Signal Processing - May 2017 - 22

Example SP Research on Portfolio
Optimization and Related Risk Modeling
In [30], a subspace formulation of MVP optimization with
risk-factor constraints and related toy examples is given.
For example, a market-neutral portfolio requires
F T x p = 0, where F is the loading weights of a single
market factor, usually estimated by a time-series regression of all security return time series with a market index,
and x p represents the desired portfolio weights. Torun et
al.'s article in IEEE Signal Processing Magazine [31]
shows an eigenfiltering method to estimate the covariance
matrix with a large number of securities. When the security pool is large, the raw covariance matrix is almost singular due to high correlations among securities. Note that
the inverse of the covariance matrix, i.e., the precision
matrix, is often necessary in MVP optimization. Therefore,
a noise regularization term is added on the eigenfiltered
covariance matrix to improve the robustness of covariance matrix estimation and its conditional number. The
same authors [32] further use an autoregressive model to
improve the stability and computational efficiency of estimation of the empirical covariance matrix of highly correlated securities. We also direct readers to an example in
[33] that uses Bayesian and regularization methods for
the estimation of the unknown observation covariance
matrix and the related MVP optimization algorithm. Also,
the authors in [34] propose to use smooth and monotone
regularization to tackle the high correlation problem in
covariance matrix estimation. The work in [35] summarizes the relative performance of different estimation strategies for minimum-variance portfolio optimization problems
using the inversion of the estimated covariance matrix or
the direct estimation of the precision matrix.
Remarks
We caution SP researchers that the risk and optimal
portfolio obtained from the covariance matrix are only
from a period of historical data, assuming that there is
a stationary joint return distribution among the assets. It
is, at most, a rough approximation of the ever-changing
financial market. The out-of-sample backtesting is necessary and helpful, but it is by no means comprehensive
and bullet proof in terms of representing the future distributions. Therefore, the economic reasoning and understanding of risk factors are always necessary in both
practice and economics theoretical development.

(by changing the weight of the risk-free asset). The idiosyncratic risk represented by the regression residual (or noise) is the
risk not rewarded (priced) by the market and therefore is the
part of the risk that investors need to try to remove by diversification. A portfolio analysis and risk-factor attribution problem
usually has the following formulation.
22

For a set of n portfolios, the time-series model of the portfolio excess return (with risk-free rate subtracted) is
R i (t) =

K

/ b si R s (t) + f i (t), i = 1, f, n,

s =1

where R s, s = 1, f, K are risk factors. Note that risk factors
are also portfolios.
The portfolio analysis or construction problems can be
1) finding common risk portfolios R s given R i; 2) given risk
factors, which are often represented by different market or
section indices, or cross-sectional portfolios, such as those
in the FF three- or five-factor models, finding the risk loadings (betas), which reflect the risk exposures of a portfolio; or
3) covariance estimation, portfolio optimization or analysis,
often subject to various constraints, such as long/short, borrowing, liquidity, and transaction cost. This is an area to which
the SP research can directly contribute due to the similarity in
problem formulation in statistical and array SP.
Specifically, practical portfolio construction and optimization usually has following steps.
■ Identify portfolio constraints. First, identify the desired
expected return or the tolerable risk represented by variance.
Note that we can either minimize the variance given the
expected return or maximize the expected return given the
variance. Second, identify the investable security pool along
with investment constraints, including short sell constraints,
liquidity constraints, transaction costs, and risk-factor exposure constraints. In practice, these constraints are often
determined by the requirements from the investment fund
stakeholders, trading systems, and regulatory bodies.
■ Identify historical data or proxy data that can be used to,
first, estimate covariance and expected returns of a security
pool and, second, conduct backtesting.
■ Find a robust algorithm to estimate the covariance matrix
(or precision matrix) and expected returns.
■ Formulate a constrained optimization problem, and find a
robust solution.
■ Conduct backtesting, and evaluate the portfolio performance
against benchmark models, usually known factor models.
In "Example SP Research on Portfolio Optimization and
Related Risk Modeling," we show SP research in portfolio
optimization. The nature of an economic system is different from that of an SP system. So far, all economic models
and theories we introduced are hypotheses. In SP, system and
signal models are often rooted in physical laws known to be
sufficiently accurate descriptions of system mechanisms. SP
researchers tend to be quite confident with their models and
believe that the same physical law applies at all times (yesterday, today, and tomorrow).
Therefore, even when the mathematical representation
in portfolio analysis, and in economic and business studies
in general, is similar to that in SP [36], the treatment can be
quite different. For example, in financial empirical studies,
when certain risk factors are identified mathematically using
the data at hand, it is actually not clear whether the identified
risk factors are indeed true risk factors or a result of statistical

IEEE Signal Processing Magazine

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May 2017

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Table of Contents for the Digital Edition of Signal Processing - May 2017

Signal Processing - May 2017 - Cover1
Signal Processing - May 2017 - Cover2
Signal Processing - May 2017 - 1
Signal Processing - May 2017 - 2
Signal Processing - May 2017 - 3
Signal Processing - May 2017 - 4
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Signal Processing - May 2017 - Cover3
Signal Processing - May 2017 - Cover4
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