Time-Variation in Diversification Benefits of Commodity, REITs, and TIPS 1

Transcription

1 Time-Variaion in Diversificaion Benefis of Commodiy, REITs, and TIPS 1 Jing-zhi Huang 2 and Zhaodong Zhong 3 This Draf: July 11, 2006 Absrac Diversificaion benefis of hree ho asse classes, Commodiy, Real Esae Invesmen Truss (REITs), and Treasury Inflaion-Proeced Securiies (TIPS), are well sudied on an individual basis and in a saic seing. In his paper, we documen ha he hree asse classes are in general no subsiues for each oher and ha all ough o be included in invesors porfolios, based on a sample of daily reurn daa from January 1999 hrough December We also find ha diversificaion benefis of he hree ho asses change subsanially over ime. For insance, benefis of TIPS were significan before 2001 bu have been decreasing gradually since hen. On he oher hand, diversificaion benefis from Commodiy and REITs flucuae significanly over he enire sample period. We show ha his observed ime-variaion in diversificaion benefi can be capured by incorporaing ime-varying reurn correlaions. To see he implicaions of his finding for asse allocaion in pracice, we examine he ou-of-sample performance of porfolio sraegies consruced based on a variey of correlaion srucures. We find ha Engle s (2002) Dynamic Condiional Correlaion model ouperforms oher correlaion srucures such as rolling, hisorical, and consan correlaions. Our findings sugges ha diversificaion benefis of he hree ho asse classes do vary subsanially over ime and ha invesors need o use appropriae correlaion esimaes in heir asse allocaion decisions o adjus for such ime variaion. 1 We hank Bill Kracaw, Hao Zhou, and seminar paricipans a he UESTC and he 2006 Journal of Banking and Finance Conference in Beijing for helpful commens and discussions. 2 Deparmen of Finance, Smeal College of Business, Penn Sae Universiy, Universiy Park, PA 16802; Tel: (814) ; jxh56@psu.edu 3 Deparmen of Finance, Smeal College of Business, Penn Sae Universiy, Universiy Park, PA 16802; Tel: (814) ; zxz126@psu.edu

2 1. Inroducion Diversificaion plays an essenial role in asse allocaion. Recenly, especially since he burs of inerne bubble, here have been many sudies on diversificaion benefis of nonradiional asse classes such as commodiy, real esae, and U.S. Treasury Inflaion- Proeced Securiies (TIPS). Rouwenhors and Goron (2005) documen ha commodiy fuure reurns are negaively correlaed wih boh equiy and bond reurns, and argue ha commodiy fuures reurns have been especially effecive in providing diversificaion of boh sock and bond porfolios (p. 28). Roll (2004) analyzes he correlaion of TIPS wih nominal bond and equiy reurns and concludes ha an invesmen porfolio diversified beween U.S. equiies and nominal bonds would be improved by he addiion of TIPS (p. 31). Chun, Sa-Aadu and Shilling (2004) show ha invesmens in real esae will pay off when consumpion growh opporuniies are low, and argue ha insiuional invesors should inves more in real esae o eliminae non-marke risk. Tha is, here is clear evidence on diversificaion benefis of he hree asse classes on an individual basis. As such, one obvious follow up quesion is if invesors should include all hree ho asse classes in heir porfolios given ha all of hem are considered o be an insrumen agains inflaion. Namely, are hese hree asse classes subsiues for each oher? Anoher relaed quesion is: Do heir diversificaion benefis change over ime and, if so, how o adjus for such variaion in asse allocaion in pracice? In his paper, we ry o shed some ligh on hese wo quesions by doing an inegraed analysis of diversificaion benefis of commodiy, real esae invesmen russ (REITs), and TIPS. More specifically, we consider seven asse classes including he hree ho ones (es asse classes hereafer) and four radiional ones (benchmark asse classes hereafer) U.S. Equiy, U.S. Bond, Inernaional Equiy, and Inernaional - 1 -

3 Bond and examine how he es asse classes benefi an invesor s porfolios using daily reurn daa from January 1999 hrough December We es for diversificaion benefis of he hree es asse classes using a variey of mehods. Specifically, we use he Gibbons, Ross, and Shanken (1989) es o examine he saisical significance of changes in he Sharpe raio. To see if a given es asse class can be spanned by oher asse classes, we consider wo spanning ess. One is Huberman and Kandel s (1987) mehod in he absence of shor-sale consrains. The oher is he spanning es developed by De Roon, Nijman and Werker (2001) ha incorporaes shorsale consrains. Finally, o quanify an asse class s diversificaion benefis, we use he increase in he angen porfolio s Sharpe raio afer he addiion of he asse (we also look a he weigh of he asse in he angen porfolio). One finding from our empirical analysis is ha he hree es asse classes are no subsiues for each oher when spanning ess are done using he full sample. However, here is evidence of subsiuion in cerain sub-periods (quarers) when spanning ess are done on a quarerly basis. In mos of such cases commodiy can be spanned by oher asses and hus becomes redundan. Based on he increase in he angen porfolio s Sharpe raio afer he addiion of he es asse class and is weigh in he angen porfolio, TIPS appear o benefi he benchmark porfolio he mos, REITs he nex, and commodiy he leas. We also find ha diversificaion benefis of he hree asse classes change subsanially over ime. For insance, TIPS improvemen in he angen porfolio is very significan before 2001 and hen begins o drop gradually since hen. The ime series behavior of he asse class s weigh in he angen porfolio ells he similar sory. We - 2 -

4 hen show ha we can capure he ime-variaion in diversificaion benefis of he hree es asse classes by using a ime-varying reurn correlaion. If correlaions are indeed ime varying and diversificaion benefis of each es asse class do change over ime, hen how should invesors rebalance heir porfolio accordingly? To answer his quesion, we implemen porfolio sraegies based on a variey of correlaion esimaes and rebalancing frequencies and do an ou-of-sample performance analysis of hese sraegies. We find ha he Dynamic Condiional Correlaion (DCC) model (Engle, 2002) ouperforms hree oher popular correlaion srucures he uncondiional correlaion, he rolling correlaion, and he consan correlaion. To summarize, our empirical findings sugges ha all hree es asse classes benefi a porfolio of radiional asse classes bu heir diversificaion benefis (and he impac) do vary subsanially over ime, and ha invesors need o use appropriae correlaion esimaes in heir asse allocaion decisions o adjus for such ime variaion. There are several relaed sudies in he lieraure. Roll (2004) and Kohari and Shanken (2004) boh find ha subsanial weighs should be given o TIPS in an efficien porfolio. Simon and Huner (2005) use a ime-varying correlaion model (no DCC hough) and show ha TIPS have superior volailiy-adjused reurn relaive o nominal bonds over ime. However, hey canno rejec he null ha TIPS can be spanning by equiies and nominal Treasury securiies in heir spanning ess. Mamun and Visalanachoi (2005) find resuls conrary o Simon and Huner (2005) s ones on spanning. Chen, Ho, Lu and Wu (2004) find ha REITs also help o improve invesors mean-variance fronier of size/book-o-marke sock porfolios in spanning ess. This paper conribues o he lieraure by providing an inegraed analysis of ime-varying - 3 -

