The Application of Asset Pricing to Portfolio Management

Transcription

1 Clemso Ecoomcs The Applcato of Asset Prcg to Portfolo Maagemet The Nature of the Problem Portfolo maagers have two basc problems. Frst they must determe whch assets to hold a portfolo, ad secod, they must determe how much of each asset to hold as a fracto of the total portfolo value. Both questos have tme dmesos: Both questos must be aswered whe the portfolo s tally formed. The, at each momet thereafter, the portfolo maager must make a decso about whether to rebalace the portfolo or smply hold the assets tally purchased. The purpose of ths dscusso s to compare dfferet portfolo costructo ad adustmet strateges theoretcally ad emprcally. We address the followg questos: For a portfolo of rsky assets, what tal portfolo allocatos are best? As the values of the assets chage, what effceces result from rebalacg the portfolo? For a portfolo composed of rsky assets, what s the value of oe addtoal asset? What rules should be used choosg the rsky assets? Alteratve Portfolo Costructo Cosder the problem of maagg a small portfolo of assets whch the vestor wshes to acheve a target rate of retur wth mmum varace. There are several approaches to ths problem that oe mght adopt. A commo textbook assumpto s that the portfolo maager puts equal dollar amouts of the portfolo captal to rsky assets. The umber ad type of assets the portfolo ad the relatve proporto of rsky assets to the rsk free vestmet determes the expected retur ad the level of rsk whe the portfolo s tally formed. Ths assumpto s used textbooks to lay the theoretcal foudato for the relato betwee rsk ad the umber of assets held a portfolo. Whle smple as a theoretcal devce, ths equal-weght strategy becomes complcated as a practcal portfolo-maagemet rule. As the value of each of the rsky assets chages, the portfolo s o loger composed of equal weghts. The questo, the, s whether ad how ofte the portfolo should be rebalaced. A slghtly more elaborate portfolo allocato rule s oe that determes portfolo weghts based o the captalzed value of the assets held the portfolo. Ths s called a value-weghted portfolo. That s, the portfolo maager chooses assets whch to vest. The total portfolo captal s the spread across the assets based o the total value of each asset relatve the total value of all assets. Say a vestor wats hold $4000 worth of wealth the stock of two telephoe compaes, BellSouth ad Nyex. I terms of equty, BellSouth s a $30 bllo compay. Nyex has a equty value of $0 bllo. To hold a value-weghted portfolo, $3000 s held BellSouth stock ad $000 Nyex securtes. The value weghted portfolo approach s a buy-ad-hold strategy. Sce the total portfolo wealth s dvded across the assets proportos based o the total captalzed value of each asset, as the values of the assets chage, so too does the share of the portfolo devoted to 6.doc; Revsed: Jauary, 000; M.T. Maloey

