Algorithm Selection:

A Quantitative Approach
JIAN YANG AND BRETT JIU

JIAN YANG he relentless pursuit of lower trans- conducting performance attribution on algo-
is a Senior Vice President
at ITG Solutions Network
in Boston, MA.
jyang@itginc.com

BRETT JIU
T action costs has led to increasing
demand for sophisticated trading
tools and algorithms, which in turn
has led to an explosion in the number of algo-
rithmic products offered in the marketplace
rithmic products. We will demonstrate how we
perform empirical analysis on the algorithm
performance and how we turn historical data-
based model parameters into forward-looking
algorithm selection criteria. Our proposed
is a Senior Research today. Yang and Borkovec [2005] predict that approach can also help investment managers
Analyst at ITG Solutions this trend will continue as more investment and traders become more proactive in selecting
Network in Boston, MA.
bjiu@itginc.com
management firms embrace best execution as algorithms that are of the highest value to them
a top priority. and help to ensure the alignment of algorithmic
Having more algorithms at their disposal trading with their investment objectives.
offers traders both opportunities and chal-
lenges. On the upside, a trader now has the ALGORITHMIC STRATEGY
opportunity to pick the suitable algorithm that SPECTRUM
will most likely achieve the trading objective
for each order. On the downside, the number The significance of conducting pretrade
of algorithm choices can be so large as to make “homework” on algorithms is well understood.
it difficult to make a quick and correct choice.1 The need to understand the nature of an algo-
Adding to the algorithm selection chal- rithm starts at the point when an algorithm is
lenge is the fact that algorithms offered by sell- offered by a third-party vendor. We begin our
side vendors usually come in the form of a discussion of algorithm choice with a look at
“black box,” with inner workings hidden to the how algorithms can be categorized.
end users. Because of this lack of transparency, At its core, a trading algorithm takes an
users may find it difficult to clearly understand order, or trade list, and structures a sequence
the performance characteristics of a particular of trades that aim to achieve the objectives
algorithm, which, in turn, further complicates of the user, for example, minimizing cost (vis-
the algorithm selection decision. à-vis a specific benchmark), maximizing fill
Instead of looking inside an algorithm, rate, or minimizing execution risk. Domowitz
we propose a systematic, quantitative approach and Yegerman [2005a] suggest that, at the most
to evaluate an algorithm’s historical performance abstract level, the different kinds of algorithms
by identifying the determining factors of rela- can be thought of as occupying a trade struc-
tive performance across alternative algorithms, ture continuum, ranging from the less-
and we present a framework for algorithmic structured to the very structured. In Exhibit 1,
selection based on these underlying factors. Our we divide this range into three categories.
methodology is easy to implement in practice On the less-structured side, we find strate-
and provides a quantitative framework for gies that can be called opportunistic, in the sense
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 EXHIBIT 1 continuously adjusts how and when each slice
Spectrum of Algorithmic Strategies of the big order is executed in order to min-
imize the impact.
While our three-part categorization of
algorithms is only a guide,2 dividing algo-
rithms into different categories is a neces-
sary first step in deciphering the nature of
the myriad strategies available. It is impor-
tant to see beyond general descriptions and
get a clear sense of what kind of strategy any
given algorithm is at its core.

