An Empirical Examination of Jump Risk and Continuous Risk in. U.S. REITs Market

Transcription

1 An Empirical Examinaion of Jump Risk and Coninuous Risk in U.S. REITs Marke Chia-Chien Chang Deparmen of Finance, Naional Kaohsiung Universiy of Applied Science, 415 Chien Kung Road, Sanmin Disric, Kaohsiung 80778, Taiwan, Chiu-Fen Kao Deparmen of Finance, Naional Sun Ya-sen Universiy No. 70, Lienhai Rd., Kaohsiung 80424, Taiwan Tsung-Li Chi Deparmen of Applied Mahemaics, I Shou Universiy No.1, Sec. 1, Syuecheng Rd., Dashu Disric, Kaohsiung 84001, Taiwan lchi@isu.edu.w Sheng-Jung, Li Deparmen and Graduae School of Insurance and Finance, Shu-Te Universiy No.59, Hengshan Rd., Yanchao Dis., Kaohsiung 82445, Taiwan i023@mail.njc.edu.w

2 An Empirical Examinaion of Jump Risk and Coninuous Risk in U.S. REITs Marke Absrac This paper uilizes recen advances in economeric heory, developed by Anderson, Bollerslev, and Diebold (2007), Barndorff-Nielsen and Shephard (2004), and Tauchen and Zhou (2006), o effecively separae he coninuous and jump componens of all REITs. By his economeric echnique, we invesigae wheher differen ypes (equiy, morgage, and hybrid ype) of U.S. REITs markes has significan jump phenomenon. In addiion, we furher decompose each of he volailiy componens ino coninuous sysemaic risk and jump sysemaic risk by exending CAPM and wo-facor models. By decomposing, we inend o discuss ha is he jump bea risk higher han coninuous bea risk? Is he jump bea risk and coninuous bea risk asymmeric? And we inend o invesigae ha is jump risk almos sysemaic or idiosyncraic (nonsysemaic)? Key words: asse price volailiy, REITs, asymmeric coninuous bea, asymmeric jump bea, idiosyncraic risk

3 1. Inroducion Asse price volailiy is he mos fundamenal elemen of risk managemen. In order o properly model he asse price process, mos of he discree ime models have been of he generalized auoregressive condiional heeroskedasic (GARCH) ype, while he coninuous ime models begin wih he jump-diffusion model of Meron (1976), which is a combinaion of a smooh and coninuous process, along wih a much less persisen jump process. A horough undersanding of boh he coninuous and jump componens of volailiy is required o manage risk effecively. The firs goal of his paper is o invesigae wheher differen ypes (equiy, morgage, and hybrid ype) of U.S. REITs markes has significan jump phenomenon by uilizing economeric echniques developed by Anderson, Bollerslev, and Diebold (2007). By using high-frequency (daily) daa, one can effecively separae he coninuous and jump componens of REITs reurns. When esimaing parameers in a jump-diffusion model, i has been difficul for financial economiss o separae jumps from he underlying diffusion process, in par because he acual jump is no readily observable from he ime-series daa of he underlying asse reurns. Mos jump parameer esimaes are based on numerical simulaions, since direc esimaes are difficul o obain in all bu a few special cases (Aı -Sahalia 2004). Tauchen and Zhou (2006) poined ou ha he main message from he empirical lieraure seems o be ha jumps are very imporan in asse pricing, bu he esimaion of jump parameers and he pricing of jump risk are no easy o implemen. This poses a serious pracical challenge o risk managers. This paper uses economeric echniques provided by Anderson, Bollerslev, and Diebold (2007), Barndorff-Nielsen and Shephard (2004), and Tauchen and Zhou (2006) o accuraely esimae he oal volailiy and he volailiy of he underlying coninuous-ime process wih measures hey call he Realized Volailiy (RV) and Bi-power

4 Variaion (BV) measures, respecively. The difference beween hese wo measures provides an unbiased esimae of he jump componen of prices. Based on his echnique, we furher invesigae wheher REITs, sock and bond markes has significan jump risk phenomenon. Moreover, many sudies (e.g., Liu e al. (1990), Peerson and Hsieh (1997), Chiang, Lee and Wisen (2004)) have invesigaed he reurn associaion beween equiy REITs and he general sock marke by using Capial Asse Pricing Model (CAPM, Sharpe 1964) and Fama-French (1993) hree-facor model. Furhermore, beas of he sock marke have responded asymmerically in differen marke condiions. REITs bea was also found o have higher correlaions wih general marke movemens in declining markes han in rising markes (Goldsein and Nelling, 1999; Sagalyn, 1990; Charah, Liang and McInosh, 2000; Chiang, Lee and Wisen, 2004). This asymmeric response of REITs reurns reflec he risk preference of invesors, who dislike downside risks. The second goal of his paper is o furher decompose each of he volailiy componens ino coninuous bea risk and jump bea risk based on he (asymmeric) CAPM model and (asymmeric) Fama-French hree-facor model. This decomposiion is ineresing because radiional sandard facor models of risk implicily assume ha an asse s sysemaic risk is uncorrelaed wih jumps in he marke (i.e., ha he asse s bea does no change on days when he marke experiences a jump). By decomposing we inend o discuss ha is he jump bea risk higher han coninuous bea risk? Is he jump bea risk and coninuous bea risk asymmeric? And we inend o invesigae ha is jump risk almos sysemaic or idiosyncraic (nonsysemaic)? The difference beween coninuous beas and jump beas has imporan implicaions for risk managemen. Wih a oal bea, one knows only he average level of sysemaic risk. However, given an asse s coninuous and jump beas, one can

