A Meta Analysis of Real Estate Fund Performance

Transcription

1 A Meta Analyss of Real Estate Fund Performance A Paper Presented at the ARES Annual Meetng Aprl 00 Naples, Florda Abstract Stephen Lee, Unversty of Readng * and Smon Stevenson, Unversty College Dubln Ths paper provdes evdence regardng the rsk-adjusted performance of 19 UK real estate funds n the UK, over the perod Usng Jensen s alpha the results are generally favourable towards the hypothess that real estate fund managers showed superor rsk-adjusted performance over ths perod. However, usng three wdely known parametrc statstcal procedures to jontly test for tmng and selecton ablty the results are less conclusve. The paper then utlses the meta-analyss technque to further examne the regresson results n an attempt to estmate the proporton of varaton n results attrbutable to samplng error. The meta-analyss results reveal strong evdence, across all models, that the varaton n fndngs s real and may not be attrbuted to samplng error. Thus, the meta-analyss results provde strong evdence that on average the sample of real estate funds analysed n ths study delvered sgnfcant rsk-adjusted performance over ths perod. The meta-analyss for the three tmng and selecton models strongly ndcatng that ths out performance of the benchmark resulted from superor selecton ablty, whle the evdence for the ablty of real estate fund managers to tme the market s at best weak. Thus, we can say that although real estate fund managers are unable to outperform a passve buy and hold strategy through tmng, they are able to mprove ther rsk-adjusted performance through selecton ablty. * Department of Land Management and Development, School of Busness, The Unversty of Readng, Readng RG6 6AW, England, Phone: , Fax: , E-mal: S.L.Lee@readng.ac.uk, Department of Bankng and Fnance, Unversty College Dubln, Blackrock, Co Dubln, Ireland, Phone: , Fax: , E-mal: smon.stevenson@ucd.e, 1

2 1. Introducton Ths paper extends the exstng lterature on the nvestment performance of real estate funds by utlsng a statstcal technque called meta-analyss. Meta-analyss s a parametrc technque for the accumulaton of results across studes, provdng estmates of the mean and standard devaton of the populaton values (Coggns and Hunter, 1987, 1993). In addton, t provdes nformaton on the proporton of the observed varaton n studes that can be explaned by samplng error. Its applcaton n the current study s concerned wth the assessment of the selecton and tmng ablty of real estate funds. In terms of ths applcaton of meta-analyss, the tmng and selecton ablty of each fund manager s vewed as a study. Thus the purpose of ths study s twofold. The frst objectve s to provde evdence on the rsk-adjusted tmng and selecton ablty of real estate funds n the UK usng three wdely used models of performance; the Treynor and Mazuy (TM) (1966) quadratc method and two specfcatons of the Henrksson and Merton (HM) (1981) dualbeta approach. However, as there are a number of potental flaws that have been ponted out wth these approaches, castng doubt on the fndngs based on these models, a second objectve of ths study s to re-examne the rsk-adjusted performance of the real estate funds usng the metaanalyss methodology of Coggns and Hunter (1987, 1993) and Hunter and Schmdt (1990), as the technque provdes a means of examnng whether the observed varaton n tmng and selectvty across funds s real or artfcal. A number of prevous studes have examned the performance of real estate funds, however, none have utlsed meta-analyss as a means of explctly comparng the results across the funds. Lee (1997) analyses 37 UK based real estate funds usng Jensen s alpha and the orgnal HM model, fndng evdence perverse market tmng ablty, but some evdence of out performance wth regard to selecton ablty. Stevenson et al (1997) and Lee and Stevenson (001) both examne the performance of Irsh based real estate funds usng conventonal CAPM based models. Stevenson et al (1997) fnds that whle real estate fund managers show no sgns of selecton ablty, there s evdence of superor market tmng ablty. Lee and Stevenson (001) extend the aforementoned paper to examne an extended perod of tme and also utlsng the Bhattacharya and Pflederer (1983) quadratc based model. The results are generally smlar, wth evdence of poor selecton ablty, whle the tmng results are not as conclusve as those reported n Stevenson et al (1997). Gallo et al (1997) examned the performance of mortgage backed securty (MBS) funds and fnd evdence of underperformance on both a selecton and tmng ablty. In contrast, Gallo et al (000) report that real estate mutual funds, whch nvest n REITs, show evdence of outperformance over the benchmark portfolo. The remander of the paper s lad out as follows. The followng secton brefly descrbes the performance evaluaton technques used. Secton three descrbes the nsttutonal characterstcs of the real estate funds as well as the data used n the study. Ths s followed by a dscusson of the meta-analyss technque. The next secton dscusses the emprcal fndngs, whle the fnal secton provdes concludng comments.. Rsk-adjusted Models of Performance The most popular measure of rsk-adjusted performance s the Jensen alpha, whch s taken as the ntercept n equaton (1), whch s a general emprcal expresson of the Captal Asset Prcng Model (CAPM). A number of recent papers examnng the ssue of tmng and selecton ablty have utlzed meta-analyss, for example Coggn and Hunter (1993), Coggn et al. (1994), Sahu et al (1998) and Stevenson (001).

