Forward-futures price differences in the UK commercial property market Arbitrage and marking-to-model explanations.

International Review of Financial Analysis 34 (2014) 177–188

Contents lists available at ScienceDirect

International Review of Financial Analysis

Forward–futures price differences in the UK commercial property
market: Arbitrage and marking-to-model explanations
Silvia Stanescu, Radu Tunaru ⁎, Made Reina Candradewi
Kent Business School, University of Kent, CT2 7PE, United Kingdom

a r t i c l e

i n f o

Article history:
Received 9 January 2014
Received in revised form 27 May 2014
Accepted 30 May 2014
Available online 12 June 2014
JEL classification:

C12
C33
G13
G19

a b s t r a c t
In this paper the differences between forward and futures prices for the UK commercial property market are
analyzed, using both time series and panel data. A first battery of tests establishes that the observed differences
are statistically significant over the study period. Further analysis considers the modeling of this difference using
mean-reverting models. The proposed models are then estimated with a number of alternative estimation
methods and second stage statistical tests are implemented in order to decide which model and estimation
method best represent the data.
© 2014 Elsevier Inc. All rights reserved.

Keywords:
Total return swaps and futures
Panel data
Mean-reversion
Markov Chain Monte Carlo

1. Introduction
The difference between forward and futures prices has been given
considerable attention in the finance literature, both from a theoretical
as well as from an empirical perspective, and for various underlying
assets. On the theoretical side, Cox, Ingersoll, and Ross (1981) (CIR)
obtained a relationship between forward and futures prices based solely
on no-arbitrage arguments.1 A series of papers subsequently tested empirically the CIR result(s). Cornell and Reinganum (1981) investigated
whether the difference between forward and futures prices in the foreign exchange market is different from zero. For several maturities
and currencies, they found that the average forward–futures difference
is not statistically different from zero. In addition, they suggested that
earlier studies identifying significant forward–futures differences for
the Treasury bill markets ought to seek explanations elsewhere than
in the CIR framework, since the corresponding covariance terms for
this market were even smaller. French (1983) reported significant

⁎ Corresponding author. Tel.: +44 1227 824608.
E-mail address: R.Tunaru@kent.ac.uk (R. Tunaru).
1
Other early studies that considered the relationship between forward and futures
prices in a perfect market without taxes and transaction costs are Margrabe (1978),

Jarrow and Oldfield (1981) and Richard and Sundaresan (1981).

http://dx.doi.org/10.1016/j.irfa.2014.05.012
1057-5219/© 2014 Elsevier Inc. All rights reserved.

differences between forward and futures prices for copper and silver.
Moreover, he conducted a series of empirical tests of the CIR theoretical
framework and concluded that his results are in partial agreement with
this theory. Park and Chen (1985) also investigated the forward–futures
differences for a number of foreign currencies and commodities and
they pointed out to significant differences for most of the commodities
that they analyzed, but not for the foreign currencies. Also, their empirical tests confirmed that the majority of the average forward–futures
price differences are in accordance with the CIR result.
Kane (1980) tried to explain the differences between futures and
forward prices based on market imperfections such as asymmetric
taxes and contract performance guarantees. Levy (1989) strongly
argued that the difference between forward and futures prices
arises from the marked-to-market process of the futures contract.
Meulbroek (1992) investigated further the relationship between forward and futures prices on the Eurodollar market and suggested that
the marked-to-market effect has a large influence. However, Grinblatt

and Jegadeesh (1996) advocated that the difference between the
futures and forward Eurodollar rates due to marking-to-market is
small. Alles and Peace (2001) concluded that the 90-day Australia
futures prices and the implied forwards are not fully supported by the
CIR model. Recently, Wimschulte (2010) showed that there is no significant statistical or economical evidence for price differences between
electricity futures and forward contracts.

178

S. Stanescu et al. / International Review of Financial Analysis 34 (2014) 177–188

Fig. 1. IPD total return swap rates (mid prices). Notes: The plotted data is from 4 February to 7 July 2009 for the five maturity dates fixed in the market calendar, for the period of study. The
total return swap rates are given as a fixed rate and not as a spread over LIBOR. A negative total return swap rate implies that the underlying commercial property market will depreciate
over the period to the horizon indicated by the maturity of the contract.

The relationship between forward and futures prices as developed
under the CIR framework makes the tacit assumption that futures are
infinitely divisible. Levy (1989) starts with the same set of assumptions
underpinning the CIR model except one. When considering interest
rates, he advocates that, if only the next day's interest rate were deterministic, a perfect hedge ratio using fractional futures positions can be

constructed to replicate the forward. Thus, for Levy (1989) it is only
the interest rate for the next day that is important and not the entire
time path of the stochastic rates. Consequently, for Levy (1989), the forward prices should be equal to futures prices and any empirical findings
regarding actual price differentials are non-systematic and they can
have only statistical explanations. On the other hand, Morgan (1981)
studied the forward–futures differential assuming that capital markets
are efficient and concluded that forward and futures prices must be different. His conclusion is mainly based on the fact that current futures
price depends on the joint future evolution of stochastic interest rates
and futures prices. Polakoff and Diz (1992) argued that due to the indivisibility of the futures contracts,2 the forward prices should be different
from futures prices even when interest rates and futures prices exhibit
zero local covariances. Moreover, they show that the autocorrelation
in the time series of the forward–futures price differences should be
expected. Hence, testing must take into consideration the presence of
autocorrelation. Polakoff and Diz (1992) offered a theoretical explanation that unifies the contradictory theoretical views originated in how
interest rates are negotiated in the model. Their main conclusion is
that it is unnecessary for futures prices and interest rates to be correlated in order to imply that forward prices should be different from futures
prices.
From the review discussed above it appears that the empirical
evidence is mixed and asset class specific. Property derivatives are an
emerging asset class of considerable importance for financial systems.

