FE670 Algorithmic Trading Strategies - …personal.stevens.edu/~syang14/fe670/presentation-fe670-lecture05.pdf · Cross-Sectional Methods for Evaluation of Factor Premiums Factor

Cross-Sectional Methods for Evaluation of Factor Premiums Factor Models Performance Evaluation of Factors Model Construction Methodologies for a Factor-Based Trading Strategy Backtesting R Time Series Factor Analysis Package tsfa

FE670 Algorithmic Trading StrategiesLecture 4. Cross-Sectional Models and Trading Strategies

Steve Yang

Stevens Institute of Technology

09/26/2013


Outline

1 Cross-Sectional Methods for Evaluation of Factor Premiums

2 Factor Models

3 Performance Evaluation of Factors

4 Model Construction Methodologies for a Factor-Based TradingStrategy

5 Backtesting

6 R Time Series Factor Analysis Package tsfa


There are several approaches used for the evaluation ofreturn premiums and risk characteristics to factors. Wediscuss four most commonly used approaches:

1 Portfolio Sorts2 Factor Models.3 Factor Portfolios.4 Information Coefficients.

* In practice, to determine the right approach for a givensituation there are several issues to consider 1). thestructure of the financial data. 2). the economic intuitionunderlying the factor. 3). validity of the underlyingassumptions of each approach.


Portfolio Sorts

The portfolios are constructed by grouping together securitieswith similar characteristics (factors). The goal of this processis to determine whether a factor earns a systemic premium.

The return of each portfolio is calculated by equally weightingthe individual stock returns. The portfolios provide arepresentation of how returns vary across the different valuesof a factor. By studying the return behavior of the factorportfolios, we may assess the return and risk profile of thefactor.

Overall, the return behavior of the portfolios will help usconclude whether there is a premium associated with a factorand describe its properties.


The construction of portfolios sorts on a factor isstraightforward:

1 Choose an appropriate sorting methodology.2 Sort the assets according to the factor.3 Group the sorted assets into N portfolios (usually N = 5, or

N = 10).4 Compute average returns (and other statistics) of the assets in

each portfolio over subsequent periods.

The standard statistical testing procedure for portfolios sortsis to use a Student’s t−test to evaluate the significance of themean return differential between the portfolios of stocks withthe highest and lowest values of the factor.


Choosing the Sorting Methodology

The sorting methodology should be consistent with thecharacteristics of the distribution of the factor and theeconomic motivation underlying its premium. Here six ways tosort factors:

Method 1:Sort stocks with factor values from the highest to lowest.

Method 2:Sort stocks with factor values from the lowest to highest.

Method 3: (Example: dividend yield factor)First allocate stocks with zero factor values into the

bottom portfolio.Sort the remaining stocks with nonzero factor values

into the remaining portfolios.Compute average returns (and other statistics) of the

assets in each portfolio over subsequent periods.



Method 4:

1). Allocate stocks with zero factor values into themiddle portfolio.

2). Sort stocks with positive factor values into theremaining higher portfolios (greater than the middle portfolio).

3). Sort stocks with negative factor values into theremaining lower portfolios (less than the middle portfolio).

Method 5: (Example: rank stocks according to earningsgrowth on a sector neutral basis)

1). Sort stocks into partitions.

2). Rank assets within each partition.

3). Combine assets with the same ranking from thedifferent partitions into portfolios.



Method 6: (Example: share repurchase factor)

1). Separate all the stocks with negative factors values.Split the group of stocks with negative values into twoportfolios using the median value as the break point.

2). Allocate stocks with zero factor values into oneportfolio.

3). Sort the remaining stocks with nonzero factor valuesinto portfolios based on their factor values.

* The portfolio sort methodology has several advantages. Theapproach is easy to implement and can easily handle stocksthat drop out or enter into the sample. The resultingportfolios diversify away idiosyncratic risk of individual assetsand provide a way of assessing how average returns differacross different magnitude of a factor.


