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True DiversificationA Optimization

Applied to Equity Index Construction

James Damschroder

Founder & Chief of Financial Engineering, Gravity Investments LLC

Jonathan Bower

Associate Director of Financial Engineering, Gravity Investments LLC

This Draft: 11/11/2010

ABSTRACT

Capitalization-weighting is the dominant method of index construction and is

the primary benchmark used for measuring market and manager performance in

the investment industry. Using the components of the S&P 500 Index, we

investigate an alternative process-driven methodology for index construction and

optimization that is centered on True Diversification®. We find that when

reconstituting the S&P 500 universe from January 1996 to July 2010 using our

methodology we can achieve annualized outperformance of 499 bps against the

benchmark index.

©2010 Gravity Investments, LLC 2

Duplication or dissemination prohibited without prior written permission.

Table of Contents

Section 1 – An Overview .........................................................................................................3

Section 2.1 – True Diversification® Optimization ..................................................................5

Section 2.2 – Settings and Process ...........................................................................................7

Section 3 – Data ........................................................................................................................9

Section 3.2 – Data and Process Exceptions ...........................................................................10

Section 4.1 – Methodology.....................................................................................................12

Section 5 – Analysis Overview ..............................................................................................14

Section 5.1 – Risk and Return Analysis ................................................................................14

Section 5.2 – Factor Analysis .................................................................................................21

Section 5.3 – Investment Analysis .........................................................................................25

Section 5.4 – Allocation Analysis ..........................................................................................30

Section 5.5 – Turnover Analysis ............................................................................................33

Section 6 – Conclusion ...........................................................................................................34



Section 1 – An Overview

Capitalization-weighted is the dominant method of index construction and is the

primary benchmark used for measuring market and manager performance in the

investment industry. There are several reasons for this choice in methodology

including less trading, low transaction costs, convenience, and large investment

capacityB. These factors have led institutions and managers to invest or benchmark

trillions of dollars in assets to these indices. Much time and effort is expended

attempting to “beat” the benchmark in its current framework, rather than focusing on

inherent structural deficiencies related to capitalization-weighted.

It is our contention that most traditional indices, by being capitalization-weighted,

provide a false sense of security because number of assets in the index is large, implying

good diversification properties. As an example, in the S&P 500, the top ten percent of

the index holdings account for approximately fifty percent of the index’s valueC. While

the S&P 500 or other capitalization-weighted indices may be good proxy for the

“market”, from a portfolio diversification perspective, they are poorly constructed.

Diversification, according to the authors is represented by both minimizing the

exposure of a portfolio to common risk factors (systemic risk) and minimizing the

idiosyncratic risk. No external benchmarks are required, which is an advantage for an

enduring index construction methodology.

Recently within the investment management industry there have been numerous

proposals for alternative methods of creating indices or developing financial products

that indirectly attempt to address the inherent weaknesses in capitalization-weighting.

Examples including GDP-weighted, equal-weight, and Fundamental WeightedD have

been proposed and are now available to investors. At the heart of these efforts is the

(correct) intuition that there is a need to create an index or product that has better

diversification properties.

Rather than address this issue of diversification indirectly we have explicitly created an

index that’s sole focus is on diversification. We believe that True Diversification®

weighting is both unique and superior to those solutions currently available in the

marketplace and importantly is not constrained to a particular market, style, or even

asset class. Also, portfolio diversification based on Gravity Investment’s philosophy

has also been shown to provide material performance improvements to both risk and

returnE. Such findings are herein reconfirmed.



One of the traditional methods of portfolio optimization and construction is mean-

variance optimization which was pioneered by Harry Markowitz in the 1950’sF. While

Markowitz models and the variants it has spawned have gained significant application

in industry they have not gained significant traction as index construction

methodologies. True Diversification® holds several advantages over mean variance and

other optimization techniques that make it better suited for index construction. Among

these attributes are consistency of time-variant allocations, relative de-sensitivity of

model outputs to small changes in inputs, no requirement to include exogenous risk

profile information to select the correct risk-efficient portfolio, the ability to include

assets with short histories, and total ex post performance.

While Markowitz is a pioneer of diversification, a mean-variance optimization is an

inefficient way to attain it. In mean-variance optimization, diversification is achieved

by combining assets with the objective function of variance minimization. As such,

diversification is inherently subservient to variance minimization. Diversification is

actually sacrificed to produce lower volatility. However, the focus on volatility fails to

exploit an asymmetric diversification/return relationshipG. Furthermore, diversification

is shown to have greater stability in a crisis and has a material impact on performanceH.

One important distinction in our index construction using True Diversification®

optimization is that we do not make any judgments about the universe of inclusions for

an index in the search for improved investment characteristics. Rather our focus is

based solely on the current components within the index and finding a more optimal

allocation of those individual assets within the context of portfolio diversification.

Thus, True Diversification® has the necessary flexibility to be applied across any index

or combinations of indices irrespective of asset class or quantity of investment options.

Our True Diversification® index construction and optimization is systematic and

process-driven. Frequently when one encounters “optimized” anything, especially in

the realm of finance, there is always a concern of data-mining to find the best solution.

As will be outlined in Section 4, there are potentially an infinite number of permutations

in the choice of settings and constraints when performing the optimization routine. For

this research we applied one, and only one, settings and constraint schema across the

approximately 600 portfolios necessary to complete the research. Also, the labor

involved in undertaking the optimization process precluded any attempt at multiple

iterations of this same research applying various settings. We have little doubt that it is

possible to find additional performance benefits by applying a different set of

constraints and/or settings, but we leave that as fertile territory for future research and

development.



While our primary focus in Section 5 is on the aggregate equity index, we present the

analysis for the ten sector indices and one aggregate index. We present the analysis of

the optimized sector indices in Sections 5.1 and 5.2 to demonstrate the consistency and

robustness of our methodology. In most cases we demonstrate statistically significant

superior investment performance relative to our benchmarks. For a couple of the sector

indices we cannot make the claim of statistical significance in our outperformance;

however, even in such cases, we can demonstrate a level of consistency in performance

that demonstrates that True Diversification® optimization consistently improves the

investment characteristics of an index.

Finally, in Section 6 we summarize our findings and make the case for True

Diversification® optimization and our index construction methodology. We believe the

implications of this research for the investment industry are potentially profound.

Section 2.1 – True Diversification® Optimization

The section provides a brief explanation of True Diversification® Optimization which is

a holistic asset allocation model and visualization platform. The process maps asset

correlations to vector angles and projects them into three dimensions. The vector

lengths in the model define the relative attractiveness of assets based on a utility

function. The model then imposes a convex hull on the asset XYZ point-cloud. Those

assets comprising the convex hull create the efficient set. Any assets that are dominated

by the convex hull are considered inefficient assets. The total portfolio value equals the

volume of the polytope formed by the convex hull. The model gives an optimal asset

allocation by dividing each asset’s pro-rata volume by the polytope volume.

Diversification Optimization models diversification as a trans-dimensional factor. It

represents diversification in every dimension, not in any single dimension. We map

asset relationships in an N-dimensional vector space. The portfolio of asset candidates

is projected to a smaller dimension using our optimization logic. The model equates

correlations (ρ) to the cosines of angles for every asset in the space where:

ρ = 1 maps to a 0 degree separating angles

ρ = 0 maps to 90-degrees separating angles

ρ = -1 maps to 180-degrees separating angles

This relationship can be seen in Exhibit 1 below.



>>>Insert picture

The model algorithm creates three matrices of equal size where:

C: an estimated correlation matrix, generally derived from historical sample data

A: a matrix of the cosines between vectors graphed in N space

E: a error matrix that gives the difference between C and A

C + E = A

The optimization then begins with a population of XYZ points for all asset candidates

and the fitness function for evaluating the correctness of this population of points is

defined by:

��

��

�

��

The length of any vector is a utility function that quantifies the relative attractiveness of

any asset. The greater the utility function’s value the longer the vector length. Greater

distances signify that the asset is more likely to extend the shell of the model, dominate

its peers, and capture a greater proportion of volume.

The enclosure imposed on the point-cloud is a convex hull and can be seen in Exhibit 2I.

Whereas mean-variance optimization creates an efficient frontier, the convex hull is

analogous to the frontier, but comprised of only the efficient assets in the portfolio.

Therefore, the convex hull is an efficient frontier of individual assets comprising one

single efficient portfolio. The shape of the hull depicts the balance of the portfolio. This

hull also determines the volume of the model portfolio, which is set equal to the capital

allocated.

>>>Insert picture

True Diversification® Optimization allocates resources based on pro-rata volume. To

determine each region’s volume we first calculate the center of gravity or “centroid” for

the modelJ.