2 Clemso Ecoomcs each. Usg ths weghtg scheme, the portfolo s always balaced as the values of the assets chage. No portfolo adustmet s requred. Ths smple buy-ad-hold, value weghtg strategy has bee the bechmark agast whch mutual fud performace has typcally bee udged: The cosesus of the academc lterature seems to be that mutual fud rebalacg adds lttle to performace relatve to the value-weghted/buy-ad-hold bechmark. Noetheless, portfolo maagers do a substatal amout of rebalacg, ad, a compettve evromet, the a pror expectato must be that survval mples effcecy. e of the goals of ths exercse s to better uderstad ths aomaly. A alteratve to equal- or value-weghtg s to choose portfolo weghts based o parameters derved from asset prcg models. The fudametal bass of asset prcg foud the theoretcal lterature s the Captal Asset Prcg Model (CAPM). I ts smplest formulato, the CAPM says that assets wll be prced by the market based o ther covarace wth other assets. Because of ths, a dvdual vestor ca mmze rsk for ay level of expected retur by dvdg hs wealth betwee a gve amout of the rsk free asset ad a portfolo of all rsky assets. Whe the dvdual vestor holds a share all rsky assets, the weghts o the assets are based o each asset's captalzed value. I ths case, the theoretcal mplcato of CAPM for portfolo maagemet s the value-weghted/buy-ad-hold approach. However, whe the umber of rsky assets held a portfolo s a subset of all rsky assets, the smple value-weghted/buy-ad-hold method of portfolo maagemet s ot ecessarly effcet. For portfolos cotag a lmted umber of assets, smple CAPM as well as more sophstcated prcg models ca be used to devse more effcet strateges for portfolo maagemet. The ext secto of ths paper shows that the parameters assocated wth these prcg models detfy portfolo weghts that are dfferet from equal- or value-weghted assgmets. Usg CAPM to Defe ptmal Portfolo Weghts Assume that the vestor wshes to hold a portfolo of rsky assets that s a subset of all possble vestmets. The vestor's obectve s to mmze rsk subect to a targeted, expected rate of retur. CAPM s oe way to detfy both rsk ad the expected retur o a rsky asset. That s, for ay rsky asset r α + r + ε m where r s the retur o asset, r m s the value weghted retur o all rsky assets, s based o the covarace betwee the retur o all rsky assets ad the retur to asset, α s equal to (- ) tmes the retur o the rsk-free asset, r f, ad ε s the resdual varato the retur to asset. I the smple CAPM the systematc relato betwee assets s captured by the lear multpler o the sgle market dex. I more advaced models other dexes are used to detfy the expected retur o a asset. We wll exted the model alog those les a momet. Frst, we develop the optmal vestmet strategy usg the smple CAPM. There s a recet paper by Davd Boothe ad Eugee Fama the Facal Maagemet Revew that does remark o the shortcomgs of the buy ad hold strategy. 6.doc; Revsed: Jauary, 000; M.T. Maloey

3 Clemso Ecoomcs The obectve fucto of the vestor ca be defed as the desre to mmze portfolo varace for a gve target portfolo retur. Ths ca be wrtte as m { ω } V ( r ) subect to : k p ω ω ω E( r ) + ω rf where V(r p ) s the varace of the portfolo retur, the ω terms are the proportos of the portfolo devoted to each asset, ad the terms refer to the varace ad covaraces of the returs o each rsky asset. The rsk free asset, whch takes a portfolo share of oe mus the sum of the shares devoted to the rsky assets, s assumed to have o varace. The obectve fucto s optmzed subect to a target portfolo retur, k. The portfolo retur s detfed by usg the CAPM; the expectato of ε s zero. The tercept of equato (), α, s replaced by ts theoretcal detty volvg ad the rsk-free retur, that s, α (- ) r f. For the costrat o the expected retur, these substtutos gve: k [( ) rf + rm ] + ω rf ω 3 By cacelg the terms volvg the sum of the portfolo weghts tmes the rsk-free retur, ths ca be smplfed to: Rewrtg gves: k Σω ( rm - r f ) + r f. 4 k - r f Σ ω φ rm - r f 5 where the rato of the target retur et of the rsk-free dvded by the market retur et of the rsk-free s labeled φ. The obectve fucto ca be rewrtte by corporatg the CAPM terms so that t reduces to: m{ ω V r p Σω ε + m Σω } ( ) 6 6.doc; Revsed: Jauary, 000; M.T. Maloey 3