ALGORITHM SELECTION:
that these strategies do not have predefined execution sched- A QUANTITATIVE FRAMEWORK
ules; instead, they utilize real-time information to actively
search for optimal times when trades can be executed. Given the availability of a basket of algorithmic
These strategies create execution schedules as they go along. strategies, attention is now turned to order-specific pre-
At the beginning of an order, a trader does not know what trade analysis. Specifically, there are two questions con-
the execution schedule will look like. An example is ITG cerning algorithm selection:
Active (formerly known as ITG activePeg® to clients), an
algorithm that employs sophisticated agent-like logic to 1. Is the order at hand suitable for algorithmic trading?
continuously search for liquidity opportunities. 2. If so, which algorithm is the optimal one for trading
At the other extreme—on the more structured this order?
end—are algorithms that follow precisely defined execu-
tion schedules; we call these algorithms schedule-driven It is well known that not all orders can be traded
strategies. The schedules are based on historical data, pre- using an algorithmic approach. This is because, essentially,
programmed into the strategy’s logic and, save for small algorithms are preprogrammed logic run on computers.
updates that incorporate real-time information, are fol- As such, algorithmic trading is not, and will never be, the
lowed precisely in optimizing trade entries. All VWAP- magic bullet that solves all transaction cost-related prob-
and TWAP-based strategies, for example, can be catego- lems. This is an important pretrade analysis issue that is
rized this way. The realized trade schedule will be similar beyond the scope of this article; instead, we focus on the
to the predefined one, absent significant, unusual changes optimal algorithm selection question.
in liquidity over the order horizon. We assume that algorithmic suitability has been
Between these two ends is a category that we call established and that the appropriate benchmark has been
evaluative strategies. Not surprisingly, these strategies com- determined. The next step is deciding which algorithm,
bine approaches of both opportunistic and schedule-driven among the many available, should be used to trade a par-
algorithms. At the macrolevel, these algorithms suggest ticular order. For example, even if VWAP is determined
how to optimally slice a large order in different time inter- to be the best strategy, there may be a number of VWAP
vals, for example, half-hour bins. At the microlevel, intel- strategies from different vendors that can be used. One still
ligent rules—often quantitative in nature—are employed needs to pick the best specific VWAP strategy.
to execute each part of the original order while balancing We propose a simple quantitative framework that
the tradeoff between cost and risk. Oftentimes these micro affords a quick but rigorous comparative analysis of algo-
rules require the input of substantial real-time informa- rithmic performance. The workflow of this pretrade analysis
tion that makes them similar to opportunistic strategies. The consists of four steps:
trader will have a good idea of what the execution trajec-
tory may look like, but the ex post trajectory may differ 1. Specify the model structure that links algorithm
little or greatly from the ex ante prediction. An example performance to a basket of factors that comes from
is ITG ACE, a highly quantitative strategy that actively order requirements, stock characteristics, or market
evaluates the potential price impact of each slice and conditions.

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 2. Derive the estimable form of the structural model This simple approach suffers one significant draw-
and obtain performance attribution parameters using back: it tells the trader nothing about how confident he
historical data. These parameters allow us to char- or she will be in achieving the cost the model says an
acterize how an algorithm’s performance responds algorithm can achieve on average. A second-degree mea-
to changes in the factors. sure is needed to augment the selection process so the
3. Forecast the performance for each candidate algo- trader can compare algorithms along two equally impor-
rithm using factor values known at order entry time. tant dimensions: cost advantage and confidence level. We
4. Finally, calculate a selection “score” for each algo- propose a simple solution in the form of estimating the
rithm. At this point, selecting the optimal algorithm cost and its variance simultaneously. In other words, we
is as simple as picking the one with the highest score. will estimate both how much an algorithm’s implementa-
tion shortfall cost will be and how closely the realized cost
We focus on the first step, model structuring, in this will come to this ex ante estimate. The structural model
section. specifies the conditional mean and variance of the cost as
While a trader’s specific trading objective may vary, functions of relevant market- and stock-specific factors.
implementation shortfall has become a popular cost bench- Our key innovation here is to propose that the cost
mark. First proposed by Perold [1988], implementation and variance functions be jointly estimated, which pro-
shortfall measures the price distance between the final, vides a set of intimately linked performance attribution
realized trade price and a pretrade decision price. In prac- parameters for each algorithm. The analyst’s job, then, is
tice this pretrade decision price can be different to different to translate the structural functions into econometrically
people, for example, the price at which a portfolio man- sound estimable specifications and employ the right econo-
ager wishes to enter or exit a position, a previous day’s metric tools to carry out the estimation. For example,
closing price, today’s open price, etc. For our purposes, one can use time series, cross-sections, or panel data to
we expand Perold’s original definition to include limit do the estimation; all that is required is that the appropriate
orders and define implementation shortfall as the differ- econometric technique be used given the chosen speci-
ence between the share-weighted average execution price and the fications. The rest of this article describes our empirical
mid-quote at the point of first entry for market or discretionary investigation, which can serve as an example to the reader
orders, and the difference between the average execution price and who wishes for a “cookbook” guide to implement quan-
the limit price of the order for limit orders.3 The nature of the titative algorithm selection.
order—limit or non-limit—is taken from the very begin- Our proposed framework is completely broker-neu-
ning; whether this nature changes during the course of tral and can be applied to any algorithm. It can be used
trading the order is not considered.4 In addition, the imple- to compare strategies across the algorithmic spectrum as
mentation shortfall is sided so that its sign is consistent for discussed earlier in the article, or even within the same
both buy orders and sell orders. (A negative value signi- general type of algorithms (e.g., VWAP from vendor X
fies a price improvement.) versus VWAP from vendor Y).
To account for the large variability in stock prices, it Next, we describe the dataset we work with before
is usual practice to express implementation shortfall in rel- discussing our empirical implementation of the general
ative terms, that is, the normalized difference between the framework.
share-weighted average execution price and the mid-quote
immediately before the order started executing for market DATA
orders and the user-specified limit price for limit orders.
As a first cut, one may imagine using a simple “horse Our aim is to obtain parameter estimates of the
race” approach to select the best algorithm: whichever structural model from historical performance data. Our
strategy that potentially achieves the lowest implementa- algorithm-related dataset contains over 100,000 single-
tion shortfall cost wins. In this approach, it is only nec- name, completely filled client orders handled by the ITG
essary to analyze the historical cost of each algorithm and SmartServer® suite of algorithmic strategies, also collec-
apply the coefficients obtained from the analysis to fore- tively known in the industry as ITG algorithms. Here, a
cast its cost for the order at hand. This approach has the client order means an explicit instruction from a user to
appeal of being very easy to implement, assuming that buy or sell a certain number of shares in a stock over a
historical trading data is readily available. prespecified period of time; it is the algorithm’s job to

