CAN WE BEAT THE BUY-AND-HOLD STRATEGY? ANALYSIS ON EUROPEAN AND AMERICAN SECURITIZED REAL ESTATE INDICES

Transcription

1 INTERNATIONAL JOURNAL OF STRATEGIC PROPERTY MANAGEMENT ISSN X prnt / ISSN onlne 2014 Volume 18(1): do: / x CAN WE BEAT THE BUY-AND-HOLD STRATEGY? ANALYSIS ON EUROPEAN AND AMERICAN SECURITIZED REAL ESTATE INDICES Edde C. M. HUI a, *, Sheung-Ch Phllp YAM b a Department of Buldng and Real Estate, The Hong Kong Polytechnc Unversty, Hung Hom, Kowloon, Hong Kong, Chna b Department of Statstcs, The Chnese Unversty of Hong Kong, Hong Kong, Chna Receved 30 January 2012; accepted 19 November 2012 ABSTRACT. The am of ths paper s to use the Shryaev-Zhou ndex to examne the performances of securtzed real estate ndces of four countres: US, UK, Canada and Germany. The result reveals that the Shryaev-Zhou ndex s a leadng ndcator and can act as a predctor on certan securtzed real estate ndces. Furthermore, our results show that the tradng strategy we constructed accordng to the Shryaev-Zhou ndex generally outperforms the buy-and-hold strategy under the assumpton of no transacton costs. The stronger the predctve power of the Shryaev-Zhou ndex s, the larger extent our tradng strategy beats the buy-and-hold strategy. Ths s useful n strategc property management that property practtoners can follow our strategy to trade real estate stocks/funds n order to ncrease ther profts. KEYWORDS: Securtzed real estate ndex; Shryaev-Zhou ndex; Buy-and-hold ; Tradng strategy; Granger-causalty test REFERENCE to ths paper should be made as follows: Hu, E. C. M.; Yam, S.-C. P Can we beat the buy-and-hold strategy? Analyss on European and Amercan securtzed real estate ndces, Internatonal Journal of Strategc Property Management 18(1): INTRODUCTION To maxmze proft s the common objectve for many nvestors. A commonly known tradng strategy s the buy-and-hold strategy,.e. one should buy a stock and hold t for a long tme. The buyand-hold strategy s based on the effcent market hypothess (EMH). However, ths hypothess may not be true. Hence some people look for a strategy to beat the buy and hold strategy. There have been studes on portfolo return optmzaton for long. The frst of such studes was done by Markowtz (1952), who ntroduced the meanvarance modern portfolo theory (MPT). Hu et al. (2009) ncorporated expert knowledge nto MPT through fuzzy set theory to obtan portfolo return optmzaton n drect real estate nvestment. Consder the followng problem: a person buys a stock at a tme and must sell t wthn a certan * Correspondng author. E-mal: edde.hu@polyu.edu.hk perod of tme, say, one year. What s the optmal tme to sell the stock? It would be deal f one can sell the stock exactly at the maxmum prce over the perod. Unfortunately, ths s mpossble as we can only know the tme the stock prce reaches the maxmum at the end of the perod. Here s a more sensble problem: to mnmze the expected relatve error between the sellng prce and the maxmum prce. Graversen et al. (2001) took a frst step to analyze smlar problems along ths drecton. They solved the problem to stop a Brownan moton so as to mnmze the square error devaton from the maxmum. Later, Shryaev (2002), after whom the Shryaev-Zhou ndex was named, nvestgated the quckest detecton problem for a change of market parameters, whle L and Zhou (2006) revealed the hgh chance of a Markowtz mean varance strategy httng the expected return target. Shryaev Copyrght 2014 Vlnus Gedmnas Techncal Unversty (VGTU) Press Technka

