alternatives to cap-weighted indices

8/6/2019 Alternatives to Cap-Weighted Indices

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Alternatives to Cap-Weighted Indices

EDHEC Institutional Days

Monaco, December 8th, 2010, 14:15-15:30

2

Lionel Martellini

Professor of Finance, EDHEC Business School

Scientific Director, EDHEC Risk Institute

[email protected]

www.edhec-risk.com


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Introduction: Beyond Cap-Weighting

In Search of Representative Indices

Cap-Weighting

Fundamental Weights

Designing Efficient Investment Benchmarks

Ad-Hoc Diversification: De-concentrating Portfolios

Scientific Diversification: Towards the Efficient Frontier

Alternative Weighting Schemes: Conditions for Optimality?

Conclusion: Concept Selection vs. Concept Diversification

Outline


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Cap-Weighting

Fundamental Weights







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A number of (index or fund) providers have recently designedand launched non cap-weighted indices.

The (non-exhaustive) list includes:

fundamental indices

equally-weighted indices minimum variance indices

efficient indices

equal-risk contribution (a.k.a. risk parity) indices

maximum diversification indices

This presentation provides a summary of the objectives of, and

assumptions behind, the various indexing concepts.

Beyond Cap Weighting

Comparing Alternatives


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Beyond Cap Weighting

Concepts versus Figures

We will not focus so much on past performance; track records (bydefinition) all look pretty good!

Instead, we propose to provide an academic perspective on theconceptual assumptions underpinning the different methods.

(Even out-of-sample) track records are sample-dependent and thusperformance figures rely on the data and time period at hand.

For long-term benchmarks, it is important that performance is drivenby a sound concept that relies on reasonable assumptionsrather than

by exploiting anomaliesin past returns data. If achieving higher risk-adjusted performance is not the focus of a

methodology, achieving it is at best a collateral benefit.

In Senecas words (circa 30 BC):If one does not know to which port one is sailing, no wind is favorable.


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The words index and benchmark are often usedinterchangeably; yet they define a priorivery different concepts.

Market perspective: an indexis a portfolio that shouldrepresent the performance of a given segment of the market.

=> focus on representativity

Investor perspective: a benchmarkis a reference portfolio thatshould represent the fair reward expected in exchange for risk

exposures that an investor is willing to accept.=> focus on efficiency

CW portfolios have long been portrayed as representative and

efficient, but have faced increased criticism on both fronts.

Beyond Cap-Weighting

Which Port do we want to Sail to: Indices versus Benchmarks


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Cap-Weighting

Fundamental Weights







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A market cap weighted scheme is the obvious default optionwhen it comes to representing a given segment of the market.

Market cap weighted indices provide by construction a fairrepresentation of the stock market;

In the end, cap-weighting is nothing but an ad-hoc weightingscheme that achieves some form of representativity.

Cap-weighted indices, however, may not provide a fairrepresentation of the underlying economic fundamentals.

Some have argued that they represent well the stock market butnot the economy.


Cap-Weighting for Representativity?


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Rather than using the market cap, fundamental indices usefirm attributes such as book value, dividends, sales or cash

flows as measures of size.

These indices aim at better representing the economy.Arnott (2007): The Fundamental Index weights companies in

accordance to their footprint in the broad economy [] you wind upwith a portfolio that mirrors the economy.

Whether or not fundamentally weighted indices betterrepresent the economy is actually an open question, if only

because representativity is not a concept that is linked to clearmeasures.


Fundamental Weighting for Representativity?


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Cap-Weighting

Fundamental Weights







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In any case, it is not clear why investors would care about theirportfolios representing the economy.

From the investors perspective, the focus should be onefficiency: obtaining fair rewards for given risk budgets.

Efficiency is intimately related to diversification: it is byconstructing well-diversified portfolios that one can achieve afair reward for a given risk exposure.

CW portfolios in fact appear to be rather inefficient and poorlydiversified portfolios, and several approaches have beendeveloped so as to improve diversification compared to cap-weighting.


Efficiency is Related to Diversification


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Cap-weighting is often believed to lead to risk/reward efficientportfolios, but that belief is not really based on firm grounds.

The belief in efficiency of CW is based on a nave interpretationof W. Sharpes Capital Asset Pricing Model (CAPM):

No need to gather any information on risk & return parametersto find optimal portfolios ... because everybody else does!

