Exploring a new technique to determine the optimal
real estate portfolio allocation
by
Tingting Fu
B.S., Economics, 2004
Xi’an Jiaotong University
Submitted to the Program in Real Estate Development in Conjunction with
the Center for Real Estate on January 17, 2014 in Partial Fulfillment of the
Requirements for the Degree of Master of Science in Real Estate
Development
at the
Massachusetts Institute of Technology
February, 2014
©2014 Tingting Fu
All rights reserved
The author hereby grants to MIT permission to reproduce and to distribute publicly
paper and electronic copies of this Thesis document in whole or in part in any medium
now known or hereafter created.
Signature of Author____________________________________________________________
Program in Real Estate Development
January 17, 2013
Certified by___________________________________________________________________
Walter N. Torous
Senior Lecturer of Center for Real Estate and Sloan School of Management
Thesis Supervisor
Accepted by___________________________________________________________________
David Geltner
Chairman, Interdepartmental Degree Program in Real Estate Development
1
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2
Exploring a new technique to determine the optimal real estate portfolio
allocation
by
Tingting Fu
Submitted to the Program in Real Estate Development in Conjunction with
the Center for Real Estate on January 17, 2014 in Partial Fulfillment of the
Requirements for the Degree of Master of Science in Real Estate
Development
Abstract
Modern Portfolio Theory has been developed over the last fifty years, and there
are several studies linking Modern Portfolio Theory with the allocation of real
estate property in multi-asset portfolios. However, in reality, most real estate
fund managers don’t use MPT as a guideline when they are structuring a
portfolio and deploying allocation strategy for a real estate fund. The main
reason for this gap between theory and reality is that the traditional meanvariance approach of MPT requires accurate data of variances, covariance and
expected return over the long term; and those data are quite difficult to collect
on an ad hoc base.
This Thesis applies a new technique to examine property asset allocation
strategies and improve the performance of a real estate investment sample
portfolio in the US. We straight-model the portfolio weight in each property type
of asset as a function of the asset’s characteristics: either physical attributes such
as property size, vacancy rate, property type, location etc.; or financial attributes
such as Cap Rate. The coefficients of this function are found by optimizing the
investor’s average utility of the portfolio’s return over a certain period of years.
The aim of this approach is to find a simple and easily modified methodology for
real estate portfolio managers when they are deciding on acquisitions and
making portfolio policies.
In general, this Thesis aims to apply the new technique to help practitioners and
other researchers improve the practical implementation of optimal portfolio
policies.
Thesis Supervisor: Walter N. Torous
Title: Senior Lecturer of Center for Real Estate and Sloan School of
Management
3
Table of Contents
ABSTRACT ................................................................................................................................................................................ 3
ACKNOWLEDGEMENTS ..................................................................................................................................................... 7
CHAPTER 1: INTRODUCTION .......................................................................................................................................... 8
1.1 THEORY REVIEW ............................................................................................................................................................ 8
1.2 STATUS QUO .................................................................................................................................................................... 8
CHAPTER 2: METHODOLOGY ........................................................................................................................................12
2.1 BASIC IDEA ....................................................................................................................................................................12
2.2 DATA ..............................................................................................................................................................................13
2.2.1 Data Sources and Portfolio Creation .....................................................................................................13
2.2.2 Variables Selections ......................................................................................................................................14
2.3 Objective function .............................................................................................................................................17
2.4 Portfolio Weight Constraints ........................................................................................................................20
2.5 Other time varying coefficients ...................................................................................................................21
CHAPTER 3: EMPIRICAL APPLICATION AND RESULTS ....................................................................................23
3.1 DATA DESCRIPTION ......................................................................................................................................................23
3.1.1 Market Value: ...................................................................................................................................................24
3.1.2 Size: ......................................................................................................................................................................25
3.1.3 Cross Sectional Average Price ..................................................................................................................26
3.1.4 Time-series total return (Quarterly) .....................................................................................................28
3.1.5 Property type ...................................................................................................................................................29
3.1.6 Geographic location ......................................................................................................................................29
3.2 STATISTICS SUMMARY .................................................................................................................................................32
3.3 RESULTS .........................................................................................................................................................................34
3.3.1 Four variables ..................................................................................................................................................34
3.3.2 Consideration of locations .........................................................................................................................36
3.3.3
More variables ..........................................................................................................................................38
3.3.4
Snapshot for each quarter ...................................................................................................................40
3.4
Extension .......................................................................................................................................................43
4
CHAPTER 4 CONCLUSION ...............................................................................................................................................44
APPENDIX 1 ...........................................................................................................................................................................45
APPENDIX 2 ...........................................................................................................................................................................49
REFERENCES .........................................................................................................................................................................52
5
6
Acknowledgements
I would like to take this opportunity to thank those people who have been
generously providing help to me since I moved to the United States.
This Thesis could not have been completed without the advice and support of
many people and organizations. Before I extend my appreciation, I would like to
sincerely thank my Thesis advisor Professor Walter Torous for his guidance and
support throughout the entire process. The entire concept of this Thesis is
inspired by Professor Torous and his fellow Professors Alberto Plazzi and
RossenValkanov. Also, I would like to thank Professor David Geltner for
generously sharing with me his comments, opinions and resources to help me
overcome barriers on the way. Last but not least, I would like to thank my
academic advisor Professor William Wheaton for sharing with me his
connections and for pushing me to become a stronger person by his wisdom,
knowledge and experience.
I also would like to thank the various real estate investment firms for sharing
with me their data necessary to complete this Thesis and portfolio managers
who accepted my interviews and shared with me their experiences and thoughts
of the industry. The names of the firms and individuals who have assisted will
not be specified so as to maintain confidentiality, however, I would like to thank
them for their generosity and willingness to provide data to me. This Thesis
definitely would not have been able to be completed without their support.
I would especially like to express my appreciation to my fellow classmates, and
the faculty and staff of MIT Center for Real Estate. Because of them, the past one
and a half years have been such a memorable and enriching experience in my
life. Many alumni of CRE helped me to explore the ideas of this Thesis and
provided valuable input, with special thanks to Pulkit Sharma, Arvind Pai,
Jonathan Richter and Peter McNally. Additional thanks to Beipeng Mu and Ermin
Wei for teaching me how to use Matlab, a new mathematics tool to me, and for
encouraging me to keep exceeding my limits.
Finally, I would like to thank my parents for their unconditional support,
encouragement and love along the way. They are my role models, and they are
the persons who teach me independence, perseverance, bravery, giving and love.
7
Chapter 1: Introduction
1.1 Theory Review
Based on traditional investment theory, we learn that investors should diversify
their investment portfolio in order to reduce total risk at a given level of return
and risk. Classic Modern Portfolio Theory (MPT) provides a perfect theoretical
framework for this process. The basic concept of MPT is that diversification is
achieved not only by investing in an increased number of investments, but also
by investing in a number of assets whose pattern of returns are distinct and
different enough from each other to partially or wholly offset each other’s
returns and therefore reduce overall portfolio volatility.
The most widely used approach generated from MPT is the “mean- variance
approach” developed by Markowitz (1952). This approach has been used to
determine an optimal portfolio allocation. An optimal portfolio of assets is
selected by combining an efficient frontier with a specification of the investor’s
preferences for risk and return. According to Getlner, Miller, Clayton and
Eichholtz’s book Commercial Real Estate Analysis and Investment, MPT makes
three major contributions: (1) it treats risk and return together in a
comprehensive and integrated manner; (2) it quantifies the investment-decisionrelevant implications of risk and return; and (3) it makes both of these
contributions at the portfolio level, the level of the investor’s overall wealth.
Experimental results of portfolio management also testify that certain
combinations of assets are more valuable than others due to their higher returns
with lower risk levels than others as far as their diversification effect is
concerned.
1.2 Status Quo
Despite the benefits of a formal portfolio optimization approach such as MPT, in
reality, such an approach is seldom implemented by fund managers regardless of
whether they are investing in traditional stocks and bonds or other alternative
investments, like real estate. The main reason of this gap between theory and
reality is that the traditional mean-variance approach requires accurate data of
8
variances, covariance and expected return over the long term, and those data are
quite difficult to collect on an ad hoc base.
