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Duke Investment Analytics Claudio Aritomi Sam Ding Mak Pitke Marcus Shaw Brian Wachob Interfractile Migration Tracking

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Summary Identify a “Primary Factor” for Use as a Basis Univariate Sort Define and Quantify Interfractile Migration (IM) IM Trending IM Volatility Study Sequential Sorts on Primary Factor, then IM Define Trading Strategies Test Out-of-Sample Conclusions Recommendations For Further Research Note that this text-intensive version of the slide deck is an extended version intended for independent study. A condensed version is intended for presentation purposes.

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Purpose of Study “Investigate whether interfractile migration tracking can improve performance in a sort- based stock selection strategy.”

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A Note To Those Reading This Slide Deck… The notes that accompany these slides (viewable in PowerPoint edit mode) contain additional information that is not entirely conveyed in the slides themselves. Please examine these notes when considering the research presented here.

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Interfractile Migration - Definition Define 2 metrics to quantify “interfractile migration” (IM) Interfractile Migration Trending (IMT) over recent periods, measure the trend of each stock’s movements through fractiles of the primary factor Interfractile Migration Volatility (IMV) over recent periods, measure the volatility of each stock’s movements through fractiles of the primary factor Define fractile resolution (with respect to primary factor) We used 10 fractiles (deciles), as segregated by Factset’s UDECILE() function.

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Identify a Basis Univariate Sort Candidates: Dividend Yield Book-to-Price Historical (Trailing) Earnings Yield Forward Earnings Yield I/B/E/S Mean Next Twelve Months I/B/E/S Mean FY1 I/B/E/S Median NTM, FY1 I/B/E/S Median FY2 Implied Cost of Capital

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Methodology FactSet quintile sorts Monthly rebalancing, 1-month holding period In-sample period: 1/31/87-11/31/01* Out-of-sample period: 12/31/01-12/31/04 Universe US-listed NYSE, NASDAQ, AMEX Top 60% by market cap Convention: Low factor values are always assigned to low-numbered fractiles When historical data necessary to evaluate the univariate sorting factor for a given stock is unavailable, that stock is excluded from the universe for that backtest date. * Note that using 31 as the last day of the month when specifying the date range in Factset is necessary—even when there is no 31st day of the specified month. If not used in this way, lagged variables may not work properly in alpha tester.

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Results of Univariate Sorts The following slides present some data evaluating the performance of selected univariate sorts. A far more detailed array of data sets and analyses evaluating these univariate sorts (and others) are contained in the Excel workbook files accompanying this PowerPoint presentation.

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Dividend Yield - Quintile Performance Value-Weighted Equal-Weighted Annualized ReturnAlpha Returns & Alpha

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Dividend Yield - Quintile Performance Value-Weighted Equal-Weighted Std. Dev. of Monthly ReturnsBeta on Market (S&P 500) Volatility & Beta

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Dividend Yield – F5-F1 Time Series, VW Cumulative

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Dividend Yield – F5-F1 Time Series, EW Cumulative

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Dividend Yield – F5-F1 Time Series Value-Weighted Equal-Weighted Year-By-YearTrailing Twelve Months All Fractiles, 12-Month Windows

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Dividend Yield – F5-F1 Returns Distributions Value-Weighted Equal-Weighted Monthly Returnsln Monthly Returns

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Dividend Yield – F5-F1 Returns Distributions Value-Weighted Equal-Weighted Monthly Returnsln Monthly Returns Summary Statistics

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Book to Price - Quintile Performance Value-Weighted Equal-Weighted Annualized ReturnAlpha Returns & Alpha

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Book to Price - Quintile Performance Value-Weighted Equal-Weighted Std. Dev. of Monthly ReturnsBeta on Market (S&P 500) Volatility & Beta

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Book to Price - F5-F1 Time Series, VW Cumulative

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Book to Price - F5-F1 Time Series, EW Cumulative

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Book to Price – F5-F1 Time Series Value-Weighted Equal-Weighted Year-By-YearTrailing Twelve Months All Fractiles, 12-Month Windows

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Book to Price – F5-F1 Returns Distributions Value-Weighted Equal-Weighted Monthly Returnsln Monthly Returns

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Book to Price – F5-F1 Returns Distributions Value-Weighted Equal-Weighted Monthly Returnsln Monthly Returns Summary Statistics

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Trailing Earnings Yield - Quintile Performance Trailing Twelve Months Earnings Yield (TEY) Value-Weighted Equal-Weighted Annualized ReturnAlpha Returns & Alpha