3 R t = α + β R + ε (1) mt t where: R t s the excess return of the fund n queston and R mt s the excess return of the benchmark ndex. As the expected value of the error term n equaton (1) s equal to zero, the ntercept can be taken to be a measure of the portfolo manager s selecton ablty. However, Fama (197) noted that the performance of fund managers could be separated nto two components: selectvty (the ablty to select undervalued assets), and tmng (the ablty to adjust securty holdngs n antcpaton of general market movements). Jensen s framework does not allow for the possblty of market tmng and as a consequence the results of the regresson analyss based on equaton 1 wll be based and any tests of sgnfcance dstorted, see Fama (197), Jensen (197), Grant (1977), Admat and Ross (1985) and Dybvg and Ross (1985) amongst others. As a result our study also uses models of rsk-adjusted performance that ncorporate both mcro (selectvty) and macro (market tmng) forecast abltes. Three alternatve models are used to estmate market tmng and selecton ablty. As stated n the ntroducton these are the quadratc model proposed by TM and two specfcatons of the HM dual-beta model. The TM quadratc model adds a quadratc term to equaton (1) to allow for market tmng ablty, and can represented as follows: R t mt mt = α + β R + γ R + ε () t The dual-beta model also comples wth the assumptons of the CAPM and as wth the quadratc models, ams to provde a means of overcomng potental bas n the measure of selectvty. The dfference between t and the quadratc model s how the two model market tmng ablty. The dual-beta model s based on the concept that a fund manager wll ether forecast that the market wll outperform the rsk-free asset, or that the rsk-free asset wll outperform the market. The orgnal dual-beta model of HM can be expressed as follows: R t 1 mt [ D t ( R mt) ] + ε t = α + β R + β (3) The dummy varable takes the value of zero when the market return s greater than that of the rsk-free asset, and 1 when the rsk-free asset s return exceeds that of the benchmark. The alternatve specfcaton, proposed by Henrksson (1984), takes nto account problems wth the return generatng process and specfcally the omsson of relevant factors and ssues concernng the choce of benchmark portfolo. Henrksson (1984) adds a second factor based on the excess return of an equally weghted portfolo of the funds analysed. The modfed HM model can be expressed as follows: R t = α + β 1 R mt + β [ D1t ( R mt )] + β3 [ R ewt βew ( R mt )] + β4 ( D1t [ R ewt βew ( R mt )]) + εt (4) Where R ewt s the excess return on the equally weghted fund portfolo, ew β s the beta of ths portfolo relatve to the benchmark ndex. The fourth expresson takes the value of max [ 0,w ( t) ] where w ( t ) equals the thrd expresson. The dummy varable takes the value of zero when the return on the equally weghted portfolo exceeds that of the rskless assets and the 1 f the reverse occurs. 3