Case and Shiller (1989, 1990) found evidence of positive serial correlation as well as inertia in house prices and excess returns, implying that
the U.S. market for single-family homes is inefficient. The use of derivatives for risk management in real estate markets has been discussed by
Case, Shiller, and Weiss (1993), Case and Shiller (1996) and Shiller and
Weiss (1999) with respect to futures and options. Fisher (2005)
discussed NCREIF-based swap products, while Shiller (2008) described
the role played by the derivatives markets in general for home prices.
For real-estate there has been a perennial lack of developments of
derivatives products that could have been used for hedging price risk.
The only property derivatives traded more liquidly in the U.S. and the

2
Although the vast majority of literature on futures is based on the assumption of infinite divisibility, Polakoff (1991) discusses the important role played by the indivisibility of
futures contracts.

U.K. are the total return swaps (TRSs), forwards and futures. In the
U.K. commercial property sector for example, all three types of contract
have the Investment Property Databank (IPD) index as the underlying.
Since February 2009 the European Exchange (Eurex) has listed the UK
property index futures. The most liquid derivatives markets on the IPD
UK index are the TRS, which is an over-the-counter market, and the

futures, both with at least five yearly market calendar December
maturities. Any portfolio of TRS contracts can be decomposed into an
equivalent portfolio of forward contracts. Hence, having data on TRS
prices and futures prices opens the opportunity to compare, after
some financial engineering, forward curves with futures curves on the
IPD index. As remarked by Polakoff and Diz (1992) it is difficult to
compare forward and futures prices on a daily basis when forwards
are traded on a non-synchronous basis. By contrast, when forwards
are derived on an implied basis from other instruments then matching
the term-to-delivery is easy.
In this paper the forward–futures price differences are investigated
for the UK commercial property market for all five December market
maturities. To our knowledge, this is the first study that considers the
forward–futures price differences for this important asset class. The
analysis of the difference is particularly important for two main reasons.
Firstly, previous literature addressing the issue for different asset classes
found that the empirical evidence was mixed and asset specific. Therefore, addressing the question for a new asset class is not an exercise of
confirming previous results, but rather a new and important question
in itself, especially since unexplained forward–futures differences can
signal arbitrage opportunities. Secondly, intrinsic characteristics of real

estate as an asset class make the contribution of this paper particularly
relevant, since the underlying (a commercial real estate index in our
case) is likely to be correlated with interest rates. According to the CIR
result, this in turn should drive significant differences between forward
and futures prices, but does this fully explain observed differences or
can these occur, at least partially, due to arbitrage also? This is essentially what our paper aims to address. Furthermore, all previous studies
relied exclusively on time series analysis, whereas in this paper we
also conduct statistical tests for panel data as well as time series tests.
To the authors' knowledge, this is the first study that considers panel
data modeling in this context. Employing panel data has a series of
advantages over basing findings on time series alone.3 For example, it
increases statistical accuracy by increasing the number of degrees of
freedom, which is particularly important for this application which
benefits from having access to a unique OTC data set, with a relatively
limited sample period, but with data available for a number of cross sections. To sum up the contribution of the paper, we analyze the forward–

3

See, for example Hsiao (2003) for a discussion of the advantages of using panel data.

S. Stanescu et al. / International Review of Financial Analysis 34 (2014) 177–188

179

Fig. 2. Eurex futures prices. Notes: The plotted data is from 4 February to 7 July 2009 for the five maturity dates fixed in the market calendar. Futures prices are given on a total return basis
so a futures price of 110 for December 2012 implies that the market expects a 10% appreciation of the commercial property in the UK at this horizon.

futures difference for a new and particularly important asset class,
employing a unique data set and providing panel test results, to our
knowledge, for the first time in this stream of literature.
In addition, several models and estimation methods for the IPD
index are investigated to try to determine which ones best capture the
IPD forward–futures price difference. Empirical properties of realestate indices suggest that the family of mean-reverting models
presented in Lo and Wang (1995) could be suitable for defining our
modeling framework. Shiller and Weiss (1999) pointed out that the
exact models advocated in Lo and Wang (1995) may not be appropriate
for real-estate derivatives since the underlying asset is not costlessly
tradable, and they advocated using a lognormal model combined with
an expected rate of return rather than a riskless rate. Later on, Fabozzi,
Shiller, and Tunaru (2012) designed a way to merge the best of the

two worlds by completing the market with the futures contracts that
are used directly to calibrate the market price of risk for the realestate index and hence, indirectly fixing also the risk-neutral pricing
measure which can be then applied for pricing other derivatives. Therefore, the first model we propose in Section 4 below will be a slight
modification of the Lo and Wang (1995) trending OU process, where
futures prices are used to calibrate the market price of risk, as pointed
out in Section 4.3.
Real-estate prices exhibit serial correlation leading to a high degree
of predictability, up to 50% R-squared for a short term horizon. Moreover, it has been documented that returns on real-estate indices are
positively autocorrelated over short horizons and negatively correlated
over longer horizons, see Fabozzi et al. (2012).
The remainder of the paper is organized as follows: Section 2 focuses
on describing the data, Section 3 contains the analysis that is real-estate
model-free and the testing methodology covering panel data as well,
Section 4 contains the modeling approach taken for the commercial
property index (including alternative estimation methods for the proposed models) and it also describes the second stage, model based
tests. Section 5 concludes.
2. Data description
For the empirical analysis of the differences between the forward
and futures prices on the IPD4 UK property index two types of tests
are performed. Firstly, it is investigated whether the observed difference

between the forward and futures prices is statistically different from
zero. Secondly, a number of established continuous time models combined with various methods of estimation are compared in order to
4
IPD stands for Investment Property Databank. A detailed description of the data is given in Section 3.1 below.

