Understanding the Profit and Loss Distribution of Trading Algorithms

Transcription

1 Undersanding he Profi and Loss Disribuion of Trading Algorihms Rober Kissell Vice Presiden, JPMorgan Robero Malamu, PhD Vice Presiden, JPMorgan February 25 Absrac Wih he adven of algorihmic rading i is essenial ha invesors become more proacive in he decision making process o ensure selecion of he mos appropriae algorihm. Invesors need o specify benchmark price, implemenaion goal, and preferred deviaion sraegy (i.e., how he opimally prescribed algorihm is o reac o changing marke condiions or prices). In his paper we describe an analyical process o assess he impac of hese decisions on he profi and loss disribuion of he algorihm. 1

2 Inroducion As financial markes have become more compeiive, invesors have sared o urn o algorihmic rading o achieve beer execuion prices. However, he uilizaion of algorihmic rading alone does no guaranee beer resuls. Traders need o become more proacive o ensure ha he underlying algorihmic sraegy is consisen wih heir invesmen objecives. Prior o he selecion of he sraegy, raders firs need o perform pre-rade analysis o assess he suiabiliy of he order and/or rade lis for algorihmic rading since no all orders are appropriaely handled via algorihms. If algorihmic execuion is deemed accepable for he order, raders subsequenly need o address macro- and micro-level issues. Macro level decisions include: specificaion of desired benchmark price, and implemenaion goal (e.g., aggressive or passive execuions). The laer decision, however, requires in-deph cos analysis o undersand he complee consequence of he decision. Micro level decisions include specifying any desired deviaion rules. This includes how he algorihm should deviae from he opimally prescribed sraegy based upon changing sock prices, marke movemen, and/or change in index or secor values, as well as changing marke condiions such as volume profiles and volailiy. Microlevel decisions also include specificaion of order submission rules such as marke or limi order, display size, wai period for new submissions, order revisions, modificaions, and/or cancellaions. Finally, i is essenial raders perform proper pos rade analysis o assess he performance of he algorihm and ensure i is consisen wih overall invesmen objecives. Traders who selec algorihms wihou knowledge of poenial oucome are ignoring heir fiduciary responsibiliies o heir invesors. To bes undersand he algorihmic decision making process, however, i is imporan o undersand he basics behind ransacion cos managemen. Seminal ransacion cos research is primarily due o Treynor (1981), Perold (1988), Berkowiz, Logue, & Noser (1988), Wagner (199), and Hasbrouck (1991). More recenly, however, Bersimas & Lo (1996), Almgren & Chriss (1999), and Kissell, Glanz, and Malamu (24) expanded his work o provide a decision making framework o manage ransacion coss. Accordingly, his work now serves as he basis for algorihmic decision making. Pre-Trade Analysis Pre-rade analysis provides he necessary daa o make informed algorihmic rading decisions. I provides invesors wih liquidiy summaries, cos & risk esimaes, as well as rading difficuly and sabiliy measures o deermine which orders can be successfully implemened via algorihmic rading and which orders require manual inervenion. No every order is well suied for algorihmic implemenaion. Pre-rade analysis provides insigh ino poenial risk reducion and hedging opporuniies o furher improve execuion. Finally, pre-rade analysis also provides invesors wih necessary daa o develop views for shor-erm price movemen and marke condiions. Afer evaluaing he suiabiliy of he order for algorihmic rading i is essenial raders develop a cusomized algorihm exacly suied o he specific order. Brokers who do no offer flexibiliy o cusomize he algorihmic sraegy are un- 2