5 explicily calculae he asse s sysemaic risk condiional on wheher or no he marke experiences a jump. This is imporan for risk managers: if REITs behaves differenly during a severe marke downurn han i does a oher imes, his informaion offers he poenial o significanly improve on calculaions such as Value a Risk (VaR). Moreover, if REITs are combined in a well-diversified porfolio, hen he REIT s sysemaic jump risk is more relevan han he REIT s oal jump risk. This highlighs he imporance of decomposing oal jump risk ino is sysemaic and idiosyncraic componens. 2. Lieraure Reviews The relaionship beween real esae securiies and general financial markes has been exensively explored, wih much of he lieraure focusing on he bivariae relaionship beween REITs and general sock markes. As reviewed below, exising sudies can be generally divided ino wo caegories, which are based on ime series analysis or asse pricing models. Some discree ime series echniques were employed o examine he dynamic relaionship beween real esae and general financial markes. Kallberg, Liu, and Pasquariello (2002) conduced srucural break ess and repored regime shifs in reurns and volailiy relaionships beween real esae and sock markes in eigh Asian markes. Based on a bivariae GARCH model, Coer and Sevenson (2006) found ha daily REIT-sock correlaions generally increased during he period from 1999 o Applying a mulivariae DCC-GARCH model o a seven asse sysem, Huang and Zhong (2006) argued ha during he period from 1999 o 2005, daily condiional correlaion beween REITs and US equiy was always posiive bu had a posiive rend and daily correlaions beween REIT and US bond flucuaed around zero. Also using a DCC-GARCH model, Case, Yang, and Yildirim (2009) examined

6 monhly condiional correlaions beween US sock and REIT markes from 1972 o 2008 and explored he implicaions for porfolio allocaion. Insead of using ime series models, many sudies have invesigaed he reurn associaion beween equiy REITs and he general sock marke based on asse pricing models. Gyourko and Linneman (1988), Gilibero (1993), Myer and Webb (1993), Han and Liang (1995), Liang, Charah and McInosh (1996), and Oppenheimer and Grissom (1998), among ohers, showed ha REITs are exposed o bea risk. The asse pricing models have been applied o invesigae inegraion versus segmenaion beween he real esae marke and he general financial markes since he firs sudy of Liu e al. (1990) on his opic. Liu e al. (1990) used a single-facor model and repored ha he US securiized real esae marke inegraes wih he sock marke, while he US privae commercial real esae marke is segmened from he sock marke. Peerson and Hsieh (1997) showed ha he risk premiums on equiy REITs are significanly relaed o hree Fama-French facors driving common sock reurns, while morgage REIT risk premiums are significanly relaed o wo bond marke facors as well as he hree sock marke facors. Using a series of commonly used muli-facor asse pricing models, Ling and Naranjo (1999) confirmed ha US REITs are inegraed wih he sock marke and he degree of such inegraion has significanly increased during he 1990s, while here is lile evidence for inegraion beween he real esae and sock markes when appraisal-based real esae reurns are used. Using a muli-facor model where sock, bond, and direc real esae reurns as proxies for underlying sae variables deermining hese asse prices, Clayon and Mackinnon (2003) repored ha while hrough 1970s and 1980s he US NAREIT reurns were driven largely by he same economic facors ha drive large cap socks, hey are more closely relaed o boh small cap sock and real esae-relaed facors in 1990s. Downs and Paerson (2005) employed a generalized asse pricing model (i.e.,

7 a discoun facor model) and showed ha US REIT reurns from 1972 o 1991 canno be fully explained by sock and bond reurns. The sandard equilibrium asse pricing model heorizes a posiive and linear rade-off beween reurn and sysemaic risks of capial asses. However, empirical evidence in small capializaion socks and REITs seems o conradic he heoreical relaionship. Beas of hese socks have responded asymmerically in differen marke condiions. REIT bea was also found o have higher correlaions wih general marke movemens in declining markes han in rising markes (Goldsein and Nelling, 1999; Sagalyn, 1990; Charah, Liang and McInosh, 2000; Chiang, Lee and Wisen, 2004). This asymmeric response of REIT reurns reflec he risk preference of invesors, who dislike downside risks. The resuls have significan implicaions for he porfolio managemen, in paricular, he allocaions of REIT in a mixed asse porfolio. The resuls imply ha REITs are no an effecive risk diversifier for a mixed asse porfolio in recessionary periods. Chahrah, Liang and McInosh (2000) argues ha asymmery in bea is caused by he combined effecs of decaying relaionships beween REIT reurns and general marke reurns and higher sock marke reurns in he recen decade. The dividend yield spread hypohesis was also rejeced, because he asymmeric bea responses were no found in uiliies socks, which share he same high dividend payou characerisics as REITs. They found similariy in he paern of asymmery in bea, bu he variance effecs (Glosen, Jagannahan and Runkel, 1993; Jagannahan and Wang, 1996) ha drive he small capializaion sock bea asymmery were no significan in REITs. Thus, he asymmeric bea hypohesis remains a puzzle.

8 3. Theoreical Framework and Empirical Mehodology 3.1 Jump Deecion Theoreical Framework This secion describes he jump deecion mehodology which is developed by Andersen, Bollerslev and Diebold (2007). Le p () denoe a logarihmic asse price a ime. The coninuous ime jump diffusion process radiionally used in asse pricing is expressed as a sochasic differenial equaion as follows: dp( ) ( ) d ( ) dw( ) ( ) dq( ), 0 T, (3.1) where () denoes a coninuous and locally bounded variaion process, () denoes a sricly posiive sochasic volailiy process, W() is a sandard Brownian moion, q () is a couning process and () is a measure of he size of he jump which condiional on a jump occurring. When jump occurs a ime, he value of dq () is equal 1; oherwise he value of dq () is equal 0. The quadraic variaion for he cumulaive reurn process, r( ) p( ) p(0), is hen [ r, r] 0 0s 2 2 ( s) ds ( s). (3.2) If he jumps do no occur ( q ( ) 0 ), hen he quadraic variaion simply equals he coninuous volailiy (inegraed volailiy) because ha he second reurn on righ-hand side disappears (discouniuious jump). Le r p( ) p( ) denoes he discreely sampled -period reurns., And we define he monhly realized volailiy (RV) by he summaion of he corresponding 1 / daily squared reurns. 1/ 2 1 ( ) r j, j1 RV. (3.3) For noaional simpliciy and wihou loss of generaliy, 1 / is assumed o be an ineger. Hence, follow Anderson and Bollersleve (1998), he realized volailiy converges uniformly in probabiliy o he incremen in he quadraic variaion process

