1 Other early studies that considered the relationshipprices in a perfect market without taxes and transactioJarrow and Oldfield (1981) and Richard and Sundaresan (
Article history:Received 9 January 2014Received in revised form 27 May 2014Accepted 30 May 2014Available online 12 June 2014
JEL classification:C12C33G13G19
Keywords:Total return swaps and futuresPanel dataMean-reversionMarkov Chain Monte Carlo
In this paper the differences between forward and futures prices for the UK commercial property market areanalyzed, using both time series and panel data. A first battery of tests establishes that the observed differencesare statistically significant over the study period. Further analysis considers the modeling of this difference usingmean-reverting models. The proposed models are then estimated with a number of alternative estimationmethods and second stage statistical tests are implemented in order to decide which model and estimationmethod best represent the data.
The difference between forward and futures prices has been givenconsiderable attention in the finance literature, both from a theoreticalas well as from an empirical perspective, and for various underlyingassets. On the theoretical side, Cox, Ingersoll, and Ross (1981) (CIR)obtained a relationship between forward and futures prices based solelyon no-arbitrage arguments.1 A series of papers subsequently tested em-pirically the CIR result(s). Cornell and Reinganum (1981) investigatedwhether the difference between forward and futures prices in the for-eign exchange market is different from zero. For several maturitiesand currencies, they found that the average forward–futures differenceis not statistically different from zero. In addition, they suggested thatearlier studies identifying significant forward–futures differences forthe Treasury bill markets ought to seek explanations elsewhere thanin the CIR framework, since the corresponding covariance terms forthis market were even smaller. French (1983) reported significant
between forward and futuresn costs are Margrabe (1978),1981).
differences between forward and futures prices for copper and silver.Moreover, he conducted a series of empirical tests of the CIR theoreticalframework and concluded that his results are in partial agreement withthis theory. Park and Chen (1985) also investigated the forward–futuresdifferences for a number of foreign currencies and commodities andthey pointed out to significant differences for most of the commoditiesthat they analyzed, but not for the foreign currencies. Also, their empir-ical tests confirmed that the majority of the average forward–futuresprice differences are in accordance with the CIR result.
Kane (1980) tried to explain the differences between futures andforward prices based on market imperfections such as asymmetrictaxes and contract performance guarantees. Levy (1989) stronglyargued that the difference between forward and futures pricesarises from the marked-to-market process of the futures contract.Meulbroek (1992) investigated further the relationship between for-ward and futures prices on the Eurodollar market and suggested thatthe marked-to-market effect has a large influence. However, Grinblattand Jegadeesh (1996) advocated that the difference between thefutures and forward Eurodollar rates due to marking-to-market issmall. Alles and Peace (2001) concluded that the 90-day Australiafutures prices and the implied forwards are not fully supported by theCIRmodel. Recently,Wimschulte (2010) showed that there is no signif-icant statistical or economical evidence for price differences betweenelectricity futures and forward contracts.
Fig. 1. IPD total return swap rates (mid prices).Notes: The plotted data is from 4 February to 7 July 2009 for the fivematurity datesfixed in themarket calendar, for the period of study. Thetotal 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 depreciateover the period to the horizon indicated by the maturity of the contract.
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The relationship between forward and futures prices as developedunder the CIR framework makes the tacit assumption that futures areinfinitely divisible. Levy (1989) starts with the same set of assumptionsunderpinning the CIR model except one. When considering interestrates, he advocates that, if only the next day's interest rate were deter-ministic, a perfect hedge ratio using fractional futures positions can beconstructed to replicate the forward. Thus, for Levy (1989) it is onlythe interest rate for the next day that is important and not the entiretime path of the stochastic rates. Consequently, for Levy (1989), the for-ward prices should be equal to futures prices and any empirical findingsregarding actual price differentials are non-systematic and they canhave only statistical explanations. On the other hand, Morgan (1981)studied the forward–futures differential assuming that capital marketsare efficient and concluded that forward and futures prices must be dif-ferent. His conclusion is mainly based on the fact that current futuresprice depends on the joint future evolution of stochastic interest ratesand futures prices. Polakoff and Diz (1992) argued that due to the indi-visibility of the futures contracts,2 the forward prices should be differentfrom futures prices even when interest rates and futures prices exhibitzero local covariances. Moreover, they show that the autocorrelationin the time series of the forward–futures price differences should beexpected. Hence, testing must take into consideration the presence ofautocorrelation. Polakoff and Diz (1992) offered a theoretical explana-tion that unifies the contradictory theoretical views originated in howinterest rates are negotiated in the model. Their main conclusion isthat it is unnecessary for futures prices and interest rates to be correlat-ed in order to imply that forward prices should be different from futuresprices.
From the review discussed above it appears that the empiricalevidence is mixed and asset class specific. Property derivatives are anemerging asset class of considerable importance for financial systems.Case and Shiller (1989, 1990) found evidence of positive serial correla-tion as well as inertia in house prices and excess returns, implying thatthe U.S. market for single-family homes is inefficient. The use of deriva-tives for risk management in real estate markets has been discussed byCase, Shiller, and Weiss (1993), Case and Shiller (1996) and Shiller andWeiss (1999) with respect to futures and options. Fisher (2005)discussed NCREIF-based swap products, while Shiller (2008) describedthe role played by the derivatives markets in general for home prices.
For real-estate there has been a perennial lack of developments ofderivatives 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 infi-nite divisibility, Polakoff (1991) discusses the important role played by the indivisibility offutures contracts.