4 However, although the coeffcents of the ordnary least squares (OLS) estmaton of equatons 1 through 4 provde consstent parameter estmates, all three may requre correcton for heteroskedastcty n the error term ε t, whch causes the parameter estmates to be neffcent, see Henrksson and Merton (1981), Chen and Stockum (1986), Lee and Rahman (1990) and Coggn et al (1993). In all cases ths t s corrected usng the methods of Hansen (198) and Whte (1980). 3. Data Indrect nvestment n real estate by UK penson funds can be made through a number of vehcles, but for a number of reasons the nearest equvalent to drect real estate nvestment s through ether a Property Unt Trust (PUT) or a Managed Property Funds (MPF), Investment Property Forum (1996). Each alternatve offerng tax-exempt nsttutons the opportunty to nvest n real estate, on an ncremental bass, wthout the need to acqure the necessary management and nvestment sklls requred to manage a real estate portfolo. Whle the pooled nature of ther structure means that PUTs and MPFs are able to offer a wder dversfed portfolo of propertes than could be held by one penson fund n solaton. Data on 19 UK real estate funds are used n ths study, consstng of 1 PUTs and 7 MPFs. All the data taken from the publcatons of Assocaton of Property Unt Trusts (APUT) as compled by the Investment Property Databank (IPD). Of the 6 real estate funds covered by APUT, seven funds were excluded as they were all only recently ncorporated nto the database, thus, they dd not have an adequate tme seres to be ncluded n the study. The remanng 19 real estate funds accountng for 77% of the 7.1 bllon aggregate value of funds covered by APUT at the end of 001. In addton, as the NAV of the funds vares wdely from 14. mllon to 1. bllon, the results should be ndcatve of real estate fund performance over ths perod. Nonetheless, the results only hold for those real estate funds that exsted throughout the sample perod. The benchmark portfolo used throughout the analyss s the Jones Lang LaSalle (JLL) UK Property Index. Snce the ndex s constructed to represent the actual performance of a typcal nsttutonal real estate portfolo n terms of fund flow and geographcal spread. The analyss performed over the perod 1991Q1-001Q3 usng quarterly data. All data used are logarthmc returns n excess of the rsk free rate, as measured by the return on 90 day Treasury Blls. 4. Meta Analyss Ths secton of the paper brefly dscusses meta-analyss. Meta analyss s a parametrc technque for the accumulaton of results across studes. However, a number of study artefacts can cause the results from one study to appear dfferent or even contradctory to those of another. Among the more obvous artefacts are samplng error and measurement error. Meta-analyss s desgned to overcome these problems and so provde estmates of the mean and standard devaton of the populaton values. Also, although meta-analyss was orgnally desgned for cross-sectonal data, the tme-seres performance models used here are dentcal n model specfcatons across the sample of real estate funds. Thus, n terms of the meta-analyss technque, each real estate fund s vewed as a study and we accumulate the results across funds. In ths way the method provdes a means of examnng whether the observed varaton n tmng and selectvty across funds s real or artfcal. In addton, t provdes nformaton on the proporton of the observed varaton that can be explaned by samplng error varaton (Coggn and Hunter, 1993). 4

5 For each study the observed values are denoted as b, the populaton values as β and e represents the samplng error. b = β + e,or,e = b β (5) The average observed value s: b = β + e (6) As the average error wll be zero across a large number of studes the above equaton can be rewrtten as: b = β. In the case of the current study we are comparng regresson results across ndvdual funds denoted by, therefore we can re-wrte equaton (6) as: b = β + e (7) As β and e wll be uncorrelated across funds, the varances of the observed values ( b ) larger than the varance of the populaton values ( σ β ) by the amount of the samplng error ( ) Therefore: σ wll be σ. e σ β = (8) σ b σ e As the varance of the samplng error can be computed, we can drectly estmate the populaton varance. Hunter and Schmdt (1990) show that n the case of a small sample sze and under the assumpton that the populaton value s constant across studes, the best estmate for t s the frequency weghted value. [ ] b N b = (9) N where b s the observed value and N s the number of observatons n the study. The varance used s the frequency weghted average square devaton. ( b b ) N N s = (10) b Whle the samplng error varance can be represented as: [ N ( standard error b )] s e = (11) N Coggn and Hunter (1993) note that f the number of studes s large there s the rsk that the null hypothess may be rejected even f there s a small amount of varaton. Ths problem can arse due to the potental stuaton where smlartes n the funds may lead to the samplng errors for the coeffcents beng non-ndependent and postvely correlated. Therefore, we take the 5

