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Penalized portfolio optimization
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Penalized portfolio optimization
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Content
PENALIZED PORTFOLIO OPTIMIZATION
by
Yemin Shi
A Thesis Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
(STATISTICS)
May 2011
Copyright 2011 Yemin Shi
ii
Acknowledgements
I am deeply grateful to my supervisor Dr. Gareth James, for his support, encouragement,
patience and guidance. For giving me the opportunity to learn the state-of-the-arts
penalized optimization research and for helping me exploring the portfolio optimization
problem, for always giving me good advice both in academics and life. Without him this
Master thesis would not have been possible. It is such a rewarding experience that my
“gross-exposure” to the interesting statistical and finance problems is maximized.
I sincerely appreciate Dr. Jianfeng Zhang and Dr. Larry Goldstein for accepting to be my
advisor, revising and quickly answering my questions and doubts.
Thanks to Dr. Xiaojiang Chen and Dr. Fengzhu Sun for their support and encouragement.
And thanks to all my friends at USC, it is their support through all these years that made
this achievement possible.
Thanks to my parents Xuejun Xie and Zhaoping Shi for encouraging me faraway from
China.
iii
Table of Contents
Acknowledgements ............................................................................................................. ii
List of Figures .................................................................................................................... iv
Abstract .............................................................................................................................. vi
Introduction ......................................................................................................................... 1
Chapter 1: Risk Optimization Problem ............................................................................... 3
Gross-exposure constrained portfolio optimization ........................................................ 5
Connection between risk minimization and regression problem .................................... 6
Chapter 2: Penalized Optimization ..................................................................................... 8
Reformulate the risk minimization problem ................................................................... 8
Penalized optimization methods...................................................................................... 9
Solving the penalized portfolio optimization with coordinate descent algorithm ........ 12
The customized coordinate descent portfolio optimization algorithm (CCDPO) ......... 17
Chapter 3: Simulation Studies .......................................................................................... 21
Fama-French three-factor model for data simulation .................................................... 21
Experimental results ...................................................................................................... 23
Table 1. The parameters for the factor loadings and the return factors ................... 23
Chapter 4: Ex-post Data Analysis ..................................................................................... 31
Data integration from Yahoo! Finance website ............................................................ 31
Experimental design and results .................................................................................... 32
Conclusion and Future Work ............................................................................................ 41
Bibliography ..................................................................................................................... 44
iv
List of Figures
Figure 2.1: the optimal solution
when
............................................................ 15
Figure 2.2: the optimal solution
when
............................................................ 16
Figure 3.1: The optimization results of a simulated portfolio with 50 assets in 1
quarter ........................................................................................................................ 26
Figure 3.2: The optimization results of a simulated portfolio with 100 assets in 1
quarter ........................................................................................................................ 27
Figure 3.3: The optimization results of a simulated portfolio with 50 assets in 2
quarters ...................................................................................................................... 28
Figure 3.4: The optimization results of a simulated portfolio with 100 assets in 2
quarters ...................................................................................................................... 29
Figure 4.1: Data integration web application. ................................................................... 32
Figure 4.2: The optimization results of the Dow-Jones composite stocks portfolio.
Using Q1 to predict Q2. ............................................................................................. 34
Figure 4.3: The optimization results of the Dow-Jones composite stocks portfolio.
Using Q2 to predict Q3. ............................................................................................. 35
Figure 4.4: The optimization results of the Dow-Jones composite stocks portfolio.
Using Q3 to predict Q4. ............................................................................................. 36
Figure 4.5: The optimization results of the Dow-Jones composite stocks portfolio.
Using Q1 and Q2 to predict Q3. ................................................................................ 37
Figure 4.6: The optimization results of the Dow-Jones composite stocks portfolio.
Using Q2 and Q3 to predict Q4. ................................................................................ 38
Figure 4.7: The optimization results of the Dow-Jones composite stocks portfolio.
Using Q1 and Q2 to predict Q3 and Q4. ................................................................... 39
v
Figure 4.8: The optimization results of the Dow-Jones composite stocks portfolio.
Using Q1, Q2 and Q3 to predict Q4. ......................................................................... 40
Figure 5.1: The constraint of minimum required return. .................................................. 43
vi
Abstract
Penalization or regularization is an important integration to the traditional regression
method to improve prediction accuracy, speed and adaptability for various problems.
Lasso type L1-regularization methods and its variants can reduce the complexity of high
dimensional data by feature selection as well as coefficient shrinkage. Fan et al. shows
that using an L1-penalty, which he calls “gross-exposure” constraint on the weights in a
portfolio, has significant advantages. In particular it can reduce risk and because the L1
penalty sets coefficients to zero, many fewer assets are needed in the portfolio, and are
therefore suitable for large-scale portfolio optimization problems. Fan et al. formulated
this constrained portfolio risk minimization problem into a convex optimization problem
and solved the problem by an efficient least angle regression (LARS) optimizer. However,
the implementation of LARS used an approximation to the true optimization criterion. To
address this problem, we propose a customized coordinate descent portfolio optimization
procedure (CCDPO). The coordinate wise updating scheme can optimize all coefficients
of the allocation vector faster than LARS. The warming-up and re-initialization steps in
CCDPO prevent the dominant coefficients from growing to extreme values. CCDPO uses
the advantage of coefficients scarcity to reduce the optimization load and achieves fast
speed. To study the performance of CCDPO, we implement a factor-based covariance
estimator for data simulation, and a data integration website for collecting real-life stock
price quotes. The optimization results on the simulated data show that CCDPO
significantly reduces the portfolio risk. And the penalty factor controls the diversity
vii
between empirical risk and actual risk in a similar fashion to the “gross-exposure”
constraint. The ex-post analysis of the real-life Yahoo! Finance data indicates that
portfolios optimized by CCDPO have significant less risk than those optimized by the
non-constrained optimizer.
1
Introduction
Portfolio optimization is one of the fundamental problems in the finance industry.
