journal of trading optimal trading algorithm selection

8/13/2019 Journal of Trading Optimal Trading Algorithm Selection

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F A L L 2 0 1 3 V O L U M E 8 N U M B E RW W W . I I J O T . C O M

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Optimal Trading AlgorithmSelection and Utilization:Traders Consensus versusReality

JINGLE L IU AND K APIL P HADNIS


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THE JOURNAL OF T RADING 9FALL 2013

Optimal Trading AlgorithmSelection and Utilization:

Traders Consensus versusReality JINGLE L IU AND K APIL P HADNIS

J INGLE L IUis a quantitative researcher

for algorithmic tradingat Bloomberg TradebookLLC in New York, NY. [email protected]

K APIL P HADNISis a quantitative researcherfor algorithmic tradingat Bloomberg TradebookLLC in New York, [email protected]

The market impact of orders cannotbe directly observed and is typi-cally inferred f rom the data. Aca-demic literature proposes various

models to describe market impact cost curvesand shows how real-world data can be f ittedinto these models. Market impact curvesversus order size have been observed as con-cave, indicating lower proportional impactfor larger orders. The market structure haschanged quite a bit in the last decade and isconstantly evolving. The U.S. equities marketis extremely fragmented, with 14 exchanges

and numerous dark pools. Market impactmodels have traditionally looked at macrofactors, including average daily volume,average spread, and volatility, among others,to explain the cost of a trade. As a lgorithmictradings popularity increases, algorithmtypes and their parameters become veryrelevant factors affecting an orders marketimpact. Institutional buy-side traders usebroker algorithms to execute orders in themarkets. These algorithms and their effect onexecution costs are the focus of this article.

Trading algorithms slice big orders intosmaller individual suborders in various stylesand route them to different venues using smartorder routing. Algorithms vary by a tradersspecif ic objectives, which could be to trade ata part icular percentage of volume, to followvolume-weighted average price (VWAP), touse volume profile, and so on. Algorithms

also try to minimize their footprints whileextracting available liquidity eff iciently byreacting to real-time market activities. Forbuy-side traders, however, using tradingalgorithms alone does not guarantee betterorder execution performance. Understandingthe characteristics and performance of varioustypes of algorithms provided by sell-side bro-kers is crucial to selecting optimal algorithmsfor certain orders under certain market con-ditions to better achieve investment objec-tives. Past studies compare the performanceof different types of algorithms (Domowitz

and Yegerman [2006]; Kissell [2007]).In this art icle, we investigate executionperformance of four types of commonly usedtrading algorithms provided by brokers. Dis-tribution of trade cost (average traded priceversus midpoint price at arrival of the order) isused to quantitatively evaluate the executionperformance. We present results of a studyon traders consensus patterns for selectingalgorithms and compare those results withactual algorithm performance. Furthermore,we explore the dependence of performanceon order size, participation rate, limit prices,and algorithm type. These results can beuseful to buy-side traders, helping them tofurther improve their algorithmic decision-making process and select optimal algorithmparameters.

The following section introduces thedataset for this study and describes termi-


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10 O PTIMAL T RADING A LGORITHM SELECTION AND U TILIZATION FALL 2013

nology used in the rest of the article for results around sta-tistics and algorithms. Next, we analyze the main resultsof our study and present data supporting our hypothesisfor optimizing algorithm usage. The final section sum-marizes the article and highlights the main conclusions.

TRADES CONSENSUSAND ACTUAL PERFORMANCE

Data Summary of Algorithm Characteristics

Our dataset consists of more than 270,000 buy-sideorders executed using trading algorithms provided byBloomberg Tradebook in the U.S. equity market. Thedataset encompasses a time frame large enough to incor-porate a variety of market conditions. All algorithmswere executed during regular market hours and excludeall stocks listed as pink sheets and the OTC BulletinBoard stocks. The dataset consists of more than 4,500tickers distributed across all market capitalizations andsectors.

We categorize the algorithms into four groups:scheduled, participation rate, dark, and implementa-tion shortfall. Scheduled algorithms are also known asVWAP algorithms because their goal is to achieve anaverage trade price as close as possible to the VWAP forthe duration of the order. Participation rate algorithmsaim to participate in the market at specified rate with

respect to the market volume. Dark algorithms restrict

participation to venues known as dark pools. Darkpools are reported to execute 14.26% of consolidatedU.S. volume as of February 2013 (Rosenblatt [2013]).Implementation shortfall algorithms aim to be oppor-tunistic with respect to arrival price benchmark while

minimizing market impact.We summarize percentage distribution of orders in

each category as shown in Exhibit 1, grouped by averagedaily volume (ADV, 30 day), order size as percentageof ADV, participation rate based on fill quantity overmarket volume in price, and algorithm duration. Wegroup the properties mentioned above into three bucketseach, where each bucket specifies a range that is deter-mined based on data properties. We describe the tabledata by grouped properties below.

