wavelet based noise dressing for reliable cardinality ...mirlabs.org/nagpur/paper01.pdfwavelet based...

Wavelet based Noise dressing for Reliable Cardinality Constrained Financial Portfolio Sets

N.C.Suganya Department of Mathematics and Computer

Applications PSG College of Technology

Coimbatore, INDIA [email protected]

G.A.Vijayalakshmi Pai Department of Mathematics and Computer

Applications PSG College of Technology

Coimbatore, INDIA [email protected]

Abstract

Cardinality constrained portfolio optimization problem deals with finding the optimal portfolios that meet the twin-objectives of maximizing expected return and minimizing risk, subject to the cardinality constraint where the investors’ restricts the portfolio to have maximum of only K assets out of N assets. The inclusion of cardinality constraint turns the problem into a complex mixed integer quadratic programming problem and hence various approaches have been attempted for their efficient solution.

Empirical covariance matrix obtained from historical time series, which is one of the key inputs for portfolio optimization problem is dominated by a high degree of noise. Hence, better estimates are required to remove the noise significantly from the covariance matrix for reliable optimal portfolios. Some of the recently developed filters are Random matrix theory (RMT) based filters and K-means cluster analysis.

In this paper, we focus on the noise dressing of the covariance matrix using a hybrid model, which is a sequential combination of wavelet shrinkage denoising method and k-means cluster analysis for the efficient solution of the cardinality constrained portfolio optimization problem. Experimental studies on measuring the reliability of the portfolio sets and a statistical analysis of the expected return of the portfolio sets have been undertaken and compared with those of the Markowitz, RMT and k-means cluster analysis filtered models. The experimental results have been demonstrated on the Bombay Stock Exchange (BSE 200 index: July 2001 to July 2006) and Tokyo Stock Exchange (Nikkei 225 index: March 2002 to March 2007) data sets.

1. Introduction

A mathematical modeling with deep economic understanding of the market is required for real investment decision. Portfolio theory is a more versatile tool in the modern investment theory. It helps in estimating risk of an investor’s portfolio, with lack of information, uncertainty and incomplete knowledge of reality, which forbids a perfect prediction of future price changes.

The distribution of a capital held by an investor over different investment options such as stocks, bonds and securities is called a portfolio. The basic process in portfolio optimization is the selection of an appropriate investment portfolio and determining the optimal proportion of capital that the investor should allocate on the different assets of the portfolio so as to meet the twin objective of maximizing expected return and minimizing risk. The solution to the portfolio optimization problem is the risk-return tradeoff curve called efficient frontier.

Markowitz [1952] laid a quantitative framework for solving the portfolio optimization problem using a mean-variance model termed as Modern Portfolio Theory (MPT) with the assumption that asset returns follow a Gaussian law. The principal inputs of Markowitz model are the expected mean and correlation (covariance) matrices obtained from historical return series.

Empirical covariance matrix has gained industry-wide applications in risk management and is also of great importance for asset allocation and investment theory. The key observation is that for any practical use of the theory, it would therefore be necessary to have reliable estimates for the correlations of the returns on the assets making up the portfolio, which are obtained from historical return series. Due to

Proceedings of the International Workshop on Machine Intelligence Research (MIR Day, GHRCE- Nagpur) © 2009 MIR Labs

1

obvious reasons, the length of empirical time series T is limited; inevitably leading to the measurement error (noise) in the covariance matrix estimates and also induces a high amount of statistical uncertainty, which can be seen as random.

The effects of estimation error in the covariance matrix are as follows. a) affects the estimation of future risks on optimized portfolios b) dominates the probability of large losses for a certain portfolio c) causes uncertainty in portfolio estimation d) leads to serious instability and a high degree of degeneracy in optimal portfolios and e) makes the entire applicability of the theory more dangerous and questionable. Hence the presence of such a high degree of noise in covariance matrix as suggested by Galluccio, 1998; Laloux Laurant et al., 1999 and Plerou et al., 1999 constitutes an intriguing paradox. Furthermore, Laloux Laurant et al., [1999] concluded in their work that “Markowitz portfolio optimization scheme based on a purely historical determination of the correlation matrix is inadequate”.

