positive (although statistically insignificant) with 0.25%, 0.23% and 0.22% respectively over the full period. Finally,
Japan has the highest idiosyncratic volatility across all markets, followed by Singapore and Hong Kong.
In Exhibit 5, Panel A, we report the implied expected excess returns for the six real estate markets. The
expected excess return in regime 1 is higher than in regime 0. In the bear market regime 0, expected excess
returns are significantly lower and there is more dispersion, with Singapore and Hong Kong having the lowest
expected excess returns (-2.30% and -2.28% respectively). The cumulative total returns on the real estate markets
over the sample period are shown in Exhibit 6. Additionally, Panel B of Exhibit 5 shows the covariance-correlation
matrices in the two regimes. The expectation is that we would be able to detect asymmetric correlations, with
correlations being higher in regime 0 (low return - high-volatility regime). This is indeed the case, with the
correlations in regime 0 being some 88% higher than in regime 1. Finally, Exhibit 7 shows the ex-ante and
smoothed regime probabilities of being in regime 1 (high return-low volatility).
(Exhibits 5, 6 and 7 here)
5.2 Mean-variance optimization models
With mean-variance portfolio construction in the presence of regimes, Exhibit 8 displays the optimal risky
portfolios in regimes 0 and 1. Furthermore, the RS allocation results are compared with (a) the conventional asset
allocation strategy which employs historical mean and variance and assumes the correlations between each pair
of markets are constant throughout the sample period (i.e. non-regime dependent); and (b) an equal-weighted
wealth strategy across the six markets.
(Exhibit 8 here)
First, in regime 0, the real estate investor places 100% of his wealth in the US real estate market. This is
expected as the USA market outperforms other markets during “recession” and consequently it becomes difficult
for the investor to diversify outside the US market. Second, in regime 1, the investment opportunity set is extended
to include other national markets. Exhibit 8 shows, in regime 1, the investor allocates almost half of her asset in
the US market and the remaining wealth is spread to the UK, Hong Kong and Australia with 17.83%, 12.84% and
21.03% respectively. Third, under the conventional mean-variance optimization, the investor will place 39.97%,
36.20% and 23.83% of her wealth in Australia, the US and the UK real estate markets respectively. On the
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contrary, the investor will place approximately 16.67% each of her wealth in all six real estate markets
(equally-weighted) to achieve international diversification benefits.
Exhibit 9 depicts the efficient frontiers for the various asset allocation strategies. The main implications of
regime switching for real estate allocation are shown. The solid line represents the frontier using the conventional
method, ignoring regime switching. The other two frontiers are the ones applicable in two different regimes. The
top frontier represents regime 1, with better risk-return trade-off here, because the expected return is high with low
volatility. The corresponding Sharpe index (SI) for the optimal risky portfolio in regime 1 is 0.1364. In the bear
market regime (i.e. regime 0), the risk-return trade-off worsens and the investor only selects a pure-US real estate
portfolio and realize a SI of 0.01with the tangency portfolio. Finally, the SI for the conventional (non-regime
dependent) portfolio is 0.0586.
(Exhibit 9 here)
5.3 Out-of-Sample Comparisons
In contrast to the in-sample portfolio results of Exhibit 8, in this section we conduct out-of-sample analyses to
examine whether the RS asset allocation strategy outperforms the conventional allocation strategy. Assume that
the investor rebalances her portfolio once a month, our first analysis uses historical data from January 1992 to
August 2003 to construct various asset allocation models as of September 2003. The RS and conventional asset
allocation models are estimated using information available only up to time t . The process is repeated every
month until August 2004. Finally, we evaluate the portfolio performance using ex-post coefficient of variation (CV),
Sharpe index (SI), Jensen α (JI) and Treynor index (TI) realized by the various strategies - RS, conventional,
world real estate market and equally-weighted allocations.
Exhibit 10 shows the portfolio weights derived under the RS strategy for each month from September 2003
to August 2004. There are few key observations. First, no wealth is placed in Singapore and Japan real estate
markets over the period from Sep 03 to Oct 04. On the contrary, the US real estate dominates the portfolio with
100% allocation, especially in May 2004, when the global real estate market was in regime 0. Second, when the
global real estate market was in regime 1, the real estate investor expands her investment opportunity set.
Accordingly she places some wealth in Hong Kong, Australia and the UK real estate markets, in addition to the US
market.
(Exhibit 10 here)
Exhibit 11 reports that over the out -sample, the RS strategy yields an average monthly return of 2.60%. Its
standard deviation is 3.89%. The return and risk performance for the other three strategies are also reported.
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Notably, the non-regime dependent (i.e. conventional) strategy’s monthly average return and standard deviation
are 1.93% and 2.91% respectively. Exhibit 12 further shows how wealth cumulates over time in these strategies.