9 by he heory of quadraic variaion, as he sampling frequency of he underlying reurns increase. Tha is RV ( ) ( s) ds ( s) 0 0s. (3.4) Obviously, he realized volailiy can be separaed ino wo processes: he coninuous sample pah process and he jump process. Thus, he quadraic variaion is consisen for he coninuous volailiy wihou jumps. Define he sandardized realized bi-power variaion (BV) as: 2 where / 2 1 1/ j, ( j1), j2 BV ( ) r r, (3.5). Then, as 0 he equaion (3.5) is possible o show, 1 BV 1 ) 0 2 ( ) ( s ds. (3.6) Hence, he conribuion o he quadraic variaion process is a resul of he jumps may be consisenly esimaed by RV BV s. (3.7) 2 1( ) 1( ) ( ) 0s This is fundamenal heory and empirical for his aricle. According o Dunham and Friesen (2008), he raio saisic defines as follow: RJ RV BV T. (3.8) RV Wih he absence of jumps, he equaion (3.8) will converge o a sandard normal disribuion. Then 1 According o Andersen, Bollerslev and Diebold (2007), he realized power variaion (for 0 ) follows ha in general for 0 p 2, RPV 1/ p p / 2 p 1 (, p) p r j, ( s) j1 ds, where 2 p / 2 (1 ( 1)) / (1/ 2) ( p p p E Z ). Therefore, he PRV 1(, p) 2 diverges o infiniy for p 2. And he impac of he disconinuous jump process disappears in he power variaion measures wih 0 p 2, while PRV 1(,2) RV 1( ) converges o he coninuous volailiy plus he jump volailiy.

10 ZJ RJ 2 1 TP {( ) 5} max(1, ) 2 2 m BV N(0,1), (3.9) where m 1/ and TP as shown in he following: 2 TP m 3 4 / 3 m m 2 m j3 r, j2 4 / 3 r, j1 4 / 3 r, j 4 / 3 1 ds 3, (3.10) 4 s where 2 2/ 3 ((4 / 3 1) / 2) / (1/ 2). 4/3 We choose a value which is significan a he 5% criical value o confirm of jumps. Hence, he acual jump calculaes as J sign( r ) ( RV BV ) I, where I is equal o one when jump occurs and zero oherwise. 3.2 Decomposing Sysemaic Risk ino Coninuous and Jump Componens The disincion beween sysemaic and nonsysemaic risk has been explicily recognized a leas since Sharpe (1964), and jumps have been explicily recognized in sochasic volailiy and opion pricing models for many years (Meron 1976; Baes 1991). To dae, lile work has examined he sysemaic and nonsysemaic characerisics of jumps for REITs marke. This secion firs develops an empirical mehodology ha decomposes oal jump risk ino sysemaic and nonsysemaic componens based on he CAPM model and wo-facor model. Nex, we es he asymmeric jump and coninuous bea effecs based on he CAPM model and 2 According o Andersen, Bollerslev and Diebold (2007), he realized power variaion (for 0 ) follows ha in general for 0 p 2, RPV 1/ p p / 2 p 1 (, p) p r j, ( s) j1 ds, where 2 p / 2 (1 ( 1)) / (1/ 2) ( p p p E Z ). Therefore, he PRV 1(, p) 2 diverges o infiniy for p 2. And he impac of he disconinuous jump process disappears in he power variaion measures wih 0 p 2, while PRV 1(,2) RV 1( ) converges o he coninuous volailiy plus he jump volailiy. 3 TP denoes he ri-power quariciy robus o jumps from Barndorff-Nielsen and shephard (2004).

11 wo-facor model. The modified CAPM model I is common o express daily reurns for an asse in erms of a facor model. Wihou loss of generaliy, consider he sandard single-facor model (such as he CAPM) for REIT i reurns: Ri i i RM i, i 1,2,...,168. (3.11) Equaion (3.11) does no disinguish beween he coninuous and jump componens of oal reurn, bu does decompose oal reurn ino sysemaic ) and ( i R M nonsysemaic ) componens. Any marke jump is embedded in R M, while ( i i any nonsysemaic jump unique o REIT i is included in he error erm. In his secion, we furher decompose each REIT s sysemaic risk ino is coninuous and jump componens. This decomposiion is ineresing because sandard facor models of risk implicily assume ha an asse s sysemaic risk is uncorrelaed wih jumps in he marke (i.e., ha he asse s bea does no change on days when he marke experiences a jump). The modified CAPM model is as follows: R i i 1 i RM JM ) 2i ( J, (3.12) M i where J M sign( R ) ( RV BV ) I is he signed magniude of he jump reurn for he sock marke, and I is equal o one when jump of sock marke occurs and zero oherwise. 1i ( RM J M ) represens coninuous sysemaic risk and 2i J M represens jump sysemaic risk of REIT i. The value of i is nonsysemaic risk. The equaion (3.12) is buil up o es wheher he reurn on REIT marke is influenced by jump bea of he sock marke. The exising lieraure views REITs as a hybrid of socks and bonds in erms of

12 reurn and risk exposure in he shor-run (e.g., Ling and Naranjo (1997), Peerson and Hseih (1997), Karolyi and Sanders (1998)), wih increased exposure o real esae revealed in longer erm price dynamics (e.g., Mei and Lee (1994), Gelner and Rodriguez (1998)). Inuiively, REIT reurns should be relaed o reurns on socks because REITs are influenced o some degree by he same macroeconomic variables ha affec sock reurns. The relaively fixed naure of he cash flows derived from income-propery wih long-erm leases and high credi qualiy enans, ogeher wih he high dividend yield REITs provide o invesors, implying ha REIT reurns and risks should also be relaed o macroeconomic variables ha affec bond reurns. Essenially, his means ha reurns o sock and bond indices can ac as proxies for he unobservable sae variables ha are common boh o REITs, socks and bonds. Hence, we expand he modified CAPM model o include he bond marke facor, as follows: R i i 1 i ( RM J M ) 2iJ M 3 i ( RB J B ) 4i J B i (3.13) where J B sign( R ) ( RV BV ) I is he signed magniude of he jump reurn for he bond marke, and I is equal o one when jump of bond marke occurs and zero oherwise. The Asymmeric Response Model Differen versions of asymmeric response models have been used by researchers o es he asymmeric bea hypohesis, (e.g., Glascock (1991) and Goldsein and Nelling (1999)). We decompose sysemaic risk ino coninuous and jump componens and include a dummy variable for declining marke sae o es he asymmery coninuous bea hypohesis. Furhermore, we include a dummy variable for down-jump signal o es he asymmery jump bea hypohesis. Based on a CAPM model, in which he reurn on a individual REIT, R i, is specified as a linear funcion of sock, as follows:

13 Ri i 1 i( RM JM ) 2iJM 3 id( RM JM ) 4iDJ JM i, (3.14) where 3i is he comparable coninuous sysemaic risk coefficien of sock markes represened by a marke dummy D, which has a value 1 when he excess marke reurn is negaive; and 0 oherwise. If he esimaed value of 3i is significanly non-zero, he asymmeric coninuous bea exiss. If he coefficien is posiive, we have he similar resuls as Goldsein and Nelling (1999), which imply a higher sysemaic risk in declining markes han in rising markes. However, if 3i is significan and negaive, we have he Glascock (1991) resuls, which imply ha REITs provide effecive risk diversificaion in non-recessionary periods. Furhermore, 4i is he comparable jump sysemaic risk coefficien of sock markes represened by a marke dummy D J, which has a value 1 when he jump magniude is negaive; and 0 oherwise. If he esimaed value of 4i is significanly non-zero, he asymmeric jump bea exiss. If he coefficien is posiive, i implies a higher jump sysemaic risk in bad signal han in good signal. The expanded empirical asymmeric response CAPM framework, in which he reurn on a individual REIT, R i, is specified as a linear funcion of sock and bond, is defined as follows: Ri i 1i ( RM J M ) 2i J M 3i D( RM J M ) 4i DJ J M 5i ( RB J B ) 6i J B 7i D( RB J B ) 8i DJ J B i (3.15) where 7i is he coninuous sysemaic risk coefficien of bond markes represened by a marke dummy, D, which has a value 1 when he excess marke reurn is negaive; and 0 oherwise. 8i is he jump sysemaic risk coefficien of bond markes represened by a marke dummy D J, which has a value 1 when he jump magniude is negaive; and 0 oherwise.

14 4. Empirical Resuls 4.1 Daa Descripion Our sample conains he U.S. hree markes: he real esae marke, he bond marke and sock marke over he period from January 1, 2000 o December 31, 2008, including 2263 daily observaions. The REITs daa of real esae marke are aken from he Cener for Research in Securiy Price (CRSP) daabase. The S&P 500 index and he Treasury bond fuures Conrac on a daily inerval over he same sample periods are used o compue he general sock marke reurns and he bond marke reurns. These daa are available in he Global Financial Daa daabase (GFD). Before esing for jump risk, we firs check wheher he series is saionary using he Augmened Dickey and Fuller mehod (ADF). The ess do no rejec he null hypohesis. Hence, all daa are convered ino he form of raes of change by calculaing hem as he difference of he naural logarihms of he daa series. 4.2 Empirical Properies of he Daa Table 1 shows he cross-secional summary saisics for realized volailiy (RV) and bi-power variaion (BV) for all samples including 168 firms of U.S. REITs, S&P 500 index and ineres rae on mid-erm bond. In panel A, he variable of RV approximaes he oal monhly reurn variance, while he BV esimaes he coninuous reurn variance. The sandard deviaions are calculaed as he square roo of hese variables. The average value of 1/ 2 RV in U.S. REITs, S&P 500 index and Treasury bond fuures reurns are 9.55, 5.24 and 2.71 percen, respecively. The resuls suggesed ha he real esae marke has more volailiy han equiy and bond markes in U.S.. This reason maybe ha he mos U.S. REITs markes are equiy ype (shown in Figure 1) and hus have he maximum value of U.S. REITs is %. Nex, in order o measure he proporion of coninuous volailiy o oal volailiy, we also

15 consruc he raio of BV/RV. For sock marke, approximaely 44.5 percen of he oal variance is due o coninuous variance. Coninuous variance conribues approximaely 45.1 percen of oal variance for he real esae marke. Panels B and C presens risk measures and he characerisics of he jumps. Toal risk is compued as he variance of he oal monhly reurn, while jump risk is he variance in monhly jump reurns 4. The difference beween jump risk and oal risk is coninuous risk, which is defined as he variance of he coninuous monhly reurn. The percenage of oal risk aribuable o jumps is calculaed as he jump variance divided by he oal variance. Jump conribues approximaely 19 percen (0.015% 0.077%) of oal variance for he real esae marke, 9 percen of oal variance for he sock marke, while 40 percen of he oal variance is due o jump componen in he bond marke. Furhermore, he raio beween jump monhs and oal monhs is he jump frequency. On days when a jump occurs, he jump size is calculaed as he square roo of he difference beween RV and he BV measure. As for he jump frequency, jumps in he real esae marke occur on 5.71 percen of rading days, while he equiy marke and he bond marke jumps occur on 3.18 and percen of rading days. Therefore, he bond marke has higher jump frequency han he oher markes in U.S.. Here we find he ineresing phenomena, alhough he bond marke has higher jump frequency bu i has smalles jump size han he oher markes. Figure 2 shows he numbers of REITs wih significan jump risk from January 1, 2000 o December 31, During he all period, we find ha he average numbers of REITs wih significan jump risk is approximaely 10 numbers for each day. Pracically, he average numbers of REIT wih significan jump risk has approximaely 20 numbers for each day during he period from year 2000 o year One possible explanaion 4 The value is equal o 1 on jump days; oherwise, he value is equal 0.

16 for his is ha he ransacion in real esae marke behaves more acive and makes he reurn and volailiy (jump risk) of REITs markes increases. Hence, he dividends of equiy REITs are higher han 10 year mauriy U.S. reasury yield from year 2000 o year 2003 (shown in Figure 3).