In practice, most asset allocation decisions must be made in the context of
incomplete information, changing estimates of return, and shifting definitions of
the acceptable investment risk. Significant amount of uncertainty about the true
underlying assets in return-generated processes makes it difficult for investors
and fund managers to execute the traditional mean-variance method. On the
other hand, even when fund managers and researchers are able to make
reasonable assumptions and collect reliable data of variances, covariance and
expected return, when the traditional mean-variance method has been
implemented to optimize portfolio weights in a large number of assets it tends to
create notoriously noisy and unstable results.
If that is the case, then, how do real estate fund managers construct their
portfolios in their daily practices? Is there an easier method existing which they
are able to apply? To answer this question, in the past five months, I interviewed
seven real estate portfolio managers who work in the real estate open-ended
fund business in the US. In terms of the interviewees’ background, four of them
work for real estate investment firms and three of them work for the real estate
management teams of investment banks. After studying and compiling the
records of the interviews, a few common results about the decision-making
processes of acquisitions and asset selection were identified, and have been
described in this Thesis.
Six of seven portfolio managers mentioned that their allocation process was
influenced by certain non-financial considerations, such as behavioral issues.
They use their own judgment, experiences and creativity to make a property
allocation investment decision. Most of the portfolio managers I interviewed
have been working in the industry for a long time, generally over 20 years. And
for them, the most common investment technique they used in their daily
practice for real estate allocation was just general experience and intuition.
Other behavioral issues involve the level of aggressiveness of investment
brokers. Two interviewees mentioned that the decisions of portfolio managers
and acquisitions managers could be heavily influenced by investment brokers’
strong intention to sell one specific property instead of another. The marketing
9
materials investment brokers provided to institutional buyers also tend to favor
a brokers’ particular preference.
Some portfolio managers determine future property allocation by benchmarking
from their current allocation. The primary reason of doing this is because of the
fact that they see their current allocation as conceptually a safe base, and it thus
becomes a benchmark from which the institution deviates as new information
becomes available.
The conclusion generated from interviews is that although there is extensive use
of hard information from the market, the use of personal “gut-feeling” as to the
state of the market and information based on the views of others is commonly
adopted in a decision-making process. Besides gut-feeling, a recent research
report in Institutional Real Estate, Inc (2013) December 1, 2013: Vol. 25, Number
11 also found that the use of asset consultants in the real estate investment
strategies for US pension funds and insurance firms is commonplace as well.
Asset consultants advise US pension funds and other institution investors on
portfolio strategy, performance monitoring and property selection. They tended
to contribute more evidently at the strategic level as well as in the allocation of
direct property investment versus listed property and listed vehicle selections
on the public market.
Meanwhile, unlike other asset types in the traditional finance market, for
example stocks or bonds, the real estate acquisition decisions tend to heavily rely
on the availability of potential opportunities in the market. Even given portfolio
policy, allocation decisions could be strictly constrained by the available
alternatives on the market. As we can imagine, all of those factors we mentioned
in last few paragraph could be barriers to keep a real estate portfolio from
achieving the optimized solution. And importantly, even in a perfect theoretical
world, the optimized solution for real estate portfolio is very difficult to measure
as well.
We learned from interviews and research that portfolio managers like to use
qualitative methodology to construct their funds, but how about quantitative
techniques of portfolio management? If MPT theory existed only in the academic
world, are there other quantitative techniques or methods that have been and
are used?
10
We received positive feedback from interviews on this question as well. A few
numerical indicators have been widely used to measure ad hoc performance of
real estate funds, and most fund managers make reference to a series of risk and
return evaluation measures when evaluating their property asset allocation
decision. Certain quantitative techniques, for example Internal Rate of Return
(IRR), are one of the most important and popular return evaluation measures.
Initial yield also has been identified as a frequently used measure of property
return. Some institutional investors also calculate cash-on-cash return.
Sensitivity analysis, debt coverage ratio and scenario analysis are also popular
quantitative risk assessment techniques.
In general, the most popular quantitative measurements for property return that
real estate portfolio managers applying practice are IRR and initial yield.
Sensitivity analysis is often used as the risk analysis tool. However, from the
interviews I conducted, I did not learn of any quantitative techniques regarding
portfolio diversification, which indicates that there might be space for applying
and developing this technique in the diversification methodology of real estate
practices.
11
Chapter 2: Methodology
2.1 Basic Idea
The quantitative technique originally comes from the traditional finance market.
To avoid exploiting facts about stocks’ characteristics, such as a firm’s book to
market ratio or its stock’s lagged return, variance and covariance as compared
with other stocks, Brandt, Santa-Clara and Valkanov developed a novel approach
in 2009 to optimizing portfolios with large numbers of assets in the stock
market.
They applied this new methodology to equity portfolio optimization based on a
firm’s characteristics. They parameterize the portfolio weight of each stock as a
function of the firm’s characteristics, and estimate the coefficients of the
portfolio policy by maximizing the average utility the investor would have
obtained by implementing the policy over the historical sample period.
(“Parametric Portfolio Policies: Exploiting Characteristics in the Cross-Section of
Equity Returns 10996”)
Commercial Real Estate as a critical asset type oftentimes triggers people to
think about whether it is possible to develop techniques similar to those used in
the financial market to quantify the effect of property characteristics on a real
estate portfolio? Instead of relying heavily on their experience and judgment
(“gut–feeling”), whether there exists an applicable methodology which fund
managers could simple and easily use in their daily work when they select a real
estate portfolio?
In past two or three years, a few pioneering researchers have explored the
possibilities for the optimization of a real estate portfolio from a mathematicallybased methodology. The major purpose of doing this type of exploration is to
simplify the portfolio allocation processes and to answer a number of unsettled
questions in the management of commercial real estate portfolios; for example,
how should investors allocate their capital across different commercial
properties based on a mathematic, scientific calculation? How does the risk12
return profiles of offices, hotels, residential and retail properties differ from one
another? How should investors alter the composition of their commercial real
estate portfolios to take advantage of movements in expected returns arising
from changing underlying macroeconomic conditions?
Plazzi, Torous, and Valkanov started exploring a new methodology to apply this
quantitative technique to the real estate market since 2010. Based on this
previous work, this Thesis aims to further apply and extend this new
methodology with the objective of optimizing real estate fund managers’
decision processes and of improving fund’s performance.
2.2 Data
2.2.1 Data Sources and Portfolio Creation
To make this quantitative technique approach more similar to investment
practices in the real world, instead of using open sources data like NCREIF NPI
index, we used the data gathered from several managed real estate funds in the
institutional investment industry to discuss and examine the proposed
techniques. The various funds that provided data to further this research came
together cooperatively and anonymously to explore the use of this new
technique, taking on the role collectively as the “client” of the Thesis, based on
their belief that they all shared common strengths, weaknesses and constraints,
as well as a similar organizational context.
Considering the similarities in the various own-account funds in terms of risk
and return requirements, in this Thesis, we are treating all of the property
investment cases as if they came from a single investment management firm for
purposes of analyzing and interpreting this property characteristics technique.
We would like to call this technique the Property Characteristics Technique
(PCT). For the sake of convenience, the source of the data pool analyzed in this
Thesis will be referred to going forward as the “Investment Portfolio”.
13
The Investment Portfolio data began in 1978 Q4 and ends at 2013 Q2. The
disaggregated information includes property’s location and asset type, property
market value, net operation income and total return. In terms of market value,
if a property is sold during a certain period, we use the actual selling price as
market value. If it is still in the portfolio, we use an appraisal value calculated by
a third party appraiser firm as the market value of the property. The total return
is calculated by investment firm’s in-house appraiser teams applying the same
methodology.
2.2.2 Variables Selections
To simplify the process of asset allocation and acquisitions decisions and to
make it easier for portfolio managers to apply in their daily practices, we have to
select intuitive parameters as variables for the PCT model. Meanwhile, in order
to be relevant, these variables have to be varied in the investment portfolio
either cross-sectionally or by time series.