When relaxing the highly unrealistic assumptions of the CAPM,financial theory does not predict that the market portfolio is efficient(Sharpe (1991), Markowitz (2005)).

If there are multiple risk factors, the mean-variance optimal portfoliois no longer CW (Merton (1971), Cochrane (1999)); in a post-CAPMmulti-factor world, CW is just an arbitrary weighting scheme.


Cap-Weighting for Efficiency?


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Cap-weighting is particularly inefficient because it leads to highconcentration: the effective number of stocksin the index is low.

Some index construction approaches simply avoid thisconcentration; such simple de-concentration strategies do notaim for optimality and are not grounded in portfolio theory.

The effective number ofstocks is the reciprocal of theHerfindhal index, a measure

of portfolio concentration.

Index Nominalnumber

Effectivenumber

S&P 500 94

NASDAQ 100 37

FTSE 100 (UK) 100 28

FTSE Eurobloc 300 104

FTSE Japan 500 103Average effective number based on quarterly assessment for the time

period 01/1959 to 12/2008 for the S&P, 01/1975 to 12/2008 for theNASDAQ, and 12/2002 to 12/2008 for the other indices .


CW leads to High Concentration


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Cap-Weighting

Fundamental Weights







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Nave de-concentration:

Equal-weighting simply gives the same weight to each of Nstocksin the index (1/Nrule).

Equal-weighting is the nave route to constructing well diversifiedportfolios.

Semi-nave de-concentration:

Equal risk contribution (ERC) takes into account contribution to risk.

Contribution to risk is not proportional to dollar contribution.

Find portfolio weights such that contributions to risk are equal(Maillard, Roncalli and Teiletche (2010)):

Ad-Hoc Approach to Well-Diversified Portfolios

Equal Weighting and Equal Risk Contribution

16 16

j

p

j

i

p

i

w

w

w

w

=


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Statistical de-concentration:

Define a diversification index and try and maximize it by utilizing the

correlations that drive the magic of diversification: The whole isbetter than the sum of its parts.

Maximum Diversification (also known as anti-benchmark) aims atgenerating portfolios with the highest possible diversification index

(Choueifaty and Coignard (2008)):

The weighted average risk(in the numerator) will be high comparedto portfolio risk(in the denominator) and thus DIwill be high if the

portfolio weights exploit well the correlations.

Ad-Hoc Approach to Well-Diversified Portfolios

Maximum Diversification Benchmarks/Anti-Benchmark

=

=

=n

ji

ijji

n

i

ii

w

ww

w

MaxDI

1,

1

17


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Cap-Weighting

Fundamental Weights







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Scientific Approach to Well-Diversified Portfolios

Towards the Efficient Frontier

19 19

Scientific diversification is based on reaching a high risk/returnobjective through portfolio construction techniques.

In practice, to get a decent proxy for efficient portfolios, one needsto use careful risk and return parameter estimates; practicalapproaches to scientific diversification make different choicesregarding the challenge of risk and return estimation.

Technology is available to generate reliable risk parameterestimates: Suitably designed factor models to mitigate the curse of

dimensionality(see also statistical shrinkage techniques). Accounting for non-stationarity: e.g., GARCHand Regime Switching

models.

On the other hand, statistics is close to useless in terms ofexpected return estimation (Merton (1980)).


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Volatility

Expected

Return

Maximum SharpeRatio (MSR) Portfolio


GMV vs. MSR

GlobalMinimumVariance

(GMV)Portfolio

The MSR provides the highest reward per unit of portfolio volatility:needed optimization inputs are expected returns, correlations andvolatilities.

The GMV provides the lowest possible portfolio volatility: neededoptimization inputs are correlations and volatilities. 20


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If you feel comfortable about estimating risk parameters the variance-covariance matrix, but not about estimating expected return

parameters, the global minimum variance (GMV) benchmark is theway to go (e.g., Amenc and Martellini (2003)).


Minimum Variance Benchmarks (GMV)

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This approach provides a low

volatility portfolio but also a lowperformance portfolio: ex-ante,MSR+cash is better than GMV.

Ex-post, MV portfolios tend to beconcentrated portfolios withoverweighting of low volatilitystocks, with a Sharpe ratio lowerthan that of EW (Garlappi et al.