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 break this order into individual trades or executions. The serious problem because while size is a significant factor
orders in the dataset cover U.S. stocks traded from Feb- for the level of IS across orders under each algorithm, it
ruary through June 2005. All the orders in our sample does not correlate significantly with cost across the strate-
were either completed or canceled by close, so there are gies. This is apparent when we examine the “Relative IS”
no multiple-day orders in our sample. We focus on three column in Exhibit 2. Even though ITG Horizon handled
particular ITG algorithmic strategies: ITG Active, an larger orders than ITG Active, it had a lower average cost.
opportunistic strategy (less structured); ITG ACE, an eval- Even though ITG ACE handled larger orders than ITG
uative strategy (middle of the road); and ITG Horizon Horizon, it had a lower median cost. Furthermore, when
Smartserver™, a VWAP strategy (more structured). we divided the range of relative order sizes into many
Each record in our dataset includes the following brackets, we found that there were statistically sufficient
order-specific variables: sample points in each bracket for all three servers; meaning,
our estimation results would not be significantly skewed
• ticker by relative sample size in different order size baskets.
• size (in number of shares) It should be emphasized that even when there is
• side (buy or sell) sample selection bias, it is an econometric estimation
• market or limit problem, not an inherent issue with our model. Econo-
• limit price (if a limit order) metric techniques exist to handle this bias. The key lesson
• starting time and ending time for the entire order here is, when performing quantitative analysis, one must
• share-weighted average fill price. take care not to fall victim to biased estimates due to the
presence of sample selection bias or other data-related issues.
Exhibit 2 gives the sample statistics of the orders data An experienced econometrician can help solve this problem
after suitability filtering. The three algorithmic strategies and provide statistically robust parameter estimates.
vary significantly in average order size, whether measured Another interesting issue with the data is the large
as a percentage of MDV (median daily volume) or dura- difference between average relative implementation short-
tion volume. ITG Active (the opportunistic strategy) tends fall and the median value for all three strategies. In the cases
to receive, on average, smaller-sized orders than ITG of ITG Active and ITG Horizon—the two “extreme-
Horizon (the VWAP strategy), which in turn gets smaller end” strategies—the average relative IS is much lower
orders than ITG ACE (the evaluative strategy). This pre- than the median. For ITG ACE, the relationship is
sents a potential sample selection bias problem across the reversed. All these suggest the presence of significant skew-
algorithms because it seems sensible that smaller orders ness in the values. We’ll come back to this issue later when
tend to have lower IS cost. If the sample selection bias is we discuss how we scale the IS value to obtain a distrib-
indeed present, the implication is that when we estimate ution that is closer to normal.
the model, ITG Active may get more favorable parameter In addition to the order-specific dataset, we also
estimates relative to the other two strategies not because obtained stock-specific data that includes essential char-
it is inherently “better” but because it handles smaller acteristics such as historical intraday volatility, histori-
“easier” orders. In our case, this turns out not to be a cal volume profile, and historical ADV. We merge these
stock-specific variables into the
orders dataset by date and stock.
EXHIBIT 2 The final dataset contains imple-
Sample Statistics mentation shortfall for each order
as well as various factor variables
taken from both order and stock
characteristics.