17 % of oal REITs marke capializaion by ype Table 1 Summary Saisics of Reurn Daa Cross-secion disribuion for U.S. REITs Sample means repored Mean Median Max Min S&P 500 Bond marke Panel A: Jump Model Parameers RV 2.091% 1.280% % 0.480% 0.388% 0.082% 1/ 2 RV 9.548% 8.307% % 6.100% 5.236% 2.707% BV 0.807% 0.562% 7.030% 0.199% 0.179% 0.037% 1/ 2 BV 6.257% 5.555% % 3.958% 3.506% 1.815% BV / RV % % % % % % ( RV 1/ 2 BV / ) % % % % % % Panel B: Risk Measures Toal risk 0.077% 0.044% 0.746% 0.017% 0.011% 0.005% Jump risk 0.015% 0.004% 0.204% 0.000% 0.001% 0.002% Coninuous risk 0.063% 0.037% 0.625% 0.016% 0.009% 0.003% Panel C: Properies of Jump Risk Jump frequenly 5.705% 5.141% % 2.298% 3.181% % Jump size % 80% Hybrid REITs Morgage REITs 60% 40% Equiy REITs 20% 0% Figure 1: Number of U.S. REITs Type from 1990 o 2008 (Source: Daa available from NAREIT)

18 Figure 2: Number of REITs wih Jump Risk from 2000 o 2008 Figure 3: Monhly Equiy REITs Dividend Yield Spread from 1990 o 2012 (Source: Graph available from news release of NAREIT) 4.3 Empirical resuls Jump bea vs. coninuous bea? The resul is shown in Table 2. Panel A repors he coefficiens of he radiional CAPM model for U.S. REITs. The oal sysemaic risk in Panel A is esimaed by regressing oal monhly reurns for REITs on he oal monhly reurn for he sock

19 marke. The average oal bea for he firms in our sample is abou 0.72, which suggess ha sysemaic risk of sock marke has a posiive effec on real esae marke. These findings are compaible wih Gilibero (1993), Han and Liang (1995) and Oppenheimer and Grissom (1998), who indicaed ha he REITs are exposed o bea risk. However, if he sock price has jump risk, radiional CAPM model will fail o capure imporan characerisics of abnormal shock even. Hence, o invesigae furher wheher he coninuous componens and jump componens of he sock marke have differen effecs on he real esae marke, i furher decomposes oal sysemaic risk ino is coninuous and jump componens, shown in Panel B. We find ha he coefficiens of 1i (he average coninuous bea) and 2i (he average jump bea) are 0.76 and 0.74, and he difference beween he coninuous and jump bea is significan a he 1% level. Therefore, i will be meaningful o explore he jump and coninuous componens of he sock marke based on he CAPM model. Nex, considering he jump bea risk, he values of 1i and 2i are saisically significan for mos U.S. REITs (wih 139 numbers of REITs, 139/168=82.74 percen), indicaing ha coninuous bea risk and jump bea risk in he sock marke has an effec on he real esae marke. Tha is, he reurns on real esae marke are correlaed wih he sock marke when he sock marke experiences a jump. Many sudies have invesigaed he reurn associaion beween equiy REITs and he general sock marke (e.g., Ambrose e al, 1992; Myer and Webb, 1993). However, one limiaion of CAPM framework is ha i ignores he common facors ha could influence he observed marke. For example, Glascock, Lu and So (2000) examines he inegraion of REITs, sock and bond reurns. They show ha REITs are coinegraed wih he bond marke. Therefore, a muli-facor reurn generaing approach is uilized in order o capure he relaionship of REITs, sock and bond reurns. Table 3 shows he coefficiens of he wo-facor model of hese variables. In

20 Panel A, he coefficiens of oal reurn bea of sock marke are significan for mos U.S. REITs (wih 137 numbers of REITs, 137/168=81.55 percen). Comparing wih sock marke, mos of he coefficiens of oal sysemaic risk are no significan in bond marke (wih 29 numbers of REITs, 29/168=17.26 percen). This indicaes ha mos of he reurns of REITs are no influenced by he U.S. bond marke. One possible explanaion is ha mos of U.S. REITs are equiy ypes, and hus have lower relaionship wih he bond marke. As menioned above, similarly we decompose each sock sysemaic risk ino is coninuous and jump componens in wo-facor model. This decomposiion is ineresing because sandard wo-facor model of risk implicily assume ha an asse s sysemaic risk is uncorrelaed wih jump in he sock marke. The difference beween he coninuous and jump bea is significan in sock marke, implying i is necessary o separae he coninuous risk and jump risk from he sysemaic risk of sock reurn in wo-facor model. This paper furher analyzes he explanaory power o he REIT reurn when we decompose bond marke. The resul is shown in Panel B. The significance of differen value beween he coninuous and jump componens in bond marke, suggesing ha coninuous bea risk and jump bea risk explains he discrepancies beween real esae and bond markes. Hence, o decompose sysemaic risk ino coninuous and jump componens is imporan for sock and bond markes. We also find ha coninuous bea risks are higher han jump bea risks. Economically, his means he REITs reurns are mos correlaed wih he sock and bond markes reurns on days when sock and bond markes do no experience a jump. Is coninuous bea asymmeric? We furher expand our model o include a dummy variable for declining marke sae o es he asymmeric coninuous bea hypohesis, and include a dummy variable

