IS
U
Yingchun Liu
TH
is a professor of finance at
California State University
in Fullerton, CA.
[email protected]
C
E
Zhenguo Lin
IT
IS
IL
LE
G
A
L
TO
R
EP
R
O
D
is an assistant professor
at Laval University in
Quebec, Canada.
[email protected]
R
FO
LE
This study takes a different approach.
Instead of tweaking the way the theory is
applied, we modify the theory itself. We
develop an alternative portfolio model that
extends the classical MPT to accommodate
the unique characteristics of real estate. Specifically, our model explicitly incorporates
three most distinct characteristics of real
estate: 1) real estate returns are not independent and identically distributed (i.i.d.) over
time; 2) real estate bears liquidity risk; and
3) real estate involves high transaction costs.
Although these characteristics have been
extensively studied in the literature in various
contexts, the cumulated knowledge has yet
to be synthesized to yield a formal model for
mixed-asset portfolio analysis. Building upon
a series of recent research in this area, the
current article develops such a model. Using
real-world data, we demonstrate that, without
resorting to any de-smoothing procedure or
imposing any ad hoc constraints, our model
produces optimal allocations to real estate in
the mixed-asset portfolio that are substantially
lower than previous studies suggest and are
more consistent with the observed practice of
institutional investors. The decades-old real
estate allocation puzzle, therefore, appears to
be caused by the inability of the classic MPT
to capture the unique characteristics of real
estate.
R
TI
he appropriate role of private real
estate in mixed-asset portfolios is
a long-standing puzzle in the real
estate literature. General consensus
among academics and practitioners dictates
that private real estate investment can bring
additional diversification benefits to traditional security-only portfolios. However,
wide disagreement remains with regard to the
optimal real estate allocation. For example,
although numerous academic studies since the
1980s have repeatedly suggested that real estate
should constitute 15% to 40%—or more—of
a diversified portfolio,1 leading institutional
investors typically have only about 3% to
5% of their total assets in real estate.2 More
recent data have shown the number increased
in recent years, to 5%–9%.3 Significant effort
has been devoted to resolving these discrepancies, but to date no resolution has gained
widespread acceptance among academics.
Previous efforts in resolving the real estate
allocation puzzle have largely been focused
on attempting to “fine-tune” the application of Modern Portfolio Theory (MPT) by
imposing additional constraints that limit the
maximum weights on real estate and/or, in
case the NCREIF data are used, inf lating the
real estate risk by some de-smoothing procedure. To a large extent, these ad hoc solutions
seem to have further complicated the puzzle
rather than solving it.
A
is an associate professor
of finance at Florida
Atlantic University in
Boca Raton, FL.
[email protected]
C
T
P ing Cheng
IN
A
N
Y
Ping Cheng, Zhenguo Lin, and Yingchun Liu
M
A
T
Is There a Real Estate
Allocation Puzzle?
The Journal of Portfolio M anagement 61
Special R eal Estate Issue 2013
Copyright © 2013
MPT AND REAL ESTATE–RELATED
LITERATURE
The elegance of MPT is based on some critical
assumptions. For example, the theory is premised on a
centralized and well-functioning asset market in which
trading is continuous and frictionless, and it assumes that
the complex characteristics of the assets can be conveniently captured with two parameters—the means and
variances of their return distributions. Real estate assets,
however, are infrequently traded in decentralized private
markets that are highly inefficient, and their returns are
neither normal nor stable over time.4 Secondly, nonvariance factors—illiquidity, immobility, heterogeneity,
indivisibility, and the like—are unique to real estate,
and the traditional mean and variance parameters are
inadequate to capture the impact of these risk factors on
real estate price (see Lusht [1988]). Lastly, while financial
assets can be inexpensively traded with ease, high transaction costs prevent real estate from trading frequently;
that is, real estate has to be held for long periods of
time in order to be a viable investment. Most researchers
acknowledge that these features of real estate do not
conform to the finance paradigm, but in the absence of
an alternative portfolio theory, these issues are often dismissed or downplayed as minor deviations. As a result,
mixed-asset portfolio analysis continues to rely on MPT,
and real estate continues to be treated in the same way
as financial assets, with no regard to its unique features.
Recent development in the literature, however, suggests
that these unique features of real estate may actually hold
the key to the real estate allocation puzzle.
ingle-Period Theory versus Multi-Period
S
Reality
MPT is essentially a single-period model, which
assumes that assets in the portfolio are to be held for one
period, and the optimal portfolio is the one that maximizes the investor’s objective over such a single-period
horizon. The reality, though, is that most assets are held
for multi-periods. The validity of single-period MPT to
the multi-period investment reality depends on a critical assumption—all asset returns are i.i.d. over time.
Early studies by Merton [1969], Samuelson [1969], and
Fama [1970] have shown that, under the i.i.d. condition,
investors’ utility maximization over multiple periods is
62 Is There a R eal Estate A llocation Puzzle?
indistinguishable from that over a single period. That
is, investment horizon is irrelevant. This is a powerful
finding, because it effectively implies that portfolio optimization needs only to be based on an asset’s singleperiod performance, regardless of investors’ expected
investment horizon. The i.i.d. condition, therefore, is
the critical link that bridges the gap between the singleperiod theory and the multi-period investment reality.
Such a link does not exist for real estate. In fact,
numerous studies have repeatedly documented that real
estate returns exhibit strong autocorrelation and serial
persistence. That is, they are not i.i.d. from period to
period.5 More recently, Cheng et al. [2011a] formally
demonstrate, from an opposite angle, that a general
mean–variance analysis cannot be simplified to the
commonly known single-period MPT without the i.i.d.
assumption. But despite our knowledge of the missing
link, there has not been, until recently, an alternative
to the i.i.d. assumption that better describes real estate
return distributions. Simply put, if real estate is not i.i.d.,
what is it?
olding-Period Dependence of Real Estate
H
Performance
The non-i.i.d. real estate returns imply that the
asset’s performance is horizon dependent. In other
words, the performance of multi-period investments
cannot be properly measured by single-period performance metrics. Rather, the appropriate performance
measure should coincide with the appropriate holding
period of the asset. This is true not only for real estate
but for other private assets, such as hedge funds (see
Clifford et al. [2001]).
So how does holding period affect the return and
variance of real estate? Lin and Liu [2008] formally
model the real estate transaction process and propose
an alternative to the i.i.d. assumption—the variance of
real estate return increases with the square of the holding time
(as opposed to linearly, with holding time under i.i.d.).
They then empirically verified that residential real
estate data is indeed more consistent with this alternative
assumption than with the i.i.d. assumption. In a subsequent study, Cheng et al. [2010a] extend the examination to the commercial real estate market and find
that the risk structure in the NCREIF indexes is even
closer to the alternative assumption than the residen-
Special R eal Estate Issue 2013
tial market.6 More recently, Cheng et al. [2013] further