2 Can we beat the buy-and-hold strategy? Analyss on European and Amercan securtzed real estate ndces 29 et al. (2008) developed the Shryaev-Zhou ndex to determne the optmal tme to buy and sell a stock n order to mnmze the average relatve error of sellng prce to maxmum prce. Meanwhle, Du Tot and Peskr (2009) provded a probablstc proof of the result. Yam et al. (2008, 2009, 2012, 2013) resolved the same problem n the bnomal tree settng and hence generalze the Shryaev- Zhou ndex over the correspondng framework. The name of the ndex s attrbuted to ts two founders, A. Shryaev and X. Y. Zhou. The Shryaev-Zhou ndex was the frst to put n practce by Hu et al. (2012) through a numercal applcaton n the Hong Kong property market. However, they just lsted the latest sellng date of each real estate stock, but dd not calculate the resultng proft. The followng queston remans: does the Shryaev-Zhou ndex really yeld a tradng strategy whch can beat the buy-and-hold strategy? Ths s the advantage of usng Shryaev-Zhou ndex that we try to llustrate. Here we use the Shryaev-Zhou ndex to construct a tradng strategy, and apply ths tradng strategy to real estate markets of dfferent countres, and see whether ths strategy can outperform the tradtonal buyand-hold strategy. Ths would be useful for nvestors to formulate a better tradng strategy to ncrease ther profts. In order to study real estate markets, we have to choose approprate real estate ndces. Most tradtonal real estate ndces are constructed from real transacton prces recorded n the market (Chau et al. 2005). Two of the examples are the Centa-Cty Leadng Index and the BRE Index (Hu, Wong 2004), the ndex was later appled by Wong and Hu (2005). However, the observatons of these tradtonal ndces cannot be done contnuously. There s always tme lag between prce change and observaton. Stock prces, on the other hand, can reflect the values of the companes contnuously. Hence they are better than tradtonal real estate ndces. Recently, research workers used econometrc methods to study relatonshp between real estate and stock markets. For example, Okunev and Wlson (1997) tested the exstence of co-ntegraton between the REIT and the S&P 500 ndces. The results ndcated that the real estate and stock markets were fractonally ntegrated. Okunev et al. (2000) conducted both lnear and nonlnear causalty tests on the US real estate and the S&P 500 ndces and concluded that there exsted undrectonal relatonshp from real estate to stock market when usng the lnear test, but there was a strong undrectonal relatonshp n the opposte drecton when usng the nonlnear test. Knght et al. (2005) constructed models of asymmetrc dependence usng the copula functon to examne the relatonshp between securtzed real estate and equty markets. They found that for both U.K. and global markets, the securtzed real estate and equty markets exhbted strong tal dependence partcularly n the negatve tal, suggestng that real estate securtes offer, at best, lmted dversfcaton protecton when other asset markets were fallng. Zhou (2010) appled the wavelet analyss to examne the comovement among nternatonal securtzed real estate markets and the cross-market comovement between the stock and securtzed real estate markets. Rehrng and Sebastan (2011) used VAR models to examne the term structure of return volatlty of UK and US drect and securtzed commercal real estate markets. They found that returns of US REIT, UK drect real estate and property shares exhbted strong mean reverson, whle US drect real estate returns show a consderable mean averson effect over short nvestment horzons. Hu et al. (2011) examned the relatonshp between real estate and stock markets n the UK and Hong Kong by the method of data mnng. They found not only a postve correlaton, but also a co-movement, between the two markets. Moreover, a number of studes showed a long-term postve correlaton between real estate and stock prces (Quan, Ttman 1999; Tse 2001; Low 2006). The above results suggest that the stock prce can be a leadng ndcator of the real estate prce. Therefore, as an alternatve, the ndces we used are securtzed real estate ndces whch have a daly frequency. Thus they can reflect the contnuous change of value, rather than chasng after the prces. Our objectve s to examne the Shryaev-Zhou ndces of securtzed real estate ndces of four countres n Europe and North Amerca: US, UK, Canada and Germany, and to verfy whether the tradng strategy we construct usng the Shryaev- Zhou ndex can beat the buy-and-hold strategy. The paper proceeds as follows: n Secton 2, we brefly descrbe the formula of the Shryaev-Zhou ndex. The statstcal method of estmatng the Shryaev-Zhou ndex s gven n Secton 3. The behavour of the Shryaev-Zhou ndex of securtzed real estate markets of the four European-Amercan countres s gven n Secton 4. Secton 5 explores the relatonshp between each securtzed real estate ndex and ts correspondng Shryaev-Zhou ndex. In partcular, we test whether the Shryaev- Zhou ndex s a leadng ndcator of ts correspond-