EMPIRICAL ESTIMATION

The structural model must

be reduced to a form that can be
statistically estimated. In deriving

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 the corresponding reduced form equations, we make the impacts of market cap and intraday volatility, we double
explicit assumption that the relative implementation short- sort our samples along the two dimensions into four sub-
fall should be scaled by the stock’s intraday volatility. There groups: large cap and small cap, and within each market-
are reasons for this transformation. First and foremost, a cap group, high volatility and low volatility. The large-cap
stock’s intraday volatility appears to be a significant driver group roughly corresponds to the Russell 1000 stocks and
for order cost.5 We can either include it as a regression the small-cap group is similar to the membership in Russell
factor (i.e., independent variable), or via a non-linear 2000 (plus some microcaps). This division implicitly assumes
specification. Earlier we mentioned that the implemen- the existence of a disjoint, “jump” effect market cap has on
tation shortfall values in our sample exhibit significant cost. The same logic applies to intraday volatility.7 One
skewness. But when we divide IS by the stock’s intraday individual model is then estimated for each algorithmic
volatility, we find the ratio having a statistical distribution trading strategy and, within each strategy, using each sub-
similar to the normal distribution. The normality of the division of the data. In all, we have three algorithmic strate-
scaled implementation shortfall is important for us in con- gies and four divisions under each strategy, giving us 12
structing a single-number rating measure of algorithms cost-variance pairs of models to estimate.
in the next section. Scaling IS by volatility also reduces One interesting pattern in our cost estimation results
or eliminates heteroskedasticity in the model. is that the impact of the relative order size factor on
We make no a priori assumption regarding whether expected cost increases in magnitude as volatility drops
the relevant factors influencing the cost and its variation from high to low. The reason for this trend has to do with
are the same or even overlap. It is possible that different how all of the algorithms presented here work their orders.
factors, some common to both, influence the two func- All three use limit orders for executions to some extent;
tions. For example, time of day may be a significant factor in fact, they employ an econometric model called the
for cost estimation but may not have any effect on the ITG Limit Order Model to forecast the probability of a
variability of this cost estimate. The choice of factors is a limit order being hit at any given time. This probability
question to be answered empirically.6 is then used by the algorithms to determine how aggres-
We also do not assume beforehand whether the cost sive or passive a limit order should be. When intraday
and its variance functions are linear or non-linear. In fact, volatility is high, the probability of a limit order being hit
as variances are often non-linear in nature, imposing a is high, therefore the individual limit-order trades are
linear functional form on cost variance would be too strict likely to be executed regardless of their sizes. In the aggre-
of a constraint and would likely produce inferior results. gate, we observe a reduced effect of total order size on the
We have run different sets of regressions using various total cost.
factors to determine the final reduced (estimable) form Our estimation results also provide empirical sup-
of the conditional cost and variance functions. A few port to our assertion that cost estimate alone does not
interesting results emerge from our analysis. determine the relative optimality of an algorithm and that
First, we found that non-linear functions of order cost variance does matter. To see this, we hypothesize a
size relative to the predicted volume over the order time utility function that is analogous to the one employed in
horizon, known as duration volume, to have worked best the mean–variance framework: it is defined as the sum
for estimating both cost and variance. In fact, this relative of cost and risk aversion-adjusted cost variance. The opti-
order size is the only factor that proves consistently sig- mization goal here is to minimize this utility function
nificant in both equations. The use of duration volume given the risk parameter by choosing over a set of avail-
contains an implicit time-of-day effect: the same order size able algorithmic strategies. Exhibit 3 plots the utility curves
and required duration have different impact depending on for ITG Active and ITG ACE over different values of risk
the time of entry, since duration volume will be different. aversion, assuming a fixed order size (set to 1% of dura-
Second, there is a non-linear, marginally decreasing tion volume), a low intraday stock volatility, and that the
effect of relative order size: the larger the order, the less stock is a small-cap name. Along the x-axis, risk aversion
increased marginal effect it has on transaction cost. increases to the right; equivalently, risk tolerance decreases
Third, even though market cap and intraday volatility to the right. The two utility curves intersect at the point
do not enter our final cost equation, we do find that both where risk aversion is equal to l*. To the left of l*, risk
market cap and intraday volatility have some effect on the aversion is low (i.e., high risk tolerance), and ITG Active
magnitude of implementation shortfall. To control the exhibits lower cost utility than ITG ACE and is therefore