4 Clemso Ecoomcs where m s the varace of the market retur. [See appedx.] Ths s a stadard expresso foud may books o portfolo theory. The portfolo varace smplfes based o the followg assumptos related to the CAPM. Frst, the covarace betwee the resdual retur (the ε 's equato ) ad the market retur s zero, ad secod, the covarace of resdual returs betwee dvdual assets s zero. 3 Importatly, the secod assumpto says that there s o expected covarace betwee ay two rsky assets except for the fact that they both wll vary wth the market retur. Ths commo respose to overall market codtos s captured ad there s o addtoal relato volved betwee the ε 's. 4 Equato (6) s optmzed subect to the costrat gve by equato (5). Usg the method of agrage, we get the followg frst order codtos: V ω ω ω + λ ( r ) 0 { ε + m - r f,, } m Σ 7 The set of equatos defed by (7) ca be solved parwse fasho to elmate λ ad the other costat terms. The rato of the portfolo shares betwee ay two assets readly reduces to: ω / ε ω / ε 8 The mpact of the costrat ca be cluded by multplyg by ad summg over. Ths gves ω ω / / From ths, the value of the costrat gve by equato (5) ca be substtuted drectly to detfy the value of each of weghts: ε ε See for example, Wm. Sharpe, Portfolo Theory, 970, McGraw-Hll. 3 These assumptos are logcally cosstet wth the CAPM so log as s ot the etre set of rsky assets. If the portfolo cluded the set of all rsky assets, the there would be a detty that operates across the set of fuctos show by equato (). That s, the sum of resduals across - assets alog wth the market retur would defe the resdual for the th asset. However, whe s a subset of all rsky assets, ths detty does ot come to play. 4 Ths assumpto may seem extreme the sese that, for stace, frms the same dustry may be expected to have returs that vary relato to oe aother a way that s more marked tha descrbed by the commo varato wth the market. However, the commo varato betwee frms the same dustry ca be ether postve or egatve. There wll be evets that cause all frms a dustry to become more valuable, but o the other had, competto meas that whe oe frm s better off, t reduces the proftablty of ts cohorts, at least margally. A pror, we do ot kow whch of these forces wll domate. I geeral, the whole dea of extedg the CAPM s to detfy ad capture commo varato returs betwee frms, ad oe exteso could well be a dustry retur factor. We wll retur to the theme of extedg the prcg model shortly. 6.doc; Revsed: Jauary, 000; M.T. Maloey 4

5 Clemso Ecoomcs φ ω 9 ε Σ ε It s mportat to recogze that the ω s are ot ecessarly postve. Whe they are egatve, t mples that the vestor takes a short posto a asset. Ths wll be the case whe oe of the assets has a egatve. Nor do the weghts sum to oe. For stace, cosder the case where φ s equal to oe. I ths case, the vestor chooses a target retur that s the market. However, gve the set of stocks held by the vestor, the market retur may oly be avalable by borrowg at the rsk free rate ad vestg ths addtoal amout of captal the portfolo of rsky assets. The geeral result s gve by summg equato (9) over : Extedg the CAPM ω φ ε Σ ε The set of portfolo weghts defed by equatos (8) ad (9) are based o the values of ad resdual varace for each asset that are derved from the asset prcg model. The effcecy of the portfolo weghts ths scheme ca be mproved by refg the prcg model. e such a exteso of the prcg model s developed recet research by Fama ad Frech. I ther model equato () becomes r α + rm + γ M s + δ M b + ε 0 where M s ad M b are so-called mmckg portfolo returs bult o frm sze ad the rato of book to market equty, respectvely. A mmckg portfolo retur s a retur that s the dfferece betwee the returs to two classes of frms. For stace, the sze mmckg portfolo retur, the returs of small frms are subtracted from the returs to bg frms. The result s a portfolo retur dfferetal that reflects the commoalty of returs to bg frms as that s dfferet from returs to small frms, ad both et of the retur to all frms, whch s held costat by the relato betwee the th frm's retur ad the market. The coeffcets o the mmckg portfolos descrbe commo factors returs across frms f the coeffcets cosstetly assocate frms wth ther peers. That s, a gve bg frm should have postve retur whe M s s postve; lkewse, whe M s s postve, a small frm should have a egatve retur. The mplcato s that γ should be postve for bg frms ad egatve for small frms f sze s a factor commo the ratoal rsk prcg of assets. Smlarly, M b s a portfolo retur dfferetal betwee hgh book-to-market frms ad low book-to-market frms. If book-to-market s a commo rsk factor, δ should be postve for hgh book frms ad 6.doc; Revsed: Jauary, 000; M.T. Maloey 5