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Trailing Earnings Yield - Quintile Performance Trailing Twelve Months Earnings Yield (TEY) Volatility & Beta Value-Weighted Equal-Weighted Std. Dev. of Monthly ReturnsBeta on Market (S&P 500)

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Trailing Earnings Yield - F5-F1 Time Series, VW Trailing Twelve Months Earnings Yield (TEY) Cumulative

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Trailing Earnings Yield - F5-F1 Time Series, EW Trailing Twelve Months Earnings Yield (TEY) Cumulative

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Trailing Earnings Yield - F5-F1 Time Series Trailing Twelve Months Earnings Yield (TEY) Value-Weighted Equal-Weighted Year-By-YearTrailing Twelve Months All Fractiles, 12-Month Windows

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Trailing Earnings Yield - F5-F1 Returns Distributions Trailing Twelve Months Earnings Yield (TEY) Value-Weighted Equal-Weighted Monthly Returnsln Monthly Returns

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Trailing Earnings Yield - F5-F1 Returns Distributions Trailing Twelve Months Earnings Yield (TEY) Value-Weighted Equal-Weighted Monthly Returnsln Monthly Returns Summary Statistics

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Forward Earnings Yield Four different definitions of forward earnings A. Mean I/B/E/S earnings forecast for “Next Twelve Months” B. Mean I/B/E/S earnings forecast for “Next Twelve Months”. If this data is unavailable, mean I/B/E/S earnings forecast for the current fiscal year is used instead. C. Median I/B/E/S earnings forecast for “Next Twelve Months”. If this data is unavailable, median I/B/E/S earnings forecast for the current fiscal year is used instead. D. Median I/B/E/S earnings forecast for forward fiscal year number 2. We backtested our univariate screening and sorting methodology using each of these definitions to contribute the numerator to our earnings yield computation.

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Forward Earnings Yield Performance across these four factor definitions is similar. Median analyst earnings estimates appear preferable to means. Definition C appears to generate the best quintile sorts. Still, closer scrutiny of the results pertaining to these four definitions is warranted and there still remains ample room for improvement in these factor definitions. We leave this for future research. We choose to focus our analysis on the backtest results using definition C for Forward Earnings Yield: Median I/B/E/S earnings forecast for “Next Twelve Months”. If this data is unavailable, median I/B/E/S earnings forecast for the current fiscal year is used instead.

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Forward Earnings Yield - Quintile Performance Forecast Next 12 Mos. Earnings Yield (FEY) Value-Weighted Equal-Weighted Annualized ReturnAlpha Returns & Alpha

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Forward Earnings Yield - Quintile Performance Forecast Next 12 Mos. Earnings Yield (FEY) Volatility & Beta Value-Weighted Equal-Weighted Std. Dev. of Monthly ReturnsBeta on Market (S&P 500)

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Forward Earnings Yield - F5-F1 Time Series, VW Forecast Next 12 Mos. Earnings Yield (FEY) Cumulative

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Forward Earnings Yield - F5-F1 Time Series, EW Forecast Next 12 Mos. Earnings Yield (FEY) Cumulative

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Forward Earnings Yield - F5-F1 Time Series Forecast Next 12 Mos. Earnings Yield (FEY) Value-Weighted Equal-Weighted Year-By-YearTrailing Twelve Months All Fractiles, 12-Month Windows

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Forward Earnings Yield - F5-F1 Returns Distributions Forecast Next 12 Mos. Earnings Yield (FEY) Value-Weighted Equal-Weighted Monthly Returnsln Monthly Returns

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Forward Earnings Yield - F5-F1 Returns Distributions Forecast Next 12 Mos. Earnings Yield (FEY) Value-Weighted Equal-Weighted Monthly Returnsln Monthly Returns Summary Statistics

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Implied Cost of Capital - Idea Idea: Base a univariate sorting factor on an estimation of implied cost of capital. Implied cost of capital should be a far more comprehensive relative valuation metric than earnings yield, dividend yield, or book-to-price. Earnings yield can be viewed as an extremely simplified expression of implied cost of equity. Common valuation models can be simplified to yield the following relations if extreme over-simplifying assumptions are made (i.e. firms have reached steady- state and will not grow) r e is the cost of equity, which can be equated to cost of capital if another extreme over- simplifying assumption is made: all firms are 100% equity financed.

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Implied Cost of Capital - Implementation A residual income (i.e. “abnormal earnings”) valuation model can serve as the basis for estimating an implied cost of equity, r e, for each firm (based on the market capitalization observed in the market). Estimates of leverage and cost of debt, r d, for each firm can be integrated with the residual income model to estimate implied cost of capital for each firm. All firms could be ranked on implied cost of capital. The firms with the highest implied cost of capital might be considered undervalued (long candidates). Those with the lowest implied cost of capital might be considered overvalued (short candidates).