Portfolio optimization is the process of searching for optimal capital weights for a basket
of assets that best meet the investment requirement (Markowitz 1959). With the
increasing number of modern financial products, portfolio monitoring, management and
optimization become difficult to be done by human intuition along, and require the help
of robust and reliable mathematical models. This work was pioneered by the ground
breaking modern portfolio theory (MPT) of Nobel laureate professor Harry Markowitz in
1952 (Markowitz 1952, Markowitz 1959). The MPT theory proposed the mean-variance
model for portfolio optimization and lead to the celebrated Capital Asset Pricing Model
(CAPM) (Sharpe 1964, Black, Jensen and Scholes 1972) and the Fama-French three-
factor model (Fama and French 1993). Under the MPT frame work, a rational investor
attempts to maximize a portfolio’s expected return for a given amount of portfolio risk, or
minimize the risk for a given level of expected return. The risk and the expected return
are measured by standard deviation and mean of the portfolio return respectively. Hence,
the portfolio optimization problem can be formulated as a classic quadratic optimization
problem.
The standard quadratic programming (QP) algorithm is typically slow when the portfolio
size becomes larger (Bertsekas 1999). Fan et al.(Fan, Zhang and Yu 2008b) had applied
the fast LARS algorithm on the portfolio risk optimization and studied the effects of
2
exposure constraints. However, the LARS algorithm left one asset out of the optimization
procedure. In this work, we reformulate the risk minimization problem into a penalized
optimization problem. Our formulation includes all the asset coefficients in the portfolio
in the optimization process. Then we propose a customized coordinate descent algorithm
to solve this problem. The algorithm optimizes the objective function by updating the
coordinates (coefficients) “one-at-a-time”, and the last coefficient can be optimized every
time after the rest of the coefficients are updated. In our formulation, the L1-norm
(Lasso-type) penalty function can reduce the coefficients to zero and drop the irrelevant
assets in the portfolio to achieve fast performance.
In the next chapter, we describe the risk optimization problem. We introduce the
penalized optimization algorithm in chapter 2. This is followed by a simulation study
based on the three-factor Fama-Franch model in chapter 3. In chapter 4, we perform an
ex-post study on the Dow-Jones composite stocks. The discussion and conclusion are in
chapter five.
3
Chapter 1: Risk Optimization Problem
The fundamental goal of portfolio management is to optimally allocate investments
between different assets. In the framework of modern portfolio theory (MPT), the return
of an asset is assumed to be normally distributed. Its mean value is used to measure the
asset’s return. And the standard deviation is defined as risk. A portfolio is modeled as a
weighted combination of assets. Since the assets are not perfectly positively correlated,
the portfolio’s standard deviation can be reduced by combining these assets with an
optimal allocation vector. Minimizing the portfolio standard deviation with a specific
expected return is called the mean-variance optimization (MVO) problem. An algorithm
used to solve this problem is called an optimizer.
Markowitz first proposed an unconstrained optimizer to solve the MVO problem
(Markowitz 1952, Markowitz 1959). However, the Markowitz optimizer is very sensitive
to the input uncertainty in practice, i.e., the errors in the estimates of the expected return
and the covariance (Fan, et al. 2008b). This sensitivity introduces problems, such as the
computational difficulty in solving large-scale quadratic optimization (Konno and
Hiroaki 1991), the extreme negative weights in the presence of dominant factors (Green
and Hollifield 1990), and the significant change in the allocation vector in response to
small changes in the input parameters (Chopra and Ziemba 1993). These problems are
more pronounced when the portfolio size is large.
4
Various efforts have been made to improve the estimated accuracy of the input. Frost and
Savarino proposed a Bayesian estimation of the means and covariance matrix (Frost and
Savarino 1986). Ledoit and Wolf suggested a shrinkage method that reduces a covariance
matrix towards either the identity matrix or the covariance matrix implied by the factor
structure (Ledoit and Wolf 2003, Ledoit and Wolf 2004). Fan illustrated that the
allocation vector estimated by the factor model significantly outperformed the vector
estimated by the sample covariance (Fan, et al. 2008b).
Another major problem of the Markowitz optimizer is the aggregated errors during the
allocation vector optimization. The aggregated error can result in a very different
allocation vector from the theoretical results. To reduce this error, several techniques
have been suggested to modify the constraints for the optimizer (De Roon, Nijman and
Werker 2001, Goldfarb and Iyengar 2003). Jaganathan and Ma imposed the no-short-sale
constraint on the Markowitz MVO problem and their simulation results outperformed the
Markowitz portfolio (Jagannathan and Ma 2003). They also demonstrated their
constrained efficient portfolio problem is equivalent to the Markowitz mean-variance
problem. Fan et al. extended the study on the no-short-sale constraint to a more relaxed
gross-exposure constraint, and found that the relaxation will improve the portfolio risk
diversification (Fan, et al. 2008b). The traditional quadratic programming algorithm
suffers from computational difficulties in solving the large-scale portfolio optimization
problems (Bertsekas 1999, Schittkowski 2010). Fan et al. adopted the LARS algorithm,
which achieved faster performance.
5
Gross-exposure constrained portfolio optimization
Given a portfolio of p assets, let
be the return vector, be
its covariance matrix, and
be the assets allocation vector, where
, is a
vector of p 1’s. The variance of the portfolio return
can be calculated by
. A
gross-exposure constraint on the portfolio weights can be implemented as
,
where c is the gross-exposure constant. The objective function of the risk minimization
problem with gross exposure can be formulated as:
(1.1)
The theoretical optimized allocation vector
and the empirical optimized allocation
vector
are defined as:
(1. )
where
is an estimated covariance. We define the theoretical minimum risk (oracle risk)
, actual risk
and empirical risk
as:
,
,
(1.2)
6
The theoretical minimum risk is the best we can achieve theoretically. The actual risk can
be estimated in the simulation study, where data are generated with a known covariance .
The empirical risk is usually the direct output from different optimizers.
The gross constraint can prevent extreme positions on the portfolio. This helps to
eliminate the MVO’s sensitivity to the input return and covariance matrix
(Jagannathan, et al. 2003). Under the gross constraint, there is no error accumulation
component in the upper boundary of the difference between empirical risk
and
theoretical risk
. When the gross-exposure constant c=1, the constraint reduces to a no-
short-sale constraint. When c=∞, there is no constraint and the pro blem reduces to a non-
constrained optimization problem.