As the key purpose of this technique is to maximum portfolio returns, the PCT
model investigates whether a portfolio allocation policy across different types of
assets can be improved by adjusting conditional variables. Therefore, one of the
essential principles of selecting variables is its relevance to the total estimated
return of a particular real estate portfolio. Based on economic theory, the study
of previous researchers’ research papers and data availability, we use the
conditional variables below to construct our model.
CAP RATE: Cap Rate is the ratio between the net operating income produced by
a property and its current market value. There are a number of researchers that
have proved the correlation between Cap Rate and return. The Cap Rate, that is,
the rent-price ratio in commercial real estate, captures fluctuations in expected
returns for apartments, retail, as well as industrial properties. For offices, by
contrast, Cap Rate does not forecast returns even though expected returns on
offices are also time-varying. (Plazzi, Torous, & Valkanov, 2010)
We use NOIi ,t to represent building i’s NOI in time t. Given that the available
data in our investment portfolio is on a quarterly basis, we calculate the
property’s Cap Rate according to the below formula:
14
4
C a p R ai ,t te
N O I
i ,t
t 1
M a r k e t Ve a l u
(1)
SIZE: A few theories in finance have demonstrated the correlation between
asset size and return. For example, in stock markets, there is a theory
called ”Small Firm Effect” which holds that smaller companies have a greater
amount of growth opportunities than larger companies. Smaller firms also tend
to have a more volatile business environment, and the correction of certain
problems - such as the correction of a funding deficiency - can lead to a large
price appreciation. Finally, stocks of small size firms tend to have lower stock
prices, and these lower prices mean that price appreciations tend to be of a
larger percentage than those among large cap stocks.
The classic Capital Asset Pricing Model (CAPM), and especially its three factor
model (The Fama-French Model), also identifies size as one of two key
determinants of long term investment performance. Likewise, researchers in the
real estate finance field also investigated the relationship between size and total
return. Pai and Gelter found that the Three Factor Model captures the historical
cross section of the NCREIF property portfolio total returns quite well.
However, the traditional model differs dramatically from similar stock market
models in that the market beta has a zero risk premium, and the Fama Frenchlike factors in the model place a return premium or price discount on larger
rather than smaller assets and on higher-tier MSA locations. The results are
exactly the opposite of the Fama-French findings in stocks market (“JPM
Article_Pai and Gelter.pdf,” n.d.). It demonstrates that larger properties
compared to smaller properties can potentially earn a return premium and have
higher returns. The empirical results came out from our model in this Thesis also
vilified this effect.
LIQUIDITY: In a further exploration, we added a liquidity factor to the model.
The idea was that we wanted to examine the connections between liquidity and
real estate portfolio performance. It is generally acknowledged that liquidity is
an important factor for asset pricing in traditional stock and bond assets.
15
Kawaguchi, Sa-Aadu, Shilling used time series data from 1982 to 1998 to show
that there is indeed an apparently substantial illiquidity effect in commercial
property returns – less liquidity implies a higher expected return for
institutional investors.
The consequence pattern that comes from their model appears just as strong as
the pattern in stock returns. This is an interesting result, given that the holding
period for commercial real estate is generally far longer than the holding period
for stocks (“Do Changes in Illiquidity Affect Investors’ Expectations”). For this
liquidity analysis, we divided properties into two groups according to their
geographic locations: Top- 6 cities and non-Top- 6 cities. Top- 6 cities include
New York, Boston, Washington DC, LA, San Francisco and Chicago.
PROPRERTY TYPE. Traditional theory on diversification strategies advocated
that property type is an important diversification criteria for institutional
investors. Historically, scholars have been exploring the optimization of property
type allocation in portfolio performance. Firstenberg, Ross and Zisler in their
paper “Real Estate: The Whole Story” (1988), use Frank Russell (FRC) AND
Evaluation Associates (EAFPI) return index series to examine how diversification
within property types and geographic locations effects estimated return. They
concluded that diversifying the composition of a portfolio among geographic
locations and property types can increase the investor’s return for a given level
of risk.
Pai and Getlner also find that property type (Office, Hotel, Apartment, Industrial,
etc.) is a very powerful predictor of long-term average investment performance
even controlling for the risk factors is considered. (“JPM Article_Pai and Gelter”)
This finding is in favor of us using property type as another variable to measure
total return.
Risk: There is a negative correlation between the risk of a real estate portfolio
and total expected return. Risk is usually measured as volatility and standard
deviation. Undoubtedly, the degree of risks that investors are willing to take
represents one of the most important factors in determining expected return.
Different investors have various risk appetites. In our Property Characteristics
Technique model, we use three numbers to represent different kinds of
investors.
16
The optimal portfolio heavily depends on the investor’s preferences. Therefore,
we assume CRRA utility function with risk aversion equal to 5 as normal risk
aversion level.
Besides  equals 5, we also report the result when  equals 9
which represent the investors who are extremely sensitive to losses, like pension
fund and other conservative investors. In additional, we also calculate the
scenario that when  equals 2, which represents the investors who have high
risk appetite like hedge funds.
The PCT model we created in this Thesis can be applied to examine the
importance of other property characteristics in commercial real estate portfolio
allocation as well. For example, the conditional variables could be extended to
transaction cost, vacancy rate, and capital expenditure etc. For most portfolio
managers, they might also like to know the importance of each market friction
that might occur. Depending on the requirements of the individual portfolio
manager, the model could be tailor-made to a more practical format.
2.3 Objective function
The basic approach of PCT is to set up a utility function in associated return with
a vector of buildings characteristics. For example, we suppose that at any
particular time (t), there is a large number (Nt ) of buildings in the portfolio. Each
building (i) has a return of ri ,t from time (t) and is associated with a vector of
the building’s characteristics ( xi ,t ) observed at time (t). The property
characteristics could be physical attributes of the buildings or financial attributes
of the assets. In this Thesis, we select Cap Rate, Size, Liquidity and Property Type
as our characteristics. The key task for a portfolio manager is to choose the
weights
i,t
to maximize the conditional expected utility of portfolio return rP ,t
To differentiate the return of an individual property from the return of the
overall portfolio, we use rP ,t to represent the return of the portfolio and rp ,t to
represent the return of the individual property.
17
Nt
maxN Et [u (rp ,t )]  Et [u ( i ,t ri ,t )]
{wi , t }i t1
(2)
t 1
The optimal portfolio weights can be parameterized as a linear function of
buildings characteristics.

i ,t
 f ( xi ,t ; )
  i ,t 
1 '
 x i ,t
Nt
(3)
 i,t is the weight of building i at time t in a benchmark portfolio. In this Thesis, I
use a market value weighted portfolio as a benchmark, which means using the
market value to calculate the weight of each individual property in the portfolio.
 is a vector of coefficients to be estimated and xi,t are the characteristics of
buildings. The vector of coefficients

reflects deviations between current
portfolio allocations with optimal portfolio allocation (QUESTION: Is this
between two different allocations?), and therefore, indicates the importance of
property selections. The intercept
benchmark portfolio while the
 i,t is the weight of the building in a
 ' xi ,t
represents the deviations of the optimal
1
portfolio weight from this benchmark. N t is a normalization that enables the
portfolio weights function to be applied to any arbitrary and time-varying
number of buildings.
Constant coefficient  is undoubtedly the most critical aspect in the
'
parameterization. It implies that the portfolio weight in each building depends
only on the building’s characteristics and not on the property’s historic return.
Constant coefficients through time mean that the coefficients that maximize the
investor’s conditional expected utility at a given date are the same for all dates
and therefore also maximize the investor’s expected utility.