6 [ s ] samplng error varance as ( r) 1 e, where r s the average correlaton between the regresson resduals. The populaton varance can therefore be estmated as: s ( 1 r) s e b = s β + (1) ( 1 r) se = ( sb s e ) rs e β = sb (13) s + Under the null hypothess that there s no varaton n parameter estmates the rato of the observed varance to the samplng error has a ch-squared dstrbuton wth k-1 degrees of freedom, where k s the number of studes (real estate funds). Rejecton of the null hypothess can then be taken as evdence of a real varaton n observed values. χ = ks b / se (14) However, f the k studes are not ndependent, then the statstcal power of the test s reduced. To correct for ths possblty equaton the standard error can be wrtten as follows: s b SE = k + rse (15) and the ch-square calculaton adjusted as follows: χ = ks b = 1 ks b ( ) ( ) 1 r s 1 r e s e (16) Fnally, we can estmate the percentage of total observed varance accounted for by samplng error, ( 1 r)s / s. 5. Emprcal Results e b Table 1 provdes the results of the ntal performance evaluaton usng Jensen s alpha and the selectvty and market tmng results usng the quadratc model the models of TM and the dualbeta models of HM. Usng Jensen s alpha the results show strong evdence of outperformance by the funds over the market benchmark, wth 15 (79%) of the funds dsplayng postve rskadjusted performance. Of these 15 funds, 10 (53%) show statstcally sgnfcant performance at the usual levels of sgnfcance. In contrast, only 4 funds (1%) show negatve alphas, wth 3 (16%) funds dsplayng sgnfcant nferor performance at conventonal statstcal levels. Ths mples that on average the 19 real estate funds analysed here out performed a passve buy and hold strategy. The results, however, do not ndcate whether ths superor rsk-adjusted performance was a consequence of tmng or selecton ablty or both. The results for the three alternatve tmng and selecton models hghlght some of the problems nherent n the Jensen measure and n partcular the potental bas that can be ntroduced n the measure due to the presence of market tmng. The quadratc model of TM produces dentcal results n relaton to selectvty, as the Jensen measure, n terms of the number of postve ntercepts and the number of those that are sgnfcant. The number of funds showng postve selecton ablty does, however, fall when the dual-beta models of HM are used. In addton, the 6

7 number of statstcally sgnfcant coeffcents falls to 8 (4%) usng the orgnal HM model and 4 (1%), when the adapted dual-beat model s used. In contrast, the number of funds showng sgnfcant nferor selecton ablty remans constant across all four models. On average, therefore, the results ndcated postve selecton ablty on the part of the 19 real estate funds analysed over ths perod. In the quadratc specfcaton only 8 (4%) funds show postve tmng ablty, the fgures for the two dual-beta models are 16 (84%) and 14 (79%) respectvely. The number of sgnfcant fndngs s, however, relatvely stable across the three models. In the case of the quadratc model the number of sgnfcant coeffcents s 4 (0%), whle the fgures for the two dual-beta models are 6 (3%) and 5 (6%) respectvely. In contrast, the number of funds dsplayng perverse market tmng changes consderably across the varous models. The quadratc model showng 11 (58%) funds wth perverse market tmng, however, the fgures for the two dual-beta models are only 3 (16%) and 5 (6%). In addton, the number of funds showng sgnfcantly perverse tmng ablty s only 3 (16%), (11%) and 1 (5%) respectvely for the three models. Overall the results ndcate that the ablty of the real estate funds to tme the market are consderably weaker than the ther ablty to select undervalued real estate. Table reports the fndngs of the meta-analyss. The am of ths analyss s to examne whether the observed varaton n tmng and selectvty across funds s real or artfcal. Followng the format of Coggn and Hunter (1993), Table 3 reports the average coeffcent (β mean) n each case; the standard devaton of the relevant coeffcent (σβ); the error term (σε); the average correlaton (ρ) between the resduals n each model; the ch-square value (χ ) for the rato of the observed varance to the samplng error varance, adjusted for the average resdual correlaton; and the last row shows the estmates of the percentage of total observed varance accounted for by samplng error, (1 ρ)σ ε/ σ β. The χ statstcs for selectvely are all hghly sgnfcant and postve across all four models of nvestment performance. Ths ndcates the varaton n results across the fnds s real and not due to samplng error. These results are also confrmed by the last row of Table 3, whch provdes evdence as to the proporton of the varance that can be accounted for by samplng error. The hghest observed varaton attrbutable to samplng error s 8% for the selectvty measure usng the adapted dual-beta model, whle the lowest s the Jensen measure at 5%. Ths lends support to the argument presented above that on average real estate funds n the UK have shown superor rsk-adjusted performance compared wth the benchmark of performance, and that ths can be attrbuted to the fund manager s selecton ablty. In contrast, the meta-analyss results about tmng provde mxed evdence regardng the ablty of real estate fund managers to successfully tme the market. The mean tmng value s negatve for the quadratc model of TM, but postve for the two dual-beta models of HM. A negatve value ndcatng that fund managers are on average unsuccessful n ther ablty to tme the market, whle a postve value mples successful tmng ablty. The χ statstcs for tmng are all hghly sgnfcant across all three models, ndcatng that these conclusons are a consequence of a real varaton n results across the funds and not a result of samplng error. A result also confrmed by the last row of Table 3, whch shows that the proporton of the varance that can be accounted for by samplng error s relatvely small and no more 7%. Ths dfference n the conclusons from the varous models ndcates the results are qute senstve to the tmng and selecton model adopted. 7

