12,647 research outputs found
Divergent estimation error in portfolio optimization and in linear regression
The problem of estimation error in portfolio optimization is discussed, in
the limit where the portfolio size N and the sample size T go to infinity such
that their ratio is fixed. The estimation error strongly depends on the ratio
N/T and diverges for a critical value of this parameter. This divergence is the
manifestation of an algorithmic phase transition, it is accompanied by a number
of critical phenomena, and displays universality. As the structure of a large
number of multidimensional regression and modelling problems is very similar to
portfolio optimization, the scope of the above observations extends far beyond
finance, and covers a large number of problems in operations research, machine
learning, bioinformatics, medical science, economics, and technology.Comment: 5 pages, 2 figures, Statphys 23 Conference Proceedin
When do improved covariance matrix estimators enhance portfolio optimization? An empirical comparative study of nine estimators
The use of improved covariance matrix estimators as an alternative to the
sample estimator is considered an important approach for enhancing portfolio
optimization. Here we empirically compare the performance of 9 improved
covariance estimation procedures by using daily returns of 90 highly
capitalized US stocks for the period 1997-2007. We find that the usefulness of
covariance matrix estimators strongly depends on the ratio between estimation
period T and number of stocks N, on the presence or absence of short selling,
and on the performance metric considered. When short selling is allowed,
several estimation methods achieve a realized risk that is significantly
smaller than the one obtained with the sample covariance method. This is
particularly true when T/N is close to one. Moreover many estimators reduce the
fraction of negative portfolio weights, while little improvement is achieved in
the degree of diversification. On the contrary when short selling is not
allowed and T>N, the considered methods are unable to outperform the sample
covariance in terms of realized risk but can give much more diversified
portfolios than the one obtained with the sample covariance. When T<N the use
of the sample covariance matrix and of the pseudoinverse gives portfolios with
very poor performance.Comment: 30 page
Generalized Pseudolikelihood Methods for Inverse Covariance Estimation
We introduce PseudoNet, a new pseudolikelihood-based estimator of the inverse
covariance matrix, that has a number of useful statistical and computational
properties. We show, through detailed experiments with synthetic and also
real-world finance as well as wind power data, that PseudoNet outperforms
related methods in terms of estimation error and support recovery, making it
well-suited for use in a downstream application, where obtaining low estimation
error can be important. We also show, under regularity conditions, that
PseudoNet is consistent. Our proof assumes the existence of accurate estimates
of the diagonal entries of the underlying inverse covariance matrix; we
additionally provide a two-step method to obtain these estimates, even in a
high-dimensional setting, going beyond the proofs for related methods. Unlike
other pseudolikelihood-based methods, we also show that PseudoNet does not
saturate, i.e., in high dimensions, there is no hard limit on the number of
nonzero entries in the PseudoNet estimate. We present a fast algorithm as well
as screening rules that make computing the PseudoNet estimate over a range of
tuning parameters tractable
Estimated Correlation Matrices and Portfolio Optimization
Financial correlations play a central role in financial theory and also in
many practical applications. From theoretical point of view, the key interest
is in a proper description of the structure and dynamics of correlations. From
practical point of view, the emphasis is on the ability of the developed models
to provide the adequate input for the numerous portfolio and risk management
procedures used in the financial industry. This is crucial, since it has been
long argued that correlation matrices determined from financial series contain
a relatively large amount of noise and, in addition, most of the portfolio and
risk management techniques used in practice can be quite sensitive to the
inputs. In this paper we introduce a model (simulation)-based approach which
can be used for a systematic investigation of the effect of the different
sources of noise in financial correlations in the portfolio and risk management
context. To illustrate the usefulness of this framework, we develop several toy
models for the structure of correlations and, by considering the finiteness of
the time series as the only source of noise, we compare the performance of
several correlation matrix estimators introduced in the academic literature and
which have since gained also a wide practical use. Based on this experience, we
believe that our simulation-based approach can also be useful for the
systematic investigation of several other problems of much interest in finance
Regularizing Portfolio Optimization
The optimization of large portfolios displays an inherent instability to
estimation error. This poses a fundamental problem, because solutions that are
not stable under sample fluctuations may look optimal for a given sample, but
are, in effect, very far from optimal with respect to the average risk. In this
paper, we approach the problem from the point of view of statistical learning
theory. The occurrence of the instability is intimately related to over-fitting
which can be avoided using known regularization methods. We show how
regularized portfolio optimization with the expected shortfall as a risk
measure is related to support vector regression. The budget constraint dictates
a modification. We present the resulting optimization problem and discuss the
solution. The L2 norm of the weight vector is used as a regularizer, which
corresponds to a diversification "pressure". This means that diversification,
besides counteracting downward fluctuations in some assets by upward
fluctuations in others, is also crucial because it improves the stability of
the solution. The approach we provide here allows for the simultaneous
treatment of optimization and diversification in one framework that enables the
investor to trade-off between the two, depending on the size of the available
data set
Seven Sins in Portfolio Optimization
Although modern portfolio theory has been in existence for over 60 years,
fund managers often struggle to get its models to produce reliable portfolio
allocations without strongly constraining the decision vector by tight bands of
strategic allocation targets. The two main root causes to this problem are
inadequate parameter estimation and numerical artifacts. When both obstacles
are overcome, portfolio models yield excellent allocations. In this paper,
which is primarily aimed at practitioners, we discuss the most common mistakes
in setting up portfolio models and in solving them algorithmically
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