U.K. are the total return swaps (TRSs), forwards and futures. In theU.K. commercial property sector for example, all three types of contracthave the Investment Property Databank (IPD) index as the underlying.Since February 2009 the European Exchange (Eurex) has listed the UKproperty index futures. The most liquid derivatives markets on the IPDUK index are the TRS, which is an over-the-counter market, and thefutures, both with at least five yearly market calendar Decembermaturities. Any portfolio of TRS contracts can be decomposed into anequivalent portfolio of forward contracts. Hence, having data on TRSprices and futures prices opens the opportunity to compare, aftersome financial engineering, forward curves with futures curves on theIPD index. As remarked by Polakoff and Diz (1992) it is difficult tocompare forward and futures prices on a daily basis when forwardsare traded on a non-synchronous basis. By contrast, when forwardsare derived on an implied basis from other instruments then matchingthe term-to-delivery is easy.
In this paper the forward–futures price differences are investigatedfor the UK commercial property market for all five December marketmaturities. To our knowledge, this is the first study that considers theforward–futures price differences for this important asset class. Theanalysis of the difference is particularly important for twomain reasons.Firstly, previous literature addressing the issue for different asset classesfound that the empirical evidence was mixed and asset specific. There-fore, addressing the question for a new asset class is not an exercise ofconfirming previous results, but rather a new and important questionin itself, especially since unexplained forward–futures differences cansignal arbitrage opportunities. Secondly, intrinsic characteristics of realestate as an asset class make the contribution of this paper particularlyrelevant, since the underlying (a commercial real estate index in ourcase) is likely to be correlated with interest rates. According to the CIRresult, this in turn should drive significant differences between forwardand futures prices, but does this fully explain observed differences orcan these occur, at least partially, due to arbitrage also? This is essential-ly what our paper aims to address. Furthermore, all previous studiesrelied exclusively on time series analysis, whereas in this paper wealso conduct statistical tests for panel data as well as time series tests.To the authors' knowledge, this is the first study that considers paneldata modeling in this context. Employing panel data has a series ofadvantages over basing findings on time series alone.3 For example, itincreases statistical accuracy by increasing the number of degrees offreedom, which is particularly important for this application whichbenefits from having access to a unique OTC data set, with a relativelylimited sample period, but with data available for a number of cross sec-tions. 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.
5 See www.eurexchange.com for more information on Eurex. IPD UK futures contractsstarted on 4 February 2009.
6 See www.ipd.com for more information on IPD.
Fig. 2. Eurex futures prices.Notes: The plotted data is from 4 February to 7 July 2009 for the fivematurity dates fixed in themarket calendar. Futures prices are given on a total return basisso 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.
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futures difference for a new and particularly important asset class,employing a unique data set and providing panel test results, to ourknowledge, for the first time in this stream of literature.
In addition, several models and estimation methods for the IPDindex are investigated to try to determine which ones best capture theIPD forward–futures price difference. Empirical properties of real-estate indices suggest that the family of mean-reverting modelspresented in Lo and Wang (1995) could be suitable for defining ourmodeling framework. Shiller and Weiss (1999) pointed out that theexactmodels advocated in Lo andWang (1995)may not be appropriatefor real-estate derivatives since the underlying asset is not costlesslytradable, and they advocated using a lognormal model combined withan expected rate of return rather than a riskless rate. Later on, Fabozzi,Shiller, and Tunaru (2012) designed a way to merge the best of thetwo worlds by completing the market with the futures contracts thatare used directly to calibrate the market price of risk for the real-estate index and hence, indirectly fixing also the risk-neutral pricingmeasure which can be then applied for pricing other derivatives. There-fore, the first model we propose in Section 4 below will be a slightmodification of the Lo and Wang (1995) trending OU process, wherefutures prices are used to calibrate the market price of risk, as pointedout in Section 4.3.
Real-estate prices exhibit serial correlation leading to a high degreeof predictability, up to 50% R-squared for a short term horizon. More-over, it has been documented that returns on real-estate indices arepositively autocorrelated over short horizons and negatively correlatedover longer horizons, see Fabozzi et al. (2012).
The remainder of the paper is organized as follows: Section 2 focuseson describing the data, Section 3 contains the analysis that is real-estatemodel-free and the testing methodology covering panel data as well,Section 4 contains the modeling approach taken for the commercialproperty index (including alternative estimation methods for the pro-posed models) and it also describes the second stage, model basedtests. Section 5 concludes.
2. Data description
For the empirical analysis of the differences between the forwardand futures prices on the IPD4 UK property index two types of testsare performed. Firstly, it is investigatedwhether the observed differencebetween the forward and futures prices is statistically different fromzero. Secondly, a number of established continuous time models com-bined with various methods of estimation are compared in order to
4 IPD stands for Investment Property Databank. A detailed description of the data is giv-en in Section 3.1 below.
identify the model or models that are able to best capture this differ-ence. Using the previously defined notation, there are: n = 5 differentmaturities and N = 71 daily observations for each maturity. The dataneeded for this study contains IPD property futures prices, the IPDtotal return swap (TRS) rates, the IPD index, and also the GBP interestrates needed to calculate discount factors. Futures prices have been ob-tained from the European Exchange (Eurex5), the property TRS data(the fixed rate) has been provided by Tradition Group, a major dealeron this market and the IPD index was sourced from the InvestmentProperty Databank (IPD6). In addition, the UK's interest rates havebeen downloaded from Datastream. Due to the limited availability ofthe property futures and TRS data, the sample period used is dailyfrom 4 February 2009 until 7 July 2009. It generates 71 property futuresdaily curves and 71 sets of TRS rates with up to five years maturity (thefirst maturity date is 31 December 2009, the secondmaturity date is 31December 2010, the third maturity date is 31 December 2011, thefourth maturity date is 31 December 2012, and the fifth maturity dateis 31 December 2013).
