Duke Investment Analytics

Claudio AritomiSam DingMak PitkeMarcus ShawBrian Wachob

Interfractile Migration Tracking

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

Annualized Return, % -- Div. Yld VW Univariate Factor Performance

-1- -2- -3- -4- -5-Fractile

Alpha, Monthly % -- Div. Yld VW Univariate Factor Performance

-1- -2- -3- -4- -5-

Fractile

Annualized Return, % -- Div.Yld. EW Univariate Factor Performance

-1- -2- -3- -4- -5-Fractile

e-Weig

Annualized Return Alpha

Alpha, Monthly % -- Div.Yld. EW Univariate Factor Performance

-1- -2- -3- -4- -5-

Fractile

Returns & Alpha

Dividend Yield - Quintile Performance

Std. Dev. of Monthly Returns -- Div.Yld. VW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Beta, on Market (S&P 500) -- Div.Yld. VW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Std. Dev. of Monthly Returns -- Div.Yld. EW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

e-Weig

Std. Dev. of Monthly Returns Beta on Market (S&P 500)

Volatility & Beta

Beta, on Market (S&P 500) -- Div.Yld. EW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Dividend Yield – F5-F1 Time Series, VWCumulative

Div.Yld. VW -- Time Series, Cumulative Performance

F1FNBmarkFN-F1

Dividend Yield – F5-F1 Time Series, EW

Div.Yld. EW -- Time Series, Cumulative Performance

F1FNBmarkFN-F1

Cumulative

Dividend Yield – F5-F1 Time Series

Fractiles, Year-By-Year Returns -- Div.Yld. VW

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

Fractile Returns, Trailing 12 Mos. -- Div.Yld. VW

1/31/1

1/31/2

tiplic

F1-Bmark

F2-Bmark

F3-Bmark

F4-Bmark

F5-Bmark

Fractiles, Year-By-Year Returns -- Div.Yld. EW

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

e-Weig

Year-By-Year Trailing Twelve Months

Fractile Returns, Trailing 12 Mos. -- Div.Yld. EW

1/31/1

1/31/2

tiplic

F1-Bmark

F2-Bmark

F3-Bmark

F4-Bmark

F5-Bmark

All Fractiles, 12-Month Windows

Dividend Yield – F5-F1 Returns Distributions

Div.Yld. VW -- FN-F1 Portfolio: Monthly Returns Distribution

.2 Inf

Monthly Return, Bin Upper Limit

Incremental

Cumulative

Div.Yld. VW -- FN-F1 Portfolio: ln Monthly Returns Distribution

ln Monthly Return, Bin Upper Limit

Incremental

Cumulative

Div.Yld. EW -- FN-F1 Portfolio: Monthly Returns Distribution

sIncremental

Cumulative

e-Weig

Monthly Returns

Div.Yld. EW -- FN-F1 Portfolio: ln Monthly Returns Distribution

.2 Inf

Incremental

Cumulative

ln Monthly Returns

Dividend Yield – F5-F1 Returns Distributions

e-Weig

Monthly Returns ln Monthly Returns

Summary Statistics

Book to Price - Quintile Performance

Annualized Return, % -- B/P VW Univariate Factor Performance

-1- -2- -3- -4- -5-Fractile

Alpha, Monthly % -- B/P VW Univariate Factor Performance

-1- -2- -3- -4- -5-

Fractile

Annualized Return, % -- B/P EW Univariate Factor Performance

-1- -2- -3- -4- -5-Fractile

e-Weig

Alpha, Monthly % -- B/P EW Univariate Factor Performance

-1- -2- -3- -4- -5-

Fractile

Returns & Alpha

Book to Price - Quintile Performance

Std. Dev. of Monthly Returns -- B/P VW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Beta, on Market (S&P 500) -- B/P VW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Std. Dev. of Monthly Returns -- B/P EW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

e-Weig

Volatility & Beta

Beta, on Market (S&P 500) -- B/P EW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Book to Price - F5-F1 Time Series, VWCumulative