5 diversificaion benefis of Commodiy, REITs and TIPS in he lieraure and demonsraing he imporance of DCC (which is saisically a beer measure of imevarying correlaions han he ad-hoc rolling correlaion) asse allocaion in he real world. The organizaion of he paper is as follows: Secion 2 discusses daa and empirical mehods used in our analysis. Secion 3 repors empirical resuls from spanning ess. Secion 4 examines ime-variaion in diversificaion benefis of Commodiy, REITs, and TIPS. We also run a horse race of four alernaive correlaion esimaes based on an ouof-sample performance analysis. Secion 5 addresses he robusness of our empirical resuls. Finally, we conclude wih secion Daa and Mehodology In his secion we describe he reurn daa and he empirical mehods used in his sudy. We firs inroduce he sandard mean-variance spanning es in he absence of shor-sale consrains (Huberman and Kandel, 1987). We hen describe he spanning es wih shor-sale consrains proposed by De Roon, Nijman and Werker (2001). Finally, we review he dynamic condiional correlaion model (Engle, 2002). 2.1 Daa We use daily reurn indexes of he following seven asse classes: Commodiy, REITs, TIPS, U.S. Equiy, U.S. Bond, Inernaional Equiy, and Inernaional Bond. In our analysis, he firs hree asse classes are used as es asse classes and he las four used as benchmark asse classes. The deailed descripions of hese reurn indexes are provided in Table 1. As can be seen from he able, he daily reurn index wih he mos - 4 -

6 recen incepion dae, January 1, 1999, is he Inernaional Equiy index. As a resul, our sample period is from January 1, 1999 o December 31, Panel A of Table 2 repors he annualized uncondiional means and sandard deviaions of he seven reurn series in our sample. As can be seen from he able, commodiy has boh he highes mean reurn (21.23%) and highes sandard deviaion (22.22%) among he seven asse classes considered here. On he oher hand, TIPS have a modes mean reurn (8.24%) and a fairly low sandard deviaion (4.70%). REITs have boh relaively high mean reurn (17.81%) and relaively high sandard deviaion (13.38%). The uncondiional mean reurns of he four convenional asse classes, namely U.S. Equiy, U.S. Bond, Inernaional Equiy, and Inernaional Bond, are 3.46%, 5.61%, 6.36% and 4.12%, respecively. The sandard deviaions of he four are 18.51%, 4.55%, 14.76% and 8.62%, respecively. Panel B of Table 2 repors he uncondiional correlaion marix. Like Rouwenhors and Goron (2005), we also find ha Commodiy is negaively correlaed wih boh U.S. Equiy (wih a correlaion coefficien of ) and U.S. Bond (wih a correlaion of ). I is worh menioning ha Rouwenhors and Goron (2005) find negaive correlaions only in low frequency daa (quarerly and annual) bu no in monhly daa. Our resul here is based on daily daa and, o some exen, complemens o heir finding. REITs are posiively correlaed wih U.S. Equiy wih a correlaion of and negaively correlaed wih U.S. Bond wih a correlaion of TIPS are negaively correlaed wih U.S. Equiy wih a correlaion of and posiively correlaed wih U.S. Bond wih a correlaion of Namely, each of he hree es asse classes is negaively correlaed wih eiher U.S. Equiy or U.S. Bond or boh

7 Anoher ineresing resul shown in Panel B is ha he hree es asses are only weakly correlaed wih each oher. In fac, he correlaion is beween Commodiy and REITs, is beween Commodiy and TIPS, and is only beween REITs and TIPS. This indicaes ha alhough all hree asses are considered o be an insrumen hedging agains inflaion, hey are no subsiues. 2.2 Spanning Tes Specificaions wihou Shor-Sale Consrains Spanning ess are a sandard mehod o sudy diversificaion benefis of a new asse. Namely, one examines wheher he efficien fronier of he es (or new) asse classes and benchmark asse classes can be spanned by he efficien fronier based on he benchmark asse classes only. A relaed es is he so called inersecion es, ha examines if he wo efficien froniers (one including es asse classes and he oher no) inersec a he same angen poin for a given risk-free rae. Figure 1 illusraes wha spanning or inersecion means. See Kan and Zhou (2001) and De Roon and Nijman (2001) for a survey on spanning ess. In he absence of shor-sale consrains, we use he following regression-based spanning es of Huberman and Kandel (1987). (See also Ferson, Foerser, and Keim (1993), De Sanis (1994), Harvey (1995), Bekaer and Urias (1996), and De Roon, Nijman and Werker (2001).) Assume v v v v r = a + ΒR + e (1) where he N 1 vecor r v denoes he reurns of N es asse classes, he K 1 vecor R v denoes he reurns of K benchmark asse classes, B is an N K marix, and a v and e v are - 6 -

8 N 1 vecors. Under his specificaion, inersecion means K benchmark asse classes inersec he K+N asse classes and he risk-free asse wih reurn 1/v, and is equivalen o he following condiions v va + Βi v K i v N = 0 v N (2) where 0 v N is an N 1 vecor of zeros, i v K is a K 1 vecor of ones and i v N is an N 1 vecor of ones. On he oher hand, spanning means K benchmark asse classes span he K+N asse classes and is equivalen o he following condiions v a 0 v v v = N and Β i K = in (3) As such, boh inersecion and spanning ess can be conduced by esing he resricions on he coefficiens in a regression-based framework. Following Ferson, Foerser, and Keim (1993), we use he generalized mehod of momens (Hansen, 1982) o implemen he ess. One advanage of he GMM is ha i conrols for heeroskedasiciy and auocorrelaion. We use he Newey-Wes (1987) 1 3 correcion wih a bandwidh of 1 n (Andrews, 1991) for a sample of n observaions 2 errors. 1 Since boh he regression model and consrains are linear, he GMM versions of he LR and LM ess have exacly he same form as he Wald es (Kan and Zhou, 2001). Therefore, we repor only he resuls of he Wald es of he coefficien resricions in he paper. 1 We also ry alernaive lags (from 4 o 20) and find ha our resuls are robus o differen lag choices