The evolution of the TRS series is depicted in Fig. 1 and one could seethat, for our period of investigation, most of the IPD TRS rates are nega-tive for the first, second, and thirdmaturity dates.We note that the totalreturn swap rates are given as a fixed rate and not as a spread overLIBOR.7 A negative total return swap rate implies that the underlyingcommercial property market will depreciate over the period to the ho-rizon indicated by the maturity of the contract. For the fourth and fifthmaturity dates, the TRS rates are mostly positive. In addition, there is adramatic increase of the fixed rate at the end of February 2009, possiblydue to the rollover off the futures contracts inMarch combinedwith thepublication of the IPD index for the year ending in December 2008. Theproperty futures prices, quoted on a total return basis, are illustrated inFig. 2. Futures prices are given on a total return basis so a futures price of110 for December 2012 implies that the market expects a 10% appreci-ation of the commercial property in the UK at this horizon.
From the daily TRS prices for the market five yearly maturities onecan reverse engineer the equivalent no-arbitrage forward prices forthe samematurities. The equivalent fair property forward prices are de-rived daily from 4 February until 7 July 2009, with maturities matchingthe futures contracts maturities.
7 The TRS rate was initially equal to LIBOR plus a spread, but due to disagreements overwhat the spread should have been equal to, the property TRS swap rate is now equal to afixed rate, which is established upfront and can be negative.
Fig. 3. The fair prices of property forwards.Notes: The plotted data is from 4 February to 7 July 2009 for the fivematurity dates fixed in themarket calendar, for the period of study. The fairproperty forward prices are reversed engineered from the corresponding portfolio of total return swaps.
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Using data on the fixed rate of total return swap, UK interest ratesand the monthly IPD all property total return index, the fair forwardprices for property derivatives can be obtained. The following are thesteps in constructing the fair prices of property forwards:
The fair prices of property forwards thus obtained are illustrated inFig. 3.
A closer examination of Figs. 2 and 3 shows that the jumps in IPDfutures and (reversed engineered) forward prices are not contempora-neous, 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 theend of March, forwards – which are reversed engineered based on TRSand UK interest rate data as explained above – jump in February dueto jumps in TRS rates, which in turn, should be due to the publicationof the new IPD index. This results in the differences between forwardand futures, depicted in Fig. 4, being higher than usual for almost onemonth.8
The descriptive statistics of the TRS rates are reported in Table 1. Themean values are mostly negative, signifying that the underlying com-mercial property market will depreciate over the period to the horizon
8 These abnormal forward–futures differences do not drive the significance of the re-sults in this paper. The results remain significant even after changing the sample and elim-inating the month of abnormally high forward futures differences.
indicated by thematurity of the contract; themean for thefirstmaturitydate is−17.80% and themeans are increasingwithmaturity. The excesskurtosis is negative for all five futures contracts and the first year TRScontract and it is positive for the remaining four series of TRS rates.The skewness values have negative signs, except for the four year fu-tures contract, implying that the distributions of the data are skewedto the left. The descriptivemoments of the differences between forwardand futures prices on the IPD commercial index are also reported inTable 1. On average, the differences for thefirst threematurities are pos-itive, while for the fourth and fifth maturities they are negative.
It can be seen in Table 1 that the futures contract for the fifthmaturity date appears to have the highest mean. The highest standarddeviation is shown in the futures contract for the second maturitydate. Similarly to TRS data, futures prices exhibit skewness and fat tailcharacteristics.
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 asso-ciated time-t forward price with maturity Ti, f(t, Ti) the time-t futuresprice with maturity Ti and D(t, Ti) the stochastic discount factor attime t for maturity Ti. Then B(t,Ti) = Et
Q(D(t,Ti)) is the time-t zero-coupon bond price, with maturity Ti, where the expectation is takenunder a risk-neutral measure Q.
There is a model-free relationship between forward and futuresprices given by9:
F t; Tð Þ− f t; Tð Þ ¼ covQt S Tð Þ;D t; Tð Þð ÞEQt D t; Tð Þð Þ
ð1Þ
which holds for any maturity T and at any time 0 ≤ t ≤ T. This funda-mental relationship opens up the first line of investigation for testingwhether the differences between forward and futures prices are statis-tically different from zero.
3.1. Testing methodology
First, the null hypothesis that the difference between themarket TRSequivalent forward prices and market futures prices is significantlydifferent from zero is tested. If this hypothesis is rejected, then in thesecond stage a series of models and estimation methods are employedfor the terms on the right hand side of the fundamental relationshipgiven by Eq. (1). The aim in the second stage is to decide on the capabil-ity of variousmodels to appropriately capture the dynamics of the indexS and the discount factor D.
9 See Shreve (2004, p. 247).
Fig. 4. The difference between forward and futures prices.Notes: The plotted data is from 4 February to 7 July 2009 for the fivematurity datesfixed in themarket calendar, for the period ofstudy. The fair property forward prices are reversed engineered from the corresponding portfolio of total return swaps.
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For the first stage analysis, the following regression model is fittedfor each maturity date Ti, i∈{1,2,…,5}:
F t; Tið Þ ¼ α0i þ β0i f t; Tið Þ þ εti ð2Þ
Table 2ADF test for the forward and futures prices.
Maturity date Forward Futures
Level First differences Level First differences
31 December 2009 −1.3453 −8.3646⁎⁎⁎ −0.8098 −5.1662⁎⁎⁎
31 December 2010 −1.9185 −8.4575⁎⁎⁎ −1.3019 −8.3168⁎⁎⁎
31 December 2011 −2.0106 −8.6799⁎⁎⁎ −1.3278 −4.3724⁎⁎⁎
31 December 2012 −3.4368⁎⁎ −5.4320⁎⁎⁎ −2.3581 −5.9891⁎⁎⁎
31 December 2013 −0.9618 −6.4863⁎⁎⁎ −1.2087 −12.1882⁎⁎⁎
Notes: Augmented Dickey–Fuller (ADF) test results for the UK IPD commercial propertyindex forward and futures prices. The test is performed for both the data in levels aswell as for the first differenced data. The optimum number of lags used in the ADF testequation is based on the Akaike Information Criterion (AIC). The data is from 4 Februaryto 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.