If we put back all the variables we selected in 2.2.2, the linear function would be
written as:
18

i ,t
 f ( xi ,t ; )
  i ,t 
=  i ,t

1 '
 x i ,t
Nt
(3)
1
[( cap  capi ,t   size  sizei ,t   top6  top 6   industrial industrial )
Nt
The key part of the modeling optimization process is to select the investors’
objective function. Unlike traditional mean variance methods, the specification of
the portfolio selection can adopt any choice of objective function as long as the
function can be specified with a unique solution. For example, (“Parametric
Portfolio Policies: Exploiting Characteristics in the Cross-Section of Equity
Returns 10996”) Brandt, Santa-Clara and Valkanov, in their paper, mentioned
that this method could be applied to behaviorally motivated utility functions
such as loss aversion, ambiguity aversion, or disappointment aversion, as well as
practitioner-oriented objective functions, including maximizing the Sharpe or
information ratios, beating or tracking a benchmark, controlling drawdowns, or
maintaining a certain value at risk.
From a practical perspective, we use the standard Constant Relative RiskAversion (CRRA) utility function in this thesis.
u ( r p ,t ) 
(1  rp ,t )1
1 
(4)
If we incorporate the expected return algorithms for this specified utility
function, the optimization problem could be revised to:
max
1 
1 T
1 T (1  rp ,t )
u
(
r
)

max
 p ,t

T t 1
T t 1 1  
Nt
rP ,t   ( i ,t   T xi ,t / N t )ri ,t
i 1
  rb,t   ra ,t
19
(5)
rb,t represents the return of each property in the benchmark portfolio and ra ,t
represents the deviation of return in the benchmark portfolio and the optimal
portfolio. The CRRA utility function as compared with other functions could
bring preferences towards higher order moments in a more discreet manner.
2.4 Portfolio Weight Constraints
To avoid using strictness constraints so as to generate an optimal solution, the
easiest constraint we could use in this optimization is non-negative weights.
There are a few alternative ways of expressing non-negative constraints in
portfolio weight. In order to be applicable to the real estate market, where
shorting is not very common, we use the below constraint:
i,t 
max[ 0, i ,t ]
Nt
 max[ 0, 
i ,t
i 1
]
(6)
Additionally, if we look back at function (2), we have to keep both the sum of
weights in the benchmark portfolio and the weight of the optimization portfolio
equal to one.

i ,t
 f ( xi ,t ;  )
  i ,t 
1 '
 x i ,t
Nt
(3)
1 '
 xi ,t
N
t


1
To meet the condition that  i ,t
and  i ,t  1, the sum of
has to
equal to zero. Therefore, it generated our second constraint that:
20
n
1
N
t 0
 ' x i ,t  0
(7)
t
2.5 Other time varying coefficients
Some of you might be wondering whether the constant coefficients of portfolio
through time is reasonable? From an academic perspective, it is a very
convenient assumption because it avoids some difficult statistics techniques.
However, there is substantial evidence showing that macroeconomic variables as
related to business cycles are highly related to forecasts of the aggregate
properties returns.
To accommodate possible time varying macroeconomic factors, we can simply
add those factors as one of the variables to the function. With it in mind, we need
to consider that macroeconomic factors have a complicated influence on each
variable in the function. To achieve this goal, we can use Kronecker Product to
reflect that the impact of the properties’ characteristics on the portfolio weight
varies with the influence of macroeconomic and business cycle variables. We can
choose the US National Statistics Bureau’s annual GDP index and other relevant
indexes as these variables.
To match our quarterly basis data and for the sake of preciseness, we choose
CFNAI as one of the macroeconomic indicators. CFNAI is the Chicago Fed
National Activity Index, which is a monthly index designed to gauge overall
economic activity and related inflationary pressure. The CFNAI is a weighted
average of 85 existing monthly indicators of national economic activity. It is
constructed to have an average value of zero and a standard deviation of one.
Since economic activity tends toward trend growth rate over time, a positive
index reading corresponds to growth above trend and a negative index reading
corresponds to growth below trend.
The 85 economic indicators that are included in the CFNAI are drawn from four
broad categories of data: production and income; employment, unemployment,
21
and hours; personal consumption and housing; and sales, orders, and
inventories. Each of these data series measures some aspect of overall
macroeconomic activity. (According to CFNAI’s official website:
http://www.chicagofed.org/webpages/publications/cfnai/)
Therefore, the linear function of weight can be revised to:

i ,t
 f ( xi ,t ;  )
  i ,t 
1 '
 ( xi ,t  zt )
Nt
  i ,t 
1
[( cap  capi ,t   size  sizei ,t   top6  top 6   industrial  industrial )  zt ]
Nt
(8)
Where we use Z to represent economics factor and ⊗ for the Kronecker Product,
which is an operation on two matrices of arbitrary size resulting in a block
matrix. It is a generalization of the outer product from vectors to matrices, and
gives the matrix of the tensor product with respect to a standard choice of basis.
(Cited from Wikipedia: http://en.wikipedia.org/wiki/Kronecker_product)
22
Chapter 3: Empirical Application and Results
3.1 Data description
To implement the theoretical idea that we illustrated in the first two chapters,
we use investment portfolio data to present an empirical application of this
technique. We use quarterly property level returns in the investment portfolio as
well as property level characteristics including market value, Cap Rate, location
information, size, NOI etc. The data set begins at 1978 Q4 and ends at 2013 Q2.
We are going to describe the data first and then release the results after running
optimization through Matlab, a numerical computing environment and
programming language that has been widely used as quantitative analysis tool
and optimization software. We are going to report the data under three different
levels of risk aversion. In this Thesis, we assume that all the properties in the
portfolio involve different level of risks and risk-free assets are not included in
this portfolio.
We construct the data set at the end of each year, which consists of four (4)
quarters of data. The number of building in the investment portfolio is
continuously increasing, although a few buildings have been sold out after
property holding period. The average annual rate of increase in the number of
buildings is 2.05% from 1978 Q4 to 2013 Q2. The average number of buildings
throughout the duration of the investment portfolio is ninety-four (94)
properties, with the fewest number twenty-one (21) in 1978 Q4, and the greatest
number two hundred forty-five (245) in 2013 Q2.
Below are a number of graphs to describe the cross sectional average and
standard deviation of Cap Rate, size and market value of the properties.
Additionally, we use graphs to illustrate the cross sectional price per square foot
of the properties. Appendix 1 provides further details about the property-level
data, including the exact definitions of the components of each variable. We use
size, Cap Rate, industrial and Top- 6 city as conditioning characteristics in the
portfolio optimization according to their relevance and data availability.
23
3.1.1 Market Value:
Figure 1 describes Total Market Value,
Average Market Value, and Market
Value by each property type and
Market Value Comparison across all
types of property.
To help us understand how large this portfolio is from 1978 to 2013, Figure 1
describes the total market value, average market value and market value for each
type of property in the portfolio. Looking at the general trend of the portfolio, the
market value is increasing throughout its duration, with a significant increase
24
since 2000. It has a downturn in 2009 and rises again ever since. The last graphs
in Figure 1 also show the proportion of market value for each type of property.
Office buildings weight the most value in the whole portfolio and hotels are the
smallest portion of the assets from a market value perspective.
3.1.2 Size:
Figure 2 describes the aggregate size
trend in each quarter from 1978 to
2013.
25
3.1.3 Cross Sectional Average Price
According to the average price analysis of five types of properties, only industrial
properties have obvious patterns that correspond with macroeconomic cycles.
Whereas with the other four types of properties, (office, retail, apartment and
hotel), in general, the average prices are increasing and could generate both
linear and exponential forecasts.
Figure 3 describes the average price by
each property type.
Let us have a close look at the average price for each single property type. The
average price of industrial property shows a significant cyclical pattern, whereas
with the other four types of properties, the earnings trend upwards generally.
Those four types of properties have significant, above-average increases from
26
2004 to 2008, and under-average price increases from 1999 to 2002 and 2009 to
2012.
27
3.1.4 Time-series total return (Quarterly)
28
Figure 4 describes quarterly total return on each type of asset and comparison
among different asset types.
Since we already know the quarterly return of individual properties from the
original data, we could use the aggregate return to calculate overall return as
well as return for each property type on a value weight basis, where the
individual property’s weight is proportional to its market value.
ri represents
the return for each individual building and rP represented the return of the
portfolio. The returns of particular property types have been donated by roffice
rhotel rindustrial rretail rapartment. By comparing these returns with certain
macroeconomic return charts, we note that it roughly matches with
macroeconomic cycles.