Hence, there is a need to devise the estimation models for removing noise from the correlation matrices constructed from measured time series. In particular, estimating correlation matrices is also an important part of traditional methods of portfolio selection. Several recent contributions are devoted to noise filter in a correlation matrix. Therefore, in order to construct an efficient portfolio using Markowitz results, the empirical covariance matrix need to be properly denoised before the application of any solution strategy.

Laloux Laurant et al., 1999 and Plerou et al., 1999 showed that the random matrix theory (RMT) can be used to investigate the properties of return correlation matrices of financial assets. Papp et al., 2005; Snarska Malgorzata and Jakub Krzych, 2006 and Laloux Laurent et al., 2007 have studied RMT filtered covariance matrices for portfolio optimization. Several researchers Mantegna, 1999; Giada and Marsili, 2001; Maslov, 2001; Bernaschi et al., 2002; Kullmann et al., 2002; Bonanno et al.,2004 and Basalto et al., 2005 have also proposed other filtering procedures based on clustering in the econophysics literature.

Following them Tola et al., [2005] demonstrated their claims only for the single linkage and average linkage clustering algorithms and also proved that clustering algorithms acted as better reliable filters when compared to RMT based filtering methods in terms of the ratio between predicted and realized risk for a given expected portfolio return. Later on Pai and Michel [2007, 2008] studied the application of k-means clustering for portfolio optimization and

concluded that k-means cluster analysis outperforms Markowitz and RMT filtered models.

Meyer [1993] has studied the application of wavelets pertaining to denoising of data, in a wide spectrum of disciplines. Wavelet shrinkage denoising methods are non-parametric and non-linear methods. Since wavelet shrinkage denoising has made significant inroads in almost many applications, we experiment this denoising method for financial covariance matrix. Wavelet denoising guarantees the reduction of high frequency noises. Several experiments were carried out on various wavelet functions and finally symlet function has been chosen as the wavelet function for denoising financial covariance matrix, because the denoised financial covariance matrix obtained using symlet shows higher correlation with the empirical covariance matrix.

In this work, we concentrate on a subclass of portfolio selection problem called cardinality constrained portfolio optimization problem. Here, in addition to basic constraint, the cardinality constraint is also added, where the investors’ decides to diversify his or her portfolios in only K out of the universe of N assets. Several theoretical and practical approaches have been suggested to handle the cardinality constraint. Initially, a brute force approach of choosing K different solutions from N assets (NCk) was suggested, resulting in inferior combinatorial solutions. Later, Farrell [1997] proposed grouping of assets based on the industry, size, geographical aspects etc forming asset classes and making a choice from each of these classes. This approach also yields inferior solutions due to the ignorance of the correlations between the assets.

Various single agent local search algorithms such as simulated annealing and threshold accepting [Winker, 2001] have been proposed. However, the solutions get stuck in local optima. Chang et al., 2001, Felix Streichert et al., 2003 and Maringer Dietmar and H Kellerer, 2003 have adopted multi agent methods such as Ant systems, Genetic algorithms and hybrid search methods and these methods have reported either near optimal or mixed results. Another approach of using Hopfield neural network has been attempted by Fernandez and Gomez [2007], where the network pruning strategy has been used to implement the cardinality constraint.

This work makes use of k-means cluster analysis to tackle the cardinality constraint. In addition, k-means cluster analysis reduces the number of design variables, facilitating the heuristic approaches used for solving the problem, thereby leading to faster convergence. Hence, for cardinality constrained portfolio optimization problem, a hybrid model has


2

been proposed which combines wavelet based filter and k-means cluster analysis, sequentially. In this model, the empirical covariance matrix is first denoised using wavelet shrinkage method and then the denoised covariance matrix along with expected return is used as input for k-means cluster analysis. In summary, the empirical covariance matrix denoised using wavelet based filter is further refined using k-means cluster analysis, in addition eliminates the cardinality constraint.

