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Real Estate in Mixed-Asset Portfolios for Various Investment Horizons

Jean-Christophe Delfim*

Martin Hoesli**

30 January 2019

*University of Geneva

Geneva School of Economics and

Management

40 boulevard du Pont-d’Arve

CH-1211 Geneva 4

Switzerland

Email: jean-christophe.delfim@unige.ch

Phone: +41 22 379 9264

Ph.D. Candidate, Research and Teaching

Assistant, University of Geneva

**University of Geneva

Geneva School of Economics and

Management

40 boulevard du Pont-d’Arve

CH-1211 Geneva 4

Switzerland

Email: martin.hoesli@unige.ch

Phone: +41 22 379 8122

Professor of Real Estate Investments at the

University of Geneva

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Abstract

This research investigates the role of real estate in a mixed-asset portfolio for various

investment horizons. Using U.S. data spanning almost three decades, we report that medium

to long term investors should allocate 20% of their portfolio to direct real estate. In contrast,

short term investors should focus on open-end core funds, which are found to be good

substitutes for direct investments. REITs are usually of limited interest as a substitute for

direct real estate, but they could be used in conjunction with direct investments for medium

and long term horizons, as they partly substitute for stocks. Value-added and opportunistic

closed-end funds are found to be imperfect substitutes for direct investments. Finally, we find

that including commodities, private equity, and hedge funds in a portfolio enhances its

performance but the allocation to real estate barely changes.

Keywords: Long Term Investments; VAR Models; Real Estate Investments; Commodity;

Private Equity; Hedge Funds; Desmoothing; Principal Component Analysis.

JEL codes: C32, C38, E44; G11, G12, G17, R33

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Introduction

The issue of asset allocation in the long run is of keen interest to institutional investors, such

as pension funds, insurance companies, and sovereign wealth funds. However, much of the

research on the role of real estate in a mixed-asset portfolio has used short time increments

and in effect the results of those studies relate to short term horizons. This is problematic as

over short time periods the impacts of the illiquidity of the asset class but also of transaction

costs will be severe and hence many of those studies have potentially overstated the role of

real estate in diversifying a portfolio, at least over the short term. It is therefore important to

address the issue of how the allocation to real estate changes over various investment

horizons. This question is largely, but not only, driven by the effects of illiquidity on returns.

Pertaining to the impacts of illiquidity on asset allocation, Anglin & Gao (2011) investigate

how liquidity can hinder the will to sell assets. They report that the optimal strategy usually

depends on the state of the economy. The other asset class characteristics held in the portfolio

particularly influence the harmfulness of illiquid assets. Long horizon investors are able to

overcome the drawbacks of illiquid assets. If real estate investments appear to be a natural

candidate for long term allocation, only limited evidence exists on the role of real estate in

long term horizon portfolios. Moreover, the existing studies usually focus only on direct real

estate investments and REITs and neglect other exposures to real estate such as non-listed real

estate funds. Real estate is also rarely used in conjunction with other alternative assets such

as private equity and hedge funds. This paper aims at filling this gap in the literature.

The choice of a relevant model is also critical for long term allocation analysis. The common

Markowitz mean-variance framework is not relevant. This is mainly due to the rather strong

assumption of i.i.d. returns which underlies this model. A common choice for portfolio

allocation analyses by investment horizon is the Campbell & Viceira (2002; 2004) framework

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which relies on VAR models1. MacKinnon & Al Zaman (2009) and Rehring (2012) are the

main reference studies that have used this framework to explore the benefits of including real

estate in a mixed-asset portfolio for various investment horizons.

MacKinnon & Al Zaman (2009), for the U.S., report that the weight of direct real estate in a

portfolio increases from 20% for a one-year horizon to 30% for a 25-year horizon, while the

allocation to stocks is stable at around 20%, and the bond allocation declines slightly from

60% to 50%. Including REITs concomitantly to direct investments is not beneficial, despite

the fact that such investments can partially substitute for direct investments. The REIT

allocation should be around 10% for one-year horizons and almost 20% for 25-year horizons.

The similar allocation to real estate as when direct holdings are considered is due to the

increasing term structure of correlations between REIT and direct investments.

In addition to return predictability realized with VAR models, Rehring (2012) also takes into

account transaction costs as well as the marketing period risk; the latter reflecting an aspect of

the illiquidity risk faced by sellers. Using U.K. data, the author concludes that predictability

and transaction costs are important, while the marketing period risk is negligible. Rehring

(2012) reports a real estate allocation ranging from zero for a one-year horizon to 87% for a

20-year horizon.

Also looking at the role of real estate investments for various investment horizons, Pagliari

(2017) concludes that low-risk investors would prefer direct real estate, while high-risk

investors should focus on listed real estate due to leverage. In general, the allocation to real

estate should not exceed 10% to 15%. Similarly, Amédée-Manesme et al. (2018) report an

allocation to real estate ranging from 10% to 20% depending on the investment horizon and

risk aversion.

1 The model is also applied in Campbell & Viceira (2005).

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The importance of return predictability for long term investors is also emphasized by Fugazza

et al. (2015). The authors report that with respect to the investment horizon, the volatility of

stocks remains constant, while that of REITs increases slightly, and that of bonds increases

strongly.

The negligible impact of marketing period risk on allocations reported by Rehring (2012) is

consistent with the results of Bond et al. (2007) who show that the impact of illiquidity risk

measured through the marketing period risk decreases sharply with the number of properties

held in the portfolio. Their analyses are performed relying on the investment risk model of

Lin and Vandell (2007) taking into account illiquidity.

Cheng, Lin, & Liu (2013) propose adjusting the Modern Portfolio Theory (MPT) framework

for explicitly taking into account the horizon-dependent performance, the liquidity risk and

the high transaction costs specific to real estate investments. They conclude that the optimal

allocation to direct real estate should lie between 3% and 9%, while the holding period should

be between two and six years (see also Cheng, Lin, & Liu, 2010). These results depend on the

assumptions regarding transaction costs and Time-On-Market (TOM). Similarly, Cheng, Lin,

& Liu (2017) report real estate allocations between 1% and 18% for a holding period of 4.5 to

6.5 years.

Sa-Aadu, Shilling, & Tiwari (2010) examine the evolution of asset class weights in a portfolio

with respect to market regimes. They first report that REITs and commodities, including

precious metals, are helpful when consumption growth is low, or its volatility is high, or when

both occur2. They conclude that REITs are particularly advantageous in both good and bad

times; the allocation to REITs is 15% and 19%, respectively.

2 Their analysis is performed by relying on Markov chain models and the Hansen-Jagannathan (1991) volatility

bounds, measuring the reduction in the standard deviation of minimum variance portfolios when an asset class is

added to the portfolio. H-J bounds are computed using the dividend yield, default spread, term spread and T-bill yield.