3 able o provide invesors wih he mos appropriae algorihms. Macro Level Decisions Benchmark Price The firs sep in he algorihmic decision making framework is specificaion of he benchmark price. The more common benchmarks can be caegorized ino pre-, inra-, and pos-rade prices. The prerade benchmark prices, also commonly referred o as implemenaion shorfall ( IS ) benchmark prices, are hose prices ha are known before or a he ime rading begins. These include he invesmen decision price, previous nigh s closing price, opening price, and arrival price (i.e., price a ime of order enry). Inraday benchmarks are comprised of hose prices ha occur during rading such as VWAP, TWAP, and average of open, high, low, and close. Pos-rade benchmarks include any price ha occurs afer or a he end of rading, he mos common of which being he day s closing price. The benchmark price is invesor specific and may in fac be differen for wo invesors wih idenical rade liss. For example, a value manager may desire execuion a heir decision price (i.e., he price used in he porfolio consrucion phase), a muual fund manager may desire execuion a he closing price o coincide wih valuaion of he fund, and a indexer may desire execuion ha achieves VWAP (e.g., o minimize marke impac) as an indicaion of fair prices for he day. Furhermore, he same invesor may specify differen benchmark prices for idenical orders (i.e., 5, shares of MSFT) on differen days if he invesmen objecives have changed. Table 1: Benchmark Prices Pre-Trade Inra Pos Decision Price VWAP Fuure Close Previous Close TWAP Opening Price OHLC Arrival Price Implemenaion Goal The nex sep in he process is o specify he inended implemenaion goal. This relaes o he level of rading aggressiveness or passiveness. Aggressive rading is associaed wih higher cos and less risk while passive rading is associaed wih lower marke impac and higher risk. This marke phenomenon gives rise o he rader s dilemma: Trading oo aggressive will lead o higher impac cos bu rading oo passively will lead o higher risk and may resul in even more cosly rades. Therefore, he implemenaion goal is o solve he rader s dilemma. The soluion is found by balancing he radeoff beween cos and risk based a he invesor specified level of risk aversion as follows: Min α Cos ( α ) + λ Risk( α ) where λ is he risk aversion parameer andα is he rading rae (defined as a percenage of volume). Solving for various levels of lambda will resul in numerous rading sraegies each wih differen cos & risk. The se of all hese poins comprises he efficien rading fronier ( ETF ) inroduced by Almgren & Chriss (1999) and provides quaniaive insigh 3

4 Figure 1: Efficien Trading Fronier. The efficien rading fronier (ETF) is he se of all opimal rading sraegies. Tha is, hose sraegies wih he lowes cos for a given quaniy of risk and he leas risk for a specified cos. Figure 1a illusraes he ETF for an arrival price benchmark. Noice he aggressive sraegies are associaed wih higher cos bu less risk while he passive sraegies are associaed wih lower cos bu higher risk. Figure 1b compares he ETF for a pre-rade decision price benchmark (such as he previous day s closing price) o he arrival price ETF. Noice ha his ETF is shifed o he righ o accoun for he addiional risk of he unknown price change from he closing price o nex day s opening price. Figure 1c compares he ETF for a fuure price benchmark such as he day s closing price o he arrival price ETF. This ETF is shifed downwards o accoun for a lower benchmark cos since he fuure price will consis of permanen marke impac whereas he order arrival price does no. Figure 1d compares he ETF for a VWAP benchmark o he arrival price ETF. The cos profile for a VWAP execuion (or any inra-rade benchmark) is unique in ha invesors can minimize boh benchmark cos and risk by paricipaing wih volume over he horizon (denoed by sraegy A ). Therefore, he VWAP ETF is upward sloping and invesors can simulaneously minimize benchmark cos and risk. ino cos consequence associaed wih level of aggressiveness. Afer compuing he ETF invesors selec he mos appropriae opimal sraegy based upon heir invesmen objecive. For example, informed raders wih expecaions regarding fuure price movemen are likely o selec an aggressive sraegy (e.g., POV=3%) wih higher cos bu more cerainy surrounding expeced ransacion prices. Indexers are likely o selec a passive sraegy (e.g., POV = 5%) or a risk neural sraegy o reduce cos. Some invesors may selec a sraegy ha balances he radeoff beween cos and risk depending upon heir level of risk aversion, and ohers may elec o paricipae wih volume hroughou he day (e.g., VWAP sraegy). 4

