Risk and Optimization in Equity
Portfolios: Is it a Match Made in
Heaven?
FOCAPO 2012
January 10th, 2012
Sebastian Ceria, CEO
Axioma, Inc.
Joint work with Anureet Saxena and Rob Stubbs,
Axioma Research
Asset Management and Optimization
• There are two type of managers of equity portfolios
– Fundamental managers (80% of total AUM/US)
– Quantitative managers (20% of total AUM/US)
• Fundamental managers build portfolios bottoms up
– Heavily research companies in their “target” list
– Choose stocks to buy or sell based on the research
– Size their positions based on simple heuristic rules
– Control risk through diversification with simple rules of thumb
• Quantitative managers build portfolios top down
– Research a broad universe of companies with statistical
techniques, based on market, balance sheet, and income
statement information, using publicly available data sets
– Construct portfolios using expectations of return, a risk model,
and a “strategy” using an optimizer, so as to maximize some
tradeoff
Copyright © 2011 Axioma
The Mean Variance Optimization Model
• Given a budget for investment (that needs to be fully utilized) into
potential assets 1, …, n
• Given expected returns i for every asset i=1, …, n
• Given a variance-covariance matrix Q of asset returns that is used
to calculate the risk of the portfolio with holdings h1, …, hn
• Given a “risk-aversion” parameter 
• Find the holdings h1 ,…, hn that optimize the following optimization
problem:
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Mean Variance Optimization Notation
Expected Return = Alpha
Risk Model
Risk Term: Portfolio Variance
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Risk is under-estimated for optimized
portfolios
4.0%
Risk Target
Realized Risk
2.00%
2.59%
Ex-post risk
3.5%
3.0%
2.5%
2.0%
1.5%
1.0%
0.5%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
Ex-ante risk
Base Model
Unbiased Risk Prediction
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Risk Models and Alpha Interaction
• Are optimal MVO portfolios “biased” with respect to certain risk
models?
• How does a risk model used in MVO affect the optimal portfolio?
• Why do risk estimates provided by risk models that were used to
construct an MVO portfolio tend to underestimate risk?
• Can a second risk model, in particular, one which was not used to
build the MVO portfolio, provide a more accurate measurement of
risk?
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Comparing Two kinds of Risk Models for
Optimized Portfolios
• Model 1: Statistical Model (PCA)
• Model 2: Fundamental Model (Cross-sectional model)
• Simple experiment
– Use the statistical model (Model 1) to optimize (and predict
risk of the optimized portfolio)
– Use the fundamental model (Model 2) to measure predicted
risk of the optimized portfolio (this risk model is not used to
optimize)
– Run an experiment on real-life data from our clients
• Go back in history (backtesting)
• For every time period, generate an optimized portfolio
• Monitor the performance of the portfolio over time
– Measure predicted vs realized risk for the experiment
– Repeat the experiment with the models “flipped”
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Statistical Model Backtest (Fundamental
Model for Measurement)
14%
12%
20-Day Realized Active Risk
Fundamental
Active Risk
10%
Fundamental Model’s
prediction is significantly
better, and “unbiased”
Statistical
8%
6%
4%
2%
0%
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
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Fundamental Model Backtest (Statistical
Model for Measurement)
12%
20-Day Realized Active Risk
Fundamental
Active Risk
10%
Statistical Model’s
prediction is significantly
better, and “unbiased”
Statistical
8%
6%
4%
2%
0%
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
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MVO: Risk Model and Alpha Interaction
• The Alpha and Risk Model are “interacting” and creating risk
underestimation
• What is driving this underestimation?
– How is alpha built?
– How is the risk model built?
• Why are MVO portfolios particularly affected by this interaction?
• How is the performance of the alphas affected by risk
underestimation?
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Factor Risk Model Notation
V = hT Q h
Q = X S XT + D 2
where
-