9 2.3 Spanning Tes Specificaions wih Shor-Sale Consrains In he case where shor-sale consrains are imposed, we use he es proposed by De Roon, Nijman and Werker (2001). Consider firs he case where shor sale consrains are imposed on he es asse classes only (bu no on he benchmark classes). Here, inersecion for a given value of v implies ha v v v v α ( v) = va + Βi K i N 0 N. (4) The Wald es saisic (Kodde and Palm, 1986) is given by ξ ( v) = min ( ˆ( α v) α( v))' Var[ ˆ( α v)] { α ( v) 0} 1 ( ˆ( α v) α( v)), (5) where ˆ α ( v) is a consisen esimae of α (v). In his case, ξ (v) follows a mixure of Chisquare disribuion and for a given value c, N 2 P( ξ ( v) c) = P( χ i c) w( N, i, Var[ ˆ( α v)]) (6) i= 0 where w( N, i, Var[ ˆ( α v)]) is he probabiliy ha i of N elemens of a vecor wih N( 0,, Var[ ˆ( α v)] ) disribuion is sricly posiive. The p-value can be calculaed by numerical simulaion as suggesed by Gourieroux, Holly, and Monfor (1982). Similarly, spanning implies ha equaion (4) holds for any v = 1/r riskfree. However, he range of v can be limied before hand if we impose cerain economic assumpions on he risk-free raes. Following De Roon, Nijman and Werker (2001), we limi he risk free rae o be beween 1 and he inercep of asympoe of he lines angen o he efficien fronier of reurns R. Therefore, if we le v min and vmax be he minimum and maximum of v, hen esing for spanning is equivalen o a join es of equaion (4) for boh v min and v max. As before, he Wald es saisics follows a mixure of Chi

10 square disribuion since all inermediae value of v will saisfy equaion (4) if boh v min and vmax do so. Consider nex he case where shor-sale consrains are imposed on boh es and benchmark asse classes. In his case, for any given v, he corresponding porfolio weighs of he opimal porfolio on he efficien fronier of he benchmark asse classes are all non-negaive. For a given v, denoe R v as he vecor of hose benchmark asses (v) wih posiive porfolio weigh only. Le a v (v) and Β be he coefficiens of v v v ( v) ( v) ( v) v ( v) r = a + Β R + e. (7) De Roon, Nijman and Werker (2001) show ha inersecion for a given value of v implies ha ( v) v v ( v) ( v) α ( v) = va + Β i v i v 0. (8) K N N This equaion can be esed in a similar fashion as equaion (4). v 1 K In erms of spanning, consider a K dimensional reurn vecor R = [ R,&&, & R ] and le ϕ = { R 1,, K &&& R }, a se of he reurn componens. Since he number of he subses of ϕ is finie, we can use hese subses o form reurn vecors, (v) v, j = 1, 2, L J. Then, all [ R j ], relevan v can be classified ino a finie number of disjoin ses V [ j] V v { v R v = R [ j] ( v) [ j] [ j] a v and }. Le v and v [ j] min [ j] Β be he coefficiens of [ j] max be he minimum and maximum of, where [ j] V, and v v v [ j] [ j] [ j] v[ j] r = a + Β R + e. (9) De Roon, Nijman and Werker (2001) show ha spanning implies [ j] v v j j [ j] v v [ ] [ ] vmina + Β ik in 0 N v v v v [ j] v a + Β i 0 [ j] max [ j] [ j] ik N N, j. (10) - 9 -

11 The Wald es saisic of he join es of he above resricions follows a mixure of Chisquare disribuion, similar o equaion (6). 2.4 Dynamic Condiional Correlaions (DCC) We use he DCC model inroduced by Engle (2002) o model he ime-varying correlaions among asse classes. The DCC model is based on he idea of esimaing volailiies by generalized auoregressive and condiional heeroskedasiciy (GARCH) processes (Bollerslev, 1986) and he correlaion marix by a GARCH-like process. Therefore, DCC is compuaionally more efficien han a mulivariae GARCH model due o is wo-sep esimaion of univariae GARCH series and GARCH-like correlaion marix. In addiion, DCC has an advanage over oher ime-varying correlaions, such as rolling correlaion (which is widely used in he indusry) because he asympoic properies of DCC are fully undersood (see Engle and Sheppard (2001), and Cappiello, Engle, and Sheppard (2003)). We follow Engle and Sheppard (2001) below. The k 1demeaned reurns series R of k asses are assumed o have he following srucure: R H 1 and Ω ~ N(0, H ) D C D (11) where D is he k k diagonal marix wih he univariae GARCH (p, q) sandard deviaion hi on he i h diagonal of H, C is he k k ime-varying correlaion marix ha follows a GARCH-like process, and The log-likelihood of his esimae is given by: Ω represens he informaion se a ime

12 L 1 T ' ) + log( C ) 1 + ε C = 1 = ( k log(2π ) + 2log( D 2 ε ) (12) where ε N(0, C ). ~ The esimaion is implemened by wo seps. In he firs sep, k univariae GARCH (p, q) are esimaed as: h i P p= 1 i Q i = ω + α r β h (13) i 2 ip i p + q= 1 iq i q for i Q i i = 1,2, L, k, and wih non-negaiviy of variance and resricions α ip + β iq < 1 P p= 1 q= 1 imposed. The second sep is he esimaion of he correlaion marix ha is assumed o follow a GARCH-like process: Q M M M ' m β n ) Q + α m ( ε mε m ) + m= 1 n= 1 m= 1 = ( 1 α β Q (14) M n= 1 n n and * 1 * 1 = Q QQ C (15) where Q is he uncondiional covariance of he sandardized residuals from he firs sage, and * Q is a diagonal marix composed of he square roo of he diagonal elemens of Wih appropriae resricions on he parameers of he k univariae GARCH processes and he correlaion s GARCH-like process, C is a posiive definie correlaion marix (see Engle and Sheppard (2001) for deails). This complees he specificaion of DCC model. In his paper, we implemen a simple version of he DCC model wih P i, Q i, M, and M Q. all se o 1, for i = 1, 2, L, k

13 3. Empirical Resuls from Spanning Tess In his secion, we repor he empirical resuls from spanning ess. We firs consider he case when here are no shor-sale consrains, and hen consider he case where shor-sale consrains are imposed. 3.1 Spanning Tess wihou Shor-Sale Consrains We presen firs some preliminary evidence on he diversificaion benefi of he hree es asse classes based on he whole sample. Figure 2 plos wo efficien froniers in each panel, one based on he four benchmark asses only (U.S. Equiy, U.S. bond, Inernaional Equiy and Inernaional Bond), and he oher based on he four benchmark asses plus one es asse class. We can see ha addiion of each es asse shifs he benchmark efficien fronier noiceably. To see if he shifs are saisically significan, we es he significance of increases in he Sharpe raio (of he angen porfolio) due o such shifs using he Gibbons, Ross, and Shanken (1989) es saisic. Tes resuls, no repored here, indicae ha he increase in he Sharpe raio is saisically significan when adding each of he hree es asses. We hen conduc he Huberman and Kandel spanning es using boh he full sample and sub-samples and repor he es resuls in Table 3. As can be seen from he able, he resuls based on he whole sample indicae ha none of he hree es asse classes can be spanned by he four benchmark asse classes. This is consisen wih he resul from he Gibbons, Ross, and Shanken es menioned earlier. However, observe from he es resuls based on sub-periods shown in Table 3 ha in cerain sub-periods especially in more recen quarers, we canno rejec he null