Connection between risk minimization and regression problem
Risk minimization allows one to improve the utility of existing portfolios, and therefore
plays an important role in portfolio tracking and asset selection. Fan et al. illustrated the
connection of the risk minimization problem to the penalized least-square problem by
formulating the risk minimization problem into a regression problem (Fan, et al. 2008b):
(1.3)
7
Where
and
(i=1,…,p-1). The optimal allocation vector is the
regression coefficients solution
that, with b, best predicts Y.
Substituting in the gross-exposure constraint with
, we have
(1.4)
However (1.4) cannot be expressed in the form of
, given a constant d.
To apply the gross-exposure constraint on the regression problem, Fan et al. used this
approximated constraint
as to substitute the true constraint
. And
the variance minimization problem (1.1) is reformulated as:
(1.5)
8
Chapter 2: Penalized Optimization
Reformulate the risk minimization problem
In the last chapter, we introduced Fan’s approximation of the gross-exposure constraint,
and their approximated risk minimization formulation. However, this formulation is not
equivalent to (1.1) since the weight of the last asset is missing in the
. The
approximated version depends on the choice of asset Y.
To solve this problem, we reformulated the objective function with the full gross-
exposure constraint:
(2.1)
Let
,
is centralized by subtracting its mean
, i=1…p. The above
equation is equivalent to:
(2.2)
The “Lagrange” version of the problem is defined as:
9
(2.3)
Where
. controls the
gross-exposure. This is a standard quadratic programming problem with inequality
constraint. When Q is a convex function, we can find a global minimum if there exists a
feasible allocation vector W satisfying the constraints. This is also known as a convex
optimization problem with inequality constraints.
The constrained convex optimization is typically carried out by a standard quadratic
programming algorithm. However, standard quadratic programming may suffered from
slow convergence and accumulated errors for large scale problems (Bertsekas 1999,
Schittkowski 2010). There are a variety of efficient methods to solve such penalized
problems.
Penalized optimization methods
Penalization methods usually outperform the prediction accuracy of the unconstrained
ordinary least squares method by decreasing the prediction variance. The objective
function of a convex optimization problem is usually penalized (regularized) to meet the
requirement of a specific problem. The L1-norm penalization method, such as LASSO
10
(Tibshirani 1996), shrinks the regression coefficient and selects the important coefficients
by dropping the rest of the coefficients to zero. This L1-norm penalization method
improves the model’s prediction accuracy and interpretability. The gross-exposure
constraint is an L1-norm penalization term, hence can be plugged into the LASSO type of
penalization method, and used to guide our assets selection procedure in portfolio
optimization.
The L1-norm penalization method was first applied in wavelets selection (Donoho and
Johnstone 1995). Tibshiriani formally introduced the LASSO for regression (Tibshirani
1996) and the same idea is used in basis pursuit (Chen, Donoho and Saunders 1998). A
typical LASSO objective function can be defined as follows:
Given observations
,
(2.4)
For a single predictor, the solution of the minimization problem is a soft thresholded
version of the OLS estimate
.
11
(2.5)
The convex objective function reduces to a few special cases, which corresponds to the
different conditions illustrated above.
In 2004, Efron et al.(Efron, Hastie, Johnstone and Tibshirani 2004) proposed a least angle
regression (LARS) algorithm to compute the entire LASSO coefficient path at a cost of a
full least-square fit. Path algorithms have been extended to various related problems, such
as the Grouped Lasso (Yuan and Lin 2006), support vector machine (Hastie, Rosset,
Tibshirani and Zhu 2004), quantile regression (Li and Zhu 2008), logistic regression and
glms (Park and Hastie 2007), and Danzig selector (Radchenko and James 2008, James
and Radchenko 2009 ). Fan et al. used the LARS algorithm for portfolio optimization
(Fan, et al. 2008b). However, Fan’s approach only approximates the optimal solution.
This motivated us to find a fully optimized algorithm for portfolio optimization with
comparable speed to LARS.
Friedman et al. proposed a coordinate-wise descent algorithm, which is capable of
solving the above problems efficiently (Friedman, Hastie, Hofling and Tibshirani 2007).
The coordinate-wise updating scheme allows one to optimize the missing weight in
LARS. Results indicate that the coordinate descent method can achieve “dramatic”
speedups over most of the competitors, including LARS, by factors of 10, 100 or even
more (Friedman, Hastie and Tibshirani 2010).
12
Solving the penalized portfolio optimization with coordinate descent
algorithm
In the last chapter, we reformulated the portfolio optimization problem with full exposure
constraints. Now we use the coordinate descent algorithm to update each coordinate
coefficient separately, while holding other coefficient fixed. The sequential updates
repeat until convergence. The detailed procedures are described in the following:
We use the first derivative to search for the solution that partially (coordinate wise)
minimizes the objective function:
(2.6)
Where
When the local minimum falls in the continuous range of
,
reaches the
local minimum when
=0, we calculate
13
(2.7)
Therefore,
(2.8)
Where,
(2.9)
When the local minimum falls into the discontinuity range of
,
is
undefined, we need to check the discontinuity points within the range to decide this
coordinate-wise optimized
. The discontinuity points corresponding to the jumping
point in the first derivative so we examine the shape of
:
(2.10)
14
Here
(2.11)
In summary, when
:
(2.12)
Similarly, when
(2.13)
The conditions in the above equations are illustrated in Figure 2.1 and Figure 2.2. On the
plane spanned by
(X-axis) and
(Y-axis), the joint of
and the X-axis is
the optimal solution
.
moves along Y-axis when the intercept
varies. Since
the line of
. Since the line of
has 5 segments, the optimal weight
has
five formats when
(Figure 2.1) and another five formats when
(Figure
2.2).
15
(a)
(b)
(c)
(d)
(e)
Figure 2.1: the optimal solution
when
.
The X-axis stands for
and Y-
axis stands for
. The
joint of
and X-axis is the
optimal solution of equation
(2.2). The 5 segments of
line are colored in
black, orange, green, red and
blue. The gray line in subpanel
(a) is the position of
when the intercept
.
When
changes its value,
have different format.