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Cumming et al. (2013, 2014) focus on portfolio optimization including alternative

investments such as private equity, commodities and hedge funds. They emphasize the

benefits of adding such investments in a portfolio, as well as the importance of reasoning

beyond the classical Markowitz’s framework. Hoevenaars et al. (2008) report similar

conclusions adopting an asset liability management (ALM) approach with the Campbell &

Viceira (2002) framework, including also REITs.

Liquidity also appears to be important for real estate funds. Hass et al. (2012) investigate the

diversification effects of including German open-end real estate funds in a mixed-asset

portfolio. They conclude that such funds help improve diversification, but the temporary

share redemption suspension pertaining to these funds implies that investors have to accept an

average 6% discount in the secondary market. The discount can reach 20% if investors think

that the fund managers will not be able to ensure liquidity within the usual two-year time

limit.

We contribute to the literature in three main ways. First, we use several different types of real

estate exposure: direct, non-listed funds, and REITs. For non-listed funds, we consider the

various strategies: core, value-added, and opportunistic. Second, we propose introducing

principal component analysis (PCA) in the Campbell & Viceira (2004) framework. Third, we

assess how the results obtained with appraisal-based indices are similar to those obtained with

transaction-based indices. Forth, we consider a wide array of alternative asset classes such as

hedge funds, commodities, and private equity. Fifth, the analyses are conducted on a long

time period covering almost three decades and including the Global Financial Crisis (GFC).

Our results show that medium to long term investors should allocate 20% of their portfolio to

direct real estate. In contrast, short term investors should focus on open-end core funds,

which are found to be good substitutes for direct investments. REITs are usually of limited

interest as a substitute for direct real estate, but they could be used in conjunction with direct

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investments over medium and long term horizons. Value-added and opportunistic closed-end

funds are found to be imperfect substitutes for direct investments. Finally, we find that

including commodities, private equity, and hedge funds in a portfolio enhances its

performance and hardly reduces the allocation to real estate investments.

The remainder of the paper is structured as follows. The next section presents our

methodology. We then turn to a discussion of the data we use, before analyzing our results.

A final section concludes.

Methodology

We rely on the Campbell and Viceira (2004) framework for asset allocation by investment

horizon. Hence, we first fit a VAR(1) model represented by the following equation:

𝑧𝑡+1 = 𝛷0 + 𝛷1𝑧𝑡 + 𝜈𝑡+1 (1)

Let’s suppose we have n variables in the model decomposed in m asset classes and n-m state

variables. Hence, 𝑧𝑡 and 𝑧𝑡+1 are the (n x 1) vectors of current and future asset class returns

and state variables, 𝛷0 is a (n x 1) vector of constants, 𝛷1, a (n x n) matrix of coefficients

containing the impacts of past performance of every asset class and state variables on the

current performance of each asset class and state variables. Finally, 𝜈𝑡+1 is the (n x n) vector

of error terms assumed to be i.i.d. normally distributed with zero mean and covariance matrix

∑𝜈 of order (n x n), such that 𝜈𝑡+1 ~ IIDN(0, ∑𝜈). In addition, 𝑧𝑡+1 decomposes in the

following way:

𝑧𝑡+1 = [

𝑟0,𝑡+1

𝑥𝑡+1

𝑠𝑡+1

] (2)

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with 𝑟0,𝑡+1, the real return on cash taken as the real three-month T-bill which is the risk free

asset, 𝑥𝑡+1, the (n-1 x 1) vector of excess returns of remaining assets to be included in the

portfolio, and 𝑠𝑡+1, the (n-m x 1) vector of state variables helping to explain asset returns3.

Hence, the term structure of risk for all variables included in the model is derived by first

computing the covariance matrix conditional to a given investment horizon k, with data at

frequency f, as4:

√𝑓

𝑘𝑣𝑎𝑟 (∑ 𝑧𝑡+𝑖

𝑘

𝑖

)

= √𝑓

𝑘 [∑𝜈 + (𝐼 + 𝛷1)∑𝜈(𝐼 + 𝛷1)′ + (𝐼 + 𝛷1 + 𝛷1

2)∑𝜈(𝐼 + 𝛷1 + 𝛷12)

′

+ ⋯

+ (𝐼 + 𝛷1 + 𝛷12 + ⋯ + 𝛷1

𝑘−1)∑𝜈(𝐼 + 𝛷1 + 𝛷12 + ⋯ + 𝛷1

𝑘−1)′]1/2

(3)

where I is the identity matrix. The diagonal elements of the matrix are the variances of the m

asset class excess returns and the n-m state variables. The off-diagonal elements are the

covariance coefficients. Note that in contrast to aforementioned studies, in order to spare

degrees of freedom, we rely on estimates obtained from a restricted version of the initial

VAR(1) we fit. The restricted VAR iteratively removes the coefficients being not significant

at a given threshold of 90% by setting them equal to zero, so that only the remaining

coefficients have to be estimated. This also allows removing the influence of insignificant

parameters in the following computation of term structures and asset allocations. Note also

that extensions of the model exist. For example, Hoevenaars et al. (2014) rely on Bayesian

3 Note that we take the aforementioned variables as expressed in terms of log in order to benefit from log

approximation properties, which tend to improve the accuracy of the estimation. Note also that if the n-m state

variables are set to be exogenous, the lower n-m lines of the 𝛷1 matrix are equal to zero, meaning that each state

variable is assumed independent of lagged asset returns and lagged values of all other state variables. 4 The frequency f is equal to 1, 4, or 12, for yearly, quarterly or monthly data, respectively.

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VAR models, which allows to better deal with parameter uncertainty and to incorporate prior

views on asset returns.

Following Rehring (2012), we take into account transaction costs. According to Hurst, Ooi,

& Pedersen (2017), we retain transaction costs of 0.06% for stocks and of 0.01% for bonds,

when buying and when selling. For direct real estate, we retain round-trip transaction costs of

6% as reported by Steverman (2014). According to the Global Property Guide (2018), these

costs are usually made of 0.5% to 1% of title search and insurance, 0.2% to 0.5% of recording

fees, and of 0.5% to 1% when buying a property. When selling, costs are made of another

0.5% to 1% in legal fees, as those are shared between buyer and seller, a property transfer tax

ranging from 1% to 1.425%, and the broker’s fee. The latter are suggested to be around 6%,

but it would lead to round-trip transaction costs of 8.7% to 10.925%. However, we maintain

that professional institutional investors face lower broker’s fees because the of the large

transaction volumes and part of the broker’s work may be undertaken by the institutional

investor himself. Thus, we retain a broker’s fee of only 2%. We hence decide to share the

round-trip costs in 2% for the buyer and 4% for the seller.

