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## Duke Investment Analytics

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**Interfractile Migration Tracking**Duke Investment Analytics Claudio Aritomi Sam Ding Mak Pitke Marcus Shaw Brian Wachob**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.**Purpose of Study**“Investigate whether interfractile migration tracking can improve performance in a sort-based stock selection strategy.”**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.**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.**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**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.**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.**Dividend Yield - Quintile Performance**Returns & Alpha Value-Weighted Equal-Weighted Annualized Return Alpha**Dividend Yield - Quintile Performance**Volatility & Beta Value-Weighted Equal-Weighted Std. Dev. of Monthly Returns Beta on Market (S&P 500)**Dividend Yield – F5-F1 Time Series, VW**Cumulative**Dividend Yield – F5-F1 Time Series, EW**Cumulative**Dividend Yield – F5-F1 Time Series**All Fractiles, 12-Month Windows Value-Weighted Equal-Weighted Year-By-Year Trailing Twelve Months**Dividend Yield – F5-F1 Returns Distributions**Value-Weighted Equal-Weighted Monthly Returns ln Monthly Returns**Dividend Yield – F5-F1 Returns Distributions**Summary Statistics Value-Weighted Equal-Weighted Monthly Returns ln Monthly Returns**Book to Price - Quintile Performance**Returns & Alpha Value-Weighted Equal-Weighted Annualized Return Alpha**Book to Price - Quintile Performance**Volatility & Beta Value-Weighted Equal-Weighted Std. Dev. of Monthly Returns Beta on Market (S&P 500)**Book to Price - F5-F1 Time Series, VW**Cumulative**Book to Price - F5-F1 Time Series, EW**Cumulative**Book to Price – F5-F1 Time Series**All Fractiles, 12-Month Windows Value-Weighted Equal-Weighted Year-By-Year Trailing Twelve Months**Book to Price – F5-F1 Returns Distributions**Value-Weighted Equal-Weighted Monthly Returns ln Monthly Returns**Book to Price – F5-F1 Returns Distributions**Summary Statistics Value-Weighted Equal-Weighted Monthly Returns ln Monthly Returns**Trailing Earnings Yield - Quintile PerformanceTrailing**Twelve Months Earnings Yield (TEY) Returns & Alpha Value-Weighted Equal-Weighted Annualized Return Alpha**Trailing Earnings Yield - Quintile PerformanceTrailing**Twelve Months Earnings Yield (TEY) Volatility & Beta Value-Weighted Equal-Weighted Std. Dev. of Monthly Returns Beta on Market (S&P 500)**Trailing Earnings Yield - F5-F1 Time Series, VWTrailing**Twelve Months Earnings Yield (TEY) Cumulative**Trailing Earnings Yield - F5-F1 Time Series, EWTrailing**Twelve Months Earnings Yield (TEY) Cumulative**Trailing Earnings Yield - F5-F1 Time SeriesTrailing Twelve**Months Earnings Yield (TEY) All Fractiles, 12-Month Windows Value-Weighted Equal-Weighted Year-By-Year Trailing Twelve Months**Trailing Earnings Yield - F5-F1 Returns**DistributionsTrailing Twelve Months Earnings Yield (TEY) Value-Weighted Equal-Weighted Monthly Returns ln Monthly Returns**Trailing Earnings Yield - F5-F1 Returns**DistributionsTrailing Twelve Months Earnings Yield (TEY) Summary Statistics Value-Weighted Equal-Weighted Monthly Returns ln Monthly Returns**Forward Earnings Yield**• Four different definitions of forward earnings • Mean I/B/E/S earnings forecast for “Next Twelve Months” • 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. • 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. • 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.**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.**Forward Earnings Yield - Quintile PerformanceForecast Next**12 Mos. Earnings Yield (FEY) Returns & Alpha Value-Weighted Equal-Weighted Annualized Return Alpha**Forward Earnings Yield - Quintile PerformanceForecast Next**12 Mos. Earnings Yield (FEY) Volatility & Beta Value-Weighted Equal-Weighted Std. Dev. of Monthly Returns Beta on Market (S&P 500)**Forward Earnings Yield - F5-F1 Time Series, VWForecast Next**12 Mos. Earnings Yield (FEY) Cumulative**Forward Earnings Yield - F5-F1 Time Series, EWForecast Next**12 Mos. Earnings Yield (FEY) Cumulative**Forward Earnings Yield -F5-F1 Time SeriesForecast Next 12**Mos. Earnings Yield (FEY) All Fractiles, 12-Month Windows Value-Weighted Equal-Weighted Year-By-Year Trailing Twelve Months**Forward Earnings Yield - F5-F1 Returns DistributionsForecast**Next 12 Mos. Earnings Yield (FEY) Value-Weighted Equal-Weighted Monthly Returns ln Monthly Returns**Forward Earnings Yield - F5-F1 Returns DistributionsForecast**Next 12 Mos. Earnings Yield (FEY) Summary Statistics Value-Weighted Equal-Weighted Monthly Returns ln Monthly Returns**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) re 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.**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, re, for each firm (based on the market capitalization observed in the market). • Estimates of leverage and cost of debt, rd, 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).**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.**Implied Cost of Capital -Industry-Normalization?**• The implied cost of capital for each firm should theoretically reflect the inherent risk of its underlying assets, ra. • 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.**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.**“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 re 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.**“Implied Cost of Capital” ≡ ICCTwo 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 E1 in the preceding slide): • ICC1: 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). • ICC2: 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 ICC1 (definition 1 for E1). • 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.**ICC1 - Quintile Performance**Returns & Alpha Value-Weighted Equal-Weighted Annualized Return Alpha**ICC1 - Quintile Performance**Volatility & Beta Value-Weighted Equal-Weighted Std. Dev. of Monthly Returns Beta on Market (S&P 500)**ICC1 - F5-F1 Time Series, VW**Cumulative**ICC1 - F5-F1 Time Series, EW**Cumulative**ICC1 –F5-F1 Time Series**All Fractiles, 12-Month Windows Value-Weighted Equal-Weighted Year-By-Year Trailing Twelve Months