The Analysis of a Momentum Model and Accompanying Portfolio Strategies

7/28/2019 The Analysis of a Momentum Model and Accompanying Portfolio Strategies

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The Analysis of a Momentum Model and Accompanying

Portfolio Strategies

Robert T. Samuel III

Draft version: 8 July 2013

Abstract

We analyze the performance of a proprietary momentum model and accompany-

ing portfolio strategies using a universe of twenty-three US Exchange Traded Funds(ETFs). For the simulation period of October 3, 2001 through May 1, 2013 we findthat our model does outperform the traditional momentum model across all strategyvariation. However, both models underperform a simple Buy & Hold strategy for theassets and time period selected. In addition to these results, we introduce a dynamicthreshold algorithm that determines ex ante the number of stocks to include in a port-folio. The results from this algorithm show that it is able to capture most of therisk-reward profile of the optimal portfolio.

1 Introduction

Momentum has historically possessed strong explanatory power within an asset pricingframework. Jegadeesh & Titman (1993) look at the performance of stock portfolios formedon a momentum rank, sometimes called cross-sectional or relative momentum, and findstatistically significant positive performance even when controlling for systemic and otherrisk factors. Specifically they find that historic time frames of a year, the formation period,in conjunction with a holding period of three-to-twelve months yield superior results. Thisextended research by de Bondt & Thaler (1985) which found winners outperforming losers,albeit only in January, and which is attributed to some of the behavioral biases discussedin Kahneman & Tversky (1982). Chan et al (1996) find that portfolios of US stocks formedbased upon returns for the past six months provide economically meaningful returns over

the next six months. In addition, these returns are not highly correlated with returns ofportfolios formed based upon earnings momentum. They go on to offer several theories forprice momentum including under-reaction to past information, but refute theories associatedwith a positive-feedback mechanism. Cahart (1997) documents the presence of a momentumfactor in addition to the traditional Fama-French 3-Factor model, in explaining the returnsof portfolios of mutual funds. However, he goes on to state that based upon the data it ap-pears that the presence of this explanatory factor is due to managers holding onto winning

Correspondence: [email protected]

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stocks and not an attempt to profit from momentum strategies. Rouwenhorst (1998) looksat the performance of momentum in international markets and find evidence that portfoliosconstructed in the manner of Jegadeesh & Titman provide positive returns for the period of1980-1995 and for all countries studied.

Jegadeesh & Titman (2001) again looked at the performance of momentum strategies and

find for the period of 1990-1998 that portfolios with a positive momentum factor loading out-perform those with a negative factor loading for up to twelve months after the formationperiod. In addition they look at the performance up to sixty months after formation and finda reversal which repudiates the claims of other authors that momentum is a measurementof the unconditional expectation of returns. This revelation adds credence to behavioraltheories which postulate that momentum is due to under/over-reaction to information bytrading agents. Fama & French (2012) analyzed the performance of the four-factor modelin explaining the excess returns of stocks globally and found that ex-Japan, momentum wasa strong explanatory factor. Baltas & Kosowski (2012) look at time-series, or absolute,momentum with futures and again find statistically significant performance for the one yeartime frame in conjunction with a one month holding period. Moskowitz et al (2012) also lookat time-series momentum using excess returns and find statistically significant positive re-turns across multiple asset classes that exhibit persistence up to twelve months. Novy-Marx(2012) found that by lagging the formation of a portfolio based upon a momentum factoryielded superior returns to just forming portfolios immediately after calculation. Lastly, An-tonacci (2013) finds that combining cross-sectional and absolute momentum yields superiorresults when utilized upon different asset classes.

Given the breadth of literature that documents the presence of a momentum factorwithin financial asset returns, we strive to develop a momentum model that encapsulatesthis empirical research. In the following sections we will cover the assets to be tested, themethod of testing and a discussion of the results1.

2 Data

We use daily Open, High, Low, Close, Adjusted Close, Volume and Dividend data fromYahoo! Finance for the ETFs listed in Table 12. We employ ETFs for our analysis primarilyto avoid having to establish point-in-time constituent members of a set of assets which wouldnormally lead to survivorship bias3. For a cash balance reference rate we use the DailyEffective Fed Funds Rate and for a risk-free rate we use the 3-Month Treasury Bill: SecondaryMarket Rate both from the from the St. Louis Federal Reserve Economic Database (FRED) 4

1Upon request, we can furnish an addendum which lists the tests, and statistics, along with their imple-

mentation.2Data from http://finance.yahoo.com/.3Survivorship Bias is the phenomena where a historic analysis is done on the current set of assets whose

membership in that set are influenced by their relative performance to other assets. In these scenarios thesimulated performance of a strategy may be influenced if the current set is used and not a point-in-timemembership.

4Data from http://research.stlouisfed.org/fred2/series/DFF and http://research.stlouisfed.org/fred2/series/DTBrespectively. In regards to the 3-Month Bill rate there were twenty-four days with no data due to ColumbusDay and Veterans Day. In this case we used an estimate for that days rate, Xt, using the time-series

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with descriptive statistics found in Table 2. In Table 3 we can see descriptive statistics onthe adjusted daily closing prices for the specified ETFs5. Jarque & Bera (1987) developed aLagrange Multiplies statistic to test for non-normality using the sample skewness and samplekurtosis and looking at our price series we see that we reject the null hypothesis of normalityfor all series. In addition, Ljung & Box (1978) developed a portmanteau test to determine

if the observations within a time series are independent by looking at lagged values and wesee that for all of our price series that the null hypothesis is rejected for a specified lag of 1.

In Table 4 we list the descriptive statistics for the returns, using the convention ofrt = log(Pt) log(Pt1), for adjusted closing prices. Mandelbrot (1963) documented thenon-normality exhibited in lognormal price changes for financial data, and we can clearlysee that all of the series exhibit non-normality. In addition, most reject the null hypothesisof independence. Bollerslev (1986) developed an extension of the the Autoregressive Condi-tional Heteroscedascity (ARCH) which he termed General ARCH (GARCH) that tests forconditional changes in the variances. In our table we see that all of series follow a GARCHprocess with a statistically significant ARCH process as well. Chow & Denning (1993) pro-posed a multiple variance test to test the null hypothesis that a time-series was a randomwalk. Given our evidence of heteroscedascity in all of the data series, we focus on the MV2statistic they developed and see that for more than half of our data series we can rejectthe null hypothesis of a random walk series. This is significant in that when our modelsidentify a trend it is not due to drift caused by past random disturbances. Lastly, Kruskal& Wallis (1952) developed a multivariate test to determine if samples come from the samedistribution, and although we do not report it, the test failed to reject the null hypothesisand therefore there is insufficient evidence to conclude that any of the returns come from adifferent distribution than the others.