Example 1: Portfolio Sorts Based on the EBITDA/EVFactor

Exhibit 7.1 contains the cross-sectional distribution of theEBITDA/EV factor. This distribution is approximately normallydistributed around a mean of 0.1, with a slight right skew. We usemethod 1 to sort the variables into five portfolios. Therefore, astrategy that goes long on portfolio 1 and short 5 appears toproduce abnormal returns.


Example 2: Portfolio Sorts Based on the Revisions Factor

Exhibit 7.2 shows that the distribution of earnings revisions isleptokurtic around a mean of about zero, withe the remaining valuessymmetrically distributed around the peak. We use method 3 tosort the variables into five portfolios. The stocks with positiverevisions are sorted into portfolio 1 and 2- while negative revisionsstocks are sorted into 4 and 5. Therefore, a strategy that goes longon portfolio 1 and short 5 appears to produce abnormal returns.


Example 3: Portfolio Sorts Based on the Share RepurchaseFactor

Exhibit 7.3 shows that the distribution of share repurchase isasymmetric and leptokurtic around a mean of about zero, withe theremaining values symmetrically distributed around the peak. We usemethod 6 to sort the variables into seven portfolios. There is a largedifference in return between the extreme portfolios. Therefore, astrategy that goes long on portfolio 1 and short 5 appears toproduce abnormal returns.


Information Ratios for Portfolio Sorts

- The information ratio (IR) is a statistic for summarizing therisk-adjusted performance of an investment strategy. It is defined asthe ratio of the average excess return to the standard deviation ofreturn.

- For actively managed equity long portfolios, the IR measures therisk-adjusted value a portfolio manager is adding relative to abenchmark.

- IR can also be used to capture the risk-adjusted performance oflong-short portfolios from a portfolio sorts.

- When comparing portfolios built using different factors, the IR is aneffective measure for differentiating the performance between thestrategies.


New Research on Portfolio Sorts

- The standard statistical testing procedure uses a Student’s t-test toevaluate the mean return differential between the two portfolioscontaining stocks with the highest and lowest values of the sortingfactor.

- However, this approach ignores important information about theoverall pattern of returns among the remaining portfolios.

- The Monotonic Relation(MR) test can reveal whether the nullhypothesis of no systemic relationship can be rejected in favor of amonotonic relationship predicted by economic theory.

- By MR it is meant that the expected returns of a factor should riseor decline monotonically in one direction as one goes from oneportfolio to another.


Factor Models

In investment management, risk is a primary concern. Themajority of trading strategies are not risk free but rathersubject to various risks. Here we describe some common risksto factor trading strategies as well as other trading strategies.

Classical financial theory states that the average return of astock is the payoff to investors for taking on risk. One way ofexpressing this risk-reward relationship is through a factormodel. A factor model can be used to decompose the returnsof a security into factor-specific and asset-specific returns:

ri ,t = αi + βi ,1f1,t + ...+ βi ,K fK ,t + εi ,t

where βi ,1, βi ,2, ..., βi ,K are the factor exposures of stocki , f1,t , f2,t , ..., fK ,t are the factor returns, αi is the averageabnormal return of stock i , and εi ,t is the residual.


Factor Models

This factor model specification is contemporaneous, that is,both left- and right-hand side variables (returns and factors)have the same time subscript, t.

For trading strategies one generally applies a forecastingspecification where the time subscript of the return and thefactors are t + h(h ≥ 1) and t, respectively. In this case, theeconometric specification becomes:

ri ,t+h = αi + βi ,1f1,t + ...+ βi ,K fK ,t + εi ,t+h

How do we interpret a trading strategy based on a factormodel? The explanatory variables represent different factorsthat forecast security returns, each factor with its associatedfactor premium.


Factor Models

Therefore, future security returns are proportional to thestock’s exposure to the factor premium

E (ri ,t+h|f1,t , f2,t , ..., fK ,t) = αi + βi ft

and the variance of future stock return is given by

Var(ri ,t+h|f1,t , f2,t , ..., fK ,t) = β′iE (ftf

′t)βi

where βi = (βi ,1, ..., βi ,K )′ and ft = (f1,t, ..., fK,t)′.