B/P VW -- Time Series, Cumulative Performance

F1FNBmarkFN-F1

Book to Price - F5-F1 Time Series, EWCumulative

B/P EW -- Time Series, Cumulative Performance

F1FNBmarkFN-F1

Book to Price – F5-F1 Time Series

Fractiles, Year-By-Year Returns -- B/P VW

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

Fractile Returns, Trailing 12 Mos. -- B/P VW

1/31/1

1/31/2

tiplic

F1-Bmark

F2-Bmark

F3-Bmark

F4-Bmark

F5-Bmark

Fractiles, Year-By-Year Returns -- B/P EW

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

e-Weig

Fractile Returns, Trailing 12 Mos. -- B/P EW

1/31/1

1/31/2

tiplic

F1-Bmark

F2-Bmark

F3-Bmark

F4-Bmark

F5-Bmark

Book to Price – F5-F1 Returns Distributions

B/P VW -- FN-F1 Portfolio: Monthly Returns Distribution

Incremental

Cumulative

B/P VW -- FN-F1 Portfolio: ln Monthly Returns Distribution

Incremental

Cumulative

B/P EW -- FN-F1 Portfolio: Monthly Returns Distribution

sIncremental

Cumulative

e-Weig

B/P EW -- FN-F1 Portfolio: ln Monthly Returns Distribution

0.2 Inf

Incremental

Cumulative

Book to Price – F5-F1 Returns Distributions

e-Weig

Summary Statistics

Trailing Earnings Yield - Quintile PerformanceTrailing Twelve Months Earnings Yield (TEY)

Annualized Return, % -- TEY VW Univariate Factor Performance

-1- -2- -3- -4- -5-Fractile

Alpha, Monthly % -- TEY VW Univariate Factor Performance

-1- -2- -3- -4- -5-

Fractile

Annualized Return, % -- TEY EW Univariate Factor Performance

-1- -2- -3- -4- -5-Fractile

e-Weig

Alpha, Monthly % -- TEY EW Univariate Factor Performance

-1- -2- -3- -4- -5-

Fractile

Returns & Alpha

Trailing Earnings Yield - Quintile PerformanceTrailing Twelve Months Earnings Yield (TEY)

Std. Dev. of Monthly Returns -- TEY VW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Beta, on Market (S&P 500) -- TEY VW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Std. Dev. of Monthly Returns -- TEY EW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Volatility & Beta

e-Weig

Beta, on Market (S&P 500) -- TEY EW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

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

TEY VW -- Time Series, Cumulative Performance

F1FNBmarkFN-F1

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

TEY EW -- Time Series, Cumulative Performance

F1FNBmarkFN-F1

Trailing Earnings Yield - F5-F1 Time Series

Trailing Twelve Months Earnings Yield (TEY)Fractiles, Year-By-Year Returns -- TEY VW

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

Fractile Returns, Trailing 12 Mos. -- TEY VW

1/31/1

1/31/2

tiplic

F1-Bmark

F2-Bmark

F3-Bmark

F4-Bmark

F5-Bmark

Fractiles, Year-By-Year Returns -- TEY EW

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

e-Weig

Fractile Returns, Trailing 12 Mos. -- TEY EW

1/31/1

1/31/2

tiplic

F1-Bmark

F2-Bmark

F3-Bmark

F4-Bmark

F5-Bmark

Trailing Earnings Yield - F5-F1 Returns Distributions

Trailing Twelve Months Earnings Yield (TEY)

TEY VW -- FN-F1 Portfolio: Monthly Returns Distribution

Incremental

Cumulative

TEY VW -- FN-F1 Portfolio: ln Monthly Returns Distribution

Incremental

Cumulative

TEY EW -- FN-F1 Portfolio: Monthly Returns Distribution

sIncremental

Cumulative

e-Weig

TEY EW -- FN-F1 Portfolio: ln Monthly Returns Distribution

Incremental

Cumulative

Trailing Earnings Yield - F5-F1 Returns Distributions

Trailing Twelve Months Earnings Yield (TEY)

e-Weig

Summary Statistics

Forward Earnings Yield

Four different definitions of forward earningsA. 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.