3 Model, Portfolio Strategies & Competing Models3.1 Model

We develop a momentum model, denoted as MM, which attempts to quantify the strengthof a trend by outputting a real value number using three parameter values which determinethe appropriate historical time period and smoothness of the model output. For our analysiswe select parameter settings so as to focus on longer-term trends, and in an effort to reduceturnover, we preference a smooth output6. Given these objectives, we develop a model andwish to perform an initial evaluation of its merits. Our proportion of interest is

MM = Mj=1Ni=2 I[sgn(Sj,i1) = sgn(Dj,i)]M(N 1) (1)model ofXt = + Xt1 + which we calculated using Ordinary Least Squares (OLS) with a sample sizeof fifteen days. We acknowledge this is a crude approximation but given the number of estimations maderelative to the overall size of the sample, we feel it is sufficient.

5We use adjusted closing prices as they adjust for price impact of ex-dividend and splits. In addition, forall of our models we use the adjusted close, or a derivation of, for calculation purposes.

6We tested two parameter settings in total, both based upon the empirical research, but chose settingsthat were based upon the research of Moskowitz et al and Baltas & Kosowksi.

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where M= 23 are the number of assets, N= 2982 is the number of observations per asset,I[] is the Indicator function, sgn() is the the Sign function, Sj,i is our model signal andDj,i = Pj,i Pj,i1 is the price differential for observation i for the jth asset using adjustedclosing prices. We estimate this proportion as MM = 0.5212, which using a standard normaldistribution, provides sufficient evidence to conclude that the proportion of interest is not

equal to 0.57

. We note that this simple proportion does not account for implementation orexecution costs but does illustrate that the model does have predictive power given our dataset.

3.2 Portfolio Strategies

In regards to strategy we focus on three variations which we define as Absolute Momentum(AM), Cross-Sectional Momentum (CM) and Cross-Sectional, Absolute Momentum(CAM). With Absolute Momentum we invest in all available assets with our momentummodel dictating whether to own the asset or be flat8. In this formulation our modified signalbecomes

SAM,MM,j,t =Buy ifSMM,j,t >= 0Flat otherwise

(3)

where SMM,j,t is the raw signal from our model for asset j at time t. Since our modelattempts to estimate the momentum of an asset, and given the literature that shows thepoor performance of losing stocks relative to winners, we close out a position when weestimate it will be a loser.

Within our Cross-Sectional Momentum strategy we subset the available set of assetsby investing in those assets whose signal rank is less than a specified threshold. In thisformulation our modified signal becomes

SCM,MM,j,t =Buy if Rank(SMM,j,t)


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in Appendix A, which in turn yields an output Ut = {1,...,M}. Given this formulation, ourCross-Sectional Momentum strategys modified signal becomes

SCM,MM,j,t =

Buy if Rank(SMM,j,t)


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In regards to portfolio turnover due to signal, we will specify a time frequency of monthlyand quarterly given that the literature shows that holding periods of 1-3 months are optimal.In addition, we specify time frequencies to rebalance the portfolio so as to align the currentpositions with the optimal positions, which for our simulations we specify as monthly andquarterly as well. Given this set-up we will test six variations of the strategies: Absolute

Momentum - Monthly & Quarterly Rebalance; Cross-Sectional Momentum - Monthly &Quarterly Rebalance; and Cross-Sectional, Absolute Momentum - Monthly & QuarterlyRebalance.

3.3 Competing Models

Our first competing, or benchmark, model is a Buy & Hold (BH) strategy where we establishlong positions in all twenty-three assets and rebalance on the above stated frequencies, whichyields Buy & Hold - Monthly Rebalance and Buy & Hold - Quarterly Rebalance. This modelshould be seen as a naive model as it conveys no information about the expected relativeperformance of an asset. From a practical standpoint, it can also be seen as the performanceof investing in an passive index fund that seeks to mimic the performance of an equally-weighted index11.

Our next competing model is a Momentum (Scaled) model (Mo) with a specified for-mation period ofT = 252 trading days as the research by Jegadeesh & Titman and othershave shown that this is the optimal formation period12. Given this set-up our Momentummodel signal for the jth asset at time t is

SMo,j,t =log(Pj,t) log(Pj,tT)

j,t(9)

where Pj,t is the adjusted closing price, and j,t is the standard deviation of the log returns

for the period T13. We scale the momentum as recommended by Moskowitz et al so as tocompare asset signals14. Given this raw signal, our modified signals become

SAM,Mo,j,t =

Buy ifSMo,j,t >= 0

Flat otherwise(10)

where Vj is a measure of dispersion. With this formulation we tested three variations of risk parity (StandardDeviation, Semi-Deviation, and Median Absolute Deviation) and all three variations under-performed theequally-weighted portfolio for all strategies, rebalancing rules and models.

11We will use an equally-weighted index, using adjusted closing prices, as our market portfolio whenperforming return attribution analysis.

12

We use 252 days as it is the average number of trading days in a year. Also note that we do not delaythe formation of the portfolio as suggested by Novy-Marx and Jegadeesh & Titman for the sake of simplicity.We note that this may not be optimal and therefore recognize the results may not be an optimal competingmodel.

13Similar to our model, we estimate Mo = 0.5320 and Mo = 0.5336 both of which provide sufficientevidence to conclude that the proportion of interest is not equal to 0.5.

14Note that due to the autocorrelation present in the log returns, the sample standard deviation estimatorwill be biased beyond the traditional bias. We do not attempt to mitigate this bias and note the shortcoming.Also note that Moskowitz et al use an exponentially-weighted volatility measure when scaling the momentumstatistic but we use the sample estimator due to simplicity.

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the prior day using the specified risk free rate minus a 0.50% brokerage fee18. Lastly, weaccrue dividends earned if that day is an ex-dividend date and also adjust our portfolioholdings for stock splits.

The purpose of including fees, interest and dividends is to provide more realistic simulatedreturns since these are impacted by our model/strategy combination. In addition assessing

management fees is crucial given its impact on the cumulative results and the purchasingpower of the hypothetical account.

4.3 Step Three: Calculate Optimal Position Weights & Quantities

The starting capital for our simulation is $25,000 and our target portfolio allocation is98%19. With these two parameter inputs we calculate the optimal position quantities, usingyesterdays closing prices, the portfolio weight scheme and our current signals.

4.4 Step Four: Determine If Trades Are Warranted

Next we determine if the current date is a trade date based upon the proposed tradingfrequency and determine if trades are warranted due to the signal or if we need to optimize aposition. If so, we create opening or closing trades within our internal blotter and then wepeek ahead to determine if we could have theoretically traded that day by seeing if, first,there is any trading volume for the day; and secondly, whether our trade quantity doesntexceed 100% of that days volume. If either of these are false, we conduct the trades butadjust the slippage amount that we discuss in the next sub-section20.

4.5 Step Five: Conduct Trades

If we need to conduct trades we do so using the opening price adjusted by the commissionrate of $0.05 per share and the trade slippage.21 In regards to trade slippage it is astatistic that quantifies the differential in price from when a signal is received and the actual

18Specifically the rate is Max[0, (FF0.005)360 ], where we use the 360 day convention. In addition, the broker-age fee used is one that is employed currently by Interactive Brokers (http://www.interactivebrokers.com)and is seen as a conservative estimate of what a hypothetical brokerage would charge.

