Portfolios - EDHEC-Risk

EDHEC-Risk Days 2012
London, March 27th, 2012, 16:45-18:00
Investing in New Forms of Equity Indices
Lionel Martellini
Professor of Finance, EDHEC Business School
Scientific Director, EDHEC Risk Institute
lionel.martellini@edhec.edu
www.edhec-risk.com
2

Inefficient i Portfolios in Efficient i Markets
• For more than 50 years, the asset management industry has
mainly focused on a single source of added-value:
outperforming commercial indices through security selection.
• This approach is based on two implicit assumptions:
– Equity markets are inefficient markets;
– Equity indices are efficient benchmarks.
• Academic research has questioned these two assumptions:
– Weak evidence of persistence in (positive) abnormal performance
(alpha) by active managers => emergence of index funds;
– Strong evidence of inefficiency of cap-weighted indices =>
emergence of alternative benchmarks.
3

Managing Investments t in Alternative ti Indices
• Once regarded as exotic curiosities, non CW equity indices have
now made it into the mainstream.
• This acceptance raises a new set of questions, and one may
expect that adopting these non CW indices as benchmarks should
require at least as much attention as investing in active managers.
• Investors face in particular the following challenges:
– Selecting alternative indices (vs. managers): in search for robustness
– ad-hoc schemes versus schemes consistent with portfolio theory;
– Diversifying alternative indices (vs. managers): different schemes
perform differently in different market conditions;
– Managing tracking error of alternative indices (vs. managers):
diversification is not effective at managing extreme relative risk.

Outline
• Alternatives to Cap-Weighted Indices
• Diversifying the Diversifiers
• Tracking the Tracking Error
5

• Alternatives to Cap-Weighted Indices
• Diversifying the Diversifiers
• Tracking the Tracking Error
6

Comparing Alternatives
• A (non-exhaustive) list of recently implemented alternative
weighting schemes :
– Equally-weighted
– Fundamentally-weighted
– Maximum-diversification-weighted
– Risk-parity-weighted
it i – Diversity-weighted
– Minimum-variance-weighted
– Efficient-weighted
• As opposed to focusing on past performance (track records -by
definition- all look pretty good), we provide a summary of the
objectives of, and assumptions behind, the various approaches.
7

Indices versus Benchmarks
• The words « index » and « benchmark » are often used
interchangeably; yet they define a priori very different concepts.
• Market perspective: an index is a portfolio that should
represent the performance of a given segment of the market.
=> focus on representativity
• Investor perspective: a benchmark is a reference portfolio that
should 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.
8

Cap-Weighting for Representativity?
• A market cap weighted scheme is the obvious default option
when it comes to representing a given segment of the market.
– Market cap weighted indices provide by construction a fair
representation of the stock market;
– In the end, cap-weighting is nothing but an ad-hoc weighting
scheme that t achieves some form of representativity.
ti it
• Cap-weighted indices, however, may not provide a fair
representation of the underlying economic fundamentals.
• Some have argued that t they represent well the stock market but
not the economy.
9

Fundamental Weighting for Representativity?
• Rather than using the market cap, fundamental indices use
firm 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 up
with a portfolio that mirrors the economy”.
• Whether or not fundamentally weighted indices better
represent the economy is actually an open question, if only
because representativity is not a concept that is linked to clear
measures.
• Conditions under which fundamental benchmarks would be
optimal are unclear.
10

Efficiency is Related to Diversification
• In any case, it is not clear why investors would care about their
portfolios representing the economy; from the investor’s
perspective, the focus should be on efficiency: obtaining fair
rewards for given risk budgets.
• Efficiency is intimately related to diversification: it is by
constructing ti well-diversified ifi d portfolios thatt one can achieve a
fair reward for a given risk exposure.
• CW portfolios appear to be rather inefficient and poorly
diversified portfolios, and several approaches have been
developed so as to improve diversification compared to CW.
• Some benchmark construction approaches simply avoid this
concentration without aiming for “optimality”; other approaches
are grounded d in portfolio theory.
11