We define low (0 to 1 million shares), medium(1 million to 10 million shares), and high (greater than10 million shares) ADV levels as shown in Exhibit 1.Scheduled, participation rate, and implementation short-fall algorithms are more popular for trading stocks thattrade 1 million to 10 million shares a day. In contrast,more than 55% of time dark algorithms are being usedfor equities of low ADV. This observation aligns withthe common belief that traders are more inclined totrade less liquid equities in dark pools.

Order size is normalized using the stocks 30-dayADV to make the statistics across different stocks morecomparable. This category is also broken into three

groups: small orders (0% to 1%), medium orders (1%

E X H I B I T 1Mean of Stock ADV, Order Size, Participation Rate, and Duration of Four Classes of Algorithms

Note: Percentage normalized within each property and algorithm type.


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to 10%), and large orders (10% to 100%). Among allalgorithms, more than 60% of all the orders have anorder size smaller than 1% of ADV, and more than 95%of all the orders have an order size smaller than 10% ofADV. This result suggests that cautious and risk-averse

traders inherently chop parent orders into smaller piecesto avoid leaking their intentions to the market duringthe execution of the parent order.

To further conf irm this argument, we counted thenumber of suborders within traders parent orders. Parentorders were constructed by combining a ll the suborderssent to a broker together based on same trader, ticker,and side during the same day. Our finding shows thatparent orders larger than 10% of ADV are typically splitinto two or three smaller orders by tradersconsistentwith the argument herein.

More than 90% of implementation shortfall algo-rithms were traded on orders smaller than 1% ADV.Thus, for small orders, traders prefer to have the orderdone quickly by extracting maximum liquidity viaimplementation shortfall a lgorithms. In the meantime,for very large orders ( > 10% of ADV), traders rely onscheduled algorithms to spread out the trade over theday to minimize the market impact or rely on dark algo-rithms to take advantage of block liquidity periodicallyavailable in dark pools.

The participation rate is the ratio of algorithmsfilled quantity to the market volume within the limit

price during that period of the execution. Participationrate is strongly related to order size and duration. Itconsists of three groups: low (0% to 5%), medium (5%to 20%), and high (20% to 100%). Dark algorithms,at 62.9%, have the highest percentage of orders fallingin the highest participation rate category, followedby implementation shortfall, participation rate, andscheduled algorithms. This finding could indicate thattraders are f inding a high percentage of the liquidity indark pools regardless of the average percentage of darkvolume. We surmise that dark liquidity begets more darkliquidity, and the average dark liquidity might not be theentire story. We are working on a study to determine ifdark liquidity tends to be autocorrelated in nature, thusleading to an understating on averages. Participationrate algorithms are more likely to be used by tradersfor targeting a certain level of participation; hence, themajority fall in the 5%20% participation rate bucket.

Algorithm duration is the time between the algo-rithms initializat ion and the time the order is completed

or canceled. Scheduled algorithms typically are used towork through relatively long periods, with 67.4% oforders lasting longer than 30 minutes. In contrast, 65.1%of orders using a dynamic algorithm are finished withinfive minutes because of the relatively smaller order size

and higher part icipation rate.

Data Summary of Trade Cost

Using a meaningful yardstick is important foralgorithmic performance analysis. In this study, we useimplementation shortfall ( IS ) as a trade cost measurebecause practitioners use it widely as a benchmark toevaluate execution performance. IS is measured as thedifference between the assets prevailing market priceat the time of the investment decision and the realizedaverage trade price. The normalized trade cost, TC , canbe expressed as

=

arrival

C

(1)

where P arrival is the midpoint quote (average of bid/askprice) at the point of order entry and P avg is the averagetraded price of the algorithm. S is the order side and hasa value of + 1 for buy orders and 1 for sell orders. Allvalues are in basis points. We will refer to th is statisticas trade cost for the remainder of the article.