19We set our capital amount at this level as our target investor is a retail investor, although the strategyis equally applicable for an institution. It should be noted that since one of our simulation checks is todetermine whether a trade could be conducted on a given day, that the lower initial capital amount affectsthe ability, and timing, of our trades. We recognize this shortcoming in our simulation but also note that itdoes not adversely impact, nor favor, any specific model. In addition, we specify a 98% target allocation soas to maintain a cash balance for the purposes of account fees and to account for the fact that there may be

a differential between todays open price and yesterdays close when determining position quantities. In thelatter case, we could theoretically assume the investor waited till the open and use the opening price, but aswe are attempting to capture the opening liquidity, we do not chose this option.

20This approach can be viewed as synonomous with working a trade over a longer time period than thatwhich we specify for a normal trade. In these cases, in lieu of scaling the volatility detailed in (13) we usethe full amount as a trade impact estimate.

21The commission rate is higher than the prevailing industry rate, and what was charged at the beginningof the simulation, but we since we do not adjust for the bid-ask spread, and we are uncertain if the customerwas charged a flat rate, we continue to use this rate.

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trade price. Normally, it is used as a measurement of trade execution and price impact inreal-time scenarios but for our simulation it is used as a price adjustment that attempts toquantify the potential trade impact, bid/ask spread and price uncertainty had we actuallytraded that day. In most simulations this is a fixed amount but for our simulation we usea localized measure based upon high and low prices. Berkowitz et al (1988) found that the

percentage range for a stock, (HighLow)/Low 100, had a statistically significant impacton the market impact costs, while controlling for commissions and trade volume, for NYSEstocks for the first quarter of 1985. Therefore we develop a statistic that uses the range toquantify the expected slippage, or implicit transaction costs, for a stock. Parkinson (1980)developed a range-based volatility estimator

t =

ni=1 log(

HiLi

)

4 log(2) n(13)

at time t and where Hi is the High Price, Li is the Low Price and n is the sample size22. We

then adjust our opening price by a scaled amount of

T

39023

. After incorporating commissionsand slippage, our final trade price is

Ot =

Ot +

t30

390+ 0.05 if Buy Order

Ot t

30

390 0.05 if Sell Order

(14)

where Ot is the Opening price at time t24.

4.6 Step Six: Calculate Profit & Loss; Adjust Account Equity,Positions and Cost Basis

Our last step is to calculate the profit & loss of the simulated portfolio using that days closingprice relative to the previous days closing price. We then adjust our account equity whichis a function of our starting capital and our accrued profits and losses. This is important asour margin levels have an impact on our trading activities. Lastly, we adjust our cost basisto reflect all incremental increases in our position(s) so as to have accurate per trade returnstatistics.

5 Discussion of Simulated Results

The earliest starting date where all ETFs trade is June 23, 2000, and with model trainingand additional days needed for back-testing implementation, our simulation start date is

22For our simulation we use n = 22 as it represents the amount of trading days in a typical month.23We use

(390) as this converts our volatility estimate into a minute frequency and then use use T = 30

as a conservative estimate of the amount of time, in minutes, that we expect that it would take to completea trade.

24There exist more advanced models for slippage that include volume in their functional argument. How-ever, we continue to use our estimate due to its simplicity and due to the fact that slippage is less of a factorin our simulations due to low turnover.

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October 3, 2001 and our final date is May 1, 2013. In Tables 5, 6, 7, 6, 9 and 10 we seethe results of our six strategy variations using differing statistics25. In what follows, we willbriefly discuss some of the statistics, their background and significance.

The annualized returns and volatility are computed in the standard manner and provideinsight into the risk-reward profiles of the strategies26. Specifically, with the annualized

returns we can see that the BH strategy outperforms all models for all strategy variationssave the CAM-Quarterly where our model outperforms. Brinson et al (1986,1991) providean analytical framework so as to decompose the returns of a strategy versus a benchmark.Within this framework we can say that our security selection is inferior to the benchmarkexcept in the case of the CAM-Quarterly strategy but we also note that when performing aKruskal-Wallis test we fail to reject the null hypothesis that the returns are from the samedistribution. Given this fact, there isnt any statistical evidence that the returns differ forany of the model/strategy combinations.

With the annualized volatility we see a marked difference when employing the AM &CAM strategies which is to be expected given their ability to invest in the relatively lowvolatility and autocorrelated cash asset. The Sharpe Ratio, which was proposed by Sharpe(1966), is a measure to quantify the reward-to-risk profile of a portfolio within the mean-variance framework that uses the excess returns from a portfolio27. For all of the strategyvariations, we can see that our model outperforms the Mo(S) model but it only outperformsthe BH model when again employing the AM & CAM strategies. However, it should be notedthat the Sharpe Ratio is sensitive to issues of non-normality and violations of iid which arerevealed via the Jarque-Bera and Ljung-Box statistics for the returns28.

Given some of the issues with the Sharpe ratio, Keating & Shadwick (2002) proposedan alternate statistic to measure the risk-reward trade-off they termed Omega that uses thecumulative distribution function (CDF) which incorporates the entire distribution of a dataseries29. We use the excess returns as our data series and a threshold, = 0 since an (0) = 1

would indicate an inability to generate returns in excess of the risk-free rate30

. We note thatall of our results demonstrate an ability to generate returns that are superior to investingin the risk-free rate but also note that there isnt much variation in the results. This is afunction of the we selected but as we wanted to contrast this statistic with the SharpeRatio, and that statistic estimates performance relative to the risk-free rate, we made thischoice of . Another risk metric is the Maximum Drawdown (MaxDD) which attempts toquantify the greatest potential run of losses for a portfolio. As with the volatility statistic,

25The Buy & Hold is strategy, which only has two variations, is listed along with the other two models forit comparable rebalancing frequency.

26Specifically, the annualized return is the average daily return multiplied by 252 and the annualizedvolatility is the daily standard deviation multiplied by the square root of 252.

27We employ the variation which uses the excess returns when calculating the standard deviation.28Ledoit & Wolf (2008) discuss methods for mitigating these issues in the context of performing hypothesis

tests between Sharpe ratios but for the sake of brevity we do not attempt to deal with this issue and reportthe Sharpe ratios as computed

29In their cited paper the authors called the statistic Gamma, but in later works they changed its nameto Omega.

30Note, that we employ a non-parametric, kernel density estimator to estimate our distribution. Also,given the non-linear characteristic of the CDF, the choice of can have a significant impact on the resultingOmega statistic.

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we can clearly see that allowing the portfolio to increase its exposure to cash greatly reducesthe MaxDD.