Regarding REITs, we apply the same transactions costs as for stocks. From INREV (2017),

we estimate that round-trip transaction costs for real estate open-end funds, such as the core

funds we consider, should not exceed 1%, and we split this percentage in 0.6% for buying

costs and 0.4% for selling costs. Regarding closed-end funds, such as the value-added and

opportunistic funds we consider, we rely on Lynn (2009) who reports round-trip costs of 3%

to 5% and we select a value of 4% that we allocate with the same proportions as for open-end

core funds in 2.4% for buying costs and 1.6% for selling costs. Turning to commodity ETFs,

we apply 0.5% of buying and selling costs as suggested by Steverman (2014). We apply

round trip transaction costs of 2% divided equally for private equity as determined from

Phalippou, Rauch, & Umber (2018). Finally, hedge funds buying and selling costs are 0.9%

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each as suggested by Steverman (2014). Because the VAR model is fitted relying on log

returns, we have to convert the transaction costs accordingly. For example, the round-trip

transaction costs for direct real estate becomes log(1 + 2%) + log(1 + 4%) = 5.9%.

Data

We use U.S. data for the period 1990Q2-2018Q2. The main asset classes we include in our

analysis are stocks, government bonds, and direct real estate investments, proxied by the U.S.

MSCI index, the U.S. Citigroup WGBI 7-10 years index, and the NCREIF TBI and NPI

indices, respectively. We take cash as the riskless asset, proxied by the three-month T-bill

rate. Then, we substitute direct real estate investment by listed real estate companies and non-

listed real estate funds. The former are proxied by the NAREIT All Equity REITs index.

Regarding non-listed funds, we consider open-end core funds using the NCREIF ODCE

index, as well as closed-end value-added and opportunistic funds using indices from

Cambridge Associates. In addition, we consider commodities, broader private equity, and

hedge funds as other alternative asset classes, relying on the commodity ETF of S&P

Goldman Sachs Commodity Indices5, the Cambridge Associate private equity closed-end

funds index for the U.S., and the Hedge Fund Research Index for North America,

respectively. All series are total returns net of management fees and expenses such as carried

interest.

The TBI index is transaction-based. As described in Geltner (2011), the initial version of this

index was built relying on a hedonic model until 2010Q4, while the current version restated

from 1984Q1 relies on a sale price appraisal ratio model (SPAR). According to Geltner

(2015), the hedonic method does not suffer from the main drawbacks affecting alternative

5 With almost 60% invested in energy in the composite index we preferred building our own commodity index

from the S&P GSCI subindices for each commodity sector. This is made by simply maximizing the Sharpe

ratio, which leads to an allocation of 31.3% in agriculture, 20.8% in energy, 31.3% in industrial metals, 6.3% in livestock and 10.4% in precious metals.

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methods. However, the current SPAR index is very close to the hedonic one, especially since

the late 1990s and early 2000s, as displayed in Figure A1 in the Appendix. Thus, we use the

hedonic return series until the end of its production in 2010Q4 and the returns from the SPAR

index for the remaining quarters until 2018Q2.

For comparison purposes, we also use the appraisal-based NPI direct real estate index treated

for the smoothing bias affecting such indices. Smoothing is similarly affecting return series

of non-listed real estate funds, private equity and hedge funds. For desmoothing return series,

we apply the method proposed by Delfim & Hoesli (2019a), relying on a regime switching

desmoothing model and robust filters devoted to identifying and treating extreme returns

often generated during the desmoothing process. The optimal estimated parameters of the

desmoothing model for direct real estate are a high regime desmoothing parameter 𝛼𝐻 = 0.88,

a low regime desmoothing parameter 𝛼𝐿 = 0.45, and a regime threshold 𝑡ℎ𝑀𝑆𝐶𝐼 = −12.7%

defined according to the MSCI excess total return. In order to be consistent in desmoothing

non-listed real estate series, we apply the same parameters as for direct. Regarding private

equity and hedge funds, we prefer re-estimating the parameters because the autocorrelation

structure differs substantially from the one observed for real estate series. The parameters are

𝛼𝐻 = 0.64, 𝛼𝐿 = 0.31 and 𝑡ℎ𝑀𝑆𝐶𝐼 = −13.1%.

Distributions of quarterly total returns, net of management fees, for the aforementioned asset

classes are presented as boxplots in Figure 1 and summary statistics are reported in Table 1.

We observe that cash is as expected almost riskless with a standard deviation of 0.55%, with

also the lowest geometric mean return (0.66%). Stocks have a geometric mean of 2.38% and

a standard deviation of 7.66%, while for government bonds these figures are 1.57% and

3.77%, respectively.

Regarding direct real estate, the transaction-based TBI index exhibits a mean of 1.91% and a

volatility of 4.43%, which places real estate in between stocks and bonds, as would be

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expected. The 5% expected shortfall of -8.24% for the TBI is also between the -15.97% and -

4.47% figures for stocks and bonds, respectively. Interestingly, the Sharpe ratio of 0.57 for

the TBI is higher than the one observed for stocks at 0.46 and almost equal to the one of

bonds which is 0.55. Desmoothed appraisal-based NPI returns are very similar to the TBI

returns with respect to both the appearance of the distribution and most of the key return

statistics. This underlines the efficiency of the method proposed by Delfim & Hoesli (2019a)

in producing desmoothed series replicating the characteristics of transaction-based series. As

would be expected, the original NPI return series deviates markedly from the previous two

series with volatility being about half that of the TBI, a very high autocorrelation of 0.81, as

well as an underestimated expected shortfall of -5.58%, and an exaggerated Sharpe ratio of

0.89.

REITs offer a slightly higher mean return than stocks (2.55%) for a larger volatility (9.47%).

The REIT drawdown of -20.58% is also larger than that of stocks and the Sharpe ratio of 0.41

is slightly lower than for stocks. The open-end core fund series appears very close to the

direct series, both before and after desmoothing. This would be expected as direct indices

consider the same types of prime properties being of interest to core funds. The higher

volatility of 5.39% for core funds compared with that of 4% to 4.5% for direct investments

can be explained by the leverage effect. Once desmoothed, both value-added and

opportunistic fund returns are close to REITs in terms of range of returns, volatility, and

expected shortfall, with value-added funds being slightly less volatile than REITs and

opportunistic funds slightly more volatile. However, the mean returns of these funds is below

the average for REITs, with values of 1.15% for value-added and 1.49% for opportunistic

funds. This can be explained by the large management fees and other carried interest usually

charged by managers of such funds. Indeed, according to Case (2015) average yearly

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management fees and expenses for value-added and opportunistic funds amount to 1.63% and

2.51%, respectively, during the period 1988-2013, compared with 0.99% for core funds.

Among other alternative investments, commodities appear as risky as stocks with a 7.62%

volatility and an expected shortfall of -16.66%, but their mean return is only 0.59%, leading to

the lowest Sharpe ratio at about zero. Once desmoothed, the volatility of private equity and

hedge fund returns is almost as large as that of stocks, with figures of 7.17% and 6.68%,

respectively. The private equity and hedge funds mean returns are 2.36% and 1.29%,

respectively. Hence, the Sharpe ratio of private equity is the same as that of stocks and the

one of hedge funds is lower (0.19). Note also that in general all return distributions are

slightly negatively asymmetric and leptokurtic.