The center of mass is ),,(321

xxxO = . To find its coordinates, xi, we use:

� = ∑�� .��



Where N is the number of assets and ji

x,

is an i-th position coordinate of the j-th asset, D

is the portfolios projected dimensionality.

Next, True Diversification® Optimization splits the assets comprising the convex hull

with a bisecting plane. The model creates a facet with an asset as the vertex and all of

the adjacent bisecting planes. This facet connects to the model centroid to create the

assets’ unique relative volume as illustrated below where:

A, B, C and P are assets

O is the center of mass

ABP and APC are facets of the convex hull

L and N are the centers of the facets

The model calculates the centroid with N=3 and j is a number of the vertex of the

facet OD and OM are the bisecting planes of the BOA and POA angles.

Thus, the bounding bisecting planes are ODL, OLM, OMN, OMK etc.

Finally, the pro-rata volume is calculated. The area of the convex hull is calculated, and

then True Diversification® Optimization calculates the volume for each region (asset)

and determines the ratio of any region relative to the entire polytope volume. This

provides a globally optimal allocation for each asset.

Section 2.2 – Settings and Process

The optimization process described above is embedded in a proprietary software

platform using a patented optimization process. Given that this software was

instrumental in conducting the research we provide an overview of the available inputs,

L

A

B

C D

O

M

N

K

P



settings, and constraints. The software is highly customizable and allows a user to

apply a series of constraints, settings, and adjustments for constructing and optimizing

a portfolio/index. Each of the sectors and then the composition of those sectors into an

aggregate index had the same settings and constraints applied to them.

The following steps highlight the settings and constraints used in the True

Diversification® index optimization process.

1. Determine the asset universe

The user can input ticker symbols directly or import a collection of ticker

symbols from an Excel spreadsheet.

2. Impose minimum and/or maximum constraints

The user can constrain all or some of the assets in the portfolio based on

allocation percentages, dollar values, or number of shares. This has the

practical benefit of ensuring an allocation consistent with policy

guidelines.

3. Import historical data for assets based on:

a. Time period length

b. Time period frequency

c. Number of samples

d. Weighting of samples

The user can select a historical sample period for the assets in the portfolio

based on daily, weekly, monthly, quarterly, or annual pricing frequency.

The user can also specify multiple samples and has the ability to

independently weight the respective samples. For example a user could

combine a six-month daily sample with a weight of 1 and a five-year

monthly sample with a weight of 2. The relevant return and statistical

calculations are the weighted average of the samples.

4. Statistical adjustment of historical time-series including:

a. Application of James-SteinK shrinkage

b. Selection of risk quantification as denominator to the utility function

c. Selection of correlation matrix

Shrinkage can be applied to the (blended) historical risk or return values

independently and can also be shrunk towards specific values. Risk can be

defined by the standard deviation, semi-variance, or maximum drawdown. All

of the adjustments made to the historical data and the selection of risk



measurement provide the optimization routine with a utility function, nominally

the Sharpe RatioL, Sortino RatioM, or the Calmar RatioN depending on the

definition of risk. The correlation matrix can be price or returns based or can be

defined by the semi-correlation. The semi-correlation is analogous to semi-

variance in capturing the downside correlation of asset co-movements.

5. Select dimensionality of computation space for optimization

The selection of dimensionality in practice affects the relative breadth of

allocation in a portfolio. Lower dimension calculations result in a more

concentrated portfolio where the better diversifying assets receive higher

allocations at the expense of the marginal diversifier.

Section 3 – Data

One of the core considerations in performing this research was to avoid survivorship

and hindsight bias. Our process is designed to maximize objectivity and avoid all

known bias possibilities. To that extent, every effort was made to ensure that when

performing the optimization of the indices we used only the data that was available up

to that point in time. The software had no visibility to any data looking forward from

the optimization date. Secondly, we wanted a large enough sample to draw statistically

meaningful results and have a sample sufficiently large that we could evaluate our True

Diversification® methodology in a variety of market conditions including the “bull

market” of the late 1990’s and the “lost decade” of the 2000’s. We used three data

sources in an effort to address these issues.

We were provided with quarterly holdings data on the S&P 500 from Standard &

Poor’s. This data gave us a snapshot of the S&P 500 on the first date of each quarter

from Q2 1995 through Q3 2010 and included the date, company name, company ticker

symbol, CUSIP, stock price, shares outstanding, market capitalization, and GICS code.

There were 953 different companies that made up the S&P 500 Index over our sample

period. We believe this data to be accurate and correct but did not cross-check it against

other sources. The only change we made to this dataset was the ticker symbol

conventions to make them compatible with our databases and software as needed.

Much time and effort was expended going through this dataset company by company

to accurately identify and match the corresponding company to the correct historical

data to account for changes in the exchange ticker symbol through time or any

associated M&A activity.



We used two databases for our historical stock price data. For currently listed stocks we

used a database that we maintain from Thompson-Reuters. We also acquired a delisted

stock database from Norgate Investor Services and integrated it into our database. The

data we used in both instances is split-adjusted but was not adjusted for dividends.

While believe the historical data to be correct and accurate, we did not cross-reference

the data against another database to confirm its validity. However, when conducting

the research we found no obvious errors or omissions in the data save the few

companies we were unable to match to their respective place in the benchmark indices

history.

Section 3.2 – Data and Process Exceptions

Despite our best efforts, there were a few companies that either we could not ascribe to

a particular ticker in our database to or we simply did not have the data for that

company at that time. Given that there are almost 31,000 observations in the S&P

dataset and we are missing approximately 250 data points it is likely that their omission

does not materially affect our results. In instances when a company could not be

correctly identified within our database it had a blank ticker assigned to it and it was

excluded from consideration in the index.

The second consideration with respect to the S&P dataset is that it was provided at a

quarterly frequency. We recognize that the S&P 500 is updated in real-time based on

corporate actions or market conditions and additions or deletions to the index can occur

within a quarter. Our choice of quarterly data was a concession to practicality when it

came to working within the confines of the software and still being true to following a

systematic and process driven index optimization. We believe our choice, while it may

not perfectly match the exact same holding as in the S&P 500 universe at every instance

in time, is true to the spirit of this research.

We did not have specific dividend data and we elected to not incorporate dividends in

our analysis. Therefore, the indices we created are price-based and we selected price-

based benchmarks for use for our comparison and analysis. It is not clear what, if any,

effect or bias may have been introduced in the optimization process by using non-

dividend adjusted price series. The dividend yield on the S&P 500 index for this period

is approximately 1.8%O. The net effect, if any is likely small, but as yet undefined.

We found a few instances where no or limited historical data was available for a

particular asset. This could occur either through a corporate action, missing data or



specifically M&A where a new company was formed and merited inclusion in the

original index, but had no history directly attributable to the new corporation. The

ideal way to deal with the scenario would be to use the historical data from prior to the

merger associated with the acquirer to calculate the appropriate risk statistics.

However, from a practical stand point this was not a realistic prospect for the research

project. To deal with this situation we assumed zero for risk, return, and correlation

calculations of the asset. Such risk and return inputs would later be normalized, having

the impact negated. The impact on correlations would generally tend to favor such

assets. If the available data if shorter than the specified historical sample period

requested we would use the entirety of available data.

In the case where there was no data available for the subsequent periods to compute a

return. This scenario could be caused by M&A activity where the acquired was no

longer listed or if a bankruptcy occurred or we did not have the data in our database.

In this case we assigned a 0% return for that asset’s allocation within the index. The

ideal way to deal with this situation would be to compute, in the case of a merger, the

allocation to the new company or acquirer and compute that return. In the case of

bankruptcy then a -99% return could be used. Again, as a matter of practicality this was

not a feasible solution when it was encountered. We believe this choice may have

created a slight downward bias in our ex-post return numbers by applying a 0% return,

but we as yet, we have not measured this potential impact.

Finally, a third case where we could have an allocation to an asset but in the following

quarter it is no longer included in the original index but data is available to compute a

return. A scenario that could cause this situation is a company’s market capitalization

declining during the prior period and being removed from the original index. In this

scenario we would still maintain an allocation to the now excluded asset and over time

it would be replaced by newer assets allocations. This is the most prominent deviation

from the original index in that we did not remove the asset until our next reconstitution.

This scenario also biases our results downward. While we have not studied exactly

how much we give-up in performance in this situation, Sui P demonstrates that recent

deletions from an index have significant negative alpha after being dropped from an

index. Also, Sui’s research indicates that recent additions have short-term positive

alpha and in our research and index construction process we have a decidedly negative

asymmetric exposure to these phenomena. That is, we hold our “losers” that dropped

from the index over a negative performing period, yet have proportionally less

available exposure to the “winner” that was added to the index since we are only

applying 25% of the addition’s potential weight by implementing an overlapping

portfolio construction process.