Next, we evaluate Jensens which was proposed by Jensen (1968) as an extensionof the Capital Asset Pricing Model (CAPM) [Sharpe (1964)], for the analysis of mutualfund performance using an appropriate market portfolio and risk-free rate. To analyze the

performance our model & strategy combination we construct an equal-weighted index ofreturns from the log returns of the assets and use the 3-Month T-Bill rate as our risk-free rateso as to create market portfolio excess returns. Given this formulation, we conduct OrdinaryLeast Squares (OLS) to estimate the parameters but also conduct diagnostic tests on ourresults. Breusch & Pagan (1979) developed a Lagrange Multiplier test for hetereoskedascityand we can see that for some of the results that it is present31. In addition to this, Breusch(1979) and Godfrey (1978) independently developed a Lagrange multiplier test to determinethe presence of autocorrelated error terms and we can see that for all of the models, save BH,that there exists statistically significant autocorrelation. Lastly we perform Jarque & Beratests on our OLS residuals to determine if normality is violated and see that in all cases thatour residuals violate our normality assumption32. We do not correct for this last violation,but given the existence of heteroscedasticity and autocorrelation, when determining thestatistical significance of our OLS parameter estimates we use adjusted standard errors usingHeteroscedasticity Consistent (HC) and Heteroscedasticity and Autocorrelation Consistent(HAC) estimators as discussed in Zeileis (2004, 2006)33. With these corrections we can seethat none of our Jensens are statistically significant except for the momentum model usingthe CM-Monthly strategy. However, all of our beta estimates are statistically significant andwe note that when employing the AM & CM strategies, we see the expected reduction incovariance with the market portfolio due to the varying exposure to this asset class.

Lastly, we list statistics related to the trading results of the individual assets and note afew details in calculation. For slippage, we report all trades inclusive of rebalancing trades

along with trades due to signal change. Berkowitz et al found the average market impactcosts, which used the percentage price differential from the volume-weighted average price(VWAP), of 0.23% for NYSE stocks for the first quarter of 1985. Jones (2002) found for theperiod of 1900-2001 that the average annual transaction costs for NYSE stocks, incorporatingbid-ask spreads, was 0.38% with a standard deviation of 0.32%. These estimate contrast withour estimates which range from 0.24%-0.29% depending on strategy or model. Given this,

31We do not use the test developed by White (1980) as it is a more general test that also tests for modelmis-specification and we are only interested in the presence of heteroscedasticity as CAPM dictates a linearrelationship.

32Gauss-Markov does not assume normality for the residuals, however, the Maximum Likelihood Estimaors(MLE) for OLS are no longer the Uniform Minimum Variance Unbiased Estimators (UMVUE) if normality

is violated.33We use a critical level of 5% from the Breusch-Pagan and Breusch-Godfrey to determine whether toreject the null hypothesis in favor of heteroscedasticity and autocorrelation respectively. In the presenceof heteroscedasticity, with no autocorrelation, we use the vcovHC() function within R as a HC estimatorusing the default settings. In the case of autocorrelation, regardless of the presence of heteroscedasticity,we use the vcovHAC() function within R as a HAC estimator using the default settings. Andrews (1991)showed that in cases of autocorrelation with homoscedasticity that the Quadratic-Spectral Kernel Estimator,the default within vcovHAC() is nearly as optimal as the most efficient estimator in that scenario, titledPARA. Therefore we feel there is minimal loss of efficiency for using it in both cases.

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we feel that our slippage estimates are adequate given the research and also note that due toour low turnover, that slippage is less of factor relative to commission costs. In addition, theNumber of Trades statistic that is reported does not include rebalancing trades but onlytrades that are due to signal. We do this as rebalancing trades are a function of the assetsthemselves and not entirely a function of the model34. We can see that all of the strategy

variations have low turnover rates which is a function of the time frame used to estimatetheir signals. Lastly, we report the mean and standard deviation of trade returns but notethat these statistics do contain all trades and not those solely based upon signal. We dothis as our rebalancing rule may facilitate the capture of gains or losses that arent entirelyreflected in a trade as dictated by signal35.

6 Conclusions & Further Research

Our results show that our model is superior to the momentum model used in the literaturebut not in statistically significant manner. In addition, rebalancing on a quarterly basis

provides superior results which was also detailed by Jegadeesh & Titman (1993). However,in almost all cases, both models underperform a simple Buy & Hold strategy. This realizationcould be a function of the assets selected and the time period analyzed. In addition, as ourinitial Kruskal & Wallis test illustrates, the assets we selected are fairly homogenous whichin turn may negate the ability of any model to differentiate the winners from the losers. Thislast fact motivates us to perform further research using our model on a wider range of assetsand for different time periods. However, our dynamic threshold algorithm appears to showsolid performance especially given that it allows for a mechanical, ex ante estimation of theproper number of assets. Further research is needed to determine if the time-frame selected,or the choice of assets, have an impact on its performance.

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[30] Keating, C. and Shadwick, W. (2002) A universal performance measure, Journal of Perfor-mance Measurement 6(3), 5984

[31] Kritzman, M., Li, Y., Page, S. and Rigobon, R. (2010) Principal Components as a Measureof Systemic Risk

[32] Kruskal, W. and Wallis, W. (1952) Use of Ranks in One-Criterion Variance Analysis, Journalof the American Statistical Association 47(260), 583-621

[33] Ledoit, O. and Wolf, M. (2003) Improved estimation of the covariance matrix of stock returnswith an application to portfolio selection, Journal of Empirical Finance 10(5), 603-621

[34] Ledoit, O. and Wolf, M. (2008) Robust performance hypothesis testing with the Sharpe ratio,Journal of Empirical Finance 15(5), 850-859

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[36] Mandelbrot, B. (1963) The Variation of Certain Speculative Prices, The Journal of Business36(4), 394-419

[37] Mann, H. and Whitney, D. (1947) On a Test of Whether one of Two Random Variables isStochastically Larger than the Other, Annals of Mathematical Statistics 18(1), 50-60

[38] Markowitz, H. Portfolio Selection: Efficient Diversification of Investments. New York: John

Wiley & Sons (1959)

[39] Moskowitz, T., Ooi, Y., and Pedersen, L. (2012) Time series momentum, Journal of FinancialEconomics 104(2), 228-250

[40] Novy-Marx, R. (2012) Is momentum really momentum?, Journal of Financial Economics103(3), 429-453

14


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[41] Opgen-Rhein, R. and Strimmer, K. (2007) Accurate Ranking of Differentially ExpressedGenes by a Distribution-Free Shrinkage Approach, Statistical Applications in Genetics andMolecular Biology 6(1)

[42] Parkinson, M. (1980) The Extreme Value Method for Estimating the Variance of the Rate ofReturn, The Journal of Business 53(1), 61-65

[43] Rouwenhorst, K. (1998) International Momentum Strategies, The Journal of Finance 53(1),267-284

[44] Sharpe, W. (1964) Capital Asset Prices: A Theory of Market Equilibrium under Conditionsof Risk, Journal of Finance 19(3), 425-442

[45] Sharpe, W. (1966) Mutual Fund Performance, The Journal of Business 39(1), 119-138

