The Conditional Distribution of Real Estate Returns: Are

higher moments time varying?

Shaun A. Bond∗and Kanak PatelDepartment of Land EconomyUniversity of Cambridge

19 Silver StreetCambridge, CB3 9EPUnited Kingdom

May 2, 2002

∗Correspondence to the first author, e-mail: sab36@cus.cam.ac.uk. The authors would like to thank Vanessa

Pearson for research assistance, and an anonymous referee, Dean Paxson, Steve Satchell, Jim Shilling, Charles Ward

and participants at the 2001 Cambridge-Maastricht Symposium for Real Estate Finance and Economics for helpful

comments. Remaining errors are, of course, the responsibility of the authors.

1

Abstract

Previous research has shown that the returns on individual properties and listed property securities are

skewed (Lizieri and Ward 2001, Young and Graff 1995 and Liu et al. 1992). This claim is investigated in

the context of listed UK property companies and US REITs. In particular, the shape of the conditional

distribution of total monthly returns is examined for a group of 20 UK companies and 20 REITS listed

continuously since 1970 and 1977, respectively. Also investigated is the claim of Young and Graff that the

skewness found in property returns varies over time. Using the model of Hansen (1994) it is found that while

a large portion of property security returns in the sample do exhibit skewness in the conditional distribution

only in a few instances is there evidence of time variation in the skewness parameter. When time varying

skewness is found there is little evidence to suggest it is associated with the economic cycle.

The link between time varying skewness models and downside risk measures is also discussed and esti-

mates of conditional downside risk are calculated for those companies exhibiting the time varying skewness

property.

Keywords: Conditional Skewness, Commercial Property, GARCH, lower partial moments

2

1 Introduction

Recent research has shown that asymmetry in investment returns is more common than is typically

assumed (Bekaert et al. 1998, Chen, et al. 2001, Perez-Quiros and Timmermann 2001). This finding

is particularly evident in the returns of small company stocks (Bond 2001), which is a category to

which many property company shares belong. A number of reasons have been suggested for this

asymmetry. For example, Hong and Stein (1999) postulate a heterogeneous agent model in which

different classes of investor have varying views on the underlying fundamental value of a company

and also where some, but not all, investors face short sale constraints. An alternative view is that

the asymmetry may be related to the economic cycle, with small firms finding it difficult to access

capital in times of recession and hence are more likely to be adversely affected by changes in the

economic cycle (Perez-Quiros and Timmermann 2000).

While the above reasons are no doubt equally relevant to the property industry as they would

be for other industry sectors, there are specific factors related to the nature of the contracts in

commercial property markets that may also give rise to the finding of skewness in the distribution

of property returns. In particular, the use of long term lease contracts in the United Kingdom

market, typically with embedded upward only rent reviews, skews the payoffs associated with holding

commercial property. This option like payoff on lease cash-flows has often been recognised by

researchers as an important feature of contracts in the property market (see for instance Ward and

French 1996 or Ambrose et al. 2001), yet little consideration has been given to the implication of this

for property company returns. It is recognised that a property company may hold many hundred or

even thousand such contracts, and the effect of any one lease contract is likely to have little impact

on the overall returns for a listed property company. However, because lease contracts of this nature

are the norm rather than the exception it is argued that there may be a systematic impact on the

returns of a property company, even one that may have a large portfolio of leases. It is also possible

that the value of the options embedded in upward-only lease contract may change over the course

of the economic cycle, for example as the volatility of the rental stream increases. Such changes are

likely to be independent of market capitalisation, hence there may be a tendency for large and small

capitalisation property companies to display skewness in returns in contrast to the argument put

forward by Perez-Quiros and Timmermann.

Given the importance of the shape of the returns distribution in financial management appli-

3

cations, this paper considers how well the distribution of listed property securities matches the

symmetric ideal often assumed in financial economics. In particular, the parametric form of the

conditional distribution of returns is estimated, with emphasis placed on testing for the presence of

asymmetry. This examination of the shape of the density functional also extends to investigation

of whether the higher order moments of the distribution are time varying. To do this, use is made

of the autoregressive conditional density function model of Hansen (1994). When time variation

in skewness appears to be present in the data, a possible link between the skewness of property

returns and the economic cycle is investigated. One of the key motivations for this research is the

importance of knowledge of the shape of the conditional density function for risk management and

trading decisions. As distribution based risk management tools such as value at risk (VaR) become

more commonly used, it is essential that important features of the data are captured. To date there

has been little consideration of how time-varying higher moments may impact upon risk measures.

As an illustration of the use of the estimated density function in risk management applications it is

shown how one measure of downside risk can be derived from the estimated time varying density

function. While there are also important extensions of this work to the multivariate case, for ex-

ample, the implications for portfolio construction, the emphasis of this paper is solely on univariate

models.