We think our choices in methodology are broadly intuitive, simple and conservative in

nature, but we cannot definitively state what the potential performance impact is from

these choices vis-à-vis another set of choices with any accuracy.

Section 4 – Methodology

The purpose of this research is to demonstrate that through the application of a

systematic, process-driven application of True Diversification® optimization we can

build an index with superior investment characteristics. We opted for a “building

block” approach where we first optimized each sector, as defined by their GICS

classification, as a stand-alone index. Our hypothesis was that we could compound

outperformance by compounding the optimization process. We took these optimized

sector indices and allocated them based on another layer of True Diversification®

optimization into the aggregate index, the “True Diversification® 500”. This aggregate

index has ostensibly the same components as contained in the S&P 500 through time.

The process for creating the indices was applied systematically and without deviation

for all optimizations and was performed as follows:

1. Collect the ticker symbols that were defined as being in the index we are

optimizing on the first business day of a quarter from the S&P dataset

2. Apply any allocation constraints to the portfolio

3. Import the relevant historical data for each ticker symbol up to the point in time

defined in (1.)

4. Apply any necessary statistical adjustments to historical data

5. Compute the optimization and save the allocation weights

6. Evaluate the obtained weights to the ticker symbols for the subsequent period

(one year)

7. Export the computed returns associated with new portfolio and create a

continuous return time series

8. Repeat this process for each quarter and sector

Applying this process for each index provided us with four time-series outputs with

each one corresponding to the beginning of a quarter and having a one-year holding

period. Each quarter’s one-year holding period return was made into a continuous

time-series. To make them continuous we took the average return on the second day of

the appropriate quarter from the new optimization and the corresponding day from the



prior optimization. This blended return is what we used as a reasonable approximation

one would get from trading and implementing the new allocation weights on the index

not including the impact of transaction costs, execution, liquidity, or other market

variables. The net result was four time-series for each index that correspond to a set of

returns attributable to the index based on an annual re-optimization and held for one

year from the beginning of Q1, Q2, Q3, or Q4. The following flow chart describes this

process:

1/2/1996 1000000

1/3/1996 1001214 0.12%

1/4/1996 999693 -0.15%

1/5/1996 991899 -0.78% 1/2/1996

. . . . . . . . . . . . 1/3/1996 0.12%

12/31/1996 1133983 -0.73% 1/4/1996 -0.15%

1/2/1997 1124645 -0.82% 1/5/1996 -0.78%

1/3/1997 1137476 1.14% . . . . . . . .

1/6/1997 1132563 -0.43% 12/31/1996 -0.73%

1/7/1997 1137294 0.42% 1/2/1997 -0.82%

1/3/1997 1.14%

1/6/1997 -0.50%

1/2/1997 1000000 1/7/1997 0.80%

1/3/1997 1012851 1.29% 1/8/1997 -0.05%

1/6/1997 1007110 -0.57% 1/9/1997 0.91%

1/7/1997 1015240 0.80%

1/8/1997 1014710 -0.05%

1/9/1997 1023948 0.91%

We chose to create these four time-series for each sector to reduce any bias resulting

from selection of a reconstitution date. Using these sets of time-series allowed us to

evaluate the performance of each index based on all combinations of quarterly, semi-

annual, and annual optimizations and holding periods including the use of both

overlapping and non-overlapping index construction. Across the sectors occasionally

we found that there were some sectors that benefited from a particular holding period

or timing. This most likely is a function of randomness, although we don’t rule out the

potential of a structural element to these findings. In aggregate no single method of

index construction was superior to another and we opted for what we feel is the best

blend of the evaluated methods.



We selected quarterly overlapping portfolios with one year holding periods as our

method of choice. Selecting this method of index construction means that each quarter

25% of an index is re-optimized and allocated. In many ways we also feel this is an

intuitive selection and does the best job of staying “on benchmark” in terms additions

or deletions of the components in the benchmark indices juxtaposed against the

potential negative effects associated with transaction costs and turnover that could

occur had we selected re-optimizing and allocating an entire index every quarter. We

have included, in the appendix, a table showing risk and return values for each of the

sectors and the True Diversification® 500 for each of the proposed index construction

methods to allow the reader to draw their own conclusions on our decision to

implement our chosen method.

Section 5 – Analysis Overview

The goal of this research was to test the idea that we could systematically apply True

Diversification® optimization to a universe of assets as defined by an index and create a

better performing investment. We have achieved the desired objective. The True

Diversification® 500 was not built to track the benchmark index and it can deviate

substantially from year to year in outperformance or underperformance. Presented

below are a collection of tables and charts that make the case, that not only is our

method superior to the benchmark index from a total return perspective, but it is also a

much more stable investment which is a desirable outcome for all investors. We begin

our analysis with the raw return numbers for all eleven indices (ten sectors and one

aggregate) that were created, followed by a factor analysis using regression to assess the

investment “characteristics” of the indices. In the third portion of the analysis section

we examine the returns of the True Diversification® 500 from an investor’s perspective

and evaluate the index’s consistency. In the next section we examine the sector

weightings used in creating the True Diversification® 500 and point to some interesting

observations about their behavior. Finally, we evaluate the turnover of the True

Diversification® 500 using two different methods. We compare these figures to other

indices and investments and then consider the potential impact of transaction costs for

implementing the True Diversification® 500.

Section 5.1 – Risk and Return Analysis

In this section we examine the basic risk and return statistics for the ten optimized

sector and True Diversification® 500 index and compare them to their respective



benchmark. We provide this analysis for the sectors to highlight the consistency of our

methodology and True Diversification® optimization. We break these statistics into

three groups. One is the full sample from January 1996 to July 2010 presented in Table

1. The other two tables are sub-samples which report the risk and return statistics for

the late 1990’s “bull market” covering January 1996 through December 1999 in Table 2.

Table 3 presents the other sub-sample which covers the “lost decade” from January

2000 through July 2010. Table 4 presents the monthly returns of the True

Diversification 500 over the full sample period. We draw your attention to several key

facts presented in the first three tables.

First, from a return perspective, all ten optimized sectors beat their respective

benchmark by an average of 3.16% on an annualized basis and ranged from 0.3% to

5.12% in outperformance. The True Diversification® 500 beat its benchmark by 4.99%

on an annualized basis. Secondly, from a risk perspective, the standard deviation of

the ten optimized sectors had volatility that was 0.26% per year higher than their

benchmarks. We consider this to be “good” volatility in that it is upside volatility. We

draw this conclusion because the average sector’s down-side volatility was reduced by

0.19% per year relative to the average benchmark. Examining the maximum drawdown

experienced by each optimized sector we again see an aggregate reduction in

drawdown, with an average reduction in maximum drawdown of 2.65% relative to the

average benchmark. The True Diversification® 500 over this sample had 0.46% less

volatility, 0.92% less downside volatility, and a 3.91% improvement in the maximum

drawdown over the benchmark index.

In the first sub-sample, the “bull market” period, only one of the optimized indices

outperformed their respective benchmark although the optimized indices on average

reduced standard deviation and downside deviation. This is not to say we had negative

returns, rather we point to the True Diversification® 500 which had annualized

performance of 17.84% during this period whereas the benchmark index returned

22.15%. The average sector underperformance during this period was of a similar

magnitude relative to the average benchmark sector, 14.35% versus 18.6% respectively.

Perhaps more important to this section from an analysis perspective are the risk and

return statistics from the “lost decade” of the 2000’s. The average sector had 5.69% in

annualized outperformance and the True Diversification® 500 outperformed by 8.04%.

Again, these returns we achieved with nearly the same or less risk than their

benchmarks.

The performance drop off between the two samples is also worthy of comparison. For

instance the True Diversification® 500 went from a 17.84% annualized return in the



“bull market” to a 5.82% annualized return during the “lost decade”. This represents a

drop-off in performance of about 12% a year, but still positive in both sub-samples.

Contrast this result to .SPX which had a drop-off in performance of more than 24% and

lost 2.22% on an annualized basis during the “lost decade”. Interestingly, the

performance of the True Diversification® 500 over the full period was 9.06% with a

standard deviation of 15.84%. This is very close to the oft-cited “historical” return of

stocks over the last century of 8% a year with 16% volatility. The traditional benchmark

during the full sample returned less than half that amount.

We believe these results are one way of pointing to the consistency and repeatability of

our True Diversification® optimization. Anecdotally, we have found that when

applying True Diversification® optimization to a portfolio of assets we typically can

improve performance. Our average sector optimized indices added about 300bps in

annualized outperformance relative to their benchmarks. The application of another

layer of True Diversification® optimization when aggregating the sectors into the True

Diversification® 500 added almost another 200bps in performance. We believe that the

performance and stability of the indices from both a risk and return perspective is an

artifact of building these indices using a diversification centric approach to investing.