3 30 E. C. M. Hu and S.-C. P. Yam ng securtzed real estate ndex, and examne the predctve power of the Shryaev-Zhou ndex. In secton 6, we compare the resultng return from tradng the securtzed real estate ndces of the four countres by two dfferent strateges: the frst strategy s formulated accordng to the Shryaev- Zhou ndces of the securtzed real estate ndces, whle the second strategy s the buy-and-hold strategy. We consder two cases: wthout transacton costs and wth transacton costs. Fnally, a concluson s drawn n Secton THE SHIRYAEV-ZHOU INDEX The Shryaev-Zhou ndex s derved by consderng the problem of mnmzng the gap between the sellng prce and the maxmum prce of the stock: S V * = max Ε[ ] 0 τ T τ, (2.1) S T where: T>0; E denotes expectaton; and ST = max Ss 0 s t s the maxmum of the stock prce. For smplcty, we assume that the contnuously compounded annual nterest rate, r = 0 n ths paper. The Shryaev-Zhou ndex s defned as (Yam et al. 2008, 2009, 2012, 2013; Hu et al. 2012): µ= ( α 0.5 σ2) / σ 2 =α/ σ2 0.5, (2.2) where: a s the annual drft or the growth rate of the stock; and s s the annual volatlty of the stock (a, s are constants). The optmal sellng tme s determned by a smple strategy. If the Shryaev-Zhou ndex s postve then the stock should be kept untl the end of the perod [0, T]. Otherwse t should be sold mmedately (Hu et al. 2012). 3. THE STATISTICAL METHOD OF ESTIMATING THE SHIRYAEV-ZHOU INDEX In the formula of the Shryaev-Zhou ndex (2.2), the drft a and the volatlty s are constants. However, n realty, these parameters can never be constant. Most mportantly, we normally do not know the exact values of a and s. Here we use the movng wndow approach to estmate ther values: for each day ( > n), we use the stock prces from day n = 1 to day to estmate the values of a and s on that day, and hence obtan the estmated value of the Shryaev-Zhou ndex on day. To facltate the statstcal estmaton of Shryaev-Zhou ndex, log(s t ) s frst transformed nto relatve rato u ( 2) S u = log S, (3.1) 1 and hence u = log( S) log( S 1), u can be regarded as the daly return of the stock on day. Hence the mean of daly return of the stock on day ( > n) can be estmated by: n 1 u = u n+ j. (3.2) n j= 1 Here we set n = 130. The estmator of a on day s smply u multpled by the number of tradng days n one year, whch we assume to be 250 days: α ˆ = 250u. (3.3) Besdes the drft term, we also estmate the daly varance on day ( >n) by: n 2 1 = n j j n 1 j= 1 ( ) 2 + s u u. (3.4) Consequently, the estmator of the varance s 2 on day s: σ ˆ = 250s (3.5) 2 2 and the estmator of the Shryaev-Zhou ndex m on day s: αˆ 0.5σˆ µ ˆ = σˆ 2 2. (3.6) 4. SHIRYAEV-ZHOU INDEX OF SECURITIZED REAL ESTATE INDICES OF DIFFERENT COUNTRIES Here we analyze the Shryaev-Zhou ndces of securtzed real estate ndces of four countres n Europe and North Amerca. We select two major economes, UK and Germany, n Europe, and two major economes, US and Canada, n North Amerca. We examne the behavour of the ndces of those countres from tme to tme. The data of the followng securtzed real estate ndces of the four dfferent countres are obtaned from Bloomberg: 1. US: S&P 500 Buldng Index; 2. UK: FTSE 350 Real Estate Index; 3. Canada: S&P/TSX Real Estate Index; 4. Germany: Prme Constructon PERF Index. All the above ndces are comprsed of stocks of real estate companes, and hence can reflect the performances of real estate markets of the correspondng countres. The perod of observaton s from January 2, 1990 to Aprl 28, 2009.