The outline of this paper is as follows. Section 2 reviews the previous empirical evidence on skew-

ness in financial markets. Section 3 introduces the models to be used in modelling the asymmetry

in returns and Section 4 outlines the data used in this study. The results are presented in Section 5

and Section 6 concludes the paper.

2 The Evidence of Asymmetry in Financial Markets

As the literature on stock price distributions has been well surveyed by many researchers (see for

instance Mittnik and Rachev 1993 and McDonald 1996) only a brief introduction is given here.

While some attention is given to the research on the unconditional distribution of stock returns, also

discussed is the more recent work on modelling the conditional distribution of returns.

It is commonly assumed in financial research that returns are normally distributed. While this

assumption is often made for convenience in theoretical models, it may be an acceptable assumption

for returns over medium to long horizons, such as quarterly or annual returns. However, it is less

4

suitable for more frequently observed data (daily, weekly or monthly). Empirical observation of

financial markets has found that large movements occur more frequently than would be expected if

returns were normally distributed (excess kurtosis) and also that on some occasions returns of either

sign are observed more frequently than returns of the opposite sign (skewness). Research into the

shape of the unconditional distribution of stock returns has a long history in finance. Mandelbrot

(1963) and Fama (1965) are both early examples of research of this type. In general, the attention

in these early research papers focused on the stable class of distributions, as the stability property of

the distribution is seen to be an important theoretical property of stock returns1 . However, Officer

(1972) raised doubts about the suitability of this assumption and this has lead to a search for other

distributions to capture the excess kurtosis and (sometimes) skewness of stock returns. Alternative

approaches considered include the student’s t distribution (Blattberg and Gonedes 1974) or a mixture

of distributions (usually a mixture of normals) as in Praetz (1972) or Kon (1984). This list is by

no means exhaustive as McDonald (1996) also mentions application of the generalised beta of the

second kind, the generalised t and the exponential generalised beta of the second kind, with the

latter distribution able to capture asymmetry as well as excess kurtosis.

Is it necessary to consider probability models which allow for skewness in financial returns?

While not being entirely conclusive, a large body of literature has shown that for some markets, and

for some time periods, returns appear to be skewed. For example, Simkowitz and Beedles (1980),

Singleton and Wingender (1986) and Badrinath and Chatterjee (1988) find evidence of skewness in

individual stock returns as well as market indices in US stock markets. This observation of skewness

is not only limited to US equity markets. Alles and King (1994), Aggarwal, Rao and Hiraki (1989)

and Theodossiou (1998) find evidence of skewness in a range of international financial markets,

including equities, bonds and currencies.

In the real estate literature, as in the other financial markets, some evidence in favour of skewness

has been presented. A cross-section study by Young and Graff (1995) investigated the return distri-

bution of individual properties in the Russell-NCREIF database. Their study strongly rejects the

use of the normal distribution to model returns and they find evidence of time-varying heteroscedas-

ticity and skewness over the period of the study. In particular, for most years in the sample period

(1980-1992) the returns on individual properties are negativity skewed. Lizieri and Ward (2001) also1An important property of the stable class of distributions is that the shape parameter of the distribution is

invariant to the sampling frequency of the data.

5

strongly reject the assumption of normality for commercial property returns in the UK. Using the

IPD all property index and a number of component indices, Lizieri and Ward assess the suitability

of a number of distributions for monthly and quarterly returns. While there appears to be no strong

overall consensus in favour of one particular distribution, there is perhaps some support for the use

of a logistic distribution. Liu, Hartzell and Grissom (1992) consider the presence of skewness (in

relation to other assets) and the implications that this has for the pricing of real estate assets. Their

study has an important difference to the others reported here, in that Liu et al. are interested in

studying co—skewness in real estate assets. A major finding is that the co-skewness of real estate is

slightly less negative than the other assets in the study. Using the three moment asset pricing model

of Kraus and Litzenberger (1976), Liu et al. conclude that investors will accept a lower expected

return on real estate assets in relation to other risky assets because of the lower negative co-skewness.

The evidence presented above is entirely in terms of the unconditional distribution of financial

returns. In many instances, such as asset pricing, risk management and performance measurement

knowledge of the shape of the conditional distribution of returns is equally important. In reviewing

the literature on conditional density functions, emphasis is given to empirical applications of con-

ditional densities implemented in the Autoregressive Conditional Heteroscedasticity (ARCH) class

of models (Bollerslev 1986, Engle 1982). The ARCH class of models are discussed in the section

below, however, to maintain consistency with the paragraphs above, the relevant work on assumed

density functions is reported here. In the original application of this class of model the assumption

of conditionally normality was typically made. If returns are conditionally normal, the ARCH model

implies that unconditional returns are leptokurtic. However, the degree of leptokurtosis introduced

in this way still does not seem to satisfactorily capture the fat tailed nature of financial data and

research has concentrated on alternative assumptions to conditional normality. Use of the student’s

t distribution was originally recommended by Bollerslev (1987) and has since been regularly applied

in the ARCH literature (Bollerslev, Chou and Kroner 1992). Another commonly used distribution is

the generalised error distribution used in Exponential Generalised ARCH (EGARCH) model (Nelson