Table 1: The True Diversification® optimized indices are listed by their sector name and the benchmark GICS sector

is listed by its relevant ticker symbol. We computed the geometrically compounded annualized return for ten the

optimized and benchmark sectors plus the aggregate True Diversification® 500 and its benchmark index for the full

sample period of this study, January 1996 – July 2010, using monthly return values. We also computed the

annualized standard deviation, the semi-variance often referred to as downside deviation, and the maximum

drawdown of the respective index. Each optimized index is matched to its benchmark index and the difference

between the relevant risk and return attributes is computed. We also computed the average of each optimized and

benchmark sector as an additional data point for comparison.

Annualized

Return

Standard

Deviation

Semi-

Variance

Maximum

Drawdown

Consumer Staples 6.58% 12.70% 8.62% 31.69%

.GSPS 5.01% 13.73% 9.34% 31.22%

difference 1.57% -1.04% -0.72% 0.48%

Consumer Discretionary 7.12% 20.44% 13.99% 61.49%

.GSPD 5.20% 19.73% 13.46% 56.24%

difference 1.92% 0.71% 0.54% 5.25%

Energy 12.13% 24.49% 15.69% 59.71%

.GSPE 8.30% 19.78% 12.69% 49.76%

difference 3.83% 4.71% 3.01% 9.94%

Financial 6.72% 22.50% 15.39% 73.95%

.GSPF 1.94% 24.16% 17.42% 80.05%

difference 4.79% -1.66% -2.03% -6.10%

Healthcare 9.51% 14.85% 10.36% 35.58%

.GSPA 5.36% 16.07% 10.97% 39.90%

difference 4.15% -1.21% -0.61% -4.32%

Industrial 4.92% 18.09% 12.60% 50.80%

.GSPI 4.63% 19.22% 13.54% 60.24%

difference 0.30% -1.13% -0.93% -9.44%

Information Technology 8.16% 30.29% 19.83% 72.52%

.GSPT 6.39% 30.08% 20.37% 80.35%

difference 1.77% 0.21% -0.54% -7.84%

Material 7.36% 21.39% 13.98% 50.68%

.GSPM 3.39% 22.20% 14.86% 57.87%

difference 3.97% -0.81% -0.88% -7.19%

Telecommunication 3.62% 25.21% 17.01% 78.24%

.GSPL -1.50% 22.45% 15.75% 76.11%

difference 5.12% 2.76% 1.26% 2.14%

Utilities 5.58% 17.13% 11.41% 49.14%

.GSPU 1.39% 17.03% 12.35% 58.58%

difference 4.19% 0.10% -0.94% -9.44%

Average Sector 7.17% 20.71% 13.89% 56.38%

Average Benchmark 4.01% 20.45% 14.07% 59.03%

difference 3.16% 0.26% -0.19% -2.65%

True Diversification® 500 9.06% 15.84% 10.81% 48.64%

.SPX 4.07% 16.30% 11.73% 52.56%

difference 4.99% -0.46% -0.92% -3.91%





optimized and benchmark sectors plus the aggregate True Diversification® 500 and its benchmark index for the

sample period of this study, January 1996 – December 1999, using monthly return values. We also computed the





Annualized

Return

Standard

Deviation

Semi-

Variance

Maximum

Drawdown


.GSPS 9.03% 17.47% 11.30% 22.27%

difference -0.73% -4.42% -3.21% 1.59%


.GSPD 22.90% 17.28% 10.79% 16.72%

difference -10.17% -0.18% 1.06% 3.85%

Energy 8.42% 21.90% 13.49% 31.73%

.GSPE 13.23% 17.86% 9.48% 20.90%

difference -4.81% 4.04% 4.02% 10.83%

Financial 15.35% 18.94% 12.05% 19.10%

.GSPF 19.63% 22.54% 14.61% 23.24%

difference -4.28% -3.60% -2.56% -4.14%

Healthcare 13.56% 16.65% 11.49% 17.87%

.GSPA 21.82% 19.21% 11.50% 15.33%

difference -8.26% -2.56% -0.01% 2.54%

Industrial 5.38% 16.96% 10.71% 24.55%

.GSPI 15.91% 17.10% 10.71% 20.41%

difference -10.54% -0.14% 0.00% 4.13%

Information Technology 34.68% 26.86% 15.00% 23.86%

.GSPT 50.94% 28.93% 14.04% 16.40%

difference -16.26% -2.08% 0.95% 7.46%

Material -0.07% 22.19% 14.64% 32.94%

.GSPM 4.62% 21.25% 13.16% 26.45%

difference -4.70% 0.94% 1.48% 6.50%

Telecommunication 34.71% 19.07% 9.49% 11.74%

.GSPL 22.17% 19.06% 10.40% 15.00%

difference 12.54% 0.01% -0.91% -3.26%

Utilities 10.50% 12.72% 7.19% 10.18%

.GSPU 5.80% 14.31% 8.59% 15.63%

difference 4.70% -1.59% -1.39% -5.45%

Average Sector 14.35% 18.54% 11.40% 21.64%

Average Benchmark 18.60% 19.50% 11.46% 19.23%

difference -4.25% -0.96% -0.06% 2.40%


.SPX 22.15% 15.61% 9.58% 15.57%

difference -4.32% -1.58% -1.12% 0.72%





optimized and benchmark sectors plus the aggregate True Diversification® 500 and its benchmark index for the

sample period of this study, January 2000 – July 2010, using monthly return values. We also computed the





Annualized

Return

Standard

Deviation

Semi-

Variance

Maximum

Drawdown


.GSPS 3.49% 12.02% 8.50% 30.35%

difference 2.43% 0.58% 0.34% 1.34%


.GSPD -0.97% 20.42% 14.40% 56.24%

difference 6.00% 1.23% 0.39% 5.25%

Energy 13.61% 25.49% 16.52% 59.71%

.GSPE 6.45% 20.53% 13.77% 49.76%

difference 7.16% 4.96% 2.75% 9.94%

Financial 3.54% 23.76% 16.55% 73.95%

.GSPF -4.22% 24.62% 18.45% 80.05%

difference 7.76% -0.86% -1.90% -6.10%

Healthcare 7.98% 14.14% 9.94% 35.58%

.GSPA -0.42% 14.39% 10.81% 39.90%

difference 8.40% -0.25% -0.87% -4.32%

Industrial 4.75% 18.58% 13.31% 50.80%

.GSPI 0.54% 19.94% 14.53% 60.24%

difference 4.21% -1.36% -1.22% -9.44%

Information Technology -0.68% 31.30% 21.47% 72.52%

.GSPT -7.14% 29.70% 22.40% 80.35%

difference 6.47% 1.60% -0.93% -7.84%

Material 10.40% 21.10% 13.77% 50.68%

.GSPM 2.92% 22.64% 15.52% 57.87%

difference 7.48% -1.54% -1.75% -7.19%

Telecommunication -6.43% 26.76% 19.19% 78.24%

.GSPL -9.42% 23.29% 17.43% 75.09%

difference 2.99% 3.47% 1.76% 3.15%

Utilities 3.73% 18.60% 12.70% 49.14%

.GSPU -0.27% 18.02% 13.57% 58.58%

difference 4.00% 0.58% -0.87% -9.44%

Average Sector 4.78% 21.40% 14.71% 56.38%

Average Benchmark -0.90% 20.56% 14.94% 58.84%

difference 5.69% 0.84% -0.23% -2.47%


.SPX -2.22% 16.26% 12.50% 52.56%

difference 8.04% 0.20% -0.88% -3.91%



Table 4: The monthly returns of the True Diversification® 500 from January 1996 through July 2010.