Econometric Considerations for Cross-Sectional FactorModels

In cross-sectional regression where the dependent variable is astock’s return and the independent variables are factors,inference problems may arise that are the result of violations ofclassical linear regression theory. The most common problems:

Measurement Problems Some factors are not explicitlygiven, but need to be estimated. These factors are estimatedwith an error. The estimation errors in the factors can havean impact on the inference from a factor model. This problemis commonly referred to as the ”errors in variables problem”.

Common Variation in Residuals The residuals from aregression often contain a source of common variation.Sources of common variation in the residuals areheteroskedasticity and serial correlation.



Common Variation in Residuals

Heteroskedasticity occurs when the variance of the residualdiffers across observations and affects the statistical inferencein a linear regression. In particular, the estimated standarderrors will be underestimated and the t-statistics will thereforebe inflated. Ignoring heteroskedasticity may lead theresearcher to find significant relationships where none actuallyexist. Several procedures have been developed to calculatestandard errors that are robust to heteroskedasticity.

Serial correlation occurs when consecutive residual terms in alinear regression are correlated, violating the assumptions ofregression theory. If the serial correlation is positive then thestandard errors are underestimated and the t-statistics will beinflated.



Multicollinearity Multicollinearity occurs when two or moreindependent variables are highly correlated. We may encounterseveral problems when this happens. First, it is difficult todetermine which factors influence the dependent variable.Second, the individual p values can be misleading – a p valuecan be high even if the variable is important. Third, theconfidence intervals for the regression coefficients will be wide.

There is no formal solution based on theory to correct formulticollinearity. The best way is by removing one or more ofthe correlated independent variables. It can be reduced byincreasing the sample size.


Fama-MacBeth Regression

To address the inference problem caused by the correlation ofthe residuals, Fama and MacBeth proposed the followingmethodology for estimating cross-sectional regressions ofreturns on factors. For notational simplicity, we describe theprocedure for one factor. The multifactor generalization isstraightforward:

First, for each point in time t we perform a cross-sectionalregression

ri ,t = βi ,t ft + εi ,t , i = 1, 2, ...,N

In the academic literature, the regressions are typicallyperformed using monthly or quarterly data, but the procedurecould be used at any frequency.

The mean and standard errors of the time series of slopes andresiduals are evaluated to determine the significance of thecross-sectional regression.


Fama-MacBeth Regression

We estimate f and εi as the average of their cross-sectionalestimates:

First, for each point in time t we perform a cross-sectionalregression

f̂ =1

T

T∑t=1

f̂t , ε̂i =1

T

T∑t=1

ε̂i ,t

The variations in the estimates determine the standard errorand capture the effects of residual correlation without actuallyestimating the correlation.

We used the standard deviations of the cross-sectionalregression estimates to calculate the sampling errors for theseestimates,

σ̂f̂ =1

T 2

T∑t=1

(f̂t − f̂ )2, σ2ε̂i =1

T 2

T∑t=1

(ε̂i ,t − ε̂i )2


Information Coefficients

To determine the forecast ability of a model, practitionerscommonly use the information coefficient (IC). The IC is alinear statistic that measures the cross-sectional correlationbetween a factor and its subsequent realized return

ICt,t+k = corr(ft , rt,t+k)

where ft is a vector of cross sectional factor values at time tand rt,t+k is a vector of returns over the time period t tot + k.

* Just like the standard correlation coefficient, the values of theIC range from −1 to +1. A positive IC indicates a positiverelation between the factor and return. A negative ICindicates a negative relation. ICs are usually calculated overan interval, for example, daily or monthly. We can evaluatehow a factor has performed by examining the time seriesbehavior of the ICs.


- An alternative specification of this measure is to make ft therank of a cross-sectional factor. This calculation is similar tothe Spearman rank coefficient. By using the rank of thefactor, we focus on the ordering of the factor instead of itsvalue. Ranking the factor value reduces the unduly influenceof outliers and reduces the influence of variables with unequalvariances.