[46] Statman, M. (1987) How Many Stocks Make a Diversified Portfolio?, Journal of Financialand Quantitative Analysis 22(3), 353-363

[47] Statman, M. (2004) The Diversification Puzzle, Financial Analysts Journal 60(4), 44-53

[48] Tversky, A., and Kahneman, D. (1974) Judgment under Uncertainty: Heuristics and Biases,Science 185(4187), 1124-1131

[49] Schafer, J., and Strimmer, K. (2005) A Shrinkage Approach to Large-scale Covariance Es-timation and Implications for Functional Genomics, Statistical Applications in Genetics andMolecular Biology 4(1)

[50] White, H. (1980) A Heteroskedasticity-Consistent Covariance Matrix Estimator and a DirectTest for Heteroskedasticity, Econometrica 48(4), 817-838

[51] Wilcoxon, F. (1945) Individual Comparisons by Ranking Methods, Biometrics Bulletin 1(6),80-83

[52] Zeileis, A. (2004) Econometric Computing with HC and HAC Covariance Matrix Estimators,Journal of Statistical Software 11(10), 117

[53] Zeileis, A. (2006) Object-oriented Computation of Sandwich Estimators, Journal of Statisti-cal Software 16(9), 116

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A Dynamic Threshold (Inverse Herfindahl-Hirschman)

We desire to develop a metric that determines our rank threshold dynamically and in a datadependent manner. Markowitz (1959) noted that as the correlation of assets increases thebenefits of diversification decreases. In addition, Statman (1987,2004) addresses the issue of

how many assets make a diversified portfolio, and extending upon Markowitzs work, definesthe problem as the marginal benefit from reduction in the portfolio standard deviation versusthe marginal cost of transaction costs. Therefore given these claims we seek a metric thatmeasures the amount of diversification present in our choice of assets. Kritzman et al (2010)mentioned the usage of a Herfindahl Index in conjunction with the eigenvalues of a PrincipalComponent Analysis (PCA) of the return data so as to measure the ability of a marketto absorb systemic shocks36. In addition Fisher & Lowrie (1970) used a Gini coefficientof concentration along with its mean difference to measure the effect of diversification of aportfolio by quantifying the amount of dispersion amongst assets. Within this forumulation,let PCA be the procedure that solves

V = V (15)

where is the covariance or correlation matrix ofX, V is the matrix of eigenvectors cor-responding to the vector of eigenvalues, , and which we estimate using the princomp()function, with an inputted covariance matrix, in R. However, due to the non-normality ofour data, we use a robust covariance matrix estimate as outputted by the cov.shrink()function in R as described in Schafer & Strimmer (2005) and Opgen-Rhein & Strimmer(2007). They proposed using a Ledoit & Wolf (2003) shrinkage estimator to estimate thecomponents of covariance matrix such that

i,j = 1s2median + (1 1)s2i,j if i = ji,jmin(1,max(0, 1 2))s2i,is2j,j if i = j (16)where i,j is the sample correlation, s

2

i,j is the sample variance, s2

median is the median of thesample variances,

1

= min(1,

Mk=1 V ar(s

2

k)Mk=1 s

2

k s2

median

) (17)

is the shrinkage estimate for the variance estimates given the number of covariates, M,

2

= i=j V ar(i,j)

i=j 2i,j (18)is the shrinkage estimate for the correlation estimates, and

V ar(s2k) = n(n 1)3ni=1

(wi,k wk)2 (19)

36It should be noted that Hirschman (1964) claimed the paternity of the index that normally bears Dr.Herfindahls name.

16


17/31

is an estimate of the standard error of s2k given the sample size n, xk =1

n

nk=1 xi,k, wi,k =

(xi,k xk)2, wk =

1

n

nk=1 wi,k and

V ar() =

n

(n 1)3

n

i=1(wk,i,j wi,j)

2 (20)

is an estimate of the standard error of given zk,i =xk,ixi

si, zi =

1

n

nk=1 zk,i, wk,i,j =

(zk,i zi)(zk,j zj), and wi,j =1

n

nk=1 wk,i,j.

Given this estimate of the covariance matrix, and the estimates of the eigenvalues fromthe PCA, let the Dynamic Threshold (Inverse Herfindahl-Hirschman) [DT(IHH)] at time t

of a data matrix, X(t), comprised of the log returns, rj,t = log(Pj,tPj,t1

), be defined as

DT(IHH)t =1 M

i=1

i,tM

i=1 i,t

1 1M

(21)

where M is the number of assets and DT(IHH) [0, 1]. We then define our dynamicthreshold value at time t as

Ut = max(DT(IHH)t M, 1) (22)

where is the rounding function and Ut = {1,...,M}. For our simulation we arbitrar-ily select a historic time-frame of 66 days when calculating our DT(IHH)t as we wish tohave enough data to estimate our covariance matrix but a shorter time-frame due to the het-eroscedasticity present in the data. It should be noted that we did test the performance of themodels and strategy combinations using the sample covariance matrix and in all cases they

had statistically inferior results versus using the robust covariance method. In addition, wedid contemplate employing a Exponentially Weighted Moving Average (EWMA) or DynamicConditional Correlation (DCC) approach to mitigate the effects of the heteroscedasticity butwe chose to save that for further research. Lastly, there didnt exist a significant differencebetween calculating DT(IHH) using the covariance matrix or the correlation matrix whichmay be due to the homogeneity of the asset returns.

In an effort to test the efficiency of our dynamic threshold we perform backtest simulationsfor fixed thresholds U = {1,...,23} for the CM & CAM strategies. In lieu of reporting allof the simulation statistics, we focus on the Omega(0) statistic using excess returns withthe results listed in Figures 1, 2, 3 and 437. In these results we can see that, although, ouralgorithm does not yield the optimal threshold it does capture the majority of the risk-reward

available for each strategy variation.In addition to this evaluation, we explore an alternate formulation of our dynamic thresh-

old that uses a Herfindahl-Hirschman index and not its inverse. Using the same definitionsas those in (21) and (22), let our alternate threshold be

Ut = max(DT(HH)t M, 1) (23)

37For reference our DT(IHH) yields a mean value of U= 10.43 for the simulation period

17


18/31

which in turn is a function of

DT(HH)t =

Mi=1

i,tM

i=1 i,t 1

M

1 1M

(24)

which is again normalized to yield an output DT(HH) [0, 1]. Given this alternate method,we perform simulations using our model for the four strategy variations that employ a dy-namic threshold and list those results in Table 11. We can see that for our CM strategythat there is little difference but there is a difference for both CAM strategy variations. Inaddition to these results we perform Mann-Whitney-Wilcoxon tests on all four simulationpairings with an alternate hypothesis that the returns from the DT(IHH)-based results aregreater than those from the DT(HH). In all four tests we find insufficient evidence to con-clude that the returns using the DT(IHH) are greater than those employing the DT(HH)variation. Lastly, we plot the threshold estimates using DT(IHH) and DT(HH) in Figures 6and 7 respectively along the benchmark index in Figure 5 for comparison. Given that the

DT(HH) estimates the amount of dispersion within a covariance matrix, we can see someof the stylized facts documented in Ang & Chen (2002) and Das & Uppal (2004). Namely,that changes in the correlation structure have an asymmetric nature which is realized as agreater increase in correlation during market declines. Therefore we see a decrease in ourthreshold using the DT(IHH) during market declines and increase in number of assets, dueto a decrease in correlation, during the market rally of 2006-2007.