5 Figure 2: Specifying Implemenaion Goal. Figure 2a illusraes how an invesor wih an implemenaion goal o achieve a cos C 1 will have differen execuion sraegies for an arrival price and previous close (decision price) benchmark. Since here is less risk associaed wih he arrival price benchmark invesors will execue following sraegy X 1 o achieve cos C 1 wih lower risk R 1. Invesors wih he previous closing price benchmark are subjec o higher risk due o poenial overnigh price movemen and will execue following X 2 o achieve cos C 1 wih higher risk R 2 (R 1 <R 2 ). Figure 2b illusraes how differen benchmark prices and differen implemenaion goals can resul in idenical execuion sraegies. Here invesors wih differen benchmark prices (arrival and day s close) as well as differen desired coss can resul in idenical sraegies X 1 and X 2 boh wih risk R 1. To solve he opimizaion problem and consruc he ETF, however, i is necessary o formulae he cos and risk of he execuion. To address his need we firs esimae he expeced average execuion price a commencemen of rading as follows 1 : P ( α ) = P + f ( X, α) X ) + ε ( α) where, X is he number of shares o rade, P is he marke price a ime of order enry, f ( X, α) is emporary impac cos due o he liquidiy demands of raders, g( X ) is he permanen impac cos due o he informaion leakage of he order, ε ( α) is price volailiy. The cos funcion is compued as follows: Cos Risk ( P ) ( α ) = E P( α ) ( α ) = σ ( ε ( α )) We nex provide he derivaion of Cos and Risk for he arrival, previous nigh s closing price, fuure closing price, and VWAP benchmarks. Arrival Price Cos ( P ) ( α ) = E P ( α ) b [ ε ( α )] = P + f ( X, α) X ) + E P = f ( X, α) X ) 2 wih E[ ε ( α )] = and [ ε ( α )] = σ ( α ) Var. The corresponding risk of he execuion is: 1 For simpliciy we express expeced execuion price only in erms of marke impac. In acualiy, he expeced execuion price will consis of boh marke impac and price appreciaion (momenum, rend, alpha) cos. R ( α ) = σ ( ε ( α )) The ETF for he arrival price benchmark 5

6 is consruced using various values for he rading raeα (i.e., percenage of volume) and is shown in figure 1a. The char illusraes ha more aggressive rading will incur higher cos bu less risk, and more passive rading incurs lower cos bu a higher quaniy of risk. The ETF allows raders o analyze and hen selec heir preferred opimal rading sraegy (i.e., he sraegy wih he lowes cos for a given level of risk and he leas risk for a specified cos). Previous Nigh s Closing Price The ETF for he previous nigh s close (denoed as P d ) is compued in a similar manner o he arrival price as follows: Cos ( P ) ( α ) = E P ( α ) = P + f ( X, α) X ) + E d [ ε ( α) ] Pd Bu since P = P d + ξ where ξ is he price change from previous close o he ime of order enry (unknown in advance) wih E [ ξ ]= and Var [ ξ ] = σ 2 ( ξ ) we have: Cos and, ( α ) = P + f X, α) X ) + E[ ε ( α) ] ( P ξ ) = f ( X, α) X ) R ( 2 2 ( α ) = σ ( ε ( α )) + σ ( ξ ) Noice ha in his case he associaed iming risk of he rade is larger han for he arrival price which is due o he fac ha here is poenial for price movemen in he sock price from he ime of he invesmen decision o he ime of he order enry. For analyical purposes, he higher iming risk causes a shif o he righ in he efficien rading fronier. This is shown in figure 1b. Therefore, wo invesors wih he idenical level of risk aversion bu one is using he arrival price benchmark and he oher is using he previous nigh s closing benchmark will specify differen algorihmic sraegies alogeher. These sraegies, however, will have he same expeced cos bu differen iming risk. This is depiced in figure 2a. Noice ha sraegies X1 & X2 have he same expeced cos C1 bu differen risk. The arrival price benchmark has a lower risk R1 han he previous closing price benchmark R2 because i is subjec o incremenal overnigh risk. Therefore, for idenical levels of risk aversion he previous nigh s closing price benchmark will be more passive han he arrival price algorihm. Fuure Closing Price To compue he cos profile for a fuure closing price i is necessary o esimae he expeced fuure closing price P T. By definiion, all fuure prices will include he permanen marke impac of he order (recall ha for simpliciy we assume no price appreciaion over he rading period) bu no emporary impac cos. Therefore, we have: P T = P X )) The cos profile is hus compued as follows: Cos ( P ) ( α ) = E P ( α ) T = P + f ( X, α) X ) + ε ( α) = f ( X, α) ( P X )) wih corresponding iming risk as follows: R ( α ) = σ ( ε ( α )) Noice ha for he fuure closing price benchmark he expeced cos is equal o 6