(a)
(b)
(c)
(d)
(e)
16
(a)
(b)
(c)
(d)
(e)
Figure 2.2: the optimal solution
when
.
The X-axis stands for
and Y-
axis stands for
. The joint
of
and X-axis is the
optimal solution of equation
(2.2). The 5 segments of
line are colored in black,
orange, green, red and blue. The
gray line in subpanel (a) is the
position of
when the
intercept
. When
changes its value,
have
different format.
(a)
(b)
(c)
(d)
(e)
Once we find the partial minimum solution
, we update the coordinate coefficient.
This procedure goes through all p-1 coordinates, and then the last coordinate p is updated
by,
17
. (2.14)
We repeat this cycle until the coefficients have stabilized.
To accelerate the speed of the coordinate descent method:
1) We start with a grid of ’s uniformly distributed on log-scale. This sample grid
concentrates on the small values, where the
change more frequently.
2) Large values are likely to push coefficients to zero, starting from larger and
saves time searching for a feasible .
3) Except for the last coefficient,
, all initial coefficients are set to zero. During
the updating procedure, the asset with largest coefficient is set as the Y in the next
coordinate updates step.
4) Current weights will be used as a warm start for the next step.
The pseudo code of the customized coordinate descent portfolio optimization algorithm is
presented below:
The customized coordinate descent portfolio optimization algorithm
(CCDPO)
Let return matrix
be a matrix. n stands for the number of
days, and p stands for the number of risk assets,
, is the allocation
vector. For a fixed :
18
1. Centralize the return
by subtracting the mean of the jth asset,
,
j=1…p.
2. Set
to zero, and set
;
3. Set
and
(j=1,…,p-1). Y is a vector and X is a
matrix.
4. For iteration , the maximum iteration number
a. For k=1 to p-1,
i.
;
.
ii.
iii.
b.
. If
, we can randomly choose another column
of
as Y and reinitialze X and repeat 4.
19
5. Reinitialize X and Y, Let
, set
, the
column with the maximun absolute weight. Set
.
Repeat step 4 and 5 until converge.
The first key advantage of our algorithm is the customized speedup steps. Our algorithm
sampled the constraint factor in its exponential space
. This setting allows more
details to be sampled in the small constraint condition, where the allocation vector W is
optimized. The sample is performed in a decreasing manner (from large value to small
value) to appreciate the scarcity of W due to the heavier penalty when is large. The
simulation results of chapter 3 (Figure 3.1 to Figure 3.4) illustrate that the number of non-
zero coefficients decreases with . The decrease is approximately linear and is
within a limited range close to zero. The optimization load under each can be
maximally reduced from to approximately .
Another speedup step is the warm-up procedure that uses the current optimized weight as
the initial weights for the next optimization cycle (with decreased). The optimized
weights save searching time for the adjacent values, therefore improving optimization
time.
The third improvement over normal coordinate descent is in its robustness and
completeness. In step 4(b), the algorithm leaves the largest coefficient being updated after
all the rest are optimized. This setting prevents the largest coefficient from growing to an
20
extreme value. Step 4(b) also helps to optimize the last coefficients
left by LARS,
hence it is the key difference between coordinate descent and LARS in solving the
portfolio optimization problem.
21
Chapter 3: Simulation Studies
In this section, we use simulated data to study the performance of our coordinate descent
algorithm. The simulated dataset is generated based on the factor model proposed by Fan
et al.(Fan, Fan and Lv 2008a). Given the true covariance matrix and the sampled
covariance matrix, we compared the actual and empirical risk generated at different
gross-exposure constraint levels, and illustrate that the divergence between actual and
empirical risk increases with the decrease of lambda. This is similar to the results with the
gross-exposure constraint in the LARS-based optimization (Fan, et al. 2008b).
Fama-French three-factor model for data simulation
Fama and French (Fama, et al. 1993) developed a three-factor model to describe the
market behavior. The three-factor model is a significant extension of the capital asset
pricing model (CAPM) (Sharpe 1964, Black, et al. 1972). In the CAPM, a capital asset’s
expected return is measured by a single factor , the expected return can be defined as:
(3.1)
Where
is the risk free return and
is the market return. Fama and French observed
that two classes of stocks perform better than the market as a whole, namely, small
capital and value stocks. They integrated these two factors into the original CAPM model
and developed the following three-factor model.
22
(3.2)
Here,
is the portfolio return,
is the factor loading for the small capital stocks, SMB
stands for small (capital assets) minus big,
is the factor loading for value stocks, HML
stands for high (B/M, book-to-market ratio) minus low, and is the idiosyncratic noise
independent of the three factors. The expected market return E(
) is constructed using
the value-weighted return on all NYSE, AMEX and NASDAQ stocks.
is the one-
month Treasury bill rate. SMB and HML are constructed by six portfolios with different
size and different B/M ratios.
The factor based covariance estimator outperforms the sample covariance in portfolio
allocation (Fan, et al. 2008a). Fan et al. proposed a factor model based covariance matrix
estimator and simulated the portfolio excessive return data from May 1, 2002 to Aug. 29
2005 (Fan, et al. 2008b). The simulation is based on the factor loadings from French’s
website in the same period of time. We adopted the same idea of covariance estimation
and data simulation.
The simulation procedure assumes we are going to generate p stocks’ excessive return in
n time periods:
1) Generate factor loadings from the normal distribution
.
2) Generate idiosyncratic noise parameters:
23
a. Noise level parameters
are generated by a gamma distribution ,
where .
b. Generated values above 0.1975 are accepted; repeat until p parameters are
accepted.
3) For each simulation:
a. Generate idiosyncratic noise with a student-t distribution with 6 degrees
of freedom, using the noise parameters
.
b. Generate n by 3 factor matrices from
.
c. The pseudo portfolio excess return is generated by .
The simulation parameters
and
are listed in Table 2.