4 Can we beat the buy-and-hold strategy? Analyss on European and Amercan securtzed real estate ndces 31 For each securtzed real estate ndex, we use the formula (3.6) to estmate the Shryaev-Zhou ndex on each day n the perod of observaton. Fgs 1 to 4 show the trends of Shryaev-Zhou ndces of securtzed real estate ndces of the four countres over the perod of observaton. Table 1 shows the descrptve statstcs of the Shryaev-Zhou ndces of securtzed real estate ndces of the four countres. From Fgs 1 to 4, we can see that the Shryaev- Zhou ndces of the securtzed real estate ndces of the four countres followed a smlar trend over a certan perod of tme. For example, the Shryaev- Zhou ndces of securtzed real estate ndces of most countres remaned negatve snce late 2007, when the subprme mortgage crss broke out and caused a global economc recesson n the followng year, showng that the crss affected most of the countres. If we get a closer look at the Shryaev-Zhou ndces of securtzed real estate ndces of US, UK and Canada, we fnd that ther trend had some smlartes: the ndces peaked n 1993, and , and reached troughs n 1990, 1994 and These countres are tradtonal economes, so ther ndces behaved normally. The Shryaev-Zhou ndex of Germany s securtzed real estate ndex behaved dfferently from the other three countres that the local events had a greater mpact on the Shryaev-Zhou ndex than the global events. From late 1995 to early 1996, Germany suffered from an unusually cold weather, causng a lot of deaths and substantal economc loss, so ts securtzed ndex fell sharply. Hence ts Shryaev-Zhou ndex plunged to a record low of On the other hand, global events had a smaller effect on Germany s Shryaev-Zhou ndex. Even though the global fnancal crss occurred n 2008, the magntude of negatvty of the ndex was relatvely small at that tme. From Table 1, we can see that US s securtzed real estate ndex s the most stable among the four ndces, as ts Shryaev-Zhou ndex has the smallest standard devaton of Canada s securtzed real estate ndex s also rather stable. The standard devaton of ts Shryaev-Zhou ndex s The securtzed real estate ndces of UK and Germany are more volatle, as ther Shryaev-Zhou ndces range from over 100 to below 100, and have a standard devaton of over 30. In comparson, the securtzed real estate markets n North Amerca are more stable than those n Europe. Shryaev ndex Shryaev ndex Shryaev ndex Shryaev ndex Shryaev ndex of S5BUILLX Index Date Fg. 1. Shryaev-Zhou ndex of US s securtzed real estate ndex Shryaev ndex of F3REAL Index Date Fg. 2. Shryaev-Zhou ndex of UK s securtzed real estate ndex Shryaev ndex of STREAL Index Date Fg. 3. Shryaev-Zhou ndex of Canada s securtzed real estate ndex Shryaev ndex of CXPO Index Date Fg. 4. Shryaev-Zhou ndex of Germany s securtzed real estate ndex

5 32 E. C. M. Hu and S.-C. P. Yam Table 1. Descrptve statstcs of the Shryaev-Zhou ndces of securtzed real estate ndces of the four countres Country US UK Canada Germany Maxmum Mnmum Mean Standard devaton RELATION BETWEEN EACH SECURITIZED REAL ESTATE INDEX AND ITS CORRESPONDING SHIRYAEV-ZHOU INDEX Ths secton explores the relatonshp between each securtzed real estate ndex and ts correspondng Shryaev-Zhou ndex. In partcular, the predctve power of the Shryaev-Zhou ndex s nvestgated. Fgs 5 to 8 show the cross correlaton functon (CCF) between each securtzed real estate ndex and ts correspondng Shryaev-Zhou ndex. Snce the values of Shryaev-Zhou ndex n the frst 130 days (January 2, 1990 to July 2, 1990) are not avalable (see Secton 3), the frst 130 observatons of each correspondng securtzed real estate ndex are removed so that the two tme seres have the same length. Fgs 5 to 8 show the results preformed by the onlne software Wessa (2012). In each fgure, Lag (k) denotes the number of days the securtzed real estate ndex lags behnd ts correspondng Shryaev-Zhou ndex. In all the four fgures, CCF > 0 for k = 0, showng a postve (but weak) correlaton between the securtzed real estate ndex and ts correspondng Shryaev-Zhou ndex for all four countres. As k reaches ts mnmum, CCF also attans the mnmum, showng that for all four countres, t s least lkely that the Shryaev-Zhou ndex lags behnd ts correspondng securtzed real estate ndex. CCF Fg. 6. CCF between UK s securtzed real estate ndex and ts correspondng Shryaev-Zhou ndex CCF Cross Correlaton Functon Lag (k) Cross Correlaton Functon Lag (k) Fg. 7. CCF between Canada s securtzed real estate ndex and ts correspondng Shryaev-Zhou ndex CCF Cross Correlaton Functon CCF Cross Correlaton Functon Lag (k) Lag (k) Fg. 5. CCF between US s securtzed real estate ndex and ts correspondng Shryaev-Zhou ndex Fg. 8. CCF between Germany s securtzed real estate ndex and ts correspondng Shryaev-Zhou ndex