7 emporary marke impac only, and i is less han he arrival price benchmark bu has he same quaniy of iming risk. This causes a downward shif in he ETF (illusraed in figure 1c). The magniude of he downward shif is equal o he quaniy of permanen marke impac. The consequence of he fuure price benchmark on he algorihmic rading decision will resul in he same sraegy bu wih lower expeced cos. This is illusraed in figure 2b. Noice ha sraegies X1 and X3 have differen expeced coss, C1 & C3 respecively, bu for he same level of risk aversion, hese sraegies will have he same quaniy of risk R1. I is imporan o reierae here ha he risk erm R1 denoes he uncerainy corresponding o he expeced cos esimae. Tha is, he uncerainy surrounding he expeced execuion price and he benchmark price. VWAP Benchmark The ETF associaed wih he VWAP benchmark (or any of he inra-day benchmarks) has a much differen shape han any of he pre-rade or pos-rade benchmarks and is an increasing funcion wih respec o risk. This is because he inra-day benchmark prices include boh emporary & permanen impac cos, and he iming risk calculaion is based upon all prices and rades over he enire rading period raher han from a specific poin in ime. For inraday benchmark prices, raders minimize boh cos (i.e., expeced gain/loss o he benchmark) and uncerainy of he expeced gain/loss (iming risk) by paricipaing wih volume. Any deviaion from his sraegy will resul in higher risk. This is depiced in figure 1d wih he only raional opimal sraegy denoed by A. Micro Level Decisions Sraegy Deviaion Rules I is imporan ha invesors specify heir preferences for he algorihmic deviaion rules. This defines how he algorihm should deviae from he originally prescribed opimal sraegy and implemenaion goal. The more common deviaion rules include changing he execuion rae (o be more or less aggressive) or POV rae based upon changing marke volumes, sock price movemen (momenum), overall marke movemen, or possibly based on changes in index values such as he S&P5 or based on changes in he sock s specific secor index. Addiionally, invesors may also choose o change he execuion sraegy based on changing marke condiions such as volume profiles due o special evens such as new announcemens, or fed indicaors, as well as changes in volailiy. In specifying he appropriae deviaion sraegy, i is imporan o have a complee undersanding of how he deviaion rule impacs he cos disribuion. For example, many price-based scaling rules (adjusing he execuion rae based on price movemen) are designed o resul in lower cos on average, bu i is accompanied wih an increase in ail risk and diminished possibiliy for large gains. Therefore, in imes of adverse momenum, he deviaion sraegy may be more cosly han a sraegy wihou any specified deviaion rule. Addiionally, i is possible o develop deviaion rules ha furher minimize he poenial for large losses wih an increase in poenial for gains, bu his comes a an increased cos. Regardless of he specified deviaion rule, i is essenial ha raders undersand he impac of he decision on he cos disribuion. 7