At the end of the sample period (i.e. August 04), the cumulative returns are 31.14% (RS), 23.12% (conventional),
25.45% (world real estate market) and 27.87% (equally-weighted). These results imply that RS allocation strategy
outperforms the non-regime dependent strategy, equally-weighted portfolio and world real estate portfolio when
return only is considered, particularly in 2003 and 2004.
(Exhibits 11 and 12 here)
Four risk-adjusted return indicators for the out-sample comparisons are also provided in Exhibit 11. As the
figures indicate, the RS strategy’s coefficient of variation (CV) is 1.50, marginally outperforms the conventional
dependent strategy (CV = 1.51) and is also much lower than the world real estate portfolio CV (2.17) and the
equally-weighted portfolio (CV = 1.62). Second, the RS allocation yields a SI of 0.64 and is marginally higher than
the SI of the conventional strategy (SI = 0.63). This is also about 45% higher than the world real estate portfolio
(SI = 0.44) and 9% higher than the equally-weighted portfolio (SI = 0.59). Third, the RS strategy also performs
better in TI (3.16), more than the TIs for the conventional portfolio (2.96), world real estate portfolio (2.03) and
equally weighted portfolio (2.88). Finally, the RS strategy does very well in its JI (0.90%), more than four times the
conventional portfolio JI (0.22%), and is also about 36% higher than the equally-weighted portfolio JI (0.66).
In all, the out-sample tests have reasonably indicated that the RS allocation strategy out-performs the
non-regime dependent strategy (conventional), the world real estate market and the equally-weighted portfolio
consistently. One possible explanation is because the RS allocation strategy helps establish a defensive portfolio
in the bear market regime (i.e. regime 0) that hedges against higher correlations and low returns in international
real estate markets. Furthermore, as the RS allocation strategy relies less on historical moments, it is likely the
resulting optimal portfolio could even be more internationally diversified (Ang and Bakaert, 2002). Consequently, it
is equally possible to add value in real estate portfolios as the presence of a bear market with high correlation
regime does not necessarily erode the benefits of international diversification in real estate. More studies that
cover other real estate markets in different historical periods are needed to document the success of the RS
allocation strategy over the non-regime dependent strategies in international real estate markets.
6.
CONCLUSION
Given the importance of securitized real estate as a real estate investment vehicle for institutional investors
to gain worldwide real estate exposure, this paper explores a regime switching allocation model in international
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real estate markets and focuses on how the presence of regimes affect portfolio composition. This is significant as
the presence of asymmetric correlations in real estate markets has cast doubt on the benefits of international
diversification in real estate assets. Hence active portfolio management should be undertaken to consider regime
changes to add value in international asset allocation.
The major findings are: (a) Strong evidence of regimes is detected in international real estate markets.
Importantly, the correlations between the various real estate markets’ returns are higher in the bear market regime
than in the bull market regime, (b) The optimal real estate portfolio in the high volatility–low return (bear market)
regime is very different from the optimal real estate portfolio in the low volatility-high return (bull market) regime. In
the bear market regime, real estate investors place 100% of their wealth in the USA real estate portfolio. In the bull
market regime, investors switch towards the high return- less volatile markets of Australia, Hong Kong, the UK and
the US. However, the model still has a significant weight (48.30%) on the US real estate market, and (c) The
out-of-sample tests suggest that the RS strategy outperforms the non-regime dependent strategy, the world real
estate portfolio and the equally-weighted portfolio from risk-adjusted return perspective. Therefore, real estate
investors are able to improve their portfolio performance under a bear market and high correlation regimes.
Finally, the results are based on the securitized real estate returns of six major real estate markets in North
American, European and Asian/Pacific economies and generalization to other developing real estate markets and
in different historical periods must be treated with caution. More studies are definitely needed to consider
regime-switching allocation strategy that include currency-hedged real estate returns, higher order moments of
returns and alternative specifications to the world real estate portfolio, such Dow Jones stock market portfolio,
MSCI world stock portfolio and MSCI world real estate portfolio.