expand the investigation to a wide range of U.S. commercial and residential indexes, and document extensive empirical evidence that the alternative assumption
is a better description of real estate risks at national,
regional, and many metropolitan levels. Their examination of the NCREIF is a vivid illustration of how far real
estate deviates from the i.i.d. assumption. As shown in
Exhibit 1, the NCREIF indexes are about as close to the
alternative assumption as the S&P 500 is to the classical
i.i.d. assumption.
Exhibit 1 suggests that real estate risk increases as
holding period becomes longer. This is in contrast to the findings of Rehring [2011], MacKinnon and Zaman [2009],
and Pagliari [2011], which suggest the variance of real
estate returns declines or “decays” in the long run. The
cause of the difference may be in the diverse tactics used
to approach the question. Whereas the previously mentioned three studies all rely on return-generating models
based on certain assumptions on autocorrelation, mean
reversion, and so on, Cheng et al. [2010b] observe the
historical data as-is and directly compute multi-period
return series from quarterly real estate indexes without
using any return-generating models.7 A similar finding is
reported in an earlier study by Collett et al. [2003, p. 207],
who directly observe U.K. commercial data and reach
the similar conclusion that “even allowing for appraisal
smoothing, private real estate looks less attractive once
more realistic and longer time horizons are considered.”
Exhibit 1
The Risk Curves of NCREIF Indexes (1978Q1–2008Q4)
Notes: This exhibit displays the standard deviations of holding-period returns of NCREIF indexes as well as the S&P 500. These standard deviations are
“scaled” so that the standard deviation of one-quarter of each index is 1. Cheng et al. [2010b] call the lines in Exhibit 1 the risk curves, by analogy to
the well-known yield curves.
Special R eal Estate Issue 2013
The Journal of Portfolio M anagement 63
Illiquidity and Ex Ante Risk of Real Estate
Proper estimation of real estate risk, however, requires
more than simply incorporating the impact of holding
period, because real estate is subject to another ­important
risk—liquidity risk. There is little dispute as to the significance of liquidity risk. A survey by Goetzmann and
Dhar [2005] finds that the majority of surveyed institutional portfolio managers consider illiquidity as the
number one risk factor for real estate investment. Additionally, the lack of an appropriate performance metric
that quantifies such risk in formal analysis is a major challenge to their portfolio decisions. These are legitimate
concerns for two reasons: Liquidity risk is substantial,
and it is systematic.
Portfolio decisions are forward-looking in nature.
Proper asset performances, therefore, should be measured with ex ante, rather than ex post, metrics. This is
particularly important for real estate because, at the time
of the acquisition decision, the future sale at the end of
the holding period for real estate is very different from
that of financial assets. While a stock investor faces only
the uncertain sales price, the real estate investor faces
an additional uncertainty—the lengthy and uncertain
time-on-market (TOM). Lin and Vandell [2007] first
proposed the concept of ex ante risk. However, their
formal analysis was still confined within the i.i.d. framework. Recent work of Cheng et al. [2010a] finds that,
under non-i.i.d. assumptions, the uncertain TOM contributes an additional 6% to 29% of the total ex ante risk
to the quarterly NCREIF index. This part of the risk is
unaccounted for by the traditional variance measure.
Liquidity risk is also a systematic risk. Generally
speaking, the expected TOM is a function of market
conditions. In hot markets, all properties sell rather
quickly, while in cold markets, the average TOM will
be substantially longer. Individual sellers may be able
to inf luence their individual selling time with listing
strategies subject to their financial constraints, but they
cannot control the average TOM under a given market
condition. A quick sale typically results in significant
price discount from the property’s fair value. While such
a (deep) discount may ref lect the degree of a seller’s
financial distress, it does not properly ref lect the trading
strategy of normal sellers who would take the time necessary under given market conditions to search until
a desirable offer arrives. In other words, liquidity risk
64 Is There a R eal Estate A llocation Puzzle?
is priced by the market. Portfolio analysis should not
ignore this component of systematic risk in determining
an asset’s optimal weight.
Transaction Cost and Holding Period
Trading real estate involves high transaction costs.
Using UK data, Collett et al. [2003] report “the roundtrip lump-sum costs” were approximately 7% –8% of
the asset value. But the effect of high transaction cost is
more than just lowering the net sales price or returns;
it also prevents frequent trading and affects investment
horizon, which in turn affects the risk of real estate
investment, because real estate performance is horizon
dependent. A number of studies have documented that,
in the presence of transaction cost, single-period utility
maximization is not the same as multi-period utility
maximization. Thus the classical single-period MPT is
not appropriate for multi-period real estate investment
when the transaction costs are high.8
Optimal Holding Period of Real Estate
The horizon-dependence of real estate performance
(as shown in Exhibit 1) suggests that real estate return
volatility (variance or standard deviation) increases faster
than that of stocks as the holding period increases. That
is, real estate becomes more risky in the long run. On the
other hand, because of liquidity risk and high transaction
cost, real estate cannot be held as short-term investment. This leads to an important question: How long is
enough to amortize the transaction cost and illiquidity,
but not too long to bear excessive price volatility? In
other words, is there an optimal holding period for real
estate? This question is important because, if real estate
performance is horizon dependent, one cannot determine the appropriate mean and variance—the inputs to
the mean–variance optimization—without first determining the appropriate holding time. Consequently, the
optimal real estate allocation in a mixed-asset portfolio is
affected by the investment horizon as well. Cheng et al.
[2010b] develop a closed-form formula for the ex ante
optimal holding period of real estate. Their results show
that the optimal holding period is longer when transaction costs are high and/or the expected time-on-market
is long. Using NCREIF data, they show that the optimal
holding periods for various types of properties to be in
Special R eal Estate Issue 2013
the range of four to six years, which is broadly consistent
with the findings by Gau and Wang [1990] and Collett
et al. [2003].
The horizon-dependent performance and the existence of an optimal holding period for real estate have
significant implications for mixed-asset portfolio optimization. In a mean–variance framework, the optimal
allocation to real estate depends on the estimates of its
mean and variance, which in turn depend on the optimal
holding period of real estate. This suggests that holding
period should be part of the optimization process for the
mixed-asset portfolio. That is, a mixed-asset portfolio
must be simultaneously optimized with regard to both the
holding period of real estate as well as the asset allocations. Such a simultaneous optimization is the heart of the
alternative multi-period model to be presented in the
next section.
A MODEL FOR THE MIXED-ASSET
PORTFOLIO
We begin this section with a brief review of the
general mean-variance analysis framework. Consider a
simple world where there are N classes of assets available
for investment. Suppose that an investor’s investment
horizon is T periods. At any given level of risk ∑2, the
investor’s objective is to choose an optimal weight wi on
asset i (i = 1,2,3,…,N) to satisfy the following:
 N