8 The results of the meta-analyss for a sample of real estate funds are smlar to the meta-analyss based fndngs of Goggn and Hunter (1993), Goggn et al (1993) and Sahu et al (1998) n the stock market and Stevenson (001) n the real estate market. Lke ths study, ther results fnd that the best funds can delver substantal rsk-adjusted performance. Also, ther results show that the superor rsk-adjusted performance s manly derved from fund managers selecton ablty rather than any tmng sklls. 6. Concluson Ths paper provdes evdence regardng the rsk-adjusted performance of 19 UK real estate funds n the UK. Ths paper dffers from prevous studes by examnng the whether the dfferences n the regresson coeffcents across funds results from sgnfcant samplng error or whether the fndngs show a real varaton n performance. Thus, the fndngs of ths study supplement those of prevous studes that have tested the selecton and tmng ablty of real estate fund managers. Usng four parametrc models of nvestment performance, one that tests overall performance, and three that conduct a jont test for the presence of tmng and selecton ablty, the results are generally favourable towards the hypothess that the sample of real estate funds analysed showed superor rsk-adjusted performance over ths perod. However, the results reveal the potental bases n the Jensen measure, wth the number of sgnfcant postve results reduced n the models that examnng selectvty and market tmng ablty smultaneously. Nonetheless, n most cases the models show a greater number of funds wth postve selecton sklls than negatve selecton ablty. In addton, the market tmng results reveal some evdence of sgnfcantly postve tmng ablty by a few real estate fund managers wth only a few funds showng sgnfcantly perverse market tmng. The paper then utlsed the meta-analyss technque to further examne the results n an attempt to estmate the proporton of varaton n regresson coeffcents s attrbutable to samplng error. The meta-analyss results reveal strong evdence, across all models, that the varaton n fndngs s real and may not be attrbuted to samplng error. Thus, the meta-analyss results provde strong evdence that on average the sample of real estate fund managers analysed n ths study delvered sgnfcant rsk-adjusted performance over ths perod. The three tmng and selecton models ndcatng that ths out performance of the benchmark resulted from superor selecton ablty by real estate fund mangers. In contrast, the results on tmng ablty are mxed. For nstance, we obtan a negatve mean tmng value usng the quadratc model of TM, but postve average tmng values usng the two HM models. The meta-analyss results ndcatng that the ablty of real estate fund managers to tme the market s at best weak. Thus we can say that although real estate fund managers are unable to outperform a passve buy and hold strategy through tmng, they are able to mprove ther rsk-adjusted performance through selecton ablty. 8