Portfolio Variance
Covariance Matrix
Factor Model
V = portfolio variance
h = N-dimensional vector of portfolio holdings
(weights)
Q = N X N covariance matrix
X = N X M matrix of factor exposures (factor
loadings)
S = M X M matrix of factor covariances
D2 = N X N matrix of security specific risk
In practice, M (factors) << N (assets)
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Factor Expected Return Notation
 Th
=EG
Expected Return
Alpha
Factor Model
Where
-

h = N-dimensional vector of portfolio holdings
(weights)
E = N X L matrix of factor exposures (factor
loadings)
G = vector of weights
In practice, L (factors) << N (assets), and the factor
models for risk and alpha do not necessarily
coincide
Copyright © 2011 Axioma
A Decomposition of Alpha
The portion of alpha
explained by the risk
factors is referred to as the
spanned component.
max  T h  2 h T Qh
h
Q = XSXT + D
     X
If alpha and risk factors are

aligned, then  = 0, or, in other
words, there is only misalignment
iff
 ≠ 0
The residual obtained by
regressing the alphas against
the factors in the risk model is
referred to as the orthogonal
component of alpha.
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A Decomposition of Alpha: Geometry
Risk Model Factors (X)
Spanned Alpha
( X )
Alpha
Orthogonal
Alpha
(  )
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Constraints: Alpha and Implied Alpha
Notation: Unconstrained Problem
Alpha
Optimal
Portfolio h*
Risk Model
Notation: Constrained Problem
Implied Alpha
Optimal
Portfolio h*
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Alpha vs Implied Alpha
Alpha
Implied
Alpha
Risk
Ellipse
Optimal
Portfolio
With
constraint
Optimal
Portfolio
No longer
feasible!
Constraint
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Correlation: Alpha and Implied Alpha
1.0
Fundamental Model
0.9
Statistical Model
0.8
Correlation
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
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The Role of Constraints
• The optimizer follows the “direction” of Implied Alpha, not Alpha
• In practice, these two could be very different
• Does it make sense to focus on alpha independently of the
constraints used in portfolio construction?
• There is no such thing as “Alpha Research” it should really be
“Implied Alpha Research”, however, we don’t have a model for
implied alpha…
• The interaction of Alpha and the Constraints is important in
understanding the structure of optimal portfolios
Copyright © 2011 Axioma
Why is Misalignment “Bad” in MVO?
max  T h  2 h T Qh
h
The optimizer sees no
systematic risk in the
orthogonal component of
alpha and is hence likely
to load up on it



No Factor Risk,
Only Specific Risk


X

Contains Factor Risk
and Specific Risk
In MVO, we are aiming to create portfolios that have an optimal riskadjusted expected return. If a portion of systematic risk is not
accounted for then the resulting risk-adjusted expected return cannot
be “optimal”.
Copyright © 2011 Axioma
Factor Alignment Problems
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Factor Alignment Problems
• Quantitative Portfolio Management brings together elements from
four different fields, namely
– Accounting – Empirical Asst pricing: Alpha model development
– Statistics – Empirical Asset pricing: Risk model development
– Optimization: Portfolio construction
– Finance
: “Efficiency” ambitions
• Each one of these streams is a mature discipline in itself, having
its own body of knowledge and operates under assumptions that
are usually well-accepted within the respective community.
• However, when concepts from these diverse fields are applied in a
common setting there are bound to be frictions between various
assumptions which get magnified due to the use of an optimizer
we call these problems Factor Alignment Problems.
Copyright © 2011 Axioma
Factor Alignment Problems: Our
Contribution
• In general, we say that risk and alpha factors are aligned if alpha
can be written as a linear combination of the risk factors
• The only research in this topic is “theoretical” and deals with the
simple unconstrained case. It suggests that aligning the risk and
alpha factors is desirable
• But what happens when we also have constraints? How do we
“align” for constraints
• We will show that there is no universal answer and the goal of the
portfolio manager should be to “manage” the (mis)alignment
Copyright © 2011 Axioma
Overweight of Orthogonal Alpha
Question: How do we measure the excess exposure of the
optimal portfolio along the orthogonal component of alpha?
Spanned Alpha
Multiplier of
Spanned
Alpha
Alpha
b
Optimal Holdings
Orthogonal
Alpha
a
We define a ratio that measures this overweight:
a