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year

1996 2.82% 0.82% 0.84% 2.66% 0.95% -1.10% -5.91% 2.72% 4.71% 1.11% 5.82% -1.21% 14.63%

1997 3.23% 0.22% -3.97% 2.09% 7.29% 3.06% 7.02% -1.27% 4.59% -3.51% 2.55% 0.33% 23.04%

1998 0.43% 6.43% 4.84% -0.16% -2.70% 2.51% -3.38% -13.01% 9.08% 4.77% 2.71% 6.35% 17.09%

1999 2.67% -4.21% 5.36% 7.69% 1.28% 4.72% -1.16% -1.19% -1.72% 2.18% 0.37% 4.57% 21.80%

2000 -2.82% -1.66% 7.61% -0.81% 1.39% -0.57% -1.61% 6.29% -0.54% 1.55% -4.42% 5.90% 9.95%

2001 0.43% -1.78% -3.35% 6.40% 0.74% -3.15% -0.67% -2.90% -8.07% 0.96% 5.13% 2.42% -4.60%

2002 -2.89% -1.60% 5.98% -3.93% -0.66% -8.06% -10.65% 2.55% -9.80% 9.10% 9.27% -3.68% -15.68%

2003 -2.46% -1.54% 0.40% 6.91% 10.42% 1.17% 0.01% 4.03% -1.13% 4.56% 1.46% 6.77% 34.17%

2004 3.06% 3.49% -0.95% -1.04% 0.28% 3.72% -1.18% -1.14% 4.47% 1.70% 6.97% 1.75% 22.83%

2005 -1.13% 4.28% -1.46% -2.81% 3.63% 3.09% 4.74% 0.65% 2.29% -3.75% 2.80% 1.16% 13.87%

2006 5.62% -1.04% 2.06% 0.79% -1.63% 1.37% 0.60% 1.42% 0.18% 2.91% 2.46% 0.70% 16.36%

2007 1.93% 0.15% 1.57% 3.56% 4.33% -1.78% -3.76% -0.06% 3.49% 1.76% -3.72% 0.98% 8.36%

2008 -7.03% -0.77% -1.67% 5.16% 4.78% -3.76% -5.71% 1.14% -12.33% -20.29% -5.79% 1.56% -38.67%

2009 -3.88% -9.93% 6.18% 11.01% 7.31% -1.43% 5.74% 2.67% 5.71% -4.70% 5.12% 6.94% 32.72%

2010 -2.82% 3.26% 5.76% 1.64% -5.26% -4.67% 6.06% 3.34%



Section 5.2 – Factor Analysis

In this section we use regression analysis in an attempt to identify the characteristics or

factors to which we can attribute our performance as outlined in Section 5.1. For the

purposes of this analysis, we use the full sample of 175 monthly returns for each index

from January 1996 through June 2010. We could not include July 2010 in the analysis as

one of the regression factors was not available for that time period. Table 5 presents the

regression of the optimized and benchmark sectors against a market beta. The market

beta we used is the excess return “RMRF” as applied in Fama-French Q and was

collected from the Ken French website. Table 6 presents the regression results from

evaluating the benchmark sector versus the optimized sector. We also ran a multiple

regression using the benchmark sectors as the market “beta”, and incorporate three

additional factors as explanatory variables; these results are in Table 7. In the multiple

regression analysis we used “SMB” as a proxy for small or large cap, and “HML” as a

proxy for value or growth as defined in Fama-FrenchR. We also incorporated “UMD”

as a proxy for momentum which was originally developed by Jegadeesh and TitmanS

and applied in this context by CarhartT as a fourth factor. The time-series for these

factors were also collected from the Ken French website.

Table 5 shows that attempting to describe or explain the returns of our optimized

indices using the “RMRF” as our market beta is not very productive. The average R2 of

the optimized and benchmark sectors is about 50% which does offers some explanation,

but as a practical matter insufficient. The R2 values indicate that, for the True

Diversification® 500 and .SPX, “RMRF” does offer a reasonable explanation of returns,

but interestingly it does a much better job in explaining the benchmark index rather

than the True Diversification® 500.

We propose that in performing a factor analysis, since we are examining the return

properties of specific sectors, a better market beta (i.e. “RMRF”) would be to use the

respective benchmark GICS sector indices as our market beta. Table 6 supports this

claim with an average R2 across the optimized indices being 82%. However, since we

do not explicitly incorporate “excess returns” by subtracting the risk free rate, the

interpretation of our intercept when we apply the four-factor regression is somewhat

different than would be classically defined in academic literature. We believe our

analysis is very much in the spirit of attempting to identify the potential drivers of our

returns.

We present the results of the four-factor regression in Table 7 and there are some

noteworthy facts. The first is that despite adding three additional variables to the



regression analysis the average R2 increased by only 3% to 85% indicating, that perhaps,

these “classic factors” do not do a good job at capturing the drivers of returns in our

optimized indices. We feel we can say this for a few reasons. One is that we report

highly statistically significant positive statistics on “SMB” with the exception of our

Telecommunications sector. This suggests that the driver of returns is explained by

small capitalization stocks. Clearly, by using the components of the S&P 500 Index as

the basis of for our universe we do not actually have any small cap stocks in the index.

The second noteworthy observation from Table 7 is that the “HML” betas have a slight

tilt towards “value” as explained by this factor. Although for half of these indices the

“HML” factor is either not statistically significant or the sector can be described as being

“growth”. We think an important take away from this result is that; despite running

the same optimization and portfolio construction process across all of the indices, this

factor cannot consistently explain the returns attributed to our optimized indices.

As with the “HML” beta, the “UMD” beta for momentum as a factor does not

consistently explain the returns attributable to our optimization and portfolio

construction process. We do have a statistically significant loading on the True

Diversification® 500 for momentum. A possible explanation for this result could be the

relatively short history we were forced to use when applying the optimization settings.

Whereas in the sector optimizations we could use up to twenty years of available asset

data, we had only a few years of our optimized sector returns data to work with when

creating the True Diversification® 500, especially in the early years of its construction.

Ex-ante one assumption might be that that however the optimization routine is being

performed there should be a set of corresponding factors that can reliably explain the

return characteristics of the optimized indices. We believe based on this factor analysis

that these traditional factors either cannot explain or do not do an adequate job of

describing the returns associated with our index optimization and portfolio

construction process. The lack of consistent explanation with these factors suggests that

our process of optimization and index construction is unique and different. We think

that in this case, being different is good.

Finally, we note in Table 7 that the beta coefficients of the optimized indices in the

regression, with the exception to our optimized Energy sector, are all below 1 and range

from 0.79 to 0.94. One possible interpretation of this statistic is that by using True

Diversification® optimization in our creating our indices we are reducing the systematic

risk in the indices through a more intelligent allocation of the assets.



Table 5: The results of the regression of the optimized and benchmark indices on the “RMRF” (market excess

returns) beta are presented below. The optimized sectors are listed by name and the benchmark indices are listed by

the appropriate ticker symbols. The test statistics are shown below their respective coefficient and are italicized. The

sample period covered is January 1996 through June 2010.

�� = �� + �� + ��

αααα

ββββrmrf R2 αααα

ββββrmrf R2

Consumer Staples 0.003 0.447 35.08% .GSPS 0.002 0.436 28.66%

1.35 9.62 0.78 8.28

Consumer Discretionary 0.001 1.022 69.88% .GSPD -0.001 1.036 77.04%

0.45 20.01 -0.32 24.07

Energy 0.007 0.811 30.83% .GSPE 0.004 0.670 32.43%

1.57 8.72 1.10 9.06

Financial 0.001 0.993 54.61% .GSPF -0.003 1.136 61.68%

0.38 14.39 -0.90 16.66

Healthcare 0.005 0.634 50.76% .GSPA 0.002 0.549 32.74%

2.08 13.32 0.74 9.12

Industrial 0.000 0.884 67.02% .GSPI -0.001 0.999 76.22%

-0.10 18.73 -0.55 23.53

Information Technology 0.001 1.556 73.38% .GSPT 0.000 1.541 72.99%

0.26 21.81 -0.14 21.60

Material 0.002 0.959 56.63% .GSPM -0.002 0.990 56.34%

0.54 15.00 -0.52 14.91

Telecommunication -0.001 1.037 47.34% .GSPL -0.005 0.856 40.90%

-0.22 12.43 -1.23 10.90

Utilities 0.002 0.506 24.82% .GSPU -0.001 0.423 17.75%

0.73 7.49 -0.16 6.03

True Diversification® 500 0.003 0.858 81.90% .SPX -0.002 0.960 97.09%

1.99 27.96 -2.59 75.98

Table 6: The results of the regression of the optimized indices on their respective benchmark index as proxy for

market a beta are presented below. The optimized sectors are listed by name. The test statistics are shown below

their respective coefficient and are italicized. The sample period covered is January 1996 through June 2010.