14 ha he es asses can be spanned. The asse ha displays such a paern mos clearly is TIPS. In his case, he null is rejeced soundly wih p-values less han 0.01% in every quarer (excep Q3 2001) in he firs half of he enire sample period bu canno be rejeced in several quarers in he second half of he period. Overall, he number of quarers wih he null rejeced a 5% level is 19 ou of 28. For commodiy and REITs, he number of quarers wih he null rejeced a 5% level is 10 and 11, respecively. These resuls provide evidence ha diversificaion benefis of he hree es asses are sensiive o ime period. Resuls repored so far in his subsecion are based on porfolios ha are consruced using benchmark asses and only one of he es asses each ime. In pracice, invesors may wan o include all hree es asses in heir porfolios. To see if he hree es asses are subsiues for each oher, we run spanning ess wih six benchmark asses, he four original benchmark asses plus wo of he hree es asses. Tha is we examine diversificaion benefis of each es asse when he oher wo es asses are already included in an invesor s porfolio. The es resuls, shown in Table 4, are similar o wha repored in Table 3. In paricular, he null ha a es asse can be spanned by he benchmark asses (six here) is rejeced based on he whole sample bu is no rejeced in cerain sub-periods. Sensiiviy of he rejecion o ime period is also similar o wha shown in Table 3. The implicaion here is ha he hree es asses are no subsiues for each oher and all ough o be included in an invesor s porfolio. Noneheless, he relaive imporance of he hree asses in he porfolio may vary over ime. As shown laer in he paper, such sensiiviy of diversificaion benefis o ime can be explained by ime-varying reurn correlaions

15 3.2 Spanning Tess wih Shor-Sale Consrains Resuls discussed in Secion 3.1 are based on he assumpion ha shor sales are always allowed. In pracice, invesors someimes face shor-sale consrains. Table 5 repors resuls from spanning ess wih shor-sale consrains on boh es and benchmark asse classes. Observe firs ha as before, he null ha he benchmark asses span a es asse is sill srongly rejeced based on he whole sample. Namely, diversificaion benefis of he hree es asses using he whole sample do no disappear when shor-sale consrains are incorporaed. We also see from he able ha resuls obained wih subsamples are weaker han hose repored in Table 3, in he sense ha he number of quarers here where he null is rejeced a 5% level is noiceably less han he one shown in Table 3 for each of he es asses. The resuls sill indicae a paern of ime varying diversificaion benefis of he here es asses. We also consider he case where shor-sale consrains are imposed on es asses only and obain similar resuls (no shown in he paper). 3.3 Inersecion Tess Secions 3.1 and 3.2 presen resuls obained from spanning ess. We also conduc inersecion ess wih or wihou shor-sale consrains. For breviy, we do no repor es resuls here. Noneheless, he main implicaions from he resuls are he same as hose from spanning ess. Namely, diversificaion benefis of es asse classes are eviden based on he whole sample bu are sensiive o ime wihin he sample period when sub-samples are used

16 4. Time-Variaion in Diversificaion Benefis In his secion we invesigae wha drives he empirical resuls presened in he previous secion, especially he ime sensiiviy of diversificaion benefis of es asses. We firs esimae ime-varying correlaions beween asse reurns using he DCC model and show ha such a correlaion srucure can explain he ime series paern of diversificaion benefis documened earlier. We hen sudy asse allocaion using correlaions based on he DCC model. In paricular, we examine how diversificaion benefis of each es asse class changes over ime using wo measures of diversificaion benefis, one based on he increase in he Sharpe raio of he angen porfolio and he oher based on he change in he angen porfolio weighs. Finally, we examine he ou-of-sample performance of porfolio sraegies consruced based on a variey of correlaion srucures in order o shed some ligh on wha correlaion esimaes o use when doing asse allocaion in pracice. 4.1 Correlaion Esimaes Based on he DCC Model Figure 3 plos correlaions beween each es asse class wih U.S. Equiy and U.S. Bond esimaed using hree alernaive mehods: he rolling correlaion wih a-100 day window (in he doed line on he figure), he uncondiional correlaion (he solid fla line), and he DCC model (he dark solid line). 2 As we can see from he figure, boh DCC and rolling correlaions display considerable deviaions from he uncondiional correlaion from ime o ime during our sample period. 2 The DCC model is esimaed using he UCSD GARCH Toolbox provided by Kelvin Sheppard. See hp://

17 For insance observe from Panel A ha alhough Commodiy has a weak negaive uncondiional correlaion wih boh U.S. Equiy and U.S. Bond ( and , respecively, from Table 1), is DCC and rolling correlaions wih he wo benchmark asses acually flucuae beween -0.4 and 0.4 over he enire sample period. A closer inspecion of he ime series behavior displayed in Panel A indicaes ha ime-varying correlaions can explain he ime variaion in diversificaion benefis of Commodiy shown in Table 3. Firs, he srong rejecion of he null in he whole sample repored in he able is consisen wih Commodiy s weak uncondiional correlaions wih U.S. Equiy and U.S. Bond. Secondly, he iming of hose sub-periods wih a srong rejecion of he null (wih he p-value less han 0.1%) coincides wih he iming of hose low DCC esimaes for he ime-varying correlaion beween Commodiy and U.S. Equiy. Panel B shows he correlaions REITs wih U.S. Equiy and U.S. Bond. Observe ha he correlaion beween REITs and U.S. Equiy is always posiive bu has a posiive rend (based on DCC and rolling correlaions) over he sample period, increasing from a bi over 0.2 in early 1991 o abou 0.7 a he end of On he oher hand, he DCC and rolling correlaions beween REITs and U.S. Bond flucuae around zero and more specifically, are iniially posiive, hen say negaive for more han wo years, and become posiive again since he end of Panel C shows he correlaions REITs wih U.S. Equiy and U.S. Bond. Like REITs and U.S. Bond, TIPS and U.S. Equiy also have a U-shape correlaion srucure ha flucuaes around zero in erms of boh DCC and rolling correlaions. The paern of TIPS s ime varying correlaions wih U.S. Bond is quie sriking. They rise gradually from around 0.41 in early 1999 o as high as 0.85 in mid 2003, and say around ha level in he remaining par of he sample period. The implicaion here is ha he marke