9 References: Admat, A.R. and Ross, S.A. (1985), Measurng Investment Performance n a Ratonal Expectatons Equlbrum Model, Journal of Busness, 58, 1-6. Bhattacharya, S. & Pflederer, P. (1983), A Note on Performance Evaluaton, Techncal Report 714, Stanford Unversty. Chen, S.R. and Stockum, S. (1986), Selectvty, Market Tmng and Random Behavour of Mutual Funds: A Generalsed Model, Journal of Fnancal Research, Sprng, Coggn, T.D., Fabozz, F.J. and Rahman, S. (1993), The Investment Performance of US Equty Penson Fund Managers: An Emprcal Investgaton, Journal of Fnance, 48, Coggn, T.D. and Hunter, J.E. (1987), A Meta-Analyss of Prcng Rsk Factors n APT, Journal of Portfolo Management, 14, 1, Coggn, T.D. and Hunter, J.E. (1993), A Meta-Analyss of Mutual Fund Performance, Revew of Quanttatve Fnance and Accountng, 3, Dybvg, P.H. and Ross, S.A. (1985), The analyss of Performance Measurement usng a Securty Market Lne, Journal of Fnance, 40, Fama, E.F. (197), Components of Investment Performance, Journal of Fnance, 7, Gallo, J.G., Buttmer, R.J., Lockwood, L.J. & Rutherford, R.C. (1997). Determnants of Performance for Mortgage-Backed Securtes Funds, Real Estate Economcs, 5, Gallo, J.G., Lockwood, L.J. & Rutherford, R.C. (000). Asset Allocaton and the Performance of Real Estate Mutual Funds, Real Estate Economcs, 8, Fama, E.F. (197), Components of Investment Performance, Journal of Fnance, 7, Grant, D. (1977), Portfolo Performance and the Cost of Tmng Decsons, Journal of Fnance, 33, Hansen, L.P. (198), Large Sample Propertes of Generalsed Method of Moments Estmates, Econometrca, 50, Henrksson, R.D. (1984), Market Tmng and Mutual Fund Performance: An Emprcal Investgaton, Journal of Busness, 57, Henrksson, R.D. and Merton, R.C. (1981), On Market Tmng and Investment Performance : Statstcal Procedures for Evaluatng Forecastng Sklls, Journal of Busness, 54, Hunter, J.E. and Schmdt, F.L. (1990). Methods of Meta-Analyss, Newbury Park CA: Sage Publcatons. Investment Property Forum (1996) Property Investment for UK Penson Funds (Report prepared by a Workng Group for the Investment Property Forum) 9

10 Jensen, M.C. (197), Optmal Utlsaton of Market Forecasts and the Evaluaton of Investment Performance, n Szego, G.P. and Shell, K. (eds). Mathematcal Methods n Investment and Fnance, Amsterdam: Elsever. Lee, C.F. and Rahman, S. (1991), New Evdence on Tmng and Securty Selecton Skll of Mutual Fund Managers, Journal of Portfolo Management, Wnter, Lee, S. (1997). The Components of Property fund Performance, Journal of Real Estate Portfolo Management, 3, Lee, S. and Stevenson, S. (001). Emprcal Evdence on the Mcro and Macro Forecastng Ablty of Real Estate Funds, workng paper, Unversty College Dubln. Sahu, A.P., Klenman, R.T. and Callaghan, J. (1998) The Tmng and Selecton Abltes of Bank Funds: Evdence Based on Meta-Analyss, Journal of Fnancal Servces Research, 13,, Stevenson, S., Knsella, R.P. & O Heala, R.S. (1997), Irsh Property Funds: Emprcal Evdence on Market Tmng and Selectvty, Irsh Busness & Admnstratve Research, 18, Stevenson, S. (001), A Performance Evaluaton of Portfolo Managers: Tests of Mcro and Macro Forecastng, workng paper, Unversty College Dubln. Treynor, J.L. and Mazuy, M. (1966), Can Mutual Funds outguess the Market, Harvard Busness Revew, 44, Whte, H. (1980), A Heteroscedastcty-Consstent Covarance Matrx and a Drect Test for Heteroscedastcty, Econome trca, 48,

11 Table 1: Jensen s Alpha, Quadratc Model and Dual-Beta Results Jensen s TM Quadratc Model HM Adapted HM Fund Number Alpha Selectvty Tmng Selectvty Tmng Selectvty Tmng Fund *** * ** Fund * * Fund Fund Fund ***.0905** *.85** Fund *** *** *** ** *** Fund *** *** *** *** *** 0.18 Fund *** *** *** *** * *** *** Fund ** ** * Fund *** *** *** *** ** *** 0.70* Fund *** *** *** ** *** * Fund 1.119***.5677*** *** 1.817*** *** *** *** Fund ** * * Fund *** ** * -.033*** *** -.577*** Fund * ** *** Fund Fund Fund ** * Fund ** * Average Coeffcent Number Postve Number Postve (Sgnfcant) Number Negatve Number Negatve (Sgnfcant)

12 Table : Meta Analyss Results Jensen Quadratc HM HM Adapted Selectvty Selectvty Tmng Selectvty Tmng Selectvty Tmng β mean σβ σε ρ χ *** *** *** 50.65*** 38.73*** *** *** (1 ρ)σ ε/ σ β