(2007)).21

0 5 10 15 20 250

2

4

6

8

10

12

14

16

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Annualiz

edexpectedreturn

Annualized volatility

Efficient frontier

Tangency line

GMV

MSR

MSR + cash


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Efficient Indexation (MSR)

Efficient Indexation is about maximizing the Sharpe ratio.

Just like in the Minimum Variance approach, EfficientIndexation exploits information on the covariance matrix ofstock returns; the approach uses suitably designed factormodels to mitigate the curse of dimensionality.

While direct estimation of expected returns from past returns isuseless, all hope on expected returns estimation is not lost!

Common sense suggests that expected return parametersshould be positively related to risk parameters (risk-returntradeoff).

Efficient Indexation uses indirect estimation of expected returns

through a stocks riskiness.


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Theory unambiguously confirms the existence of a positiverisk/return relationship:

Systematic risk is rewarded (APT);

Specific risk is also rewarded (Merton (1987)) (*);

Total volatility (model-free) should therefore be rewarded;

Higher moment risk is also rewarded (many references).

Use the risk-return relationship to build efficient portfolios: magicof diversification is about mixing high-risk-and-therefore-high-

return stocks in a smart way so as to generate low risk portfolios!

(*) See also Barberis and Huang (2001) Malkiel and Yu (2002), Boyle, Garlappi, Uppal and Wang (2009) .


On the Risk-Return Relationship


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iv Puzzle VW Portfolios over Short Horizons

Ang, Hodrick, Xing and Zhang (2006,2009): iv puzzle12 Month idiosyncratic volatility

1 Month realized return10 VW PortfoliosValue Weighted Portfolio returnsNegative RelationshipHigh-Low returns mainly driven by high

iVol portfolio

Value Weighted Portfolios: Short Horizon (iVol)

0.00

0.01

0.10

1.00

10.00

100.00

1000.00

64' 67' 70' 73' 76' 79' 82' 85' 88' 91' 94' 97' 00' 03' 06' 09'

Valueof1$investedin

1964

Low 2 3 4 5 6 7 8 9 High

Value Weighted Portfolios: Short Horizon (iVol)

-10.0%

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

0% 5% 10% 15% 20% 25% 30%

Average Risk over Cross-Section

AveragePortfolio

Retur

n

and

Standard

ErrorBoun

ds

Ten VW portfolios containing an equal number ofstocks (extracted from the CRSP data base) are builtevery month after sorting the stocks based on somerisk measure, here idiosyncratic volatility w.r.t. FF

model (calculated using daily data for last 12months); the returns of each of these portfolios are

calculated subsequent one-month periods andaveraged across the portfolio formation date.


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No iv Puzzle EW Portfolios over Short Horizons

Negative relationship disappears whenEW used.Extremely low return of High-Volatilityportfolio disappears.We still do not have a positive relationship.

Return reversal : Huang, Liu, Rhee, and Zhang (2009) Extreme winners and losers (over the past month)typically have high iVol over the last 1 month

In high iVol portfolios: # past winners is almost equalto # past losers, but average weight of past winners issubstantially larger. Short-term return reversal effect: past-month winnerstend to under perform in subsequent month . So, VW lowers the portfolio return compared to other

portfolios and EW does not.

Equally Weighted Portfolios: Short Horizon (iVol)

0.10

1.00

10.00

100.00

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10000.00

64' 67' 70' 73' 76' 79' 82' 85' 88' 91' 94' 97' 00' 03' 06' 09'Valueof1$investedin

1964

Low 2 3 4 5 6 7 8 9 High

Equally Weighted Portfolios: Short Horizon (iVol)

-10.0%

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

0% 5% 10% 15% 20% 25% 30%


AveragePortfolio

Retur

n

and

Standard

ErrorBoun

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Positive risk-return relationship across allportfolios.

Not only the extreme portfolios.Intuition: long-horizon realized returns areless susceptible to local events and hencebetter proxies for expected returns.compared to short-horizon realized returns.