1991). Asymmetry has also been investigated in the shape of the conditional density function, and

as discussed above there appears to be some evidence in favour of skewness. Using non-parametric

and semi-parametric approaches Hsieh (1989), Gallant, Hsieh and Tauchen (1991) and Engle and

Gonzalez-Rivera (1991) find evidence of asymmetry in financial data. Parametric investigation of

asymmetry have been undertaken by using versions of the skewed t distribution by Harvey and Sid-

6

dique (1999) and Hansen (1994). Indeed these authors extend their model to capture time variation

in the conditional third moment. Lee and Tse (1991) have investigated conditional skewness using a

Gram-Charlier expansion, although they failed to find evidence of skewness in the data they exam-

ined. Bond (2000) has used a double-gamma distribution to investigate skewness in the conditional

density of exchange rates but found mixed evidence for the presence of skewness.

3 Distributional modelling issues

In this paper, the ARCH class of models is used to investigate the asymmetric properties of UK real

estate companies stock returns and when asymmetry is found to be present, estimates of downside

risk (or the second lower partial moment) are calculated and compared to a symmetric measure of risk

such as standard deviation. In particular, the GARCH model with student’s t distribution and the

Hansen (1994) model with a skewed student’s t distribution are estimated. GARCH models are well

reviewed in the literature with recent surveys by Palm (1996) and Shephard (1996), complementing

more extensive surveys by Bollerslev, Chou and Kroner (1992) and Bollerslev, Engle and Nelson

(1994). The basic form of the model expresses a time series variable xt as the product of a scale

variable and standardised innovation term, such that for the GARCH(1,1) model

xt = σtzt (1)

where

σ2t = ω + αx

2t + βσ

2t−1 (2)

and zt ∼ iid (0, 1) with it typically assumed that zt ∼ NID (0, 1) . Non-negativity constraints are

also imposed on the parameters of equation (2) to ensure that σ2t ≥ 0. While ω,α,β ≥ 0 is usually

sufficient to ensure the conditional variance is positive, Nelson and Cao (1992) have shown that

a broader range of parameter values are permissible. Estimation of the model is often performed

using maximum likelihood techniques. If dynamic effects are present in the mean of the series, a

conditional mean equation may be specified, such that for an AR(1) process

xt = a0 + a1xt−1 + et (3)

and

σ2t = ω + αe

2t + βσ

2t−1. (4)

7

Of course more complex forms of the conditional mean equation are possible. Another common

extension is to include ‘threshold’ or ‘leverage’ effects in the conditional variance equation (see, for

example, Glosten, Jagannathan and Runkle 1991 and Nelson 1991).

In the previous section the choice of conditional density function was discussed. The chosen

distributions for this analysis are the student’s t density and the skewed t density of Hansen (1994).

This choice results from a review paper by Bond (1999) in which the model by Hansen, while not

without limitations, was found to perform well in comparison to the alternative parameterisation of

the skewed t distribution by Harvey and Siddique (1999). The standardised t distribution has the

form

f (zt|Ωt−1, η) =Γ

¡η+1

2

¢pπ (η − 2)Γ ¡

η2

¢ µ1 +

z2t

(η − 2)¶−(η+1

2 )(5)

with 2 < η <∞. Hansen’s parameterisation to allow for skewness is given by

f (zt|Ωt−1, η,λ) = bc

Ã1 +

1

η − 2µbzt + a

1− λ¶2

!− (η+1)2

zt < −ab

(6)

= bc

Ã1 +

1

η − 2µbzt + a

1 + λ

¶2!− (η+1)

2

zt ≥ −ab

(7)

where

a = 4λc

µη − 2η − 1

¶(8)

b2 = 1 + 3λ2 − a2 (9)

c =Γ

¡η+1

2

¢pπ (η − 2)Γ ¡

η2

¢ (10)

and −1 < λ < 1, where λ is a parameter to control for the skewness of the distribution. It can

be readily verified that if λ = 0, the distribution will collapse to the standardised t distribution

with η degrees of freedom. To capture time variation in the skewness parameter, Hansen suggests a

quadratic law of motion such as

λt = γ0 + γ1et + γ2e2t . (11)

At first sight this is not necessarily the most intuitive model, as a model containing a cubic power

of the error term seems consistent with the general specification of the conditional variance. Harvey

8

and Siddique (1999) and Bond (1999) suggest alternative forms of the skewness parameter, with the

most obvious being

λt = γ0 + γ1e3t + γ2λt−1. (12)

To ensure that the λt variable stays in the range of [-1,1], either a constrained estimation technique

could be employed or a logathrimic transformation of an unconstrained variable be used.