3.1.5 Property type
There are five property types included in the investment portfolio we are
analyzing in this Thesis: Industrial, Office, Apartment, Retail and Hotel. The
predominant type of property in the portfolio is Industrial. Therefore we choose
Industrial versus non-Industrial property as one of the standardized variables in
the portfolio function to examine how the variation of property type may cause
changes in the return of the portfolio.
3.1.6 Geographic location
To better illustrate the geographic location distributions of this portfolio and
examine the impact of these distributions on the performance of the portfolio,
we only select properties which are located in the US. We use Excel to make a 3D data visualization model of these properties and mark out all the location
distributions of each asset by size and filtered with property type. According to
the regional divisions used by the US Census Bureau, we divided all the states into
four areas: Northeast, Midwest, South and West.
Northeast includes: Maine, New Hampshire, Vermont, Massachusetts, Rhode
Island, Connecticut, New York, Pennsylvania, and New Jersey
29
Midwest includes: Wisconsin, Michigan, Illinois, Indiana, Ohio, Missouri, North
Dakota, South Dakota, Nebraska, Kansas, Minnesota, and Iowa
South includes: Delaware, Maryland, District of Columbia, Virginia, West Virginia,
North Carolina, South Carolina, Georgia, Florida, Kentucky, Tennessee,
Mississippi, Alabama, Oklahoma, Texas, Arkansas, and Louisiana
West include: Idaho, Montana, Wyoming, Nevada, Utah, Colorado, Arizona, New
Mexico, Alaska, Washington, Oregon, California, and Hawaii
30
31
3.2 Statistics Summary
All Return Industrial return Apartment return Office return Retail return Hotel return
Mean 0.0237
0.0208
0.0244
0.0176
0.0195
0.0225
Std Dev 0.0273
0.0719
0.0560
0.0729
0.0562
0.0599
Var
0.0007
0.0052
0.0031
0.0053
0.0032
0.0036
AR(1) -0.3317
-0.3520
-0.2020
-0.1296
-0.1998
-0.0533
Cap Rate
0.0714
0.0338
0.0011
-0.8501
Size
5.2508
0.5664
0.3228
-0.9922
CFNAI
-0.2142
0.9303
0.8654
-0.9954
Exhibit 1. Panel A Statistics Summary for the value weighted property return
series.
This panel demonstrates basic descriptive parameters regarding investment
portfolios, time-series mean standard deviations, variances and first order
autoregressive coefficients for value weighed return series to all properties
during the period from 1978 Q4 to 2013 Q2. We described data as per property
type: office, industrial, apartment, hotel and retail in addition to Cap Rate, size,
and CFNAI.
Average returns of all of property types don’t vary dramatically when looking at
low standard deviations and variances of all property type returns. Multi-family
housing / apartments on average have slightly higher returns than other types of
property. Industrial has the second highest average return. By examining the
standard deviation, we learned that office property returns have the highest
cross sectional variability. The second-highest are from industrial types of
property. We got a similar conclusion by checking the variances that industrial
and office properties have relatively higher variability than other properties.
We also calculated one order-lagged autocorrelation. We noticed that the
autocorrelation of five property types are quite low, which means that the data
sample is reliable. Cap Rate, size and CFNAI’s autocorrelation coefficients are
very high. This is consistent with the fact that size and Cap Rate are more
constant and persistent over time, while CFNAI macroeconomics have a high
correlation with the last period (QUESTION: Last period of what?), as opposed to
being constant and persistent.
By any means, we have to acknowledge that real estate investment decisions are
highly effected by economic conditions, and real estate investment opportunities
also rely on market conditions.
32
All Return
Industrial
Apartment
Office
Retail
Hotel
Cap Rate
Size
CFNAI
All Return
1.0000
0.6267
0.3837
0.4977
0.2758
0.1359
0.0211
0.0069
0.2192
Industrial
Apartment
Office
Retail
Hotel
Cap Rate
Size
CFNAI
1.0000
-0.0343
-0.0196
-0.0202
-0.0100
0.2074
0.2194
0.1065
1.0000
-0.0227
-0.0233
-0.0115
0.3510
0.3566
0.1373
1.0000
-0.0133
-0.0066
0.1813
0.2067
0.0995
1.0000
-0.0068
0.2948
0.3111
0.0962
1.0000
0.4013
0.3410
0.0457
1.0000
0.0152
0.0383
1.0000
-0.0334
1.0000
Panel B displays the time-series correlation coefficients during the sample
period.
To understand the correlations among time series variables, we calculated
correlations among each pair of lagged variables. Panel B shows that Industrial
property’s return is the most highly related to overall return, and then Office,
followed by Apartment/multifamily house. Each type of property itself is less
related to other property types because we screened out information according
to type, which means that the information of each pair of property type are
independent and less relevant.
Apartment’s returns are the most highly correlated with their size, and Hotels
placed second in terms of correlation with size. The correlation coefficient of size
and over-all return is fairly low.
Apartments have the highest correlation with economic activity in general, and
all types of properties have positive correlations with Cap Rate. Hotels, in
particular, have the highest coefficient with Cap Rate.
33
3.3 Results
3.3.1 Four variables
The results of optimal portfolio policy coefficients with non-negative weights
have been listed in Exhibit 1. In this Exhibit, we use four (4) variables
incorporated in a linear function under three levels of risk aversions.

`
i ,t
 f ( xi ,t ;  )
  i ,t 
1 '
 ( xi ,t  zt )
Nt
  i ,t 
1
[( cap  capi ,t   size  sizei ,t   top6  top 6   industrial  industrial )  zt ]
Nt
(8)
All
αcap
αsize
αtop6
αind
Base Case
-
0.20961 0.1837 0.46404
0.24746 0.24782 0.24548
1.6313 1.82427 3.25464
-0.104 -0.5666 -0.7901
maxR
minR
0.0867
-0.502
0.10631 0.08318 0.07401
-0.0344 -0.0345 -0.0351
max N
min N
249
21
249
21
249
21
249
21
M(Rp)
M(Rp-Rf)
Std(Rp)
SR
0.0974
0.04705
0.08783
0.53563
0.11303
0.06268
0.09449
0.66334
0.11134
0.06099
0.08899
0.68531
0.11129
0.06099
0.08899
0.68531
γ=2
γ=5
γ=9
Exhibit 2. Optimal portfolio policy coefficients in four variables model with nonnegative weights estimated for all properties.
Column 1represents the results for the base case, in which we listed out the
maximum and minimum returns, and the range of number of buildings in a
sample portfolio. The mean and standard deviation of portfolio return, the mean
34
of the difference between portfolio return and 3 months Treasure bills and the
Sharpe Ratio of sample investment portfolio.
Column 2 to 4 describe the optimal portfolio policy coefficients with nonnegative weights estimated for all properties. To compare how the coefficients
varied among different investors, we listed out three scenarios with different
risk aversion levels.
From Exhibit 2, we learned that, generally speaking, as compared to current
market-value weighted portfolios, the optimal portfolio estimate for all types of
property leaned more towards a liquid market, a Top- 6 city, and larger size
properties measured by market value. In terms of property types with all other
factors being equal, the allocation of the optimal portfolio tended to invest less in
Industrial properties.
Considering risks effect, we are able also to observe that the coefficient of
liquidity matters most for conservative investors. The increasing Top- 6
coefficient demonstrated that they should tilt more towards a highly liquid
market and increase their investments in Top- 6 markets to hedge illiquidity risk.
The coefficients of size did not change very much between different types of
investors, but it did suggest that we should slightly increase the investment in
larger buildings in general. Cap Rate’s coefficient increased slightly in general as
well. It demonstrated the fact that in order to achieve higher returns, the
portfolio should invest more in those properties with a higher Cap Rate.
From a statistic standpoint, we noticed that for a relatively smaller value of  ,
the coefficients on Cap Rate, size and other variables might have smaller absolute
values and be statistically significant. The coefficients will approach larger
numbers when the risk factor becomes greater and investors become risk averse.