8 Figure 3: Deviaion Rule Cos Disribuions. Figure 3 illusraes he rading cos disribuion associaed wih alernaive deviaion sraegies (i.e., how he algorihm will adap o changing marke condiions such as price rends). Figure 3a illusraes a sandard rading cos disribuion for a specified iniial rading rae expressed as a percenage of volume ( POV ). In his siuaion he expeced cos is C 1 wih he poenial of unfavorable prices denoed as Risk in he righ hand ail. Figure 3b illusraes he disribuion for a Plus sraegy ha maximizes he likelihood ha he realized price will be more favorable han he benchmark price. In his siuaion he algorihm will be aggressive in imes of favorable price and passive in imes of adverse price movemen o limi excess marke impac cos. The overall resul will be a lower cos on average C 3 <C 1 bu higher poenial for unfavorable prices if he adverse rend coninues. This higher risk is denoed as Risk in he righ hand ail. Addiionally, he Plus algorihm limis he poenial o realize beer prices if he favorable rend persiss. Figure 3c illusraes he disribuion for a Wealh sraegy ha becomes aggressive in imes of adverse price movemen o limi losses and passive in imes of favorable prices o ake advanage of he beer prices. The overall resul is a higher cos on average C 4 >C 1 caused by being aggressive in imes of adverse price movemen bu he benefi is lower risk of high coss and higher poenial for gains. Figure 3d illusraes he cos disribuion for a Srike sraegy ha will coninuously adap o changing marke condiions such ha expeced realized price will be equal o he benchmark price. In his siuaion he algorihm will be mos aggressive in imes of favorable prices o lock ino he benchmark price and passive in imes of adverse price movemen in order o limi unnecessary marke impac cos. The overall resul here is lower cos on average C 2 <C 1 bu wih increased exposure o unfavorable prices if he adverse rend coninues. The srike deviaion sraegy also limis he poenial for beer price if he favorable rend persiss. To bes illusrae his poin we will discuss he impac of deviaion rules on he cos disribuions resuling from price based scaling schemes 2. For example, one price-based scaling scheme may be o become more aggressive (i.e., increase 2 In addiion o sock price movemen, price-based scaling schemes can also be developed based on marke movemen, index values, secor values, or indusry values, or any oher momenum measure. he POV rae) when prices are more favorable han he benchmark and become less aggressive (i.e., decrease he POV rae) when prices are less favorable han he benchmark. An alernaive deviaion scheme is o become more passive when prices are favorable and more aggressive when prices are unfavorable o limi poenially higher coss and coninue o paricipae wih he beer prices wih he 8

9 hope of even beer execuion prices. There are hree (3) main caegories of price-based scaling echniques: i) Srike (associaed wih he lowes mean and highes risk compared o he saic POV sraegy and alernaive price-based scaling schemes), ii) Plus (associaed wih a lower mean and higher risk han he saic POV sraegy), and iii) Wealh (associaed wih a higher mean bu lower risk compared o he saic POV sraegy) 3. These disribuions are shown in figures 3a 3d. Each echnique adjuss he algorihm in a differen manner o become more or less aggressive based upon changing prices and marke condiions. Bu before invesors/raders specify he preferred deviaion sraegy, i is essenial o have a clear undersanding of he consequences of he rule on he cos disribuion. These are described below: Srike: Min α () E ( P ( α ) P ) 2 The srike deviaion rule dynamically adjuss he POV rae o minimize he quadraic cos funcion wihou regards o risk. Here, he algorihm will become more aggressive in imes of favorable prices and less aggressive in imes of unfavorable prices. For example, for a buy order he srike algorihm will become more aggressive when prices are less han he specified benchmark price ( in-hemoney ) and less aggressive when prices are higher han he benchmark ( ou-ofhe-money ). This deviaion scheme provides lower cos on average bu comes a he expense of increased risk on he cos side and reduced poenial o achieve very favorable prices because he order 3 Price-based scaling is only one ype of sraegy deviaion rule. Oher rules can be developed based on volumes and volume profiles, volailiy, news and informaion, ec. b will likely be compleed prior o hese prices arising. This cos disribuion for he srike is shown in figure 4a. The expeced cos C2 is less han ha for he saic POV sraegy wih expeced cos C1. I is lower on average because he algorihm will ake advanage of favorable prices when hey arise. The disribuion, however, is skewed o he cos side. This is because by becoming more passive when prices are unfavorable here is a higher chance of realizing exremely unfavorable prices if adverse price rends persis. Similarly, his deviaion rule does no provide poenial opporuniy o achieve beer prices if favorable rends persis because i is likely he order will already be compleed by he ime hese prices arise. Therefore, he cos curve is also runcaed on he benefi side. Plus: Max α () E ( P ( α ) Pb ) R( α ) The plus deviaion rule dynamically adjuss he POV rae o maximize he likelihood ha he algorihm will ouperform he specified benchmark price. Noice ha his formulaion is idenical o ha of maximizing he Sharpe raio. The plus deviaion rule behaves similar o he srike algorihm by becoming more aggressive when in-he-money and less aggressive when ou-of-he-money. Bu unlike he srike, he plus is risk sensiive (noice he risk erm in he denominaor) so i does no expose invesors o he same degree of fa ail risk. I does, however, provide opporuniy o achieve somewha beer prices if favorable rends persis. Unforunaely, his comes a a slighly higher cos on average C 3 >C 2. Figure 4b compares he disribuion of he plus deviaion rule o he saic POV rule. By coninuously adaping o 9