�� = �� + �� + ��

αααα

ββββsector R2 αααα

ββββsector R2

Consumer Staples 0.002 0.814 77.88% Information Technology 0.002 0.936 86.46%

1.50 24.66 0.82 33.22

Consumer Discretionary 0.002 0.981 89.54% Materials 0.004 0.915 89.61%

1.23 38.46 2.34 38.62

Energy 0.003 1.141 84.46% Telecommunications 0.005 0.982 76.36%

1.33 30.65 1.82 23.62

Financial 0.004 0.864 86.33% Utilities 0.004 0.930 85.49%

2.14 33.03 2.48 31.91

Healthcare 0.004 0.752 66.36% True Diversification® 500 0.004 0.878 81.46%

2.35 18.44 2.96 27.55

Industrials 0.001 0.860 83.05%

0.54 29.09



Table 7: The results of the regression of the optimized indices on their respective benchmark index, as proxy for the

market beta, and two of the three Fama-French factors (SMB and HML) plus the Carhart momentum factor (UMD) is

presented below. The optimized sectors are listed by name. We include the regression results for .SPX using RMRF

as the market beta for comparison purposes. The test statistics are shown below their respective coefficient and are

italicized. The sample period covered is January 1996 through June 2010.

�� = �� + �� + �� + � �! � �! � + �"�#�"�# � + ��

αααα

ββββsector ββββSMB ββββHML ββββUMD R2

Consumer Staples 0.002 0.794 0.084 0.104 -0.021 80.56%

1.30 24.70 2.24 2.59 -0.70

Consumer Discretionary 0.001 0.925 0.228 0.253 0.036 92.97%

0.50 38.68 6.24 6.49 1.16

Energy 0.002 1.127 0.195 0.072 -0.021 85.60%

1.05 31.08 3.15 1.08 -0.42

Financial 0.003 0.859 0.115 0.216 0.125 87.42%

1.40 30.68 2.17 3.76 2.86

Healthcare 0.003 0.803 0.357 0.101 0.001 74.49%

1.61 21.38 6.86 1.85 0.02

Industrial 0.000 0.819 0.239 0.217 0.030 87.01%

-0.23 28.94 5.52 4.64 0.82

Information Technology 0.001 0.873 0.525 0.116 -0.046 90.85%

0.34 28.67 8.50 1.46 -0.78

Material 0.003 0.888 0.217 0.094 0.003 91.19%

1.88 38.10 5.14 2.07 0.09

Telecommunication 0.006 0.938 0.128 -0.282 -0.231 79.22%

2.35 22.58 1.65 -3.31 -3.54

Utilities 0.003 0.925 0.156 0.029 -0.038 87.15%

2.27 32.78 3.83 0.65 -1.17

True Diversification® 500 0.003 0.889 0.186 0.201 0.094 84.20%

2.03 27.31 4.44 4.31 2.56

.SPX -0.001 0.993 -0.177 0.052 -0.009 99.32%

-4.09 143.13 -19.57 5.19 -1.11



Section 5.3 – Investment Analysis

Up to this point our analysis and the presentation of our findings had focused on

annualized return numbers and a more “academic” approach. In this section we

present four separate cases that our True Diversification® 500 is a superior investment

relative to its benchmark and represents “better beta”. We feel this is what an investor

could expect – nay, should demand – when undertaking a well diversified long-only

large-cap equity investment. We evaluate and present the monthly returns of our True

Diversification® 500 and the benchmark index by comparing them head-to-head both

monthly and annually. We also evaluate the monthly performance using a rolling

window analysis; this allows us to compare an investment regardless of the starting

point of an investment which can be severely biased depending on the timing of initial

or subsequent investment. We also attempt to remove the timing bias by looking at the

growth of $1000 through time and compare our index to the benchmark with various

starting points. Finally, present an analysis of the returns of what a “401k investor”

might experience when investing on a regular schedule in both indices.

In Table 8 we present the monthly and annual head-to-head numbers for both the True

Diversification® 500 and the benchmark index. A 4.99% annualized outperformance on

its face could be misconstrued that we somehow outperform the index each year by that

amount; that is not the case. On a month-to-month basis the True Diversification® 500

outperforms the benchmark 56% of the time. That may not appear to be particularly

interesting. However, we have a positive asymmetric response to this fact. In months

when we do outperform it is by an average 1.78% and when we underperform it is by

an average of 1.39%. This could be referred to as positive skew, and it is a desirable

investment trait. It is this small incremental compounding effect that we believe is one

of the primary drivers of our outperformance. On a year-to-year basis we beat the

benchmark 73% of the time, and again with a positive asymmetric response.

Remember, the True Diversification® 500 is not intended to track the benchmark index

and this can be seen in Table 8 in the minimum underperformance and the maximum

outperformance. We do not see this as a drawback.

Rolling window analysis is a non-traditional method of analysis, but is frequently used

in the evaluation of alternative investments. Using rolling windows of monthly returns

removes the timing bias of an initial investment and provides a more complete analysis

of an investment’s risk characteristics. We present a six-month, one-year, three-year,

and five-year rolling window analysis in Table 9. To compute the relevant statistics we

used a monthly return series. For each window of analysis we calculate the compound

return experienced during that window and created a time-series of those compounded



returns. This gives us a time-series of each window’s returns that we can use to then

compute the relevant statistics that are presented.

The good way to envision this type of analysis is that it answers the following question:

“Over any consecutive (one-year) window, what are the ranges of possible outcomes for

this investment from a particular statistical perspective?” For example, we found that

the True Diversification® 500 index over every possible consecutive one-year sample

period in the data had a positive return 73.62% of the time which is 6.14% more often

than the benchmark index. A careful examination of Table 9 will show that the

performance of the True Diversification® 500 is consistent and stable. Also, by nearly

every measure available, the True Diversification® 500 was shown to be a superior

method of gaining exposure to a group of large capitalization stocks relative to the

benchmark. An important note on the statistics that are presented in Table 9; they are

presented a raw format and are not converted to an annualized number. For instance,

in the five-year analysis the standard deviation is reported to be 40.37%. This number

could be annualized (approximately) by taking 40.37% and dividing by the square root

of 5 to get an 18% annualized standard deviation.

In Chart 1a we present the growth of a $1000. Chart 1b is the same data as in Chart 1a

but on a log scale to keep the percentage changes equal through time. This method of

analysis can be biased by the choice of starting point. In Table 10 we compute and

compare the total return experienced by an investor in each index starting at the

beginning of each year in our sample through time. Regardless of the timing of an

initial investment, the True Diversification® 500 outperforms the benchmark index

every single time.

In our final analysis of the True Diversification® 500 index as an investment we

examined what a “401k” investor might expect as a return if they made regular

contributions to their investment these indices. In this analysis we assume that at the

beginning of every month the investor adds the same dollar value, in this case $100, to

their investment holdings. Using monthly data we compute the total value of the

investment that would be experienced by each investor. After a total contribution of

$17,500 over the full sample period an investor in the benchmark index would have an

additional $554. Contrast this to an investor in the True Diversification® 500 index; who

after the same $17,500 contribution has accumulated an additional $12,342. This

analysis is presented in Chart 2.



Table 8: These are the results of comparing monthly and annual calendar returns over the sample period of January

1996 through July 2010.

Monthly Annually

Observations 174 15

Percent Outperformed 56.32% 73.33%

Mean Outperformance 1.78% 8.44%

Mean Underperformance -1.39% -5.84%

Max Outperformance 7.45% 20.09%

Min Underperformance -4.72% -9.61%

Table 9: These are the computed results of a rolling window analysis through time over the full sample period from

January 1996 through July 2010. The six-month, one-year, three-year, and five-year results are presented. An

individual rolling window return is defined as the compound return observed during the defined window. That

return is aggregated into a new time-series and the calculated statistics apply to the time series of calculated window

returns. The difference between the True Diversification® 500 and benchmark index computed and presented to the

right is italicized. The risk ratios are raw and not presented in the traditional annualized format.

Six-Months One-Year

True Diversification® 500 .SPX True Diversification® 500 .SPX

Observations 169 169 163 163

Percent Positive 74.56% 65.68% 8.88% 73.62% 67.49% 6.14%

Mean Return 5.21% 2.84% 2.37% 10.65% 5.79% 4.86% Mean Positive Return 10.62% 10.06% 0.55% 20.11% 17.66% 2.45%

Mean Negative Return -10.63% -10.98% 0.35% -15.75% -18.86% 3.11% Maximum Return 35.35% 38.84% -3.49% 53.82% 50.25% 3.57%

Minimum Return -42.11% -42.70% 0.59% -42.44% -44.76% 2.32%

Standard Deviation 12.40% 13.00% -0.60% 19.33% 20.38% -1.05% Semi-Variance 11.83% 9.85% 1.98% 12.63% 11.06% 1.58%

Sharpe Ratio 0.42 0.22 0.20 0.55 0.28 0.27 Sortino Ratio 0.44 0.29 0.15 0.84 0.52 0.32

Three-Years Five-Years

True Diversification® 500 .SPX True Diversification® 500 .SPX

Observations 139 139 115 115

Percent Positive 74.10% 58.99% 15.11% 98.26% 46.96% 51.30%

Mean Return 34.45% 14.68% 19.77% 60.03% 13.84% 46.19% Mean Positive Return 51.79% 41.08% 10.71% 61.26% 44.40% 16.86%

Mean Negative Return -15.16% -23.31% 8.14% -9.05% -13.21% 4.16%

Maximum Return 115.29% 107.62% 7.67% 174.62% 114.71% 59.91% Minimum Return -35.94% -43.40% 7.46% -12.21% -35.77% 23.56%

Standard Deviation 37.43% 40.62% -3.18% 40.37% 35.71% 4.66% Semi-Variance 8.51% 11.41% -2.90% 4.47% 6.80% -2.33%

Sharpe Ratio 0.92 0.36 0.56 1.49 0.39 1.10

Sortino Ratio 4.05 1.29 2.76 13.43 2.04 11.40



Chart 1a: The growth of $1000 in the True Diversification® 500 index and the benchmark index from January 1996

through July 2010.

