Alternatives to Cap-Weighted Indices
-
Upload
diego-liechti -
Category
Documents
-
view
228 -
download
0
Transcript of Alternatives to Cap-Weighted Indices
-
8/6/2019 Alternatives to Cap-Weighted Indices
1/43
-
8/6/2019 Alternatives to Cap-Weighted Indices
2/43
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
www.edhec-risk.com
-
8/6/2019 Alternatives to Cap-Weighted Indices
3/43
3
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
-
8/6/2019 Alternatives to Cap-Weighted Indices
4/43
4
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
-
8/6/2019 Alternatives to Cap-Weighted Indices
5/43
5
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
-
8/6/2019 Alternatives to Cap-Weighted Indices
6/43
6
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.
-
8/6/2019 Alternatives to Cap-Weighted Indices
7/43
7
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
-
8/6/2019 Alternatives to Cap-Weighted Indices
8/43
8
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
-
8/6/2019 Alternatives to Cap-Weighted Indices
9/43
9
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.
In Search of Representative Indices
Cap-Weighting for Representativity?
-
8/6/2019 Alternatives to Cap-Weighted Indices
10/43
10
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.
In Search of Representative Indices
Fundamental Weighting for Representativity?
-
8/6/2019 Alternatives to Cap-Weighted Indices
11/43
11
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
-
8/6/2019 Alternatives to Cap-Weighted Indices
12/43
12
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.
Designing Efficient Investment Benchmarks
Efficiency is Related to Diversification
-
8/6/2019 Alternatives to Cap-Weighted Indices
13/43
13
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.
Designing Efficient Investment Benchmarks
Cap-Weighting for Efficiency?
-
8/6/2019 Alternatives to Cap-Weighted Indices
14/43
14
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 .
Designing Efficient Investment Benchmarks
CW leads to High Concentration
-
8/6/2019 Alternatives to Cap-Weighted Indices
15/43
15
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
-
8/6/2019 Alternatives to Cap-Weighted Indices
16/43
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
=
-
8/6/2019 Alternatives to Cap-Weighted Indices
17/43
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
-
8/6/2019 Alternatives to Cap-Weighted Indices
18/43
18
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
-
8/6/2019 Alternatives to Cap-Weighted Indices
19/43
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)).
-
8/6/2019 Alternatives to Cap-Weighted Indices
20/43
Volatility
Expected
Return
Maximum SharpeRatio (MSR) Portfolio
Scientific Approach to Well-Diversified Portfolios
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
-
8/6/2019 Alternatives to Cap-Weighted Indices
21/43
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)).
Scientific Approach to Well-Diversified Portfolios
Minimum Variance Benchmarks (GMV)
21
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
18
Annualiz
edexpectedreturn
Annualized volatility
Efficient frontier
Tangency line
GMV
MSR
MSR + cash
-
8/6/2019 Alternatives to Cap-Weighted Indices
22/43
22
Scientific Approach to Well-Diversified Portfolios
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.
-
8/6/2019 Alternatives to Cap-Weighted Indices
23/43
23
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) .
Scientific Approach to Well-Diversified Portfolios
On the Risk-Return Relationship
-
8/6/2019 Alternatives to Cap-Weighted Indices
24/43
Scientific Approach to Well-Diversified Portfolios
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$investedin
1964
Low 2 3 4 5 6 7 8 9 High
Value Weighted Portfolios: Short Horizon (iVol)
-10.0%
-5.0%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
0% 5% 10% 15% 20% 25% 30%
Average Risk over Cross-Section
AveragePortfolio
Retur
n
and
Standard
ErrorBoun
ds
Ten VW portfolios containing an equal number ofstocks (extracted from the CRSP data base) are builtevery month after sorting the stocks based on somerisk measure, here idiosyncratic volatility w.r.t. FF
model (calculated using daily data for last 12months); the returns of each of these portfolios are
calculated subsequent one-month periods andaveraged across the portfolio formation date.
-
8/6/2019 Alternatives to Cap-Weighted Indices
25/43
Scientific Approach to Well-Diversified Portfolios
No iv Puzzle EW Portfolios over Short Horizons
Negative relationship disappears whenEW used.Extremely low return of High-Volatilityportfolio disappears.We still do not have a positive relationship.
Return reversal : Huang, Liu, Rhee, and Zhang (2009) Extreme winners and losers (over the past month)typically have high iVol over the last 1 month
In high iVol portfolios: # past winners is almost equalto # past losers, but average weight of past winners issubstantially larger. Short-term return reversal effect: past-month winnerstend to under perform in subsequent month . So, VW lowers the portfolio return compared to other
portfolios and EW does not.