6 Can we beat the buy-and-hold strategy? Analyss on European and Amercan securtzed real estate ndces 33 Table 2. The result of the Granger causalty test Number of nonseasonal tme lags n test P-value of F-test US UK Canada Germany Another common feature of the four CCF graphs s that for k < 0, the CCF s strctly ncreasng. However, for k 0, the four graphs dffer. For UK and Canada, CCF keeps strctly ncreasng. Ths shows that for these two countres, ther Shryaev- Zhou ndces clearly lead ther correspondng securtzed real estate ndces. However, for Germany, the CCF remans nearly constant, so t s unclear whether ts Shryaev-Zhou ndex leads ts correspondng securtzed real estate ndex. For US, CCF peaks at around k = 0 and decreases slghtly when k further ncreases. It s more lkely that US s securtzed real estate ndex and ts correspondng Shryaev-Zhou ndex are postvely correlated rather than one ndex leadng the other. Ths result reveals that for all four countres, f the Shryaev-Zhou ndex attans a peak (or trough), the correspondng securtzed real estate ndex wll also attans a peak (or trough) sooner or later. In partcular, UK and Canada s Shryaev-Zhou ndces are clear leadng ndcators of ther correspondng securtzed real estate ndces. Next, we examne the predctve power of the Shryaev-Zhou on ts correspondng securtzed real estate ndex. For each of the four countres, we conduct the Granger causalty test aganst the followng null hypothess: H 0 : the Shryaev-Zhou does not Granger-cause ts correspondng securtzed real estate ndex. As before, the frst 130 observatons of each securtzed real estate ndex are removed. Table 2 shows the result preformed by Wessa (2012). From Table 2, for UK and Canada, the Granger causalty test gves p-values of <0.05 for all cases, except when the number of non-seasonal tme lags n test s 7 for UK, and 5 for Canada. In most cases, at 5% sgnfcance level, the null hypothess H 0 s rejected,.e., there s sgnfcant evdence that the Shryaev-Zhou ndex Granger-cause ts correspondng securtzed real estate ndex. Ths shows that for UK and Canada, ther Shryaev- Zhou ndces are good predctors of ther correspondng securtzed real estate ndces. However, how US and Germany, the Granger causalty test gves p-values of <0.05 only when the number of non-seasonal tme lags n test s 10 or 11 for US, and 2 for Germany. For most cases, at 5% sgnfcance level, the null hypothess H 0 s not rejected,.e., there s no sgnfcant evdence that the Shryaev-Zhou ndex Granger-cause ts correspondng securtzed real estate ndex. Hence the predctve power of US and Germany s Shryaev-Zhou ndces on ther correspondng securtzed real estate ndces s smaller than that of UK and Canada s Shryaev-Zhou ndces. Comparng the result of Table 2 wth Fgs 5 to 8, we can see that f the Shryaev-Zhou ndex s a clear leadng ndcator of ts correspondng securtzed real estate ndex, then the predctng power of the Shryaev-Zhou ndex s stronger. 6. THE RESULTING RETURN ON BUYING AND SELLING THE INDICES ACCORDING TO THEIR SHIRYAEV-ZHOU INDICES Here we construct a tradng strategy of the four securtzed real estate ndces by usng ther Shryaev-Zhou ndces, compute the return by usng ths strategy to trade the ndces, and compare the result wth the return by usng the buy-and-hold strategy. The Shryaev-Zhou ndex of each securtzed real estate ndex on each day s estmated by the method descrbed n Secton 3. We make the followng assumptons: 1. We start wth an nfnte amount of money. 2. The transacton prce (buyng and sellng prce) s the closng prce of that day. 3. There are no transacton costs. For each of the four ndces, we construct a tradng strategy as follows: 1. From Day 1 to Day 130 (January 2 to July 2, 1990), do not take any acton as the value of µ ˆ s not avalable yet. 2. On Day 131 (July 3, 1990), f µ ˆ 0, buy one unt of the ndex. Otherwse, do not take any acton.