Chart 1b: The growth of $1000 in the True Diversification® 500 index and the benchmark index from January 1996

through July 2010 with the vertical axis in a log base 10 format where each tick on the vertical axis represents $1000.

$0

$500

$1,000

$1,500

$2,000

$2,500

$3,000

$3,500

$4,000

$4,500

$5,000

1996 1998 2000 2002 2004 2006 2008 2010

Growth of $1000 January 1996 - July 2010

True Diversification 500

.SPX

$1,000

1996 1998 2000 2002 2004 2006 2008 2010

Growth of $1000 January 1996 - July 2010


.SPX



Table 10: A comparison of the total return of $1000 invested in the respective indices at various starting points

through time. The returns are calculated on a monthly return series with the investment being made on the first day

of each year.

True Diversification® 500 .SPX

1996 254.07% 78.85% 175.22%

1997 208.89% 48.72% 160.18%

1998 151.05% 13.52% 137.53%

1999 114.40% -10.38% 124.78%

2000 76.02% -25.02% 101.05%

2001 60.10% -16.56% 76.66%

2002 67.82% -4.05% 71.87%

2003 99.02% 25.21% 73.82%

2004 48.34% -0.93% 49.27%

2005 20.77% -9.10% 29.87%

2006 6.06% -11.75% 17.81%

2007 -10.58% -23.41% 12.83%

2008 -15.89% -24.98% 9.09%

2009 37.16% 21.96% 15.20%

2010 3.34% -1.21% 4.55%

Chart 2: The accumulation of investing $100 on the first every month in the respective indices from January 1996

through July 2010. The total contribution by the investor is $17,500 over this period and should be subtracted to

compare total investment return.

$0

$5,000

$10,000

$15,000

$20,000

$25,000

$30,000

$35,000

1996 1998 2000 2002 2004 2006 2008 2010

Growth of Regular $100 Monthly Contributions January 1996 - July 2010


.SPX



Section 5.4 – Allocation Analysis

We present in Chart 3 the sector allocations within the True Diversification® 500. There

are a myriad of interesting stories represented by the facts within this chart.

Throughout our process we have applied a diversification centric approach to

constructing our index based on an existing and pre-defined universe. As such, one can

see that through time allocations to various sectors within the True Diversification® 500

shift and are not “traditional” compared to capitalization weighting. We believe we are

adding value to the process by allowing the optimization to allocate to the sectors in

diversification centric manner when we create the True Diversification® 500. This fact is

borne out in Table 1 where we added near 200bps in the sector aggregation process

relative to an equal weighting of the sectors. Also, bear in mind when examining Chart

3 our methodology as outlined in Section 4.1. When aggregating the sectors into the

True Diversification® 500 the optimization process has no future knowledge of a

particular sectors performance. You may be inclined to believe otherwise based on

some of the allocation behaviors presented.

We could have elected to start the True Diversification® 500 index in 1997 since we need

at least one year of historical data from the optimized sectors to implement the

allocation optimization process. Instead, we implemented a “naïve” equal-weight

diversification strategy during the first year of the index to give twelve additional data

points when comparing the True Diversification® 500 to the benchmark index. Our

index construction methodology was quarterly overlapping portfolios; therefore, each

quarter 25% of the sector weight within the index was allocated based on the most

recent optimization. To compute the allocation weights presented in Chart 3 for each

sector through time we calculate the average allocation to each sector over a four

quarter period.

A few interesting observations that can be gleaned from Chart 3; the first is the average

allocation applied to each sector through time. The allocations are reasonably stable,

but do move to over- or under-weight relative to their “typical” allocation. You can

also see that some sectors consistently get higher allocations than others. If you were to

cross-reference these allocation weights to the relative optimized sector performance in

Table 1 you would find that the True Diversification® optimization routine had a strong

tendency to allocate consistently to what ex-post turned out to be the best performing

sectors through time.

Secondly, as we examine the sector allocations as we move toward and through the

“tech bubble” in 2000 there are some interesting things to note. One is that the two



highest allocations were to Utilities (24%) and Energy (17%) near the peak in Q4 1999.

Many might considered such an aggressive allocation to Utilities to be “defensive” and

to Energy as a “diversifier”. Also, note the allocation to Information Technology which

shrank in the “tech bubble” run-up from 19% in beginning of 1998 to 10% at the end of

1999. Looking at the Consumer Staples sector during the ensuing bear market, the

optimization continued to allocate to a “defensive” sector increasing its allocation

weight to more than 12% during the year 2000. Again, we note that this is “interesting”

behavior that resulted from the True Diversification® optimization process.

Was the True Diversification® optimization able to repeat this “interesting” behavior for

the financial crisis in 2008? In a word, yes. Throughout 2007 and 2008 the sector

allocation to Utilities increased from less than 10% more than 17% taking a “defensive”

posture ahead of the market peak. We also see the same behavior as before in the

allocation to Consumer Staples with a larger and larger allocation as the crisis unfolded.

How about the Financial sector during this period? In Q4 2006 we had an allocation of

11%, by Q4 2007 the True Diversification® 500 had less than a 4% allocation.

Of course True Diversification® optimization didn’t get everything right. The Energy

sector was the highest allocation percentage in the index when crude oil went from $140

to $35/barrel from mid-2008 to early 2009 taking the performance of the sector down

with it. While these stories describing the sector allocations within the True

Diversification® 500 are meant to be anecdotal in nature, they help present the story of

how True Diversification® optimization can consistently add incremental value to an

investment. We also note here that the sector allocation changes through time are not

necessarily predictive, but rather are descriptive of an investment behavior centered on

diversification.



Chart 3: The optimized sector allocations through time within the True Diversification® 500 index. The sector

weights are the average optimized sector weights over a four quarter window to capture the overlapping portfolio

construction process that was implemented. The sample begins with an equal weight allocation in Q1 1996 and is

fully dynamic consistent with the True Diversification® optimization allocations from Q4 1997 through Q3 2010.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1996 1998 2000 2002 2004 2006 2008 2010

Sector Allocation Weights True Diversification 500

Utilities

Telecommunications

Material

Information Technology

Industrial

Healthcare

Financial

Energy

Consumer Discretionary

Consumer Staples



Section 5.5 – Turnover Analysis

Turnover is important consideration when creating an index or investment product as it

has a direct negative impact on performance due to the associated costs of trading.

Some of these trading costs include brokerage commissions, bid/ask spread and market

slippage and are often considered the “hidden cost” of implementing an investment

strategy. Numerous studies have examined these costs that are borne by an investor in

mutual funds. Mark CarhartU and John BogleV are two studies that were done in the

1990’s that calculated the negative impact of turnover and found that the cost for every

100% in turnover to be 0.95% and 1.2% respectively. A more recent study by David

Blanchett from 2001 to 2006 found substantially lower costs associated with turnover

which is consistent with a long-term industry trend of lower commissions, improved

liquidity, and efficient execution algorithms. Blanchett found that the trading costs per

100% of turnover for large capitalization equities to be 0.19% and across all

capitalizations, including international equities, to be 0.48%W.

We computed the portfolio turnover of the True Diversification® 500 in an attempt to

quantify the implementation costs associated with this index based on our index’s

turnover. For the purposes of this analysis we define turnover as the absolute change in

percentage allocation of an asset from one period to the next. Given our choice of index

construction, which includes two layers of True Diversification® optimization and

quarterly overlapping portfolios, we computed turnover using two different methods in

an effort to establish a range of estimates.

The first method computed the absolute percent change in allocation for an asset from

one quarter to the next for each sector. If an asset’s optimized allocation moved from

1% to 2% between quarters we record this as a 1% change. If the asset is no longer

listed in the subsequent quarter then the percent change is just the asset’s prior

allocation. In the case where a new asset is added in the subsequent quarter then the

percent change is the new allocation percent. Summing the total of these absolute

changes for a sector gives us a turnover figure from one quarter to the next for a sector.