Table 2: The parameters for the factor loadings and the return factors
Parameters for the factor loadings
0.7828 0.02914 0.02387 0.010184
0.518 0.02387 0.05395 -0.00697
0.41 0.01018 -0.00696 0.086856
Parameters for the return factors
0.02355 1.2507 -0.035 -0.2042
0.01298 -0.035 0.3156 -0.0023
0.02071 -0.2042 -0.0023 0.193
Experimental results
To study the performance of CCDPO, we simulated the assets return data from different
scales (50 and 100 portfolios) and in different time periods (1 quarter and 2 quarters). The
24
results are plotted in the following figures (Figure 3.1 to Figure 3.4). We compared our
results with the risk of the un-optimized portfolio (equal weights portfolio) and the risk of
the portfolio that optimized with non-constrained optimizer (NCO). The former does not
change with and is plotted as a horizontal line. The non-constrained optimization result
is a special case ( ) of our constrained optimizer, hence is illustrated at the left end
of our risk curves, where is close to zero. We use the actual risk to demonstrate the
performance.
Our simulation studies indicate that CCDPO improves the risk performance with respect
to both equal weight portfolio and non-constraint portfolio. Figure 3.1 illustrates the
results of a quarterly assets return simulation of 50 assets in 62 days (1 quarter). Figure
3.1(a) shows the overall trend of the risk with the increase of the constraint. When
starts to increase from the non-constrained condition on the left side of the figure ( =
), the risk starts to drop until
, where the CCDPO reaches the
minimum risk. When increases further, the actual risk begins to increase. We set the
maximum on the right side, where the risk still increases. The minimum risk
achieved by CCDPO is 0.5% and is about 63% of the NCO risk which is 0.8%. The L1-
penalty of CCDPO maximally reduced about 36% of the actual risk in comparison with
the results from the unconstrained optimization. The CCDPO achieves very accurate
results in risk approximation. The divergence between actual risk and empirical risk
begins to converge when increases above
. Figure 3.1(a) also indicates that the
25
optimal risk resulted in a reduction of the equal weights portfolio risk (0.95%) by 53%.
Figure 3.1(b) demonstrates that the number of non-zero weights increases in the inverse
ratio of . Nearly 90% of weights become “active” within
, while most of
the weights are zero when
. The decreasing optimization strategy in CCDPO uses
this fact to reduce computational cost. Figure 3.1(b) also shows that most weights are
positive when
, below a critical point,
, the number of positive and negative
weights increase simultaneously, this increase results in short-sale assets in the portfolio,
and the increasing divergence between actual risk and empirical risk. Figure 3.1(c)
illustrates the coefficient, i.e. weights pathways between
. Most of the asset
weights are positive and constrained close to zero when
, once passed the critical
point, the gross-exposure constraint is weak enough so that more stocks become “active”
and most of them enter into the portfolio between
. Since the minimum actual
risk is located at
, the candidate for a low risk portfolio with a limited percentage
of short-sale assets falls within
. For briefness purpose, we call this the candidate
of optimal , and this region the optimal region in the future description.
26
(a)
(b)
(c)
Figure 3.1: The optimization results of a simulated
portfolio with 50 assets in 1 quarter (63 days). (a)
Actual risk, empirical risk and the equal weights risk.
(b) The number of non-zero weights (blue), positive
weights (green) and negative weights (red). (c) 2D
allocation vector profile.
Next, we perform a quarterly simulation on a large scale portfolio (100 assets in 62 days).
CCDPO reaches a similar result to that of the medium sized portfolio above. Figure 3.2(a)
shows the equal weighted risk is 0.79%. The minimum actual risk is about 0.33%, and
the unconstrained optimization risk is 0.47%. Hence CCDPO reduced 58% of the actual
risk from the equal weighted portfolio and reduced 30% of the risk from the optimized
portfolio without the L1-penalty. The critical region of the constraint factor is still
-8 -6 -4 -2 0 2 4 6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
log(lambda)
Volatility (%)
Risk approximation
Empirical Risk
Actural Risk
Equal Weights Risk
-8 -6 -4 -2 0 2 4 6
0
5
10
15
20
25
30
35
40
45
50
log(lambda)
Number of stocks in the portfolio
Number of stocks in the portfolio
Non-zero weights
Positive weights
Negative weights
-8 -6 -4 -2 0 2 4 6
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
log(lambda)
Allocation vector weights
Allocation vector coefficients profile
27
within
, most of the weights are positive when
, since the minimum
actual risk is at the point of
, the candidate of optimal is between
.
(a)
(b)
(c)
Figure 3.2: The optimization results of a simulated
portfolio with 100 assets in1 quarter (63 days). (a)
Actual risk, empirical risk and the equal weights risk.
(b) The number of non-zero weights (blue), positive
weights (green) and negative weights (red). (c) 2D
allocation vector profile.
We also simulated the two quarters (126 days) of stocks returns for the medium size
portfolio (50 stocks) and a larger size portfolio (100 stocks). Figure 3.3 presents the
optimized results for the 50-stock portfolio: The optimized actual risk of the 50-stock
portfolio is 0.43% and 0.38% for the unconstrained optimizer and the constrained
-8 -6 -4 -2 0 2 4 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
log(lambda)
Volatility (%)
Risk approximation
Empirical Risk
Actural Risk
Equal Weights Risk
-8 -6 -4 -2 0 2 4 6
0
10
20
30
40
50
60
70
80
log(lambda)
Number of stocks in the portfolio
Number of stocks in the portfolio
Non-zero weights
Positive weights
Negative weights
-8 -6 -4 -2 0 2 4 6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
log(lambda)
Allocation vector weights
Allocation vector coefficients profile
28
CCDPO, the equal weights constraint is 0.91%. The actual risk reduced from the
unconstrained optimal portfolio is 12%, the reduced actual risk from equal weights
portfolio is 58%. The critical region of the constraint factor is also within
,
most of the weights are positive when
, the minimum actual risk is at the point of
and the candidate of optimal is between
.
(a)
(b)
(c)
Figure 3.3: The optimization results of a simulated
portfolio with 50 assets in 2 quarters (126 days). (a)
Actual risk, empirical risk and the equal weights risk.
(b) The number of non-zero weights (blue), positive
weights (green) and negative weights (red). (c) 2D
allocation vector profile.