identify the model or models that are able to best capture this difference. Using the previously defined notation, there are: n = 5 different
maturities and N = 71 daily observations for each maturity. The data
needed for this study contains IPD property futures prices, the IPD
total return swap (TRS) rates, the IPD index, and also the GBP interest
rates needed to calculate discount factors. Futures prices have been obtained from the European Exchange (Eurex5), the property TRS data
(the fixed rate) has been provided by Tradition Group, a major dealer
on this market and the IPD index was sourced from the Investment
Property Databank (IPD6). In addition, the UK's interest rates have
been downloaded from Datastream. Due to the limited availability of
the property futures and TRS data, the sample period used is daily
from 4 February 2009 until 7 July 2009. It generates 71 property futures
daily curves and 71 sets of TRS rates with up to five years maturity (the
first maturity date is 31 December 2009, the second maturity date is 31
December 2010, the third maturity date is 31 December 2011, the
fourth maturity date is 31 December 2012, and the fifth maturity date
is 31 December 2013).
The evolution of the TRS series is depicted in Fig. 1 and one could see
that, for our period of investigation, most of the IPD TRS rates are negative for the first, second, and third maturity dates. We note that the total
return swap rates are given as a fixed rate and not as a spread over
LIBOR.7 A negative total return swap rate implies that the underlying
commercial property market will depreciate over the period to the horizon indicated by the maturity of the contract. For the fourth and fifth
maturity dates, the TRS rates are mostly positive. In addition, there is a
dramatic increase of the fixed rate at the end of February 2009, possibly
due to the rollover off the futures contracts in March combined with the
publication of the IPD index for the year ending in December 2008. The
property futures prices, quoted on a total return basis, are illustrated in
Fig. 2. Futures prices are given on a total return basis so a futures price of
110 for December 2012 implies that the market expects a 10% appreciation of the commercial property in the UK at this horizon.
From the daily TRS prices for the market five yearly maturities one
can reverse engineer the equivalent no-arbitrage forward prices for
the same maturities. The equivalent fair property forward prices are derived daily from 4 February until 7 July 2009, with maturities matching
the futures contracts maturities.

5
See www.eurexchange.com for more information on Eurex. IPD UK futures contracts
started on 4 February 2009.
6
See www.ipd.com for more information on IPD.
7
The TRS rate was initially equal to LIBOR plus a spread, but due to disagreements over
what the spread should have been equal to, the property TRS swap rate is now equal to a
fixed rate, which is established upfront and can be negative.

180

S. Stanescu et al. / International Review of Financial Analysis 34 (2014) 177–188

Fig. 3. The fair prices of property forwards. Notes: The plotted data is from 4 February to 7 July 2009 for the five maturity dates fixed in the market calendar, for the period of study. The fair
property forward prices are reversed engineered from the corresponding portfolio of total return swaps.

Using data on the fixed rate of total return swap, UK interest rates
and the monthly IPD all property total return index, the fair forward
prices for property derivatives can be obtained. The following are the
steps in constructing the fair prices of property forwards:
1. Calculate the adjusted discount factor.
adj discount factor ¼ discount factor

ðT−t Þ
:
intðT−t Þ þ 1

2. The gross mid TRS is obtained from:
gross mid TR ¼ sum of adj df  the fixed rate of TRSðT; t Þ:
3. The projected calendar mid TR is calculated (in percentages):
projected calendar mid TR ¼ Δgross mid TR 

adj df
:
100

4. We can now obtain the projected index level as:
projected index level ¼ previous index level
 ð1 þ projected calendar midÞ:
5. Finally, the fair forward prices are generated (quoted in bp).
fair forward pricesðT; tÞ ¼

projected index levelðTÞ
 100:
projected index levelðT−1Þ

The fair prices of property forwards thus obtained are illustrated in
Fig. 3.
A closer examination of Figs. 2 and 3 shows that the jumps in IPD
futures and (reversed engineered) forward prices are not contemporaneous, but rather that that futures prices jump up at the end of March,
whereas the upward jump in February is visible for the forward prices.
While futures prices jump due to the roll-over of the contract at the
end of March, forwards – which are reversed engineered based on TRS
and UK interest rate data as explained above – jump in February due
to jumps in TRS rates, which in turn, should be due to the publication
of the new IPD index. This results in the differences between forward
and futures, depicted in Fig. 4, being higher than usual for almost one
month.8
The descriptive statistics of the TRS rates are reported in Table 1. The
mean values are mostly negative, signifying that the underlying commercial property market will depreciate over the period to the horizon
8
These abnormal forward–futures differences do not drive the significance of the results in this paper. The results remain significant even after changing the sample and eliminating the month of abnormally high forward futures differences.

indicated by the maturity of the contract; the mean for the first maturity
date is −17.80% and the means are increasing with maturity. The excess
kurtosis is negative for all five futures contracts and the first year TRS
contract and it is positive for the remaining four series of TRS rates.
The skewness values have negative signs, except for the four year futures contract, implying that the distributions of the data are skewed
to the left. The descriptive moments of the differences between forward
and futures prices on the IPD commercial index are also reported in
Table 1. On average, the differences for the first three maturities are positive, while for the fourth and fifth maturities they are negative.
It can be seen in Table 1 that the futures contract for the fifth
maturity date appears to have the highest mean. The highest standard
deviation is shown in the futures contract for the second maturity
date. Similarly to TRS data, futures prices exhibit skewness and fat tail
characteristics.
3. Model-free analysis of forward–futures differences
Let S(t) be the spot value of the IPD index at time-t, F(t, Ti), the associated time-t forward price with maturity Ti, f(t, Ti) the time-t futures
price with maturity Ti and D(t, Ti) the stochastic discount factor at
time t for maturity Ti. Then B(t,Ti) = EQ
t (D(t,Ti)) is the time-t zerocoupon bond price, with maturity Ti, where the expectation is taken
under a risk-neutral measure Q.
There is a model-free relationship between forward and futures
prices given by9:

F ðt; T Þ−f ðt; T Þ ¼

covQt ðSðT Þ; Dðt; T ÞÞ
E Qt ðDðt; T ÞÞ

ð1Þ

which holds for any maturity T and at any time 0 ≤ t ≤ T. This fundamental relationship opens up the first line of investigation for testing
whether the differences between forward and futures prices are statistically different from zero.
3.1. Testing methodology
First, the null hypothesis that the difference between the market TRS
equivalent forward prices and market futures prices is significantly
different from zero is tested. If this hypothesis is rejected, then in the
second stage a series of models and estimation methods are employed
for the terms on the right hand side of the fundamental relationship
given by Eq. (1). The aim in the second stage is to decide on the capability of various models to appropriately capture the dynamics of the index
S and the discount factor D.
9

See Shreve (2004, p. 247).

