Performance attribution involves

Transcription

1 STUART MORGA s an analyst at Wngate Asset Management n Melbourne, Australa. stuart.morgan@wngategrou. com.au Performance Attrbuton of Otons: Defnng Sngle-Stock Oton Exosures and Understandng the Brnson-Fachler Effects STUART MORGA Performance attrbuton nvolves breakng u an nvestment ortfolo nto ts consttuent comonents n order to analyze the drvers of the ortfolo s erformance. Ths artcle examnes what nformaton a erformance analyst may gan when analyzng a ortfolo contanng (sngle-stock otons. In artcular, the artcle dscusses the followng four basc oton strateges: Long calls aked uts Covered calls Protectve uts There are countless ways n whch to aggregate the ndvdual securtes or comonents of a artcular ortfolo n order to analyze a ortfolo s erformance. The dea behnd ths s to grou securtes nto sectors, however they are defned, that tend to be somewhat correlated. A erformance analyst wll then be able to determne the degree and sgn of a ortfolo s relatve erformance that s due to exosures to artcular securtes wthn the sectors, known as stock selecton, as well as the degree and sgn of a ortfolo s relatve erformance that s due to beng over- or under-weghted n the sectors themselves, known as asset allocaton. Some of the more common aggregatons for ths tye of ortfolo are: Indvdual securtes (.e., no aggregaton Asset tye (.e., cash, equtes, uts, and calls Industry groung (e.g., GICS sectors The frst ste n erformance attrbuton s to calculate the contrbutons to return for each comonent, where,, n and there are n comonents n the ortfolo. It s therefore necessary to determne each comonent s weght or exosure w, as well as the return r from that comonent. These two factors yeld the comonent s contrbuton to return CTR : CTR w r ( When the ortfolo contans dervatve nstruments, t s sometmes the case that the defntons of exosures and returns are lldefned and/or non trval. The erformance analyst therefore has to choose whch defnton to use. The overarchng rncle of ths artcle s that all choces made n the calculaton of erformance attrbuton should ref lect the ortfolo manager s nvestment style. If a ortfolo contans sngle-stock otons, then the choce of defnton of exosure and return should deend on how the ortfolo manager uses those otons. IT IS ILLEGAL TO REPRODUCE THIS ARTICLE I AY FORMAT WITER 24 THE JOURAL OF PORTFOLIO MAAGEMET 3 Coyrght 24

2 Ths artcle has two uroses: the frst s to dscuss how sngle-stock otons should be treated n attrbuton analyses, ncludng a roosal for a new method of defnng exosures for otons. The second urose s to dscuss how the varous attrbuton effects are affected by the choce of comonent aggregaton. For smlcty, the tme frame wll always be taken to be a sngle day, and the ortfolo wll be taken to be a sngle-currency ortfolo (U.S. dollars, wth no external flows over the erod. Ths means that the arthmetc attrbuton effects wll add wthout the need to emloy smoothng algorthms. RELATIE ATTRIBUTIO In relatve attrbuton, the exosures and returns, as well as an arorate benchmark, need to be defned for the ortfolo. I wll use Latn letters to refer to the ortfolo and ts securtes (wth the exceton of delta, used for otons n the usual way, and the corresondng Greek letters to refer to the benchmark. The exosures and returns for the benchmark wll therefore be denoted by ω and ρ, resectvely. If comonents are combned arthmetcally, the overall return r for the ortfolo and ρ for the benchmark s the sum of the contrbutons to return: r wr ; ρ ωρ The comonents are themselves made u of one or more securtes. The contrbuton to return of each comonent can lkewse be determned from the exosures and returns of these securtes. The exosure for a comonent can be determned by smly summng the exosures of the securtes wthn that comonent. The return of the comonent can be determned by ether calculatng the weghted-average return of the securtes, or smly dvdng the contrbuton to return by the exosure of the comonent. The exosures and returns of the comonents of a ortfolo can be comared to that of the benchmark n order to calculate varous attrbuton effects. The most common such method s due to Brnson and Fachler [985], hereafter referred to as the BF attrbuton. Ths model calculates three attrbuton effects: Allocaton A A, Selecton S S and Interacton I I (where, agan,, n and n s the number of comonents. A ( ( (2a S ( (2b I ( ( (2c The sum of the three BF attrbuton effects for the whole ortfolo s equal to the relatve erformance of the ortfolo aganst the benchmark: r ρ S I+ ωρ S + ( ω ( ρ + ( ω ρ S+ I + A ( ω ( ρ + ρ + ( ω The fnal term of the second-to-last lne s equal to zero, snce the sum of all over(underweghts of a ortfolo comared to a benchmark s zero. Ths term can then be combned wth the other remanng term by brngng the benchmark s total return ρ nsde the sum, yeldng the Allocaton effect. If the ρ term was not ntroduced nto ths lne, then the remanng term of the second-to-last lne s actually equal to the Allocaton effect as defned by Brnson et al. [986]. Securtes can be aggregated n any number of dfferent ways to form the comonents or sectors of a ortfolo. In general, the BF effects wll dffer for each choce of aggregaton. For examle, f a ortfolo has n number of sectors and, n securtes n each sector, then the Allocaton effect for the th sector wll stll be defned as er Equaton (2a. However, the weghts w and ω for the ortfolo and benchmark are the total exosure to the sector n queston. In terms of the sector s consttuent securtes: w n w (3a n ω ω (3b The return for the sector to the benchmark s the weghted average return of the sector s consttuent securtes: 4 PERFORMACE ATTRIBUTIO OF OPTIOS WITER 24

3 ρ n n ωρ ω (3c If the Allocaton effect dd not deend uon the aggregaton, then, wth,, n securtes n the sector, the aggregated Allocaton effect would be equal to the sum of the Allocaton effects for each securty n that sector: n A ( ( (4 where A s defned n Equaton (2a and the weghts and returns are defned n Equatons (3. Equaton (2a becomes: A n n w w n ωρ ω ρ n ω n ωρ n ρ n ω n n ωρ The last two terms have attaned the form of Equaton (4 as requred. However, t s not true, n general, that: n w n n ωρ n ω ρ w ( Therefore, the assumton that the Allocaton effect does not deend uon the choce of aggregaton does not hold n general, and so the rooston that the Allocaton effect does deend uon the choce of aggregaton has been roven by contradcton. Smlar arguments hold for the other two BF effects: they wll vary wth the choce of aggregaton. I wll henceforth ncororate the Interacton nto the Selecton effect. However, all arguments below can be smly modfed so as to exlctly nclude the Interacton effect, wthout changng the substance of the arguments, f one were to choose to do so. The modfed Selecton wll be denoted S * and s defned as: S w ( w (5 OPTIO EXPOSURES Stannard [996] outlnes two ways to calculate the exosures of sngle-stock otons. The frst way s to use the drect value of the oton tself, wth no reference to the underlyng securty. In other words: for otons contracts (n contracts of sze share er contract and oton rce, the drect exosure n a ortfolo of value s gven by: w For oton rces at the end of day zero and at the end of day one, the return s smly r and the contrbuton to return for comonent s the roduct of the exosure and the return, whch s equal to: CTR ( (6 Stannard argues aganst usng ths aroach. I wll argue below that the drect exosure method may be arorate, deendng uon how the ortfolo manager uses the otons. Stannard s second way to calculate exosures for otons s to combne the oton oston wth a cash exosure. Ths exosure s equal to the oton s delta δ multled by the rce of the underlyng S : w S Stannard calls ths the full exosure, but I shall call t the delta exosure or delta oston. A corresondng (vrtual securty must be ncluded n the ortfolo n order that the sum of all exosures n the ortfolo s equal to one. Ths vrtual securty, called offsettng cash for the oton, s equal to the dfference between the delta exosure and the drect exosure: ( cash offset (7 The easest way to calculate the return s to realze that the dollar return wll be the same as t was for the drect exosure calculaton. Then: WITER 24 THE JOURAL OF PORTFOLIO MAAGEMET 5

