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## The Gory Details: Bumpiness Coefficient and Computational Cost

The computation for the bumpiness coefficient for a single company with 12 years of non-decreasing dividend history is:

=100/11*

SQRT(SUM((div2010/div2009-1-DGR)^2;

(div2009/div2008-1-DGR)^2;

(div2008/div2007-1-DGR)^2;

(div2007/div2006-1-DGR)^2;

(div2006/div2005-1-DGR)^2;

(div2005/div2004-1-DGR)^2;

(div2004/div2003-1-DGR)^2;

(div2003/div2002-1-DGR)^2;

(div2002/div2001-1-DGR)^2;

(div2001/div2000-1-DGR)^2;

(div2000/div1999-1-DGR)^2

))

Where DGR is:

=(div2010/div1999)^(1/11)

On the other hand, the AADGR is:

=

AVERAGE(div2010/div2009-1;

div2009/div2008-1;

div2008/div2007-1;

div2007/div2006-1;

div2006/div2005-1;

div2005/div2004-1;

div2004/div2003-1;

div2003/div2002-1;

div2002/div2001-1;

div2001/div2000-1;

div2000/div1999-1)

Computing the BC requires fourteen divisions, twelve exponents, a multiplication and a square root.

Computing the AADGR requires twelve divisions.

When examining the computation cost of a procedure:

Additions and subtractions are ignored because they are "computationally cheap".

Multiplications and divisions are a bit more expensive but are ignored.

Exponents and roots tend to be very expensive and they drive the "computational cost".

The number of exponents for computing the BC is equal to one plus the number of years of dividend increases.

The number of divisions for computing the AADGR is equal to one plus the number of years of dividend increases.

As the number of years of dividend payments increase the computational cost of the BC compared to the AADGR increases.

One further point, AADGR can be viewed as "a rule of thumb", an easily learned and easily computed procedure for approximating a score to predict the bumpiness of a company's non-decreasing dividend history.

## The Gory Details: How Much Bumpiness Do You Need

The Gory Details of the AnalysisWe are looking for factors that “correlate” well with dividend bumpiness. If the correlation is linear then R

^{2 }or the “coefficient of determination” will be near one and R or the “Pearson product-moment correlation coefficient” will be either near one (for a positive correlation, i.e. as the factor increases the bumpiness increases) or near minus one (for a negative correlation i.e. as the factor increase the bumpiness decreases).Fortunately, these kind of calculations no longer have to be worked out by hand, nor do we have to buy a sophisticated software statistics-package. Free tools are now available on-line. For this problem I used the “Linear Correlation and Regression” tool provided by Richard Lowry of Vassar College, which can be found here: faculty.vassar.edu/lowry/corr_big.html. The tool has a companion e-text in basic statistics “

”. I found Chapter 3 “Concepts & Applications of Inferential Statistics” to be lucid and well-written. The complete e-text is here: faculty.vassar.edu/lowry/webtext.html.Introduction to Linear Correlation and RegressionMr. Schwartz defines dividend bumpiness as the (scaled) standard deviation about the “Dividend Computed Annual Growth Rate” (also known as the Dividend Growth Rate or DGR). As you might recall, standard deviation shows the degree of variation from an expected value. Thus, a low standard deviation means the data tends to cluster very close to the expected value, while a large standard deviation means the data tends to be scattered over a much larger range of values.

The 31 May 2011 version of the Champions spreadsheet provides a dividend history table that ranges from 2010 back to 1999 (columns AP through BA, 12 years of dividend history or 11 years of dividend increases). I also reanalyzed the data after I augmented the dividend table with the projected 2011 dividends (13 years of dividends or 12 years of dividend increases).

As augmenting the data with the projected 2011 dividends made no appreciable difference to the results, the discussion below is for the unaugmented data.I also added the following new columns:

• Average Annual Dividend

(average of all dividend payments).

• Average Annual Dividend Growth Rate

or AADGR (average of all the

annual dividend growth rates).

• Data Points (the number of years

of dividend increases).

• Dividend Growth Rate or DGR

(also known as the Dividend

Compounded Annual Growth Rate).

• Bumpiness Coefficient

(100 x Standard Deviation

computed about the DGR).

In the above list of new columns, the term “all dividends” means all monotonically increasing dividends, which means dividends with “no decreases” and “no-zeros” (mathematically, dividing by zero is a “bad-thing”). For the Champions that is not an issue. For the Contenders and Challengers it means some of the dividend history is discarded.

Why discard some of the dividend history?For example, the Raytheon Company (

RTN) has 12 years of dividend payment history listed in the spreadsheet. In 1999, 2000, and2001 the company paid an annual $0.0008 dividend. It also paid a $0.80 annual dividend in 2003 and 2004. Those series of “flat” dividend payments are why it is a Challenger instead of a Contender. However, as there was “no-decrease” in the dividend payment those dividends are used for computing values for the new columns listed above.In the other hand, Alliant Energy Corporation (

LNT) paid a $2.00 annual dividend for the years 1999 through 2002. However, in 2003, the dividend was cut to $1.00. Thereafter the company consistently raised its dividend. In order to maintain a monotonically increasing data set, the dividend payments for 2002 and prior years were discarded. Discarded data were not used in any calculation.Fortunately, modern spreadsheets include functions that make computing such otherwise cumbersome and tedious calculations rather easy.