S. Stanescu et al. / International Review of Financial Analysis 34 (2014) 177–188

181

Fig. 4. The difference between forward and futures prices. Notes: The plotted data is from 4 February to 7 July 2009 for the five maturity dates fixed in the market calendar, for the period of
study. The fair property forward prices are reversed engineered from the corresponding portfolio of total return swaps.

For the first stage analysis, the following regression model is fitted
for each maturity date Ti, i∈{1,2,…,5}:
Fðt; T i Þ ¼ α0i þ β0i f ðt; T i Þ þ εti

ð2Þ

Table 1
Descriptive statistics for total return swap rates, Eurex futures prices and the forward–
futures differences.
Maturity dates
31-Dec-09

31-Dec-10

31-Dec-11

31-Dec-12

31-Dec-13

Total return swaps
Mean
Standard deviation
Skewness
Excess kurtosis

−0.178
0.0217
−0.6925
−0.5131

−0.0971
0.0548
−1.3581
0.1383

−0.0389
0.0521
−1.4446
0.2633

−0.0056
0.0401
−1.4177
0.2135

0.0138
0.0336
−1.3889
0.1585

Futures prices
Mean
Standard deviation
Skewness
Excess kurtosis

81.1982
2.6558
−0.0992
−1.5313

94.275
10.6358
−0.4889
−1.7844

106.1732
5.1369
−0.5411
−1.6426

111.7035
1.2792
0.164
−0.6302

113.5915
3.7762
−0.2771
−1.6544

Forward–futures differences
Mean
1.1018
Standard deviation
1.3846
Kurtosis
0.3093
Skewness
1.2702

4.2432
7.2574
0.9912
1.6562

2.0765
3.7344
0.8006
1.5582

−1.599
1.0573
0.0824
0.3179

−3.7103
3.0863
−1.6454
0.2274

F ðt; T i Þ ¼ α0 þ β0 f ðt; T i Þ þ εti

Notes: The descriptive statistics are of the total return swap rates, futures prices and
forward–futures differences on the IPD UK All Property index. Daily mid prices are used
for calculation for the period 4 February 2009 to 7 July 2009 for the five market calendar
maturities, namely December 2009, December 2010, December 2011, December 2012
and December 2013. The forward prices used here are the synthetic fair prices derived
from total return swap rates, synchronous with the futures prices.

Table 2
ADF test for the forward and futures prices.
Maturity date

31 December 2009
31 December 2010
31 December 2011
31 December 2012
31 December 2013

(with Ti fixed for each of the five time series regressions) and test
whether α0i = 0 and β0i = 1. If the null hypothesis cannot be rejected,
then one can conclude that the difference between forward and futures
prices is due to noise. If, however, the null is rejected, we then proceed
to the second stage of our analysis. The same econometric analysis described above from a times series point of view, can also be performed
using panel data. Using panel data has a series of advantages.10 Firstly,
it enables the analysis of a larger spectrum of problems that could
not be tackled with cross-sectional or time series information alone.
Secondly, it generally results in a greater number of degrees of freedom
and a reduction in the collinearity among explanatory variables, thus
increasing the efficiency of estimation. Furthermore, the larger number
of observations can also help alleviate model identification or omitted
variable problems.
The regression equation in Eq. (2) is rewritten for our panel data as:

Forward

Futures

Level

First differences

Level

First differences

−1.3453
−1.9185
−2.0106
−3.4368⁎⁎

−8.3646⁎⁎⁎
−8.4575⁎⁎⁎
−8.6799⁎⁎⁎
−5.4320⁎⁎⁎
−6.4863⁎⁎⁎

−0.8098
−1.3019
−1.3278
−2.3581
−1.2087

−5.1662⁎⁎⁎
−8.3168⁎⁎⁎
−4.3724⁎⁎⁎
−5.9891⁎⁎⁎
−12.1882⁎⁎⁎

−0.9618

Notes: Augmented Dickey–Fuller (ADF) test results for the UK IPD commercial property
index forward and futures prices. The test is performed for both the data in levels as
well as for the first differenced data. The optimum number of lags used in the ADF test
equation is based on the Akaike Information Criterion (AIC). The data is from 4 February
to 7 July 2009 for the five maturity dates given in the first column.
⁎ Denotes significance at the 10% level.
⁎⁎ Denotes significance at the 5% level.
⁎⁎⁎ Denotes significance at the 1% level.

ð3Þ

with i∈{1,2,…,5} and t∈{1,2,…,71}.
More variations of a panel regression exist, the simplest one being
the pooled regression, described above in Eq. (3), which implies estimating the regression equation by simply stacking all the data together,
for both the explained and explanatory variables. Furthermore, the
fixed effects model for panel data is given by:
F ðt; T i Þ ¼ α0 þ β0 f ðt; T i Þ þ αi þ υit

ð4Þ

where αi varies cross-sectionally (i.e. in our case it is different for each
maturity date Ti), but not over time. Similarly, a time-fixed effects
model can be formulated, in which case one would need to estimate:
F ðt; T i Þ ¼ α0 þ β0 f ðt; T i Þ þ λt þ υit

ð5Þ

where λt varies over time, but not cross-sectionally. The fixed effects
model and the time-fixed effects model, as well as a model with both
fixed effects and the time-fixed effects, will be analyzed. One can test
whether the fixed effects are necessary using the redundant fixed
effects LR test.
For the panel data random effects model the regression specification
is given by:
F ðt; T i Þ ¼ α0 þ β0 f ðt; T i Þ þ εi þ υit

ð6Þ

where εi is now assumed to be random, with zero mean and constant
variance σ 2ε , independent of υit and f(t,Ti). Similarly, a random timeeffects model can be formulated in the context of this paper as:
F ðt; T i Þ ¼ α0 þ β0 f ðt; T i Þ þ εt þ υit :
10

See also Baltagi (1995), Hsiao (2003).