Previous studies indicate that an orders size andaggressiveness play important roles in driving tradecost performance (Almgren and Neil [2001]). Smallerorders tend to outperform larger orders, and participa-tion exhibits a complicated inf luence on implementationshortfall. To make a fair comparison of implementationshortfall performance for the four common algorithmslisted here, we compare them by each pair of order sizebucket and participation rate bucket. Therefore, wehave, in total, 3 3 = 9 categories if we divide ordersize and participation rate into three buckets, respec-tively, as shown in Exhibit 2.

Order size increases from left to right, and partici-pation rate increases from bottom to top. At each ordersize and participation rate category, we calculate usage-based percentages, which are the ratios of order countsfor each algorithm to the total count for all algorithms,along with the mean and standard deviation of trade cost.Exhibit 2 summarizes those numbers. Mean of trade costmeasures expected algorithm performance against arrival


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price, and standard deviation of trade cost is an indicatorof pricing risk, or the probability that the average priceis away from the expected value. Both should be consid-ered for performance evaluation and comparison.

For small orders ( < 1% of ADV), traders tend to usescheduled algorithms with a low aggressive level (lowparticipation rate) or use participation algorithms with amid/high aggressive level (mid/high participation rate)while choosing dark algorithms to search for large fil lsand quickly f inish the order.

For medium-size orders (1% to 10% of ADV), tradersprefer scheduled or participation algorithms with low ormid aggressiveness (low/mid participation rate) whilepreferring to leverage block liquidity by using dark algo-rithms to achieve a decent trade rate.

For large orders ( > 10% of ADV), traders rely onthe scheduled algorithms (34.5% for high participationrate and 60.9% for medium participation rate) or blocktrade opportunities from the dark pools to minimizetheir footprints and stay away from predatory (48.1% forhigh part icipation rate). Large orders with low participa-tion rates have few data points because, naturally, most

of those large orders have > 5% participation rate in themarket to get the order done by the end of day.

The usage percentage statistics in Exhibit 2 representa consensus of traders opinions about which algorithm(s)to use based on nature of order itself and urgency level.Interestingly, the results of trade cost performance do notnecessarily agree with the consensus of algorithm usage.For example, 77.2% of small orders with low participation

level were done using scheduled algorithms even thoughimplementation shortfall algorithms provide better averageperformance and smaller pricing risk. Compared with theplanned allocation scheme used by scheduled algorithms,implementation shortfall algorithms can better adjust sub-order allocation and aggressiveness based on real-timemarket price movement and liquidity availability and,therefore, can achieve lower trading cost, on average.Implementation shortfall algorithms can immediatelyextract continuous liquidity from lit exchanges, but dark

algorithms depend on the discrete liquidity in dark pools.The continuous filling profile leads to a small variationof the trade cost of implementation shortfall algorithms.After finding the matching opposite sides in the darkpools, however, dark algorithms can trade at a higher par-ticipation rate without moving the market compared withalgorithms relying on lit exchanges. This result mightexplain why dark algorithms are able to achieve loweraverage trade cost at mid/high participation rates but notat the smallest variance.

For medium-size orders (1% to 10% of ADV), darkalgorithms perform the best, while scheduled algo-

rithms perform the worst for mid/high participationrate; participation algorithms perform the worst for lowparticipation rate. Best performance of algorithm over-laps with most-used algorithms for high participationrate but not for mid/low participation rate.

For large orders ( > 10% of ADV), dark algorithmshave the highest trade cost performance and smalleststandard deviation, while scheduled algorithms turn in

E X H I B I T 2Trade Cost Performance for Four Groups of Algorithms for Various Order Sizes and Participation Rates

Notes: Algorithm groups in each category with sampling size of less than 200 are marked in gray and a re not taken into consideration for comparison.Highest-usage-based percentage and best per formance in each category are marked in bold.


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the worst mean performance. The poor performance ofscheduled algorithms for large, highly aggressive ordersis likely attributable to the fact that the imposed schedulereduces the chance of participating in the liquidity eventwhile increasing the chance of trading too much during

inactive market periods.High-frequency trading activity in the frag-

mented markets is also considered to play a role in theperformance of a brokerages algorithms. Such preda-tory strategies estimate the probability of forthcomingbig orders or those under way by watching the tapeand front-running those orders to amplify the marketimpact. Scheduled algorithms, which trade in a predict-able fashion, tend to be more easily spotted by preda-tory gamers and, therefore, become more susceptibleto technological adverse selection (Agatonovic et al.[2012]). Opportunistic and dynamic algorithms areless susceptible to predatory strategies because of thenature of high dynamics and unpredictability, whichmight partly explain the lower market impact of darkand implementation shortfall algorithms compared withscheduled and participation algorithms.