18 differeniaes TIPS from he nominal bonds markedly during he inerne bubble period and ha afer he bubble burs, he wo insrumens become much more similar as he inflaion risk is perceived o be low. This observaion also explains he empirical resul repored earlier in Table 3 ha he null (ha TIPS can be spanned) is srongly rejeced in he firs half of he sample period bu is no so in he second half of he period. In summary, correlaions beween each of he hree es asses and U.S. Equiy or U.S. Bond have subsanial ime variaions during he sample period, and such ime varying correlaions appear o drive he ime sensiiviy of he es asses diversificaion benefis observed earlier. 4.2 Time-Variaion of Sharpe Raios We now use he increase in he Sharpe raio of he angen porfolio due o he addiion of a es asse o he (four) benchmark asses as a measure of diversificaion benefis of he paricular es asse. This allows us o quanify he diversificaion benefi of each es asse and examine is variaion over ime. More specifically, we firs esimae he mean reurn for each asse and DCC beween each es asse and he benchmark asses using he whole sample. (We do no consider rolling correlaions here as hey are ad-hoc and heir asympoic properies are no known.) Nex, we calculae he Sharpe raio of he angen porfolio on a daily basis using daily DCC and volailiy series. This is done for porfolios of four benchmark asses firs and hen for porfolios of he benchmark asses plus each of he hree es asses. This gives us four ime series of he Sharpe raio of he angen porfolio, one for he benchmark porfolio, and hree for he benchmark porfolio wih he addiion of each of he es asses. We hen calculae he increase in he Sharpe raio relaive o he series

19 of he benchmark Sharpe raio for each es asse and obain hree ime series of he increase in he Sharpe raio. Figure 4 illusraes he hree ime series of he increase in he Sharpe raio. We can see ha he addiion of commodiy can increase he Sharpe raio by he value of 0.1 o 0.5 over he sample period. Compared o commodiy, REITs provide a larger diversificaion benefis as i can raise he Sharpe raio by as high as 1.5 in some period. While he TIPS can increase he Sharpe raio by as high as 3 early in he sample period, he diversificaion benefis decrease gradually afer This paern is consisen wih ime variaion in TIPS s diversificaion benefis shown in Table 3. Again, he underlying facor here is he behavior of TIPS s ime-varying correlaions wih U.S. Bond illusraed in Figure 3c. In any case, such dramaic changes of diversificaion benefis as in he case of TIPS indicae he imporance of sudying how diversificaion benefis of nonradiional asse classes vary over ime. Finally, noice from Figure 4 ha diversificaion benefis (as measured by he increase in he Sharpe raio) of he hree es asse classes show a endency of convergence o some exen, a reflecion of he fac ha correlaions among he es asses are geing higher. Table 6 provides he summary saisics of increases in he Sharpe raio for each es asse class. In erms of he average increase over he whole sample period, TIPS provide he larges diversificaion benefi wih an average increase of REITs are he second wih , and Commodiy ranks he las wih Also, he -es saisics show ha he increase in he Sharpe raio is significanly greaer han zero for all hree es asse classes

20 4.3 Time-Variaion of Tangen Porfolio Weighs Anoher way o look a he ime variaion in diversificaion benefis of es asse classes is o examine how heir weighs in he angen porfolio change over ime. The implemenaion here is similar o wha is done in Secion 4.2. However, here we include all seven asses when consrucing porfolios since invesors should have access o all hree es asse classes in pracice. For illusraion, we do no calculae he weighs on a daily basis bu raher ake 18 snap-shos of hese weighs (assuming ha invesors rebalance heir porfolio every 100 days). Resuls shown in Figure 5 indicae ha TIPS make up he larges porion of he angen porfolio in he beginning of sample period bu he porion become smaller in he laer period. In conras, REITs ake up a significan weigh across he enire sample period. The weighs of Commodiy are relaively sable bu are much lower relaively o hose of REITs and TIPS. This paern is consisen wih wha observed in he resuls from spanning ess, ime-varying correlaions, and increases in he Sharpe raio. 4.4 Performance of Porfolio Sraegies Using Alernaive Correlaion Esimaes and Rebalancing Frequencies We have documened ha diversificaion benefis of each es asse class have subsanial ime variaions and so do opimal porfolio weighs. However, he resuls presened so far, in paricular, hose on he angen porfolio s Sharpe raio and weighs are based on in-sample esimaes. In realiy, invesors have o esimae parameers using hisorical daa available a he ime of esimaion. In his secion, o see he implicaions of our analysis for asse allocaion in he real world, we consruc he opimal porfolio using only he informaion available a he ime of he consrucion, allow porfolio

21 rebalances o ake ino accoun ime-varying correlaions, and examine he ou-of-sample performance of such porfolio sraegies. In paricular, we run a horserace among four alernaive esimaes of reurn correlaions. Again, invesors are assumed o have access o all seven asse classes considered here. We now describe how o form our porfolios. Firs, we use he iniial 1,000 observaions o consruc he firs porfolio for a pre-specified (arge) expeced reurn by a proper mix of he angen porfolio and he risk-free asse. Afer ha, he porfolio is rebalanced regularly. Again, only he pas informaion up o he ime of rebalancing is used o form a new porfolio. Given a paricular porfolio sraegy, we calculae is realized sandardized deviaion from he argeed reurn in each rebalancing period and hen average hese realized sandard deviaions. We hen evaluae he performance of a given porfolio sraegy based on is average realized sandard deviaion. This evaluaion crierion is suggesed by Engle and Colacio (2003), We consider four alernaive correlaion esimaes in his exercise. They include DCC, rolling correlaion (wih a 100-day window), hisorical correlaion, and consan correlaion (which is equal o he average of all pair-wise hisorical correlaions). 3 Levels of he arge reurn considered are 5%, 10%, 15%, and 20%. Porfolios are assumed o rebalance every 20, 50, or 100 days. Table 7 repors he average realized sandard deviaion of porfolio sraegies based on each of he four correlaion srucures. One observaion is ha he higher he rebalancing frequency, he beer he performance, regardless of he correlaion srucure used. The inuiion here is ha more rebalancing leads o more updaed adjusmen for 3 The consan correlaion is sudied in Elon and Gruber (1973), Elon, Gruber, and Ulrich (1978) and Elon, Gruber, and Spizer (2005)

22 ime-varying correlaions. Anoher observaion is ha porfolios consruced using DCC have he smalles (average) realized sandard deviaion from he argeed reurns in mos cases. Furhermore, he advanage of DCC is larger when he rebalancing frequency is higher. This is no surprising given he naure of DCC. To ge a beer sense on he impacs of correlaion esimaes on asse allocaion, we now look a he angen porfolio s weighs. Panels A hrough D in Figure 6 illusrae he weighs of he angen porfolio under each of he four alernaive correlaion srucures wih rebalancing every 100 days. I is easy o see ha resuls obained from using alernaive correlaion esimaes differ significanly. This finding suggess ha invesors need o use appropriae correlaion esimaes o adjus for he ime-variaion in diversificaion benefis of he es asse classes. 5. Robusness Checks Resuls repored so far are all based on daily reurn daa. To examine he robusness of our es resuls on he frequency of he reurn daa used, we repea spanning ess using boh weekly and monhly daa. The resuls, shown in Tables 8 and 9, respecively, indicae ha he main findings obained using daily daa sill hold. For insance, none of he hree es asses can be spanned by he benchmark asses in he whole sample and diversificaion benefis are sill ime sensiive. In addiion, TIPS are shown o be more imporan in he early period han in he laer period. However, i is worh menioning ha he power of spanning ess decreases noiceably when he frequency of daa is lower. Kan and Zhou (2001) show ha he power of spanning ess increases wih he number of observaions used in he es. Therefore, i is possible ha our resuls based on daily daa have more power han he