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Exhibit 1
A Regime-Switching Model for the World Real Estate Market
Time t
Time t+1
Regime 0
Realized
Regime
(Shaded)
Regime 0
1-q
q
Regime 1
Expected
Return
qu w (1) + (1 − q)u w (0)
u1w
Conditional q (σ w (1)) 2 + (1 − q )(σ w ( 0 )) 2
+ q (1 − q )[ u w (1) − u w ( 0 )] 2
Variance
(σ 1w ) 2
Regime 1
(1− p)u w (1) + puw (0)
u 0w
(1 − p )(σ w (1)) 2 + p (σ w ( 0 )) 2
+ p (1 − p )[ u w (1) − u w ( 0 )] 2
(σ 0w ) 2
14
Exhibit 2
Real Estate Price Indices
400
300
200
100
World real estate
Japan
US
Note:
Australia
Singapore
Jul-04
Jan-04
Jul-03
Jan-03
Jul-02
Jan-02
Jul-01
Jan-01
Jul-00
Jan-00
Jul-99
Jan-99
Jul-98
Jan-98
Jul-97
Jan-97
Jul-96
Jan-96
Jul-95
Jan-95
Jul-94
Jan-94
Jul-93
Jan-93
Jul-92
Jan-92
0
Hong Kong
UK
All data are Dow Jones real estate stock indexes (US dollar) and the sample period is from
January 1992 till August 2004.
Exhibit 3
Descriptive Statistics of Monthly Real Estate Returns (1992:01 to 2004:08)
Mean
Maximum
Minimum
Std Dev
(%)
(%)
(%)
(%)
0.38
41.90
-46.83
Singapore
0.18
48.93
Japan
-0.07
28.41
Australia
0.55
UK
US
World real estate
Hong Kong
Skewness
Kurtosis
11.74
0.012
6.09
-41.07
9.62
0.028
6.62
-20.39
10.61
0.33
3.31
13.77
-15.15
5.05
-0.35
3.67
0.56
14.37
-15.60
6.71
-0.35
3.14
0.49
9.00
-16.80
3.47
-0.90
5.72
0.35
19.32
-21.91
4.49
-0.26
5.23
15
Exhibit 4
Parameter Estimates under Regime Switching
Transition Probabilities
P
Q
Estimate
0.8921
0.9722
Std error
0.02
0.08
World Real Estate Excess Market Return (DJWRE) (%)
u w (0)
u w (1)
σ w (0)
σ w (1)
Estimate
-1.46
0.44
8.11
4.12
Std error
1.49
0.21
2.76
0.65
Country real estate beta ( β )
Hong Kong
Singapore
Japan
Australia
UK
US
Estimates
1.82
1.68
0.62
0.49
0.54
0.42
Std error
0.09
0.12
0.12
0.06
0.08
0.05
Jensen abnormal return index α (%)
Estimates
0.00
-0.20
-0.41
0.22
0.23
0.25
Std error
0.45
0.62
0.67
0.33
0.40
0.21
4.99
3.11
__
Idiosyncratic volatilities
Estimates
5.58
7.64
8.31
σ
(%)
4.12
w
w
Notes: The mean return is u ( 0 ) in state 0 and u (1) in state 1. The standard deviation of returns is
σ w (0)
in state 0 and
σ w (1)
in state 1. The transitional probabilities are P (state 0) and Q (state 1)
16
Exhibit 5
Regime-Dependent Expected Returns and Covariance / Correlations
Panel A: Regime-Dependent Excess Return (%)
Hong Kong
Singapore
Japan
Australia
UK
US
Regime 0
-2.28
-2.30
-1.19
-0.40
-0.46
-0.28
Regime 1
0.71
0.45
-0.17
0.40
0.44
0.41
Panel B: Regime-Dependent Covariance / Correlation coefficients
Regime 0
Hong Kong
2.27
[0.81]
[0.47]
[0.64]
[0.61]
[0.68]
Singapore
1.85
2.29
[0.43]
[0.59]
[0.56]
[0.63]
Japan
0.69
0.63
0.92
[0.34]
[0.33]
[0.37]
Australia
0.54
0.50
0.18
0.31
[0.44]
[0.49]
UK
0.60
0.55
0.20
0.16
0.43
[0.47]
US
0.47
0.43
0.16
0.13
0.14
0.20
Regime 1
Hong Kong
0.50
[0.75]
[0.33]
[0.50]
[0.46]
[0.55]
Singapore
0.56
1.10
[0.21]
[0.31]
[0.29]
[0.34]
Japan
0.21
0.19
0.76
[0.14]
[0.13]
[0.15]
Australia
0.16
0.15
0.05
0.21
[0.19]
[0.23]
UK
0.18
0.17
0.06
0.05
0.30
[0.21]
US
0.14
0.13
0.05
0.04
0.04
0.13
Notes:
We report the regime-dependent means and covariance of excess returns for six real estate markets. All
numbers are listed in percentages and are annualized. For the covariance matrix, the correlation coefficients are
placed in the upper-right triangular matrix in square brackets.