4 ( r S δ S The harder way to calculate the return s to comare the delta exosure at the end of day one to the end of day zero. One mght be temted to calculate the return based smly uon these two exosures. However, ths does not take nto account what haens to the cash offset. For examle, f the value of delta changes so that less(more cash s needed to offset the delta exosure, then ths contrbutes to the delta oston as an ncome(exense amount. The thrd, new, way to calculate exosures s to use the gross exosure. Let us frst defne four basc oton strateges: Long calls: Calls are bought n order to gan exosure to the usde of a stock. Broadly seakng, cash s turned nto stock as the underlyng share rce rses, and ths s a bullsh strategy. aked uts: Puts are sold short n order to receve the oton remum. Broadly seakng, cash s turned nto stock as the underlyng share rce falls, and ths s a bullsh strategy. Covered calls: Calls are sold, whch cover shares n the ortfolo. Broadly seakng, stock s turned nto cash as the share rce rses, and ths s a bearsh strategy. Protectve uts: Puts are bought, whch rotect shares n the ortfolo. Broadly seakng, stock s turned nto cash as the share rce falls, and ths s a bearsh strategy. Other oton strateges can be defned as combnatons of these. For examle, a call oton sold short wth no covered shares can be thought of as a combnaton of covered calls and short shares. Long uts can be thought of as rotectve uts and short shares. Alternatvely, the methods dscussed below can be modfed so as to exlctly account for these tyes of oton strateges. The four basc oton strateges can be searated nto two grous, bullsh and bearsh, for whch the gross exosure s calculated dfferently. The bullsh strateges are backed by cash, and the gross exosure s defned as: ( w (8 The varable l ± deendng uon whether the oton s a call (ostve or a ut 2. ote that wll be ostve or negatve deendng uon whether the oton s long or short. Snce the oton s backed by cash, a cash-offset needs to be ncluded n a smlar way to the delta exosure examle. However, n ths case, the oton value s exlctly ncluded n the oton oston rather than n the offsettng cash as t s for the delta exosure. The offsettng cash wll not change, snce t s based on the strke rce, whch s a constant. Ths means that the return calculaton s based urely on the oton oston (whch ncludes the gross exosure for the cash. The return s: X r X (9 Exlctly, the return for a long call s X ; for X a short ut t s X. X The bearsh strateges are backed by shares, and so the gross exosure s: ( w ( Rather than usng an offsettng cash oston, the bearsh strateges wll have an offsettng equty oston. The underlyng equty oston declnes by the absolute value of, and the value of these shares shfts nto the oton oston. The return s: S r S ( Consder a covered call that s n the money. If the underlyng equty ncreases n value, then the ntrnsc value of the (short call oton wll ncrease by the same amount. Therefore, f there s no change n tme value of the oton, the return wll be zero. In other words, the change of the oton oston n ths case s smly the change n tme value of the oton. 6 PERFORMACE ATTRIBUTIO OF OPTIOS WITER 24

5 The followng four examles llustrate the dfferences among the three exosure measures. Consder a ortfolo that begns wth $2, n cash and 2, shares wth a value of $5 er share. By the end of day zero, the ortfolo has ether bought or sold contracts, of contract sze, of call or ut otons for $8, whch s the rce of the oton at the end of day zero. At the end of day one, the underlyng ncreases to $5 er share. For call otons, the value ncreases to $8, and the delta ncreases from.4 at the end of day zero to.5 at the end of day one. For ut otons, the value decreases to $3, and the delta decreases from.5 at the end of day zero to.4 at the end of day one. The four examles consder each of the basc oton strateges. In all examles, the ortfolo has a begnnng value of $4,. The frst examle s that of a long call, where $, s shfted from the cash comonent to the oton comonent. On day one, the ortfolo benefts from the stock rce ncreasng as well as the oton rce ncreasng, such that the value ncreases to $44, a gan of $3,, or 7.32%. The stock comonent has a return of 9.52%, wth an exosure of 5.22%, for a contrbuton to return of 4.88%. The oton comonent return deends uon the choce of exosure. Exhbt detals the exosures and returns for the three dfferent aroaches. The second examle s that of a naked ut. Ths strategy shfts $, from the cash comonent to the oton comonent n the same way as the long call examle. On day one, the ortfolo benefts from the stock rce ncreasng as well as the oton lablty decreasng, such that the ortfolo ncreases n E XHIBIT Long Call Examle value to $435,, a gan of 6.