23 resuls based on weekly and monhly daa merely because we have more observaions in daily daa. A beer undersanding of his issue is cerainly of grea ineres bu is beyond he scope of his paper. Spanning ess considered in Secion 4 are uncondiional ess. Here we repea our empirical sudies using condiional spanning ess. There are several mehods ha can incorporae condiioning informaion in spanning ess. (See De Roon and Nijman (2001) for a survey on his subjec.) For example, Cochrane (1996) and Behaer and Urias (1996) use scaled reurns. Alernaively, Shanken (1990) and Ferson and Schad (1996) assume ha he regression coefficiens a v and Β in Eq. (1) are a linear funcion of insrumens. We follow he firs approach in our condiional spanning ess. Table 10 repors he resuls from condiional spanning ess. The insrumens used in he ess are also described here. Noice ha sub-periods used here are no longer quarers as he number of daily observaions in a quarer is no high enough o do a sensible condiional spanning es. We can see ha resuls shown in he able are similar o hose from uncondiional ess (repored in Table 3). Namely, he main findings are robus under condiional spanning ess. Finally, we consider an exended sample period. As menioned earlier, he sample period January 1999 o December 2005 is chosen because he daily reurn index of Inernaional Equiy is inceped on January 2, Since he TIPS were firs inroduced in he early 1997, mos sudies ha involve TIPS have sample periods saring from For beer comparison wih exising sudies, we use he MSCI World Equiy Price Index (excluding U.S.) as a proxy for Inernaional Equiy and, as a resul, are able o exend our sample period o March We hen repea he analysis using his exend sample. We find ha he uncondiional mean and sandard deviaion of indexes

24 reurns are quie sensiive o he lengh of sample period. However, he main findings of his paper he paerns of ime-varying diversificaion benefis of es asse classes sill hold. In paricular, diversificaion benefis of TIPS change significanly afer The only excepion is ha he overall diversificaion benefi of Commodiy is lower using he exended daa. To some exen, his resul acually reinforces he imporance of sudying ime-variaion in diversificaion benefis of asse classes. 6. Conclusions In his paper, we examine diversificaion benefis of hree ho asse classes, Commodiy, REITs, and TIPS, using daily reurn daa from January 1999 hrough December We es for evidence of diversificaion benefis using Gibbons, Ross, and Shanken (1989) es and he Huberman and Kandel (1987) mean-variance spanning es. We quanify an asse s diversificaion benefis using he increase in he angen porfolio s Sharpe raio afer he addiion of he asse. We focus on answering wo quesions: Are he hree ho asse classes subsiues for each oher, given ha hey are all considered o be an insrumen agains inflaion? Do diversificaion benefis of he hree asse classes change over ime and if so, how o ake i ino accoun in asse allocaion in pracice? We find ha he hree asse classes are no subsiues for each oher based on he analysis in a full sample. However, here is evidence on he subsiuion effec in cerain sub-periods (quarers) when he analysis is done on a quarerly basis. In mos of such cases commodiy is redundan. Based on he increase in he angen porfolio s Sharpe raio afer he addiion of an asse and he asse s weigh in he angen porfolio, TIPS dominae REITs which in urn dominae commodiy

25 We find ha diversificaion benefis of he hree asse classes change subsanially over ime. For insance, TIPS s improvemen in he angen porfolio is very significan before 2001 and hen begins o drop gradually since hen. The ime series behavior of he asse class s weigh in he angen porfolio ells he similar sory. We show ha he ime variaion in diversificaion benefis of he hree ho asse classes can be explained by using a ime-varying reurn correlaion. Finally, based on an ou-of-sample performance analysis, we find ha he Dynamic Condiional Correlaion (DCC) model (Engle, 2002) ouperforms hree oher popular correlaion srucures he uncondiional correlaion, he rolling correlaion, and he consan correlaion. Tha is, in order o ake ino accoun ime varying diversificaion benefis, he bes correlaion esimae o use in asse allocaion is he DCC esimae

26 References: Andrews, D. (1991), Heeroskedasiciy and Auocorrelaion Consisen Covariance Marix Esimaion, Economerica, 59, Bekaer, G., and Urias, M. (1996), Diversificaion, Inegraion, and Emerging Marke Closed- End Funds, Journal of Finance, 511, Bollerslev, T. (1986), Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, 31, Cappiello, L., Engle, R., and Sheppard, K. (2003), Asymmeric Dynamics in he Correlaions of Global Equiy and Bond Reurns, ECB Working Paper No. 204 Chen, H., Ho, K., Lu, C., and Wu, C. (2004), An Asse Allocaion Perspecive of Real Esae: he Case of Real Esae Invesmen Truss, Yuan Ze Universiy working paper Chun, G., Sa-Aadu, J., and Shilling, D. (2004), The Role of Real Esae in an Insiuional Invesor's Porfolio Revisied, Journal of Real Esae Finance and Economics, 29, Cochrane, J. (1996), A Cross-secional Tes of An Invesmen-based Asse Pricing Model, Journal of Poliical Economics, 104, De Roon, F., Nijman, T. (2001), Tesing for Mean-Variance Spanning: a Survey, Journal of Empirical Finance, 8, De Roon, F., Nijman, T., and Werker, B. (2001), Tesing for Mean Variance Spanning wih Shor Sales Consrains and Transacion Coss: he Case of Emerging Markes, Journal of Finance, 56, De Sanis, G. (1994), Asse Pricing and Porfolio Diversificaion: Evidence from Emerging Financial Markes, in Mike Howell, Ed.: Invesing in Emerging Markes, Euromoney Books, London Elon, E. and Gruber, M. (1973), Esimaing he Dependence Srucure of Share Prices Implicaion for Porfolio Selecion, Journal of Finance, 28, Elon, E., Gruber, M., and Ulrich T. (1978), Are Beas Bes? Journal of Finance, 23, Elon, E., Gruber, M., and Blake, C. (2004), The Adequacy of Invesmen Choices Offered by 401(k) Plans, Journal of Public Economics, Forhcoming Elon, E., Gruber, M., and Spizer, J. (2005), Improved Esimaes of Correlaion Coefficiens and Their Impac on he Opimum Porfolios, New York Universiy working paper Engle, R. (2002), "Dynamic Condiional Correlaion - A Simple Class of Mulivariae GARCH Models," Journal of Business and Economic Saisics, 20, Engle, R., and Colacio, R. (2003), Tesing and Valuing Dynamic Correlaions for Asse Allocaion, New York Universiy working paper