The measurement of reliability (ratio between the predicted and realized risk) of the portfolio set obtained for various models such as hybrid model, k-means cluster analysis, RMT based filter and statistical analysis of the expected returns obtained for k-means cluster analysis and hybrid model using paired t-test have been demonstrated on the Bombay Stock Exchange (BSE 200 index: July 2001 to July 2006) and Tokyo Stock Exchange (Nikkei 225 index: March 2002 to March 2007) data sets.

This paper is organized as follows. In section 2, we describe briefly the cardinality constrained portfolio optimization problem, the notations and mathematical formulations. In section 3, we summarize the importance of historical estimate of covariance matrix in cardinality constrained portfolio optimization problem. In section 4, we review the approaches of recently introduced random matrix theory based filter and k-means cluster analysis for the estimation of covariance matrix. In section 5, we describe the application of wavelet based filter for financial covariance matrix. In section 6, measuring the reliability of wavelet denoised portfolio sets is carried out. In section 7, the hybrid mix of wavelet based filter and k-means cluster analysis is applied for the cardinality constrained portfolio optimization problem. Finally in section 8, we summarize our results. 2. Mean-Variance Portfolio Selection

The Mean-Variance approach introduced by Harry Markowitz [1952] is the most universally accepted approach to portfolio selection. According to Modern Portfolio Theory (MPT), an investment strategy that seeks to construct an optimal portfolio should diversify the wealth among many different assets instead of selecting the dominant stocks. The portfolio optimization problem is a multi-objective optimization problem with two competing objectives of minimizing the risk of the portfolio and maximizing the return of the portfolio. Since, risk of the portfolio is influenced by the correlations between different assets; the

portfolio selection process represents a complex optimization problem. 2.1. Cardinality Constrained Portfolio Optimization Problem

Portfolio optimization problem is a quadratic optimization problem comprising of computationally effective algorithms. In contrast, this problem turns very complex when real-world constraints such as cardinality constraint are added to the problem model [Maringer, 2005]. The cardinality constrained portfolio optimization problem is a bi-objective problem, which can be formulated as a single criterion problem by using the weighting parameter 1,0 . Studies on the

cardinality constrained portfolio optimization problem have been made by Chang et al., 2000; Fernandez and Gomez, 2007 and Suganya and Pai, 2008. The constraints incorporated in the portfolio optimization problem are as follows. Basic Constraint

Each asset weights lie between 0 and 1 and in addition the sum total of all the asset weights adds to 1, for various values of . Cardinality Constraint

The portfolio is restricted to contain no more than K different assets out of N assets. Let,

N the total number of assets in the universe. Cij the covariance between the returns of assets i

and j.

iμ the mean return (per period) of each asset i.

the weighting parameter lie between 0 and 1. xi the cardinality constant i.e., the value of xi is

restricted to 0 or 1. i.e., xi=1, if ith asset is included in the portfolio and xi=0, if ith asset is not included in the portfolio.

The cardinality constrained portfolio optimization problem, which is an extension of Markowitz’s Mean-Variance model, has the following mathematical formulation.

Minimize

N

1i

N

1j

N

1i)iμipλ)((1)jpijcipλ( (1)

Subject to (Basic Constraint) (2) 1pN

1ii

(Cardinality Constraint)

(3)

KxN

1i

i


3

1,2,3...Ni10,x i This weighted formulation of the objective function

is suggestive of the fact that for those values of close to 0, the weights shift towards stocks yielding high returns and for those values of close to 1, the weights shift towards stocks yielding low volatility in the efficient set. Equations (1-2) define the Markowitz model and solution for this problem could be easily attained through quadratic programming. However, the inclusion of the cardinality constraint (equation 3) turns the problem into a mixed integer-quadratic programming problem, making it complex for direct solving by analytical or numerical methods.