The details for each new column can be found in the

at the end of this blog.Appendix AWhat are the results of the analysis?For each proposed predictive factor, I examined the range of values then computed the R and R

^{2}values with thetool. The tool also computes the Y-intercept and Slope of the resulting data trend line.Linear Correlation and RegressionYears of available data.

The Champions'

BCranges from a high of 64.42 for CenturyLink Inc. (CTL) to a low of 0.00 for Tootsie Roll Industries (TR).The Contenders'

BCranges from a high of 92.87 for Flowers Foods (FLO) to a low of 0.1344 for Atmos Energy (ATO).The Challengers'

BCranges from a high of 2,261 for the Raytheon Company (RTN) to a low of 0.2471 for ROC Resources (ROCO).All the Champions have 12 years of available data, yet the

BCranges across two orders of magnitude (from 0.00 to 64.42). Thus, years of available data does not predict dividend bumpiness. Thetool confirms this.Linear Correlation and RegressionConclusiondoes notAverage Annual Dividend.The Champions' average annual dividend ranges from a high of $2.27 for Consolidated Edison (

ED) to a low of $0.16 for Lowe's Companies (LOW).The Contenders' average annual dividend ranges from a high of $3.29 for NuStar Energy LP (

NS) to a low of $0.10 for SEI Investments Company (SEIC).The Challengers' average annual dividend ranges from a high of $3.45 for ONEOK Partners LP (

OKS) to a low of $0.03 for Aaron's Inc (AAN).The

tool computes an R of -0.097 and an RLinear Correlation and Regression^{2}of 0.0094 for the Champions'Average Annual DividendandBC. In other words,Average Annual Dividendpredicts about one percent of the variation in dividend bumpiness.ConclusionAverage AnnualDividend

does notdividend bumpiness.

DGR.The Champions'

DGRranges from a high of 0.2875 (~29%) for CenturyLink Inc (CTL) to a low of 0.0084 (~1%) for California Water Service (CWT).The Contenders'

DGRranges from a high of 0.6969 (~70%) for Fastenal Company (FAST) to a low of 0.0176 (~2%) for Atmos Energy (ATO).The Challengers'

DGRranges from a high of 1.1285 (~113%) for Crestwood Midstream Partners LP (CMPL) to a low of 0.0079 (~1%) for OGE Energy Corp (OGE).The

tool computes an R of 0.5622 and an RLinear Correlation and Regression^{2}of 0.3161 for the Champions'DGRandBC. In other words, changes inDGRpredicts about 32% of the variation in dividend bumpiness.ConclusionDGRpredicts only a thirdof dividend bumpiness.

AADGR.The Champions'

AADGRranges from a high of 0.7310 (~73%) for CenturyLink Inc (CTL) to low of 0.0084 (~1%) for California Water Service (CWT).The Contenders'

AADGRranges from a high of 1.2568 (~126%) for Flowers Foods (FLO) to low of 0.0176 (~2%) for Atmos Energy (ATO).The Challengers'

AADGRranges from a high of 23.0323 (~2,303%) for the Raytheon Company (RTN) to low of 0.0079 (~1%) for OGE Energy Corp (OGE).The

tool computes an R of 0.8742 and an RLinear Correlation and Regression^{2}of 0.7642 for the Champions'AADGRandBC. In other words, changes inAADGRpredicts about three quarters of the variation in dividend bumpiness. These results seem promising.Repeating the analysis with the full set of 449 data points gives an R of 0.9964 and an R

^{2}of 0.9928. This is very significant.The corresponding linear regression data trend line has a slope of 97.179754 and a Y-intercept of -10.969416.

The maximum

AADGRthat results in zero predicted bumpiness is ~11%. The computational details are in.Appendix BHowever, when we graph the complete data set for the 449 companies, we notice something interesting.

There are two extreme outliers that warp the trend line:

• the Raytheon Company (

RTN)(23.0323, 2261.7913) has an

AADGRof 2,303% and aBCof 2,262• Waste Management (

WM)(6.7766, 667.8622) with an

AADGR

of 678% and aBCof 668.If we drop the two outliers and re-run the analysis for the remaining 447 data points, the R and R

^{2}are still very good at 0.9608 and 0.923 respectively.The resulting maximum

AADGRthat results in zero predicted bumpiness is 8%.:ConclusionAADGRis an excellentpredictor of dividend bumpiness.

Appendix AThe new columns for the analysis are shown below along with a brief description and (where possible) an Excel Spreadsheet formula.

A note about the formulas...

A symbolic reference such as “div2010” means the spreadsheet cell which contains a specific company's dividend for 2010 instead of the actual cell reference which might be something like “AP7”. A symbolic reference retains its meaning while an actual cell reference might become invalid if columns are added to or deleted from the spreadsheet.