27 Engle, R., and Sheppard, K. (2001), Theoreical and Empirical Properies of Dynamic Condiional Correlaion Mulivariae GARCH, NBER Working Paper Ferson, W., Foerser, S., and Keim, D. (1993), General Tess of Laen Variable Models and Mean-Variance Spanning, Journal of Finance, 48, Ferson, W., and Schad, R. (1996), Measuring Funds Sraegy and Performance in Changing Economic Condiions, Journal of Finance, 51, Gibbons, M., Ross, R., and Shanken, J. (1989), A Tes of he Efficiency of A Given Porfolio, Economerica, 57, Goron, G., and Rouwenhors, K. (2005), Facs and Fanasies abou Commodiy Fuures, Financial Analys Journal, Forhcoming Gourieroux, C., Holly, A., and Monfor, A. (1982), Likelihood Raio Tes, Wald Tes, and Kuhn-Tucker Tes in Linear Models wih Inequaliy Consrains on Regressions Parameers, Economerica, 50, Harvey, C. (1995), Predicable Risk and Reurns in Emerging Markes, Review of Financial Sudies, 8, Hansen, L. Large Sample Properies of Generalized Mehod of Momens Esimaors, Economeria, 50, Huberman, G., and Kandel, S., (1987), Mean-Variance Spanning, Journal of Finance, 42, Kan, R., and Zhou, G. (2001), Tess of Mean-Variance Spanning, Washingon Universiy working paper Kodde, D., and Palm, F. (1986), Wald Crieria for Joinly Tesing Equaliy and Inequaliy Resricions, Economeria, 54, Kohari, S., and Shanken, J. (2004), Asse Allocaion wih Inflaion-Proeced Bonds, Financial Analyss Journal, 60, Mamun, A., and Visalanachoi, N. (2004), Diversificaion Benefis of Treasury Inflaion Proeced Securiies, Massey Universiy working paper Newey, W. and Wes, K. (1987), A Simple Posiive Semi-Definie, Heeroskedasiciy, Auocorrelaion Consisen Covariance Marix, Economeria, 55, Roll, R. (2004), Empirical TIPS, Financial Analyss Journal, 60, Shanken, J. (1990), Ineremporal Asse Pricing: An Empirical Invesigaion, Journal of Economerics, 45, Simon, D., and Huner, D. (2005), Are TIPS he Real Deal: A Condiional Assessmen of heir Role in a Nominal Porfolio, Journal of Banking and Finance, 29,

28 Table 1: Daa Descripions This able describes he sources and incepion daes of he daily oal reurn indexes used in he paper. Asse Class Descripion Frequency Source Incepion Dae End Dae U.S. Equiy S&P500 Index Daily Bloomberg 01/04/ /30/2005 U.S. Bond U.S. Broad Invesmen Grade (USBIG) Bond Index Daily Daasream 12/31/ /30/2005 In l Equiy MSCI World Index (Excluding U.S.) Daily Bloomberg 01/01/ /30/2005 In l Bond MSCI Sovereign Deb Indices (Excluding U.S.) Daily Daasream 12/31/ /30/2005 Commodiy Goldman Sachs Commodiy Index (GSCI) Daily Daasream 12/31/ /30/2005 REITs Dow Jones Wilshire REIT Index Daily Daasream 01/31/ /30/2005 TIPS U.S. Inflaion-Linked Securiies Index (ILSI) Daily Daasream 03/03/ /30/

29 Table 2: Summary Saisics on Reurns and Correlaions This able repors he summary saisics of he seven daily daa series used in his paper. Sample period is from January 1999 o December Panel A: Uncondiional Mean and Sandard Deviaion (annualized percenage) Mean Sandard Deviaion U.S. Equiy U.S. Bond Inernaional Equiy Inernaional Bond Commodiy REITs TIPS Panel B: Uncondiional Correlaion Marix U.S. Equiy U.S. Bond In l Equiy In l Bond Commodiy REITs TIPS U.S. Equiy U.S. Bond In l Equiy In l Bond Commodiy REITs TIPS

30 Table 3: Spanning Tess wih Four Benchmark Asse Classes: U.S. Equiy, U.S. Bond, Inernaional Equiy and Inernaional Bond This able repors he Wald-es p-values of spanning ess. The asse class being esed is lised in he op of each column. The benchmark asse classes are U.S. Equiy, U.S. Bond, Inernaional Equiy and Inernaional Bond. We conduc spanning ess for each quarer in our sample period. The las row repors he spanning es resuls for he whole sample period. Newey-Wes sandard errors are used o conrol for heeroskedasiciy and auocorrelaion. Daily daa from January 1999 o December 2005 are used. Tes Period p-values for Each Tes Asse Class Commodiy REITs TIPS 1999 Q < Q < Q3 < < Q < Q < Q <.0001 < Q < Q < Q <.0001 < Q < Q Q Q < Q < Q Q Q Q Q Q Q Q Q Q Q <.0001 < Q Q Q Whole Sample <.0001 <.0001 <.0001 Number of quarers wih p < 5% Percenage of quarers wih p < 5% 36% 39% 68%

31 Table 4: Spanning Tess wih Six Benchmark Asse Classes: U.S. Equiy, U.S. Bond, Inernaional Equiy, Inernaional Bond, and Two of he Three Tes Asse Classes This able repors he Wald-es p-values of spanning ess. The asse class being esed is lised in he op of each column. The benchmark (spanning) asse classes are U.S. Equiy, U.S. Bond, Inernaional Equiy, Inernaional Bond, and wo of he hree es asse classes (for example, when Commodiy is he es asse class, we also include REITs and TIPS in he spanning asse classes). We conduc spanning ess for each quarer in our sample period. The las row repors he spanning es resuls for he whole sample period. Newey-Wes sandard errors are used o conrol for heeroskedasiciy and auocorrelaion. Daily daa from January 1999 o December 2005 are used. Tes Period p-values for Each Tes Asse Class Commodiy REITs TIPS 1999 Q < Q < Q3 < < Q < Q < Q < Q < Q < Q < Q < Q Q Q < Q < Q Q Q Q Q Q Q Q Q Q Q < Q Q Q Whole Sample <.0001 <.0001 <.0001 Number of quarers wih p < 5% Percenage of quarers wih p < 5% 21% 39% 68%

32 Table 5: Spanning Tess wih Shor-Sale Consrains This able repors he p-values of spanning ess wih shor-sale consrains on boh es asse classes and benchmark asse classes. The asse class being esed is lised in he op of each column. The benchmark asse classes are U.S. Equiy, U.S. Bond, Inernaional Equiy, Inernaional Bond. We conduc spanning ess for each quarer in our sample period. The las row repors he es resuls using he whole sample. p- values are calculaed using numerical simulaion. Daily daa from January 1999 o December 2005 are used. Tes Period p-values for Each Tes Asse Class Commodiy REITs TIPS 1999 Q Q Q Q Q < Q Q Q Q Q Q Q Q Q < Q Q Q Q Q Q Q Q Q Q Q Q Q Q Whole Sample <.0001 <.0001 Number of quarers wih p < 5% Percenage of quarers wih p < 5% 14% 18% 18%