3.1 Downside Risk Measurement

One of the stated aims of this paper is to highlight the construction of downside risk measures if

the series are found to exhibit skewness. This is because if returns are found to be skewed standard

risk measures may not be appropriate. In addition, the construction of a portfolio based solely on

mean and variance may no longer be consistent with expected utility maximisation. An alternative

measure of risk in the presence of asymmetry was put forward by Markowitz (1991). The target

semi-variance (also known as the second lower partial moment) of a security or portfolio around a

target rate of return (τ) is given by

sv (x) = LPM (x; 2, τ) =

τZ−∞

(x− τ)2 f (x) dx. (13)

The target rate of return is considered as a minimum benchmark that the portfolio should achieve. It

is often specified as a performance benchmark, in the case of funds management, a risk-free interest

rate or some other desirable performance target. As the emphasis of this paper is on conditional

models of returns the conditional semi-variance has an analogous definition of

sv (xt|Ωt−1) = LPM (xt; 2, τ) =

τZ−∞

(xt − τ)2 f (xt|Ωt−1) dxt. (14)

The information set available at time t is denoted by Ωt−1. In calculating the conditional semi-

variance from the estimated conditional distribution, Bond (1999) has shown that the following

expression for a GARCH process, of the type outlined in equations (3) and (4) above, can be

evaluated numerically to provide an estimate of the conditional semi-variance,

sv (xt|Ωt−1) = µ2t

−µtσtZ

−∞f (zt|Ωt−1) dzt + 2µtσt

−µtσtZ

−∞zt f (zt|Ωt−1) dzt + σ

2t

−µtσtZ

−∞z2t f (zt|Ωt−1) dzt, (15)

9

where E (xt|Ωt−1) = µt, σ2t is the variance of the unstandardised conditional density and zt is the

standardised innovation discussed in the sub-section above.

Two points should be noted. Firstly, the term ‘risk measure’ and semi-variance (or its square

root, the semi-standard deviation) are used interchangeably regardless of the form of the underlying

security returns. This usage may be inappropriate when the variable of interest, as in this paper, is

a single security (or set of securities), as it is well known that in a one-factor equilibrium setting the

measure of risk of an individual security is its covariance with the market or beta. It is possible to

consider the construction of lower partial moment betas to estimate an equivalent beta measure of

the downside risk of a security (see Hogan and Warren 1974, Harlow and Rao 1989). However, for the

purposes of this paper we continue to focus on the importance of the semi-variance of a security as

some measure of the riskiness of that security even though the term may be more accurately thought

of as being valid only in the case of a portfolio. If nothing else, this study provides some information

on the nature of the time varying moments (including lower partial moments) of property returns.

Secondly, each security is modelled on an individual basis. No consideration is given to modelling

the joint density of property company returns, even though this may provide interesting insight into

common, industry specific movements in the return process.

An alternative measure of downside risk that is often employed for capital adequacy requirements

is the Value at Risk (vaR). The VaR is simply a quantile of the conditional density function and

is interpreted as the maximum portfolio loss that could occur with a given confidence level. The

modelling approach outlined in this paper could certainly be used in calculating the VaR. However,

the semi-variance was chosen as it is the asymmetric counterpart of the conditional volatility, a

frequently referred to measure in financial economics (particularly with reference to GARCHmodels).

The semi-variance is also a risk measure consistent with a quadratic loss aversion utility function,

which once again is an asymmetric version of the frequently used quadratic utility function, and

may allow for the results of this research to be extended to the task of portfolio optimisation.

4 Data

In order to study the conditional distribution of real estate returns, attention will be focused on

securitised real estate markets (consistent with the work of Young and Graff 1995). This provides

for a larger set of observations than is possible if appraisal based performance measures of the real

10

estate market were used and allows for possible time variation in the higher moments of the density

function to be examined over a range of economic cycles. It also avoids the need to unsmooth

appraisal based indices and hence the uncertainty surrounding the most appropriate method for

correcting the downward bias in volatility observed in these indices.

At present there are around 60 companies included in the real estate sector of the London Stock

Exchange (excluding the AIM-listed stocks) and over 150 equity REITs in the US. However, few

of these companies have been continuously listed for an extensive period. In choosing the data set

a trade-off existed between ensuring a sufficient number of observations were available for analysis

while also ensuring a sufficiently large sample of companies were used to enable valid conclusions to

be drawn. The complete data set chosen for this study is made up of a sample of 20 UK property

companies which have been continually listed since January 1970 and 20 REITS continually listed

since January 1977. This selection of 40 companies provided a reasonable trade-off between the

length of the data (30 years for the UK and 23 years for the US) and number of companies that could

be studied. The UK companies were chosen from those listed in the property section of the Financial

Times industry groupings and the US REITS were selected from those publicly traded companies

listed on the CRSP database which are also members of NAREIT. The names of the companies and

the summary statistics of the continuously compounded monthly returns are shown in the appendix.

The UK data series extend from January 1970 to March 2000, a total of 362 observations. The

REIT series begin in January 1977 and continue until December 2000 (288 observations). The first

three sample moments are reported in Tables 1a and 1b, along with a measure of trading inactivity.