18


19/31

Ticker DescriptionEWA iShares MSCI Australia IndexEWC iShares MSCI Canada Index

EWG iShares MSCI Germany IndexEWH iShares MSCI Hong Kong IndexEWI iShares MSCI Italy Capped IndexEWJ iShares MSCI Japan IndexEWS iShares MSCI Singapore IndexEWT iShares MSCI Taiwan IndexEWW iShares MSCI Mexico Capped Investable MarketEWY iShares MSCI South Korea Capped IndexIWM iShares Russell 2000 IndexMDY SPDR S&P MidCap 400QQQ PowerShares QQQ

SPY SPDR S&P 500XLB Materials Select Sector SPDRXLE Energy Select Sector SPDRXLF Financial Select Sector SPDRXLI Industrial Select Sector SPDR

XLK Technology Select Sector SPDRXLP Consumer Staples Select Sector SPDRXLU Utilities Select Sector SPDRXLV Health Care Select Sector SPDRXLY Consumer Discretionary Select Sector SPDR

Table 1: Ticker symbols and their descriptions for the ETFs used in the analysis. Informationobtained from Yahoo! Finance

Effective Fed Funds 3Month T-BillNumber of Observations 2,914 2,914Mean 0.0047% 0.0044%Minimum 0.0000% 0.0000%

Maximum 0.0139% 0.0140%Standard Deviation 0.0050% 0.0046%

Table 2: Descriptive statistics Effective Fed Funds (EFF) rate and 3-Month Treasury Bill:Secondary Market (3MTB) rate for the period of October 3, 2001 through May 1, 2013. Note

these statistics are based upon the daily rate using the 360 day convention to convert the quoteddatum.

19


20/31

EWA

EWC

EWG

EWH

EW

I

EWJ

EWS

EWT

NumberofObservat

ions

3,232

3,232

3,232

3,232

3,232

3,232

3,232

3,232

MeanPrice

14.52

19.59

18.69

12

17.33

9.739

7.768

10.41

MinimumPrice

4.68

7.07

7.22

5.17

8.75

5.68

2.69

5.28

MaximumPrice

28.1

32.99

32.87

20.53

30.76

14.28

14.44

15.09

StandardDeviation

6.74

7.61

5.78

4.19

5.66

1.91

3.44

2.3

JBa

276.87***

291.68***

89.23***

217.68***

405.23***

59.91***

302.35***

98.69***

Qb

3,223.62***

3,227.43***

3,223***

3,222.37***

3,225.5

9***

3,213.88***

3,225.45***

3,204.32***

EWW

EWY

IWM

MDY

QQQ

SPY

XLB

XLE

NumberofObservat

ions

3,232

3,232

3,232

3,232

3,232

3,232

3,232

3,232

MeanPrice

35.33

37.76

58.2

120.6

43.4

106.6

26.43

46.18

MinimumPrice

9.57

10.17

28.47

61.27

18.89

62.28

13.29

16.94

MaximumPrice

76.71

71.67

94.53

211

96.64

159.7

41.13

83.66

StandardDeviation

19.47

16.87

15.39

35.24

13.85

20.11

7.84

19.11

JB

282.4***

263.97***

139.85***

155.37***

740.56***

54.81***

250.59***

248.55***

Q

3,226.08***

3,224.48***

3,218.16***

3,220.11***

3,209.6

3***

3,214.86***

3,220.46***

3,223.68***

XLF

XLI

XLK

XLP

XLU

XLV

XLY

NumberofObservat

ions

3,232

3,232

3,232

3,232

3,232

3,232

3,232

MeanPrice

20.28

27.3

21.68

22.76

25.12

28.23

30.04

MinimumPrice

5.77

14.01

10.27

14.26

10.62

18.55

15.08

MaximumPrice

33.35

42

50.93

41.43

41.43

48.19

54.61

StandardDeviation

5.96

6.16

6.17

5.41

6.77

4.96

7.44

JB

91.62***

111.9***

4497.31***

757.24***

132.37***

1195.52***

430.84***

Q

3,223.99***

3,217.89***

3,201.97***

3,218.43***

3,222.4

5***

3,208.39***

3,213.8***

Table3:DescriptivestatisticsoftheadjustedclosepricesfortheETFswithintheana

lysisfortheperiodof6/23/00through5/1/13.

***significantat1%level,**

significantat5%leveland*

significantat10%level.

aJarque-Berastatistic.

bLjung-Boxstatistic.