DGRas used in the formulas means the company specificDGRthat is computed as one of the new columns.Bold upper case terms (such as

AVERAGE) are spreadsheet functions.Average Dividend.The average of all dividend payments.

Formula:

=

AVERAGE(div2010; div2009; div2008; ...)AADGR.The average of all annual dividend growth rates.

Formula:

=

AVERAGE(div2010/div2009-1; div2009/div2008-1; ...)Data Points.Determined by inspection.

For the Champions page, the resulting

Data Pointscolumn was all 11s. For the Contenders page the Data Points column ranged from a high of 11 to a low of 8. For the Challengers page, the Data Points column ranged from a high of 11 to a low of 3.Why?The Champions page lists companies that have consistently raised dividends for 25 or more years. Thus, with only the most recent 12 years of data, you would expect (and inspection shows) that the dividend data “monotonically” increases. Why a 11 when there are 12 years of data? Because a growth rate requires two years of data. For example, the year 2000 annual dividend growth rate is div2000/div1999 – 1. The 1999 annual dividend growth rate can't be computed from this data set as no data is listed for 1998.

The Contenders page lists companies that have consistently raised dividends for 10 to 24 years. With 12 years of data a company may have recently made it onto the list due to a zero dividend or a dividend cut prior to 2002. And it takes nine years of data to give 8 data points.

The Challengers page lists companies that have consistently raised dividends for 5 to 9 years. With 12 years of data a company may have recently made it onto the list due to a zero dividend or a dividend cut prior to 2007. And it takes four years of data to give 3 data points.

DGR.The DGR is computed from the first and most recent year of monotonically increasing dividend payments.

DataPoints

Formula11 =(div2010/div2000)^(1/11)

10 =(div2010/div2001)^(1/10)

9 =(div2010/div2002)^(1/9)

8 =(div2010/div2003)^(1/8)

7 =(div2010/div2004)^(1/7)

6 =(div2010/div2005)^(1/6)

5 =(div2010/div2006)^(1/5)

4 =(div2010/div2007)^(1/4)

3 =(div2010/div2008)^(1/3)

Bumpiness Coefficient.The Bumpiness Coefficient (or BC) is 100 times the Standard Deviation about the dividend CAGR.

DataPoints

Formula11 =100/11*

SQRT(SUM((div2010/div2009-1-DGR)^2;

(div2009/div2008-1-DGR)^2;

…

(div2001/div2000-1-DGR)^2))

10 =100/10*

SQRT(SUM((div2010/div2009-1-DGR)^2;

(div2009/div2008-1-DGR)^2;

…

(div2002/div2001-1-DGR)^2))

9 =100/9*

SQRT(SUM((div2010/div2009-1-DGR)^2;

(div2009/div2008-1-DGR)^2;

…

(div2003/div2002-1-DGR)^2))

8 =100/8*

SQRT(SUM((div2010/div2009-1-DGR)^2;

(div2009/div2008-1-DGR)^2;

…

(div2003/div2002-1-DGR)^2))

7 =100/7*

SQRT(SUM((div2010/div2009-1-DGR)^2;

(div2009/div2008-1-DGR)^2;

…

(div2004/div2003-1-DGR)^2))

6 =100/6*

SQRT(SUM((div2010/div2009-1-DGR)^2;

(div2009/div2008-1-DGR)^2;

…

(div2005/div2004-1-DGR)^2))

5 =100/5*

SQRT(SUM((div2010/div2009-1-DGR)^2;

(div2009/div2008-1-DGR)^2;

…

(div2006/div2005-1-DGR)^2))

4 =100/4*

SQRT(SUM((div2010/div2009-1-DGR)^2;

(div2009/div2008-1-DGR)^2;

…

(div2007/div2006-1-DGR)^2))

3 =100/3*

SQRT(SUM((div2010/div2009-1-DGR)^2;

(div2009/div2008-1-DGR)^2;

(div2008/div2007-1-DGR)^2))

Appendix BIf you took basic algebra, you might remember the equation for a straight line as:

Y := a + b•X

Where

Xis the independent variable,Yis the dependent variable,ais the Y-intercept andbis the slope.For the linear regression trend line, the

AADGRis our X and the bumpiness coefficient orBCis our Y.With the

tool computed (449 company) Y-intercept of -10.969416 and slope of 97.179754, the trend line equation for predictingLinear Correlation and RegressionBCis:[

BC] := a + b•[AADGR]With the restriction that the bumpiness coefficient can never be negative.

Solving for the maximum

AADGRthat yields zeroBCgives:0 := a + b•[

AADGR]-a := b•[

AADGR][

AADGR] := -a / b[

AADGR] :=-(-10.969416) / 97.179754

[

AADGR] := 0.112877 => ~11%Repeating the process for the reduced (447 company) data, the

tool computed Y-intercept of -5.965385 and slope of 74.523808, the trend line for predictingLinear Correlation and RegressionBCis:[

BC] := -5.965385+ 74.523808•[

AADGR]Solving for the maximum

AADGRthat yields a zeroBCgives:[

AADGR] := 0.0800467 => 8%