Table 1Descriptive statistics for total return swap rates, Eurex futures prices and the forward–futures differences.
Notes: The descriptive statistics are of the total return swap rates, futures prices andforward–futures differences on the IPD UK All Property index. Daily mid prices are usedfor calculation for the period 4 February 2009 to 7 July 2009 for the five market calendarmaturities, namely December 2009, December 2010, December 2011, December 2012and December 2013. The forward prices used here are the synthetic fair prices derivedfrom total return swap rates, synchronous with the futures prices.
(with Ti fixed for each of the five time series regressions) and testwhether α0i = 0 and β0i = 1. If the null hypothesis cannot be rejected,then one can conclude that the difference between forward and futuresprices is due to noise. If, however, the null is rejected, we then proceedto the second stage of our analysis. The same econometric analysis de-scribed above from a times series point of view, can also be performedusing panel data. Using panel data has a series of advantages.10 Firstly,it enables the analysis of a larger spectrum of problems that couldnot be tackled with cross-sectional or time series information alone.Secondly, it generally results in a greater number of degrees of freedomand a reduction in the collinearity among explanatory variables, thusincreasing the efficiency of estimation. Furthermore, the larger numberof observations can also help alleviate model identification or omittedvariable problems.
The regression equation in Eq. (2) is rewritten for our panel data as:
F t; Tið Þ ¼ α0 þ β0 f t; Tið Þ þ εti 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 esti-mating the regression equation by simply stacking all the data together,for both the explained and explanatory variables. Furthermore, thefixed effects model for panel data is given by:
F t; Tið Þ ¼ α0 þ β0 f t; Tið Þ þ αi þ υit ð4Þ
where αi varies cross-sectionally (i.e. in our case it is different for eachmaturity date Ti), but not over time. Similarly, a time-fixed effectsmodel can be formulated, in which case one would need to estimate:
F t; Tið Þ ¼ α0 þ β0 f t; Tið Þ þ λt þ υit ð5Þ
where λt varies over time, but not cross-sectionally. The fixed effectsmodel and the time-fixed effects model, as well as a model with bothfixed effects and the time-fixed effects, will be analyzed. One can testwhether the fixed effects are necessary using the redundant fixedeffects LR test.
For the panel data random effects model the regression specificationis given by:
F t; Tið Þ ¼ α0 þ β0 f t; Tið Þ þ εi þ υit ð6Þ
where εi is now assumed to be random, with zero mean and constantvariance σ ε
2, independent of υit and f(t,Ti). Similarly, a random time-effects model can be formulated in the context of this paper as:
F t; Tið Þ ¼ α0 þ β0 f t; Tið Þ þ εt þ υit : ð7Þ
10 See also Baltagi (1995), Hsiao (2003).
Table 3F-test for time series data.
F-test t-test t-TestMaturity date α0i = 0 and β0i = 1 α0i = 0 β0i = 1 Durbin–Watson statistic
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 maturitydates 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.
12 In addition, the diagnostic statistics for these regressions are investigated and theDurbin–Watson test statistic results are reported in Table 3. For all but the fifth maturitydate there is no autocorrelation in the regression errors. Furthermore, the individual t-
Table 4t-Statistics for the differences between forward and futures prices.
Notes: The values of the t-test are computed for thedifferences between forward and futures prices on theUK IPD commercial property index, using data from4 February to 7 July 2009 andfrom 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.
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Again, random effects and random time-effects models, as well as atwo-way model which allows for both random effects and randomtime-effects, can be estimated. Furthermore, it is important to testwhether the assumption that the random effects are uncorrelatedwith the regressors is satisfied.
For the second stage analysis several models are employed for thedynamics of the IPD index S. If the analysis is conditioned on knowingthe bond prices, the RHS of identity (1) can be expressed as:
covQt S Tð Þ;D t; Tð Þð ÞEQt D t; Tð Þð Þ ¼ S tð Þ
B t; Tð Þ−EQt S Tð Þð Þ: ð8Þ
Based on Eqs. (1) and (8), it is evident that for testing purposes thefollowing regression is useful:
F t; Tð Þ− f t; Tð Þ ¼ αþ βS tð Þ
B t; Tð Þ−EQt;m S Tð Þð Þ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}f m t;Tð Þ
264
375þ ut;m ð9Þ
and testwhetherα=0 and β=1, for eachmodelm. For eachmodelm,failing to reject the null hypothesis implies that this particular model issuitable for describing the dynamics of the underlying IPD index. Uponestimation of all the parameters of eachmodelm,11 the regression givenin Eq. (9) is fitted. The competingmodels andmethods of estimation arecompared with respect to whether β is significant and also consideringthe R2 measure of goodness-of-fit.
3.2. Model-free analysis
Having both series of forward and futures prices available allowsdirect testing of whether the forward–futures difference time seriesdiverges significantly away from zero. Before running the regressionsin Eqs. (2)–(7), the forward and futures price series are tested forstationary using the Augmented Dickey–Fuller (ADF) test. The resultsare reported in Table 2.
11 The specific stochastic models and parameter estimation methods employed in thispaper are described in Section 4 below.
As it can be seen in Table 2, most of the ADF results show that theforward series for the first, second, third, and fifth maturity dates arenon-stationary while the forward series for the fourth maturity date isstationary at the 5% significance level. In addition, the ADF test indicatesthat the futures series for all maturity dates are non-stationary. Further-more, the stationarity of the first differenced data is also investigated.According to Table 2, the forward and futures series for all maturitydates are stationary in the first differences.
Sincemost of the data is found to be non-stationary in levels and sta-tionary in the first differences, the remaining analysis is performed onthe first differenced data. The null hypothesis H0: α0i = 0 and β0i = 1vs. H1: α0i ≠ 0 or β0i ≠ 1 is tested using an F-test and the results canbe found in Table 3.