10 Figure 4: Comparison of Price-Based Scaling Cos Disribuions. Figure 4a compares he Srike algorihm o he POV sraegy. The Srike algorihm will incur a lower cos on average bu exposes he rade o grea risk of unfavorable prices and limis he poenial o paricipae wih favorable prices. Figure 4b compares he Plus sraegy o he POV sraegy. The Plus sraegy incurs also lower cos on average bu wih increased risk of unfavorable prices and decreased poenial for paricipaing in beer prices if favorable rends coninue. Figure 4c compares he Wealh sraegy o he POV sraegy. The Wealh sraegy limis he poenial o incur unfavorable prices and increases poenial o paricipae wih beer prices in imes of favorable rends, bu his comes a a slighly higher cos on average han a POV sraegy. Figures 4d-4e compare he cos disribuion of each of he deviaion sraegies. I illusraes o invesors he change in coss and shif in disribuion for he differen deviaion algorihms so ha invesors can choose he algorihm mos inline wih overall invesmen objecive. 1

11 marke condiions, i will realize lower coss on average C 3 <C 1. However i does increase risk of higher poenial coss and does no allow for he full array of beer prices, alhough i does provide addiional proecion over he srike as well as opporuniy for addiional gains. Figure 4d compares he plus algorihm o he srike algorihm. Wealh: Max α () log E [ P ( α )] { H } The wealh deviaion rule is mos closely relaed o he radiional economic syle risk aversion uiliy funcion H inroduced by Pra (1964) and laer formalized by Arrow (1971), and i is inended o maximize invesor wealh in presence of uncerainy. This deviaion rule behaves in he opposie manner of Srike and Plus by becoming more passive in imes of favorable prices and more aggressive in imes of unfavorable prices as a means of limiing poenial large losses if adverse condiions persis. The cos disribuion of he wealh deviaion rule is illusraed in figure 3c. Noice ha i is skewed o he lef and runcaed on he righ showing he larger poenial for large gains and ample proecion from adverse marke condiions. Bu i does come a an increase in expeced cos on average wih C 4 > C 1. Figure 4e and 4f compare he wealh cos disribuion o he srike and plus cos disribuion curves respecively. A final poin wih regards o hese price based scaling deviaion rules is ha hey are designed for individual order execuion. For lis rading, i is imporan ha deviaion rules be specified based on how he adjusmens affec he cos & risk profile of he enire rade lis. For example, invesors selecing he srike deviaion rule for hedged lis (i.e., buy-sell bea neural lis) will find ha he algorihms execue orders in-he-money and hold ono orders ou-of-he-money which will resul in higher coss (if rends do no reverse) han an algorihmic sraegy wihou any price-based scaling schemes. For lis rading, price-based scaling and deviaion rules need o be deermined based on he consequence o he enire rade lis. In lis rading siuaions, an appropriaely specified price based scaling scheme should only deviae when doing so would resul in lower cos, beer prices, and overall reduced oal rade lis risk. Some examples of proper rade lis deviaion sraegies are described in Malamu (22) and Kissell & Glanz (25). Order Submission Rules Order Submission Rules refer o he acual marke pricing schemes (e.g., marke or limi order), share quaniies, wai period beween order submissions, revisions, and cancellaion. The more common pricing rules include marke and limi orders (all variaions) as well as floaing prices ha are pegged o a reference price such as he bid, ask, or midpoin and change wih he reference price. Varying hese order ypes allows he algorihm o adhere o he opimally prescribed sraegy by execuing aggressively (i.e., marke orders) and/or passively (i.e., limi orders) when needed. The use of an algorihm when submiing markeable orders affords a degree of anonymiy and does no highligh he ype of order being enered. In mos siuaions i is appropriae o combine limi, marke, floas and reserve orders. For example, suppose he 11