6 Clemso Ecoomcs egatve for low book frms. Fama ad Frech show that sze ad book-to-market are deed commo rsk factors. For the purpose of optmal portfolo balacg, the exteded CAPM gve equato (0) complcates matters somewhat. The varace of the portfolo retur becomes m{ ω V p Σω ε + m Σω + s Σω γ + b Σω δ } ( r ) Ths formulato s based o several assumptos. Aga, we treat the covarace betwee the resduals of the prcg equatos across assets as zero ad assume that the resduals are ot correlated wth the market ad rsk factor dexes (.e., the market retur ad the mmckg portfolo returs o sze ad o book-to-market equty). Furthermore, we assume that the dexes are ot themselves correlated. Thus, the portfolo varace for the assets s gve by equato (). Eve though the portfolo varace becomes more complex whe we expad the prcg model, the costrat s uchaged. Because the mmckg portfolo returs are dffereces betwee the returs to frms based o the characterstcs of sze ad book-to-market, ther expectato s zero. For stace, lookg ahead we have o reaso to expect that bg frms wll outperform small frms or vce versa. Therefore, whe we compute the costrat o the expected portfolo retur, the mmckg portfolo dexes fall out. The frst order codtos for the optmzato problem usg the exteded prcg equatos reduce to equato () terms of the relato betwee ay two portfolo weghts: / ω / ε ε + ω s ε Σ k γ γ b k - + ω γ k ε Σ k δ k k - δ ω δ,,. The costrat mposes the addtoal requremet o the overall sze of the weghts; t s the same as the smple CAPM as gve by equato (8). Notce that the soluto value for equato () s complcated by the fact that the sum of the weghts tmes the mmckg portfolo coeffcets shows up for both dexes. Because of ths, the formula for the portfolo weghts gve by () do ot have the smple soluto descrbed by equatos (8) ad (9) where we used the basc CAPM. Noetheless, a uque soluto s attaable usg umercal methods. Prelmary Emprcal Results Equatos (8), (9), ad () defe rules for costructg portfolos. The rules apply where portfolos cota a lmted umber of assets. The questo s whether these rules actually mprove portfolo performace. The bechmark s to compare the performace of portfolos costructed usg these rules to the performace of value-weghted portfolos. I order to asses the value of the portfolo costructo strateges descrbed above, we smulated returs to 00-stock portfolos usg NASDAQ securtes. We used returs to 6.doc; Revsed: Jauary, 000; M.T. Maloey 6

7 Clemso Ecoomcs securtes traded the NASDAQ Natoal Market System (NMS) over the years 984 through 99. Usg these returs we estmated for each stock the smple CAPM as well as the exteded prcg models lke those descrbed by Fama ad Frech for each of the years 984 through 990. (The estmato procedure ad dagostc aalyss s descrbed elsewhere.) Gve the prcg model estmates, we formed portfolos for the followg year. The returs ad varaces of these portfolos were the compared to the returs ad varaces of valued-weghted portfolos of the same securtes. The prelmary fdgs of ths research suggests that the portfolo balacg approach based o prcg model parameters does, fact, produce superor performace compared to a value-weghted/buy-ad-hold strategy. Further work s eeded to assess the magtude ad sestvty of ths result. More Evdece o the Effect of Portfolo Dversfcato The followg s a table cocerg the effects of portfolo dversfcato: Table : ptmal v. Sub-optmal Portfolos Market Varace Number of Assets Varace of ptmzed Portfolo Ieffcecy of ptmzed Portfolo Ieffcecy % 5.56% 4.8% 0.85%.95% Sub-optmal Portfolo Varace Ieffcecy of Sub-optmal Portfolo Ieffcecy % 3.8% 87.34% 46.9% 3.5% Reducto Ieffcecy 58.8% 5.% 54.97% 48.40% 6.doc; Revsed: Jauary, 000; M.T. Maloey 7

8 Clemso Ecoomcs % of ptmal Portfolo held Rsky Assets 55.5% 6.7% 7.5% 84.% Ths table s costructed from the aother mutual fud data set, whch s avalable from my hard dsk the subdrectory C:\pub. The fle s 855b.wk. ths fle, the betas ad R s for each fud are gve alog wth the te year average retur ad stadard devato. These data come from Morgstar. I costructed the stadard error of the resdual of the market model ad cluded t the spreadsheet. The formula s smple: ( R ) ε r Thus gve the stadard devato of the fud ad the goodess of ft of t to the market model, we ca compute the resdual varace. Next I solved for the optmal portfolo weghts usg portfolos of varous szes. The optmal weghts are foud from the formula that s dscussed ecture 6. They are foud by takg the rato of beta to resdual varace for asset ad dvdg ths by the sum of rato of beta squared dvded by resdual varace across all. That s, ω N M Σ φ ε Q P ε 3 where φ s set equal to uder the assumpto that the vestor seeks to ear a expected retur equal to the market. Part of the terest we have Table above s to determe how effcet s the optmal portfolo ad also to assess how ths effcecy chages as the umber of assets (mutual fuds) grows. The frst row Table s the market varace. Market varace s computed from the average retur across all mutual fuds. The market varace s the varace of ths average mutual fud retur (.3%) across the last te years. (Note that the average s low because the set of mutual fuds cludes corporate ad govermet bod fuds addto to commo stock fuds. The average aual value weghted retur to NYSE-AMEX stocks was 4% over the same te year perod, wth a stadard devato of 3%.) Portfolo varace s equal to: V ( r p ) ω ε + mφ 4 Whe φ s set equal to, portfolo varace s equal to the market varace plus the varace attrbutable to the resdual varaces of the assets the portfolo. Ths term, ω ε, s the 6.doc; Revsed: Jauary, 000; M.T. Maloey 8