ð7Þ

182

S. Stanescu et al. / International Review of Financial Analysis 34 (2014) 177–188

Table 3
F-test for time series data.

Maturity date

F-test
α0i = 0 and β0i = 1

t-test
α0i = 0

t-Test
β0i = 1

Durbin–Watson statistic

31/12/2009
31/12/2010
31/12/2011
31/12/2012
31/12/2013

83.5072⁎⁎⁎
49.4309⁎⁎⁎
23.8320⁎⁎⁎
31.1717⁎⁎⁎
158.2534⁎⁎⁎

2.2788⁎⁎
1.3366
1.1492
0.1737
0.2345

−12.9217⁎⁎⁎
−9.9426⁎⁎⁎
−6.9033⁎⁎⁎
−7.8919⁎⁎⁎
−17.7483⁎⁎⁎

1.9905
2.0595
2.0613
2.2589
2.7247

Notes: F-Test t-test and Durbin–Watson statistic results for the regression in the Eq. (2) for the property forward and futures data from 4 February until 7 July 2009 for the five maturity
dates given in the first column. For the F-test, the null hypothesis is that the difference between the forward and futures prices is just noise (i.e. α0i = 0 and β0i = 1).
⁎ Denotes significance at the 10% level.
⁎⁎ Denotes significance at the 5% level.
⁎⁎⁎ Denotes significance at the 1% level.

Table 4
t-Statistics for the differences between forward and futures prices.
t-Statistic

Maturity dates
31-Dec-09

31-Dec-10

31-Dec-11

31-Dec-12

31-Dec-13

4 Feb–7 Jul 2009
2 Apr–7 Jul 2009

6.7060⁎⁎⁎
6.695⁎⁎⁎

4.9265⁎⁎⁎
6.826⁎⁎⁎

4.6852⁎⁎⁎
4.612⁎⁎⁎

−12.741⁎⁎⁎
−20.003⁎⁎⁎

10.1291⁎⁎⁎
−20.261⁎⁎⁎

Notes: The values of the t-test are computed for the differences between forward and futures prices on the UK IPD commercial property index, using data from 4 February to 7 July 2009 and
from 2 April to 7 July, respectively, for the five maturity dates given in the second row.
⁎ Denotes significance at the 10% level.
⁎⁎ Denotes significance at the 5% level.
⁎⁎⁎ Denotes significance at the 1% level.

Again, random effects and random time-effects models, as well as a
two-way model which allows for both random effects and random
time-effects, can be estimated. Furthermore, it is important to test
whether the assumption that the random effects are uncorrelated
with the regressors is satisfied.
For the second stage analysis several models are employed for the
dynamics of the IPD index S. If the analysis is conditioned on knowing
the bond prices, the RHS of identity (1) can be expressed as:
covQt ðSðT Þ; Dðt; T ÞÞ
EQt ðDðt; T ÞÞ

¼

Sðt Þ
Q
−Et ðSðT ÞÞ:
Bðt; T Þ

ð8Þ

Based on Eqs. (1) and (8), it is evident that for testing purposes the
following regression is useful:
2

3

Q
6 Sðt Þ
7
−Et;m ðSðT ÞÞ 5 þ ut;m
F ðt; T Þ− f ðt; T Þ ¼ α þ β4
Bðt; T Þ |fflfflfflfflfflffl
ffl{zfflfflfflfflfflfflffl}

ð9Þ

f m ðt;T Þ

and test whether α = 0 and β = 1, for each model m. For each model m,
failing to reject the null hypothesis implies that this particular model is
suitable for describing the dynamics of the underlying IPD index. Upon
estimation of all the parameters of each model m,11 the regression given
in Eq. (9) is fitted. The competing models and methods of estimation are
compared with respect to whether β is significant and also considering
the R2 measure of goodness-of-fit.
3.2. Model-free analysis
Having both series of forward and futures prices available allows
direct testing of whether the forward–futures difference time series
diverges significantly away from zero. Before running the regressions
in Eqs. (2)–(7), the forward and futures price series are tested for
stationary using the Augmented Dickey–Fuller (ADF) test. The results
are reported in Table 2.
11
The specific stochastic models and parameter estimation methods employed in this
paper are described in Section 4 below.

As it can be seen in Table 2, most of the ADF results show that the
forward series for the first, second, third, and fifth maturity dates are
non-stationary while the forward series for the fourth maturity date is
stationary at the 5% significance level. In addition, the ADF test indicates
that the futures series for all maturity dates are non-stationary. Furthermore, the stationarity of the first differenced data is also investigated.
According to Table 2, the forward and futures series for all maturity
dates are stationary in the first differences.
Since most of the data is found to be non-stationary in levels and stationary in the first differences, the remaining analysis is performed on
the first differenced data. The null hypothesis H0: α0i = 0 and β0i = 1
vs. H1: α0i ≠ 0 or β0i ≠ 1 is tested using an F-test and the results can
be found in Table 3.
The F-test results presented in Table 3 show that the null hypothesis
for all maturity dates could be rejected at the 1% significance level. This
implies that the difference between forward and futures is not just
noise.12 The same conclusion is reached if we analyze the values of the
t-statistics for the forward–futures differences reported in Table 4. The
results in Table 4 show that the significance of the forward–futures
difference is not driven by the month of abnormally high differences: indeed, the difference is still significant when a shorter sample period – 2
April to 7 July 2009 – is considered, which now eliminates the abnormal
sub-period from end February to end March.
As a robustness check of our time-series results, particularly important given the relatively limited sample available for the time-series
analysis (i.e. 71 observations), we also test whether the differences
between forward and futures prices are significant using a panel regression. To choose an appropriate specification for the panel regression, we
first test whether the fixed effects are necessary using the redundant
fixed effects LR test. The results of this test are reported in Table 5.
From the test results reported in Table 5, it appears that a model with
fixed time effects only is most supported by the data in this research.
Furthermore, a random effects model may be appropriate and this is
tested using the Hausman test; the results of this test are also reported
12
In addition, the diagnostic statistics for these regressions are investigated and the
Durbin–Watson test statistic results are reported in Table 3. For all but the fifth maturity
date there is no autocorrelation in the regression errors. Furthermore, the individual ttests for the individual hypotheses α0i = 0 (vs. α0i ≠ 0) and β0i = 1 (vs. β0i ≠ 1) are also
reported. We find that β is always different from 1 and that α is generally not different
from zero (with the exception of the first maturity).