20


21/31

EWA

EWC

EWG

EWH

EW

I

EWJ

EWS

EWT

NumberofObservatio

ns

3,231

3,231

3,231

3,231

3,231

3,231

3,231

3,231

MeanReturn

0.05%

0.02%

0.01%

0.03%

0.00%

0.00%

0.03%

0.00%

MinimumReturn

-13.18%

-11.65%

-12.01%

-13.08%

-11.17%

-11.06%

-11.90%

-12.38%

MaximumReturn

18.79%

11.68%

18.05%

15.66%

14.25

%

14.65%

16.52%

13.26%

StandardDeviation

1.87%

1.60%

1.86%

1.83%

1.91%

1.54%

1.86%

2.15%

JBa

11,359.75***

3,941.04***

6,354.91***

6,592.13***

3,818.66***

6,081.59***

5,446.96***

2,031.58***

Qb

41.22***

3.77*

8.77***

88.98***

17.78***

41.9***

77.06***

34.71***

1c

0.09***

0.07***

0.08***

0.07***

0.07***

0.09***

0.10***

0.07***

1

0.89***

0.92***

0.90***

0.92***

0.92***

0.89***

0.89***

0.92***

MV1

d

6.43***

1.94***

2.97***

9.44***

4.22***

6.47***

8.78***

5.90***

MV2

3.38***

1.15

1.97

5.26***

2.80**

3.90***

5.42***

4.24***

EWW

EWY

IWM

MDY

QQQ

SPY

XLB

XLE

NumberofObservatio

ns

3,231

3,231

3,231

3,231

3,231

3,231

3,231

3,231

MeanReturn

0.06%

0.03%

0.02%

0.03%

-0.01%

0.01%

0.03%

0.03%

MinimumReturn

-13.24%

-18.03%

-11.93%

-12.39%

-9.36%

-10.37%

-13.26%

-15.61%

MaximumReturn

19.45%

20.24%

8.28%

11.33%

15.55

%

13.55%

13.15%

15.26%

StandardDeviation

1.92%

2.43%

1.64%

1.50%

1.87%

1.33%

1.68%

1.87%

JB

7,647.6***

8,969.91***

2,215.51***

7,665.15***

4,099.01***

12,962.42***

3,689.51***

11,181.49***

Q

0.61

14.46***

18.05***

5.79**

2.96*

16.46***

2.3

10.87***

1

0.08***

0.07***

0.08***

0.09***

0.07***

0.09***

0.08***

0.07***

1

0.91***

0.93***

0.90***

0.90***

0.93***

0.90***

0.91***

0.92***

MV1

2.26*

3.96***

4.27***

3.69***

3.65***

4.75***

2.48**

4.66***

MV2

1.34

2.48**

2.56**

1.83

2.24*

2.38*

1.53

2.33*

XLF

XLI

XLK

XLP

XLU

XLV

XLY

NumberofObservatio

ns

3,231

3,231

3,231

3,231

3,231

3,231

3,231

MeanReturn

0.00%

0.02%

-0.01%

0.02%

0.02%

0.02%

0.03%

MinimumReturn

-18.21%

-9.90%

-9.06%

-6.24%

-8.91%

-10.30%

-12.37%

MaximumReturn

15.19%

10.18%

14.92%

6.65%

11.43

%

11.38%

9.38%

StandardDeviation

2.14%

1.46%

1.76%

0.94%

1.26%

1.17%

1.53%

JB

20,270.92***

3,568.17***

4,993.46***

2,566.15***

12,536.5***

10,912.25***

3,619.71***

Q

38.72***

3.1*

5.28**

28.78***

16.88***

0.94

2.27

1

0.09***

0.09***

0.07***

0.08***

0.12***

0.10***

0.08***

1

0.91***

0.91***

0.92***

0.91***

0.87***

0.89***

0.92***

MV1

6.23***

1.96

3.51***

5.38***

4.11***

2.72**

2.57**

MV2

2.73**

1.32

2.20*

3.57***

1.98

1.33

1.57

Table4:Descriptivestatisticsofthelogreturnsofad

justedclosepricesfortheETFswithintheanalysisfortheperiodof6/24/00

through

5/1/13.***significantat1%

level,**significantat5%leveland*significantat10%level.

aJarque-Berastatistic.

bLjung-Boxstatistic.

cThis,andbelow,areGARCHstatistics.

21


22/31

Momentum Model Buy & Hold Momentum (Scaled)

Terminal Wealth $48,612.21 $52,892.72 $45,714.72

Total Dividends $4,812.95 $7,530.09 $4,876.19

Total Interest $1,019.14 $146.79 $702.23

Total Fees Paid ($8,520.04) ($8,884.08) ($8,278.81)

Annualized Return 6.47% 8.42% 5.95%

Annualized Volatility 12.02% 21.31% 12.13%

Sharpe Ratio 0.45 0.34 0.40

Omega 1.15 1.16 1.14

MaxDD -17.92% -58.03% -21.53%

Jarque-Bera 2737.24*** 10035.85*** 4009.89***

Ljung-Box 16.82*** 20.05*** 13.09***

ACF(1) -0.08 -0.08 -0.07

Alpha (Annualized) 2.72% 0.46% 2.17%

Beta 0.37*** 0.96*** 0.37***

Breusch-Pagan 0.65 2.95* 1.68

Breusch-Godfrey 17.92*** 0.26 13.73***

Jarque-Bera (Residuals) 5375.36*** 129174755.55*** 5031.28***

Trading Profit & Loss $26,300.16 $29,099.91 $23,415.11Total Commissions Paid ($2,105.8) ($2,369.1) ($1,252.45)

Mean Slippage -0.25% -0.29% -0.24%

StDev Slippage 0.15% 0.22% 0.14%

Number of Trades 294 23 133

Average Holding Period 227.57 4228.00 495.89

Mean Trade Return 19.16% 35.65% 30.24%

StDev Trade Return 24.73% 42.15% 35.32%

Table 5: Performance statistics for the Absolute Momentum (AM) strategy with monthly

rebalancing and $25,000 starting capital for the period of October 3, 2001 through May 1, 2013.*** significant at 1% level, ** significant at 5% level and * significant at 10% level.

22


23/31


Terminal Wealth $46,391.61 $53,743.67 $39,389.95

Total Dividends $5,032.48 $7,624.32 $4,371.57

Total Interest $914.71 $137.75 $591.59

Total Fees Paid ($8,430.38) ($8,987.34) ($7,729.26)




Omega 1.15 1.16 1.14

MaxDD -24.58% -57.81% -26.65%

Jarque-Bera 1966.49*** 9721.38*** 6687.39***

Ljung-Box 8.51*** 19.96*** 15.27***

ACF(1) -0.05 -0.08 -0.07


Beta 0.37*** 0.96*** 0.38***

Breusch-Pagan 0.04 2.53 3.57*

Breusch-Godfrey 12.64*** 0.28 16.2***

Jarque-Bera (Residuals) 4365.3*** 117015161.65*** 6482***

Trading Profit & Loss $23,874.81 $29,968.94 $17,156.04Total Commissions Paid ($1,164) ($2,240.4) ($858.5)

Mean Slippage -0.26% -0.27% -0.25%






Table 6: Performance statistics for the Absolute Momentum (AM) strategy with quarterly


23


24/31


Terminal Wealth $38,583.03 $52,892.72 $33,325.94

Total Dividends $5,982.29 $7,530.09 $5,938.25

Total Interest $98.07 $146.79 $103.03

Total Fees Paid ($7,621.41) ($8,884.08) ($7,096.18)




Omega 1.12 1.16 1.16

MaxDD -57% -58.03% -62.19%

Jarque-Bera 3796.16*** 10035.85*** 7613.2***

Ljung-Box 21.32*** 20.05*** 16.52***

ACF(1) -0.09 -0.08 -0.08

Alpha (Annualized) -1.81% 0.46% -3.1%*

Beta 0.90*** 0.96*** 0.89***

Breusch-Pagan 5.6** 2.95* 0.36

Breusch-Godfrey 27.85*** 0.26 13.6***


Trading Profit & Loss $15,124.09 $29,099.91 $9,380.84Total Commissions Paid ($8,007.45) ($2,369.1) ($5,387.9)

Mean Slippage -0.27% -0.29% -0.27%






Table 7: Performance statistics for the Cross-Sectional Momentum (CM) strategy with monthly


24


25/31


Terminal Wealth $44,276.25 $53,743.67 $42,005.43

Total Dividends $6,799.67 $7,624.32 $6,634.06

Total Interest $89.18 $137.75 $97.03

Total Fees Paid ($8,420.81) ($8,987.34) ($7,885.52)