maxN
E  ∑ w i R i ,T   ( w, ,w 2 ,...,w N , ∑ wi =1) 
  i =1
 
i =1
(1)


St. Var  ∑ w i R i ,T  = Σ 2
 i =1

N
where R i ,T is the total return on individual asset i over
N
T periods and E  ∑ i =1 w i R i ,T  is the expected hold

ing-period (total) return of the portfolio. Mathematically, ­Equation (1) is equivalent to solving the following
Lagrangian function,
N
L (w , , w 2 ,..., w N , ∑ w i = 1, λ )
i =1


 N

N

= E  ∑ w i R i ,T  − λ Var  ∑ w i R i ,T  − Σ 2  (2)
 i =1



 i =1

Special R eal Estate Issue 2013
For a given level of risk, there exists a unique λ
in Lagrangian function (Equation (2)), which λ essentially captures the degree of the investor’s risk aversion.
Mathematically, Lagrangian function (Equation (2)) is
equivalent to maximizing the portfolio’s risk-adjusted
return for a given risk aversion parameter λ.
For mathematical simplicity, we assume that there
are only three assets available to investors: an illiquid
real estate asset, a liquid financial asset, and a risk-free
asset. By directly applying Modern Portfolio Theory,
we can obtain the optimal asset allocations by solving
the following problem:
w RE uRE + w L uL + (1 − w RE − w L )r f − 
max 
2 2 
w RE ,w L λ  w 2 σ 2 + 2w
  RE RE
RE w L σ RE σ L ρ + w L σ L 
 