ALGORITHMIC PERFORMANCE FACTORS

Order Size

The persistence of order f low is overwhelmingly

attributable to order splitt ing (Toth et al., [2011]). Order

E X H I B I T 3Trade Cost Dependence on Order Size

E X H I B I T 3 (Continued)

Note: In Panel C, the vertical line indicates the mean of the distribution.


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size is a major factor contributing to trade costs becausethe market impact generated by the executed sharesduring the early part of the order might affect the laterpart of the order execution. Other factors mentionedearlier have a more complex dependency on order size.

For example, duration depending on order size can affecttrade costs but is also algorithm-dependent.

In this section, we study the conditional distribu-tion of trade cost with respect to order size as well as thetype of algorithm. Panels A and B of Exhibit 3 show themean and standard deviation of trade cost against ordersize as a percentage of 30-day average daily volume.We plot trade cost versus grouped percentage ordersize in six intervals given by [0%, 1%], [1%, 3%], [3%,10%], [10%, 20%], [20%, 40%], and [40%, 100%]. Theboundaries of intervals are selected based on having arelatively uniform distribution of data density for eachgroup as well as enough data resolution to show cost sen-sitivity in relation to order size. We find that the ordersize dependence can be f itted by an exponent functionmean (TC ) O , where O is the order size and a is theexponent. The result of least square regression based onempirical data is = 0.4761, which agrees with previousstudies that show market impact to be a concave functionof order size (Almgren et al. [2005]). Pricing r iskthatis, the standard deviation of the trade costis also fitby an exponent function sd (TC) O , where the bestf it value for is 0.21.

We also provide a unique way to look at the distri-bution of trade cost with varying order sizes. Exhibit 3,Panel C illustrates the distr ibution profiles of the tradecost for order size intervals. From bottom to top, eachdensity represents the trade cost for an interval oforder sizes. The vertical line indicates the distributionsmeanwe can clearly see it shift higher with order size.We also observe that the density distribution becomesless peaked as we go from bottom to top or as percentageof order size increases. This observation indicates, andwe verify, a substantial change in the third and fourthmomentsthat is, a decrease in kurtosis and an increasein the positive skew of the distribution along with anincrease in standard deviation. This dynamic indicatesthat trading costs become quite uncertain as the ordersize goes up. We confirm by estimating standard errorsfor these statistics that are robust estimates. The distri-bution profile changes quickly as order size increases to20% of ADV and changes less significantly as the ordersize rises above 20%.

Exhibit 4 shows the relationship between order sizeand estimated mean and standard deviation of trade costdistribution by algorithm type. Scheduled, participa-tion rate, dark, and implementation shortfall algorithmsexhibit monotonically decreasing performance with

order size at different rates. Among these algorithms,scheduled algorithms have the worst performance andtheir trade cost curve falls off the fastest compared with

E X H I B I T 4Trade Cost Dependence on Order Size for EachAlgorithm Class


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the other algorithms. In part, this occurs becausescheduled algorithms typically have more fixed share- allocation schedules, which does not allow for f lexibilityto adjust the participation rate over the course of tradingand, therefore, cannot capture the occasional excessive

surging liquidity. Another reason is that traders typi-cally set schedule algorithms to longer periods, leadingto longer algorithm duration and higher pricing risk.Therefore, for medium and large orders, traders shouldbe discouraged from using scheduled algorithms whentrade costs are measured with respect to arrival price.

Participation rate algorithm trade cost falls offbefore that of dark and implementation shortfall algo-rithms. Using this curve, we can also estimate a favor-able order size to use for participation rate algorithms onthe impact curve. Implementation shortfall algorithmsexhibit greater ability to incur lower trade costs becauseof the al location design, which balances market impactand pricing risk.

Finally, dark algorithms are much less dependenton order size than the other three and exhibit consis-tently lower trade cost across the whole range of ordersize. This result suggests that trading in dark pools couldbe very beneficialespecially for very large orders,which would not incur the large trade cost of otheralgorithms. If certain equity has a decent number ofshares traded in dark historically, traders should stronglyconsider using the liquidities in dark pools before or

at the same time that they trade in a lit exchange. Wecaution that this result applies only for Tradebooks darkalgorithms. Because dark algorithms are not standard-ized across brokersthat is, their logic and access todark pools is highly broker-dependentresults may varyacross broker/dealers.