33 Table 6: Summary Saisics on Difference in Sharpe Raios This able repors he summary saisics of he difference in Sharpe raios wih and wihou each es asse class. For each es asse class, we use he DCC esimaes and mean reurns o calculae he Sharpe raio of angen porfolios of (1) benchmark asse classes only and (2) benchmark asse classes and he es asse class. Difference in Sharpe Raio is he difference of hese wo Sharpe raios (wih and wihou each es asse class). We use 3-monh U.S. Treasury yield as he risk-free rae. Daily daa from January 1999 o December 2005 are used. Tes Asse Mean Sandard Deviaion Median Min Max Commodiy *** REITs *** TIPS *** *** Significanly larger han zero a one percen level

34 Table 7: Empirical Performance of Porfolio Sraegies using Alernaive Correlaion Esimaes and Rebalancing Frequencies This able repors he average realized volailiies (annualized percenage) of porfolio sraegies using alernaive correlaion esimaes and rebalancing frequencies. Correlaion esimaes used are DCC esimaes, rolling correlaion (100-day), hisorical correlaion, and consan correlaion (all correlaion is equal o he average pair-wise hisorical correlaion). The firs 1000 observaions are used o make iniial esimaion and form he firs invesmen porfolios. The porfolios are hen rebalanced every 20, 50, or 100 days. All hisorical daa from he sar of sample period o he ime of rebalancing are assumed o be available in making rebalancing decisions. All seven asse classes are included in each sraegy. Daily daa from January 1999 o December 2005 are used. Panel A: Realized Sandard Deviaion Rebalancing every 20 days Targeed Reurn (annualized %) DCC Average Realized Sandard Deviaion (annualized %) Rolling Correlaion Hisorical Correlaion Consan Correlaion Panel B: Realized Sandard Deviaion- Rebalancing every 50 days Targeed Reurn (annualized %) Average Realized Sandard Deviaion (annualized %) DCC Rolling Hisorical Correlaion Correlaion Panel C: Realized Sandard Deviaion - Rebalancing every 100 days Consan Correlaion Average Realized Sandard Deviaion (annualized %) Targeed Reurn Rolling Hisorical Consan DCC (annualized %) Correlaion Correlaion Correlaion

35 Table 8: Spanning Tess Using Weekly and Monhly Daa No Shor-Sale Consrains This able repors he Wald-es p-values of spanning ess. The asse class being esed is lised in he op of each column. The benchmark asse classes are U.S. Equiy, U.S. Bond, Inernaional Equiy and Inernaional Bond. In Panel A, he daa frequency is weekly. We conduc spanning ess for each year in our sample period. The las row repors he spanning es resuls for he whole sample period. In Panel B, he daa frequency is monhly. Newey-Wes sandard errors are used o conrol for heeroskedasiciy and auocorrelaion. The sample period is from January 1999 o December Panel A: Resuls based on Weekly Daa Tes Period p-values for Each Tes Asse Class Commodiy REITs TIPS < < < Whole Sample <.0001 Panel B: Resuls based on Monhly Daa Tes Period p-values for Each Tes Asse Class Commodiy REITs TIPS Whole Sample

36 Table 9: Spanning Tess Using Weekly and Monhly Daa wih Shor-Sale Consrains This able repors he p-values of spanning ess wih shor-sale consrains on boh es asse classes and benchmark asse classes. The asse class being esed is lised in he op of each column. The benchmark asse classes are U.S. Equiy, U.S. Bond, Inernaional Equiy, Inernaional Bond. In Panel A, he daa frequency is weekly. We conduc spanning ess for each year in our sample period. The las row repors he spanning es resuls for he whole sample period. In Panel B, he daa frequency is monhly. p-values are calculaed by numerical simulaion. The sample period is from January 1999 o December Panel A: Resuls based on Weekly Daa Tes Period p-values for Each Tes Asse Class Commodiy REITs TIPS < Whole Sample <.0001 <.0001 Panel B: Resuls based on Monhly Daa Tes Period p-values for Each Tes Asse Class Commodiy REITs TIPS Whole Sample

37 Table 10: Condiional Spanning Tess This able repors he p-values of condiional spanning ess. The asse class being esed is lised in he op of each column. The benchmark asse classes are U.S. Equiy, U.S. Bond, Inernaional Equiy and Inernaional Bond. The model is: v v v e = r ΒR v v Β i K = i N where N 1 vecor r v denoes he reurns of N es asse classes, he K 1 vecor R v denoes he reurns of K benchmark asse classes, B is an N K marix, and e v is an N 1 vecor. The orhogonaliy condiion is v v v E( e [ R, Z 1 ]) = 0, where Z v 1 is a L 1 vecor of insrumenal variables. The insrumens we used are: a consan, he lagged reurns of benchmark asse classes, he U.S. shor-erm risk-free rae (3-monh U.S. Treasury consan mauriy) and he yield curve slope of U.S. Treasury raes (10-year minus 2-year). We conduc he Hansen s (1982) over-idenificaion es using daa for each year in our sample period. The las row repors he spanning es resuls for he whole sample period. Daily daa from January 1999 o December 2005 are used. Tes Period p-values for Each Tes Asse Class Commodiy REITs TIPS < < < Whole Sample <.0001 <.0001 <

38 R 1/v I III II --- Mean-variance efficien fronier of benchmark asse classes I. Benchmark asse classes span es asse classes and benchmark asse classes II. Benchmark asse classes inersec es asse classes, benchmark asse classes, and risk-free asse, 1/v III. Benchmark asse classes do no span es asse classes and benchmark asse classes σ Figure 1. Illusraions of Spanning and Inersecion Tess. The doed line represens he fronier of benchmark asse classes only. The solid lines denoed by I, II, III, represen differen cases of froniers generaed by he es and benchmark asse classes

39 Froniers wih and wihou Commodiy* Froniers wih and wihou REITs*** Froniers wih and wihou TIPS*** Figure 2. Efficien Froniers wih and wihou Each Tes Asse Class. We use 3-monh U.S. Treasury yield as he risk-free rae. Daily daa from January 1999 o December 2005 are used. * GRS es saisic for he increase in he Sharpe raio due o he addiion of a es asse class is significan a 5% level. *** GRS es saisic for he increase in he Sharpe raio due o he addiion of a es asse class is significan a 1% level

40 Figure 3a. DCC, Rolling Correlaions (wih a 100-day window), and Uncondiional Correlaions of Commodiy wih boh U.S. Equiy and U.S. Bond over Time. Daily daa from January 1999 o December 2005 are used

41 Figure 3b. DCC, Rolling Correlaions (wih a 100-day window), and Uncondiional Correlaions of REITs wih boh U.S. Equiy and U.S. Bond over Time. Daily daa from January 1999 o December 2005 are used

42 Figure 3c. DCC, Rolling Correlaions (wih a 100-day window), and Uncondiional Correlaions of TIPS wih boh U.S. Equiy and U.S. Bond over Time. Daily daa from January 1999 o December 2005 are used