The F-test results presented in Table 3 show that the null hypothesisfor all maturity dates could be rejected at the 1% significance level. Thisimplies that the difference between forward and futures is not justnoise.12 The same conclusion is reached if we analyze the values of thet-statistics for the forward–futures differences reported in Table 4. Theresults in Table 4 show that the significance of the forward–futuresdifference is not driven by themonth of abnormally high differences: in-deed, the difference is still significant when a shorter sample period – 2April to 7 July 2009 – is considered, which now eliminates the abnormalsub-period from end February to end March.
As a robustness check of our time-series results, particularly impor-tant given the relatively limited sample available for the time-seriesanalysis (i.e. 71 observations), we also test whether the differencesbetween forward and futures prices are significant using a panel regres-sion. To choose an appropriate specification for the panel regression,wefirst test whether the fixed effects are necessary using the redundantfixed effects LR test. The results of this test are reported in Table 5.
From the test results reported in Table 5, it appears that amodelwithfixed time effects only is most supported by the data in this research.Furthermore, a random effects model may be appropriate and this istested using the Hausman test; the results of this test are also reported
tests for the individual hypothesesα0i=0 (vs.α0i≠ 0) and β0i = 1 (vs. β0i≠ 1) are alsoreported. We find that β is always different from 1 and that α is generally not differentfrom zero (with the exception of the first maturity).
Table 6The F-test for panel data.
Test statistic Value
F-statistic 237.3960⁎⁎⁎
Durbin–Watson statistic 2.1419
Notes: F-test andDurbin–Watson statistic results for the regression in Eq. (6) for the prop-erty forward and futures panel data from4 February to 7 July 2009, using the cross-sectionrandom effects specification. For the F-test, the null hypothesis is that the differencebetween 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.
15 The continuously compounded τ-period returns, computed at time t, are defined asrτ(t)= p(t)− p(t− τ). The autocorrelation function of the returns process employedhereis the same as in Lo and Wang (1995) – see their expression (A3) – namely:
for any t1,t2, and τ such that t1 ≤ t2 − τ, to ensure that returns are non-overlapping.16
Table 5Tests for determining the most suitable panel regression model.
Test Value
Redundant fixed effects testCross-section F 1.0925Cross-section Chi-square 5.5181Period F 2.2062⁎⁎⁎
Period Chi-square 154.1939⁎⁎⁎
Cross-section/period F 2.1450⁎⁎⁎
Cross-section/period Chi-square 157.7444⁎⁎⁎
Hausman testCross-section random 3.0341⁎
Period random 0.0003Cross-section and period random 0.1690
Notes: The redundantfixed cross-section effects test has a panel regressionwith fixed time(period) effects only under the null. Both the F and the Chi-square version of the test arereported. The redundant fixed time (period) effects have a panel regression with fixedcross-section effects only under the null. Both the F and the Chi-square version of thetest are reported. For the random effects test (i.e. Hausman test) the null hypothesis inthis case is that the random effect is uncorrelated with the explanatory variables. Thepanel 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.
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in Table 5. Based on these results, the random effect model is to bepreferred in this case.
Next, the F-test statistic for multiple coefficient hypotheses is com-putedusing the panel regression randomeffect specification; the resultsare reported in Table 6.
According to Table 6, the F-values are significant at the 1% level. Thenull hypothesis (H0: α0 = 0 and β0 = 1) can be strongly rejected andtherefore the differences between forward and futures are not justnoise in the panel data.13
4. Analysis with parametric models
Having shown in the previous section that IPD forward–futuresprice differences are statistically significant, one-factor and two-factor mean-reverting models that seem suitable for this asset classare studied in this section. The models are subsequently coupledwith two different methods of parameter estimation – maximumlikelihood (ML) as well Markov Chain Monte Carlo (MCMC) forwhich various quantiles from the distribution of parameters are esti-mated – the model parameters are calibrated to IPD index data, themodel futures and forward prices are thus calculated and finally, anumber of statistical tests are implemented in order to see whichmodel and estimation method best captures the empirical evolutionof the IPD forwards and futures.
4.1. Mean-reverting models
Here a slight variation of the trending (mean-reverting) OUprocess 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 dynamicsof q(t) under the physical measure P are described by the equa-tion:
dq tð Þ ¼ −γq tð Þdt þ σdW tð Þ ð10Þ
13 In addition, the values of the Durbin–Watson test show that there is no autocorrela-tion in the panel regression errors.14 To simplify notation, we suppress model subscripts, univ and bivar for the univariateand bivariate models, respectively, unless where absolutely necessary.
where γ ≥ 0,σ ≥ 0,μ0,μ∈R. Solving the SDE in Eq. (10) leads to theclosed form solution
p tð Þ ¼ μ0 þ μt þ p 0ð Þ−μ0½ � exp −γtð Þ
þ σZtv¼0
exp −γ t−vð Þð ÞdW vð Þ ð11Þ
for any 0≤ t≤ T, where themodel parameters will be estimated (usingboth maximum likelihood and MCMC) using the IPD index data, asdetailed below. The model-implied (theoretical) futures price can benow be derived in closed-form as outlined in Appendix A:
f univ 0; Tð Þ ¼ σ2
4γexp
�μ0 þ μT−μ0 exp −γTð Þ−ησ
γ
þ exp −γTð Þ ln S 0ð Þð Þ þ ησγ
� �Þ 1− exp −2γTð Þð Þ
ð12Þ
where η is the market price of risk, which shall be calibrated followingstandard practice, by minimizing the squared difference between themarket and model (theoretical) futures prices.