The number in column 5 of each table shows the proportion of all company observations which

record a monthly return of zero. Four UK companies and four REITs have a large element of

non-trading, these companies, Cardiff Properties, Bolton Group International, Jermyn Investments,

Mountview Estates, Pittsburgh and West Virginia, Tarragon Realty Investors, Presidential Realty

Corp (New) and BRT Realty Trust each record at least ten percent of the sample having a zero

monthly return. Ten of the UK companies (half the sample) and 16 REITs record an unconditional

skewness coefficient that is significantly different from zero at a ten percent level of significance. Of

the significant skewness coefficients, six out of the ten UK companies are positive and all of the

REITs are positive2 . However, it is noted that having a high number of non-trading observations2 It is noted that Peiró (1999) is critical of such significance tests, as the rejection may be because the data do not

conform to a normal distribution rather than because the data are skewed. Peiró provides alternative critical values

11

may distort the skewness test results, with greater probability mass occurring at zero than expected

for an actively traded company. To remove this distortive effect companies with a high degree of

non-trading observations are omitted from the analysis. The revised significant skewness figures are

eight UK companies from the 16 remaining in the sample and 12 of the remaining 16 US REITs.

A question which immediately arises is how representative is this group of 40 companies of

listed property vehicles as a whole? Clearly many more companies are currently listed and were

previously listed over this time period, and this limitation must be borne in mind when examining

the conclusions of this paper. A related point is the issue of survivorship bias. As the companies

were chosen on the basis of having been listed continuously over a 30 year period (23 years for

REITs), the characteristics of this group of companies may be somewhat different from the group

of property companies taken as a whole, as the set of all possible companies will include those that

were taken over or failed. In particular, it is expected that the results in this paper may be biased

toward finding a greater level of positive skewness (and correspondingly lower downside risk).

5 Results

The results from applying the models outlined in Section (3) to the data on individual property

company share returns are not shown in this paper because of space limitations but are available on

request from the authors. The models are estimated using maximum likelihood estimation imple-

mented in the GAUSS econometric software package using the MAXLIK routine (Aptech 1996). For

each series a GARCH model with a student’s t distribution, a skewed student’s t distribution with

the skewness parameter remaining constant over the sample periods and the time varying skewness

model are estimated. Unlike in Hansen’s original paper (Hansen 1994), a model which also endo-

genised the degrees of freedom parameter was not estimated. The functional form of the skewness

parameter which proved most effective to estimate was the quadratic form, given in equation (11).

The cubic form of the equation, equation (12), while successful for some of the series, generally

of the sample skewness coefficient based on the t distribution. While the critical values do not cover the sample size

of the present dataset and the appropriate degrees of freedom for the t distribution are unknown in each series, the

results which are presented suggest the five percent critical value for the sample skewness statistic, for the present

study, is around 0.8 or 0.9. If this were correct fewer companies could be considered as having a skewed unconditional

distribution than the ten originally reported.

12

proved difficult to estimate and did not converge in many instances. Because of this only the results

for the quadratic form of the model are reported. Convergence problems were also experienced in

those models where the underlying data series had a high level of zero returns. The eight companies

with at least ten percent of the monthly returns equal to zero were dropped from the sample. Esti-

mation difficulties were also encountered in series nine of the UK data set (Hampton Trust), where

for much of the early part of the sample the company was infrequently traded. To overcome this

limitation the first 159 observations were removed from the sample for this company and the models

estimated over the remaining 203 observations.

The specification of the conditional mean equation for each UK series was modelled as an AR(1)

form. An ARMA(1,1) was tested but found to be unnecessary. For the US data set an ARMA(1,1)

equation was used for the conditional mean, with the MA(1) term an important component of the

model. This may suggest microstructure differences between the two international markets in the

way shares are traded. It is recognised that such parsimony in the specification of the conditional

mean equation may prove inappropriate should the returns process take a more complicated form.

In light of this any findings arising from this study must be interpreted with the caveat that further

work on an acceptable form the conditional mean equation needs to be undertaken. Given that a

simple form for the conditional mean was adopted, large shifts in the conditional mean could not

be explained by the model and a dummy variable was used for any observation greater than four

standard deviations from zero3.

The results from the models are generally consistent with previous applications of GARCH

models to financial data. The sum of the α and β parameters is close to one, indicating a high

level of persistence in volatility. When the results were numerically indistinguishable from one,

the IGARCH restriction4 was imposed and the model was re-estimated. An IGARCH model was

implemented for series 10 and 20 (Helical Bar and Smart & Co.).

In terms of the acceptability of the skewness hypothesis, the first two columns of Table 2 show 13

companies of the 32 remaining in the sample reject the null hypothesis of a conditional t distribution

in favour of the constant skewed t distribution (six UK and seven REITs). This implies that for just3 In fact two dummy variables were used, one for deviations greater than four standard deviations below zero and

one for the equivalent deviations above zero. Despite the large movements in the series, very few observations fell this

far from zero.4An IGARCH restriction imposes that α+ β = 1.