The VAR models we apply require additional variables for better explaining the performance

of asset classes. Campbell & Viceira (2005) suggest using the nominal yield on T-bills, the

stock dividend yield and the 5-year to 3-month yield spread, while REIT returns and the real

estate cap rate are in addition considered by MacKinnon & Al Zaman (2009) and Rehring

(2012). In addition to the equity yield from the MSCI, the cap rate from NCREIF and the 10-

year to 3-month term spread (TS), we propose supplementing the model with the credit spread

(CS), computed as the difference between the AAA and the BAA corporate bond yield from

Moody’s, the industrial production (IP) growth, the Chicago Board Leading Economic

Indicator, the CPI inflation, the M2 money supply growth, the construction cost growth

computed with RSMeans6, the unemployment rate, the real 10-year interest rate, and the

CBOE VIX index. These macroeconomic variables are reported as having an influence on

real estate investments by Delfim & Hoesli (2019b). In addition, the SmB, HmL, MoM, and

6 As the RSMeans index is available on a yearly basis we estimates its intermediary changes relying on the Shelter CPI.

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Pastor & Stambaugh liquidity factor (PSLiq) are also included7 (see Fama & French 1993,

Carhart 1997, and Pastor & Stambaugh 2003).

[Figure 1 about here]

[Table 1 about here]

The summary statistics for the macroeconomic and financial variables are reported in Table 2.

The main observations are the very low volatility, around 1%, and high autocorrelation, over

0.8, of most series (usually over 0.8). This is true except notably for the last four portfolio

factors that have an intermediate volatility of around 3% and a low autocorrelation. Despite

the fact that we have 114 observations for each series, which is a relatively high number when

considering real estate, adding all the sixteen aforementioned variables in the model could use

too many degrees of freedom. Hence, we perform a principal component analysis (PCA) on

the macroeconomic and financial series, and retain only the necessary components for

explaining a minimum cumulative variance of 80%. The representation of the dataset as

principal components (PC) also allows solving the multicollinearity issue that can sometimes

be an issue in econometric analysis. We conclude that nine principal components are

necessary to reach the threshold of 80% of variance explained, which allows to reduce by

almost one half the number of additional variables in the model. The factor loadings

corresponding to each principal component and the resulting factor series are reported in

Table A1 and Figure A2 of the Appendix, respectively.

[Table 2 about here]

7 These four factors are sourced from the WRDS database. Aforementioned macroeconomic series are sourced from Thomson Reuters Datastream if no other source is mentioned.

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Results

We start our analysis by fitting a VAR model including stocks, government bonds and direct

real estate, in addition to cash and the nine factor series derived from the PCA. We report in

Table 3 the estimated parameters and adjusted R2 for asset classes from the matrix 𝛷 =

[𝛷0, 𝛷1]. The explanatory power is good in comparison with that reported in the main

aforementioned studies. In particular, direct real estate and cash have a high adjusted R2.

Table 4 reports the matrix of VAR residual series correlations, with the standard deviation

coefficients for each residual series on the diagonal. The information summarized in the

aforementioned two tables allows us to compute the term structures defined in Equation (3)

and then perform the asset allocations by investment horizon.

[Table 3 about here]

[Table 4 about here]

The term structures of annualized expected real total returns are presented in Figure 2 for

investment horizons from 1 to 100 quarters. We observe that term structures are very stable at

around 0%, 7.5%, and 3.75% for cash, stocks, and bonds, respectively. Conversely, the

structure is strongly negative during the first year for the TBI, due to high transaction costs,

before converging quickly to around 5%. Turning to Figure 3 and the term structures of

annualized volatility,it can be seen that the risk is actually increasing with the investment

horizon, except for bonds for which it remains stable at around 7%. In particular, the

volatility increases from about 7% for real estate for a one-quarter horizon to 13% for a 25-

year horizon. The figures for stocks are 16% and 20%, respectively. Increasing term

structure volatility suggests mean aversion, a finding that is not reported in most past studies.

This result is likely due to the fact that we consider a long time period covering the long bull

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market of the last decade, while previous studies usually stop around the GFC8. Hence, we

believe that the time span considered in this research is more comprehensive and

representative of the various environments that can be experienced by investors than those

examined in past studies.

The term structures of correlations between asset classes are reported in Figure 4. We first

observe that the stocks-bonds correlation is stable at around -40%, as is the cash-stocks

correlation at around zero, while the cash-bonds correlation is gradually increasing from about

15% to 45% for a 25-year horizon. Regarding correlations with direct real estate, we report a

decreasing relationship with cash from about zero to -40%. The behavior of the correlation

between real estate and bonds is more complex as it starts increasing from around 20% to

40% until two years and then it progressively decreases to -20%. The real estate-stocks

correlation is rapidly increasing, going from zero to 50% after three years and then

converging toward 60%. The term structures of correlations also bears the seal of the last

decade of bullish stock and real estate markets, linking these two asset classes and making

them diverge from the bond market, which underperformed during the same period.

[Figures 2 to 4 about here]

Based on the figures discussed above we are now able to build portfolios for various

investment horizons. We present in Figure 5 the allocation to stocks, bonds, and direct real

estate corresponding to a portfolio having the maximum expected Sharpe ratio. The results

suggest that, for short-term investors having a horizon of less than 2.5 years, there should be

no direct real estate in the portfolio while the shares of stocks and bonds should be around

25% and 75%, respectively. Real estate enters the portfolio once the investment horizon

exceeds 2.5 years, and its share gradually increases to reach 10% for a five-year horizon, 16%

8 We verified if the increasing term structures were not caused by a computational artifact by reproducing the

volatility term structures presented in Campbell & Viceira (2005) and Rehring (2012) using the estimated

parameters they report for the 𝛷 and ∑𝜈 matrices. We obtained the same results as in these two studies.

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for a 10-year horizon, and finally converges toward 20% for a 25-year horizon. The share of

bonds decreases only slightly to converge at about 65%, while the weight of stocks diminishes

slightly from 18% for a five-year horizon to 13% for a 25-year horizon. The annualized

expected real total returns of these portfolios is very stable whatever the investment horizon at

around 4.7%, while volatility converges from 5.5% toward 5%. The Sharpe ratio in turn

increases from 0.82 to 0.89.

[Figure 5 about here]

We are now interested in investigating what would happen if we had no transaction-based

information for direct real estate and hence would have to rely on a desmoothed appraisal-

based index. We thus estimate another model replacing the TBI series with the desmoothed

NPI one. The term structures of annualized expected real total returns presented in Figure 6

display features identical to those obtained with the TBI (Figure 2). While the patterns of

term structures of annualized volatility in Figure 7 are also very similar to those with the TBI

(Figure 3), they now tend to converge to slightly higher values for stocks and real estate. The

general features of correlation term structures from Figure 8 are very close to the ones

presented in Figure 4 with the TBI. This is especially true for the correlations involving real

estate. Then, as expected from the last three figures, the allocations obtained using desmooth

direct real estate returns (Figure 9) are almost identical to those we got with the transaction-

based series. These results are encouraging are they suggest that using properly desmoothed

returns allows reaching accurate conclusions concerning asset allocation whatever the

investment horizon selected. Hence, we are confident in using desmoothed series for core,

value-added, and opportunistic real estate funds, as well as private equity and hedge funds in

the subsequent analyses.