As remarked in Lo andWang (1995), although this specification is avalid modeling starting point, it has an important disadvantage in thatthe autocorrelation coefficients of continuously compounded τ-periodreturns can only take negative values.15 A more flexible approach, alsoproposed 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 suitthe application in this paper:
where dWS(t)dWr(t) = ρdt and the second stochastic factor on whichthe log-price of the underlying depends is the short interest rate r(t).16
Following the derivations and risk neutralization performed inAppendix B, the futures price is obtained in closed form:
f bivar 0; Tð Þ ¼ exp C2 Tð Þ þ σ2y Tð Þ2
!ð15Þ
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 bivariatemodel, unlike the univariate model outlined above, ismore flexible and can allow for bothpositive and negative autocorrelations.
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 repeatedusing a larger sample, namely Dec 1986 to Oct 2010, with an increased sample size of287 monthly observations. The parameter estimates do not change much.
Table 7Models and estimation methods.
Name Model Estimation method
univ_ML Univariate time-trending OU: dq(t) = −γq(t)dt + σdW(t) Exact maximum likelihooduniv_MCMC_mean Markov Chain Monte Carlo (MCMC), mean parameter estimatesuniv_MCMC_2.5q MCMC, using the 2.5th quantile of the distribution of estimatesuniv_MCMC_97.5q MCMC, using the 97.5th quantile of the distribution of estimatesbivar_ML Bivariate time-trending OU:
Exact maximum likelihoodbiv_MCMC_mean MCMC, mean parameter estimatesbiv_MCMC_2.5q MCMC, using the 2.5th quantile of the distribution of estimatesbiv_MCMC_97.5q MCMC, using the 97.5th quantile of the distribution of estimates
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) + (μ0 + μt). r(t) denotes the short in-terest rate. For both models, the futures price is obtained as: fm(0,T) = E0,m
Q (S(T)) where m = univ or bivar, for the two models, respectively, Q is the martingale pricing measure, andT is the futures maturity time.
Notes: Parameter estimates for the univariate OUmodel, obtained using likelihood (ML) (column 2), andMarkov ChainMonte Carlo (MCMC) (columns 3–5), withmean (column3), 2.5thquantile (column 4) and 97.5th quantile (column 5) parameter estimates. The data used estimating themodel parameterswith these alternative estimationmethods containsmonthly logprices on the IPD index, observed over the period between December 1986 and January 2009, and totalling 266 historical observations.
184 S. Stanescu et al. / International Review of Financial Analysis 34 (2014) 177–188
with
C2 Tð Þ ¼ μ0 þ μT−ση1
γþ ln p 0ð Þð Þ þ ση1
γ
� �exp −γTð Þ
þ λC1
γ1− exp −γTð Þð Þ þ r 0ð Þ−C1
γ−δexp −δTð Þ− exp −γTð Þð Þ
� �
whereC1 ¼ μr− σrδ ϱη1 þ
ffiffiffiffiffiffiffiffiffiffiffiffiffi1−ϱ2
qη2
� �, η=(η1η2)T is themarket price
of risk (now bi-dimensional) and
σ2y Tð Þ ¼ σ2
2γ1− exp −2γTð Þð Þ þ λσ2
r
γ−δð Þ2"1− exp −2δTð Þ
2δ
þ 1− exp −2γTð Þ2γ
−2 1− exp − γþ δð ÞTð Þð Þγþ δ
#
þ 2ρλσσr
γ−δ1− exp − γþ δð ÞTð Þ
γþ δ−1− exp −2γTð Þ
2γ
" #:
4.2. Estimation
In order to be able to use themodels enumerated above their param-eters should be calibrated first. The parameters of the continuous timemodels specified in Eq. (10) and Eqs. (13)–(14) can be estimated fromthe monthly log prices on the IPD index, observed over the period be-tween December 1986 and January 2009, and totalling 266 historicalobservations. The estimates then will be carried forward for analyzingthe differences between the forward and futures on IPD starting fromFebruary 2009.
4.2.1. Maximum likelihoodWhen feasible, parametric inference for diffusion processes from
discrete-time observations should employ the likelihood function,given its generality and desirable asymptotic properties of consistencyand efficiency (Phillips & Yu, 2009). The continuous time likelihoodfunction can be approximated with a function derived from discrete-time observations, obtained by replacing the Lebesgue and Ito integralswith 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 likeli-hood estimator can be strongly biased in finite samples.17
First de-trend the log price data by estimating the regression:
ptk ¼ μ0 þ μtk þ utkð16Þ
and subsequently work with the residuals from this equation, wherek = 1, 2, …, 266 and tk = kτ, with τ ¼ 1
12 for monthly returns. The(exact) discretization of Eq. (11) leads to:
utk¼ cutk−1
þ εtk ð17Þ
where c = exp(−γτ) and εtk ¼ σ ∫tk
tk−1
exp −γ tk−sð Þð ÞdW sð Þ; εtk � N
0; σ2
2γ 1− exp −2γτð Þð Þ�
.
Maximum likelihood estimation of the discrete-time model inEq. (17) gives18 c= 0.995086 and the standard deviation ofεtk as 0.011.
The exact discretization of the two-factor model given byEqs. (B.7)–(B.8) (given in Appendix B for lack of space) is:
qtk ¼ αq þ βqqtk−1þ φrtk−1
þ εq;tkrtk ¼ αr þ βrrtk−1
þ εr;tkð18Þ
where, for reasons of space, the expressions for the parameters as wellas the distribution of the error terms in Eq. (18) are only given inAppendix C.