- The subsequent realized returns to a factor typically vary overdifferent time horizons. For example, the return to a factorbased on price reversal is realized over short horizons, whilevaluation metrics such as EBITDA/EV are realized over longerperiods. It therefore makes sense to calculate multiple ICs fora set of factor forecasts whereby each calculation varies thehorizon over which the returns are measured.


- Information coefficients can also be used to assess the risk offactors and trading strategies. The standard deviation of thetime series (with respect to) of ICs for a particular factor canbe interpreted as the strategy risk of a factor. Examining thetime series behavior of std(ICt,t+k) over different time periodsmay give a better understanding of how often a particularfactor may fail.

- The expected tracking error can be used to understand theactive risk of investment portfolios. Qian and Hua defined anexpected tracking error as:

σTE = std(ICt,t+k)√Nσmodeldis(Rt)

where N is the number of stocks in the universe (breath),σmodel is the risk model tracking error, and dis(Rt) isdispersion of returns defined by

dis(Rt) = std(r1,t , r2,t , ..., rN,t)


Example: Information Coefficients

Exhibit 7.4 displays the time-varying behavior of ICs for each one ofthe factors EBITDA/EV, growth of fiscal year 1 and 2 earningsestimates, revisions, and momentum. The graph depicts theinformation horizons for each factor. The EBITDA/EV factor earnshigher returns. The overall pattern shows that the return realizationpattern to different factors varies.


Factor Portfolios

- Factor portfolios are constructed to measure the informationcontent of a factor. The objective is to mimic the returnbehavior of a factor and minimize the residual risk. Similar toportfolio sorts, we evaluate the behavior of these factorportfolios to determine whether a factor earns a systematicpremium.

- Typically, a factor portfolio has a unit exposure to a factorand zero exposure to other factors. Construction of factorportfolios requires holding both long and short positions. Wecan also build a factor portfolio that has exposure to multipleattributes, such as beta, sectors, or other characteristics.Portfolios with exposures to multiple factors provide theopportunity to analyze the interaction of different factors.


A Factor Model Approach

- By using a multifactor model, we can build factor portfoliosthat control for different risks. We decompose return and riskat a point in time into a systematic and specific componentusing the regression:

r = Xb + u

where r is an N vector of excess returns of the stocksconsidered, X is an N by K matrix of factor loadings, b is a Kvector of factor returns, and u is a N vector of firm specificreturns (residual returns).

- We assume that factor returns are uncorrelated with the firmspecific return. Further assuming that firm specific returns ofdifferent companies are uncorrelated, the N by N covariancematrix of stock return V is given by:

V = XFX′ + ∆


where F is the K by K factor return covariance matrix and ∆is the N by N diagonal matrix of variances of the specificreturns.

- We can use the Fama-MacBeth procedure discussed earlier toestimate the factor returns over time. Each month, weperform Generalize Least Square - GLS regression to obtain

b = (X′∆−1X)−1X′∆−1r

OLS would give us an unbiased estimate, but since theresiduals are heteroskedastic the GLS methodology is preferredand will deliver a more efficient estimate. The resultingholdings for each factor portfolio are given by the rows of(X ′∆−1X )−1X ′∆−1.


An Optimization-Based Approach

- A second approach to build factor portfolios usesmean-variance optimization. Using optimization techniquesprovide a flexible approach for implementing additionalobjectives and constrains. We would like to construct aportfolio that has maximum exposure to one targeted factorfrom X (the alpha factor), zero exposure to all other factors,and minimum portfolio risk. Let us denote the alpha factor byXα and all the remaining ones by Xσ. Then the resultingoptimization problem takes the form:

maxw

{w′Wα −

1

2λw′Vw

}s.t.w′Xσ = 0

- The analytical solution is given by:

h∗ =1

λV−1

[I− Xσ(X′σV−1Xσ)−1X′σV−1

]


Performance Evaluation of Factors

Analyzing the performance of different factors is an importantpart of the development of a factor-based trading strategy. Aresearcher may construct and analyze over a hundred differentfactors, so the means to evaluate and compare these factors isneeded. Most often this process starts by trying to understandthe time-series properties of each factor in isolation and thestudy how they interact with each other.