Equally Weighted Portfolios: Long Horizon (iVol)

0.10

1.00

10.00

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10000.00

64' 67' 70' 73' 76' 79' 82' 85' 88' 91' 94' 97' 00' 03' 06' 09'Valueof1$invested

in

1964

Low 2 3 4 5 6 7 8 9 High

Equally Weighted Portfolios: Long Horizon (iVol)

-10.0%

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

0% 5% 10% 15% 20% 25%


AveragePortfolio

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Standard

ErrorBoun

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What iv Puzzle ? EW Portfolios over Long Horizons

Ten EW portfolios containing an equal number ofstocks (extracted from the CRSP data base) are builtevery month after sorting the stocks based on somerisk measure, here idiosyncratic volatility w.r.t. FF

model (calculated using daily data for last 12months); the returns of each of these portfolios are

calculated subsequent 24-months periods andaveraged across the portfolio formation date.


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Blitz and Vliet (2007)12 Month total volatility

1 Month realized return10 PortfoliosValue Weighted Portfolio returnsNegative risk-return relationship High-Low returns mainly driven by

low tVol portfolio

Value Weighted Portfolios: Short Horizon (tVol)

0.01

0.10

1.00

10.00

100.00

1000.00

64' 67' 70' 73' 76' 79' 82' 85' 88' 91' 94' 97' 00' 03' 06' 09'

Valueof1$investedin

1964

Low 2 3 4 5 6 7 8 9 High

Value Weighted Portfolios: Short Horizon (tVol)

-10.0%

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

0% 5% 10% 15% 20% 25% 30%


AveragePortfolio

Retur

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and

Standard

ErrorBoun

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tv Puzzle VW Portfolios over Short Horizons

Ten VW portfolios containing an equal number ofstocks (extracted from the CRSP data base) are builtevery month after sorting the stocks based on somerisk measure, here total volatility (calculated using

daily data for last 12 months); the returns of each ofthese portfolios are calculated subsequent one-month periods and averaged across the portfolio

formation date.


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12 Month total volatility

1 Month realized return10 PortfoliosEqually Weighted Portfolio returns

Again, Negative relationship disappearswhen EW used. Extremely low return of High-Volatilityportfolio disappears.We still do not have a positive relationship.

Equally Weighted Portfolios: Short Horizon (tVol)

0.10

1.00

10.00

100.00

1000.00

64' 67' 70' 73' 76' 79' 82' 85' 88' 91' 94' 97' 00' 03' 06' 09'Valueof1$invested

in

1964

Low 2 3 4 5 6 7 8 9 High

Equally Weighted Portfolios: Short Horizon (tVol)

-10.0%

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

0% 5% 10% 15% 20% 25% 30%


AveragePortfolio

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and

Standard

ErrorBoun

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No tv Puzzle EW Portfolios over Short Horizons


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12 Month total volatility

24 Month realized return10 EW Portfolios

Positive risk-return relationshipacross all portfolios.Not only the extreme portfolios.Results are valid even tVol iscalculated using larger period.

Equally Weighted Portfolios: Long Horizon (tVol)

0.10

1.00

10.00

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10000.00

64' 67' 70' 73' 76' 79' 82' 85' 88' 91' 94' 97' 00' 03' 06' 09'Valueof1$investedin

1964

Low 2 3 4 5 6 7 8 9 High

Equally Weighted Portfolios: Long Horizon (tVol)

-10.0%

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

0% 5% 10% 15% 20% 25%


AveragePortfolio

Retur

n

and

Standard

ErrorBoun

ds


What tv Puzzle? EW Portfolios over Long Horizons

Ten EW portfolios containing an equal number ofstocks (extracted from the CRSP data base) are builtevery month after sorting the stocks based on somerisk measure, here total volatility (calculated using

daily data for last 12 months); the returns of each ofthese portfolios are calculated subsequent 24-monthperiods and averaged across the portfolio formation

date.


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Evidence that stock downside risk is related to expected returns


Downside Risk & Expected Returns

Authors Risk Measure MomentsZhang (2005) Skewness Skew

Boyer, Mitton andVorkink (2009)

Skewness Skew

Tang and Shum (2003) Skewness Skew

Connrad, Dittmar andGhysels (2009)

Skewness Skew

Ang et al. (2006) Downside correlation Vol, Skew, Kurt

Huang et al (2009) Value-at-Risk (EVT) Vol, Skew, Kurt

Bali and Cakici (2004) Value-at-Risk(Historical)

Vol,Skew, Kurt

Chen et al. (2009) Semi-deviation Vol, Skew, Kurt

Estrada (2000) Semi-deviation Vol, Skew, Kurt


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Total Semi-Deviation EW Decile Portfolios Long Horizon