-8 -6 -4 -2 0 2 4 6
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
log(lambda)
Volatility (%)
Risk approximation
Empirical Risk
Actural Risk
Equal Weights Risk
-8 -6 -4 -2 0 2 4 6
0
5
10
15
20
25
30
35
40
45
50
log(lambda)
Number of stocks in the portfolio
Number of stocks in the portfolio
Non-zero weights
Positive weights
Negative weights
-8 -6 -4 -2 0 2 4 6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
log(lambda)
Allocation vector weights
Allocation vector coefficients profile
29
Figure 3.4 shows the optimized results for the 100-stock portfolio. The optimized actual
risk of the 100-stock portfolio is 0.96% and 0.40% for the unconstrained optimizer and
the constrained CCDPO, the equal weights constraint is 0.97%. The reduced risk from
unconstrained optimal portfolio is 58%, the reduced risk from equal weights portfolio is
59%. The candidate of optimal is between
.
(a)
(b)
(c)
Figure 3.4: The optimization results of a simulated
portfolio with 100 assets in 2 quarters (126 days). (a)
Actual risk, empirical risk and the equal weights risk.
(b) The number of non-zero weights (blue), positive
weights (green) and negative weights (red). (c) 2D
allocation vector profile.
-8 -6 -4 -2 0 2 4 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
log(lambda)
Volatility (%)
Risk approximation
Empirical Risk
Actural Risk
Equal Weights Risk
-8 -6 -4 -2 0 2 4 6
0
10
20
30
40
50
60
70
80
90
100
log(lambda)
Number of stocks in the portfolio
Number of stocks in the portfolio
Non-zero weights
Positive weights
Negative weights
-8 -6 -4 -2 0 2 4 6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
log(lambda)
Allocation vector weights
Allocation vector coefficients profile
30
Our simulation study shows that CCDPO significantly reduces the portfolio risk. The
results outperform those from an unconstrained optimizer and the equal weights portfolio.
The minimum risk point is located around =1.0. A candidate of optimal is usually
within the optimal region
. On the right boundary of the region, the optimized
allocation vector has no negative weights. A close to this side corresponds to the no-
short-sale constraint condition. On the left side of the region, there are more negative
weights. A close to this side corresponds to the condition without gross-exposure
constraint.
31
Chapter 4: Ex-post Data Analysis
In this section, we will use CCDPO to perform the ex-post analysis on the real-life data
set, such as Yahoo! Finance dataset. CCDPO can significantly reduce the actual risk
robustlly.
Data integration from Yahoo! Finance website
The Yahoo! Finance website provides instant stock quotes and portfolio management
resources. It is convenient for accessing abundant information on a single stock. However
the data integration software is usually for commercial users. In this project, we
implemented our own CGI-based web application for data integration. As a user provides
the date range and symbol list, the application will automatically integrate the feasible
data from the Yahoo! Finance website. The integrated stock quotes will be displayed in
the web browser with a download link at the bottom of the data list (Figure 4.1) .
32
(a)
(b)
Figure 4.1: Data integration web application. Implemented by python CGI scripts on an Apache server.
(a) The interactive interface. Link: helicase.usc.edu/ppo/ppo.html. (b) The downloading page.
Experimental design and results
To study the portfolio optimization performance, we used 12 months of stock price
quotes of the Dow Jones Index composite for ex-post risk performance analysis. In order
to study the prediction power, we design the following experiments:
1. Use 1 quarter to predict next quarter (Q1-Q2, Q2-Q3, Q3-Q4)
2. Use 2 quarters to predict next 1 quarter (Q1Q2-Q3,Q2Q3-Q4)
3. Use 2 quarters to predict next 2 quarters (Q1Q2-Q3Q4)
4. Use 3 quarters to predict 1 quarter (Q1Q2Q3-Q4)
Where Q1-Q2, indicates we use the data of quarter 1 to train the model and predict the
diversity in quarter 2, Q2-Q3, Q3-Q4 are defined in a similar way. Q1Q2-Q3 means to
use the first 2 quarter’s data to train the model and predict the diversity of Q3. Q2Q3-Q4
and Q1Q2Q3-Q4 are defined using a similar scheme. The data period ranges from Jan 1
st
33
2010 to Jan 1
st
2011. The price quotes are processed into percentage of return based on
the previous closing price. And these returns are then centralized by subtracting the mean
of the return. The covariance of the stock returns during the period to be predicted are
generated by basic sampling. The results are illustrated in Figure 4.2 to Figure 4.8.
Figure 4.2 to Figure 4.4 respectively present the results of Q1-Q2, Q2-Q3 and Q3-Q4.
The results illustrate that in these three prediction scenarios, the CCDPO reduces actual
risk by 0.2%, 0.2% and 0.3% in comparison with the results of the non-constrained
optimizer. In other words, CCDPO reduces 22%, 20% and 33% of the risk obtained by
the non-constrained optimizer respectively. The minimum actual risk falls
between
. The candidate of the optimal for non-short-sale portfolio locates
between
.
34
(a)
(b)
(c)
Figure 4.2: The optimization results of the Dow-Jones
composite stocks portfolio. Using Q1 to predict Q2.
(a) Actual risk, empirical risk and the equal weights
risk. (b) The number of non-zero weights (blue),
positive weights (green) and negative weights (red).
(c) 2D allocation vector profile.
-8 -6 -4 -2 0 2 4 6
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
log(lambda)
Volatility (%)
Risk approximation
Empirical Risk
Actural Risk
Equal Weights Risk
-8 -6 -4 -2 0 2 4 6
0
5
10
15
20
25
30
log(lambda)
Number of stocks in the portfolio
Number of stocks in the portfolio
Non-zero weights
Positive weights
Negative weights
-8 -6 -4 -2 0 2 4 6
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
log(lambda)
Allocation vector weights
Allocation vector coefficients profile
35
(a)
(b)
(c)
Figure 4.3: The optimization results of the Dow-Jones
composite stocks portfolio. Using Q2 to predict Q3.
(a) Actual risk, empirical risk and the equal weights
risk. (b) The number of non-zero weights (blue),
positive weights (green) and negative weights (red).
(c) 2D allocation vector profile.