We compute this turnover each quarter and for each sector. We then compute the

portion of turnover associated with each sector based on the sector weights within the

True Diversification® 500 by multiplying the sector turnover by the sector weight.

Because we are using quarterly overlapping portfolios, we compute a four quarter

average of each sector’s turnover within the True Diversification® 500. Summing these

turnover figures gives us the average turnover in the True Diversification® 500 over the

preceding four quarters and from this we can create a time-series of these average

turnover figures. We found that the turnover computed in this manner averaged 62%

per year with a range of 58% to 66%.



The second method we used to quantify portfolio turnover is very similar to the first

however in the case where an asset is present in each quarter we take one-fourth of the

absolute percentage change in allocations to mimic the effects of applying quarterly

overlapping based index construction method. However, in the case of additions or

deletions we still take the full percentage change during that quarter. In both methods

of computing our turnover this severely biases the turnover rate higher relative to how

we computed the returns for the index. The returns assume that additions and

deletions are added incrementally through time, whereas these methods of turnover

computation overstate the turnover at each quarter. As before, we compute and sum

the absolute change in percent allocations from one quarter to the next for each sector

and multiply those turnover figures by the respective weight of each sector in the True

Diversification® 500. Adding these quarterly sector turnover figures gives us a total

turnover figure for that quarter. As before, we compute a time-series of rolling one year

periods, this time by summing rather than averaging over four quarters. We found that

turnover computed using this methodology to be on average 68% per year and the

range of turnover was 74% to 63%.

We intentionally attempted to conservatively state our turnover figures by penalizing

additions and deletions as they occurred rather than smooth their turnover over a one-

year period which would be more consistent with our index construction. The turnover

rate at 60-70% per year is remarkably consistent through time which is not always the

case with capitalization-weighted indices. This turnover figure is comparable to the

median turnover of a mutual fundX. Our turnover is higher than the typical turnover

reported in capitalization index funds. From 2002 – 2008 the SPY ETF reported

turnover of 4% but the range in turnover was 1% to 9% per year. Other alternative

equity indices report higher turnover. For example over the same period the equal

weight S&P 500 ETF, EW, had turnover of 22% per yearY. We believe that the superior

investment results of the True Diversification® 500 more than make up for the potential

costs associated with this level of turnover.

Section 6 – Conclusion

The application of Diversification Optimization to a defined universe, herein the S&P

500, was able to consistently generate greater returns with commensurate, but slightly

lower risk measurements. The results of this research support the efficacy of

diversification as an index construction methodology for an enduring, systematic

investment product, or series of products based on True Diversification®. The relative

superiority of True Diversification during the “lost decade” is of particular consequence

for investors seeking exposure to equities during periods of lower estimated returns.



The ex-post 499 basis points of incremental performance over the index, we believe

constitute the best estimation of forward, relative performance of this methodology.

The consistency and flexibility of the model lend it to numerous applications and

investment products.

The raw database results of this research are available to select enterprise clients and

partners under applicable agreements.



Appendix

This table shows the variation in risk and return values based on the method of portfolio construction, optimization

frequency, and holding period length. All ten sectors and the True Diversification 500 are included using the full

sample data from January 1996 through July 2010. We highlight the construction process in yellow for ease of

comparison. The annualized returns presented were computed from daily returns by taking the average return and

multiplying by 252. Likewise the standard deviation was compute from daily returns and multiplied by the square

root of 252 to annualize it. This method is not as accurate as computing the geometrically compounded return and

therefore is not directly comparable to the returns presented in the body of the paper, but these calculations are

useful as a quick comparison and demonstrates regardless of method of index construction True Diversification®

optimization generated superior performance and we did not selectively choose the “best” construction method.

Overlapping Portfolio No No No No Yes Yes Yes No No No

Optimization Quarter 1 2 3 4 1 , 3 2 , 4 1 , 2 , 3 , 4 1 , 3 2 , 4 1 , 2 , 3 , 4

Holding Period Annual Annual Annual Annual Annual Annual Annual Semi Semi Quarterly

Return

Consumer Staples 7.86% 7.53% 7.76% 6.89% 7.83% 7.21% 7.52% 7.01% 7.13% 7.35%

Consumer Discretionary 9.64% 10.37% 9.25% 8.30% 9.48% 9.33% 9.41% 9.51% 10.00% 9.70%

Energy 17.20% 14.60% 15.96% 17.10% 16.59% 15.85% 16.22% 16.64% 15.81% 16.45%

Financial 10.77% 11.82% 12.33% 8.49% 11.57% 10.16% 10.86% 12.06% 9.19% 9.06%

Healthcare 10.97% 10.65% 11.02% 11.07% 11.02% 10.93% 10.91% 11.31% 10.89% 12.25%

Industrial 5.94% 8.41% 5.47% 7.49% 5.73% 7.95% 6.84% 6.54% 7.66% 7.54%

Information Technology 10.61% 12.99% 11.98% 13.86% 11.33% 13.43% 12.38% 11.57% 14.22% 12.25%

Material 9.54% 10.76% 9.30% 9.23% 9.48% 10.00% 9.74% 9.60% 10.22% 10.56%

Telecommunications 6.50% 7.41% 7.71% 7.08% 7.12% 7.24% 7.18% 5.79% 6.21% 5.31%

Utilities 7.63% 6.37% 6.84% 8.78% 7.25% 7.57% 7.41% 8.63% 6.77% 7.72%

True Diversification® 500 9.66% 10.54% 11.65% 10.33% 10.66% 10.43% 10.59% 10.94% 10.28% 10.21%

Standard Deviation

Consumer Staples 15.36% 15.26% 15.25% 15.34% 15.19% 15.19% 15.12% 15.57% 15.34% 15.57%

Consumer Discretionary 22.82% 22.67% 22.61% 22.29% 22.61% 22.38% 22.46% 22.86% 22.74% 22.95%

Energy 31.26% 30.40% 30.96% 31.14% 31.00% 30.65% 30.79% 31.42% 31.31% 31.69%

Financial 29.74% 30.00% 30.79% 29.53% 29.83% 29.48% 29.53% 32.48% 30.22% 31.37%

Healthcare 19.46% 19.05% 19.53% 19.08% 19.30% 18.88% 19.02% 19.65% 19.19% 19.57%

Industrial 20.25% 19.90% 20.25% 20.83% 20.15% 20.22% 20.14% 20.47% 20.64% 20.88%

Information Technology 30.16% 29.85% 30.45% 31.00% 30.13% 30.22% 30.10% 30.69% 30.83% 31.10%

Material 22.83% 23.61% 22.72% 23.51% 22.64% 23.39% 22.93% 22.89% 23.98% 23.94%

Telecommunications 27.66% 27.48% 26.59% 27.71% 26.82% 27.28% 26.95% 28.21% 28.12% 28.79%

Utilities 20.63% 20.14% 19.42% 20.36% 19.84% 20.05% 19.88% 20.41% 20.83% 21.03%

True Diversification® 500 20.27% 19.59% 19.72% 20.22% 19.93% 19.86% 19.51% 19.87% 19.85% 19.91%

Sharpe Ratio

Consumer Staples 0.51 0.49 0.51 0.45 0.52 0.47 0.50 0.45 0.46 0.47

Consumer Discretionary 0.42 0.46 0.41 0.37 0.42 0.42 0.42 0.42 0.44 0.42

Energy 0.55 0.48 0.52 0.55 0.54 0.52 0.53 0.53 0.51 0.52

Financial 0.36 0.39 0.40 0.29 0.39 0.34 0.37 0.37 0.30 0.29

Healthcare 0.56 0.56 0.56 0.58 0.57 0.58 0.57 0.58 0.57 0.63

Industrial 0.29 0.42 0.27 0.36 0.28 0.39 0.34 0.32 0.37 0.36

Information Technology 0.35 0.44 0.39 0.45 0.38 0.44 0.41 0.38 0.46 0.39

Material 0.42 0.46 0.41 0.39 0.42 0.43 0.42 0.42 0.43 0.44

Telecommunications 0.23 0.27 0.29 0.26 0.27 0.27 0.27 0.21 0.22 0.18

Utilities 0.37 0.32 0.35 0.43 0.37 0.38 0.37 0.42 0.32 0.37

True Diversification® 500 0.48 0.54 0.59 0.51 0.53 0.53 0.54 0.55 0.52 0.51



A True Diversification® is a registered trademark of Gravity Investments, LLC. James

Damschroder, the founder of Gravity Investments, LLC, holds several patents awarded

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applied to equity index construction - gravity investments€¦ · true diversification a...

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