21 for negaive jump size o es he asymmeric jump bea hypohesis. The resuls are shown in Panel C of Table 2 and Table 3. In Table 2, he coefficiens of 3i are used o infer he asymmeric in coninuous risk exposures; In Table 3, he coefficiens of 3i and 7i are used o infer he asymmeric in coninuous risk exposures. The resuls of asymmeric coninuous bea ess in Panel C are esimaed similarly using Glascock (1991) approach ha includes a dummy variable for bear marke, which is defined by negaive excess marke reurns, in he sandard CAPM framework. We find he significan posiive value of 3i in percen of REITs (75 168=44.64%) in Table 2 and percen (63 168=37.50%) in Table 3. In he bond marke, here are approximaely 47 percen in U.S. REITs o have significan posiive 7i value in Table 3. Furhermore, we assumed ha 1 i 3 i and 5i 7i o es he asymmeric relaionship of coninuous bea. The resul shows he difference beween he symmery coninuous and asymmery coninuous beas are significan a he 1% level in sock and bond marke, suggesing ha he resuls suppor he asymmeric coninuous bea hypohesis and compaible wih Sagalyn (1990), Goldsein and Neling (1999), ha claim he REITs reurns more rack he sock reurns in declining marke han in rising marke. Is jump bea asymmeric? Excep he business cycle, he good news and bad news of he marke could influence he relaionship beween differen markes. This concep of leverage effec is firs proposed by Black (1976). Many sudies have widely used his concep o analyze he financial markes, e.g., Koumos, 1998; Wu and Xiao, 2002; Chen, e al., 2003; Mohany, When he markes was suffered by he posiive (good news) and he negaive (bad news) of he informaion shock, if he bad news of he informaion affec he volailiy of asse price more large han he good news of he informaion,

22 namely he exisence of he leverage effec in his marke. For his reason, we wan o examine wheher he asymmeric jump bea exiss when markes suffer he shocks in his subsecion. The resuls of asymmeric jump bea ess are shown in Panel C of Table 2 and Table 3. In Table 2, he coefficiens of 4i are used o infer he asymmeric in jump risk exposures; In Table 3, he coefficiens of 4i and 8i are defined as he asymmeric in jump risk exposures. We find ha he values of 4i in sock marke in Table 2 are significan and posiive in 72 numbers (42.86 percen) of U.S. REITs. In Table 3, i sill has 77 numbers (45.8 percen) of REITs o have significan posiive 4i, indicaing he jump bea risk wih bad news is higher han wih good news. This resul appears o be consisen wih Huang, Lee and Tzou (2009), ha finds evidence of a srong asymmeric effec wih respec o he impac of pas good and bad news in he U.S., Ausralia and Japan EREITs. In he bond marke, he significan value of 8i are approximaely 46 percen in U.S. REITs. Moreover, by applying a es o es he asymmeric relaionship beween bad news and good news for sock marke and bond marke, i is assumed ha 2i 4i and 6i 8i. The resuls show he difference beween he symmery jump and asymmery jump bea are significan a he 1% level in sock and bond marke, suggesing ha here is a suppor for he asymmeric jump bea hypohesis. I indicaes ha imporan pieces of bad news end o increase he volailiy in U.S. excess REITs reurns. Hence, separaing he posiive jump size and negaive jump size is imporan when sudying he impac of news on financial markes. The bad shocks probably limi he wealh and invesmen plans of he invesors. Ignoring his phenomenon, he invesor maybe undervalues he relaionship beween asses and o work ou he wrong decisions. Furhermore, our resul implies ha when pricing he derivaives of he REITs, i is necessary o disinguish beween he posiive and he negaive of jump size of he REITs and hus jump diffusion model maybe fail o capure he dynamic process of

23 REITs. Is jump risk nonsysemaic? In he equaion (3.13), i separaes sysemaic risk ino coninuous and jump componens. However, all nonsysemaic risks do no separaed coninuous and jump risk. Therefore, in his secion, we decompose nonsysemaic risk ino coninuous and jump componens. As menioned above, he value of i is oal nonsysemaic risk. Define he nonsysemaic jump reurn ( NonJump ) and nonsysemaic coninuous reurn ( NonCon ) as follows: ˆ ˆ J, NonJump Ji 2iJiR 4i ib NonCon v J ˆ J ˆ J ). i ( i 2i ir 4i ib Hence, i can decompose oal risk ino sysemaic (jump and coninuous) and nonsysemaic risks (jump and coninuous). This sudy defines oal risk as he variance of oal monhly reurns and he sysemaic coninuous (jump) risk as squared coninuous (jump) sysemaic risk muliplied by he variance of coninuous (jump) marke reurns. Nonsysemaic jump risk is defined as he sample variance of NonJump and nonsysemaic coninuous risk is defined as he sample variance of NonCon. The resul is shown in Panel B of Table 2 and Table 3. I finds ha mos jump risk is nonsysemaic, suggesing ha accouning for jump risk is mos imporan in a non-diversified conex, where nonsysemaic risk is presen. In Summary, our sudy resuls in several imporan findings. Firs, he coninuous bea is higher han he jump bea in U.S. REITs marke, hus he coninuous bea is he mos relevan measure of co-movemen wih he marke on days when sock and bond markes do no experience a jump. The U.S. REITs seem o behave differenly during a severe marke downurn han i does a oher imes, and his informaion offers he poenial o significanly improve on calculaions such as Value a Risk (VAR).

24 Second, he asymmeric coninuous beas are posiive in sock and bond markes, indicaing he coninuous beas are higher during down markes and lower during up markes. Such behavior may imply ha, REITs reurns would be more affeced during periods of significan marke decline. Third, his evidence indicaes ha here is asymmeric/leverage effec in risk premia of sock and bond effecs on REITs reurns. Finally, mos jump risk is nonsysemaic, wih sysemaic jump risk conribuing less han 6% of oal reurn variance. This would sugges ha accouning for jump risk is mos imporan in a non-diversified conex where nonsysemaic risk is presen.

25 Table 2: Resuls of decomposing sysemaic risk ino coninuous and jump componens for CAPM model Mean Median Max Min Sd Dev. No.(Percen) Panel A: Tradiional CAPM R R i i 1 i M i Toal reurn bea(s) (80.36%) Panel B: Modify CAPM_ Coninuous and Jumps Beas R ( R J J i i 1 i M M ) 2i M i Coninuous bea(s) (82.74%) Jump bea(s) (81.55%) Differece(Con-jump) *** Percenage oal risk Sysemaic jump risk(s) 6.00% 5.09% 16.54% % Sysemaic coninuous risk(s) 4.53% 4.01% 12.47% 0.001% 3.73% Nonsysemaic jump risk(s) 45.39% 45.70% 52.93% 34.41% 5.12% Nonsysemaic coninuous risk(s) 44.08% 44.83% 50.42% 36.34% 3.63% Panel C: Modify CAPM_ Asymmery Coninuous and Jumps Beas R ( R J ) J D( R J ) D J i i 1i M M 2i M 3i M M 4i J M i If RM 0, hen D 1 If J 0, hen D 1 M J Esimaed parameer Coninuous bea (8.33%) Jump bea (8.33%) Coninuous bea (44.64%) Jump bea (42.86%) Differece(Con1-jump2) Differece(Con3-jump4) *** Differece(Con1-con3) *** Differece(Jump2-jump4) *** Noe: Aserisks denoe saisical significance a he 1% (***), 5% (**), or 10% (*) level, respecively. The S denoes he sock marke.