(3)
where
uRE and σRE are the period return and risk of the
real estate asset;
uL and σL are the period return and risk of the
financial asset;
ρ is the correlation between the real estate and
financial asset;
r f is the period return for the risk-free asset;
wRE and wL are the weights on the real estate and
financial assets, respectively; and
λ is the investor’s risk-averse parameter.
Therefore, we can calculate the optimal weight on
real estate as follows:
*
w RE
uRE − r f
 uL − r f  − ρ
σ RE
 σ L 
=
2λ(1 − ρ2 )σ RE
(4)
A number of studies such as Hartzell et al. [1986],
Fogler [1984], Firstenberg et al. [1988], and HudsonWilson et al. [2005], among others, by directly applying
MPT, have suggested that any diversified portfolio
should contain 15% to 40% in real estate.
Next, we extend the classical MPT to accommodate multi-period utility maximization as well as the
unique features of real estate. Exhibit 2 illustrates the
chain of events occurring in the sale of a portfolio.
Suppose that the investor acquires a portfolio
at time 0 and wishes to sell the entire portfolio after
The Journal of Portfolio M anagement 65
holding it for T periods.9 The financial asset can be sold
immediately at time T, but the real estate will have to
wait to be sold at T + t , where time-on-market (t ) and
the eventual sales price are both uncertain and not fully
within the control of the investor.10 Note that the investor’s actual holding period for the real estate is T + t .
Since t is random, the decision to choose the optimal
holding period for the investor is essentially to choose
the optimal T *.
Now, we incorporate the three features of real
estate: 1) the horizon-dependent performance; 2)
liquidity risk; and 3) transaction cost.
First, for the financial asset in the portfolio, the
periodic risk-adjusted return can be readily obtained as
rLAdj (w L ) = (w L uL − λw L2 σ L2 ) (5)
But for the illiquid real estate asset, the computation of the periodic risk-adjusted return is much more
complex than the financial asset. As discussed earlier,
there are two sources of risk associated with real estate
transaction, as well as a high transaction cost (C).
Exhibit 2 indicates that, by the time the property
is sold, the eventual holding period of the investment is
(T + t ), which is uncertain in ex ante. The total risk­adjusted return for the real estate can then be expressed
as
Adj
rRE
(w RE ) = E ex ante [ w RE R T +t ,RE ] − λ
× Var ex ante (w RE R T +t ,RE ) − w REC (6)
Given the fact that the time-on-market t and

RT +t , RE |t can be regarded as two stochastic variables,
by applying the conditional variance formula for any two
stochastic variables, the ex ante variance in Equation (6),
Var ex ante (w RE R T +t ,RE ), can be computed as
(
Var ex ante (w RE R T +t ,RE ) = Var E w RE R T +t ,RE t 
(
)
)
+ E Var w RE R T +t ,RE t  (7)
Without specifying a particular distribution of
time-on-market t , we denote its mean and variance as
2
2
tTOM and σTOM
, respectively. Both tTOM and σTOM
essentially capture the degree of real estate illiquidity. In
terms of ex post return (R T +t ,RE | t ), recent studies by
Lin and Liu [2008] and Cheng et al. [2010a] find that
the total return upon sale follows the distribution with
mean (T + t )uRE and standard deviation (T + t )σ RE . We
can thus simplify Equation (7) as
Var ex ante (w RE R T +t ,RE )
2
2
2
2
2
= w RE
+ ( σ RE
+ uRE
)σTOM
{(T + tTOM )2 σ RE
}
(8)
Equation (8) integrates two source of risks in real
estate investment, price risk and liquidity risk (tTOM ,
2
), into one unified risk metric.
σTOM
For the ex ante expected return—the first term of
Equation 6, first calculate the expectation of the ex post
expected return conditional on time-on-market (t ), and
then use the Law of Iterated Expectations:
Exhibit 2
The Sequence of Events in the Sale of the Portfolio
66 Is There a R eal Estate A llocation Puzzle?
Special R eal Estate Issue 2013
E ex ante w RE R T +t ,RE  = E  E w RE R T +t ,RE t  
= E [ w RE (T + t )uRE ]
(9)
= w RE (T + tTOM )uRE
Substituting Equations (8) and (9) into Equation (6)
yields
Adj
2
2
rRE
(w RE ) = w RE (T + tTOM )uRE − λw RE
(T + tTOM )2 σ RE
2
2
2
+ σTOM
(uRE
+ σ RE
) − w REC
(10)
Therefore, the periodic risk-adjusted return for the
illiquid real estate is
Adj
2
rRE
(w RE ) = w RE uRE − λw RE