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 EXHIBIT 3 EXHIBIT 4
Algorithm Utility as a Function of Risk Aversion Calculating Probability of Keeping IS Below x

does one measure one’s own risk aversion parameter and

in what unit should this parameter be measured?
the better algorithm to follow. To the right of l*, risk Instead, we propose another quantitative method
aversion is high (low risk tolerance), and ITG ACE has that yields a concrete “score” for each algorithm: we can
the lower cost utility value and is therefore the winning calculate a Value at Risk (VaR)-like number that reflects
strategy in this example. Exhibit 3, in effect, demonstrates the probability of keeping the implementation shortfall
the interaction between cost and cost variance in deter- cost for an order under a certain threshold. Empirically
mining which algorithmic strategy is optimal. our volatility-adjusted implementation shortfall cost
exhibits a sample distribution close to a normal. From the
FROM ESTIMATION TO SELECTION estimates of the cost as well as its variance, the normal
cumulative distribution curve can be used to compute the
Now that we have constructed the models and probability of achieving an implementation shortfall that
obtained parameter estimates for each algorithm, we can is lower than the specified maximum threshold. This VaR-
perform comparative analysis and predict, given a specific like value answers the question “What is the probability
order, which algorithm will be the best candidate for that implementation shortfall cost can be kept under a spe-
trading this order. cific target?” Exhibit 4 illustrates this application, with c
First, we forecast the cost and cost variance of each being the volatility-adjusted forecast of implementation
algorithm using the factor values known at the time of shortfall, derived from the model’s parameter estimates,
order entry. To summarize, the factors that act as input and x the volatility-adjusted maximum target. (For
to our estimated models are 1) number of shares in the example, we base our target on a maximum of 10 basis
order; 2) the stock’s historical duration volume as calcu- points in cost.) The intersection of the accumulative prob-
lated from a) expected or required order start time and end ability curve and the dashed line represents the probability
time and b) the stock’s historical volume profile and ADV; of achieving a (volatility-scaled) cost less than or equal to
and 3) the stock’s historical intraday volatility. We simply x. When we calculate the VaR probabilities for all the can-
feed these numbers into the reduced-form models for didate algorithms, the strategy that achieves the highest
each algorithm and obtain the predicted values of cost probability is the optimal strategy for trading the order at
and cost variance for that algorithm. hand: by using this strategy to trade the order at hand, we
Exactly how these forecast numbers can be used is know it has the best chance of not incurring a cost higher
up to the user. One obvious approach is using the cost than the maximum we are willing to tolerate.
utility function we mentioned in the previous section. Of course, this is only one example of an algorithm
The problem with this approach is the additional com- selection criterion, and this is not meant to be a com-
plexity introduced by the risk aversion parameter: how plete guide for such an analysis. Our framework is flex-