26 Table 3: Resuls of decomposing jump risk ino sysemaic and nonsysemaic componens for wo-facor model Mean Median Max Min Sd Dev. No.(Percen) Panel A: Tradiional Three-Facor Model R R R i i 1 i M 2i B i Toal reurn bea 1(S) (81.55%) Toal reurn bea 2(B) (17.26%) Panel B: Modify Three-Facor Model _ Coninuous and Jumps Beas R ( R J ) J ( R J J i i 1 i M M 2i M 3i B B ) 4i B Esimaed parameer Coninuous bea1(s) (80.36%) Jump bea2(s) (79.76%) Coninuous bea3(b) (19.64%) Jump bea4(b) (19.64%) Difference(Con1-jump2) *** Difference(Con3-jump4) *** Percenage oal risk Sysemaic jump risk(s) 5.55% 5.28% 15.89% 0.01% 4.50% Sysemaic coninuous risk(s) 4.20% 4.15% 11.62% 0.00% 3.42% Sysemaic jump risk(b) 0.88% 0.29% 9.28% 0.00% 1.43% Sysemaic coninuous risk(b) 0.40% 0.16% 4.29% 0.00% 0.65% Nonsysemaic jump risk 45.13% 45.31% 52.78% 35.15% 4.97% Nonsysemaic coninuous risk 43.85% 44.12% 49.18% 35.50% 3.46% Panel C: Modify Three- Facor Model _ Asymmery Coninuous and Jumps Beas R ( R J ) J D( R J ) D J ( R J ) J D( R J D J i i 1 i M M 2i M 3i M M 4i J M 5i B B 6i B 7i B B ) 8i Esimaed parameer Coninuous bea1(s) (8.93%) Jump bea2 (S) (8.93%) Coninuous bea3 (S) (37.50%) Jump bea4 (S) (37.50%) Coninuous bea5 (B) (48.21%) Jump bea6 (B) (47.02%) Coninuous bea7 (B) (46.43%) Jump bea8 (B) (45.83%) Difference (Con1-jump2) ** Difference (Con3-jump4) *** Difference (Con5-jump6) *** Difference (Con7-jump8) *** Difference (Con1-con3) *** Difference (Con5-con7) *** Difference (Jump2-jump4) *** Difference (Jump6-jump8) *** Noe: Aserisks denoe saisical significance a he 1% (***), 5% (**), or 10% (*) level, respecively. The S denoes he sock marke and he B denoes he bond marke. i J B i

27 5. Conclusions This paper uses an economeric echnique which is developed by Andersen, Bollerslev and Diebold (2007) o verify he exisence of jump in REITs, sock and bond markes in U.S.. This es shows ha jump conribues approximaely 62.4 percen of oal variance for all REITs in Taiwan. Nex, previous sudies only know he average level of sysemaic risk (oal bea). We exend he CAPM and hree-facor models o decompose he sock and bond marke s sysemaic risk ino he coninuous bea risk and jump bea risk. Our empirical resuls find ha jump bea risks are higher han coninuous bea risks. Economically, his means he REITs co-move wih he sock marke much more on days when sock marke experiences a jump. This implies ha REITs behave differenly during a severe marke downurn han i does a oher imes, his informaion offers he poenial o significanly improve on calculaions such as Value a Risk (VAR). The R-squared of he model ha decomposes jump risk ino sysemaic and nonsysemaic componens improves in Taiwan REITs, compared wih he radiional CAPM and hree-facor model. Third, mos of REITs in Taiwan do no have significanly asymmeric coninuous bea effec and asymmeric/leverage effec. Forh, we decompose nonsysemaic risk of sysemaic reurn in he sock and bond marke ino coninuous and jump componens, and we find ha he mos jump risk is nonsysemaic. This suggess ha accouning for jump risk is mos imporan in a non-diversified conex where nonsysemaic risk is presen.

28 References 1. Aı T-Sahalia, Y., (2004), Disenangling Diffusion from Jumps, Journal of Financial Economics, 74, Anderson, T., and T. Bollerslev, (1998), Answering he Skepics: Yes, Sandard Volailiy Models Do Provide Accurae Forecass, Inernaional Economic Review 39, Anderson, T., T. Bollerslev and F. Diebold, (2007), Roughing I Up: Disenangling Coninuous and Jump Componens in Measuring, Modeling and Forecasing Asse Reurn volailiy, Journal of Economics and Saisics, 89, Ambrose, B. W., E, Ancel and M. D. Giffihs (1992), The Fracal Srucure of Real Esae Invesmen Trus Reurns: The Search for Evidence of Marke Segmenaion and Nonlinear Dependency, Real Esae Economics, 20, Baek, C., (2006), A Comparison of Alernaive Opion Pricing Models: Sysemaic vs. Mixed Sysemaic/Diversifiable Jump Diffusion, Ph.D. disseraion, Universiy of Nebraska Lincoln. 6. Barndorfe-Nielsen, O., and N. Shephard, (2004), Power and Bi-power Variaion wih Sochasic Volailiy and Jumps, Journal of Financial Economerics, 2, Baes, O., (1991), Jumps and Sochasic Volailiy: Exchange Rae Processes Implici in Deusche Mark Opions, Review of Financial Economerics, 9, Black, F. (1976), Sudies in Sock Price Volailiy Changes, Proceeding of he 1976 Meeings of he Business and Economics Saisics Secion, American Saisical Associaion, Booh, G. G., and J. P. Broussard, (2002), The Role of REITs in Asse Allocaion, Finance, 23, Case, B., Yang, Y. & Yildirim, Y., (2009), Dynamic Correlaions among Asse Classes: REIT and Sock Reurns, Journal of Real Esae Finance and Economics, forhcoming. 11. Chen, W. S. C., T. C. Chiang and M. K. P. So, (1992), Asymmerical Reacion o US Sock Reurn News: Evidence from Major Sock Markes Based on a Double-Threshold Model, Journal of Economics and Business, 55, Charah, A., Y. Liang and W. McInosh, (2000), The Asymmeric REIT-Bea Puzzle, Journal of Real Esae Porfolio Managemen, 6, Chiang, K. C. H., M.-L. Lee and C. H. Wisen, (2004), Anoher Look a he Asymmeric REIT-Bea Puzzle, Journal of Real Esae Research, 26, Clayon, J. and G. Mackinnon, (2003), The Relaive Imporance of Sock, Bond

