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Sparse and robust normal and t-portfolios bypenalized Lq-likelihood minimization
Davide Ferrari1, Margherita Giuzio2, Sandra Paterlini2
1University of Melbourne, AU2EBS Universitat fur Wirtschaft und Recht, GER
Universita degli Studi di Bergamo
Seminar 07.04.2014
Portfolio Selection
Task: Select the ”optimal” assets and their ”optimal” weights
• Ideal Characteristics
• Good Out-of-sample Performance• Stable weights• Robustness• Cheap to implement and maintain
• Main Issues• Normality assumption• Highly correlated returns• Sensitivity of Weights to Estimation Errors and Model Misspecification• High-dimensionality
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 2 / 44
Main Issue
State-of-art Methods
• Robust and Shrinkage Estimators =⇒ Estimation ErrorsHuber (1981)Ledoit and Wolf (2004)
• Penalized Least Squares =⇒ Model Misspecifications and SparsityDeMiguel, Garlappi, Nogales, Uppal (2007)Fan, Zhang, Yu (2012)Fastrich, Paterlini, Winker (2012)
• Minimum Divergence Methods =⇒ Model MisspecificationsFerrari, Yang (2010)Ferrari, Paterlini (2010)Ferrari, La Vecchia (2012)
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 4 / 44
Our Proposal
Targets:• Robustness to both Estimation Errors and Model Misspecification• Sparsity: automatically select a small number of relevant active
positions
How can we achieve that?Generalized Description Length Approach
• based on q-Entropy• using priors as sparsifying operator
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 5 / 44
Maximum Likelihood
• If observations are i.i.d. from a probability distribution f (x ; θ0),maximizing
log L(θ0) = logn∏
i=1f (Xi | θ) =
n∑i=1
log f (Xi ; θ)
we obtain the maximum log-likelihood estimator of θ0.
n∑i=1
log f (Xi ; θ)→ E [log f (Xi ; θ)],
as n→∞• Properties
• Consistency, Efficiency• Asymptotic normality• Easy to implement
Shannon Entropy
• Information Theory (1940s): how to measure uncertainty with respectto a probability distribution f (x)
H(X ) = −E [log f (x)]
• − log f (x) is the information included in the observation x• H(X ) represents the average uncertainty removed after observing the
outcome of the variable X .
RelationshipGiven n i.i.d. observations, the Maximum Likelihood Estimator can beseen as a minimization of the Shannon entropy
maxn∑
i=1log f (Xi ; θ) = min−E [log f (Xi ; θ)]
q-Entropy
• Havrda and Charvat (1967) proposed a generalized measure ofentropy of order q, then used in physics, biology and finance
H(X ) =E [1− f (x)q−1]
1− q , q > 0
• Tsallis (1988) arrived to the following specification:
H(X ) = −E [Lqf (x)],
where
Lq =
{(u1−q − 1)/(1− q), q 6= 1,
log(u), q = 1,
Maximum Lq Likelihood
• Given n i.i.d. observations from from a probability distributionf (x ; θ0), maximizing
n∑i=1
Lq[f (Xi ; θ)]
we obtain the Maximum Lq-Likelihood Estimator of θ0.
• Properties• Consistency and Fisher-consistency (Ferrari, La Vecchia, 2009)• Asymptotic normality• Changing q we balance the trade-off between bias and variance,
robustness and efficiency• When q → 1, MLqE → MLE
Asset Returns
Generalized Description Length criterion (GDL)
Let:
• X = (X1, . . . ,Xp)T a p-dimensional random vector such thatE (X) = µ and Var(X) = Σ
• Y = βT X a portfolio where βT is the weights vector
• the portfolio mean, µ∗ is a fixed target
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 11 / 44
Generalized Description Length criterion (GDL)
AssumptionsDistribution of the Data: f (y ;µ, σ2) = f (xT
i β;µ, σ2)Prior Distribution of Portfolio Weights: π(βj ;λ) for the jth weight βj
β = argminβ
−n∑
i=1Lq
{f(
xTi β − µ∗
σ
)}−
p∑j=1
Lq {π(βj ;λ)}
, (1)
with q-entropy function
Lq =
{(u1−q − 1)/(1− q), q 6= 1,
log(u), q = 1
• Information from the data given the model• Information from the model itself
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 12 / 44
Penalized q-entropy minimizationComputing the first derivative
0 =n∑
i=1wi (xi ,β, σ)∇ log f (σ−1(xT
i β − µ∗)) +p∑
j=1vj(βj , λ)∇ log π(βj ;λ).