σ 2 (u 2 + σ 2RE ) 
×  (T + tTOM )σ 2RE + TOM RE

T + tTOM


C
(11)
− w RE
T + tTOM
For mathematical simplicity, we assume that the
correlation between the real estate and financial assets
is zero, i.e., ρ = 0. Although this assumption may seem
strong, it is not entirely unreasonable, as it has been
well documented in the empirical studies that real estate
historically has low correlation with financial assets. For
example, using annual U.S. data from 1947 to 1982,
Ibbotson and Siegel [1984] found real estate’s correlation with S&P stocks to be –0.06. Worzala and Vandell
[1993] estimate the correlation between the NCREIF
quarterly index from 1980 to 1991 and stocks of the
same period to be about –0.0971. Eichholtz and Hartzell
[1996] document correlations between real estate and
stock indexes to be –0.08 for the United States, –0.10 for
Canada, and –0.09 for the United Kingdom. Quan and
Titman [1999] examine the correlation for 17 countries
and, in contrast to earlier studies, they find a generally
positive correlation pattern in most countries, but for the
United States such positive correlation is insignificant.
Using the quarterly NCREIF and S&P 500 indexes
during the period of 1978Q1 to 2008Q3, we compute
the correlations over different investment horizons. The
results are displayed in Exhibit 3.
From Exhibit 3, we can conclude that the magnitudes of the correlations are actually quite consistent with
previous studies in that all coefficients are rather small.
There is somewhat a trend that the correlations seem to
slightly decline as holding period increases. This result,
along with the findings by previous studies, reaffirms that
real estate has low correlations with stocks and provides
some reasonable justifications to the assumption of zero
correlation between real estate and stocks in this study.
Although it may seem extreme to assume a zero
correlation between real estate and financial assets, given
that the above suggest that real estate has low but mostly
positive correlations with financial assets, optimization
under zero correlation would therefore set the upper
bound of the optimal weights for assets. This of course
also has the added benefit of simplifying the math derivation. But it should be noted that the model developed
in this study can be readily extended to the case of nonzero correlation.
Under the assumption of ρ = 0, the portfolio’s riskadjusted return is the sum of the risk-adjusted return of
each asset in the portfolio—the financial asset, the illiquid
real estate, and the risk-free asset. We can thus express the
periodic risk-adjusted return for the portfolio as,
Adj
Adj
rPort
= rLAdj (w L ) + rRE
(w RE ) + (1 − w L − w RE )r f (12)
Exhibit 3
Long-Run Correlation Coefficients between NCREIF and S&P 500
Special R eal Estate Issue 2013
The Journal of Portfolio M anagement 67
The optimization problem for the investor is
*
to choose the optimal weights (w L* and w RE
) and the
*
Adj . By suboptimal holding period T to maximize rPort
stituting Equations (5) and (11) into Equation (12), we
can thus formulate the optimal asset allocations among
the three assets at time zero as the following optimization problem:

( w u − λw σ ) + (1 − w − w )r + w

max 
2
−λ w (T + t )σ 2 + σTOM
TOM
RE


T + tTOM

L
L
w RE , w L ,T
2
RE
2
2
L
L
RE
L
f
RE

C
u RE − T + t

TOM

(u + σ ) 

2
RE
2
RE






(13)
The important difference between this model
and the classical MPT is that the optimal policy in this
model involves not only the optimal weights (w L* and
*
w RE
) but also the investor’s optimal holding period T *,
whereas the classical MPT is a single-period model. In
addition, the alterative model treats the real estate differently from the financial asset in three aspects: 1) the
model distinguishes the price behavior of real estate from
that of financial assets by recognizing the fact that real
estate returns are not i.i.d.; 2) the real estate performance requires a different measure from that of financial
assets—the ex ante measure, which integrates price risk
with real estate liquidity risk, is the appropriate real
estate performance measure; and 3) the model explicitly
incorporates the high transaction cost of real estate.
Now suppose that real estate can be traded instantly
with trivial time-on-market risk, i.e., tTOM ≈ 0 and
2
σTOM
≈ 0, and further suppose that the transaction cost
is negligible when selling real estate (C ≈ 0). Equation
(13) can then be rewritten as
w RE uRE + w L uL + (1 − w RE − w L )r f
max 
w RE ,w L ,T −λ  w 2 Tσ 2 + w 2 σ 2 
L L
  RE RE
 

(14)
This is still different from the classical MPT, as in
Equation (3) with ρ = 0, because of the holding period
T in Equation 14. However, we can readily show that
the classical MPT holds only when real estate returns
are i.i.d.
We next solve the optimization problem in Equation (13). First, we take the partial derivative of Equation
(13) with respect to T, to obtain
68 Is There a R eal Estate A llocation Puzzle?
2
2
 2
σTOM
(uRE
+ σ 2RE ) 
w REC
2
−λw RE
σ
−
=0
 RE
+
2
2
(T + tTOM )

 (T + tTOM )
(15)
Similarly, the partial derivative of Equation (13)
with respect to wRE results in
uRE − r f −
C
− 2λ w RE
T + tTOM
2


σTOM
2
2
2
uRE
+ σ RE
× (T + tTOM ) σ RE
+
(
)
=0
T + tTOM


(16)
Equations (15) and (16) are two equations with two
unknown variables: wRE and T. We can solve the optimal
*
real estate allocation (w RE
) and the optimal holding period
*
(T ) simultaneously. The existence of the optimal holding
period suggests that the optimal mixed-asset allocation is
related to the investor’s holding period. In contrast, the
classical MPT solves only a single-period optimization
problem and implies that the optimal asset allocation is
independent of holding period. To obtain a closed-form
solution to Equations (15) and (16) is not easy, but we
can examine some important properties of the optimal
holding period (T *).
From Equation (15), we have
2
2
2
λw RE σTOM
(uRE
+ σ RE
)+C
2
= λw RE σ RE
2
(T + tTOM )
(17)
Hence,
T * + tTOM =
2
2
2
λw RE σTOM
+ σ RE
(uRE
)+C
2
λw RE σ RE
(18)
Therefore,
∂T *
=
∂C
1
2λw RE σ
2
RE
2
2
2
λw RE σTOM
(uRE
+ σ RE
)+C
2
λw RE σ RE
>0
(19)
Special R eal Estate Issue 2013
*
Theorem 2. The optimal weight on real estate (w RE
)
is given by
∂T *
2
∂σTOM
1
=
2
2
2
2
(uRE
)+C
λw RE σTOM
+ σ RE
2
λw RE σ RE
2
2
 (uRE
)
+ σ RE
2