%. The stock return and exosure s the same as n the long call examle. Exhbt 2 detals the exosures and returns for the oton comonent for the three dfferent aroaches for ths examle. The thrd examle s that of a covered call. Ths strategy shfts, of the shares out of the stock comonent and nto the oton comonent. On day one, the ortfolo benefts from the ncrease n the stock rce, but ths beneft s lmted to the uncovered shares, whch acheve a return of 9.52% on an exosure of 25.6%, for a contrbuton of 2.44%. As mentoned earler, snce ths oton s n-the-money, the gan for the covered shares s elmnated by a corresondng loss n ntrnsc value for the coverng call otons. Exhbt 3 detals the exosures and returns for the oton comonent for ths examle. Ths ortfolo ncreases n value to $42, at the end of day one, whch s a gan of 2.44%. The fourth and fnal examle s that of a rotectve ut. Ths strategy also shfts, of the shares out of the stock comonent and nto the oton comonent. On day one, the ortfolo benefts from the ncrease n value of the shares, but mtgatng ths gan s a loss n (tme value of the ut oton. The stock comonent, wth an exosure of 25.6%, ncreases n value by 9.52%, for a contrbuton of 2.44%. However, the oton comonent, wth an exosure of 27.56%, ncreases n value by ust 4.42%, for a contrbuton of.22%. Overall the ortfolo ncreases n value to $425, at the end of day one, whch s a gan of 3.66%. E XHIBIT 3 Covered Call Examle E XHIBIT 2 aked Put Examle E XHIBIT 4 Protectve Put Examle WITER 24 THE JOURAL OF PORTFOLIO MAAGEMET 7

6 CHOOSIG THE EXPOSURE The choce of exosure for sngle-stock otons n the attrbuton analyss should roerly ref lect the nvestment style of the ortfolo manager. As an examle, the Wngate Asset Management equty strategy uses exchange-traded sngle-stock otons to enter and ext long-only large ca equty ostons. Instead of buyng the underlyng shares, Wngate wll sell a naked ut oton to gan exosure to that stock. The cash needed to buy the shares s set asde n the case that the otons are assgned to the fund. In other words, Wngate makes an allocaton decson over the whole of that cash (gross exosure. Ths s n contradstncton to a ortfolo manager who allocates only the delta-adusted exosure to that stock oston. Ths latter tye of ortfolo has a dynamc cash-offset fgure whch changes n ste wth changes n delta. It would clearly be more arorate to use the delta-adusted exosure n the attrbuton analyss for ths tye of ortfolo. A thrd ortfolo manager mght choose to allocate only the bare mnmum amount of cash to the short ut oston the collateral requred by the manager s clearer or the oton s exchange. In ths case, t would be most arorate to substtute delta for the collateral factor the roorton of the gross exosure that s requred for collateral. A fourth ortfolo manager mght trade n otons for whch she does not need to ost collateral, or only needs to ost a mnmal amount of collateral. Ths ortfolo manager consders the otons themselves to be assets. For ths ortfolo, a erformance analyst should choose to use the drect exosure method n any erformance attrbuton. In all four of these cases, the contrbuton to return for naked ut otons over the tme erod remans unaffected by the choce of exosure. The exosure decson affects only the exosures and the resultng returns to the ortfolo for each securty or comonent. Ths s arorate, snce the roorton of offsettng cash chosen by the ortfolo manager s an allocaton decson, as shall be seen below. The same s true for long calls. For the bearsh strateges the covered call and the rotectve ut the otons are backed by shares rather than cash, and the contrbuton to return s macted by the choce of exosure n the attrbuton analyss. The gross exosure n ths case s smlar n concet to the drect exosure, excet n the latter the underlyng shares are exlctly ncluded n the otons category. One should be aware that erformance of the underlyng shares under the drect exosure method wll overstate ostve erformance and understate negatve erformance n the case of a covered call (and the ooste n the case of a rotectve ut due to the asymmetrc nature of the coverng call