12 Month Total Semi-Deviation

24 Month realized return10 PortfoliosEqually Weighted Portfolio returns

Equally Weighted Portfolios: Long Horizon (sem i-deviation)

1.00

10.00

100.00

1000.00

10000.00

64' 67' 70' 73' 76' 79' 82' 85' 88' 91' 94' 97' 00' 03' 06' 09'

Valueof1$invested

in

1964

Low 2 3 4 5 6 7 8 9 High

Equally Weighted Portfolios: Long Horizon (sem i-deviation)

-10.0%

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

0% 5% 10% 15% 20%


AveragePortfolio

Retur

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and

Standard

ErrorBoun

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Positive risk-return relationship

across all portfolios.Not only the extreme portfolios.Results are valid even semi-deviation is calculated using largerperiod.


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The average cumulative return for portfolios sorted on semi-deviation.

0%

10%

20%

30%

40%

50%

60%

70%

80%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Month after portfolio formation

Port Low

Port 2

Port 3

Port 4

Port 5

Port 6

Port 7

Port 8

Port 9

Port High

Ten portfolios containing an equal number of stocks (extracted from the CRSP data base) are built every month after sorting the stocks

based on their semi-deviation (calculated using daily data for last 30 months); the cumulative returns of each of these portfolios are

calculated for various holding periods and averaged across the portfolio formation date.


Downside Risk and Expected Returns


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Long-Term Results

Index

Ann.

average

return

Ann. std.

Deviation

Sharpe

Ratio

Information

Ratio

Tracking

Error

Efficient Index 11.63% 14.65% 0.41 0.52 4.65%

Cap-weighted 9.23% 15.20% 0.24 0.00 0.00%

Difference (Efficientminus Cap-weighted)

2.40% -0.55% 0.17 - -

p-value for difference 0.14% 6.04% 0.04% - -

The table shows risk and return statistics portfolios constructed with using the same set of constituents as the cap-weighted S&P 500 index.

Rebalancing is quarterly subject to an optimal control of portfolio turnover (by setting the reoptimisation threshold to 50%). Portfolios areconstructed by maximising the Sharpe ratio given an expected return estimate and a covariance estimate. The expected return estimate is

set to the median total risk of stocks in the same decile when sorting on total risk. The covariance matrix is estimated using an implicit factormodel for stock returns. Weight constraints are set so that each stock's weight is between 1/2N and 2/N, where N is the number of index

constituents. P-values for differences are computed using the paired t-test for the average, the F-test for volatility, and a Jobson-Korkie testfor the Sharpe ratio. The results are based on weekly return data from 01/1959. We use a calibration period of 2 years and rebalance the

portfolio every three months (at the beginning of January, April, July and October).

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Cap-Weighting

Fundamental Weights







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Each of the aforementioned weighting methods makes differentmethodological choices.

However, portfolio theory tells us that there is only one optimalportfolio: the tangency (MSR) portfolio.

Question: Under which conditions would the portfolio constructionchoices of different index weighting schemes be truly optimal?

KIS(BNTS) principle: robustness of a method may justify simpleassumptions but is important that assumptions also remainreasonable; if the conditions are too restrictive, we are unlikely toobtain optimal portfolios.

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Conditions for Optimality

Keep it Simple But Not Too Simple


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Volatility

ExpectedReturn

The true tangency portfolio is a function of the(unknown) trueparameter values

OptimalPortfolio

ijiiMSRfw ,,=

Implementable proxies depend onassumptions aboutparameter values

ijiiMSR fw ,, =

Cap-weighted index

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Assumptions about Parameter Values


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Indices aiming at Representativity

Cap-weighting:

One simply turns to the market, and hope that everyone else hasdone a careful job at estimating risk and return parameters anddesigning efficient benchmarks so we simply do not have to itourselves!

This would be a very nave belief in the CAPM.

Fundamental weighting:

Conditions under which this weighting scheme would be optimalare not clear.

As an example, it would be optimal if risk parameters areidentical and expected return is proportional to the fourfundamental variables used for the weighting.

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De-Concentration Approaches

Equal-weighting:

Optimal if and only if one assumes all stocks have the sameexpected return and

the same volatility and

the same pairwise correlations!