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Implied Cost of Capital - Limitations It is obviously false to assert that the implied cost of capital for all firms should be equivalent in expectation. However, the assertion is similarly flawed for the other valuation metrics previously examined (earnings yield, dividend yield, book-to-price). Still, an advantageous informational advantage seems to have been found (for forward earnings yield, for example) Differing expected future growth rates and patterns, payout ratios, and capital structures are sources of differing expected earnings yield. Implied cost of capital can take all of these firm-specific features into account.

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Implied Cost of Capital - Industry-Normalization? The implied cost of capital for each firm should theoretically reflect the inherent risk of its underlying assets, r a. Thus, it probably makes more sense to compare any given firm’s implied cost of capital against that of other firms in the same industry. Of course, by the same logic, industry normalization might improve performance of other valuation metrics such as earnings yield, dividend yield, and book-to- price.

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Implied Cost of Capital - Implementation Challenges Estimating even implied cost of equity (let alone implied cost of capital) for each firm requires numeric methods. FactSet’s Alpha Testing module does not appear capable of implementing the necessary algorithms. Time limits did not permit us to write our own code to replicate the functionality of FastSet’s Alpha Testing and implement numeric methods to solve for implied cost of capital. However, we believe we have determined that the implementation of this backtest is possible with FQL (FactSet Query Language) and even in Excel via Visual Basic and the FastSet Excel Plug-In.

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“Implied Cost of Capital” - Ours Is A Weak Approximation Though we present the idea here, we did not implement a strong evaluation of implied cost of capital as a univariate sorting factor. We did implement an extreme simplification of the idea using the following relation to grossly approximate implied cost of equity: Note that we chose 5.5% as the nominal terminal growth rate, g ∞, for all firms. This relation could be implemented in FactSet’s Alpha Testing because it is solvable for r e by the quadratic equation. Note that though we have called this “implied cost of capital,” it is in truth a highly over-simplified implementation of what is typically meant by “implied cost of capital.” This implementation does little more than achieve a reasonable integration of forward earnings yield and book-to-price into one metric.

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“Implied Cost of Capital” ≡ ICC Two methods of calculation Recognizing that our “implied cost of capital” had become little more than an integration of forward earnings yield and book- to-price into a single metric, we experimented with two definitions of forward earnings (denoted E 1 in the preceding slide): 1. ICC 1 : Median I/B/E/S earnings forecast for “Next Twelve Months”. If this data is unavailable, median I/B/E/S earnings forecast for the current fiscal year is used instead (as in FEY definition C). 2. ICC 2 : Median I/B/E/S earnings forecast for forward fiscal year number 2 (as in FEY definition D). (Idea/justification: Use the most forward earnings forecast to extrapolate into perpetuity even though this earnings estimate should be discounted more heavily.) The following slides focus on the backtest results using ICC 1 (definition 1 for E 1 ). This definition was chosen because backtest results were similar across both definitions for ICC, but definition 1 is more theoretically valid in the highly simplified valuation expression presented in the previous slide.

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ICC 1 - Quintile Performance Value-Weighted Equal-Weighted Annualized ReturnAlpha Returns & Alpha

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ICC 1 - Quintile Performance Value-Weighted Equal-Weighted Std. Dev. of Monthly ReturnsBeta on Market (S&P 500) Volatility & Beta

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ICC 1 - F5-F1 Time Series, VW Cumulative

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ICC 1 - F5-F1 Time Series, EW Cumulative

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ICC 1 – F5-F1 Time Series Value-Weighted Equal-Weighted Year-By-YearTrailing Twelve Months All Fractiles, 12-Month Windows

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ICC 1 – F5-F1 Returns Distributions Value-Weighted Equal-Weighted Monthly Returnsln Monthly Returns

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ICC 1 – F5-F1 Returns Distributions Value-Weighted Equal-Weighted Monthly Returnsln Monthly Returns Summary Statistics

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Univariate Sorts - Summary FEY_C ICC 2 ICC 1 FEY_D FEY_B TEY FEY_A B/P Div.Yld.

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Selected Primary Factor Forward Earnings Yield Definition C was chosen Even though our backtesting results slightly favor ICC as a univariate sort factor (especially in value-weighted portfolios), we selected FEY as the primary (basis) factor upon which to experiment with interfractile migration tracking. FEY was selected because its performance was similar to that of ICC, but its interpretation is more intuitive and its use in financial analyses more widespread.