7 34 E. C. M. Hu and S.-C. P. Yam 3. From Day 132 (July 4, 1990) onwards, trade the ndex accordng to the followng strategy (see Table 3). Table 3. Our tradng strategy from Day 132 onwards µ ˆ 1 µ ˆ Acton 0 0 No acton (keep holdng one unt of the ndex) 0 <0 Sell the entre one unt of the ndex we hold <0 0 Buy one unt of the ndex <0 <0 No acton (keep holdng entre cash) 4. On the last day of the perod (Aprl 28, 2009), sell the entre one unt of the ndex f one s stll holdng one unt of the ndex. Table 4 shows the resultng return of the securtzed real estate ndces of the four countres by applyng the tradng strategy above. Table 5 shows the resultng return of the securtzed real estate ndces of the four countres by holdng one unt of each ndex throughout the whole perod (.e. the buy-and-hold strategy). From Tables 4 and 5, we can see that for the securtzed real estate ndces of UK, Canada and Germany, we can obtan a much greater ndex return and percentage return by followng the tradng strategy accordng to the Shryaev-Zhou ndex. If we hold the ndex throughout the whole perod, we would suffer from a loss, but f we follow the strategy accordng to the Shryaev-Zhou ndex, then we can earn a proft, whch s much better than the result when we use the buy-andhold strategy. The only excepton s US s ndex, of whch we can only get a slghtly larger ndex return usng the tradng strategy accordng to the Shryaev-Zhou ndex, but the percentage return s much lower because we have to buy the ndex several tmes, makng the total cost much larger. In overall, the tradng strategy accordng to the Shryaev-Zhou ndex outperforms the buy-andhold strategy. However, n real lfe stuatons, transacton costs do exst. Ths may affect the results. In the followng, we compare the return of the two tradng strateges on the four securtzed real estate ndces. We assume that the transacton cost of buyng one unt of each ndex s equal to 0.5% of ts transacton prce, and there are no transacton costs of sellng any of the four ndces. The followng Tables 6 and 7 show the resultng return of the securtzed real estate ndces of the four countres by applyng the two tradng strateges. Comparng Tables 5 and 6 wth Tables 3 and 4, we can see that wth the presence of 0.5% buyng cost, the returns by usng the buy-and-hold strategy decrease slghtly only. However, f we use the tradng strategy accordng to the Shryaev-Zhou ndex, the returns dmnsh by a larger extent. For US s securtzed real estate ndex, the buy-andhold strategy even outperforms the tradng strategy accordng to the Shryaev-Zhou ndex. For the securtzed real estate ndces of the other three countres, the tradng strategy accordng to the Shryaev-Zhou ndex stll provdes a greater return than the buy-and-hold strategy. Ths s because the buy-and-hold strategy requres nvestors to Table 4. The return of the securtzed real estate ndces of the four countres by the tradng strategy accordng to the Shryaev-Zhou ndex Countres US UK Canada Germany Index return Percentage return 0.05% 5.75% 1.03% 0.38% Table 5. The return of the securtzed real estate ndces of the four countres by holdng one unt of each ndex throughout the whole perod Countres US UK Canada Germany Index return Percentage return 6.74% 13.91% 53.25% 8.88% Table 6. The return of the securtzed real estate ndces of the four countres by the tradng strategy accordng to the Shryaev-Zhou ndex, wth the presence of 0.5% buyng cost Countres US UK Canada Germany Index return Percentage return 0.45% 5.23% 0.52% 0.12%

8 Can we beat the buy-and-hold strategy? Analyss on European and Amercan securtzed real estate ndces 35 Table 7. The return of the securtzed real estate ndces of the four countres by holdng one unt of each ndex throughout the whole perod, wth the presence of 0.5% buyng cost Countres US UK Canada Germany Index return Percentage return 6.21% 14.34% 53.49% 9.33% Table 8. No. of tmes of buyng the securtzed real estate ndces of the four countres when applyng the strategy accordng to the Shryaev-Zhou ndex Countres US UK Canada Germany No. of tmes of buyng the ndex purchase and sell the stock/stock ndex one tme only, so the transacton cost s much smaller. However, when we use the tradng strategy accordng to the Shryaev-Zhou ndex, we have to purchase