Misalignment Bias
b
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Is Misalignment Bad?
• When we work in “theory”, we assume that the orthogonal
component of alpha is all noise
– In this case, we do not want exposure to orthogonal alpha
(Hence, Misalignment is Bad)
• However, in practice, the orthogonal component of alpha may
have positive information (IC = Information Coefficient), so, a
positive exposure to the orthogonal component of alpha will
increase returns (Hence, Misalignment is good)
• But, what if the orthogonal component of alpha also contains
systematic risk? (Then, we should be managing the tradeoff given
by the misalignment)
• In practice, however, we can’t look at alpha, we need to look at
IMPLIED alpha
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How Do We Manage this Tradeoff?
• Exposure to the orthogonal implied alpha can be rewarding in
terms of return (IC), but it may also lead to increased risk
• Even though the risk of the orthogonal implied alpha appears to
be low, it is weighted so heavily in the optimal portfolio that the
risk of the orthogonal alpha should still be considered
• How can we manage our exposure to the orthogonal implied alpha
in the portfolio construction process?
1. Add the orthogonal implied alpha as a risk factor
2. Penalize exposure along the orthogonal implied alpha
• Reducing exposure to the orthogonal implied alpha will also
reduce the Misalignment Bias (it is the numerator of the ratio)
Copyright © 2011 Axioma
1. Add the Orthogonal Implied Alpha as
a Risk Factor
• We really need the implied alpha, not alpha, which means, this
process would require us to solve optimization problems to
identify the implied alpha
• We would need to re-estimate the whole risk model with the new
factor
• We would need to iterate this process, until the orthogonal
component of the implied alpha is zero
• Not practical, may not converge -- not recommended
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2. Penalizing Exposure to the Orthogonal
Implied Alpha: Iterative Refinement
Optimal Portfolio
Determine Implied
Alpha (  *)
Solve Rebalancing
Penalize
*
exposure to  
Does this process reach an equilibrium?
Not necessarily…
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The Alpha Alignment Factor
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The “Alpha Alignment Factor” Methodology
Problem:
•
To address this systematic problem, we need a new methodology
Solution:
•
Axioma introduced in 2006 a new methodology for dealing with risk
underestimation, we called it the “Alpha Factor” (Patented)
•
Methodology identifies an additional factor (the “Alpha Factor”) that is
added to the risk model to correct for (factor) risk underestimation in
optimized portfolios
•
A portfolio’s exposure to the Alpha Factor is portfolio specific
– Identified through an optimization process (error maximization)
•
The Alpha Factor is “dependent” on the optimized portfolio
– In turn, the optimized portfolio is “dependent” on the alpha process and
investment strategy
•
The Alpha Factor must be orthogonal to the other factors used in a
generic risk model
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Finding the Alpha Factor
• We solve an optimization problem to find an additional factor that
increases risk the most, given a fixed portfolio (w)
Q’
v2
XT f = 0
|| f || = 1
a fixed constant (new factor volatility)
new factor is “orthogonal” to existing factors
normalize new factor exposures
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Integrating the Alpha Factor and MVO
Maximize Expected Return
αth
αth
Traditional MVO
MVO with Alpha Factor
Risk Target
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From the “Alpha Factor” to the “Alpha
Alignment Factor”
• When you do the math, the “Alpha Factor” (f) is the orthogonal part
of implied alpha
• Adding the Alpha Factor to the optimization problem as we
proposed before is, in fact, penalizing the exposure to the
orthogonal implied alpha
• Hence, the Alpha Factor Method, “automatically” solves Problem 2.