Hence, the set of portfolio of assets that offer a minimum risk for an expected level of portfolio return traces a curve called the efficient frontier. The efficient frontier is the collection of all efficient portfolios. The efficient frontier traced for the Bombay Stock Exchange (BSE 200 index: July 2001 to July 2006) and Tokyo Stock Exchange (Nikkei 225 index: March 2002 to March 2007) data sets with short series assets eliminated from their respective data sets and obtained using MATLAB’s optimization toolbox are shown in figures 1(a) and 1(b) respectively.

3. Historical Estimates

In the business world, companies interact with one another and are developing the complex system. Also it is hard to understand the companies’ interaction details. However, they are reflected in the correlations of stock prices. These correlations play a crucial role in several branches of finance such as investment theory, risk management or capital allocation. Therefore, it is necessary to have reliable estimates for the correlations of the returns on the assets making up the portfolio, which are usually obtained from historical return series. Correlation matrices serve as inputs to the portfolio optimization problem in the classical Markowitz portfolio theory. Moreover, Kevin Terhaar et al., [2003] acknowledged that ‘…for conventional assets which are frequently traded and have observable prices in the market, the looking-period-forward is assumed to be adequately proxied by historical data’.

The covariance matrix determined from empirical financial time series contains a high amount of noise, which is regarded as random. In particular, the smallest eigen values of this matrix (corresponding to the smallest risk portfolios) are dominated by noise. For practical reasons, the length of available time series is limited for the estimation of covariance matrix. This finite length T of the time series inevitably leads to the appearance of noise (measurement error) in the

covariance matrix estimates. If N is the number of assets, then the covariance matrix has N(N-1)/2 N 2/2 distinct entries. In order to have small error on the covariance, one needs time series of length T>>N, which is not practical.

Figure 1(a). Efficient frontier for the BSE200 data sets obtained

by the Markowitz Mean-Variance model

Figure 1(b). Efficient frontier for the Nikkei225 data sets

obtained by the Markowitz Mean-Variance model

As a consequence, covariance matrix induces

serious instability and a high degree of degeneracy in optimal portfolios. Hence, it is important to develop methods capable of filtering the covariance matrix which is less likely to be affected by statistical uncertainty.

4. Existing Noise Filters for Financial

Covariance Matrix – a Review

There have been several contributions in the literature devoted to noise filter in a correlation matrix. Initially, various models such as single and multi-index models, grouping by industry sector or macroeconomic factor models have been introduced on economic grounds in order to impose some structure on the covariance matrix to decrease the effective number of


4

parameters to be estimated [Elton and Gruber, 1995]. However, Jorion 1986; Frost and Sararino, 1986; Chopra et al., 1993 and Ledoif and Wolf, 2003 have used pure statistical estimation methods such as Principal Component Analysis or Bayesian Shrinkage estimation.

This section presents a review on the different filtering procedures such as Random Matrix Theory Approach (RMT), k-means cluster analysis and their reliability measures based on Bombay Stock Exchange and Tokyo Stock Exchange data sets. 4.1. Random Matrix Theory based Filter

A random matrix is a matrix of given type and size whose values consist of random numbers from some specified distribution [M L Metha, 1990]. The unusual and most active applications of random matrix theory have been to describe phenomena in nuclear physics, mesoscopic quantum mechanics and wave phenomena. In many applications, the statistical behavior of systems exhibiting correlations involving Eigen values and Eigen states have been quantitatively modeled using ensembles of matrices with completely random and uncorrelated entries.

The returns from different stocks are assumed to be Gaussian random variables and also any set of correlated Gaussian random variables can always be decomposed into a linear combination of independent Gaussian random variables and the converse is also

true. Therefore, with zero mean and variance , the eigen value spectral density of the covariance matrix is given by

2

)λλ)(λ(λλ2π

Tρ(λ) minmax2

(4)

where T is the length of the time series for N assets

with , and a fixed ratio T N 1N

TQ , then

is the eigen value such that with

and satisfying,

λ

λ

maxmin λ,λλmin maxλ

Q

12

Q

11σλ 2max

min (5)

In order to denoise the correlation matrix, first the Eigen values are ranked such that .