[Figures 6 to 9 about here]

18

We now fit four new VAR models where we in turn replace direct real estate by each

alternative type of exposure to real estate. The term structure of expected returns for these

exposures are presented in Figure 10 along with the one obtained earlier for the TBI. We

observe first that expected returns for REITs are very close to those observed in Figure 2 for

stocks. Value-added and opportunistic fund returns are more akin to direct real estate returns

due notably to high transaction costs. However, fund returns converge to a lower value than

TBI returns despite the leverage they benefit from. This is due to the high management fees

that such funds command. Core funds have positive returns even for a one-quarter horizon

thanks to their low transaction costs. Their returns remain larger than those of other non-

listed exposures over longer horizons.

Term structures of volatility are shown in Figure 11. The figure highlights the similarities

between REITs and the three kinds of non-listed funds. The volatility term structure of REITs

is also very similar to the one presented for stocks in Figure 7. The increase in the term

structure of core funds can seem counter-intuitive, but is likely due to leverage. As would be

expected, core fund risk remains lower than the risk of the other indirect exposures.

[Figures 10 to 11 about here]

Figure 12 depicts the allocation to each kind of real estate exposure when they are

individually included in a portfolio with stocks and bonds. First, we observe that there is a

positive allocation to REITs from a one-quarter horizon. The allocation grows from 1% to

about 4% after five years, 5.5% after 10 years and to almost 6.5% for a 25-year horizon. This

allocation path is in general two to three times lower than what we observed for direct real

estate, which suggests that REITs are only a partial substitute for direct real estate even in the

long run. The decrease in the real estate allocation when REITs are considered is

compensated by an almost equal increase in the stock and bond allocations.

19

Core funds are included in the portfolio even for short term horizons (weight of about 10%)

and their allocation increases to 16% for a 25-year horizon. Despite the fact that the core fund

allocation does not reach the 20% weight observed for direct investments, they appear to be

the closest substitute to direct holdings. In addition, due to the relatively low transaction costs

and flexibility to invest through open-end funds, compared with direct real estate, core funds

allow taking advantage of being exposed to real estate market even for short investment

horizons, while not suffering the large correlation with stocks affecting REITs at such

horizons. In comparison to core funds, both value-added and opportunistic funds are clearly

less attractive, mainly due to their high transaction costs. Hence, allocations are zero until 2.5

years for opportunistic and 5.5 years for value-added funds. They converge at 3% for longer

horizons.

[Figure 12 about here]

The next analysis we perform consists of including in the model, with stocks and bonds, direct

real estate, as proxied by TBI returns, and each indirect exposure in turn. The term structure

of the correlations between the TBI and indirect exposures are presented in Figure 13. The

conclusions we can draw also help explaining the results discussed in the paragraph above.

REITs are found to have the lowest correlation with direct real estate, starting at zero and

converging rapidly, after around three years, toward 50%. In contrast, the correlations of TBI

returns with non-listed fund returns are very high: 30-40% for a one-quarter horizon; more

than 80% for core and value-added funds, and 70% for opportunistic funds, respectively, after

three years; and eventually 90% for core and value-added, and more than 80% for

opportunistic funds, respectively, for a 25-year investment horizon.

[Figure 13 about here]

20

Regarding the resulting asset allocations, value-added and opportunistic funds are not

included in portfolios containing already direct investments. REITs and core funds, however,

impact allocations of portfolios containing also direct investments. This is reflected in Figure

14, displaying the evolution of the total allocation to real estate, using either only the TBI, the

TBI and REITs, or the TBI and core funds. The figure also displays the proportion of each

indirect exposure (core funds or REITs) in the total allocation to real estate. Regarding first

the case of direct real estate with listed investments, we observe that thanks to REITs the

allocation to real estate is positive (2.5%) even for short horizons. The weight grows to

almost 7% for a two-year horizon, before transaction costs of direct real estate are sufficiently

dampened in order to be included directly in the portfolio. Then the total allocation increases

rapidly, to reach 20% for a four-year horizon and 28% for a 25-year horizon. Once direct real

estate starts being included in the portfolio, the proportion of REITs in the total allocation to

real estate decreases sharply to 20% for a horizon of six years and converges toward 16%

thereafter. In addition to the role played by REITs in short term investment horizons, the total

share allocated to real estate is larger than the one obtained when direct investments only are

considered. This suggests that in addition to the partial role as substitutes for direct real estate

played by REITs over short horizons, such investments also are useful in conjunction with

direct investments over longer horizons.

Core funds are substitutes for direct investments over short horizons, with weights of around

12%, until almost a six-year horizon, when direct investments start being included in the

portfolio. The allocation over short horizons is shorter than is the case when REITs are

considered. The inclusion of direct real estate also occurs later when core funds are

considered rather than REITs. We also observe that the total share of the portfolio allocated

to real estate is the same as the one prevailing when only direct investments are considered.

Another difference from the results obtained with REITs is the large allocation to core funds

21

even for long investment horizons. In fact, the core fund allocation (up to two thirds) remains

higher than the allocation to direct real estate.

[Figure 14 about here]

The final analysis we report focuses on the case where other alternative asset classes

(commodities, general private equity, and hedge funds) are included in the portfolio with

stocks, bonds, and direct real estate. A VAR model is fitted with all the aforementioned asset

classes. The term structures of annualized expected real returns are reported in Figure 15.

The figure indicates negative returns for the three alternative classes for a one-quarter

horizon, converging rapidly toward 4%, 6.5% and 4.5% for commodities, private equity, and

hedge funds, respectively. Then, on Figure 16, term structures of volatilities are depicted.

The volatility of commodities appears very close to that of stocks, starting at 14% and

converging to 22%. For private equity, the structure is rather flat, increasing slightly from

about 12% to almost 15%. Finally, hedge funds interestingly display a structure decreasing

sharply from 13% to 11% for horizons of a few quarters. The term structures of correlations

between direct real estate and each other asset class are shown in Figure 17. The linkages

between private equity and real estate increase rapidly toward 60%, while commodities

appear to be positively correlated with stocks (coefficients grow from about zero to 50%). In

contrast, similarly to the correlation of real estate with cash and bonds, the TBI-hedge funds

correlation is generally negative, evolving from about zero to almost -30%.