13

under half of the companies analysed, a constant level of skewness appears to be a feature of the

conditional density function. For a further three of those six UK companies, the constant skewness

model is rejected in favour of a model with time variation in the skewness parameter. In addition,

there is weak evidence of time varying skewness in one additional series (Land Securities). For the

US REITS, four companies display time varying skewness when no skewness is detected using a

constant skewness model. Hence, in total 18 companies (seven UK and 11 REITs) display either a

constant degree of skewness in their conditional density function or show an element of predictability

in the skewness of the density function. For the companies which display stronger evidence of time

variation in the third moment, plots of the time varying skewness parameter are shown in Figures 1

and 2.

A possible link between the economic cycle and the skewness parameter is suggested in the

first two charts of Figure 1. The first period of extreme negative skewness occurs from January

1973 to January 1975 (the months numbered 40 to 64 on the horizontal axis). The second occurs

during the recession in the early 1990s (January 1990 is month number 244 and January 1993 is

month number 280). Notice that the effect of the October 1987 stock market crash on the skewness

parameter (month 219) is relatively minor compared to the subsequent recession in the property

market. Large negative values of the skewness parameter have also been recorded in the most recent

data for these two companies. The plot of the time varying skewness parameter is of a slightly

different form in the third plot (Land Securities) shown in Figure 1. The recession in the early 1970s

is clearly visible as is the rapid recovery in the year following the recession. The crash in 1987 also

results is a large skewness parameter but the subsequent property crash in the early 1990s is not as

readily detectable. Another slightly unusual series is shown in plot four in the figure. The skewness

parameter for McKay securities is always positive and while the peaks in the series still correspond

to the economic cycles the direction of the skewness parameter is different from the other series. For

the US REIT series (Figure 2), some cyclical variation appears common between the first two charts

shown (for Presidential and Washington Real Estate).

While the evidence for time variation in the third moment appears minor, this may be partly a

function of the specification of the model in this study. As mentioned above, an alternative specifi-

cation for the pattern of skewness in the data takes a cubic form with an allowance for an element

of time dependence (equation 12). When this model was applied to series 12 (Land Securities), a

14

likelihood ratio test rejected the null of the symmetric t density in favour of the time varying model5,

whereas when the alternative was the quadratic form of the model the null hypothesis could not

be rejected (at the five percent level of significance). However, as convergence difficulties with the

cubic form of the model were encountered, the model was not applied to the full set of companies. If

alternative forms of the specification for the law of motion of the skewness parameter were applied,

the level of support for time variation in higher moments may be different from that found in this

study.

The resulting downside risk estimates calculated for the time varying skewness series are shown

in Figures 2 and 3. Once again the correspondence between the increase in downside risk and the

course of the economic cycle is evident from the individual charts. The semi-standard deviation,

which is the square root of the semi-variance, increases around the time of the early 1970s recession

and also again at the end of the 1980s and into the 1990s. This is particularly evident in the chart

of Creston.

5.1 Skewness and Economic Conditions

To investigate the linkage between the economic cycle and the skewness parameter that was noted

above, the conditional skewness equation (11) was extended to include an industrial production

variable. The rationale being that more favourable economic conditions may lead to a reduction

in the downside risk associated with securitised property returns. Similarly a fall in industrial

production may be associated with increased downside risk (that is the distribution becomes more

negatively skewed). Other researchers have also considered the link between economic variables

and skewness. For example, Bekaert et al. (1998) found GDP growth to be negatively related to

skewness in a large sample of emerging stock markets. Perez-Quiros and Timmermann (2001) find

that small firms exhibit negative skewness from the late expansion to early recession stage of the

economic cycle.

The results obtained from including industrial production in the skewness equation of the model

are not convincing (results not reported). In only three cases out of the entire sample, was the

coefficient of industrial production significant (Hammerson, Land Securities and McKay Securities).

In each of these cases the sign of the coefficient was negative, which is consistent with the previous5The test results are not reported here.

15

literature in this area. However, given the low number of significant responses there appears to

be little or no evidence to support a possible link between the economic cycle and the skewness

coefficient.

When the coefficient of the industrial production term was significant, the industrial produc-

tion variable was included in the conditional mean equation to investigate whether the significant

response to the variable in the conditional skewness equation was not just detecting a misspecifi-

cation of the expected returns equation. On the whole there was no evidence to suggest this, the

industrial production variable was generally not significant when included in the conditional mean

equation (this did not depend of whether industrial production was also included in the conditional

skewness equation). However this does not negate the need for further attention to be devoted to

the appropriate specification of expected returns. Also the conclusion reached here may be sensitive

to alternative specifications for introducing a measure of the economic cycle into the model.