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Interfractile Migration - Definition Define 2 metrics to quantify “interfractile migration” (IM) Interfractile Migration Trending (IMT) over recent periods, measure the trend of each stock’s movements through fractiles of the primary factor Interfractile Migration Volatility (IMV) over recent periods, measure the volatility of each stock’s movements through fractiles of the primary factor Define fractile resolution (with respect to primary factor) We used 10 fractiles (deciles), as segregated by Factset’s UDECILE() function.

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Interfractile Migration - Definition IM Trending, IMT = 0 IM Volatility, IMV = 0 Red Dot Stock

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Interfractile Migration - Definition IMT is highly positive IMV is moderate Green Circle Stock

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Interfractile Migration - Definition IMT is only slightly positive IMV is moderate Black X Stock

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Interfractile Migration - Definition IMT is negligible IMV is very high Blue Square Stock

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Interfractile Migration - Quantification We implemented two variants of each metric IMT SMA3(PFD)-SMA12(PFD) Trailing triangular-weighted average of previous 11 ΔPFD’s IMV Mean of previous 11 |ΔPFD|’s Trailing triangular-weighted average of previous 11 |ΔPFD|’s PFD i,t – “Primary Factor Decile”; the decile into which a stock is binned when sorted on the primary factor Note that we used some special techniques to estimate IMT and IMV for stocks that were missing primary factor (forward earnings yield) data in some periods within the last 12 months. Refer to our report for more details.

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Interfractile Migration - Illustration To illustrate, let’s focus on these two particular variants of our IM metrics: IMT: SMA3(PFD)-SMA12(PFD) IMV: Mean of previous 11 |ΔPFD|’s Note that we also applied our alternate definitions of IMT and IMV, but at least upon a first look at charts illustrating performance across the 15 sub-fractiles, there does not appear to be significant additional information contributed by these alternate definitions.

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Interfractile Migration - Illustration IMT Equal-WeightedIMT Value-Weighted IMV Equal-WeightedIMV Value-Weighted We studied both equal-weighted and value-weighted portfolios. Findings are roughly similar across both intra-fractile weighting schemes. Thus, in these slides we focus on value-weighted portfolios. Liquidity issues tend to make these more easily implementable. Refer directly to Excel source files for details of performance in equal-weighted portfolios.

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Interfractile Migration - Implementation We analyze IM factor performance within each Forward Earnings Yield (FEY) quintile: Two-step sequential sort 1 st Sort: Quintiles on primary factor (FEY) 2 nd Sort: Sub-Trintiles on IMT or IMV

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Interfractile Migration Trending Raw Mean Geometric Annualized Return

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Interfractile Migration Trending Market Risk-Adjusted Monthly Alpha

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Interfractile Migration Trending Standard Deviation of Monthly Returns (Sigma)

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Interfractile Migration Trending High correlations between high and low IMT fractiles (1 vs. 3; 4 vs. 6) suggest low variance spread trade strategy. Lower correlations between high and low FEY fractiles (1,2,3 vs. 13,14,15) suggest higher variance spread trade strategy. Interfractile Correlation

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IMT – Trading Strategies Control Portfolio Long the 3 high FEY fractiles (13,14,15) Short the 3 low FEY fractiles (1,2,3) IMT Isolation for Low FEY Portfolio Long the low IMT fractiles in low FEY quintiles (1,4) Short the high IMT fractiles in low FEY quintiles (3,6) Hybrid Portfolio Long fractiles 13,14,15,1 Short fractiles 2,3,6 When building simulated portfolios that combine fractiles, value-weighting was used within fractiles. However, within a multi-fractile long (or short) portfolio, each fractile was equally weighted with monthly rebalancing. This methodology was implemented solely for computational convenience and could be reconsidered in an alternate analysis.

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IMT – Hybrid Portfolio Composition Various combinations of the 15 sub-fractile portfolios were considered and examined as candidate “hybrid portfolios.” Qualitative justification of long{13, 14, 15, 1} / short{2, 3, 6} Fractiles 2, 3, and 6 showed the lowest raw mean returns and alphas in-sample. Fractile 1 is highly correlated with fractiles 2, 3, and 6, but with a higher mean return. It is more highly correlated with the aggregated short fractiles than fractile 4 (and less correlated with its fellow long fractiles 13, 14, and 15). Of course, low correlations among all-long or all-short portfolio positions result in desirable lower overall volatility. The same is true for high correlations between hedge portfolio positions. Refer to source spreadsheets for more detail pertaining to assorted experimental candidate hybrid portfolios. Note that we experimented with portfolio optimization techniques in search of an optimal hybrid portfolio definition, paying special attention to the non-normal nature of monthly fractile return distributions. More focus could be given to these methods in future studies.