Omega 1.14 1.16 1.18

MaxDD -58.98% -57.81% -60.75%

Jarque-Bera 4149.34*** 9721.38*** 7415.53***

Ljung-Box 17.49*** 19.96*** 17.18***

ACF(1) -0.08 -0.08 -0.08

Alpha (Annualized) -0.68% 0.60% -1.12%

Beta 0.92*** 0.96*** 0.89***

Breusch-Pagan 6.50** 2.53 0.27

Breusch-Godfrey 27.44*** 0.28 16.33***


Trading Profit & Loss $20,808.21 $29,968.94 $18,159.85Total Commissions Paid ($5,538) ($2,240.4) ($4,089.3)

Mean Slippage -0.28% -0.27% -0.26%






Table 8: Performance statistics for the Cross-Sectional Momentum (CM) strategy with quarterly


25


26/31


Terminal Wealth $53,747.15 $52,892.72 $45,507.12

Total Dividends $6,358.33 $7,530.09 $5,887.12

Total Interest $458.16 $146.79 $395.17

Total Fees Paid ($9,105.93) ($8,884.08) ($8,043.96)




Omega 1.13 1.16 1.14

MaxDD -26.10% -58.03% -31.13%Jarque-Bera 1958.27*** 10035.85*** 2432.56***

Ljung-Box 21.33*** 20.05*** 14.38***

ACF(1) -0.09 -0.08 -0.07


Beta 0.51*** 0.96*** 0.46***

Breusch-Pagan 1.64 2.95* 0.41

Breusch-Godfrey 26.19*** 0.26 14.64***


Trading Profit & Loss $31,036.59 $29,099.91 $22,268.8

Total Commissions Paid ($6,219.5) ($2,369.1) ($3,581.05)

Mean Slippage -0.25% -0.29% -0.24%






Table 9: Performance statistics for the Cross-Sectional, Absolute Momentum (CAM) strategywith monthly rebalancing and $25,000 starting capital for the period of October 3, 2001 through

May 1, 2013. *** significant at 1% level, ** significant at 5% level and * significant at 10%level.

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Terminal Wealth $59,867.75 $53,743.67 $44,931.13

Total Dividends $7,118.02 $7,624.32 $5,639.73

Total Interest $424.18 $137.75 $349.07

Total Fees Paid ($9,858.6) ($8,987.34) ($7,825.25)




Omega 1.14 1.16 1.15

MaxDD -26.13% -57.81% -30.27%Jarque-Bera 2033.82*** 9721.38*** 1714.99***

Ljung-Box 14.91*** 19.96*** 13.83***

ACF(1) -0.07 -0.08 -0.07


Beta 0.51*** 0.96*** 0.46***

Breusch-Pagan 0.62 2.53 0.00

Breusch-Godfrey 22.07*** 0.28 12.54***


Trading Profit & Loss $37,184.16 $29,968.94 $21,767.58

Total Commissions Paid ($3,452) ($2,240.4) ($2,148.9)

Mean Slippage -0.26% -0.27% -0.25%






Table 10: Performance statistics for the Cross-Sectional, Absolute Momentum (CAM) strategywith quarterly rebalancing and $25,000 starting capital for the period of October 3, 2001 through

May 1, 2013. *** significant at 1% level, ** significant at 5% level and * significant at 10%level.

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Figure 1: Omega(0) statistics of varying thresholds, U= {1,...,23}, for the Cross-SectionalMomentum (CM) strategy with monthly rebalancing and employing our momentum model. The

trendline represents the Omega(0)=1.1216 for the strategy using our DT(IHH) algorithm.

Figure 2: Omega(0) statistics of varying thresholds, U= {1,...,23}, for the Cross-SectionalMomentum (CM) strategy with quarterly rebalancing and employing our momentum model. The

trendline represents the Omega(0)=1.1359 for the strategy using our DT(IHH) algorithm.

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Figure 3: Omega(0) statistics of varying thresholds, U= {1,...,23}, for the Cross-Sectional,Absolute Momentum (CAM) strategy with monthly rebalancing and employing our momentum

model. The trendline represents the Omega(0)=1.1290 for the strategy using our DT(IHH)algorithm.

Figure 4: Omega(0) statistics of varying thresholds, U= {1,...,23}, for the Cross-Sectional,Absolute Momentum (CAM) strategy with quarterly rebalancing and employing our momentum

model. The trendline represents the Omega(0)=1.1439 for the strategy using our DT(IHH)algorithm.

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CM

Monthly

CM

Quarterly

CAM

Monthly

CAM

Quarterly

DT(IHH)

DT(HH)

DT(IHH)

DT(HH)

DT

(IHH)

DT(HH)

DT(IHH

)

DT(HH)

AnnualizedReturn

5.6

9%

5.6

2%

6.9

6%

6.8

9%

7.9

0%

6.8

1%

8.8

4%

6.6

3%

AnnualizedVolatilit

y

20.6

8%

21.2

3%

21.0

3%

21.2

3%

15

.98%

15.2

5%

16.0

5%

15.1

2%

SharpeRatio

0.2

2

0.2

1

0.2

8

0.2

7

0.4

2

0.3

7

0.4

8

0.3

6

Omega

1.1

2

1.1

4

1.1

4

1.1

4

1.1

3

1.1

5

1.1

4

1.1

5

MaxDD

-57.0

0%

-57.0

3%

-58.9

8%

-58.7

4%

-26.1

0%

-24.4

5%

-26.1

3%

-29.3

2%

Alpha(Annualized)

-1.8

1%

-2.1

6%

-0.6

8%

-0.9

0%

3.1

6%

2.3

1%

4.0

8%

2.1

6%

Beta

0.9

0***

0.9

4***

0.9

2***

0.9

4***

0.51***

0.4

8***

0.5

1***

0.4

7***

NumberofTrades

369

384

202

194

333

378

175

182

AverageHoldingPe

riod

119.4

8

139.9

6

217.9

9

277.6

7

111.1

7

111.0

4

206.70

220.6

3

MeanTradeReturn

8.0

8%

6.2

1%

9.3

2%

8.2

5%

8.5

8%

7.7

5%

10.7

0%

10.5

0%

StDevTradeReturn

15.7

8%

17.3

2%

21.0

7%

20.7

3%

13

.69%

12.8

6%

19.5

8%

17.7

8%

Table11:Comparisonofrank-basedstrategiesusing

themomentummodelwhicheitheremploytheDynamicThres

hold(Inverse

Herfindahl-Hirschman

)[DT(IHH)]orDynamicThreshold(Herfindahl-Hirschman)[DT(HH)]todeterminetherankthreshold.***

significantat1%level,**s

ignificantat5%leveland*sig

nificantat10%level.

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Figure 5: Equally-weighted index values for the period of October 3, 2001 through May 1, 2013with a starting value of 1.

Figure 6: Threshold values using the Dynamic Threshold (Inverse Herfindahl-Hirschman) at amonthly frequency for the period of October 3, 2001 through May 1, 2013.

Figure 7: Threshold values using the Dynamic Threshold (Herfindahl-Hirschman) at a monthlyfrequency for the period of October 3, 2001 through May 1, 2013.

The Analysis of a Momentum Model and Accompanying Portfolio Strategies

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