(2)
Define wi and vj as
• Weights on obs y wi (xi ,β, σ) = f (σ−1(xTi β − µ∗))1−q
• Weights on β vj(βj , λ) = π(βj ;λ)1−q
Role of q- Changing q, we modify the role of unusual observations xi and sparsifythe parameters βj- When q < 1 we gain robustness to models misspecification
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 13 / 44
GDL Penalty Function
−p∑
j=1Lq {π(βj ;λ)}
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 14 / 44
GDL vs Other Penalties
−2 −1 0 1 20
0.2
0.4
0.6
0.8
1GDL Penalty
Parameter β
−L q π
(β)
q = 0.1q = 0.3q = 0.5q = 0.7q = 0.9
−2 −1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4Lasso Penalty
Parameter β
−L q π
(β)
−2 −1 0 1 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8SCAD Penalty
Parameter β
Pen
alty
g(β
)
−2 −1 0 1 20
2
4
6
8
10Logarithm Penalty
Parameter β
−L q π
(β)
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 15 / 44
Model 1: Normal portfolios with Laplace prior
Model SpecificationY ∼ N1(µ, σ2)
π(βj ;λ) = λ exp {−λ|βj |}/2
• GDL N
β(s)
= argminβ
n∑
i=1w (s−1)
i12
(µ∗ − xT
i β
σ(s−1)
)2
+ λp∑
j=1v (s−1)
j |βj |
(3)
• Lasso
β(s)
= argminβ
n∑
i=1(µ∗ − xT
i β)2 + λp∑
j=1|βj |
(4)
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 16 / 44
Model 2: t-portfolios with Laplace prior
Model SpecificationY ∼ t-Student fν(y ;µ, σ2) = N(µ, σ2Z−1
i ), Zi ∼ Γ(ν/2, ν/2)
π(βj ;λ) = λ exp {−λ|βj |}/2
• GDL t
argminβ,σ
{−(ν + 1
2
) n∑i=1
w (s−1)i log
{1 +
(xiβT − µ∗)2
νσ2
}+ λ
p∑j=1
v (s−1)j |βj |
}(5)
where νq = qν + (q − 1).
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 17 / 44
Behaviour of wi – Observations Weights
w (s)i = f ((xi β
(s−1)− µ∗)/σ(s−1))1−q (6)
−10 −5 0 5 100
0.2
0.4
0.6
0.8
1
1.2
x 10−3
Xβ
w
−10 −5 0 5 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
−3
Xβw
q = 0.5 q = 0.9
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 18 / 44
Behaviour of vj – Parameters Weights
v (s)j = π(β
(s−1)j ;λ)1−q. (7)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
β
v
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
β
vq = 0.5 q = 0.9
Minimizing the GDL criterion• Modify the role of extreme observations and parameters by varying q• Minimize the divergence between the hypothetical and the tranformed version of
the density and prior distributionsFerrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 19 / 44
Selection of λ
Select λ (for a given q) by minimizing
• ”Standard” AIC and BIC• Robust AIC and BIC (Ronchetti, 1997 and Machado, 1993)
AICq = −2n∑
i=1Lq
{f(
xTi βq,λ − µ∗
σq,λ
)}+ 2k, (8)
BICq = −2n∑
i=1Lq
{f(
xTi βq,λ − µ∗
σq,λ
)}+ log(n)k, (9)
where k is the number of active portfolio positions
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 20 / 44
Geometric Interpretation of GDL
ExampleLet’s assume two asset returns have bivariate normal distribution(
X1X2
)∼ N2
((01
),
(2 1/
√2
1/√
2 1
)),
When comparing Sharpe Ratios, the second asset is preferable
• Compare the optimal portfolios obtained with GDL N and LASSO• Contaminated vs not contaminated data
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 21 / 44
Geometric Interpretation of GDL
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 22 / 44
Geometric Interpretation of GDL
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 23 / 44
Simulation Study
Empirical Setup• n = 200 observations, p = 50 assets, k = 10 optimal positions• Four settings: ρ = 0.2, 0.4, 0.6, 0.8, 50 simulations• q = 0.9, target return µ∗ = k = 10
Model 1: X ∼ Np(µ,Σ) Model 2: X ∼ tp(µ,Σ, ν)
µj =
{1, j ≤ k,0, j > k,
Σij =
{1, i = j ,ρ, i 6= j ,
ν = 6.