Equally Weighted Portfolios: Short Horizon (iVol)
0.10
1.00
10.00
100.00
1000.00
10000.00
64' 67' 70' 73' 76' 79' 82' 85' 88' 91' 94' 97' 00' 03' 06' 09'Valueof1$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%
Average Risk over Cross-Section
AveragePortfolio
Retur
n
and
Standard
ErrorBoun
ds
-
8/6/2019 Alternatives to Cap-Weighted Indices
26/43
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
100.00
1000.00
10000.00
64' 67' 70' 73' 76' 79' 82' 85' 88' 91' 94' 97' 00' 03' 06' 09'Valueof1$invested
in
1964
Low 2 3 4 5 6 7 8 9 High
Equally Weighted Portfolios: Long Horizon (iVol)
-10.0%
-5.0%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
0% 5% 10% 15% 20% 25%
Average Risk over Cross-Section
AveragePortfolio
Retur
n
and
Standard
ErrorBoun
ds
Scientific Approach to Well-Diversified Portfolios
What iv Puzzle ? EW Portfolios over Long Horizons
Ten EW portfolios containing an equal number ofstocks (extracted from the CRSP data base) are builtevery month after sorting the stocks based on somerisk measure, here idiosyncratic volatility w.r.t. FF
model (calculated using daily data for last 12months); the returns of each of these portfolios are
calculated subsequent 24-months periods andaveraged across the portfolio formation date.
-
8/6/2019 Alternatives to Cap-Weighted Indices
27/43
Blitz and Vliet (2007)12 Month total volatility
1 Month realized return10 PortfoliosValue Weighted Portfolio returnsNegative risk-return relationship High-Low returns mainly driven by
low tVol portfolio
Value Weighted Portfolios: Short Horizon (tVol)
0.01
0.10
1.00
10.00
100.00
1000.00
64' 67' 70' 73' 76' 79' 82' 85' 88' 91' 94' 97' 00' 03' 06' 09'
Valueof1$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%
Average Risk over Cross-Section
AveragePortfolio
Retur
n
and
Standard
ErrorBoun
ds
Scientific Approach to Well-Diversified Portfolios
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.
-
8/6/2019 Alternatives to Cap-Weighted Indices
28/43
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$invested
in
1964
Low 2 3 4 5 6 7 8 9 High
Equally Weighted Portfolios: Short Horizon (tVol)
-10.0%
-5.0%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
0% 5% 10% 15% 20% 25% 30%
Average Risk over Cross-Section
AveragePortfolio
Retur
n
and
Standard
ErrorBoun
ds
Scientific Approach to Well-Diversified Portfolios
No tv Puzzle EW Portfolios over Short Horizons
-
8/6/2019 Alternatives to Cap-Weighted Indices
29/43
12 Month total volatility
24 Month realized return10 EW Portfolios
Positive risk-return relationshipacross all portfolios.Not only the extreme portfolios.Results are valid even tVol iscalculated using larger period.
Equally Weighted Portfolios: Long Horizon (tVol)
0.10
1.00
10.00
100.00
1000.00
10000.00
64' 67' 70' 73' 76' 79' 82' 85' 88' 91' 94' 97' 00' 03' 06' 09'Valueof1$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%
Average Risk over Cross-Section
AveragePortfolio
Retur
n
and
Standard
ErrorBoun
ds
Scientific Approach to Well-Diversified Portfolios
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.
-
8/6/2019 Alternatives to Cap-Weighted Indices
30/43
Evidence that stock downside risk is related to expected returns
Scientific Approach to Well-Diversified Portfolios
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
-
8/6/2019 Alternatives to Cap-Weighted Indices
31/43
Scientific Approach to Well-Diversified Portfolios
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%
Average Risk over Cross-Section
AveragePortfolio
Retur
n
and
Standard
ErrorBoun
ds
Positive risk-return relationship
across all portfolios.Not only the extreme portfolios.Results are valid even semi-deviation is calculated using largerperiod.
-
8/6/2019 Alternatives to Cap-Weighted Indices
32/43
32
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.
Scientific Approach to Well-Diversified Portfolios
Downside Risk and Expected Returns
-
8/6/2019 Alternatives to Cap-Weighted Indices
33/43
Scientific Approach to Well-Diversified Portfolios
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).
33
-
8/6/2019 Alternatives to Cap-Weighted Indices
34/43
34
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
-
8/6/2019 Alternatives to Cap-Weighted Indices
35/43
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.
35
Conditions for Optimality
Keep it Simple But Not Too Simple
-
8/6/2019 Alternatives to Cap-Weighted Indices
36/43
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
36
Conditions for Optimality
Assumptions about Parameter Values
-
8/6/2019 Alternatives to Cap-Weighted Indices
37/43
Conditions for Optimality
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.
37
-
8/6/2019 Alternatives to Cap-Weighted Indices
38/43
Conditions for Optimality
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.
38
-
8/6/2019 Alternatives to Cap-Weighted Indices
39/43
Conditions for Optimality
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.
39
-
8/6/2019 Alternatives to Cap-Weighted Indices
40/43
40
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
-
8/6/2019 Alternatives to Cap-Weighted Indices
41/43
41
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.
-
8/6/2019 Alternatives to Cap-Weighted Indices
42/43
42
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.
-
8/6/2019 Alternatives to Cap-Weighted Indices
43/43
43
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.