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)

Annualized Return, % -- FEY VW Univariate Factor Performance

-1- -2- -3- -4- -5-Fractile

Alpha, Monthly % -- FEY VW Univariate Factor Performance

-1- -2- -3- -4- -5-

Fractile

Annualized Return, % -- FEW EW Univariate Factor Performance

-1- -2- -3- -4- -5-Fractile

e-Weig

Returns & Alpha

Alpha, Monthly % -- FEW EW Univariate Factor Performance

-1- -2- -3- -4- -5-

Fractile

Forward Earnings Yield - Quintile PerformanceForecast Next 12 Mos. Earnings Yield (FEY)

Std. Dev. of Monthly Returns -- FEY VW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Beta, on Market (S&P 500) -- FEY VW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Std. Dev. of Monthly Returns -- FEW EW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Volatility & Beta

e-Weig

Beta, on Market (S&P 500) -- FEW EW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

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

FEY VW -- Time Series, Cumulative Performance

F1FNBmarkFN-F1

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

FEY EW -- Time Series, Cumulative Performance

F1FNBmarkFN-F1

Forward Earnings Yield - F5-F1 Time Series

Forecast Next 12 Mos. Earnings Yield (FEY)Fractiles, Year-By-Year Returns -- FEY VW

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

Fractile Returns, Trailing 12 Mos. -- FEY VW

1/31/1

1/31/2

tiplic

F1-Bmark

F2-Bmark

F3-Bmark

F4-Bmark

F5-Bmark

Fractiles, Year-By-Year Returns -- FEY EW

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

e-Weig

Fractile Returns, Trailing 12 Mos. -- FEY EW

1/31/1

1/31/2

tiplic

F1-Bmark

F2-Bmark

F3-Bmark

F4-Bmark

F5-Bmark

Forward Earnings Yield - F5-F1 Returns Distributions

Forecast Next 12 Mos. Earnings Yield (FEY)

FEY VW -- FN-F1 Portfolio: Monthly Returns Distribution

Incremental

Cumulative

FEY VW -- FN-F1 Portfolio: ln Monthly Returns Distribution

Incremental

Cumulative

FEY EW -- FN-F1 Portfolio: Monthly Returns Distribution

sIncremental

Cumulative

e-Weig

FEY EW -- FN-F1 Portfolio: ln Monthly Returns Distribution

Incremental

Cumulative

Forward Earnings Yield - F5-F1 Returns Distributions

Forecast Next 12 Mos. Earnings Yield (FEY)

e-Weig

Summary Statistics

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.

)1)(())(1(

1000101

rgrBrEg

rBrEBP

“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):

1. 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).

2. 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

Annualized Return, % -- ICC VW Univariate Factor Performance

-1- -2- -3- -4- -5-Fractile

Alpha, Monthly % -- ICC VW Univariate Factor Performance

-1- -2- -3- -4- -5-

Fractile

Annualized Return, % -- ICC EW Univariate Factor Performance

-1- -2- -3- -4- -5-Fractile

e-Weig

Returns & Alpha

Alpha, Monthly % -- ICC EW Univariate Factor Performance

-1- -2- -3- -4- -5-

Fractile

ICC1 - Quintile Performance

Std. Dev. of Monthly Returns -- ICC VW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Beta, on Market (S&P 500) -- ICC VW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

Std. Dev. of Monthly Returns -- ICC EW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