The allocations to the various asset classes are depicted in Figure 18. Compared to the base

case exposed in Figure 5, the allocation to bonds is lower as it now rapidly converges to

almost 50% instead of 70%. The allocation to stocks collapses, moving from 27% for a one-

quarter horizon to zero for horizons of four years or more. The weight of direct real estate is

only slightly diminished: it emerges from a two-year horizon, reaches 11% for a five-year

horizon, and converges toward almost 17% for a 25-year horizon. The weight is 20% if only

22

stocks, bonds, and real estate are combined, as shown earlier. It makes real estate the second

most important asset class for the longest horizon portfolios, behind bonds and slightly ahead

of private equity. Finally, the total allocation to the other alternative asset classes is generally

around one third of the portfolio. More specifically, the weight allocated to commodities is

small (about 4.5%). Private equity and hedge funds are included for horizons of at least three

quarters and their allocation grows sharply to converge toward almost 15% for private equity,

with a peak at 18% for a four-year horizon, and to 9% for hedge funds, with a peak at 15% for

a three-year horizon.

[Figures 15-18 about here]

Conclusion

In this study, we have investigated asset allocation in the long run applying the Campbell &

Viceira (2004) framework, which relies on VAR models. We focus on the role of real estate

investments, either direct or indirect, in diversifying portfolios of stocks and bonds. We

consider REITs and both open-end (core) and closed-end (value-added and opportunistic)

funds. We also examine the impacts of including other types of alternative asset classes, such

as commodities, general private equity, and hedge funds.

First, we analyzed the role of direct real estate investments in a portfolio also containing

stocks and bonds. For investment horizons of less than two years and a half, the allocation to

real estate, in a portfolio maximizing the Sharpe ratio, is zero due to the high transaction costs

associated with the asset class. Over such horizons, the allocation to stocks and bonds is 20-

30% and 70-80%, respectively. The weight of real estate increases to 10%, 15%, and 20% for

5-, 10-, and 25-year horizons, respectively. The allocations to stocks and bonds decrease to

reach 13% and 68%, respectively, for a 25-year horizon.

23

Second, we addressed the issue of smoothing affecting appraisal-based series for direct real

estate investments and non-listed funds, but also private equity and hedge funds. We

desmoothed these series following the method proposed by Delfim & Hoesli (2019a). Our

analyses suggest that the desmoothing method we apply is able to produce series returning the

same conclusions in terms of portfolio allocations as when transaction-based series are used.

Third, we compared allocations in real estate if REITs or any of the three kinds of non-listed

real estate funds are used in lieu of direct investments, and then together with direct

investments. We reported that open-end core funds reproduce very well the behavior of direct

real estate such that they are a good substitute for direct real estate in a portfolio. In addition,

over short investment horizons of less than three years, where direct investments are not

interesting due to their high transaction costs, core funds are effective in getting exposure to

the real estate market. Over such horizons, the allocation to core funds is around 12%. In

contrast, REITs are found to have the lowest correlation with direct investments, even in the

long run, and hence are poor substitutes for direct investments. This is true except for short

horizons, where direct real estate is not appealing. We also conclude that REITs can be used

in conjunction with direct investments over long horizons, but their weight remains low at

around 5%. Despite exhibiting high correlations with direct real estate, value-added and

opportunistic funds are poor substitutes for direct investments, with allocations of zero over

short horizons and below 4% over longer ones. Closed-end funds are also found to be useless

if direct investments are also included in the portfolio.

Finally, we included other alternative asset classes in a portfolio of stocks, bonds, and direct

real estate. We concluded that real estate is still one of the most attractive asset classes, with

an allocation of 10-15%, except over short investment horizons. The allocation to stocks is

almost zero for horizons of four years or more, while the weight exceeds 15% over horizons

of one year or less. Bonds are still the most important class, with an allocation moving from

24

more than 70% to about 50% in the long run. Commodities are of limited but constant

interest with an allocation of roughly 5% whatever the investment horizon. Private equity

constitutes an appealing asset class, except in the very short term; its allocation is almost 20%

in the medium term and about 15% in the long run. Hedge funds follow a rather similar

pattern as private equity, with its allocation peaking at 15% for a three-year horizon.

25

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28

Tables and Figures

Table 1. Summary Statistics of Total Returns – 1990Q2:2018Q2

Cash Stocks Gov.

Bonds

Direct

RE TBI

Direct

RE NPI

dsmth

Direct

RE NPI

REITs Core RE

Funds

dsmth

Core RE

Funds

Value-

Added

RE

Funds

dsmth

Value-

Added

RE

Funds

Opportu

-nistic

RE

Funds

dsmth

Opportu

-nisitc

RE

Funds

Commo-

dity

Private

Equity

dsmth

Private

Equity

Hedge

Funds

dsmth

Hedge

Funds

Max 1.88% 21.90% 12.00% 18.81% 11.92% 5.12% 33.28% 19.15% 5.22% 23.96% 7.34% 36.22% 8.63% 26.83% 17.50% 15.74% 15.69% 11.98%

q75 1.19% 6.62% 3.81% 4.34% 4.11% 2.90% 8.20% 5.54% 3.29% 6.45% 2.96% 6.97% 3.38% 4.96% 7.32% 4.96% 6.78% 3.81%

Median 0.69% 3.37% 1.34% 2.00% 1.46% 2.14% 3.14% 2.18% 2.16% 1.79% 2.12% 1.79% 2.46% 1.68% 3.32% 2.90% 1.69% 1.99%

Average 0.67% 2.67% 1.63% 2.01% 1.84% 1.67% 3.02% 2.17% 1.60% 1.50% 1.32% 2.13% 1.65% 0.89% 2.61% 2.63% 1.51% 1.49%

Geom. mean 0.66% 2.38% 1.57% 1.91% 1.76% 1.65% 2.55% 2.03% 1.56% 1.15% 1.26% 1.49% 1.59% 0.59% 2.36% 2.53% 1.29% 1.42%

q25 0.05% -0.23% -0.93% -0.52% -0.26% 1.28% -1.16% -0.44% 1.05% -2.34% 0.58% -3.38% 1.05% -3.43% -1.90% 0.42% -4.30% -0.83%

Min 0.00% -22.22% -5.59% -16.89% -12.13% -8.39% -38.80% -16.20% -13.90% -27.42% -16.11% -41.35% -13.63% -30.65% -15.47% -16.27% -11.98% -14.53%

St.dev. 0.55% 7.66% 3.37% 4.43% 4.08% 2.24% 9.47% 5.39% 2.95% 8.17% 3.52% 11.32% 3.45% 7.62% 7.17% 4.48% 6.58% 3.87%

rob. Skew. 0.15 -0.17 0.10 -0.01 0.11 -0.53 0.02 -0.12 -0.58 -0.18 -0.37 0.16 -0.55 -0.09 -0.14 -0.06 0.00 -0.08

rob. Kurt. -1.51 1.52 -0.32 0.19 1.00 3.21 0.99 0.56 2.72 1.16 3.39 1.87 3.58 0.08 0.18 1.20 2.20 3.12

Autocor. 0.99 0.06 0.02 -0.01 0.09 0.81 0.11 0.44 0.87 -0.14 0.71 -0.18 0.81 0.21 -0.29 0.30 -0.30 0.15