From one aspect, it shows that the properties characteristics are highly related to
both average returns and risk.
35
Form another aspect, it also shows that when the level of risk aversion is
increasing, the investor weighs the contribution of property characteristics to
alleviating the risk more heavily. For example, conservative investors tend to
value core assets with a lower Cap Rate located in high liquidity market i.e. (Top6 cities) as compared to other cities.
The average properties characteristics show very similar patterns in different
risk levels. When we compare property size’s coefficients, we realize that size
actually does not matter very much among different kinds of investors. However,
location and Cap Rate do.
If we look at the statistical numbers of the portfolio, we can see in the
optimization portfolio that the return improved with an increase in the Sharpe
Ratio. Looking at the Sharpe ratio of optimal allocation, it increased from 0.53 to
0.66. For the conservative investors, the Sharpe ratio even increased to 0.68. This
demonstrated that the higher returns generated from optimal allocation do not
come with too much additional risk. Considering both means of portfolio return
and Sharpe ratio, the optimal portfolio has better risk-adjusted performance. To
take ratios, we use the three-month Treasury Bill rate for each quarter and then
calculate 1) the mean of the difference between portfolio return and the free risk
return, and 2) the standard deviation of portfolio returns.
3.3.2 Consideration of locations
Depending on the different needs of analysis, we can always add more variables
to the liner function. For example, geographic location is an essential element for
real estate business, and we can easily extend a locations consideration into our
model with information as to the zip codes where the properties are located. If
we applied the geographic location classification method in 3.1.6 to include four
(4) location dummies in the specification to consider how the changes of location
distribution effect optimal portfolio allocation, the linear function could be
revised to:
36

i ,t
 f ( xi ,t ;  )
  i ,t 
1 '
 ( xi ,t  zt )
Nt
  i ,t 
1
[( cap  capi ,t   size  sizei ,t   top6  top 6   industrial industrial   Northeast  Northeast  West  West   Midwest  Midwest   South  South )  zt ]
Nt
(9)
All
αcap
αsize
αtop6
αind
αne
αw
αmw
αs
maxR
minR
max N
min N
M(Rp)
M(RpRf)
Std(Rp)
SR
Base
Case
-
γ=2
γ=5
γ=9
0.0873
0.394
0.6697
-0.1
-0.249
2.1202
0.0928
-1.607
0.1006
0.3012
0.7955
-0.142
-0.334
1.9085
-0.167
-1.087
0.0669
0.238
0.8413
-0.093
0.0423
1.1011
-0.224
-0.786
0.0867
-0.502
0.0872
-0.027
0.0901
-0.03
0.0881
-0.033
249
21
249
21
249
21
249
21
0.0974
0.1163
0.1158
0.114
0.0471
0.066
0.0655
0.0637
0.0878
0.5356
0.0876
0.7528
0.0901
0.7264
0.0899
0.7087
Exhibit 3. Optimal portfolio policy coefficients with non-negative weights applied
in eight (8) variables models with consideration of locations dummies estimated
for all types of properties.
From Exhibit 3, we note that the Industrial and Northeast area coefficients are
negative, while the coefficient West coefficient becomes very strong especially
for aggressive investors. The Cap Rate coefficients were above 0.66 for all types
of investors.
37
In terms of the liquidity effect, emphasizing the Top- 6 cities of New York,
Washington DC, San Francisco, Los Angles, Chicago or Boston could definitely
improve returns. However, more specific location diversification strategy were
also reflected from the coefficients of each locations. For example, the policy
indicated that investment in the West should be increased and investment in the
South should be decreased. Especially for aggressive investors, they should
invest more in the West region, and meanwhile decrease their investment in the
South. Increasing investment in the Northeast region will bring marginal returns
for conservative investors.
From a statistical perspective, we noticed that doing optimal allocation in the
risk level of 2, the mean of the portfolio return increased significantly from 9.3%
to 11.63% while the Sharpe ratio decreased to 0.75. From this model, we actually
are clearly able to see the impact that locations diversification contributes to
return. Certainly, the most aggressive investors get the highest return, but they
also enjoy the highest Sharpe ratio in those scenarios where they give strong
consideration to the location factors in the portfolio management processes.
3.3.3
More variables
Let’s consider a more complex situation of applying this model by examining
both geographical locations and property types. Besides the Cap Rate, size and
liquidity which we have been considering since the first function, the variables in
this new function have been increased to twelve with the additional
consideration of locations and property types. As with our other analyses, we
would like to see the optimal solutions in different risk levels.
38
All
αcap
αsize
αtop6
αind
αoff
αapt
αrtl
αhtl
αne
αw
αmw
αs
Base Case
-
γ=2
γ=5
γ=9
0.06503
0.3198
0.61272
-0.0074
-0.1043
-0.6844
0.82568
0.02566
-0.1708
1.72685
0.06392
-1.3208
0.07461
0.30976
0.87714
0.04048
0.24271
-0.8413
0.72911
-0.0029
-0.0967
1.64956
0.03571
-1.3142
0.09399
0.3276
1.40384
0.09662
0.56401
-0.8519
0.76774
0.0448
0.02733
1.80541
-0.1284
-1.4699
maxR
minR
0.0867
-0.502
0.0998
-0.0353
0.10928 0.10042
-0.0367 -0.0365
max N
min N
249
21
249
21
M(Rp)
M(Rp-Rf)
Std(Rp)
SR
249
21
249
21
0.0974
0.1186 0.11811 0.11604
0.04705 0.06825 0.06776 0.06569
0.08783 0.09409 0.09578 0.09347
0.53563 0.72532 0.70744 0.70278
Exhibit 4. Optimal portfolio policy coefficients with non-negative weights
estimated when the conditioning variables are interacted with dummy variables
of locations and property types. Quarterly rebalancing is assumed.
By examining the results in this Exhibit, we learned that the general investment
policy about cap size and liquidity are consistent with the last two models. Large
buildings with a higher Cap Rate and in liquid markets could benefit the portfolio
return. However the locations selections and property type are varied by
investors’ risk appetite.
39
For aggressive investors, they should increase their investment in Retail,
somewhat increase their investments in Hotels, and decrease their Industrial,
Office and Apartment investments. Like what we analyzed in the model, they
should increase investment in the West.
For normal investors, they could keep Industrial as what it is in the portfolio and
increase investment in Office and Retail. The West area is still the favorable
investment location selection for these investors. On the contrary, the
conservative investors could keep the percentage of Industrial and perhaps even
increase it slightly. Conservative investors enjoy the highest coefficient for office
buildings, which mean they should increase the percentage of office buildings in
their portfolio
The statistical numbers also reflect the same positive information. Portfolio
returns of all three kinds of investors improved with the increase of Sharpe ratio
from 0.53 to 0.70.
3.3.4
Snapshot for each quarter
We have been discussing portfolio policy in general in this Thesis; however, fund
managers might be interested in knowing in each time point, how the coefficient
varied and how should they adjust their portfolios. To answer this question, we
use the four variable model as an example to make time series coefficient graphs
to show the changes of coefficients at each time point.
40
41
Exhibits 5. Time series coefficient
graphs to show the changes of
coefficients at each time point in
three different levels of risk aversion.
42
3.4
Extension
This model could be easily be extended to examine other factors that are
relevant to fund managers and that they care about. Because it is an easilyimplemented and practical approach to improve a fund’s risk adjusted
performance, it can be used by practitioners and researchers to test the
importance of each kind of properties characteristics in real estate portfolio
allocation.
For example, we could examine how variables change when CFNAI is a positive
value that corresponds to a market expansion as compared to when CFNAI is a
negative value that corresponds to a market contraction. Also, it might be
interesting to know the sensitivity of those results when transaction costs and
other market frictions are included; and we might be able to further break down
geographical locations into more detailed regions. Furthermore, we even could
use different objective functions and different parameterizations to
accommodate short sale constraints. We will not cover those topics in this paper,
but will leave investigating these and other issues to future research.