e-Weig

Volatility & Beta

Beta, on Market (S&P 500) -- ICC EW As Univariate Sort Factor

-1- -2- -3- -4- -5-Fractile

ICC1 - F5-F1 Time Series, VWCumulative

ICC VW -- Time Series, Cumulative Performance

F1FNBmarkFN-F1

ICC1 - F5-F1 Time Series, EWCumulative

ICC EW -- Time Series, Cumulative Performance

F1FNBmarkFN-F1

ICC1 – F5-F1 Time Series

Fractiles, Year-By-Year Returns -- ICC VW

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

Fractile Returns, Trailing 12 Mos. -- ICC VW

1/31/1

1/31/2

tiplic

F1-Bmark

F2-Bmark

F3-Bmark

F4-Bmark

F5-Bmark

Fractiles, Year-By-Year Returns -- ICC EW

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

e-Weig

Fractile Returns, Trailing 12 Mos. -- ICC EW

1/31/1

1/31/2

tiplic

F1-Bmark

F2-Bmark

F3-Bmark

F4-Bmark

F5-Bmark

ICC1 – F5-F1 Returns Distributions

ICC VW -- FN-F1 Portfolio: Monthly Returns Distribution

Incremental

Cumulative

ICC VW -- FN-F1 Portfolio: ln Monthly Returns Distribution

Incremental

Cumulative

ICC EW -- FN-F1 Portfolio: Monthly Returns Distribution

sIncremental

Cumulative

e-Weig

ICC EW -- FN-F1 Portfolio: ln Monthly Returns Distribution

0.2 Inf

Incremental

Cumulative

ICC1 – F5-F1 Returns Distributions

e-Weig

Summary Statistics

Univariate Sorts - Summary

Div.Yld.

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.

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.

IM Trending, IMT = 0IM Volatility, IMV = 0

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

Lagged Months

Red Dot Stock

IMT is highly positiveIMV is moderate

Green Circle Stock

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

Lagged Months

IMT is only slightly positiveIMV is moderate

Black X Stock

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

Lagged Months

IMT is negligibleIMV is very high

Blue Square Stock

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

Lagged Months

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

PFDi,t – “Primary Factor Decile”; the decile into which a stock is binned when sorted on the primary factor

101,, )(

)11(2(n

nnini PFDPFD

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.

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.

Interfractile Migration - Illustration

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractile

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractile

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractile

IMT Equal-Weighted IMT Value-Weighted

IMV Equal-Weighted IMV Value-Weighted

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractiles

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.

Interfractile Migration - Implementation

We analyze IM factor performance within each Forward Earnings Yield (FEY) quintile:

Two-step sequential sort 1st Sort: Quintiles on primary factor (FEY) 2nd Sort: Sub-Trintiles on IMT or IMV

Interfractile Migration Trending

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractile

Raw Mean Geometric Annualized Return

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractile

Market Risk-Adjusted Monthly Alpha

Standard Deviation of Monthly Returns (Sigma)

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractile

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

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.

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.

IMT – Trading Strategies

Dec-87 Dec-88 Dec-89 Dec-90 Dec-91 Dec-92 Dec-93 Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec-00 Dec-01

s for S

P500 O

13,14,15 - 1,2,3; Control: FEY Long/Short

1,13,14,15 - 2,3,6; Hybrid

1,4 - 3,6; IMT Isolation

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractile

Interfractile Migration Volatility

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractile

Standard Deviation of Monthly Returns (Sigma)

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractile

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

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.

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.

IMV – Trading Strategies

s or S

3,13,14,15 - 1,2,4; Hybrid

3,6 - 1,4; IMV Isolation

In-Sample Summary

IM Trading Strategies

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractile

rnOut of Sample – IMT

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractile

Out of Sample – IMT

Dec-87

Dec-88

Dec-89

Dec-90

Dec-91

Dec-92

Dec-93

Dec-94

Dec-95

Dec-96

Dec-97

Dec-98

Dec-99

Dec-00

Dec-01

Dec-02

Dec-03

Dec-04

500 On

1,13,14,15 - 2,3,6; Hybrid

1,4 - 3,6; IMT Isolation

OUT OF SAMPLE

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractile

Out of Sample – IMV

-1- -2- -3- -4- -5- -6- -7- -8- -9- -10- -11- -12- -13- -14- -15-

Fractile

Dec-87

Dec-88

Dec-89

Dec-90

Dec-91

Dec-92

Dec-93

Dec-94

Dec-95

Dec-96

Dec-97

Dec-98

Dec-99

Dec-00

Dec-01

Dec-02

Dec-03

Dec-04

500 On

3,13,14,15 - 1,2,4; Hybrid

3,6 - 1,4; IMV Isolation

OUT OF SAMPLE

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.

Conclusions

Out of sample, these trading strategies underperformed.

Perhaps the patterns observed in-sample were mere data artifacts.

Alternately, perhaps the 2002-2004 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.

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

101,0, )(

)11(2(n

nnii PFDPFD

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.

Duke Investment Analytics

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