S. Stanescu et al. / International Review of Financial Analysis 34 (2014) 177–188
Table 5
Tests for determining the most suitable panel regression model.
Test

183

Table 6
The F-test for panel data.
Value

Redundant fixed effects test
Cross-section F
Cross-section Chi-square
Period F
Period Chi-square
Cross-section/period F
Cross-section/period Chi-square

1.0925
5.5181
2.2062⁎⁎⁎
154.1939⁎⁎⁎
2.1450⁎⁎⁎
157.7444⁎⁎⁎

Hausman test
Cross-section random
Period random
Cross-section and period random

3.0341⁎
0.0003
0.1690

Notes: The redundant fixed cross-section effects test has a panel regression with fixed time
(period) effects only under the null. Both the F and the Chi-square version of the test are
reported. The redundant fixed time (period) effects have a panel regression with fixed
cross-section effects only under the null. Both the F and the Chi-square version of the
test are reported. For the random effects test (i.e. Hausman test) the null hypothesis in
this case is that the random effect is uncorrelated with the explanatory variables. The
panel data is from 4 February to 7 July 2009 for five maturities, namely December 2009,
December 2010, December 2011, December 2012 and December 2013.
⁎ Denotes significance at the 10% level.
⁎⁎ Denotes significance at the 5% level.
⁎⁎⁎ Denotes significance at the 1% level.

Test statistic

Value

F-statistic
Durbin–Watson statistic

237.3960⁎⁎⁎
2.1419

Notes: F-test and Durbin–Watson statistic results for the regression in Eq. (6) for the property forward and futures panel data from 4 February to 7 July 2009, using the cross-section
random effects specification. For the F-test, the null hypothesis is that the difference
between the forward and futures prices is just noise (i.e. α0 = 0 and β0 = 1).
⁎ Denotes significance at the 10% level.
⁎⁎ Denotes significance at the 5% level.
⁎⁎⁎ Denotes significance at the 1% level.

where γ ≥ 0,σ ≥ 0,μ0 ,μ ∈ R. Solving the SDE in Eq. (10) leads to the
closed form solution
pðt Þ ¼ μ 0 þ μt þ ½pð0Þ−μ 0 Š expð−γt Þ
Zt
expð−γðt−vÞÞdW ðvÞ
þσ

ð11Þ

v¼0

for any 0 ≤ t ≤ T, where the model parameters will be estimated (using
both maximum likelihood and MCMC) using the IPD index data, as
detailed below. The model-implied (theoretical) futures price can be
now be derived in closed-form as outlined in Appendix A:
2

in Table 5. Based on these results, the random effect model is to be
preferred in this case.
Next, the F-test statistic for multiple coefficient hypotheses is computed using the panel regression random effect specification; the results
are reported in Table 6.
According to Table 6, the F-values are significant at the 1% level. The
null hypothesis (H0: α0 = 0 and β0 = 1) can be strongly rejected and
therefore the differences between forward and futures are not just
noise in the panel data.13
4. Analysis with parametric models
Having shown in the previous section that IPD forward–futures
price differences are statistically significant, one-factor and twofactor mean-reverting models that seem suitable for this asset class
are studied in this section. The models are subsequently coupled
with two different methods of parameter estimation – maximum
likelihood (ML) as well Markov Chain Monte Carlo (MCMC) for
which various quantiles from the distribution of parameters are estimated – the model parameters are calibrated to IPD index data, the
model futures and forward prices are thus calculated and finally, a
number of statistical tests are implemented in order to see which
model and estimation method best captures the empirical evolution
of the IPD forwards and futures.

f univ ð0; T Þ ¼


σ
ησ
exp μ 0 þ μT−μ 0 expð−γT Þ−
4γ
γ


ησ
þ expð−γT Þ ln ðSð0ÞÞ þ
Þð1− expð−2γT ÞÞ
γ

where η is the market price of risk, which shall be calibrated following
standard practice, by minimizing the squared difference between the
market and model (theoretical) futures prices.
As remarked in Lo and Wang (1995), although this specification is a
valid modeling starting point, it has an important disadvantage in that
the autocorrelation coefficients of continuously compounded τ-period
returns can only take negative values.15 A more flexible approach, also
proposed in Lo and Wang (1995), is the bivariate trending OU process,
a natural extension of the univariate version above.
We propose the following version of their bivariate model, to suit
the application in this paper:
dqðt Þ ¼ ½−γqðt Þ þ λr ðt ފdt þ σdW S ðt Þ

ð13Þ

drðt Þ ¼ δðμ r −rðt ÞÞdt þ σ r dW r ðt Þ

ð14Þ

where dWS(t)dWr(t) = ρdt and the second stochastic factor on which
the log-price of the underlying depends is the short interest rate r(t).16
Following the derivations and risk neutralization performed in
Appendix B, the futures price is obtained in closed form:

4.1. Mean-reverting models
f bivar ð0; T Þ ¼ exp C 2 ðT Þ þ
Here a slight variation of the trending (mean-reverting) OU
process presented in Lo and Wang (1995) is considered as follows:
let p(t) = ln(S(t)); p(t) = q(t) + (μ0 + μt),14 where the dynamics
of q(t) under the physical measure P are described by the equation:
dqðt Þ ¼ −γqðt Þdt þ σdW ðt Þ
13

In addition, the values of the Durbin–Watson test show that there is no autocorrelation in the panel regression errors.
14
To simplify notation, we suppress model subscripts, univ and bivar for the univariate
and bivariate models, respectively, unless where absolutely necessary.