• But you still have to manage the tradeoff, fortunately, it is only one
parameter v, the “volatility” of the alpha factor
• Because the Alpha Factor solves the alignment problem we call it
“The Alpha Alignment Factor” (AAF)
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Client Backtests: Alpha Alignment Factor
Improvements (Fixed v for all tests)
Fundamental Risk
Model
Statistical Risk
Model
No. of Backtests
24
24
Alpha Factor improves IR
17
19
Average % Improvement in IR
7.93%
8.54%
Average % Improvement in
Misalignment Bias
39.8%
28.6%
IR = Information Ratio = Return/Risk
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The Alpha Factor also Improves
Performance (Information Ratios)
7
6
5
4
3
2
1
0
-5%
0%
5%
10%
15%
20%
25%
30%
35%
40%
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Alternatively, We Could Penalize the
Orthogonal Alpha (rather than Implied)
• Bender, Stefek and Lee propose penalizing the orthogonal alpha in
order to improve risk underestimation and misalignment
• Much simpler optimization process (just need to compute the
orthogonal alpha which does not depend on the optimal portfolio
or the constraints)
• Much less effective in practice
• Since the orthogonal alpha does not depend on the constraints or
the optimal solution, we can just compute the orthogonal alpha a
priori, and add a penalty to the objective function
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Comparison of the Alpha Factor Method
and Penalizing Orthogonal Alpha
Fundamental Risk Model
Statistical Risk Model
Alpha Factor
7.93% (17/24)
8.54% (19/24)
Penalizing
Orthogonal
Alpha
1.55% (16/24)
0.66% (13/24)
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Improvement IR Histogram Comparison
Between Alpha Factor and Penalizing
Orthogonal Alpha
8
7
Penalizing Orthogonal Component of Alpha
6
Alpha Factor
5
4
3
2
1
0
-40%
-30%
-25%
-20%
-15%
-10%
-5%
0%
5%
10%
15%
20%
25%
30%
35%
40%
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Summary
• The understanding of Factor Alignment Problems is an essential
component of empirical asset management
• The misalignment of alpha and risk model factors can be solved
with the Alpha Alignment Factor methodology
• The Alpha Alignment Factor methodology also improves ex-post
performance
• The theoretical foundation behind the Alpha Factor Methodology
can be used to
• In general, the optimizer will tend to overweight the orthogonal
(implied) alpha, so, if there is no information present in the
orthogonal alpha, penalizing it will yield better performance
• The Alpha Factor Method solves this problem directly and
efficiently
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Bibliography
• MacKinlay, A. C., 1995, “Multifactor models do not explain
deviations from the CAPM”, Journal of Financial Economics, 38,
3—28.
• MacKinlay, A. C., and L. Pastor, 2000, “Asset Pricing Models:
Implications for Expected Returns and Portfolio Selection”, The
Review of Financial Studies, 13(4), 883—916.
• Lee, J. H., and D. Stefek, 2008, “Do Risk Factors Eat Alphas?”,
The Journal of Portfolio Management, 34(4), 12—25.
• Bender, J., J. H. Lee, and D. Stefek, 2009, “Refining Portfolio
Construction When Alphas and Risk Factors are Misaligned”, MSCI
Barra Research Paper, No. 2009-09.
Copyright © 2011 Axioma
Bibliography (cont)
• S. Ceria, A. Saxena, and R. A. Stubbs. Factor alignment problems
and quantitative portfolio management. Journal of Portfolio
Management, To Appear, 2012.
• A. Renshaw, R. A. Stubbs, S. Schmieta, and S. Ceria. Axioma
alpha factor method: Improving risk estimation by reducing risk
model portfolio selection bias. Technical report, Axioma, Inc.
Research Report, March 2006.
• A. Saxena and R. A. Stubbs. Alpha alignment factor: A solution to
the underestimation of risk for optimized active portfolios.
Technical report, Axioma, Inc. Research Report #015, February
2010a.
• A. Saxena and R. A. Stubbs. Pushing frontiers (literally) using
alpha alignment factor. Technical report, Axioma, Inc. Research
Report #022, February 2010b.
Copyright © 2011 Axioma