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 24 / 44
Simulation Study
Goals:• Portfolio with low risk and high return, compared to a target• Sparsity
In each simulation:
• Given a grid of λ, compute estimates of β with GDL N, GDL t,Lasso, Zhang, Scad, Log, Lq
• Choose the model with the lowest BIC• Compare β to the optimal vector of β∗
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 25 / 44
Simulation Study
• Assess the whole performance by the Monte Carlo mean squared error
MSE =1
50
50∑b=1
µT βb − µ?√β
Tb Σβb
2
, (10)
where:
- µ?/σ? → target Sharpe ratio with σ? =
√β
Tb Σβb
- µT βb/
√β
Tb Σβb → portfolio Sharpe ratio
• Verify the selection of the k active assets by the F-measure:
F-measure = 2 |supp(β∗)| ∩ |supp(β)||supp(β∗)|+ |supp(β)|
, (11)
where supp(β) = {j : |βj | ≥ τ}.
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 26 / 44
Simulation Results – Sparsity
0.2 0.4 0.6 0.8
5
10
15
20
25
30
ρ
Num
ber
Act
ive
Pos
ition
s
GDL NGDL tLassoZhangLogLq
0.2 0.4 0.6 0.8
5
10
15
20
25
30
ρN
umbe
r A
ctiv
e P
ositi
ons
Model 1 Model 2
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 27 / 44
Simulation Results – F-measure
0.2 0.4 0.6 0.8
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ρ
F−
mea
sure
GDL NGDL tLassoZhangLogLq
0.2 0.4 0.6 0.8
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ρF
−m
easu
re
Model 1 Model 2
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 28 / 44
Simulation Results – MSE
0.2 0.4 0.6 0.80
0.5
1
1.5
2
2.5
ρ
Mea
n S
quar
ed E
rror
GDL NGDL tLassoZhangLogLq
0.2 0.4 0.6 0.80
1
2
3
4
5
ρM
ean
Squ
ared
Err
or
Model 1 Model 2
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 29 / 44
Financial Data
Index Trackingy returns vector of a Financial IndexX returns matrix of the Index Componentsβ vector of asset weights to be estimated
Data n p r σ S KF&F 100 1401 100 13.97 18.24 -0.03 5.27S&P 200 1401 200 10.70 26.94 -0.23 10.57S&P 500 1401 500 9.80 28.33 -0.30 13.09
Period from 23.08.2002 to 27.03.2008
Goal: build sparse portfolios able to track as closely as possible the index
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 30 / 44
Financial Data
−0.05 0 0.050
50
100
150
200
250
300
350
Log−returns
Abs
. Fre
quen
cyF&F 100
F&F 100Normalt−Student
−0.05 0 0.050
50
100
150
200
250
300
350
Log−returns
Abs
. Fre
quen
cy
S&P 200
S&P 200Normalt−Student
−0.05 0 0.050
50
100
150
200
250
300
350
Log−returns
Abs
. Fre
quen
cy
S&P 500
S&P 500Normalt−Student
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 31 / 44
Performance Evaluation
• Rolling Window of 250 obs, stepsize = 1 obs• Compare GDL N and GDL t with q = 0.5, 0.9,
Lasso and Eq. Weighted (1/N) Portfolios
Performance measures• Risk-return performance:
Information Ratio IR = ERTEV