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 ible enough to accommodate the construction of any sta- range (in shares, in basis points relative to daily
tistic a practitioner wishes to derive from the estimated volume, or in basis points relative to duration
models. Our feeling is that the criteria should be easy to volume)? Does the algorithm handle extraordinarily
understand and also easy to implement. low or high volatilities? Is the algorithm time-of-day
dependent?
CONCLUSION 4. Choice of benchmark. Traders often have less flexi-
bility in selecting the benchmarks as benchmarks
The rising popularity of algorithmic trading has led are usually part of the desk’s trading policy, but it
to the mushrooming of algorithm products in the is still worth asking whether a given benchmark
marketplace today. A buy-side trader often has a large is the appropriate one under the circumstances.
array of algorithmic choices available. Some of these algo- Additionally, it is important to have a good idea of
rithms may have come from in-house R&D whereas how the benchmark is actually calculated inside
others could have been acquired from a third-party vendor the algorithm.
and are likely to be of the “black box” type.
As we have amply demonstrated in this article, using For the algorithm selection problem, we propose a
algorithms is not a simple task. The main advantages of quantitative approach that requires no knowledge of the
using an algorithm, when used correctly, are twofold: internal mechanisms of the algorithm. Our approach
first, it gives the trader a systematic, disciplined way to focuses on performance attribution using historical data
trade an order that is consistent with the trading objec- and provides parameters that help forecast the potential
tive; second, it generates an optimal trading trajectory performance of the algorithms in the context of the
that can maximize the chance of achieving the trading specific order and the prevailing market circumstances.
objective. To ensure algorithms are properly used, a trader Our proposed framework is general and is broker-
must keep the following checklist of issues in mind when neutral. We demonstrate, by example, how to turn the
considering the use of algorithmic trading: framework into a reduced form that can be estimated and
how to use the estimation results in algorithm selection.
1. Nature of algorithmic strategy. A thorough analysis The key takeaway is, it is not enough to just consider the
should be done on the nature of each algorithm comparative point estimates of the performance measure
before the algorithm is ever used. At a minimum, among the algorithm candidates. By considering the per-
a trader should cast the algorithm in the three- formance variance one can gain additional insight into
category paradigm we described. This paradigm how well each algorithm will likely perform given the
helps the trader conceptualize the underpinnings of order at hand. In our study, we estimate a simple single-
each strategy so that he or she can later quickly call factor model that is both intuitive and easy to implement;
on the appropriate strategies for an order. it also requires little computation time to generate useable
2. Suitability of algorithmic trading. Some orders are less ex ante selection scores for the algorithms.
suitable for execution via an algorithm and may be Quantitative pretrade analysis of algorithms is an
better handled (and closely monitored) by humans. essential part of algorithmic trading and should not be
These are typically very large orders, orders for stocks omitted from the trader’s algorithmic toolkit. The extra
with difficult liquidity conditions, or those with time and effort needed to conduct the analysis will more
very specific requirements. than pay for itself for each trade, and in the long run, by
3. Fit between order and algorithms. Even if an order is helping to ensure that the best algorithm be used in
a “normal” one and can be algorithmically traded, achieving the trading objective.
the trader must determine which available algorithms
are suitable for this particular order. Algorithms are ACKNOWLEDGMENTS
not all the same. Some are better under certain
circumstances while others prevail under other cir- The authors wish to thank Milan Borkovec, Gabe
cumstances. When offered an algorithmic trading Butler, Vitaly Serbin, Xiangyang Wang, James Wong,
product, the trader must question the vendor re- Henry Yegerman, and Ian Domowitz, all of ITG Inc., as
garding the “optimal” operating conditions of the well as Yingchuan Wang, for their support and comments.
product. For instance, what is the tradable order size The information contained herein is for informational

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 purposes only. Nothing herein is investment advice as REFERENCES
defined by the Investment Advisers Act of 1940. ITG Inc.
does not guarantee its accuracy or completeness and ITG Domowitz, I., and H. Yegerman. “Measuring and Interpreting
Inc. does not make any warranties regarding results from the Performance of Broker Algorithms.” ITG Inc. Research
Report, August 2005a.
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subject to change without notice and may not reflect the Comparative Performance.” In Brian Bruce, ed., Algorithmic
opinion of ITG Inc. This communication is neither an Trading: Precision, Control, Execution, New York: Institutional
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ENDNOTES
1
In addition to the problem of choosing from a large
To order reprints of this article, please contact Dewey Palmieri at
number of algorithms, one must also consider whether the
dpalmieri@iijournals.com or 212-224-3675.
order at hand is suitable for algorithmic trading. We will not
address this second issue in this article. The interested reader can
see, for example, Domowitz and Yegerman [2005a, 2005b].
2
Yang and Borkovec [2005], in contrast, use a two-cate-
gory approach and characterize evaluative algorithms as a spe-
cial case of structured strategies.
3
Some practitioners call the measure vis-à-vis mid-quote
at entry (which is Perold’s original definition) “realized market
[or price] impact.”
4
For example, opportunistic and evaluative strategies may
dynamically adjust the order type of each trade to liquidity con-
ditions.
5
This may simply be a feature specific to the strategies
we study; it is possible that some algorithmic strategies in the
marketplace can stay volatility neutral.
6
Any factor that is found to be significant empirically
should also have sound economic justification behind it.
7
In addition, this grouping approach can also be taken
with regard to discrete factors such as exchange membership
or industry sector classification.

34 ALGORITHM SELECTION: A QUANTITATIVE APPROACH SPRING 2006

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