43
Chapter 4 Conclusion
There are a number of obvious advantages of using the technique described in
this Thesis to determine optimal real estate portfolio allocation, and also
demonstrated a few highlights of this methodology. Although this technique
originally comes from the traditional financial industry and has been applied
more frequently in stock market, benefits of this technique can be applied in the
real estate market. First, it enables a fund manager to avoid the very difficult
steps of modeling the joint distribution of returns and property characteristics.
Instead, in this technique we can just focus directly on the key point most fund
managers care about: property weight and how to adjust that weight.
Second, the Markowitz method in traditional financial market requires a
complicated modeling of (N+1)*N/2 second moments of returns. However, in
this new approach we only need to concentrate on the coefficients of variables
and N buildings for each time point, which significantly reduces the workload
and increases the simplicity of implementation. More importantly, this approach
directly captures the relationship between property characteristics, returns, and
covariance and provides a reliable forecast for making portfolio policy.
By applying this real estate portfolio allocation approach, we are able to more
easily incorporate building characteristics information into commercial real
estate portfolio management and strategic policy making. The results of this
approach are very encouraging as well, and by using it to provide diversification
of a real estate portfolio across property types and geographical locations, the
risk adjusted performance of that portfolio has been improved significantly. In
addition, this technique also tests the importance of property characteristics’
contributions to returns and helps to explain the deviations of optimal portfolio
weight from a current market valued portfolio.
44
Appendix 1
Row
Labels
19784
19791
19792
19793
19794
19801
19802
19803
19804
19811
19812
19813
19814
19821
19822
19823
19824
19831
19832
19833
19834
19841
19842
19843
19844
19851
19852
19853
19854
19861
19862
19863
19864
19871
19872
19873
19874
Sum of MV
24,726,067
38,932,879
45,293,632
62,656,564
64,979,242
108,054,273
110,340,439
139,765,014
144,629,162
174,560,860
205,115,327
231,357,895
240,245,342
248,834,119
279,727,155
291,669,347
287,341,406
448,446,507
525,439,354
516,851,303
529,907,576
564,341,590
601,410,552
693,407,198
736,916,572
772,268,098
840,435,754
833,182,499
847,431,396
867,233,661
889,601,997
844,867,641
843,709,418
1,024,312,960
1,098,479,472
1,112,461,750
1,200,532,793
Sum of SqFt
Apartment
838,006
1,538,086
1,671,858
2,214,461
2,214,461
3,572,814
3,583,831
3,994,486
4,099,416
4,575,119
6,041,000
6,268,526
6,356,290
6,576,586
7,467,120
7,811,684
7,649,041
10,632,964
11,184,277
11,097,481
11,097,481
11,577,265
11,834,891
12,605,174
13,139,099
13,932,642
14,501,156
13,937,447
13,583,918
14,350,299
14,259,499
12,794,659
12,835,035
14,769,360
14,731,998
14,938,806
15,704,797
45
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
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4
4
6
7
7
8
8
8
Indus
trial
19
22
22
24
24
27
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58
59
Offic
e
0
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0
0
0
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4
4
4
6
6
7
7
7
7
11
14
13
13
13
14
16
17
18
19
19
19
19
20
15
14
17
18
18
19
Retail
2
3
5
7
7
13
13
15
15
18
20
20
20
20
21
20
19
24
24
24
24
24
24
24
21
21
20
19
18
17
16
15
14
15
15
12
11
Hot
el
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
19881
19882
19883
19884
19891
19892
19893
19894
19901
19902
19903
19904
19911
19912
19913
19914
19921
19922
19923
19924
19931
19932
19933
19934
19941
19942
19943
19944
19951
19952
19953
19954
19961
19962
19963
19964
19971
19972
19973
19974
19981
19982
1,175,871,164
1,220,944,528
1,403,547,617
1,356,671,034
1,427,422,524
1,516,634,210
1,332,688,743
1,331,153,931
1,324,142,079
1,309,797,943
1,418,212,273
1,468,635,174
1,448,957,483
1,418,530,066
1,388,471,813
1,283,089,087
1,205,675,470
1,186,547,543
1,160,132,943
1,070,505,260
1,026,909,489
978,495,707
927,458,271
917,748,671
780,396,209
761,536,085
769,969,683
676,340,280
699,431,512
710,082,855
711,733,536
765,753,110
981,444,594
1,139,925,649
1,163,685,053
1,192,033,572
1,213,066,821
1,279,599,571
1,341,070,015
1,537,244,601
1,918,302,896
2,046,588,420
15,142,268
15,842,488
17,302,941
17,062,797
17,384,210
16,650,924
14,426,863
14,234,930
13,865,693
13,600,019
14,909,746
16,120,558
15,911,515
15,999,213
15,999,213
16,020,332
15,413,292
15,494,764
14,908,066
14,471,764
13,966,941
13,702,441
13,258,620
13,258,620
11,199,786
10,622,917
10,470,848
10,553,299
11,080,349
11,080,349
10,634,168
11,494,647
14,904,937
17,536,168
17,536,168
17,009,118
17,009,118
17,457,627
17,628,758
19,261,776
20,676,658
21,360,554
46
8
8
9
8
9
9
9
9
9
9
9
11
10
9
9
9
9
9
9
9
9
9
8
8
7
7
7
8
8
8
8
10
19
20
20
20
20
20
20
22
22
25
59
59
61
61
61
61
59
59
59
59
59
60
19
19
19
20
19
17
17
16
16
15
14
14
12
10
10
10
10
10
10
10
10
13
13
13
13
15
15
17
17
17
19
20
21
22
22
21
22
21
20
19
21
22
22
23
23
21
20
21
20
19
17
17
16
16
15
15
14
14
14
14
14
14
15
18
18
18
18
18
19
20
22
22
10
11
12
11
11
9
5
4
4
4
5
5
5
5
5
5
5
5
5
5
4
4
4
4
3
3
3
2
3
3
3
3
4
4
4
3
3
3
3
3
4
4
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
19983
19984
19991
19992
19993
19994
20001
20002
20003
20004
20011
20012
20013
20014
20021
20022
20023
20024
20031
20032
20033
20034
20041
20042
20043
20044
20051
20052
20053
20054
20061
20062
20063
20064
20071
20072
20073
20074
20081
20082
20083
20084
2,136,890,690
2,562,827,281
2,877,119,104
2,947,448,219
3,006,313,177
3,096,351,684
3,127,474,438
3,249,007,227
3,362,196,792
3,403,304,350
3,379,875,866
3,611,781,670
3,523,895,825
3,548,445,046
3,567,881,813
3,520,519,619
3,711,323,508
3,889,718,314
4,188,005,078
4,393,675,772
4,501,835,531
4,670,086,751
4,989,763,708
5,124,155,981
5,291,560,440
5,627,001,179
6,034,187,229
6,360,370,354
6,620,444,935
6,835,295,471
7,415,328,112
7,953,578,395
8,855,780,839
9,485,621,276
9,861,125,315
10,016,168,937
9,969,340,015
10,655,220,188
11,203,891,177
11,204,531,755
11,164,733,053
10,399,647,153
21,909,908
25,138,435
27,280,597
27,637,281
28,042,900
28,202,900
28,050,300
28,155,278
28,687,678
28,640,704
28,640,704
30,073,204
29,674,260
30,916,208
30,582,630
29,928,374
31,406,808
31,933,484
34,563,561
37,175,929
37,599,234
38,468,763
39,861,353
40,201,611
40,696,838
41,533,306
42,869,062
43,269,062
43,760,942
43,400,090
46,283,235
48,698,604
51,092,732
53,639,989
54,943,851
54,764,471
53,436,168
55,465,454
53,204,477
54,132,093
53,805,686
53,033,276
47
27
31
33
34
35
35
35
35
35
35
35
37
37
38
38
36
38
37
37
38
39
38
39
39
40
42
42
41
42
40
42
45
46
47
48
46
46
51
52
51
51
51
18
18
19
19
20
20
20
20
21
21
21
21
20
21
20
20
20
20
21
24
24
24
24
24
24
24
24
25
24
24
24
24
23
26
27
26
25
38
38
39
38
38
22
24
26
26
26
27
26
28
28
28
28
29
29
30
30
31
32
32
32
32
32
33
35
36
36
36
38
38
35
35
34
33
35
36
35
31
30
30
28
30
30
31
4
5
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
5
5
5
5
5
5
6
15
19
21
21
23
23
23
23
24
24
24
24
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
6
6
6
6
6
6
6
6
6
6
6
7
7
7
6
7
7
7
7
7
7
7
6
6
6
20091
20092
20093
20094
20101
20102
20103
20104
20111
20112
20113
20114
20121
20122
20123
20124
20131
20132
Grand
Total
9,295,326,206
8,747,595,024
8,298,904,089
7,849,579,000
7,867,729,000
8,103,839,197
8,431,372,000
8,597,405,000
8,998,688,000
9,245,996,000
9,628,907,000
10,302,577,000
11,571,262,000
12,032,167,070
12,476,963,000
12,746,994,000
12,877,580,000
13,753,868,000
490,646,389,15
2
55,473,765
55,762,620
55,699,761
56,087,321
55,199,944
53,836,384
54,528,321
54,981,905
57,100,811
56,158,624
56,878,754
58,730,852
62,583,628
67,963,097
68,076,986
70,504,826
71,475,791
77,164,914
3,691,456,2
40
50
50
50
48
47
46
46
46
49
49
51
51
51
51
52
51
54
54
3123
38
39
38
39
40
41
41
41
40
40
40
40
63
114
117
120
123
126
4892
31
31
31
31
31
31
31
31
31
31
31
32
34
34
33
34
33
34
305
0
24
24
24
24
23
22
22
22
22
22
22
22
22
22
22
23
22
25
1658
Appendix 1, the number of buildings by each property type, total market value
and total size in quarterly basis.