4.2.2. Markov Chain Monte Carlo (MCMC)Despite its desirable asymptotic properties, ML estimates of the pa-
rameters of a continuous-time model based on discrete-sampled data
t-Test for β 31.9967⁎⁎⁎ 17.6248⁎⁎⁎ 25.8627⁎⁎⁎ 15.4195⁎⁎⁎
R_squared 0.9369 0.8182 0.9065 0.7751
Notes: For the regression in Eq. (9), we report the value of the F-statistics for the nullα = 0 and β = 1, the value of the t-statistic for the beta coefficient and the R-squared of the regres-sion, where the RHS, independent variable is based on the univariate OUmodel with parameters estimated using maximum likelihood (ML) in column 2 andMarkov Chain Monte Carlo(MCMC) in columns 3–5,withmean (column 3), 2.5th quantile (column 4) and97.5th quantile (column 5) parameter estimates. The data used forfitting themodel parameterswith thesealternative estimation methods contains monthly log prices on the IPD index, observed over the period between December 1986 and January 2009, and totalling 266 historical observa-tions. 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, December2011, December 2012 and December 2013.⁎ Denotes significance at the 10% level.⁎⁎ Denotes significance at the 5% level.⁎⁎⁎ Denotes significance at the 1% level.
t-Test for β 10.7141⁎⁎⁎ 11.3375⁎⁎⁎ 10.6135⁎⁎⁎ 6.8171⁎⁎⁎
R_squared 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 nullα = 0 and β = 1, the value of the t-statistic for the beta coefficient and the R-squared of the regres-sion, where the RHS, independent variable is based on the univariate OUmodel with parameters estimated using maximum likelihood (ML) in column 2 andMarkov Chain Monte Carlo(MCMC) in columns 3–5,withmean (column 3), 2.5th quantile (column 4) and97.5th quantile (column 5) parameter estimates. The data used forfitting themodel parameterswith thesealternative estimation methods contains monthly log prices on the IPD index, observed over the period between December 1986 and January 2009, and totalling 266 historical observa-tions. The panel forwards and futures data used for the testing reported in this table is from 4 February until 7 July 2009, for fivematurities, namely December 2009, December 2010, De-cember 2011, December 2012 and December 2013.⁎ Denotes significance at the 10% level.
⁎⁎ Denotes significance at the 5% level.⁎⁎⁎ Denotes significance at the 1% level.
185S. Stanescu et al. / International Review of Financial Analysis 34 (2014) 177–188
are biased in finite samples. An alternative estimation technique that isapplied here in order to circumvent this problem is the Markov ChainMonte Carlo (MCMC) methodology (see Tsay, 2010).
MCMC techniques19 are based on a Bayesian inference theoreticalsupport and offer an elegant solution to many problems encounteredwith other estimation methods, at the cost of computational effort.The main advantage of employing this type of inferential mechanismis the capability to produce not only a point estimate but an entire pos-terior distribution for parameters of interest. Selecting various statisticsfrom this distribution provides a more informed view on the plausiblevalues of the parameters. Hence, for estimation purposes the mean,the 2.5% quantile and the 97.5% quantile of the posterior distributionof the mean reversion parameter are going to be used. The estimatesfor the discretized version of the mean-reverting model given inEq. (10) are reported in Table 8. One great advantage of the MCMC
19 For an excellent introduction see Tsay (2010). All MCMC inference in this paper hasbeen produced withWinBUGS 1.4, from a sample of 100,000 iterations after a burn-in pe-riod of 500,000 iterations.
approach is that all parameters are estimated easily from the sameoutput without additional computational effort.
4.3. The calibration of the market price of risk
To calibrate the market price of risk η, we compute the differences(i.e. pricing errors) between observed and model futures prices. Subse-quently, standard practice is followed and the value of η is chosen suchthat it minimizes the mean squared error function. This optimizationexercise is performed for each day in our sample and for each of theestimation methodologies described above. All parameter estimatescan be determined now and then the theoretical model can be usedfor producing property futures prices. Table 7 gives a list of the modelsinvestigated in this paper with various methods of estimation. InTable 8 we report the parameter estimation results for first half ofthese models based on our data.20
20 Due to lack of space, the empirical implementation of the bivariate models is left forfuture research.
186 S. Stanescu et al. / International Review of Financial Analysis 34 (2014) 177–188
4.4. Empirical results
The empirical analysis considers here a more refined investiga-tion looking at several models for the underlying IPD index dynam-ics, coupled with a model for interest rates, but also consideringseveral estimation methods. From a financial economics point ofview, it has been established that even if the interest rates are con-stant then futures prices can differ from the associated forwardprices. Assuming that interest rates are stochastic leads directly tothe conclusion that the two series will diverge significantly overtime. In this part of the paper the research question is “whichmodel and estimationmethodmost likely support the observedmar-ket differences?” Furthermore, an additional level of complexity isgenerated from employing panel data tests.
Tables 9 and 10 summarize the results of our model comparison, forthe time series and panel data, respectively. The MCMC methodsperform better than the ML method. The R-square seems to increasewith maturity overall hinting that stationarity problems may be moreacute for near maturities. Please note that since maturities are fixed inthe calendar by the market, the time to maturity of our series getsprogressively smaller, across all five contracts.
The results in Table 10 reveal that for panel data analysis all modelsemployed here arewell specified and the beta t-test is highly significant.All models apart from theMCMC approach for the 97.5th quantile yieldsimilar values for the R-squared, of around 20%.
5. Conclusions
In this paper the differences between forward and futures pricesare analyzed on commercial real-estate, using a battery of models,estimation methods and tests. The forward prices have been reversedengineered from total return swap rates using standardmarket practice.Testing is donenot only on individual time series data but also in a paneldata framework.
The results obtained in this paper provide evidence of significantdifferences between the implied forward and futures prices for theIPD UK index. One possible explanation could be the period ofstudy, several months during 2009, in the aftermath of the subprimecrisis.
Although the overall conclusion is that, for the period 4 February2009 to 7 July 2009, the forward prices were different from futuresprices, there is substantial variation in the strength of these resultsacross contract maturities, methods of estimation and testing frame-works. Given the significance of these results on a model-free basis ini-tially, a model race that best explains the relationship betweensynthetic forward prices derived from daily total return swap ratesand the daily futures prices was performed. Themodels were generatedby using various methods of estimation for the mean-reverting OUcontinuous time process assumed for the underlying IPD index. Themodels employed in the paper provided significant explanatory powerfor the relationship between forward and futures prices on commercialreal-estate index in the UK but the analysis of the error terms showsthat there is more that can be explained. From a theoretical point ofview our study can be expanded to more advanced two-factor modelsas detailed in the paper. The difficult question to answer then is whatconstitutes the second factor for commercial-real estate. This wouldbe the subject of a future research.