5.2 Skewness and Capitalisation

An important discussion at the beginning of this document raised the link between skewness and

market capitalisation. The findings of Perez-Quiros and Timmermann (2000) that skewness over the

economic cycle is most commonly associated with small capitalisation companies because of difficulty

in accessing capital in times of credit rationing, were reported. An alternative view discussed in the

introduction is that skewness may arise in the returns of commercial property companies (particularly

those in the United Kingdom) because of the option-like payoff associated with long-term, upward

only lease contracts. In this case any skewness found in the data will be independent of company size.

The findings of this paper appear to support the second rather than the first explanation of skewness

in returns. In Table 2a, of the four companies which displayed time variation in skewness, two are

large capitalisation companies and the remaining two are small capitalisation companies, with a

market capitalisation below £60m (a similar result is found if only the constant skewness model

results are examined. For the US REITs, there is a similar lack of association between capitalisation

and the finding of time varying skewness. Interestingly, if only the unconditional skewness measures

reported in Table 1a for the UK had been referred to, the finding would have supported the first

explanation. In Table 1a of the ten companies having a significant unconditional skewness coefficient,

seven of them have market capitalisations below £60m ($90m). However, when the US REIT data

16

is examined, there does not appear to be a link between the unconditional skewness parameter and

capitalisation.

However a point of caution needs to be made at this point, and this refers to the need for further

methodological work on the possible association between the nature of upward-only lease contracts

in the United Kingdom and the finding of skewness in commercial property returns.

6 Conclusion

In this study, the shape of the conditional density function for 20 UK listed property companies and

20 US REITs has been investigated. This was in response to recent findings in the general finance

literature on skewness in stock price returns and also a belief that the nature of lease contracts may

give rise to skewness in commercial property returns. Such investigations are important for issues

such as risk management, with extensive use now being made of Value at Risk measures, knowledge

of the empirical distribution of returns and how this may change over time are vital ingredients in

these calculations.

In order to investigate these issues parametric models of the conditional density function of

securitised property returns were estimated. The use of securitised returns provided a larger and

more consistent dataset than appraisal or valuation based returns series would have allowed. It also

avoided problems associated with the known downward bias in volatility associated with appraisal

based indices. The parametric models chosen are based on the student’s t distribution. While all

tests of symmetry conducted will be sensitive to the base model chosen, the use of the student’s t

distribution is very common in financial modelling and allows for a greater probability mass to be

assigned to the tails of the distribution (than the normal distribution). The student’s t distribution,

a skewed version of the student’s t distribution, and a model which allowed the skewness parameter

in the skewed student’s t distribution to vary over time were applied to 30 years and 23 years of

data, respectively, on the total monthly returns of 20 UK listed property companies and 20 REITs.

Satisfactory time series models could be estimated for 16 of the UK companies and 16 REITs. Of

these companies, the symmetric version of the student’s t distribution was rejected in 13 instances

(11 at five percent significance). Using a five percent significance level, the skewed student’s t model

was rejected in favour of a time varying skewness model in eight cases. The finding of skewness

in real estate assets is broadly consistent with previous work in this field (Lizieri and Ward 2001),

17

however, the limited evidence of time varying skewness appears at odds with the study by Young

and Graff (1995). When the transmission mechanism linking the skewness of property returns to the

economic cycle was examined the changes in industrial production were found to be on the whole

negatively related to skewness. However, in only three companies was the coefficient for industrial

production significant at a five percent level. It was observed that the capitalisation of the property

companies were not related to the finding of skewness being associated with the economic cycle.

This is more in keeping with the option-based arguments relating skewness to the economic cycle

rather than the credit rationing model proposed by Perez-Quiros and Timmermann (2000).

In drawing out the implications of the findings of this study caution must be applied in inferring

too much from a small sample of companies. However, given that over half of the companies, for

which a satisfactory time series model could be obtained, showed either skewness in the conditional

density or time variation in skewness, it implies that risk managers and investors may need to pay

more attention to the distributional characteristics of listed property companies. Furthermore, it

also suggests that the methods by which investment managers construct portfolios of companies

need to be evaluated in light of these findings (however multivariate extensions were not considered

in this paper). In addition, for those stocks which appear to be skewed, it suggests that traditional

theoretical approaches to asset pricing such as the CAPM may not be suitable, and alternative such

as those put forward by Kraus and Litzenberger (1976) or Hogan and Warren (1974) should be

considered. In this regard the paper provides some evidence in support of the findings of Liu et al.

(1992).