48
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
6
352
Appendix 2
Code for optimization model in Matlab:
global stats k penalty_equ;
load('timeIndex');
filename = 'Grace_Fu_Thesis_Data_MASKED - Copy2_V2.xlsx';
sheetname = 'NPI_Returns0';
% stats.CFNAI = xlsread(filename,sheetname,'BA2:BA140');
% stats.YYYYQ2 = xlsread(filename,sheetname,'AT2:AT140');
stats.YYYYQ = xlsread(filename,sheetname,'B2:B13128');
stats.Weight = xlsread(filename,sheetname,'AM2:AM13128');
stats.AdjustedCapRate = xlsread(filename,sheetname,'U2:U13128');
stats.SqFt = xlsread(filename,sheetname,'AR2:AR13128');
stats.TotalReturn = xlsread(filename,sheetname,'T2:T13128');
stats.Top6City = xlsread(filename,sheetname,'AH2:AH13128');
stats.Northeast = xlsread(filename,sheetname,'AI2:AI13128');
stats.West = xlsread(filename,sheetname,'AJ2:AJ13128');
stats.Midwest = xlsread(filename,sheetname,'AK2:AK13128');
stats.South = xlsread(filename,sheetname,'AL2:AL13128');
stats.Industrial = xlsread(filename,sheetname,'AC2:AC13128');
stats.Numberofbuilding = xlsread(filename,sheetname,'AP2:AP13128');
stats.timeindex=timeIndex;
% remove bad points
badpoints = find(stats.Weight==0 | stats.SqFt==0);
stats.YYYYQ(badpoints)=[];
stats.Weight(badpoints)=[];
stats.AdjustedCapRate(badpoints)=[];
stats.SqFt(badpoints)=[];
stats.TotalReturn(badpoints)=[];
stats.Top6City(badpoints)=[];
stats.Northeast(badpoints)=[];
stats.West(badpoints)=[];
stats.Midwest(badpoints)=[];
stats.South(badpoints)=[];
stats.Industrial(badpoints)=[];
stats.Numberofbuilding(badpoints)=[];
%% penalize equality constraints
penalty_equ = 0;
49
% %% build timeindex
% timeindex=zeros(138,1);
% timeindex(1)=19784;
% index=2;
% for year=1979:2012
% for quarter=1:4
% timeindex(index) = str2double(strcat(int2str(year),int2str(quarter)));
% index=index+1;
% end
% end
% timeindex(index)=20131;
% timeindex(index+1)=20132;
% clear year index quarter;
%% compute constraint matrix
Z=[];
For i=1:139
indice=find(stats.YYYYQ==stats.timeindex(i));
Z=[Z; ones(size(indice))*stats.CFNAI(i)];
End
Z=Z./stats.Numberofbuilding;
A = -[stats.AdjustedCapRate.*Z stats.SqFt.*Z stats.Industrial.*Z stats.Top6City.*Z
stats.Northeast.*Z stats.West.*Z stats.Midwest.*Z stats.South.*Z];
b = stats.Weight;
% Aeq = [];
% for i=1:139
% indice=find(stats.YYYYQ==stats.timeindex(i));
% At = [mean(stats.AdjustedCapRate(indice)) mean(stats.SqFt(indice))
mean(stats.Top6City(indice)) mean(stats.Industrial(indice))];
% Aeq = [Aeq; At];
% end
% beq=zeros(139,1);
%
options = optimset('Algorithm','interior-point');
k = 2;
a02 = fmincon(@objective,zeros(8,1),A,b,[],[],[],[],[],options)
%
k = 5;
50
a05 =
fmincon(@objective,zeros(8,1),A,b,[],[],[],[],[],options) %,[],[],[],[],[],options
k = 9;
a09 = fmincon(@objective,zeros(8,1),A,b,[],[],[],[],[],options)
functionob=objective(a)
% a = [0 0 0 0]'
global stats k penalty_equ;
ob=0;
for i=1:139
indice=find(stats.YYYYQ==stats.timeindex(i));
wBar= stats.Weight(indice);
N =length(indice);
% stats.CFNAI1 = stats.CFNAI1(2:140);
Z=stats.CFNAI(find(stats.YYYYQ2==stats.timeindex(i)));
r=stats.TotalReturn(indice);
X = [stats.AdjustedCapRate(indice) stats.SqFt(indice) stats.Industrial(indice)
stats.Top6City(indice) stats.Northeast(indice) stats.West(indice)
stats.Midwest(indice) stats.South(indice)];
Rp=(wBar+1/N*X*a*Z)'*r;
% ob=ob+(1+Rp)^(1-k);
% penalize equlity constraint
ob=ob+(1+Rp)^(1-k)/(1-k) - penalty_equ*sum(1/N*X*a*Z)^2;
end
51
References
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(2007)
Stocks are from Mars, Real Estate is from Venus. Arvind Pai and David Gelter.
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Parametric Portfolio Policies: Exploiting Characteristics in the Cross-Section of
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Expected Returns and Expected Growth in Rents of Commercial Real Estate.
Plazzi, A., Torous, W., & Valkanov, R. (2010). Review of Financial Studies, 23(9),
3469–3519. doi:10.1093/rfs/hhq069
Applying MPT to Institutional Real Estate Portfolios: The Good, the Bad and the
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Exploiting Property Characteristics in Commercial Real Estate Portfolio
Allocation. Alberto Plazzi, Walter Torous and Rossen Valkanov (2011)
Real Estate is Not Normal: A Fresh Look at Real Estate Return Distributions.
Michael S. Young, Richard A. Graff (1995)
Real Estate Return Correlations: Real-World Limitations on Relationships
Inferred from NCREIF Data. Richard A Graff, Michael S. Young.(1996)
Determining the Current Optimal Allocation to Property: A Study of Australian
Fund Managers. Wejendra Reddy. (2012)
Property-Level Performance Attribution: Demonstrating a Practical Tool for Real
Estate Investment Management Diagnostics. Tony Feng. (2010)
The Vacancy Rate and Rent Levels in the Commercial Office Market.
Frew and G. Donald Jud (1988)
James R.
Vacancy, Search and Prices in a Housing Market Matching Model. William C.
Wheaton(1990).
52
53