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IMT – Trading Strategies

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Interfractile Migration Volatility Raw Mean Geometric Annualized Return

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Interfractile Migration Volatility Market Risk-Adjusted Monthly Alpha

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Interfractile Migration Volatility Standard Deviation of Monthly Returns (Sigma)

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Interfractile Migration Volatility High correlations between high and low IMT fractiles (1 vs. 3; 4 vs. 6) suggest low variance spread trade strategy. Lower correlations between high and low FEY fractiles (1,2,3 vs. 13,14,15) suggest higher variance spread trade strategy. Interfractile Correlation

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IMV – Trading Strategies Control Portfolio Long the 3 high FEY fractiles (13,14,15) Short the 3 low FEY fractiles (1,2,3) IMV Isolation for Low FEY Portfolio Long the high IMV fractiles in low FEY quintiles (3,6) Short the low IMV fractiles in low FEY quintiles (1,4) Hybrid Portfolio Long fractiles 3,13,14,15 Short fractiles 1,2,4 When building simulated portfolios that combine fractiles, value-weighting was used within fractiles. However, within a multi-fractile long (or short) portfolio, each fractile was equally weighted with monthly rebalancing.

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IMV – Hybrid Portfolio Composition Various combinations of the 15 sub-fractile portfolios were considered and examined as candidate “hybrid portfolios.” Qualitative justification of long{3, 13, 14, 15} / short{1, 2, 4} Fractiles 1, 2, and 4 showed the lowest raw mean returns and alphas in-sample. Fractile 3 is highly correlated with fractiles 1, 2, and 4, but with a higher mean return. It is more highly correlated with each of these short fractiles than fractile 6 (and less correlated with its fellow long fractiles 13, 14, and 15). Of course, low correlations among all-long or all-short portfolio positions result in desirable lower overall volatility. The same is true for high correlations between hedge portfolio positions. Refer to source spreadsheets for more detail pertaining to assorted experimental candidate hybrid portfolios. Note that we experimented with portfolio optimization techniques in search of an optimal hybrid portfolio definition, paying special attention to the non-normal nature of monthly fractile return distributions. More focus could be given to these methods in future studies.

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IMV – Trading Strategies

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In-Sample Summary IM Trading Strategies

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Out of Sample – IMT Raw Mean Geometric Annualized Return

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Out of Sample – IMT Market Risk-Adjusted Monthly Alpha

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Out of Sample – IMT

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Out of Sample – IMV Raw Mean Geometric Annualized Return

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Out of Sample – IMV Market Risk-Adjusted Monthly Alpha

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Out of Sample – IMV

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Conclusions Among the valuation-based univariate screens examined, Forward Earnings Yield and ICC were the best. For unrebalanced FEY long F5 - short F1, monthly alpha =.87%. A secondary sequential sort on IM trending (or volatility) appeared to add information (in-sample) regarding returns in low FEY quintiles. IM-based enhanced trading strategies looked promising in-sample. Improved variance and total returns over the control strategy looked possible.

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Conclusions Out of sample, these trading strategies underperformed. Perhaps the patterns observed in-sample were mere data artifacts. Alternately, perhaps the period happens to be a poor period for these IM- based strategies within FEY sorts. Perhaps the pattern observed in-sample will re- emerge in future months.

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Recommendations For Future Research Monitor continuing out-of-sample IMT and IMV performance for Forward Earnings Yield Examine IM-based sorts in other Primary Factors (besides FEY– perhaps on an improved implied cost of capital factor, or an industry-normalized factor, or a non-valuation-based factor) Examine sensitivity to IM metric definitions (i.e. lengths of trailing periods, weightings, and fractile resolution). Study a combination of trending and volatility elements of IM into a single sorting factor. Study a third IM metric definition: IM Permanence– weighted (?) trailing average of differences with current fractile membership. Maybe: Incorporate transaction costs into analysis

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Recommendations For Future Research Rigorously implement backtesting of implied cost of capital as a univariate factor sort. Introduce industry-normalization to definitions of basis univariate sorting factors (seems especially pertinent for these valuation- ratio-grounded metrics). Study more closely any rebalancing effects on these long/short portfolios. Apparent rebalancing effects observed in this study suggest that there may exist “factor momentum” where factor portfolio performance in a given month is predictive of factor portfolio performance in the subsequent month. Implementation of these analyses in a regression-based framework rather than fractile sorts would enable better integration into multivariate stock selection models as well as additional factor performance diagnostics. Also, refer to slide notes for specific suggested improvements to our screening and sorting methodologies as executed in this analysis.

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