18

7 Data Appendix

Table 1aStatistical Summary

Series Mean Std Dev. Skew p value Non-trades Market Cap∗.(%) (%) ($m)

CNC Properties 0.665 11.529 -1.069 0.00 7 74.42Cardiff Property 1.100 9.370 0.269 0.20 11 16.87Cathay Intl Holdings 0.254 13.265 0.466 0.07 6 24.19Creston 0.721 14.031 0.699 0.01 7 23.55B.P.T 1.448 8.774 -0.150 0.32 1 520.04Bolton Group Intl -0.021 17.913 2.877 0.00 15 9.54Brixton Estate 1.203 9.386 -0.033 0.46 1 835.38Hammerson 0.795 9.169 -0.614 0.03 0 1,854.46Hampton Trust 0.958 11.504 0.595 0.03 2 25.40Helical Bar 2.087 15.007 0.865 0.00 3 253.94Jermyn Investments 0.700 11.353 0.432 0.09 19 82.47Land Securities 0.950 8.046 -0.120 0.35 0 5,805.14Ldn Merchant Securities 1.165 11.262 -0.098 0.38 2 709.88MEPC 0.671 10.304 -0.105 0.37 0 2,415.98McKay Securities 1.247 7.499 -0.623 0.02 7 88.98Mountview Estates 1.620 10.012 0.189 0.27 10 164.90Mucklow (A&J) GP 1.335 8.129 0.216 0.25 1 253.62Saville Gordon Estates 1.238 10.682 -0.415 0.09 3 199.64Slough Estates 1.037 9.034 -0.043 0.45 1 2,185.50Smart, J & Co. 1.426 7.465 -0.193 0.27 5 44.92

* - Market capitalisation at 28 December 2000.

19

Table 1bStatistical Summary

Series Mean Std Dev. Skew p value Non-trades Market Cap∗.(%) (%) ($m)

Pittsburgh & West Virginia RR 0.940 6.219 3.940 0.00 15 10.66Tarragon Realty Investors Inc 1.918 30.069 12.285 0.00 17 77.52Cousins Property Inc 2.470 10.023 1.111 0.00 4 1,369.61Vornado Realty Trust 2.227 10.508 1.141 0.00 2 3,327.36Presidential Realty Corp New 1.328 10.186 1.987 0.00 10 3.08Presidential Realty Corp 1.491 10.194 0.610 0.04 7 20.36Alexanders Inc 1.486 11.724 1.905 0.00 3 338.51Thackeray Corp 0.842 13.208 1.282 0.00 8 11.49Urstadt Biddle Properties Inc 0.908 5.946 0.835 0.01 7 38.42First Union Real Estate Eq & MG Inv 0.784 8.640 0.249 0.24 5 108.83Pennsylvania Real Estate Inv Tr 1.406 6.048 0.200 0.29 5 255.32Washington Real Estate Inv Tr 1.491 5.655 0.172 0.31 3 844.22IRT Property Co. 1.448 6.816 0.538 0.06 5 255.93Lomas & Nettleton Mtg Inv 1.013 13.695 1.666 0.00 7 59.50Starwood Hotels & Rest Wldwd Inc 1.389 12.693 1.558 0.00 5 6,828.77HMG Courtland Properties Ltd 0.930 10.794 0.997 0.00 5 9.05BRT Realty Trust 1.435 13.482 1.063 0.00 13 57.33Federal Realty Investment Trust 1.306 5.977 0.003 0.50 3 750.03Archstone Communities Trust 1.769 7.177 1.365 0.00 7 3,153.99Rouse Company 1.578 8.845 0.496 0.08 1 1,752.64

* - Market capitalisation at 29 December 2000.

20

Table 2Hypothesis Test Results - Time Varying Skewness Model

Company Null Symmetry Null Constant Skew[p-value] [p-value]

UK Property CompaniesCNC Prop. 0.45 0.39Cathay Intl. 0.24 0.17Creston 0.02 0.01B.P.T 0.57 0.87Brixton Est. 0.04 0.23Hammerson 0.00 0.01Hampton Trust 0.01 0.75Helical Bar 0.27 0.74Land Sec. 0.14 0.06Ldn Merchant Sec. 0.03 0.57MEPC 0.14 0.41McKay Sec. 0.01 0.07Mucklow 0.18 0.30Saville Gordon Est. 0.99 0.21Slough Est. 0.36 0.25Smart 0.16 0.10US REITsCousins Property Inc 0.00 1.00Vornado Realty Trust 0.10 0.05Presidential Realty Corp 0.30 0.00Alexanders Inc 0.00 0.32Thackeray Corp 0.00 0.28Urstadt Biddle Prop Inc 0.36 0.28First Union Real Estate 0.49 0.91Pennsylvania Real Estate 0.06 0.12Washington Real Estate 0.85 0.00IRT Property Co. 0.23 0.31Lomas & Nettleton Mtg In 0.85 0.05Starwood Hotels & Rest 0.07 0.56HMG Courtland Properties 0.00 0.22Federal Realty Inv Trust 0.50 0.76Archstone Communities Tr 0.00 0.20Rouse Company 0.21 0.58

21

Figure 1: Time Varying Skewness Parameter, Selected UK Property Companies.22

Figure 2: Time Varying Skewness Parameter and Semi-standard Deviation Measure, Selected US

REIT stocks. 23

Figure 3: Semi-standard Deviation Measure, Selected UK Property Companies.24

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