oton. The contrbuton to return for the otons comonent consders the oton wthout reference to the underlyng shares, whereas ths s exlctly ncluded n the gross exosure method. The delta exosure method has the same contrbuton to return as the drect exosure method, but the exosure s modfed by delta. The contrbuton to return should be comared aganst the contrbuton to return of the underlyng shares n order not to exaggerate the contrbuton of the underlyng shares. If ths s done, then the contrbuton to return for the otons ncludng the underlyng shares wll be the same under the delta exosure method as t wll be for the gross exosure method. The dfference s that the delta exosure method has an offsettng vrtual cash oston that ncreases as the oton goes more n the money. However, n most cases ths cash wll not be avalable for the ortfolo manager to nvest elsewhere, and so there wll be an effectve overallocaton to cash n the attrbuton analyss. Ths roblem s mnmzed n ortfolos that use delta hedgng, for examle. Other ssues wth usng the delta exosure nclude: Comutatonal comlexty, esecally wth regard to calculatng a cash-offset amount that changes on a daly bass Incomlete hstorcal data records of delta The change n value of the share rce may ndeed be a good aroxmaton for the change n value of the delta oston, but ths s only true f the change n oton rce (or share rce s small. If ths aroxmaton does not hold, then there s no advantage n usng delta n the frst lace. SECURITY ATTRIBUTIO The Selecton effect of Equaton (5 has two factors: the weght of the securty n the ortfolo and the relatve erformance of that securty comared to the 8 PERFORMACE ATTRIBUTIO OF OPTIOS WITER 24

7 benchmark. If a ortfolo contans a artcular securty but does not trade n the securty over the tme frame of the attrbuton, then the Selecton effect at the securty level wll be dentcally zero: the securty has dentcal erformance n the ortfolo as t has n the benchmark. The corollary to ths s: f there s a Selecton effect for a artcular securty, then there has been tradng done n the securty over the attrbuton s tme frame. As such, the Selecton effect for a erformance attrbuton by securty (.e., no aggregaton actually gves a measure of the tradng effectveness of the ortfolo manager. On the other hand, the Allocaton effect of Equaton (2a has two factors: the actve weght of the securty comared to the benchmark, and the relatve erformance of that securty comared to the benchmark s return as a whole. Ths means that the Allocaton effect at the securty level s actually a measure of securty selecton. The ndvdual Allocaton effects can gve some ndcaton of whch artcular allocatons to securtes rovded large magntudes of relatve return. For ths reason, t s useful to exlctly nclude all nonbenchmark securtes n the attrbuton analyss. The sum of the allocaton effects for the ortfolo has only lmted usefulness n erformance analyss, snce t s smly the ortfolo s out(undererformance net of any tradng effects. A near alternatve to attrbuton by ndvdual securty s to aggregate the securtes by arent comany. Consder a fund that conssts of cash and naked ut otons. The erformance analyst could use the delta exosure for ths analyss, but ths wll not reveal much about the effectveness of the otons strategy, snce the attrbuton wll rovde the same results as f the fund were made u of cash and a delta-adusted number of (long shares n the underlyng stock of the ut otons. In other words, ths analyss s agnostc about whether the erformance was acheved by shares or by otons. That beng sad, the Selecton effect wll rovde a measure of tradng effectveness whch could be useful n determnng the cost of ortfolos that emloy dynamc delta hedgng, for examle. A fund that uses naked ut otons or long calls to gan exosure to a stock could nstead use the gross exosure method. The resultng Allocaton effect from the attrbuton by arent comany ndcates how well the ortfolo manager selected stocks. The Selecton effect, on the other hand, s the sum of two thngs: the ortfolo manager s tradng effectveness and the effectveness of the otons strategy vs-à-vs a ortfolo that was smly long n the shares of the underlyng stock. Thus, the attrbuton analyss exlctly accounts for the otons strategy. A fund that uses covered calls or rotectve uts would