!
2
σ y ðT Þ
2

ð15Þ

15
The continuously compounded τ-period returns, computed at time t, are defined as
rτ(t) = p(t) − p(t − τ). The autocorrelation function of the returns process employed here
is the same as in Lo and Wang (1995) – see their expression (A3) – namely:

corruniv ðr τ ðt 1 Þ; r τ ðt 2 ÞÞ ¼ −

ð10Þ

ð12Þ

1
2
exp½−γðt 2 −t 1 −τފ½1− expð−γτފ ≤0;
2

for any t1,t2, and τ such that t1 ≤ t2 − τ, to ensure that returns are non-overlapping.
16
The expression for the correlation of τ-period returns corrbivar(rτ(t1),rτ(t2)) for any t1,
t2, and τ such that t1 ≤ t2 − τ, is mathematically more complex and thus excluded here.
However, it can be shown that for certain values of the model parameters, the bivariate
model, unlike the univariate model outlined above, is more flexible and can allow for both
positive and negative autocorrelations.

184

S. Stanescu et al. / International Review of Financial Analysis 34 (2014) 177–188

Table 7
Models and estimation methods.
Name

Model

Estimation method

univ_ML
univ_MCMC_mean
univ_MCMC_2.5q
univ_MCMC_97.5q
bivar_ML
biv_MCMC_mean
biv_MCMC_2.5q
biv_MCMC_97.5q

Univariate time-trending OU: dq(t) = −γq(t)dt + σdW(t)

Exact maximum likelihood
Markov Chain Monte Carlo (MCMC), mean parameter estimates
MCMC, using the 2.5th quantile of the distribution of estimates
MCMC, using the 97.5th quantile of the distribution of estimates
Exact maximum likelihood
MCMC, mean parameter estimates
MCMC, using the 2.5th quantile of the distribution of estimates
MCMC, using the 97.5th quantile of the distribution of estimates

dqðt Þ ¼ ½−γqðt Þ þ λr ðt ފdt þ σdW S ðt Þ
Bivariate time-trending OU: drðt Þ ¼ δðμ r −r ðt ÞÞdt þ σ r dW r ðt Þ
dW S ðt ÞdW r ðt Þ ¼ ρdt

Notes: q(t) is the de-trended log price process for the underlying S(t), the IPD UK commercial property price index: p(t) = ln(S(t)); p(t) = q(t) + (μ 0 + μt). r(t) denotes the short interest rate. For both models, the futures price is obtained as: fm(0,T) = E Q
0,m(S(T)) where m = univ or bivar, for the two models, respectively, Q is the martingale pricing measure, and
T is the futures maturity time.

Table 8
Parameter estimates.
Parameter/model

univ_ML

univ_MCMC_mean

univ_MCMC_2.5q

univ_MCMC_97.5q

μ0
μ
γ
σ
η (average)

–
0.4117
0.0591
0.0384
−37.2076

4.221
0.3121
1.559
0.0178
−962.8572

1.35
0.2657
0.1892
0.0131
−259.2317

4.463
0.7051
1.979
0.0238
−944.3792

Notes: Parameter estimates for the univariate OU model, obtained using likelihood (ML) (column 2), and Markov Chain Monte Carlo (MCMC) (columns 3–5), with mean (column 3), 2.5th
quantile (column 4) and 97.5th quantile (column 5) parameter estimates. The data used estimating the model parameters with these alternative estimation methods contains monthly log
prices on the IPD index, observed over the period between December 1986 and January 2009, and totalling 266 historical observations.

with




σ η1
σ η1
þ ln ðpð0ÞÞ þ
expð−γT Þ
C 2 ðT Þ ¼ μ 0 þ μT−
γ
γ


C
r ð0Þ−C 1
ð expð−δT Þ− expð−γT ÞÞ
þ λ 1 ð1− expð−γT ÞÞ þ
γ
γ−δ

qffiffiffiffiffiffiffiffiffiffiffiffiffi 
where C 1 ¼ μ r − σδr ϱη1 þ 1−ϱ2 η2 , η = (η1η2)T is the market price
of risk (now bi-dimensional) and
2
σ y ðT Þ

"
2
2
σ
λσ r
1− expð−2δT Þ
ð1− expð−2γT ÞÞ þ
¼
2γ
2δ
ðγ−δÞ2
#
1− expð−2γT Þ 2ð1− expð−ðγ þ δÞT ÞÞ
−
þ
2γ
γþδ
"
#
2ρλσσ r 1− expð−ðγ þ δÞT Þ 1− expð−2γT Þ
−
:
þ
γ−δ
γþδ
2γ

4.2. Estimation
In order to be able to use the models enumerated above their parameters should be calibrated first. The parameters of the continuous time
models specified in Eq. (10) and Eqs. (13)–(14) can be estimated from
the monthly log prices on the IPD index, observed over the period between December 1986 and January 2009, and totalling 266 historical
observations. The estimates then will be carried forward for analyzing
the differences between the forward and futures on IPD starting from
February 2009.
4.2.1. Maximum likelihood
When feasible, parametric inference for diffusion processes from
discrete-time observations should employ the likelihood function,
given its generality and desirable asymptotic properties of consistency
and efficiency (Phillips & Yu, 2009). The continuous time likelihood
function can be approximated with a function derived from discretetime observations, obtained by replacing the Lebesgue and Ito integrals
with the Riemann–Ito sums. Remark that this approach gives reliable

results only when the observations are spaced at small time intervals.
When the time between observations is not small the maximum likelihood estimator can be strongly biased in finite samples.17
First de-trend the log price data by estimating the regression:
pt k ¼ μ 0 þ μt k þ ut k

ð16Þ

and subsequently work with the residuals from this equation, where
1
k = 1, 2, …, 266 and tk = kτ, with τ ¼ 12
for monthly returns. The
(exact) discretization of Eq. (11) leads to:
ut k ¼ cutk−1 þ εt k

ð17Þ
tk

where c = exp(− γτ) and ε tk ¼ σ ∫ expð−γðt k −sÞÞdW ðsÞ; εt k  N
t k−1



2
0; σ2γ ð1− expð−2γτÞÞ .