ES 5% 0.01% -15.97% -4.47% -8.24% -7.17% -5.58% -20.58% -11.03% -8.18% -20.51% -9.99% -23.47% -9.19% -16.66% -12.82% -8.80% -10.54% -8.10%

SR 0.13 0.46 0.55 0.57 0.54 0.89 0.41 0.51 0.61 0.12 0.35 0.15 0.56 0.02 0.46 0.87 0.19 0.40

This table reports main statistics of quarterly total returns for the asset classes considered in this study. The 1 st and 3rd quartiles are denoted q25 and q75, respectively, while the ES 5% is the expected shortfall at

5% and SR is the annualized Sharpe ratio. Note also that we use robust measures of skewness and excess kurtosis, denoted rob. Skew and rob. Kurt, respectively. Indeed, as reported by Bonato (2011), who

proposes alternative robust measures for these statistics, the conventional measures for 3rd and 4th moments are strongly influenced by the most extreme observations. Hence, for skewness we retain a generalization

of the Bowley (1920) measure, as suggested by Hinkley (1975), with parameter α = 2.5% in order to only cut the influence of most extreme observations. Regarding the kurtosis, we rely on the Crow & Siddiqui

(1967) measure with α = 2.5% and β = 25%.

29

Table 2. Summary Statistics of Macroeconomic and Financial Variables – 1990Q2:2018Q2

Max q75 Median Average Geom.

mean

q25 Min St.dev. rob. Skew. rob. Kurt. Autocor.

Equity Yield 3.73% 2.38% 2.08% 2.18% 2.17% 1.86% 1.11% 0.60% 0.26 4.72 0.99

Cap Rate 8.86% 8.22% 6.66% 6.82% 6.81% 5.57% 4.52% 1.41% 0.00 1.59 1.00

TS 3.82% 2.76% 1.93% 1.86% 1.85% 1.01% -0.37% 1.05% -0.15 2.09 0.93

CS 3.00% 1.08% 0.88% 0.96% 0.96% 0.71% 0.57% 0.38% 0.57 3.49 0.87

IP 2.28% 1.11% 0.66% 0.47% 0.46% 0.21% -5.69% 1.15% -0.33 4.20 0.79

CB Leading Idx 3.13% 1.47% 0.87% 0.44% 0.43% -0.10% -7.81% 1.74% -0.52 4.15 0.81

Inflation 1.57% 0.79% 0.65% 0.63% 0.63% 0.46% -1.00% 0.35% -0.08 4.77 0.46

Money Supply 4.15% 1.65% 1.39% 1.33% 1.33% 0.99% 0.04% 0.60% -0.13 3.25 0.80

Constr. Costs 2.17% 0.88% 0.66% 0.77% 0.77% 0.54% 0.04% 0.45% 0.39 5.25 0.81

Unempl. Rate 9.65% 6.85% 5.59% 5.95% 5.94% 4.78% 3.95% 1.52% 0.43 2.65 0.99

10 Y Real Int. Rate 8.57% 6.04% 4.51% 4.57% 4.55% 2.84% 1.57% 1.89% 0.16 2.04 0.98

VIX 61.86% 24.01% 17.34% 19.63% 19.38% 13.61% 8.65% 7.98% 0.47 2.45 0.77

SmB 13.86% 2.56% 0.77% 0.65% 0.60% -0.71% -16.87% 3.12% 0.07 3.03 -0.37

HmL 7.37% 1.50% 0.14% 0.21% 0.17% -1.23% -9.86% 2.72% 0.15 3.96 -0.09

MoM 16.60% 3.49% 1.47% 1.79% 1.71% -0.50% -11.38% 4.04% 0.21 4.24 0.00

PsLiq 9.10% 2.10% 0.19% -0.13% -0.20% -2.20% -11.93% 3.53% -0.08 3.35 0.26

This table reports main statistics of the macroeconomic and financial series included in the VAR model. The growth rate, or change figures are reported at the quarterly frequency, except for

expected inflation and the inflation surprise, reported on a year-on-year basis. The 1st and 3rd quartiles are denoted q25 and q75, respectively, while the ES 5% is the expected shortfall at 5% and SR

is the annualized Sharpe ratio. Note also that we use robust measures of skewness and excess kurtosis, denoted rob. Skew and rob. Kurt, respectively. Indeed, as reported by Bonato (2011), who

proposes alternative robust measures for these statistics, the conventional measures for 3rd and 4th moments are strongly influenced by the most extreme observations. Hence, for skewness we retain

a generalization of the Bowley (1920) measure, as suggested by Hinkley (1975), with parameter α = 2.5% in order to only cut the influence of most extreme observations. Regarding the kurtosis, we

rely on the Crow & Siddiqui (1967) measure with α = 2.5% and β = 25%.

30

Table 3. VAR Estimated Parameters

Cash MSCI Treas. TBI

Constant

0.0357 0.009 0.0176

Cash lag1 0.7631

-1.9725

MSCI lag1

Treas. lag1 -0.2334

TBI lag1

-0.2856

PC 1 lag1

PC 2 lag1 0.9334

0.0084 -0.7148

PC 3 lag1 0.1153

PC 4 lag1

PC 5 lag1

PC 6 lag1

PC 7 lag1

-0.2939

PC 8 lag1

0.2916

PC 9 lag1

R2 adj. 0.63 0.17 0.11 0.43

31

Table 4. VAR Residual Correlation Matrix with Standard Deviations

Cash MSCI Treas TBI PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9

Cash 0.0032 -0.0125 0.0778 0.0774 0.0728 -0.6601 -0.2202 0.2591 0.0553 -0.2204 -0.5824 0.0258 0.3735

MSCI -0.0125 0.0791 -0.3730 0.0281 0.0874 -0.3393 0.1696 -0.1862 -0.0116 0.0847 -0.0080 0.1729 0.1931

Treas 0.0778 -0.3730 0.0350 0.0557 -0.1877 0.4001 -0.3132 0.2380 0.0805 -0.2275 -0.1188 0.0046 0.2254