29 and Real Esae Facors in Explaining REIT Reurns, Journal of Real Esae Finance and Economics, 27(1), Coer, J. & Sevenson, S., (2006). Mulivariae Modeling of Daily REIT Volailiy. Journal of Real Esae Finance and Economics, 32, Downs, D.H. and G.A. Paerson, (2005), Asse Pricing Informaion in Vinage REIT Reurns: An Informaion Subse Tes, Real Esae Economics, 33, 1, Gilibero, S. M. (1993), Measuring Real Esae Reurns: The Hedged REIT Index, Journal of Porfolio Managemen, 19, Glascock, J. L.,(1991), Marke Condiions, Risk, and Real Esae Porfolio Reurns: Some Empirical Evidence, Journal of Real Esae Finance and Economics, 4, Glascock, J. L., C. Lu and R. So, (2000), Furher Evidence on he Inegraion of REIT, Bond and Sock Reurns, Journal of Real Esae Finance and Economics, 20 (2), Glosen, L.R., R. Jagannahan, and D.E. Runkle, (1993), On he Relaion beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks, Journal of Finance, 48(5), Goldsein, A. and E. F. Nelling, (1999), REIT Reurn Behavavior in Adancing and Declining Sock Markes, Real Esae Finance, 15, Gyourko, J. and P. Linneman, (1988), Owner-Occupied Homes, Income-Producing Properies, and REITs as Inflaion Hedges: Empirical Findings, Journal of Real Esae Finance and Economics, 1, Han, J. and Y. Liang, (1995), The Hisorical Performance of Real Esae Invesmen Truss, Journal of Real Esae Research, 10, Huang, J. & Zhong, Z., (2006). Time-Variaion in Diversificaion Benefis of Commodiy, REITs, and TIPS, Working Paper, Pennsylvania Sae Universiy. 25. Huang, S. S., Y. H. Lee and Y. P. Tzou, (2009), Risk Premium and Asymmeric Volailiy of Equiy REITs, 2009 Conference on Financial Innovaion and Inerdisciplinary, Hsinchu, Taiwan. 26. Jagannahan, R. and Z. Wang, (1996), The Condiional CAPM and he Cross-Secion of Expeced Reurn, Journal of Finance, 51(1), Kallberg, J. G., Liu, C. H. & Pasquariello, P., (2002), Regime Shifs in Asian Equiy and Real Esae Markes, Real Esae Economics, 30, Kim, Y. S., Rachev, S. T., Bianchi, M. L., Fabozzi, F. J., (2008), Financial marke models wih L evy processes and ime-varying volailiy, Journal of Banking and Finance, 32, Koumos, G., (1998), Asymmeries in he Condiional Mean and he Condiional Variance: Evidence from nine Sock Markes, Journal of

30 Economerics and Business, 50, Liang, Y., A. Charah and W. McInosh, (1996), Aparmen REITs and Aparmen Real Esae, Journal of Real Esae Research, 11, Ling, D. and A. Naranjo, (1999), The Inegraion of Commercial Real Esae Markes and Sock Markes, Real Esae Economics, 27, Liu, C., Harzell, D., Greig, W. & Grissom, T., (1990), The Inegraion of he Real Esae Marke and he Sock Marke: Some Preliminary Evidence, Journal of Real Esae Finance and Economics, 3, Lizieri Colin, Sachell Sephen, Zhang Qi, 2007, The underlying reurn-generaing facors for REIT reurns: An applicaion of independen componen analysis, Real Esae Economics, 35, Meron, R. C., (1976), Opion Pricing when Underlying Sock Reurns are Disconinuous, Journal of Financial Economic, 3, Mohany, P. (2006), A Sudy of Asymmeric Volailiy in he Indian Equiy Marke: A GARCH Approach, Decison, 33, Myer, F. C. N. and J. R. Webb, (1993), Reurn Properies of Equiy REITs, Common Socks and Commercial Real Esae: A Comparison, Journal of Real Esae Research, 8, Oppenheimer, P. and T. V. Grissom, (1998), Frequency Space Correlaion beween REITs and Capial Marke Indices, Journal of Real Esae Research, 16, Peerson, J. and C. Hsieh, (1997), Do Common Risk Facors in he Reurns on Socks and Bonds Explain Reurns on REITs?, Real Esae Economics, 25, Sagalyn, L., (1990), Real Esae Risk and Business Cycle: Evidence from Securiy Markes, Journal of Real Esae Research, 18, Sharpe, W., (1964), Capial Asse Prices: A Theory of Marke Equilibrium under Condiions of Risk, Journal of Finance, 19, Tauchen, G., and H. Zhou, (2006), Realized Jumps on Financial Markes and Predicing Credi Spreads, Finance and Economics Discussion Series, Wu, G. and Z. Xiao, (2002), A Gerneralized Parially Linear Model of Asymmeric Volailiy, Journal of Empirical Finance, 9, Young M. S. and R. A. Graff, (1995), Real-Esae is no Normal - A fresh look a Real-Esae Reurn Disribuions, Journal of Real Esae Finance and Economics, 10,