The standard deviation of the trade cost of sched-uled and participation rate algorithms increases muchfaster than trade costs incurred when using dark andimplementation shortfall algorithms. For order sizeslarger than 20% of ADV, the standard deviations of thetrade cost of dark and implementation shortfall algo-rithms are more than 30% lower than that of scheduledand participation rate algorithms, as shown in Exhibit 4,Panel B. Because of the relatively fixed time scheduleor participation rate, schedule and participation ratealgorithms tend to last longer and have more pricinguncertaintythat is, the effect of trade cost standarddeviation is conditional on other factors.

Participation Rate

Participation rate is another important factor thataffects algorithm performance. Higher participation ratetakes away more liquidity from the market, leading to

a larger market impact that will affect the executionprice of the orders remaining shares. Exhibit 5 showsthe trade cost distr ibution at dif ferent participation rateranges for all the algorithms. The profile width decreasesas the participation rate increases from 0% to 20% andthen stays relatively the same at above 20% part icipationrate. Kurtosis (peakedness) increases as the participationrate increases.

Exhibit 6 illustrates mean and standard deviationof execution performance for each individual algorithmtype. Scheduled algorithms performance shows much

E X H I B I T 5Distribution of Trade Cost Performance againstArrival Price for Various Participation Rate Ranges


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greater sensitivity to participation rate than that of theother algorithms. Again, this high sensitivity can beattributed to the rigid share-allocation schedule imposedby the scheduled algorithm. Interestingly, no signif icantpattern of relationship appears between part icipation rateand variance of the performance. The participation ratedependence of performance standard deviation is rela-tively flat for all of the algorithms.

Limit Price

Traders typically use limit prices in algorithms tomanage price risk when routes are sliced from largerorders to brokers. In our dataset, more than 90% of

algorithms are used with a limit price. Limit price affectsboth participation rate and trade costs. Our hypoth-esis is that setting a too aggressive or marketable limitprice would make the execution faster but provide verylittle price improvement. On the other hand, setting atoo passive or unmarketable limit price would lead to abig price improvement but could not guarantee the fullf ill. We analyze the trade-off between participation ratewith marketable limit price and trade cost, and we f indoptimal curves for the trader to choose limit price basedon the level of order urgency.

All the algorithms with a user-set limit price aregrouped into buckets based on the normalized differ-ence between arrival price and limit, P lmt to arrPx calcu-lated as

P

P r x

a rPx where the sign of buy order is positive

and the sign of sell order is negative. The more positivethe difference, the more marketable the order. Panel Aof Exhibit 7 plots the trade cost distribution of all thealgorithms with limit price. When limit price is passiveor unmarketable, the distribution has positive averageperformance with a long negative tail. When limit priceis set close to arrival price, the distribution has slightlyworse average performance with smaller standard devia-tion and larger positive skewness, which indicates thattrade cost is tightly concentrated around arrival price.When the limit price is aggressive or marketable, distri-bution widens again, with a long positive tail and worseaverage performance. Panel B of Exhibit 7 presents themean of the trade cost within each bucket. The trade costperformance gradually decreases as P lmt to arrPx becomeslarger, but not at a constant rate. It increases quickly aslimit price changes from passive to neutral, and slowlyas limit price changes from neutral to aggressive.

Panel A of Exhibit 8 shows how the mean of tradecost changes with limit price for each algorithm type.

In the unmarketable zone, all the algorithms exhibitsimilar dependence. In the marketable zone, dark algo-rithm performance decreases the fastest with limit price,while scheduled algorithm performance decreases theslowest. In other words, the average performance ofthe dark algorithm is more sensitive to the limit price.

E X H I B I T 6Trade Cost Dependence on Participation Rate


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The standard deviation of the performance is the lowestas limit price is set close to arrival and increases l inearlyas the price moves away from arrival in either direc-tion as shown in Exhibit 8, Panel B. Therefore, if a traderwants to reduce the variance of his or her algorithmicperformance, setting limit price close to arrival pricewill help achieve this objective.