 > 0
σ
*
w RE
=
RE
(20)
∂T *
=
∂σ 2RE

2
2λ (T * + tTOM )σ RE

*
(22)
1
2
2
2
λw RE σTOM
(uRE
+ σ RE
)+C
2
2
λw RE σ RE
 σ2 u2

C
×  − TOM4 RE −
< 0 (21)
4 
σ RE
λw RE σ RE 

These results can be summarized in Theorem 1.
Theorem 1. Real estate transaction cost, liquidity risk,
and price risk play a crucial role in determining the optimal
holding period. In particular, we have
C
T + tTOM

σ2
2
2
)
+ * TOM (uRE
+ σ RE
T + tTOM

uRE − r f −
∂T *
∂T *
∂T *
<0
>
> 0;
0;
2
∂σ 2RE
∂σTOM
∂C
Other things being equal, a less liquid market (a larger
2
) with a higher transaction cost (higher C) implies a
σTOM
longer optimal holding period. However, a higher price volatility
(σ 2RE ) implies a shorter optimal holding period.
The important insight revealed by Theorem 1 is
that the optimal holding period is determined by both
systematic and non-systematic factors—market condition
2
(σTOM
and C) and property-specific risk (σ 2RE ). This is consistent with the findings by a recent study of Cheng
et al. [2010b], which concludes that the optimal holding
period of real estate is affected by both market conditions and property-specific performance. Theorem 1 is
also consistent with the findings by Mayshar [1979] and
Constantinides [1986], both of which find that even a
small transaction cost results in a longer holding period
with fewer assets (as opposed to the market portfolio).
With the optimal holding period (T *), we can
*
readily solve the optimal weight of real estate (w RE
)
by using Equation (16), which can be summarized in
Theorem 2.
Special R eal Estate Issue 2013
*
In addition, w RE
is negatively related to real estate
*
liquidity
risk
and
its
transaction
cost, i.e., a) ∂∂wCRE < 0 ; b)
*
∂ w RE
∂σTOM < 0 .
Two conclusions can be drawn from Theorem 2.
First, MPT, which implicitly assumes that all assets have
zero time-on-market risk, is not directly applicable to
mixed-asset portfolios that include real estate. Second,
by taking the liquidity risk, high transaction cost, and
non-i.i.d. nature of real estate returns into consideration,
we demonstrate that optimal real estate allocation is
affected by liquidity risk, investor time horizon, and real
estate transaction cost. In particular, the optimal weight
on the real estate asset is always less than that suggested
by the classical MPT. In other words, the optimal real
estate allocation suggested by pervious literature using
MPT is exaggerated.
A NUMERICAL EXAMPLE
In this section, we use an example to study how
we can numerically solve both the optimal real estate
allocation and the optimal holding period.
From Equation (22), we have
*
w RE
=
C
T * + tTOM