-8 -6 -4 -2 0 2 4 6
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
log(lambda)
Volatility (%)
Risk approximation
Empirical Risk
Actural Risk
Equal Weights Risk
-8 -6 -4 -2 0 2 4 6
0
5
10
15
20
25
30
log(lambda)
Number of stocks in the portfolio
Number of stocks in the portfolio
Non-zero weights
Positive weights
Negative weights
-8 -6 -4 -2 0 2 4 6
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
log(lambda)
Allocation vector weights
Allocation vector coefficients profile
36
(a)
(b)
(c)
Figure 4.4: The optimization results of the Dow-Jones
composite stocks portfolio. Using Q3 to predict Q4.
(a) Actual risk, empirical risk and the equal weights
risk. (b) The number of non-zero weights (blue),
positive weights (green) and negative weights (red).
(c) 2D allocation vector profile.
Figure 4.4. and Figure 4.5 present the results of Q1Q2-Q3 and Q2Q3-Q4. In these two
scenarios, CCDPO reduces 0.03% and 0.13% actual risks in comparison with the results
of the non-constrained optimizer. These results stand for 4% and 17% risk reduction from
the risk obtained by the non-constrained optimizer. The minimum actual risk falls
between
. The candidate of the optimal for non-short-sale portfolio is
between
.
-8 -6 -4 -2 0 2 4 6
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
log(lambda)
Volatility (%)
Risk approximation
Empirical Risk
Actural Risk
Equal Weights Risk
-8 -6 -4 -2 0 2 4 6
0
5
10
15
20
25
30
log(lambda)
Number of stocks in the portfolio
Number of stocks in the portfolio
Non-zero weights
Positive weights
Negative weights
-8 -6 -4 -2 0 2 4 6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
log(lambda)
Allocation vector weights
Allocation vector coefficients profile
37
(a)
(b)
(c)
Figure 4.5: The optimization results of the Dow-Jones
composite stocks portfolio. Using Q1 and Q2 to predict
Q3.
(a) Actual risk, empirical risk and the equal weights
risk. (b) The number of non-zero weights (blue),
positive weights (green) and negative weights (red).
(c) 2D allocation vector profile.
-8 -6 -4 -2 0 2 4 6
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
log(lambda)
Volatility (%)
Risk approximation
Empirical Risk
Actural Risk
Equal Weights Risk
-8 -6 -4 -2 0 2 4 6
0
5
10
15
20
25
30
log(lambda)
Number of stocks in the portfolio
Number of stocks in the portfolio
Non-zero weights
Positive weights
Negative weights
-8 -6 -4 -2 0 2 4 6
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
log(lambda)
Allocation vector weights
Allocation vector coefficients profile
38
(a)
(b)
(c)
Figure 4.6: The optimization results of the Dow-Jones
composite stocks portfolio. Use Q2 and Q3 to predict
Q4.
(a) Actual risk, empirical risk and the equal weights
risk. (b) The number of non-zero weights (blue),
positive weights (green) and negative weights (red).
(c) 2D allocation vector profile.
Figure 4.7. presents the results of Q1Q2-Q3Q4. CCDPO reduces 0.03% actual risks in
comparison with the optimal risk of the non-constrained optimizer. CCDPO reduces 4%
risk from the result of non-constrained optimizer. The minimum actual risk falls
between
. The candidate of the optimal for non-short-sale portfolio is
between
.
-8 -6 -4 -2 0 2 4 6
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
log(lambda)
Volatility (%)
Risk approximation
Empirical Risk
Actural Risk
Equal Weights Risk
-8 -6 -4 -2 0 2 4 6
0
5
10
15
20
25
30
log(lambda)
Number of stocks in the portfolio
Number of stocks in the portfolio
Non-zero weights
Positive weights
Negative weights
-8 -6 -4 -2 0 2 4 6
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
log(lambda)
Allocation vector weights
Allocation vector coefficients profile
39
(a)
(b)
(c)
Figure 4.7: The optimization results of the Dow-Jones
composite stocks portfolio. Use Q1 and Q2 to predict
Q3 and Q4.
(a) Actual risk, empirical risk and the equal weights
risk. (b) The number of non-zero weights (blue),
positive weights (green) and negative weights (red).
(c) 2D allocation vector profile.
Figure 4.8. presents the results of Q1Q2Q3-Q4. CCDPO reduces 0.08% actual risks in
comparison with the optimal risk of the non-constrained optimizer. CCDPO reduces 12%
risk from the results of the non-constrained optimizer. The minimum actual risk falls
between
. The candidate of the optimal for non-short-sale portfolio is
between
.
-8 -6 -4 -2 0 2 4 6
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
log(lambda)
Volatility (%)
Risk approximation
Empirical Risk
Actural Risk
Equal Weights Risk
-8 -6 -4 -2 0 2 4 6
0
5
10
15
20
25
30
log(lambda)
Number of stocks in the portfolio
Number of stocks in the portfolio
Non-zero weights
Positive weights
Negative weights
-8 -6 -4 -2 0 2 4 6
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
log(lambda)
Allocation vector weights
Allocation vector coefficients profile
40
(a)
(b)
(c)
Figure 4.8: The optimization results of the Dow-Jones
composite stocks portfolio. Use Q1, Q2 and Q3 to
predict Q4.
(a) Actual risk, empirical risk and the equal weights
risk. (b) The number of non-zero weights (blue),
positive weights (green) and negative weights (red).
(c) 2D allocation vector profile.
The real life risk prediction results indicate CCDPO’s can optimize the stock actual risk
for the next future period within one year. The improvement in performance is more
pronounced given a shorter period of time, such as one quarter. The minimum actual risk
usually sits within
, the optimal region is usually between
and the
non-short-sale portfolio allocation vector can be obtained by setting to the right
boundary of optimal region.
-8 -6 -4 -2 0 2 4 6
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
log(lambda)
Volatility (%)
Risk approximation
Empirical Risk
Actural Risk
Equal Weights Risk
-8 -6 -4 -2 0 2 4 6
0
5
10
15
20
25
30
log(lambda)
Number of stocks in the portfolio
Number of stocks in the portfolio
Non-zero weights
Positive weights
Negative weights
-8 -6 -4 -2 0 2 4 6
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
log(lambda)
Allocation vector weights
Allocation vector coefficients profile
41
Conclusion and Future Work
In this thesis, we have addressed the incomplete optimization problem in the LARS
algorithm and formulated the portfolio risk optimization problem into a constrained
convex optimization problem. We proposed a fast customized coordinate descent
portfolio optimization procedure (CCDPO) to solve this problem. CCDPO updates each
coordinate separately, and therefore is able to optimize the whole allocation vector.