have a ostve Allocaton effect wth the gross exosure method f the stock outerforms the market, and a negatve one f the stock undererforms the market, thus ndcatng whether or not the stock was selected well. et of any tradng effects, the Selecton effect wll be ostve f the share rce decreases (ndcatng a benefcal otons strategy and negatve f the share rce ncreases (ndcatng the otons strategy hndered erformance. ASSET TYPE ATTRIBUTIO An asset tye attrbuton for a ortfolo contanng shares and otons aggregates securtes by the followng sectors: Equtes Puts Calls Cash ote that the delta exosure would effectvely remove the uts and calls asset tyes, snce these would smly fall nto the equtes category (wth resect to ther resectve delta-adusted exosures. The gross exosure method s therefore useful n answerng the queston of whether or not the otons strategy added value. ote also that the bearsh strateges would shft some of the equtes exosure out of the equtes category and nto the calls category under the gross exosure method. The calculaton of the exosure and return numbers n the equtes sector s trval. If the benchmark s an ndex, then the Allocaton effect wll be dentcally zero, snce the return to the benchmark for the equtes sector s the same as the total return of the benchmark. The total attrbuton effect s therefore gven by the Selecton effect. Ths effect smly ndcates how well the ortfolo s equtes erformed comared to the benchmark. The return calculaton for each ut oton s gven n Equaton (9, such that the return for the ut sector s WITER 24 THE JOURAL OF PORTFOLIO MAAGEMET 9

8 the weghted average return of all of the ut securtes (weghted by the gross exosure. The exosure and the return of the uts sector can then be used to calculate the attrbuton effects. However, a queston stll remans: what s the return ρ to the benchmark for the uts sector? Ths s not well defned, snce there are no uts n the benchmark. There are two obvous ossbltes: The frst ossblty s to use the return of the uts sector to the ortfolo as the return to the benchmark. The Selecton (and nteracton, f one nssts effect s then dentcally zero. Ths leaves only the Allocaton effect, whch reduces to: A w ( (2 snce the uts sector has zero weght n the benchmark. ote that ths looks very smlar to a Selecton effect. The only dfference s that ρ n Equaton (5, the return to the benchmark for the sector, has been relaced by ρ, the total return of the benchmark. Ths Allocaton tye effect s also the total effect for the uts sector. It smly ndcates the effectveness of the ortfolo manager s ut otons strategy. ote that ths dffers from the Selecton effect n the arent comany attrbuton only n the sense that the latter assumes the ortfolo manager would have otherwse bought the equtes of the underlyng stock nstead of sellng the uts; the former assumes the ortfolo manager would have bought the ndex rather than the ndvdual equtes. The second ossblty s to use the total return of the benchmark as the return of the benchmark n the uts sector. In ths case, the Allocaton effect s dentcally zero. The Selecton effect reduces to the same form as the Allocaton effect n Equaton (2, whch s also the total attrbuton effect for the sector. It s not easly determned whether ths effect has been caused by allocaton or selecton, so I roose to name t the Put effect nstead. The exosure for the calls sector s calculated as the sum of drect exosures for all the call otons n the ortfolo. The return s the weghted average return of the call otons. The Allocaton and Selecton effects have the same ssues as the ut sector. Therefore there s only one effect, the total attrbuton effect, whch shall be called the Call effect. The exosure to the cash sector for the ortfolo s calculated by addng and subtractng all oton cash offsets from the cash balance. If there s no currency effect, then there wll also be no Selecton effect, and the resultng Allocaton effect wll also be the total attrbuton effect whch shall be called the Cash effect. The return to an ndex benchmark for the cash sector s defned to be zero. Therefore, assumng the ortfolo s not leveraged, the Cash effect wll be ostve f the benchmark s negatve over the erod, and negatve f the benchmark s ostve. Ths wll be reversed f the ortfolo s leveraged (n other words, f t