• Sparsity and stability:k, average turnover (TO)
• Tracking ability:OOS correlation and beta wrt Index
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 32 / 44
Performance Evaluation
Index GDL N GDL N GDL t GDL t Lasso 1/Nq = 0.9 q = 0.5 q = 0.9 q = 0.5
F&F 100ER (%) 0.338 0.302 0.170 0.186 1.030 0.929
TEV (%) 0.624 0.659 0.492 0.527 2.117 3.854IR 0.542 0.458 0.346 0.352 0.486 0.241
S&P 200ER (%) 0.319 0.639 -2.421 -0.196 4.760 5.937
TEV (%) 4.500 4.961 4.897 4.614 7.267 2.518IR 0.071 0.129 -0.494 -0.042 0.655 2.357
S&P 500ER (%) 2.906 -1.989 1.192 2.287 2.986 5.113
TEV (%) 6.966 8.100 9.018 8.859 10.315 3.107IR 0.417 -0.245 0.132 0.258 0.289 1.646
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 33 / 44
Performance Evaluation
Index GDL N GDL N GDL t GDL t Lasso 1/Nq = 0.9 q = 0.5 q = 0.9 q = 0.5
F&F 100k 37.749 38.244 32.241 32.121 65.939 98
TO 0.068 0.060 0.066 0.064 0.017 -S&P 200
k 36.950 43.627 28.431 28.011 66.532 200TO 0.399 0.346 0.520 0.481 0.037 -
S&P 500k 44.564 45.736 27.770 26.944 66.407 500
TO 0.605 0.322 0.811 0.801 0.053 -
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 34 / 44
Tracking ability - Normalized trends
0 200 400 600 800 1000 120080
100
120
140
160
180
200
Time Window
Out
−O
f−S
ampl
e R
etur
n
F&F 1001/NLassoGDL t q=0.5
0 200 400 600 800 1000 120080
100
120
140
160
180
200
Time Window
Out
−O
f−S
ampl
e R
etur
n
S&P 2001/NLassoGDL N q=0.5
0 200 400 600 800 1000 120080
100
120
140
160
180
200
Time Window
Out
−O
f−S
ampl
e R
etur
n
S&P 5001/NLassoGDL N q=0.9
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 35 / 44
Conclusion
Penalized q-entropy minimization helps to
• select a sparse portfolio• handle highly-correlated variables• handle outliers and control for estimation errors
On-going work...• test on other financial data• extend to mean-variance framework• develop a method to optimally choose parameters q and λ
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 36 / 44
Appendix
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 37 / 44
Other Penalties
• LASSO
λp∑
i=1ρ(βi ) = λ‖β‖1.
• SCAD
λp∑
i=1ρ(βi ) =
p∑i=1
{λ|βi |, |βi | ≤ λ,
−β2i +2aλ|βi |−λ2
2(a−1) , λ < |βi | ≤ aλ,(a+1)λ2
2 aλ < |βi |
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 38 / 44
Other Penalties
• Zhang
λp∑
i=1ρ(βi ) =
p∑i=1
{λ|βi |, |βi | ≤ η,λη, η < |βi |
• Lq
λp∑
i=1ρ(βi ) = λ‖β‖qq, 0 < q < 1.
• Log
λp∑
i=1ρ(βi ) = λ
p∑i=1
(log(|βi |+ Φ)− log(Φ)).
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 39 / 44
General Re-weighting Algorithm
Given q, λ and µ∗
0. At Step s = 0, compute initial parameter values β(s) and σ(s).
1. Set s = s + 1, and update the data weightsw (s)
i = f ((xi β(s−1) − µ∗)/σ(s−1))1−q, i = 1, ..., n, and the penalty
weights v (s)j = π(β
(s−1)j ;λ)1−q, j = 1, ..., p.