Scientific Approaches: Towards the Efficient Frontier
• Scientific diversification aims at reaching a high risk/return
objective through portfolio construction techniques.
• Two remarkable portfolios on the efficient frontier:
– The MSR generates the highest reward per unit of portfolio
volatility: needed optimization inputs are expected returns,
correlations and volatilities.
– The GMV generates the lowest possible portfolio volatility: needed
optimization inputs are correlations and volatilities.
• In what follows, we focus on these two approaches, and discuss
some of the challenges involved in robust optimization.
• Ad-hoc approaches can generally be regarded as specific cases
under some particular assumptions.
12 12

MSR versus GMV
Expected
Return
w
−1
( μ − re)
( μ − re)
Σ
= ≡
e'
Σ
( μ , σ , ρ )
MSR −1
i i ij
f
Maximum Sharpe
Ratio (MSR) Portfolio
Global
Minimum
Variance (GMV)
Portfolio
●
−1
1
●
●
( )
Σ e
w GMV = ≡ g σ ,
1
e'
e
i ρ
−
Σ
ij
●
Equally-weighted index
Cap-weighted index
Volatility
13

Implementing the Diversifiers
• Estimating covariance matrix parameters is a serious
challenge, but a number of statistical techniques can prove
effective in addressing this challenge.
• If statistical techniques are useful for estimating risk
parameter, statistics on the other hand does not allow us to
estimate expected returns reliably.
• Direct estimation of expected returns from past returns is close
to useless (Merton (1980)).
• What can we do in practice?
– Give up “naïvely” on expected return estimation
– Give up “smartly” on expected return estimation
14

Giving Up Naïvely – Global Minimum Variance Portfolios
• The only scientifically diversified portfolio that relies only on risk
parameters is the minimum risk portfolio (GMV in MV setting).
• In its purest form, “magic” of diversification is not working
properly with GMV portfolios because low volatility stocks are
not penalized and therefore over-weighted.
• We need to penalize low volatility stocks to avoid concentration
in such stocks through weight constraints (flexible norm
constraints versus rigid weight constraints – DeMiguel,
Garlappi, Nogales and Uppal (2009)).
15

Giving Up Smartly – MSR
• How to penalize low volatility stocks based on economic theory?
• Theory unambiguously confirms the existence of a positive
risk/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).
• This justifies the use of the risk-return relationship to build
efficient portfolios: magic of diversification is about mixing highrisk-and-therefore-high-return
stocks so as to generate low risk
portfolios through a smart use of correlations.
16

Ad-Hoc Approaches: EW and ERC (Risk-Parity)
• Naïve de-concentration:
– Equal-weighting simply gives the same weight to each of N stocks
in the index (“1/N rule”)
).
– Equal-weighting is the naïve route to constructing well diversified
portfolios
– EW is optimal (e.g., MSR) if all stocks are identical.
• Semi-naïve de-concentration:
ti
– 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 (volatility) are
equal (Maillard, Roncalli and Teiletche (2010)).
– ERC is optimal (e.g., MSR) if all Sharpe ratios and all pairwise
ERC is optimal (e.g., MSR) if all Sharpe ratios and all pairwise
correlations are identical.
17 17

Ad-Hoc Approaches: Maximum Diversification
• Statistical de-concentration:
– Define a diversification index and try and maximize it.
– Maximum Diversification aims at generating portfolios with the
highest possible diversification index (Choueifaty and Coignard
(2008)):
⎛ n
⎞
⎜
⎟
⎜∑
wiσ
i
⎟
DI = Max⎜
i=
1
⎟
w ⎜
n ⎟
⎟
⎜ ⎜ ∑ w i w jσ
ij
⎟
⎝
i,
j= 1 ⎠
– MD is optimal (e.g., MSR) if all Sharpe ratios are identical.
The weighted average risk (in the numerator) will be high compared
to portfolio risk (in the denominator) and thus DI will be high if the
portfolio weights exploit well the correlations.
18

• Alternatives to Cap-Weighted Indices
• Diversifying the Diversifiers
• Tracking the Tracking Error
19