Maximum likelihood estimation of the discrete-time model in
Eq. (17) gives18 c = 0.995086 and the standard deviation of εt k as 0.011.
The exact discretization of the two-factor model given by
Eqs. (B.7)–(B.8) (given in Appendix B for lack of space) is:
qt k ¼ αq þ βq qt k−1 þ φr t k−1 þ εq;t k
r t k ¼ αr þ βr rt k−1 þ εr;t k

ð18Þ

where, for reasons of space, the expressions for the parameters as well
as the distribution of the error terms in Eq. (18) are only given in
Appendix C.
4.2.2. Markov Chain Monte Carlo (MCMC)
Despite its desirable asymptotic properties, ML estimates of the parameters of a continuous-time model based on discrete-sampled data

17
Further discussion is given in Dacunha-Castelle and Florens-Zmirou (1986), Lo
(1988), Florens-Zmirou (1989), Yoshida (1990) and Phillips and Yu (2009).
18
To check the stability of the parameter estimates, the estimation above is repeated
using a larger sample, namely Dec 1986 to Oct 2010, with an increased sample size of
287 monthly observations. The parameter estimates do not change much.

S. Stanescu et al. / International Review of Financial Analysis 34 (2014) 177–188

185

Table 9
Model comparison — time series data.
Test/model

univ_ML

univ_MCMC_mean

univ_MCMC_2.5q

univ_MCMC_97.5q

1st maturity
F-test
t-Test for β
R_squared

120,757.2⁎⁎⁎
1.4369
0.0291

112,810.4⁎⁎⁎
−0.8494
0.0103

121,757.8⁎⁎⁎
1.5724
0.0346

99,747.38⁎⁎⁎
−1.9758⁎
0.0535

2nd maturity
F-test
t-Test for β
R_squared

5946.6⁎⁎⁎
2.0079⁎⁎
0.0552

2705.565⁎⁎⁎
0.5759
0.0048

6043.822⁎⁎⁎
2.0585⁎⁎
0.0579

411.4135⁎⁎⁎
0.5991
0.0052

3rd maturity
F-test
t-test for β
R_squared

30,397.9⁎⁎⁎
4.2573⁎⁎⁎
0.208

4200.665⁎⁎⁎
4.2845⁎⁎⁎
0.2101

31,643.03⁎⁎⁎
4.3118⁎⁎⁎
0.2123

63.6275⁎⁎⁎
15.7831⁎⁎⁎
0.7831

4th maturity
F-test
t-Test for β
R_squared
5th maturity
F-test
t-Test for β
R_squared

216,988.5⁎⁎⁎
2.2969⁎⁎
0.071
21,886.5⁎⁎⁎
31.9967⁎⁎⁎
0.9369

2426.505⁎⁎⁎
7.6797⁎⁎⁎
0.4608
24,054.33⁎⁎⁎
17.6248⁎⁎⁎
0.8182

214,077.2⁎⁎⁎
0.9275
0.0123
14,288.12⁎⁎⁎
25.8627⁎⁎⁎
0.9065

2354.060⁎⁎⁎
8.1801⁎⁎⁎
0.4923
2822.825⁎⁎⁎
15.4195⁎⁎⁎
0.7751

Notes: For the regression in Eq. (9), we report the value of the F-statistics for the null α = 0 and β = 1, the value of the t-statistic for the beta coefficient and the R-squared of the regression, where the RHS, independent variable is based on the univariate OU model with parameters estimated using maximum likelihood (ML) in column 2 and Markov Chain Monte Carlo
(MCMC) in columns 3–5, with mean (column 3), 2.5th quantile (column 4) and 97.5th quantile (column 5) parameter estimates. The data used for fitting the model parameters with these
alternative estimation methods contains monthly log prices on the IPD index, observed over the period between December 1986 and January 2009, and totalling 266 historical observations. The forwards and futures data used for the testing reported in this table is from 4 February to 7 July 2009 for five maturities, namely December 2009, December 2010, December
2011, December 2012 and December 2013.
⁎ Denotes significance at the 10% level.
⁎⁎ Denotes significance at the 5% level.
⁎⁎⁎ Denotes significance at the 1% level.

Table 10
Model comparison — panel data.
Test/model

univ_ML

univ_MCMC_mean

univ_MCMC_2.5q

univ_MCMC_97.5q

F-test
t-Test for β
R_squared

62,789.3⁎⁎⁎
10.7141⁎⁎⁎

26,358.31⁎⁎⁎
11.3375⁎⁎⁎

64,035.43⁎⁎⁎
10.6135⁎⁎⁎

10,284.18⁎⁎⁎
6.8171⁎⁎⁎

0.2454

0.2669

0.2419

0.1163

Notes: For the regression in Eq. (9), we report the value of the F-statistics for the null α = 0 and β = 1, the value of the t-statistic for the beta coefficient and the R-squared of the regression, where the RHS, independent variable is based on the univariate OU model with parameters estimated using maximum likelihood (ML) in column 2 and Markov Chain Monte Carlo
(MCMC) in columns 3–5, with mean (column 3), 2.5th quantile (column 4) and 97.5th quantile (column 5) parameter estimates. The data