To better understand the time variation of the performance ofthese factors, we calculate rolling 24-month mean returns andcorrelations of the factors.


Model Construction Methodologies for a Factor-BasedTrading Strategy

The key aspect of building a model is to (1) determine whatfactors to use out of the universe of factors that we have, and(2) how to weight them.

- We describe four methodologies to combine and weightfactors to build a model for a trading strategy. Thesemethodologies are used to translate the empirical work onfactors into a working model.

It is important to be careful how each methodology isimplemented. In particular, it is critical to balance the iterativeprocess of finding a robust model with good forecasting abilityversus finding a model that is a result of data mining.


The Data Driven Approach

- A data driven approach uses statistical methods to select andweight factors in a forecasting model. This approach usesreturns as independent variables and factors as the dependentvariables. There are a variety of estimation procedures, suchas neural nets, classification trees, and principal components,that can be used to estimate these models.

- Many data driven approaches have no structural assumptionson potential relationships the statistical method finds.Therefore, it is sometimes difficult to understand or evenexplain the relationship among the dependent variables usedin the model.


The Factor Model Approach

- The goal of the factor model is to develop a parsimoniousmodel that forecast returns accurately. One approach is forthe researcher to predetermine the variables to be used in thefactor model based on economic intuition. The model isestimated and then the estimated coefficients are used toproduce the forecasts.

- A second approach is to use statistical tools for model section.In this approach we construct several models - often byvarying the factors and the number of factors used - and havethem compete against each other, and then choose the bestperforming model.

- Factor model performance can be evaluated in three ways.We can evaluate the fit, forecast ability, and economicsignificance of the model.


The Heuristic Approach

- Heuristics are based on common sense, intuition, and marketinsight and are not formal statistical or mathematicaltechniques designed to meet a given set of requirements. Theresearcher decides the factors to use, creates rules in order toevaluate the factors, and chooses how to combine the factorsand implement the model.

- There are different approaches to evaluate a heuristicapproach. Statistical analysis can be used to estimate theprobability of incorrect outcomes. Another approach is toevaluate economic significance.

- There is no theory that can provide guidance when makingmodeling choices in the heuristic approach. Consequently, theresearcher has to be careful not to fall into the data miningtrap.


The Optimization Approach

- In this approach, we use optimization to select and weightfactors in a forecasting model. An optimization approachallows us flexibility in calibrating the model andsimultaneously optimize an objective function specifying adesirable investment criteria.

- There is substantial overlap between optimization use inforecast modeling and portfolio construction. The factorsprovide a lower dimensional representation of the completeuniverse of the stocks considered. Besides the dimensionalityreduction, which reduces computational time, the resultingoptimization problem is typically more robust to changes inthe inputs.


Backtesting

- Model scores are converted into portfolios and then examinedto assess how these portfolios perform over time. Thebacktest should mirror as closely as possible the actualinvesting environment incorporating both the investment’sobjectives and the trading environment.

- In-sample backtesting is referred to use the same data sampleto specify, calibrate and evaluate a model.

- Out-sample backtesting is where the researcher uses a subsetof the sample to specify and calibrate a model, and thenevaluates the forecasting ability on a different subset of thedata (split-sample method and recursive out-of-samplemethod).


- This example illustrates the steps for estimating a factormodel using as an example the data and process which led toresults reported in Gilbert and Meijer (2006). The backgroundtheory is reported in Gilbert and Meijer (2005)

Figure : Data Explained by a 4 Factor Model

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FE670 Algorithmic Trading Strategies - …personal.stevens.edu/~syang14/fe670/presentation-fe670-lecture05.pdf · Cross-Sectional Methods for Evaluation of Factor Premiums Factor