Selection Risk – Relative Returns
• Depending on market conditions, different strategies work
differently; he worst performing strategy in one sub-period can
turn out to be the best performer in subsequent sub-period
period.
Sub­period
MSCI USA Minimum
Volatility Index
FTSE EDHEC Risk
Efficient US Index
2003 (Jan­Jun) ‐4.72% 2.46%
2003 (Jul­Dec) ‐2.93% 3.46%
2004 (Jan­Jun) 0.05% 3.28%
2004 (Jul­Dec) 2.64% 3.18%
2005 (Jan­Jun) 4.58% 3.95%
2005 (Jul­Dec) ‐3.85% 2.04%
2006 (Jan­Jun) ‐0.70% 0.82%
2006 (Jul­Dec) ‐2.54% ‐2.30%
2007 (Jan­Jun) ‐1.63% 2.06%
2007 (Jul­Dec) 1.51% ‐3.31%
2008 (Jan­Jun) 0.38% 1.46%
2008 (Jul­Dec) 10.10% ‐2.40%
2009 (Jan­Jun) ‐4.10% 7.58%
2009 (Jul­Dec) ‐3.75% 6.96%
2010 (Jan­Jun) 1.67% 3.73%
2010 (Jul­Dec) ‐2.06% 2.26%
2011 (Jan­Jun) 4.01% 3.39%
2011 (Jul­Dec) 6.36% ‐2.32%
The table shows excess return over the cap‐weighted index (the S&P 500 index) of MSCI USA Minimum Volatility Index and FTSE EDHEC Risk Efficient US Index computed over
16 half yearly sub‐periods. Weekly return data from 3 January 2003 to 30 December 2011 is used for the analysis.
20

Selection Risk – Relative Perspective
• Periods of very pronounced underperformance occurred for each
of these indices, as can be seen from extremely high values for
max relative drawdowns and extreme annual tracking error.
Risk Measures
MSCI USA
Minimum
Volatility Index
FTSE EDHEC
Risk Efficient US
Index
Max relative drawdown 12.23% 8.43%
Start date 21‐nov‐08 21‐apr‐06
End date 23‐apr‐10 21‐nov‐08
Annualised excess Return over
cap­weighted index
1.21% 3.75%
Tracking Error (TE) 5.69% 3.82%
Extreme Tracking Error (95 th
percentile of rolling one year TE)
7.42% 6.72%
21

Diversifying the Diversifiers – Long-Term Analysis
• Both diversifiers add value, as can be seen from the improvement
in Sharpe ratio (without impact on extreme risk levels), but not
necessarily under the same market conditions.
Cap­Weighted
Minimum
Volatility
Maximum Sharpe
Ratio
Ann Return 9.60% 10.89% 11.02%
Ann Std 15.46% 14.10% 14.53%
Ann Semi Dev 11.28% 10.39% 10.73%
Tracking Error 0.00% 2.76% 2.77%
Market β 1.00 0.90 0.93
Sharpe Ratio 0.27 0.39 0.39
Sortino Ratio 0.39 0.55 0.55
Information Ratio ‐ 0.47 0.51
Treynor Ratio 0.04 0.06 0.06
Max Drawdown 53.83% 50.42% 51.72%
95% VaR 3.31% 3.02% 3.14%
99% VaR 7.77% 7.64% 7.77%
Skewness ‐0.34 ‐0.50 ‐0.52
Kurtosis 8.06 9.09 8.82
95% Value­at­Tracking at Error Risk
(VaTER)
‐
3.89% 3.96%
The period of analysis is from 2nd January 1959 to 31st December 2010. All statistics are annualized and performance ratios that involve the average returns are
based on the geometric average.
22

Relative Performance Across Market Conditions
• Robust proxies for the GMV portfolio provide defensive exposure
to equity that does well in adverse market conditions, while robust
proxies for Max Sharpe Ratio portfolios provide a higher access
to the upside of equity markets.
Diversified (50% Minimum
Minimum Volatility Maximum Sharpe Ratio Volatility + 50% Maximum
Sharpe Ratio)
Conditioning Variable Regime Excess Return over S&P 500 index
Annualised Equity Market Excess
Returns (Market –risk free rate)
Low 3.20% 2.06% 2.63%
2 2.62% 1.95% 2.29%
3 0.78% 0.98% 0.88%
4 ‐0.50% 0.94% 0.22%
High ‐2.62% ‐0.10% ‐1.36%
Conditioning Variable Regime Excess Return over S&P 500 index
Annualised Market Volatility (of
Excess Market Return)
Low 1.40% 2.53% 1.96%
2 1.01% 1.03% 1.02%
3 ‐0.10% ‐0.07% ‐0.08%
4 1.64% 2.05% 1.85%
High 2.24% 1.58% 1.91%
23