TBI 0.0774 0.0281 0.0557 0.0360 0.0394 -0.0100 0.0201 0.0915 0.1413 0.1591 -0.0417 0.0421 -0.0580

PC1 0.0728 0.0874 -0.1877 0.0394 1.0849 0.0926 -0.3055 0.2184 -0.1443 -0.0675 -0.0129 -0.0211 -0.0679

PC2 -0.6601 -0.3393 0.4001 -0.0100 0.0926 1.1453 -0.0797 0.2095 -0.0524 -0.0438 0.4623 -0.0631 -0.2641

PC3 -0.2202 0.1696 -0.3132 0.0201 -0.3055 -0.0797 0.7485 0.2242 0.2544 -0.0207 0.0381 -0.1335 -0.1309

PC4 0.2591 -0.1862 0.2380 0.0915 0.2184 0.2095 0.2242 0.9010 0.1076 -0.5563 0.0400 -0.1313 0.1722

PC5 0.0553 -0.0116 0.0805 0.1413 -0.1443 -0.0524 0.2544 0.1076 0.7650 -0.3193 -0.1542 0.0435 -0.1902

PC6 -0.2204 0.0847 -0.2275 0.1591 -0.0675 -0.0438 -0.0207 -0.5563 -0.3193 0.4972 -0.0723 -0.0540 -0.2214

PC7 -0.5824 -0.0080 -0.1188 -0.0417 -0.0129 0.4623 0.0381 0.0400 -0.1542 -0.0723 0.6000 0.1484 -0.3240

PC8 0.0258 0.1729 0.0046 0.0421 -0.0211 -0.0631 -0.1335 -0.1313 0.0435 -0.0540 0.1484 0.6597 0.2283

PC9 0.3735 0.1931 0.2254 -0.0580 -0.0679 -0.2641 -0.1309 0.1722 -0.1902 -0.2214 -0.3240 0.2283 0.5796

32

Figure 1. Boxplots of Total Returns Distributions – 1990Q2:2018Q2

-15%

-10%

-5%

0%

5%

10%

15%

20%

33

-20%

-15%

-10%

-5%

0%

5%

10%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

Figure 2. Term Structure of Expected Returns with TBI

Cash MSCI Treas TBI

0%

5%

10%

15%

20%

25%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

Figure 3. Term Structure of Volatility with TBI

Cash MSCI Treas TBI

-60%

-40%

-20%

0%

20%

40%

60%

80%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

Figure 4. Term Structure of Correlations (TBI)

Cash-MSCI Cash-Treas Cash-TBI

MSCI-Treas MSCI-TBI Treas-TBI

-10%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

Figure 5. Portfolio Allocation by Horizon with TBI

MSCI Treasury TBI

34

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

0 4 8

12

16

20

24

28

32

36

40

44

48

52

56

60

64

68

72

76

80

84

88

92

96

100

Figure 6. Term Structure of Expected Returns with

Desmoothed NPI

Cash MSCI Treas Desmoothed NPI

0%

5%

10%

15%

20%

25%

30%

0 4 8

12

16

20

24

28

32

36

40

44

48

52

56

60

64

68

72

76

80

84

88

92

96

100

Figure 7. Term Structure of Volatility with Desmoothed NPI

Cash MSCI Treas Desmoothed NPI

-60%

-40%

-20%

0%

20%

40%

60%

80%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

Figure 8. Term Structure of Correlations (Desmoothed NPI)

cash-MSCI cash-Treas cash-dsmth NPI

MSCI-Treas MSCI-dsmth NPI Treas-dsmth NPI

-10%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

Figure 9. Portfolio Allocation by Horizon with Desmoothed NPI

MSCI Treas Desmoothed NPI

35

-20%

-15%

-10%

-5%

0%

5%

10%

15%

0 4 8

12

16

20

24

28

32

36

40

44

48

52

56

60

64

68

72

76

80

84

88

92

96

100

Figure 10. Term Structure of Expected Returns for each RE

Exposure

TBI NAREIT Core Value-Added Opportunistic

0%

5%

10%

15%

20%

25%

30%

35%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

Figure 11. Term Structure of Volatility for each RE Exposure

TBI NAREIT Core Value-Added Opportunistic

-5%

0%

5%

10%

15%

20%

25%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

Figure 12. Alloc. to each RE Exposure with Stocks & Bonds

TBI NAREIT Core Value-Added Opportunistic

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

Figure 13. Term Structure of Correlations with TBI by RE Exposure

NAREIT Core Value-Added Opportunistic

36

Figure 14. Total Real Estate Allocation and Shares of Indirect Investments

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

TBI alone TBI+NAREIT TBI+Core

Core in RE NAREIT in RE

37

-20%

-15%

-10%

-5%

0%

5%

10%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

Figure 15. Term Structure of Expected Returns, TBI & All Asset

Classes

Cash MSCI Treas. TBI Commo. PE HF

0%

5%

10%

15%

20%

25%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

Figure 16. Term Structure of Volatility, TBI & All Asset Classes

Cash MSCI Treas. TBI Commo. PE HF

-60%

-40%

-20%

0%

20%

40%

60%

80%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

Figure 17. Term Structure of Correl. of TBI with each Asset Class

Cash-TBI MSCI-TBI Treas.-TBI

TBI-Commo TBI-PE TBI-HF

0%

10%

20%

30%

40%

50%

60%

70%

80%

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

Figure 18. Portfolio Allocation by Horizon with TBI & All Asset Classes

MSCI Treas. TBI Commo. PE HF

38

39

Appendix

Figure A1. TBI Hedonic vs. TBI SPAR Total Returns - 1984Q2:2010Q4

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

25%

TBI SPAR TBI Hedonic

40

Figure A2. Factor Series Obtained by PCA - 1990Q2:2018Q2

41

Table A1. Factor Loadings of Principal Components

PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9

Equity Yield 0.11 0.10 0.10 0.00 0.06 0.18 0.07 0.05 0.30

Cap Rate -0.02 0.05 0.15 0.02 -0.10 -0.03 -0.20 0.08 -0.05

TS 0.17 0.03 0.23 0.18 0.09 0.35 0.28 -0.11 0.29

CS -0.02 -0.01 -0.08 -0.08 0.04 0.04 0.16 -0.12 -0.01

IP 0.08 -0.10 0.20 0.04 -0.27 0.10 -0.63 0.51 0.02

CB Leading Idx 0.14 -0.07 0.15 0.28 -0.06 0.24 -0.10 0.06 0.29

Inflation 0.01 0.08 0.06 0.23 -0.09 -0.33 0.08 -0.05 -0.29

Money Supply -0.19 -0.18 -0.29 -0.64 0.13 0.38 -0.09 0.05 -0.10

Constr. Costs 0.07 -0.31 -0.39 0.03 0.21 -0.47 -0.34 -0.18 0.52

Unempl. Rate -0.11 0.17 -0.08 0.03 -0.18 -0.38 0.11 0.21 -0.18

10 Y Real Int. Rate 0.29 -0.81 0.30 -0.09 -0.15 -0.11 0.19 -0.10 -0.22

VIX -0.08 -0.02 0.03 -0.11 -0.08 0.09 -0.10 0.06 -0.20

SmB -0.22 -0.23 0.00 0.39 0.74 0.09 -0.03 0.30 -0.28

HmL 0.73 0.22 -0.20 0.02 0.19 0.12 -0.30 -0.27 -0.39

MoM -0.44 -0.02 0.31 0.12 -0.02 0.10 -0.40 -0.67 -0.12

PsLiq -0.12 -0.23 -0.61 0.47 -0.42 0.35 0.06 -0.05 -0.11

Cumulative variance

explained 13.5% 25.2% 36.2% 46.9% 55.5% 63.5% 70.8% 77.4% 82.9%