17
Exhibit 6
Cumulative Total Returns in Six Real Estate Markets
%
150
100
50
0
92
-50
-100
Hong Kong(nominal)
Japan
Singapore
Australia
UK
US
Exhibit 7 Ex-Ante and Smoothed Probabilities of Global Real Estate Market (Regime 1)
1
0.5
0
Ex-ante Prob
Notes:
This exhibit shows the ex-ante probabilities (using information up to time t-1) and the smoothed probabilities (using
all sample information) of being in world regime 1(high return-low volatility).
18
Jul-04
Jan-04
Jul-03
Jan-03
Jul-02
Jan-02
Jul-01
Jan-01
Jul-00
Jan-00
Jul-99
Jan-99
Jul-98
Jan-98
Jul-97
Jan-97
Jul-96
Jan-96
Jul-95
Jan-95
Jul-94
Jan-94
Jul-93
Jan-93
Jul-92
Jan-92
Smoothed Prob
Exhibit 8
Optimal Portfolio Weights for Different Strategies (in-sample)
Hong Kong
Singapore
Japan
Australia
UK
US
Regime 0
0
0
0
0
0
100%
Regime 1
12.84%
0
0
21.03%
17.83%
48.30%
Conventional
0
0
0
39.97%
23.83%
36.20%
Equal Weights
16.67%
16.67%
16.67%
16.67%
16.67%
16.67%
Notes:
We report the mean-variance efficient real estate portfolios for different strategies. The monthly risk-free rate is
0.32%. “Regime 0” portfolio is constructed by taking the current month as in regime 0 and then employing our
regime switching asset allocation model to estimate. It is the similar case for “Regime 1”. . “Conventional
“(non-regime dependent) method uses historical average return and variance as expected return and variance to
construct the portfolio. “Equally-weighted” portfolio construction method allocates same amount of wealth across
the six real estate markets.
Exhibit 9
Mean-Variance Efficient Frontiers of the RS and Conventional Models
0.01
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
0.028
0.03
0.032
regime0
0.034
conventional
0.036
0.038
0.04
regime1
Notes:
We plot the mean-variance frontier of regime 0 (bear market), regime 1(bull market) and the conventional
(non-regime) mean-variance frontier. The optimal risky portfolios for each frontier are also marked.
19
Exhibit 10
Optimal Portfolio Weights with the Regime Switching Real Estate Model (Out-of Sample)
Month
Hong Kong
Singapore
Japan
Australia
UK
US
Sep 03
34.66%
0.00%
0.00%
31.62%
0.00%
33.72%
Oct 03
29.97%
0.00%
0.00%
33.27%
0.22%
36.54%
Nov 03
0.80%
0.00%
0.00%
42.32%
10.16%
46.72%
Dec 03
5.06%
0.00%
0.00%
35.34%
15.05%
44.55%
Jan 04
25.43%
0.00%
0.00%
32.70%
14.30%
27.57%
Feb 04
21.04%
0.00%
0.00%
34.35%
14.43%
30.18%
Mar 04
15.84%
0.00%
0.00%
28.03%
18.83%
37.30%
Apr 04
10.50%
0.00%
0.00%
37.30%
27.10%
25.10%
May 04
0.00%
0.00%
0.00%
0.00%
0.00%
100.00%
Jun 04
16.03%
0.00%
0.00%
23.71%
26.84%
33.42%
Jul 04
13.77%
0.00%
0.00%
28.83%
24.51%
32.89%
Aug 04
24.21%
0.00%
0.00%
27.46%
21.73%
26.60%
Notes
The real estate portfolio is rebalanced once a month. The optimal portfolio weights are calculated by the RS asset
allocation model.
Exhibit 11
Out-of Sample Portfolio Performance Evaluation for Different Asset Allocation Strategies
Regime
Switching
Mean Return (%)
2.60
Conventional
(Non-regime
dependent)
1.93
Standard Deviation (%)
3.89
2.91
4.60
3.76
Coefficient of Variation
(CV)
1.50
1.51
2.17
1.62
Sharpe Index (SI)
0.64
0.63
0.44
0.59
Treynor Index (TI)
3.16
2.96
2.03
2.88
α (%) (JI)
0.90
0.22
0.00
0.66
Jensen
World real estate
market
Equally-weighted
2.12
2.32
Notes:
We report the mean, standard deviation, CV, SI, TI and JI of out-of-sample returns following the respective asset
allocation strategies.
20
Exhibit 12 Out-of Sample Cumulative Returns for Different Asset Allocation Models (%)
40.00
Regime Switching
Conventional
DJWI World RE
Equal Weight
30.00
20.00
10.00
0.00
1
2
3
4
5
6
7
8
9
10
11
12
Notes:
This plot shows the cumulative total returns from Sep 2003 (period 1) to Aug 2004 (period 12) for our four portfolio
asset allocations (RS, no-regime dependent, world real estate market portfolio and equally-weighted portfolio).
21