σ2
2
2
)
+ * TOM (uRE
+ σ RE
T + tTOM

uRE − r f −

2
2λ (T * + tTOM )σ RE

(23)
and from Equation (18), we have
T * + tTOM =
*
2
2
2
λw RE
σTOM
+ σ RE
(uRE
)+C
*
2
λw RE σ RE
(24)
The Journal of Portfolio M anagement 69
Substituting Equation (24) into Equation (23), we
can obtain an equation of only one unknown variable
*
*
(w RE
). To obtain a closed-form solution for w RE
is difficult. However, we can find a numerical solution if we
have all the model parameters: 1) real estate illiquidity,
2
i.e., tTOM and σTOM
; 2) the periodic return and risk of the
real estate asset, uRE and σRE ; and 3) the transaction cost
(C), the risk-free rate (rf ), and the risk aversion parameter
(λ). Now we discuss these parameters in turn.
1.Real estate illiquidity. Although not required for
model development, the distribution of TOM
needs to be reasonably assumed in order to esti2
mate tTOM and σTOM
. Following Cheng et al.
[2010a], we assume the distribution of TOM to be
negatively exponential for demonstration purposes.
The negative exponential distribution has a simple
mathematical property: Its variance is equal to the
square of its mean. Therefore, the variance of the
time-on-market (TOM) is equal to the square of
2
2
the expected TOM, i.e., σTOM
. In addition,
= tTOM
based on information from the National Association of Realtors, the average TOM for the US
residential market during the period from 1982
to 2008 was about six months. Considering the
fact that average marketing periods in commercial
markets are often longer than those in residential
markets, we thus choose tTOM ranging from four to
12 months.
2.The periodic return (uRE ) and risk (σRE ) of the real
estate. Using the annualized NCREIF property
index during 1978–2008, we obtain an average
annual return of 9.1% and a standard deviation of
8.3%, and use them as an estimate for uRE and σRE .
For demonstration purposes, we make no attempt
to correct any smoothing bias in the NCREIF
data. Although smoothing is a known issue with
the NCREIF index, recent studies have found that
the effect of smoothing is rather modest.11,12
3. The real estate transaction cost. According to Collett
et al. [2003], the “round-trip lump-sum costs” of
real estate can be approximately 7% to 8% of the
value of an asset. In order to see how transaction
cost affects the optimal holding period and real estate
allocation, we choose C ranging from 3% to 9%. In
addition, we choose rf = 4.5% based on the estimation
of the risk-free rate for the corresponding period.
70 Is There a R eal Estate A llocation Puzzle?
4.The risk aversion parameter (λ). The presence of
a risk aversion parameter in the model suggests
that optimal asset allocation is investor specific.
However, although the market is full of investors
with various degrees of risk aversion, the competitive force of the marketplace ensures that
only the highest bidder gets the deal, and these
highest bidders are likely to be investors who have
a ­particular degree of risk aversion such that the
property’s expected risk-adjusted return is the
highest to them. The degree of risk aversion of
those highest-bidding investors, therefore, can be
implied from the observed market data. As mentioned before, studies in the past, using the classical
MPT, have reported real estate allocations to be
between 15% and 40%.13 If we take the middle
point of this range, 27% for instance, it will imply a
risk ­aversion parameter λ = 12, according to Equation (4), with ρ = 0.
With the model parameters at hand, we can
*
obtain numerical solutions to w RE
and T *. The results
are reported in Exhibits 4 and 5.
From Exhibit 4, the expected optimal holding
period for real estate ranges from 2.1 to 6.3 years,
depending on market condition (degree of illiquidity)
and transaction cost. The optimal holding periods are
longer when transaction costs are high and/or the timeon-market is expected to be longer, which is consistent
with the findings of previous studies in finance literature. The numerical results in Exhibit 4 are also in line
with the empirical finding by Gau and Wang [1990] and
reasonably close to the finding by Collett et al. [2003].
The main point of interest, of course, is the optimal
real estate allocation displayed in Exhibit 5, which
ranges from 2.86% to 8.71%. Compared to previous
findings based on the classical MPT, these real estate
allocations are much more in line with institutional
portfolio reality. Clearly, the optimal allocation to real
estate is determined by investment horizon, liquidity
risk, and transaction cost. Holding other things equal,
higher transaction cost and higher liquidity risk (longer
TOM) correlate with lower real estate allocations. These
results suggest that the alternative model is able to provide a viable solution to the long-standing real estate
allocation puzzle.
Special R eal Estate Issue 2013
Exhibit 4
different autoregression model, Pagliari
[2011] finds that although real estate
The Optimal Holding Period of Real Estate
variance “decays” over the long run,
the variance of financial assets decays
faster, thus making real estate relatively
more risky as the holding period gets
longer. This finding, accompanied by
what he observes to be increased correlation between private and public assets
in the long run, causes real estate to be
Notes: This exhibit displays the optimal holding period (T * + tTOM ) under various assumptions
less appealing and thus carry less weight
of transaction cost and TOM. All numbers are in years unless noted. Exhibit 4 is quite consistent
with the finding by Cheng et al. [2010b].
in mixed-asset portfolios. He suggests
about 12% allocation in real estate for
investors with a four-year investment
Exhibit 5
horizon, which is still much higher
The Optimal Weights for Real Estate
than the reported 3% –5% institutional
reality. None of these three studies
addresses real estate liquidity risk. In
an effort to incorporate illiquidity into
mixed-asset portfolio decisions, Anglin
and Gao [2011] develop a model to discuss the impact of liquidity and liquidity
shock on portfolios, but they do not
­recommend what the optimal real estate
Notes: This exhibit displays the optimal weights of real estate given the corresponding optimal
holding periods in Exhibit 4.
weight should be.
The differences between our
approach
and
these
aforementioned
studies are obvious.
It may be of interest to compare our approach and
First, the approaches of MacKinnon and Zaman [2009],
results with a few recent studies that have attempted
Rehring [2011], and Pagliari [2011] essentially remain
to resolve the real estate allocation puzzle. MacKinnon
within the realm of empirically modifying the way
and Zaman [2009] examine the effect of long investMPT is applied to real estate. Specifically, they attempt
ment horizon on optimal real estate allocation, using the
to “fine-tune” the way to use MPT, not MPT itself. Our
VAR model developed in Campbell and Viceira [2002].
approach, on the other hand, is to modify the theory
Applying the model to US property data, they find
by extending it to explicitly accommodate the unique
that real estate does exhibit long-run mean reversion;