Equal Risk Contribution (Maillard et al. (2010)):

Optimal if and only if one assumes all stocks have sameSharpe ratios and

the same pairwise correlations.

Maximum Diversification (Choueifaty and Coignard (2008)):

Optimal if and only if one assumes all stocks have sameSharpe ratios.

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Efficient Frontier Approaches

Minimum Variance:

Only optimal if one assumes that all stocks have the sameexpected returns, hardly a neutral/reasonable choice.

Efficient Indexation:

Optimal if one assumes that expected returns between stocksare different, and positively related to downside risk.

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Cap-Weighting

Fundamental Weights







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Conclusion

Cap-weighted indices are not efficient or well-diversifiedportfolios because they were never meant to be.

While alternative weighting schemes typically improveperformance, they have different objectives and more or lessstrong assumptions need to be made before one canconclude that they are truly optimal portfolios.

Investors beyond assessing performance need toconsider whether assumptions and objectives behind eachconcept are compatible with their views and needs.

An outstanding question, which we do not address in thispresentation, is that of concept diversification versus conceptselection.


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References

Amenc, N., F. Goltz, L. Martellini, and P. Retkowsky, 2010, Efficient Indexation: An Alternative toCap-Weighted Indices," Journal of Investment Management, forthcoming. Bali, Turan G., and Nusret Cakici, 2004, Value at Risk and Expected Stock Returns. Financial

Analysts Journal, 60(2), 57-73. Barberis, N., and M. Huang, 2001, Mental Accounting, Loss Aversion and Individual Stock Returns,Journal of Finance, 56, 1247-1292. Barberis, N. and M. Huang, Stocks as lotteries: The implications of probability weighting forsecurity prices, 2007, working paper. Boyer, B., and K. Vorkink, 2007, Equilibrium Underdiversification and the Preference for Skewness,

Review of Financial Studies, 20(4), 1255-1288. Boyer, B., T. Mitton and K. Vorkink, 2009, Expected Idiosyncratic Skewness, Review of FinancialStudies, forthcoming. Chen, D.H., C.D. Chen, and J. Chen, 2009, Downside risk measures and equity returns in theNYSE, Applied Economics, 41, 1055-1070. Connrad, J., R.F. Dittmar and E. Ghysels, Ex Ante Skewness and Expected Stock Returns, 2008,

working paper. Choueifaty, Y., and Y. Coignard, 2008, Toward Maximum Diversification, The Journal of PortfolioManagement, 35, 1, 40-51. Cochrane, John H., 2005, Asset Pricing (Revised), Princeton University Press Estrada, J, 2000, The Cost of Equity in Emerging Markets: A Downside Risk Approach, EmergingMarkets Quarterly, 19-30.

Grinold, Richard C. Are Benchmark Portfolios Efficient?, Journal of Portfolio Management, Fall1992.


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References

Haugen, R. A., and Baker N. L., The Efficient Market Inefficiency of Capitalization-weighted StockPortfolios, Journal of Portfolio Management, Spring 1991. Malkiel, B., and Y. Xu, 2002, Idiosyncratic Risk and Security Returns, working Paper, University of

Texas at Dallas. Maillard,, S., T. Roncalli and J. Teiletche, 2010, The Properties of Equally Weighted RiskContribution, Journal of Portfolio Management. Markowitz, H. M., Market efficiency: A Theoretical Distinction and So What?, Financial AnalystsJournal, September/October 2005. Merton, Robert, 1987, A Simple Model of Capital Market Equilibrium with Incomplete Information,

Journal of Finance, 42(3). Schwartz, T., 2000, How to Beat the S&P500 with Portfolio Optimization, DePaul University,working paper. Sharpe, W.F., 1991 , Capital Asset Prices with and without Negative Holdings, Journal ofFinance, 46. Tang, Y., and Shum, 2003, The relationships between unsystematic risk, skewness and stock

returns during up and down markets, International Business Review. Tinic, S., and R. West, 1986, Risk, Return and Equilibrium: A revisit, Journal of Political Economy,94, 1, 126-147. Tobin, J., 1958, Liquidity Preference as Behavior Towards Risk, Review of Economic Studies, 67,65-86. Zhang, Y., 2005, Individual Skewness and the Cross-Section of Average Stock Returns, Yale

University, working paper.

alternatives to cap-weighted indices

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