12 specified macro level opimal sraegy is a POV rae of 15%. Here a micro-level algorihm may submi limi orders o he marke for execuion beer han he midquoe for as long as he acual POV rae is consisen wih 15% of marke volume, bu once he algorihm sars lagging behind he specified rae i would submi appropriaely sized and spaced marke orders o be more aggressive. A reserve order could also be used o auomaically replenish limi orders a favorable prices. More advanced micro pricing sraegies uilize real-ime daa, prices and quoes, and recen rading aciviy o forecas very shor-erm price rends while providing probabilisic esimaes surrounding he likelihood ha a limi order will execue wihin a cerain period of ime. For example, a limi order model will provide probabilisic esimaes regarding he likelihood ha specified number of shares will be execued wihin a desired amoun of ime. When he likelihood of compleion is oo low, invesors would be beer served via a marke order raher han he desired limi order. Pos Trade Analysis Algorihmic pos rade analysis is a wo par process ha consiss of cos measuremen and algorihm performance analysis. Firs, cos is measured as he difference beween he acual realized execuion price and he specified benchmark price. This allows invesors o criique he accuracy of he rading cos model o improve fuure cos esimaes and macro sraegy decisions, and i provides managers wih higher qualiy price informaion o improve invesmen decisions. Second, algorihmic performance is analyzed o assess he abiliy of he algorihm o adhere o he opimally prescribed sraegy, is abiliy o achieve fair and reasonable prices, and deermine if he algorihm deviaes from he opimally specified sraegy in an appropriae manner. Invesors mus coninuously perform pos-rade analysis o ensure brokers are delivering as adverised and quesion hose execuions ha are ou of line wih pre-rade cos esimaes. Summary Algorihmic rading has recenly become a popular vehicle for efficien and low cos execuions. However, proper specificaion of algorihmic rules requires invesors o become more proacive during implemenaion by aking greaer conrol of heir execuion decisions. Accordingly, invesors are required o specify decision on he macro level, i.e., specificaion of benchmark price and implemenaion goal, as well as on he micro level, i.e., appropriae deviaion schemes (e.g., price-based scaling, volume paerns, volailiy, ec.) and order submission rules. I is essenial ha raders no only develop cusomized algorihms so ha expeced ransacion coss (e.g., marke impac and iming risk) are consisen wih heir overall invesmen objecives, bu ha hey also analyze and compare alernaive algorihms o deermine he mos appropriae algorihm. Moreover, i requires invesors o selec brokers who can bes cusomize algorihms wih invesor implemenaion and invesmen goals. Only hen will invesors have highes chances o achieve heir invesmen goals. 12

13 References Arrow, K.J., Essays in he Theory of Risk-Bearing, Norh Holland, Amserdam, Almgren, R., and N. Chriss, (1999). "Value Under Liquidaion," J. Risk 12(12), Almgren, R., and N. Chriss, (2). "Opimal Execuion of Porfolio Transacions." J. Risk 3(2), Berkowiz, S., D. Logue, and E. Noser (1988). The Toal Cos of Transacions on he NYSE. Journal of Finance 41, pg Bersimas, D., and A. Lo, (1998). "Opimal Conrol of Liquidaion Coss," Journal of Financial Markes, Vol 1. Hasbrouch, J. (1991). Measuring he informaion conen of sock rades, Journal of Finance 46, page Kissell, R. and M. Glanz (23). "Opimal Trading Sraegies," AMACOM, Inc., NY, 23. Kissell, R., M. Glanz, and R. Malamu, (24). A Pracical Framework for Esimaing Marke Impac and Achieving Bes Execuion, Finance Research Leers I pg Kissell, R., and M. Glanz (25). Managing Trading Risk. Forhcoming Elsevier Science. Malamu, R. (22), Muli-Period Opimizaion Techniques for Trade Scheduling, QWAFAFEW, April 22. O Hara, M. (1995). Marke Microsrucure Theory. Cambridge, MA: Blackwell 1995 Perold, A.F., (1988). "The Implemenaion Shorfall: Paper versus Realiy," Journal of Porfolio Managemen, Spring Pra, J.W., Risk Aversion in he Small and he Large, Economerica, Jan-Apr, 1964, page Treynor, J. L., (1981). "Wha Does I Take o Win he Trading Game," Financial Analyss Journal, January/February 1981, pg Wagner, W. (199). Transacion Coss: Measuremen and Conrol, Financial Analyss Journal, Sepember/Ocober