real estate features based on multi-period utility maxihowever, the behavior is weaker than equities, which
mization. Second, while the studies mentioned focus on
means real estate is almost as risky as equities in the long
exploring better empirical approaches for estimating the
run. But despite this discovery, they find the optimal
input data (long-run mean and variances) to MPT, we
allocation to real estate remains rather high (17% –31%)
focus on developing an alternative model by explicitly
and has the tendency to be higher as the investment
incorporating the unique features of real estate.
horizon gets longer. Rehring [2011] adopts the same
Campbell–Viceira method to examine U.K. commercial
property data. Unlike MacKinnon and Zaman [2009],
CONCLUSIONS
he finds that the model-predicted real estate returns
MPT is a single-period asset allocation model. Its
exhibit decreasing variance in the longer horizon, which
validity on multi-period portfolio decisions hinges on
is consistent with his finding that the optimal real estate
a critical assumption—an efficient market where asset
allocation increases rather rapidly as holding period proreturns are independent and identically distributed over
longs. Therefore, neither study is able to resolve the real
time. Because real estate does not fit in this paradigm,
estate allocation puzzle. In contrast, using a somewhat
Special R eal Estate Issue 2013
The Journal of Portfolio M anagement 71
the classical MPT needs to be extended to accommodate
the more complex features of real estate and mixedasset portfolio analysis. Building upon a series of recent
research on real estate illiquidity and performance metrics, this article synthesizes some of the latest advances
in the literature to develop an alternative model that
extends the classical MPT for mixed-asset portfolio
analysis. Unlike many previous efforts, which attempt
to empirically solve the real estate allocation puzzle with
ad hoc solutions, we provide a formal model that explicitly incorporates the three most unique features of real
estate—horizon-dependent performance, liquidity risk,
and high transaction cost—into a multi-period meanvariance analysis. Using commercial real estate data, the
alternative model produces a range of optimal real estate
allocations that are in line with the reality of institutional
portfolios.
It should be acknowledged that, although our model
is able to succeed where the conventional approach has
failed in resolving the long-standing real estate allocation puzzle, this work should be viewed only as a first
step in the search for a more general portfolio theory for
mixed-asset portfolio analysis. Several important issues
still remain. First, we have implicitly assumed that the
investors are “normal” sellers who are not under any
liquidity shock to force liquidation of real estate. A more
general theory should incorporate seller heterogeneity
and the possibility of liquidity shock into the model.
Second, we do not consider the indivisibility or partial
sale of real estate assets. We assume the entire real estate
holding will be sold together, although in reality, investors facing liquidity shock may need to liquidate only
part of their real estate holdings to satisfy the need for
cash. This issue is discussed in Anglin and Gao [2011] to
some extent. Third, while the alternative model breaks
away from the efficient-market paradigm, in which
asset prices are assumed to follow a random walk, it
is still confined within the mean–variance framework,
which implicitly assumes that investors have a symmetric
aversion to price volatility. The reality, though, is that
investors’ risk perception is often asymmetric. To the
extent that asset returns are not jointly normally distributed, the concept of downside risk is an appealing
(and more complex) risk metric. This is, of course, a
much broader issue, which also pertains to nearly all
mainstream finance theories. Despite these simplifications, the empirical analysis and theoretical exploration
accomplish the two modest objectives of this article: to
72 Is There a R eal Estate A llocation Puzzle?
extend the MPT for mixed-asset portfolio analysis and
to suggest a solution to the decades-old real estate allocation puzzle. While this line of work is certainly at the
stage of early exploration, the prospect is promising.
ENDNOTES
1
See, for example, Hartzell et al. [1986], Fogler [1984],
Firstenberg et al. [1988], and Hudson-Wilson et al. [2005].
2
See Goetzmann and Dhar [2005] and an earlier survey
by Pension & Investments in 2002, which reported that the top
200 defined-benefit pension funds had only about 3% of their
total assets in private real estate equities.
3
See report by Pension Real Estate Association (PREA)
http://www.prea.org/research/plansponsorsurvey_2012.pdf
4
See, for example, Young and Graff [1995], Young et al.
[2006], and Young [2008].
5
The literature on the subject is too large to be reviewed
here fully. A few examples include Case and Shiller [1989],
Young and Graff [1995], Englund et al. [1999] and Gao et al.
[2009], among others.
6
A brief description of their approach is attached in
Appendix A.
7
For detail descriptions of their computation process, see
Cheng et al. [2011b]. An alternative non-bootstrap method is
discussed in Cheng et al. [2010a].
8
See, for example, Mayshar [1979], Constantinides
[1986], Amihud and Mendelson [1986], and Atkins and Dyl
[1997].
9
For mathematical simplicity, we assume that there is
no partial sale.
10
Immediate sale of real estate (as in the case of imminent
foreclosure) would characterize the behavior of an extremely
distressed seller. While the likely (deep) price discount ref lects
the degree of distress of the seller, it does not properly ref lect
the illiquidity of the real asset under normal situations. Put
differently, since normal sellers do not attempt immediate sale
of their real estate, the price discount that would be necessary
for such immediate sale is not an appropriate measure of the
asset’s illiquidity. Lippman and McCall [1986] define such
illiquidity as the expected time until an asset is successfully
exchanged for cash under optimal policies.
11
See, for example, Cheng et al. [2011b].
12
For interested readers, we also examined the transaction-based NCREIF TBI index, which, due to a much shorter
history, is not directly comparable with the original NCREIF
index. That being said, we find TBI exhibit comparable returns
with the original NCREIF index but higher volatility. So
other things being equal, TBI data should yield a lower real
estate weight than what we presented in the article. That is,
our results are the upper bound of the real estate weight.
Special R eal Estate Issue 2013
Numerous academic studies since the 1980s often conclude that real estate should constitute 15% to 40% or more of
a diversified portfolio. See, for example, Hartzell et al. [1986],
Fogler [1984], Firstenberg et al. [1988], and Hudson-Wilson
et al. [2005].
13
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74 Is There a R eal Estate A llocation Puzzle?
Special R eal Estate Issue 2013