has a negatve cash exosure. GICS SECTOR ATTRIBUTIO A GICS sector tye attrbuton s a very common method of attrbuton, and the rocess s well known. The only comlexty for a ortfolo of equtes, otons, and cash s to roerly defne the exosures and returns for the searate GICS sectors. The delta exosure method wll yeld the same results as f a delta-adusted number of shares had been used n the ortfolo nstead of the otons. For the gross exosure method, each ut and call oton must be allocated to the GICS sector of the underlyng stock. The exosure for long calls and naked uts s the gross exosure, and the exosure for covered calls and rotectve uts s the drect exosure (whch wll be the same as the gross exosure n ths case. The return for each sector s the weghted average return of all securtes, where the erformance of equty securtes s calculated n the usual manner. The erformance of bullsh otons s calculated as er Equaton (9, and the erformance of bearsh otons s calculated as the drect return of the otons themselves. The Allocaton effect ndcates the ortfolo manager s effectveness n allocatng to GICS sectors. The Selecton effect ndcates how well the ortfolo s chosen stocks erformed wthn the sector, together wth the effectveness of the otons strategy. The tradng effectveness and effectveness of the otons strategy can be searated by aggregatng the securtes by asset tye wthn each GICS sector. COCLUSIO One mght argue that t would be better to use the delta exosures when calculatng exosures of the resectve comonents (for examle, GICS sectors, PERFORMACE ATTRIBUTIO OF OPTIOS WITER 24

9 regardless of what the ortfolo manager ntended when enterng the oton oston, snce the delta exosure gves a more accurate cture of the ortfolo s exosure to each sector. However, such an attrbuton would commonly overestmate a ortfolo s allocaton to cash. Ths s less obvous wth naked uts and long calls than t s for rotectve uts and covered calls, snce wth the latter two tyes of otons there s no actual cash freed u by an ncrease n (magntude of delta. Unless the ortfolo s, say, a hedge fund that has the faclty to borrow aganst ths vrtual cash, there s no cash that the ortfolo manager can use n other oortuntes. I argue, however, that ust as the whole of the underlyng equty oston s often used as collateral for a covered call, so too the whole of the underlyng cash oston (the gross exosure the amount of cash needed to buy the shares s often used as the collateral or offsettng cash for a naked ut oston. In these cases, t s more arorate to consder the exosure to the oton oston as the gross exosure, snce that s how much of the ortfolo has been allocated to the oston. In any case, a erformance analyst can answer artcular questons about the effectveness of oton strateges by calculatng the erformance attrbuton wth arorate aggregatons. To obtan the fullest cture of a ortfolo s erformance, the analyst should calculate the BF attrbuton effects, where the securtes are aggregated by GICS sector (along wth any other aggregaton n whch the analyst s nterested, such as market catalzaton as well as calculatng the BF attrbuton effects for two aggregatons: arent comany and asset tye. The Selecton effect from the arent comany aggregaton wll ndcate the effectveness of the ortfolo manager s otons strategy (along wth the tradng effectveness. The Put effect and the Call effect from the asset-tye aggregaton yeld mortant nformaton about the effectveness of the ortfolo manager s otons strategy. And the Cash effect ndcates the oortunty cost or beneft of not beng fully nvested n the market (or of beng overnvested n the market for a leveraged ortfolo. EDOTES If one desred to exlctly nclude the oton contract sze C n these calculatons, then substtute for C. 2 It s the sgn of δ. REFERECES Brnson, G.P., and. Fachler. Measurng on-u.s. Equty Portfolo Performance. The Journal of Portfolo Management, ol., o. 4 (985, Brnson, G.P., L.R. Hood, and G.L. Beebower. Determnants of Portfolo Performance. Fnancal Analysts Journal, ol. 42, o. 4 (986, Stannard, J.C. Measurng Investment Returns of Portfolos Contanng Futures and Otons. The Journal of Performance Measurement, ol., o. (996, To order rernts of ths artcle, lease contact Dewey Palmer at dalmer@ournals.com or WITER 24 THE JOURAL OF PORTFOLIO MAAGEMET