and sell the stock/stock ndex several tmes. Therefore, the transacton cost accumulates and may even offset the gan. Nevertheless, the tradng strategy accordng to the Shryaev-Zhou ndex s stll superor on the whole as ths strategy yelds a greater return than the buy-and-hold strategy does, except for US s securtzed real estate ndex. If we get a closer look at the number of tmes we buy the securtzed real estate ndces when we trade them usng the strategy we constructed n ths paper, we can see why transacton cost dmnshes the return when applyng ths strategy. From Table 8, we can see that by applyng the tradng strategy we constructed usng the Shryaev-Zhou ndex, the number of tmes of buyng U.K. s securtzed real estate ndex s the least among the four ndces (38 tmes). Ths s why the presence of transacton cost just reduces the return slghtly from to (see Tables 4 and 6). For the other three ndces, we have to buy the ndces more frequently (66 to 84 tmes), makng the total cost much larger. Hence transacton cost dmnshes the return by a larger extent (see Tables 4 and 6). For Canada and Germany s ndces, wthout transacton costs, our strategy outperforms the buy-and-hold strategy by a larger extent, so even wth 0.5% buyng cost, our strategy s stll superor. However, for U.S. s ndex, our strategy only yelds a slghtly larger ndex return than the buy-and-hold strategy does. Hence wth 0.5% buyng cost, our strategy s outperformed by the buy-and-hold strategy. Comparng the results of Tables 4 to 7 wth Table 2, we can see a relatonshp between the Shryaev-Zhou s predctve power on ts correspondng securtzed real estate ndex and the return by applyng the correspondng tradng strategy n comparson to the buy-and-hold strategy. For UK and Canada, Table 2 shows that the predctve power of ther Shryaev ndces on ther correspondng securtzed real estate ndces s stronger. Tables 4 to 7 show that no matter the 0.5% buyng cost exsts or not, the tradng strategy we constructed usng the Shryaev-Zhou ndex greatly outperforms the buy-and-hold strategy for UK and Canada s securtzed real estate ndces. On the other hand, US and Germany s Shryaev ndces have a weaker predctve power (see Table 2). From Tables 4 to 7, our tradng strategy just merely beats the buyand-hold strategy for Germany s securtzed real estate ndex n the presence of 0.5% buyng cost. For US s securtzed real estate ndex, the return s even worse: wthout transacton costs our strategy only slghtly outperforms the buy-and-hold strategy. Wth 0.5% buyng cost, our strategy even yelds a lower return than the buy-and-hold strategy. Hence f the Shryaev-Zhou has a stronger predctve power, the correspondng tradng strategy wll beat the buy-and-hold strategy by a larger extent. There s stll an error n real lfe stuatons. Our method use the stock prces from day n + 1 to day to estmate the Shryaev-Zhou ndex m on day. Hence our estmated value µ ˆ n fact lags behnd the true value m. If n s larger, then the laggng effect wll be greater. However, f n s too small, the error n estmatng the drft a and the volatlty s wll be too large. Therefore, we should choose a sutable movng wndow sze n, and t turns out that t s practcally effectve to choose n = 130 (Hu et al. 2012). 7. CONCLUSIONS From the above statstcs of the Shryaev-Zhou ndces of securtzed real estate ndces of four countres, a number of conclusons can be drawn. Frstly, the Shryaev-Zhou ndces of securtzed real estate ndces of the four countres can reflect

9 36 E. C. M. Hu and S.-C. P. Yam mportant events whch affect the real estate markets. Some of the events are worldwde and affect almost all countres, lke the subprme mortgage crss and the global fnancal crss startng from late 2007, whch caused the worst global economc recesson snce World War II. Ths s reflected by the Shryaev-Zhou ndex of securtzed real estate ndces of most countres, whch stayed negatve snce late Some events, on the other hand, occur locally and affect a partcular country only. For example, n early 1996, an unusually cold wnter ht Germany, httng ts economy hard. Thus ts Shryaev-Zhou ndex struck a hstorcal low at 170. However, the Shryaev-Zhou ndex of securtzed real estate ndces of most other countres remaned relatvely stable at that tme. The