To characterize traders behavior while settinglimit prices, and limit prices influence on participationrate, we measure the ratio of missing interval volumeto volume in price, which can be calculated as V V V i nP x a ll inPx

,

where V all and V inPx are total volume within order exe-cution horizon and volume in price, respectively. Thismeasure indicates how much liquidity traders wouldmiss by setting a limit price on the algorithms. We callthis liquidity loss ratio. Panel C of Exhibit 7 plots theliquidity loss ratio as a function of distance of limit pricefrom arrival in percentage points. A sharp increase isobserved as the limit price approaches the arrival price,and then the liquidity loss ratio increases slowly all the

way to 0 as the limit price is set more marketable. Toachieve optimal balance between price improvement andliquidity loss ratio, the util ity objective function shouldbe maximized with respect to limit price, as shown inthe following equation:

)max (2)

E X H I B I T 7Trade Cost Performance against Arrival Price forVarious Limit Price Ranges



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for the implementation shortfall algorithm, the traderis wil ling is to give up a lot of liquidity to get fil ls at hisor her price. This is less so for participation rate algo-rithms (i.e., only when they become marketable), andeven less so for dark and scheduled algorithms. Tradersare more inclined to give up liquidity for opportunisticalgorithmsthat is, they are hopeful of getting a favor-able price.

CONCLUSION

Quantitative analysis of algorithms performanceplays an important role in algorithmic trading. We ana-lyze trader usage of execution algorithms and extractconsensus patterns. We use these patterns to offerinsights that can help traders make optimal decisionswhen selecting algorithms and determining their param-eters to achieve reduced implicit trade costs.

Analyzing more than 150,000 trading algorithms

executed via Bloomberg Tradebook in varied marketconditions, we provide a comprehensive picture of buy-side traders usage patterns. By looking at statistics aroundalgorithm and trade cost parameters, we describe thetype of algorithms and the conditions the traders selectto execute their orders. Statistics we comprehensivelystudy are ADV, size, participation rate, and duration,as well as implicit trade costs for four dif ferent types of

where L (P lmt ) is the liquidity loss ratio and is a constantthat depends on the traders risk aversion or the orderurgency level.

Panel C of Exhibit 8 plots trader behavior as atrader compares normalized limit price (versus arrival

price) with liquidity loss ratio. Here, we characterize howlimit prices are set by traders for different algorithms potentially giving us insight into the traders perceptionof the different algorithms. Interestingly, we see that

E X H I B I T 8Trade Cost Dependence on Limit Price for EachAlgorithm Class



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algorithms. By comparing this empirical dataset of usagepatterns with implementation shortfall performance, weoffer insights on using optimal parameters for differentalgorithms.

We f ind that our implicit trade cost curve is con-

cave, scaling as power law with an exponent of closeto 0.5 for all strategies. When examined by strategytype, trade cost curves are conditional on strategy type.Scheduled algorithms have the largest exponent, anddark and implementation shortfall strategies have smallerexponents. Standard deviation of the estimate of tradecost also increases with order size and differs for differentstrategy types.

We describe different ways of looking at trade costdistribution versus participation rate, including usinghigher moments of trade cost and conditional histo-grams for various buckets of participation rates. Sched-uled algorithms exhibit high trade costs as par ticipationrates increase, but surprisingly, other algorithms showf latter curves as participation rates increase. We intendto research this dynamic further, but we anticipate thatother factors are confounding elements in these curves,including duration, volatility, and the microstructure ofindividual stocks.

We construct a metric called the liquidity loss ratiobased on limit prices set on various types of algorithms.This metric analyzes the usage of limit prices with respectto loss in volume over the duration of the algorithm. We

will quantify this relationship in a future article, but wepoint out the importance of setting limit prices and theirperformance impact for different algorithms.

REFERENCES

Agatonovic, M., V. Patel, and C. Sparrow. Adverse Select ionin a High-Frequency Trading Environment. The Journal ofTrading , Vol. 7, No. 1 (2012), pp. 18-33.

Almgren, R., and C. Neil. Optimal Execution of Port-folio Transactions. Journal of Risk, Vol. 3, No. 2 (2001), pp.5-39.

Almgren, R., T. Chee, H. Emmanuel, and L. Hong. EquityMarket Impact. Risk, ( July 2005), pp. 57-62.

Domowitz, I., and H. Yegerman. The Cost of AlgorithmicTrading: A First Look at Comparative Performance. The

Journal of Trading , Vol. 1, No. 1 (2006), pp. 33-42.

Kissell, R. Statistical Methods to Compare Algorithmic

Performance. The Journal of Trading , Vol. 2, No. 2 (2007),pp. 53-62.

Rosenblatt Securities. Rosenblatts Monthly Dark LiquidityTracker. March 2013.

Toth, B., I. Palit, F. Lillo, and J.D. Farmer. Why Is OrderFlow So Persistent? arXiv: 1108.1632 (2011). http://arxiv.org/abs/1108.1632

To order reprints of this article, please contact Dewey Palmierat dpalmieri @iijournals.com or 212-224-3675.

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