21 See also proposition 4.19 from Bjork (2009).
Acknowledgement
We are very grateful to Eurex for supporting our research andproviding us with the data on IPD UK futures. We would also like tothank an anonymous referee for very helpful suggestions which helpedimprove the paper. All mistakes are obviously ours.
Appendix A. The derivation of the futures price for the univariateprocess
Using Eq. (10), the corresponding equation for p(t) under the real-measure P is:
where η is the market price of risk and WQ(t) is a Q-Brownian motion.The solution to this modified equation is similar to Eq. (11):
p tð Þ ¼ μ0 þ μt−μ0 exp −γtð Þ−ησγ
þ exp −γtð Þ p 0ð Þ þ ησγ
� �
þ σZtv¼0
exp −γ t−vð Þð ÞdWQ vð Þ
for any 0 ≤ t ≤ T. Given the normality of p(t), the expression for thetheoretical futures prices can be obtained in closed-form, as given inEq. (12):
f univ 0; Tð Þ ¼ EQ0;univ S Tð Þð Þ ¼ EQ0;univ exp p Tð Þð Þð Þ
¼ σ2
4γexp
μ0 þ μT−μ0 exp −γTð Þ−ησ
γ
þ exp −γTð Þ ln S 0ð Þð Þ þ ησγ
� �!1− exp −2γTð Þð Þ:
Appendix B. The derivation of the futures price for the bivariateprocess
The solution to Eq. (14) is:
r tð Þ ¼ μ r þ exp −δtð Þ r 0ð Þ−μrð Þ þ σr
Ztv¼0
exp −δ t−vð Þð ÞdWr vð Þ ðB:1Þ
for any 0 ≤ t ≤ T. Combining Eqs. (B.1) and (13) gives the analyticalsolution for the log of the underlying index value:
p tð Þ ¼ μ0 þ μt þ exp −γtð Þ p 0ð Þ−μ0ð Þ þ μ rλγ
1− exp −γtð Þð Þ
þ λγ−δ
r 0ð Þ−μ rð Þ exp −δtð Þ− exp −γtð Þ½ �
þ λσr
γ−δ
Ztv¼0
exp −δ t−vð Þð Þ− exp −γ t−vð Þð Þ½ �dWr vð Þ
þ σZtv¼0
exp −γ t−vð Þð ÞdWs vð Þ:
ðB:2Þ
Next, in order to obtain the risk neutral Q-dynamics of the system inEqs. (13) and (14) where dWS(t)dWr(t)= ρdt first the system is rewrit-ten under the physical measure P, but depending solely on the non-correlated Brownians21 W1 andW2, where, for any t
Thus, the system in Eqs. (13)–(14) can be re-written as follows:
dp tð Þdr tð Þ� �
¼ μ þ γ μ0 þ μtð Þδμ r
� �þ −γ λ
0 −δ
� �p tð Þr tð Þ� � �
dt
þ σ 0ϱσr σr
ffiffiffiffiffiffiffiffiffiffiffiffiffi1−ϱ2
q" #dW1 tð ÞdW2 tð Þ� �
: ðB:5Þ
The risk neutral Q-dynamics of this system is given by:
dp tð Þdr tð Þ� �
¼(
μ þ γ μ0 þ μtð Þδμr
� �−
σ 0ϱσr σr
ffiffiffiffiffiffiffiffiffiffiffiffiffi1−ρ2
q" #η1η2
� �
þ −γ λ0 −δ
� �p tð Þr tð Þ� �)
dt þ σ 0ϱσr σr
ffiffiffiffiffiffiffiffiffiffiffiffiffi1−ϱ2
q" #dWQ
1 tð ÞdWQ
2 tð Þ
" #:
ðB:6Þ
The above system can be solved leading to the solution:
r tð Þ ¼ C1 þ r sð Þ−C1½ � exp −δ t−sð Þð Þ þ σr
Ztv¼s
exp −δ t−vð Þð ÞdWQr vð Þ:
ðB:7Þ
Or, slightly simplified,
r tð Þ ¼ C1 þ r 0ð Þ−C1½ � exp −δtð Þ þ σr
Ztv¼0
e −δ t−vð Þð ÞdWQr vð Þ ðB:7bisÞ
withC1 ¼ μr− σrδ ϱη1 þ
ffiffiffiffiffiffiffiffiffiffiffiffiffi1−ϱ2
qη2
� �, anddWQ
r tð Þ ¼ ϱWQ1 tð Þ þ
ffiffiffiffiffiffiffiffiffiffiffiffiffi1−ϱ2
qWQ
2 tð Þ. Also,
p tð Þ ¼ μ0 þ μt−ση1
γþ p 0ð Þ−μ0 þ
ση1
γ
� �exp −γtð Þ
þ σZtv¼0
exp −γ t−vð Þð ÞdWQs vð Þ
þ λ
C1
γ1− exp −γtð Þ½ � þ r 0ð Þ−C1
γ−δexp −δtð Þ− exp −γtð Þ½ �
þ σr
γ−δ
Ztv¼0
exp −δ t−vð Þð Þ− exp −γ t−vð Þð Þ½ �dWQr vð Þ
8>>>><>>>>:
9>>>>=>>>>;:
ðB:8Þ
The futures price formaturity T can be easily derived now as: fbivar(0,T)=E0,bivarQ (S(T)) and calculations of this expectation yield the expression in
Eq. (15).
Appendix C. The discretization of the bivariate model
The complete specification of the discretization for the two-factormodel used in the paper is given as
188 S. Stanescu et al. / International Review of Financial Analysis 34 (2014) 177–188
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