9 Clemso Ecoomcs effcecy of the portfolo. It represets the crease varace of retur that could be elmated f the vestor were able to hold a perfectly dversfed portfolo. Table shows the effceces of portfolos of szes 5, 5, 45, ad 85. These portfolos are costructed from the mutual fuds that had the hghest retur (regardless of rsk) over the past te years. Ieffcecy falls from 50% to % as the umber of assets held the portfolo creases. Table also shows sub-optmal portfolo costructo. The sub-optmal portfolos were costructed from the same assets as the optmal portfolos. However, the weghts were set equal across all assets held the portfolos. Equally weghted portfolos become more effcet as the umber of assets creases. However, there s always a substatal crease portfolo varace due to effcet portfolo balacg. e ssue that we have omtted thusfar our dscusso of portfolo balacg s the questo of whch assets to choose for cluso a portfolo. ur CAPM model sheds some lght o ths ssue. If we combe equatos (3) ad (4) above we see that portfolo varace ca be wrtte as: M NM V ( r p ) φ + M / The mplcatos for asset choce are clear. Pck assets wth hgh ratos of beta squared to resdual varace. Also ote from (5) that as the umber of assets the portfolo grows the sum of the rato s ever creasg. As approaches fty, the effcecy of portfolo varace goes to zero. Ieffcet portfolo varace ca be terpreted terms of lost retur by choosg φ a way that equates portfolo varace wth market varace. I other words, we have bee assumg that the vestor s shootg for a target retur equal to that expected from the market. Ieffcecy portfolo costructo meas that the vestor bears more rsk tha would be the case a trasactos-costless world. I cotrast, we ca ask the questo, how much retur wll the vestor lose off of the market retur f the vestor refuses to bear the addtoal rsk. To get at ths we ca force equato (5) to equal market varace: Collectg terms ad smplfyg gves: M NM ε V ( r p ) φ + M / m QP m m ε P Q P doc; Revsed: Jauary, 000; M.T. Maloey 9

10 Clemso Ecoomcs φ$ N M m / ε + Q P / 7 We relabel φ as φ $ to sgfy that the vestor s chagg the expectato of retur order to bear o more tha the market level of rsk. The rato term m / ε s the portfolo effcecy show Table. That s, uder the assumpto that φ s equal to, m / ε ω ε m As a example, gve the portfolo of 5 assets, the effcecy s 5% ad the $ φ that yelds varace equal to the market varace s Recall what φ stads for: NM / QP k rf φ r r where k s the expected portfolo retur. If we set $ φ.8, r m., ad the rsk free retur equal to.05, the k.06. Ths meas that effcet portfolo dversfcato costs the vestor.4% expected retur f the vestor s uwllg to bear more rsk tha the market level. Note that whe the portfolo effcecy s reduced to %, $ φ goes to.945 ad ths retur shortfall decles to aroud a quarter of a pot. m Fally we tur to the questo of portfolo rebalacg. As asset values chage, portfolo composto chages ad the ew portfolo weghts are ot ecessarly optmal. For stace, f oe asset creases value because of a chage formato that has o effect o beta or resdual varace, the the default portfolo holdgs of that asset after the value crease are too large. The f. 8 6.doc; Revsed: Jauary, 000; M.T. Maloey 0