Shryaev-Zhou ndex provdes convncng sgnals whch truly reflect dfferent/dverse stuatons and trends n the four countres under nvestgaton. Secondly, the Shryaev-Zhou ndex has some mportant relatonshps wth ts correspondng stock (or stock ndex). From Secton 5, there s a postve (but weak) correlaton between the securtzed real estate ndex and ts correspondng Shryaev-Zhou ndex for all four countres. Ths shows that each securtzed real estate ndex and ts correspondng Shryaev-Zhou ndex move n the same drecton n general. A further analyss of the CCF between each securtzed real estate ndex and ts correspondng Shryaev-Zhou ndex shows that for all four countres, the Shryaev-Zhou ndex does not lag behnd ts correspondng securtzed real estate ndex. In partcular, UK and Canada s Shryaev-Zhou ndces are clear leadng ndcators of ther correspondng securtzed real estate ndces. For these two countres, the Granger causalty test shows that ther Shryaev-Zhou ndces have a stronger predctve power on ther correspondng securtzed real estate ndces, too. We can see that the Shryaev-Zhou ndex s a good predctor of ts correspondng stock (or stock ndex) when the CCF between the stock prce (or stock ndex) s strctly ncreasng,.e. the Shryaev-Zhou s a leadng ndcator of ts correspondng stock (or stock ndex). Moreover, the Shryaev-Zhou ndex can also provde a strategy for buyng and sellng stocks as follows: an nvestor buys a stock as soon as the ndex turns postve, and holds the stock untl the ndex turns negatve, when the nvestor should sell the stock mmedately (as descrbed n Secton 5). Our result n Secton 6 shows that ths strategy generally outperforms the buy-and-hold strategy under the assumpton of no transacton costs. However, the presence of transacton costs reduces the return substantally when we use the tradng strategy accordng to the Shryaev-Zhou ndex. Ths strategy may even underperform the buy-and-hold strategy f the transacton costs are large enough. In our case, ths strategy s stll superor on the whole even wth the presence of 0.5% buyng cost. In real lfe stuatons, the transacton costs are relatvely small n general, so t s expected that ths strategy would outperform the buy-and-hold strategy n most cases. Therefore, t s stll worthwhle to follow ths tradng strategy n general. The Shryaev-Zhou ndex can help nvestors to formulate a better tradng strategy to ncrease ther proft. Further analyss shows that the return by applyng our tradng strategy constructed usng the Shryaev-Zhou ndex s related to the Shryaev- Zhou ndex s predctve power on ts correspondng stock (or stock ndex). For UK and Canada, where the Granger causalty test shows that ther Shryaev-Zhou ndces have a stronger predctve power on ther correspondng securtzed real estate ndces, our strategy yelds a much larger return than the buy-and-hold strategy does. For the other two ndces wth smaller predctve power, out strategy only merely beats the buyand-hold strategy, or, even worse, s outperformed by the buy-and-hold strategy, as n the case of US s securtzed real estate ndex wth the presence of 0.5% buyng cost. Hence before applyng our tradng strategy, t s preferable to conduct the Granger causalty test frst. If the test shows that the Shryaev-Zhou ndex s a good predctor of ts correspondng stock (or stock ndex), then our tradng strategy s lkely to beat the buy-and-hold strategy overwhelmngly. Otherwse, our tradng strategy may only outperform the buy-and-hold strategy by a smaller extent. In the worst case, when the predctve power of the Shryaev-Zhou s weak and the transacton costs are hgh, t may be better to stck wth the buy-and-hold strategy. In applyng our tradng strategy, there s stll an error whch comes from the movng wndow sze n (no. of days n estmatng the drft and volatlty) we choose, whch affects the estmated value µ ˆ of the Shryaev-Zhou ndex m. In the future, one may nvestgate the effect of choosng dfferent szes of n on the resultng return on tradng the stocks (or stock ndces) usng the strategy accordng to the Shryaev-Zhou ndces of the stocks (or stock ndces). To conclude, the Shryaev-Zhou ndex can reflect local and worldwde events affectng the economy and serve as a predctor for certan stocks/

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