2. Find the parameter values β and σ by minimizing
n∑i=1
wi log f ((xTi β − µ∗)/σ) +
p∑j=1
vj log π(βj ;λ), (12)
3. Compute β(s) and σ(s) by solving
f ((xTi β − µ∗)/σ) ∝ f (xT
i β − µ∗)/, σ)q for β and σ.4. Repeat Steps 1 and 2 until a stopping criterion is satisfied.
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 40 / 44
Model 1: Normal portfolios with Laplace prior
The weights wi and vj are updated using estimates obtained in Step s − 1as follows:
w (s−1)i =
1√2πσ2(s−1)
exp
−(µ∗ − xT
i β(s−1)
)2
2σ2(s−1)
1−q
, (13)
v (s−1)j =
[λ
2 exp{−λ|β(s−1)
j |}]1−q
. (14)
The portfolio variance is also updated using estimates from Step s − 1 as
σ2(s) =
∑ni=1 w (s−1)
i
(µ∗ − xT
i β(s−1)
)2
q∑n
i=1 w (s−1)i
. (15)
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 41 / 44
Model 2: t-portfolios with Laplace priorGiven w (s−1)
i and v (s−1)i , set the initial mixture weights zi = 1/n.
Then, obtain the updated estimates β(s) and σ(s) by iterating the
following expectation-maximization steps
• M-Step:
β′ = argminβ
{ n∑i=1
w (s−1)i z (s−1)
i12
(xT
i β − µσ(s−1)
)2
+ λ
p∑j=1
v (s−1)j |βj |
}, (16)
σ′2
=
∑ni=1 w (s−1)
i z (s−1)i
(xT
i β − µ)2
∑ni=1 w (s−1)
i z (s−1)i
× ν
(ν + 1)q − 1 . (17)
• E-Step:
zi =(νq + 1)σ′
2
νqσ′2 + w (s−1)
i (xTi β′ − µ)2
, i = 1, . . . , n, (18)
where νq = (ν + 1)q − 1.Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 42 / 44
References I
• Huber, P. J., 1981. Robust Statistics. John Wiley & Sons.• Ledoit, O. and Wolf, M., 2004. Honey, I shrunk the sample covariance
matrix. Journal of Portfolio Management 30, Volume 4, 110-119• Fan, J., Zhang, J., Yu, K., 2012. Vast Portfolio Selection With
Gross-Exposure Constraints. Journal of the American StatisticalAssociation, 107 (498), 592-606
• DeMiguel, V., Garlappi, L., Nogales, J., Uppal, R., 2009. AGeneralized Approach to Portfolio Optimization: ImprovingPerformance By Constraining Portfolio Norms. Management Science,Volume 55 (5), 798-812
• Ferrari, D., Yang, Y., 2010. Maximum Lq-likelihood estimation.Annals of Statistics, 38 (2), 753-783.
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 43 / 44
References II
• Ferrari, D., La Vecchia, D., 2012. On robust estimation viapseudo-additive information. Biometrika, 99 (1), 238-244.
• Fastrich, B., Paterlini, S., Winker, P., 2012. Constructing OptimalSparse Portfolios Using Regularization Methods. Available at SSRN:http://ssrn.com/abstract=2169062 orhttp://dx.doi.org/10.2139/ssrn.2169062
• Ferrari, D., Paterlini, S., 2010. Efficient and robust estimation forfinancial returns: an approach based on q-entropy. Available atSSRN: http://ssrn.com/abstract=1906819 orhttp://dx.doi.org/10.2139/ssrn.1906819
• Ronchetti, E.,1997. Robustness aspects of model choice. StatisticaSinica, 7, 327-338
• Machado, J., 1993. Robust model selection and m-estimation.Econometric Theory, 9, 478-478
Ferrari, Giuzio, Paterlini (AU & GER) Penalized Lq-likelihood minimization Seminar 07.04.2014 44 / 44