11 Clemso Ecoomcs dversfyg vestor eeds to sell off some of ths asset ad buy more of the others. 5 Portfolo rebalacg volves costly trasactos, however, ad some margal aalyss of the costs ad beefts of rebalacg s approprate. We ca get a clue about the margal value of rebalacg from equato (7) above. I equato (7) above, $ φ tells us the proporto of the rsk premum (r m - r f ) that the vestor ears as a cosequece of portfolo effcecy; (- $ φ ) s the proportoal cost of effcecy. Rewrtte terms of the weghts $ φ ca be expressed as: φ$ NM ω ε m + QP / 8 et s defe devatos from the optmal weghts absolute value as $ω. Dfferetatg $ φ wth respect to oe of the weghts we have: φ$ ω$ φ ω $ ε 3 9 m Substtutg from (3) for ω we have: φ ω$ φ$ 3 N M Σ φ$ ε Q P ε ε m φ$ 4 m Σ ε 0 where m Σ ε s the portfolo effcecy. So we have: φ$ ω$ φ$ ( φ$ ) 5 Harold Mulher ad I expla stock splts o ths bass. Stock splts usually occur followg a ru up asset value. We argue that splttg s a optmal way for the frm to allow ts dversfyg shareholders to sell off a porto of ther terest the frm. 6.doc; Revsed: Jauary, 000; M.T. Maloey

12 Clemso Ecoomcs Pluggg some real umbers, f the portfolo effcecy s %, the the chage $ φ for a {.} chage a dvdual weght o a asset wth a beta of oe s.095%. Ivestmet bakers talk about hudredths of percetage pots as bass pots. The commsso o a average trade for them s aroud 80 bass pots. Equato () suggests that f eght assets a portfolo (wth average betas of oe) were off by. each, ts probably tme to rebalace. 6.doc; Revsed: Jauary, 000; M.T. Maloey

13 Clemso Ecoomcs Table : Comparsos of Portfolos of Commo Stocks Average Aual Retur o the Market 8.06% Stadard Devato of Aual Market Retur over 5 yrs.00% Average Aual Retur o Equally Weghted Portfolo 0.78% Stadard Devato of Aual Retur over 5 yrs o Equally Weghted Portfolo 9.93% Average Aual Retur o Beta Weghted Portfolo 0.54% Stadard Devato of Aual Retur over 5 yrs o Beta Weghted Portfolo 7.55% Stadard Devato across the 5 yr average Aual Retur o Equally Weghted Portfolo 3.47% Stadard Devato across the 5 yr average Aual Retur o Beta Weghted Portfolo 3.3% 5yr Compoud Retur No Rebalacg 5.96% Stadard Devato across Portfolos.0% 5 yr Compoud Retur Aual Rebalacg 57.88% Stadard Devato across Portfolos.80% e hudred NYSE-AMEX stocks wth hghest rato of beta to stadard error; broke to te portfolos. Market model estmated from mothly returs over 5 years, 85-89; returs calculated over doc; Revsed: Jauary, 000; M.T. Maloey 3

14 Clemso Ecoomcs APPENDIX: The varace of a portfolo of assets ca be wrtte as: V ( r ) E[ r E( r )] F H G I N M K J E ω r E r ω Q P E NM ω r E r ω ( ) QP E NM ω [ r E( r )] QP p p p Usg the CAPM model to defe the retur to the th asset gves: [ r E( r )] α + rm + ε α E( rm ) [ r E( r )] + ε m m Substtutg to the portfolo varace expresso, we have: NM V ( rp ) E ω( [ rm E( rm )] + ε ) d m d QP E [ r E( r )] ω + ω ε m m ω V ( r ) + ω V ( ε ) m + ω ω Cov( r, ε ) + ω ω Cov( ε, ε ) ω V ( r ) + ω V ( ε ) m Sce the covarace of the market retur wth the resdual varato of every asset s assumed to be zero as s the covarace betwee the resdual varato of all dvdual assets, the portfolo varace s oly a fucto of the market varace ad the resdual varato of each asset. Ths s the basc prcple of the CAPM. Assets are prced relato to oe aother based o ther covarace wth each other, whch s captured beta. 6.doc; Revsed: Jauary, 000; M.T. Maloey 4