Tracking Error Across Market Conditions
• A combination of both approaches is expected to lead to a
smoother conditional performance, and a lower relative risk with
respect to the cap-weighted index.
Minimum Volatility
Maximum Sharpe Ratio
Diversified (50% Minimum
Volatility + 50% Maximum
Sharpe Ratio)
Conditioning Variable Regime Tracking Error with S&P 500 index
Annualised Equity Market Excess
Returns (Market –risk free rate)
Low 3.28% 3.30% 3.23%
2 2.12% 2.17% 2.10%
3 1.81% 1.82% 1.77%
4 2.21% 21% 2.18% 2.14%
High 2.37% 2.54% 2.41%
Conditioning Variable Regime Tracking Error with S&P 500 index
Annualised Market Volatility (of
Excess Market Return)
Low 1.55% 1.72% 1.59%
2 1.78% 1.78% 1.73%
3 1.99% 2.08% 1.99%
4 2.45% 2.50% 2.42%
High 3.97% 3.89% 3.87%
24

Extreme Tracking Error Across Market Conditions
• In 7 out of 10 cases, the diversified strategy does not lower
VaTER compared to the individual strategy with the lowest
VaTER; this is hardly surprising: diversification is not meant to
address extreme risk, and we should turn to hedging.
Diversified (50% Minimum
Minimum Volatility Maximum Sharpe Ratio Volatility + 50% Maximum
Sharpe Ratio)
Conditioning Variable Regime 95% Value at Tracking Error Risk
Annualised Equity Market Excess
Returns (Market –risk free rate)
Low 5.13% 4.99% 5.13%
2 2.92% 3.11% 2.89%
3 3.18% 3.03% 2.99%
4 3.65% 3.50% 3.52%
High 4.95% 4.66% 4.71%
Conditioning Variable Regime 95% Value at Tracking Error Risk
Annualised Market Volatility (of Excess
Market Return)
Low 2.39% 2.83% 2.54%
2 3.02% 2.86% 2.86%
3 3.33% 3.59% 3.50%
4 4.06% 4.05% 3.90%
High 7.10% 6.77% 7.11%
25

• Alternatives to Cap-Weighted Indices
• Diversifying the Diversifiers
• Tracking the Tracking Error
26

Tracking the Tracking Error
• Even with a combination of a minimum volatility portfolio and a
maximum Sharpe ratio portfolio, extreme tracking error risk
remains substantial.
• This extreme relative risk is a severe concern for a Chief
Investment Officer who has made the choice of adopting an
alternative weighting scheme.
When such underperformance occurs with active managers, the failure of a
third-party manager, which is a risk inherent to the very logic of the delegation
process of portfolio management, typically translates into the termination of
the manager.
In the case of underperformance of an alternative ti equity index, however, it
would be difficult for the CIO to blame anyone but himself/herself for the
selection of this index.
27

Tracking Error Constraints
• If we subject active managers to TE control, why not expect the
same requirements from alternative benchmarks?
• To control tracking error, one may perform a maximum Sharpe
ratio or minimum variance portfolio optimization subject to
tracking error constraints.
• This is an inefficient approach, inconsistent with the fund
separation theorems that lie at the foundation of asset pricing
theory, and which suggests instead to use a core-satellite
approach.
• In fact, the two approaches are not strictly mutually exclusive.
28

“Fund Separation Theorem” Approach to TE Control
• Fund separation theorems recognize that t risk and performance
are conflicting objectives that are best managed when managed
separately.
• In simple words, the optimal solution in the presence of TE
constraints is a mixture of some proxy for a well-diversified MSR
portfolio and the CW benchmark, here the risk-free asset:(*)
w
⎛ ⎞
= −
CW MSR
⎟
w
t = c
t w
t + 1
⎝ γσ
14243 144
243
⎠ 4
* λt
MSR λt
t w
t +
1
γσ
⎜
t
t
speculative demand
HD w.r.t. benchmark risk
( − c
CW
t ) w
t
• The risk-aversion parameter γ is calibrated to as to reach a
target TE level; for a given γ, the weight allocated to the
improved vs. CW portfolio is a function of market conditions.
29
(*) Result based on constrained maximization of expected CRRA utility of terminal wealth relative to CW benchmark.