CCDPO updates the maximum vector weights at the end of each updating sweep, and
prevents the dominant weight from reaching extreme values. During the optimization,
CCDPO appreciates the vector sparsity. A decreasing searching strategy of the penalty
factor reduces the computational load. And a customized warm-up step passes the
optimized coefficients to the next iteration and saves computational time. All these
customized steps help to improve the robustness and efficiency of our CCDPO.
To study the performance of CCDPO, we have implemented a factor based covariance
matrix estimator to simulate the assets return data in the market. We optimize these data
with CCDPO and show that the optimized risk is significantly lower than that of the non-
constrained optimizer. The improvement ranges between 12% to 58%. All the results
optimized by CCDPO outperform the equal weights portfolio. The penalty factor
corresponding to the minimum actual risk usually falls within a narrow range
between
. An optimal for the non-short-sale portfolio is close to the right side
of this range. The difference between the actual risk and the empirical risk convergences
42
as passes the critical point around
, this indicates that the CCDPO is very accurate
in risk approximation.
We studied the performance of CCDPO on 12 months of the Dow-Jones composite stock
return data, downloaded from the Yahoo! Finance website. The real life data ex-post risk
estimation (prediction) demonstrates that our CCDOP can decrease the actual risk of the
portfolio in the next period, such as one or two quarters. Our study also shows that
CCDPO reduces risk more significantly when the prediction period is about one quarter.
Longer training time such as two or three quarters does not necessary improve the
prediction performance. This might be thanks to the similar covariance matrix between
two adjacent quarters.
There are several possible directions in which one can further improve the CCDPO. 1) A
constraint can be applied on the optimization iteration to ensure a minimum return. For
example, during the iteration, when the expected return decreases to hit the minimum
required return, we can either stop and output the current allocation vector, or iterate a
few more steps to check if the return rebounds back (Figure 5.1). If the expected return
decreases further below a threshold, we can stop and trace back for an optimal solution. If
the expected return rebounds back, we can keep on to do further optimization. 2) The
CCDPO can be further speedup. For example, during the optimization iteration for a
specific , it is not necessary to update all the coordinate weights when the allocation
vector is sparse. One can only iterate through non-zero weights and sweep the vector
43
weights in the last step to see if there are any new non-zero weights come into the vector.
If so, the optimization for this is done. Otherwise, one can involve all the weights and
do iteration again. In most cases, there won’t any new non -zero weights appear, hence the
CCDPO can be speedup. 3) The optimal weights are changing dynamically with the
stock market due to the change of the stock covariance matrix. One might not want the
optimal weights to change too often. In order to smooth the variation of the optimal
weights, one can use the fussed Lasso to constrain the variance of the weights. 4) It is
also possible to use grouped-lasso to include the section constraint to limit the percentage
of stocks from the same industry.
Figure 5.1: The constraint of minimum required return.
When the expected return decreases to hit the minimum expected return
(green dash line), we can let do a few more iterations to see if the return
bounds back. If it decreases further, (e.g. hits the red dash line), we can go
back to chose an optimal allocation vector that meet the minimum expected
return requirement.
1 2 3 4 5 6 7 8 9 10 11
0.22
0.23
0.24
0.25
0.26
0.27
0.28
Iteration
Expected Return
Constraint on iterations
44
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Abstract (if available)
Abstract
Penalization or regularization is an important integration to the traditional regression method to improve prediction accuracy, speed and adaptability for various problems. Lasso type L1-regularization methods and its variants can reduce the complexity of high dimensional data by feature selection as well as coefficient shrinkage. Fan et al. shows that using an L1-penalty, which he calls “gross-exposure” constraint on the weights in a portfolio, has significant advantages. In particular it can reduce risk and because the L1 penalty sets coefficients to zero, many fewer assets are needed in the portfolio, and are therefore suitable for large-scale portfolio optimization problems. Fan et al. formulated this constrained portfolio risk minimization problem into a convex optimization problem and solved the problem by an efficient least angle regression (LARS) optimizer. However, the implementation of LARS used an approximation to the true optimization criterion. To address this problem, we propose a customized coordinate descent portfolio optimization procedure (CCDPO). The coordinate wise updating scheme can optimize all coefficients of the allocation vector faster than LARS. The warming-up and re-initialization steps in CCDPO prevent the dominant coefficients from growing to extreme values. CCDPO uses the advantage of coefficients scarcity to reduce the optimization load and achieves fast speed. To study the performance of CCDPO, we implement a factor-based covariance estimator for data simulation, and a data integration website for collecting real-life stock price quotes. The optimization results on the simulated data show that CCDPO significantly reduces the portfolio risk. And the penalty factor controls the diversity between empirical risk and actual risk in a similar fashion to the “gross-exposure” constraint.
Linked assets
University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Shi, Yemin
(author)
Core Title
Penalized portfolio optimization
School
College of Letters, Arts and Sciences
Degree
Master of Science
Degree Program
Mathematics
Publication Date
05/02/2011
Defense Date
05/01/2011
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
asset management,coordinate descent,feature selection,L1-penalty,LARS,lasso,OAI-PMH Harvest,penalized optimization,portfolio optimization,regularized regression
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Goldstein, Larry (
committee chair
), James, Gareth (
committee member
), Zhang, Jianfeng (
committee member
)
Creator Email
biostanley@gmail.com,yeminshi@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m3839
Unique identifier
UC1195564
Identifier
etd-SHI-4566 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-456828 (legacy record id),usctheses-m3839 (legacy record id)
Legacy Identifier
etd-SHI-4566.pdf
Dmrecord
456828
Document Type
Thesis
Rights
Shi, Yemin
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
asset management
coordinate descent
feature selection
L1-penalty
LARS
lasso
penalized optimization
portfolio optimization
regularized regression