What Separation Theorems Say and Do not Say
• This approach is sometimes known as core-satellite investing,
and in essence strictly similar to LDI when the benchmark is a
liability-driven benchmark.
• Now, this approach works even better when the TE of the
satellite portfolio is “well-behaved”; a concern over VaTER in
particular would lead to a severe opportunity cost.
• Therefore, when designing the alternatively weighted satellite
part, it is still useful to maximise the Sharpe ratio subject to
suitably defined (relatively loose, e.g., 5%) TE constraints.
• Adding beta constraints (with respect to a number of betas
that may vary as a function of market conditions) is
particularly effective in controlling for VaTER.
30

Impact of Relative Risk Control
• Extreme tracking error is markedly lower for the relative risk
controlled portfolios (ex-ante TE target: 3%).
Minimum
Volatility
Without Relative Risk Control
Maximum Sharpe
Ratio
Diversified (50%
Minimum
Volatility + 50%
Maximum Sharpe
Ratio)
Minimum
Volatility
With Relative Risk Control
Maximum Sharpe
Ratio
Diversified (50%
Minimum
Volatility + 50%
Maximum Sharpe
Ratio)
Expected Return over CW 1.29% 1.42% 1.35% 0.79% 1.18% 0.99%
Information Ratio 0.47 0.51 0.50 0.44 0.52 0.54
Modified IR 0.23 0.28 0.27 0.25 0.29 0.30
Median Relative Return 1.06% 1.46% 1.35% 0.67% 1.14% 1.12%
5% Relative Return ‐4.49% ‐4.35% ‐4.32% ‐2.94% ‐3.60% ‐3.15%
Min Relative Return ‐12.01% ‐10.57% ‐11.29% ‐8.45% ‐7.05% ‐7.68%
Median Tracking Error 2.08% 2.15% 2.05% 1.53% 1.87% 1.51%
95% Tracking Error 5.60% 5.09% 5.07% 3.09% 4.03% 3.26%
Max Tracking Error 8.69% 8.36% 8.51% 4.48% 5.20% 4.77%
Max Relative Drawdown 22.96% 21.05% 21.83% 15.78% 14.65% 14.88%
Start t Date 16‐sept‐93 24‐mar‐94 7‐oct‐93 16‐sept‐93 22‐sept‐94 24‐mar‐94
End Date 24‐mar‐00 24‐mar‐00 24‐mar‐00 24‐mar‐00 24‐mar‐00 24‐mar‐00
Weekly Value­at­Tracking
Error Risk (VaTER)
‐0,54% ‐0,55% ‐0,54% ‐0,38% ‐0,45% ‐0,38%
Max TUW (weeks) 340 313 337 340 287 313
Prob of Outperformance (1Y
Rel Return)
Prob of Outperformance (3Y
Rel Return)
66.00% 65.03% 67.09% 64.65% 64.95% 67.47%
75.96% 75.84% 76.86% 76.31% 70.84% 76.39%
31

Impact of Relative Risk Control
Panel A: Relative Drawdown of Diversified Strategy with and
without relative risk control
0%
Panel B: Rolling One­year Tracking Error of Diversified Strategy
with and without relative risk control
9%
8%
‐5%
‐10%
‐15%
‐20%
7%
6%
5%
4%
3%
2%
1%
‐25%
0%
Diversified ifi (with RR control) Diversified ifi (without t RR control) Diversified ifi (with RR control) Diversified ifi (without t RR control)
The plot shows relative drawdown of diversified portfolios (50% Minimum Volatility + 50% Maximum Sharpe Ratio) with and without relative risk
control as compared to the S&P 500 index. Panel B ‐ The plot shows 1‐year trailing tracking error for Diversified (50% Minimum Volatility + 50%
Maximum Sharpe Ratio) portfolios with and without relative risk control as compared to the S&P 500 index. The period of analysis ranges from 2nd
January 1959 to 31st December 2010.
32

Improved Benchmarks in Investors’ Portfolios
• CW indices are macro-consistent, relatively transparent,
systematic and come with low-cost, and will arguably remain the
ultimate references; on the other hand, they provide severely
inefficient risk/reward properties due to their high concentration.
• Active managers come with less transparency and higher h costs
but have difficulties in persistently outperforming even inefficient
CW indices due to a focus on stock selection.
• More efficient equity benchmarks can be designed, which may
be used as substitutes for active managers in attempts to add
value with respect to the CW index, while allowing for an explicit
control of downside relative risk through diversification and
hedging.
33