Secondly, the variance of the part not explained by the

highest Eigen value is computed as

1kk λλ

N

λ1σ 12

λ

and this

value is used in eqn (5) to compute and

.Then a filtered diagonal matrix is constructed by

setting all the Eigen values smaller than to zero

and leaving the rest unaltered. Finally, the filtered correlation matrix is obtained by transforming the filtered diagonal matrix in the original basis and setting the diagonal elements of the filtered correlation matrix to one.

min

maxλ

maxλ

4.2. Clustering algorithms – a report

A most common practice in multivariate data analysis is clustering [K V Mardia et al., 1979]. Cluster analysis gives a meaningful partition of a set of N variables in groups according to their characteristics. R N Mantegna; 1999 and G Bonanno et al., 2001 showed that correlation based clustering have been used to infer the hierarchical structure of a portfolio of stocks from its correlation coefficient matrix. Correlation based clustering transforms a matrix by retaining a smaller number of distinct elements, which may be seen as a filtering procedure.

Tola et al.,[2005] claimed that clustering algorithms act as better filters when compared to random matrix theory based filtering methods. They also verified that single linkage and average linkage clustering algorithms reported improved portfolio reliability in terms of predicted and realized risk for a given expected portfolio return. Following their footsteps, Pai and Michel [2007] analysed that k-means clustering based on the mean and covariance of the expected returns is also in no means less a noise filters, yielding portfolios with better reliability when compared to RMT based filters and also serves to eliminate the cardinality constraint from the portfolio optimization problem model.

K-means cluster analysis for portfolio optimization

Cluster analysis deals with algorithms which can classify objects into clusters depending on the maximum degree of association shared by objects belonging to the same cluster. Unlike, other clustering algorithm, k-means clustering is the only algorithm that partitions the universe into K clusters for a specified value of K. Furthermore, allowing the investor to make his or her choice of one asset from each of the clusters, ensuring both selection and diversification, sufficient to tackle the cardinality constraint.

After experimenting with several possible parameters, the mean and covariance of daily returns have been decided as the inputs for k-means clustering of assets. If the investor decides to have maximum of K assets out of N assets, then the universe of N assets


5

is grouped into K clusters i.e. .

Hence, an investable universe is defined as such that each a

K1,2,3,...,i,A i

iii Aa1,2...K,/iaI i is randomly

or preferentially chosen from each cluster Ai, i=1, 2…K. Hence the solution set consists of only K assets. 5. Wavelet Based Filter for Financial Covariance Matrix One of the recent developments in the area of applied mathematics is wavelet analysis, which is emerging as one of the fastest growing field with application ranging from seismology to astrophysics. Wavelets are a special kind of functions which exhibits a damped oscillation for a very limited duration of time. In many real world applications, applied scientists and engineers have studied the presence of noise in the data. Since the noise corrupts the original data, these must be removed in order to recover the original data and to proceed with further data analysis.

Wavelet shrinkage denoising method is the more efficient method of denoising than most other methods, attempting to remove whatever noise is present and retain whatever signal is present regardless of the frequency content of the signal. This method is considered as a non-parametric method, distinct from parametric methods of estimating parameters for a particular model that is assumed apriori. Although, wavelet theory, which is richer in mathematical content, is employed in almost many areas and applications, there is no literature that describes the application of wavelet to denoise financial covariance matrix.

This work demonstrates the wavelet denoising procedure experimented to denoise financial covariance matrix and its efficiency is quantified by its better portfolio reliability between predicted and realized risk for a given expected portfolio return when compared with k-means cluster analysis, RMT based filter and Markowitz model. 5.1. Wavelet Coefficient Shrinkage technique for denoising Let the noisy covariance matrix is given as,

1,2...Nj1,2...N,i(t),ε(t)C(t)S ijijij

(6)

where is the noisy covariance matrix which

are assumed to come from zero-mean normal distribution, is the original data and is the

independent standard normal random variable ~ N(0,1), representing the noise.

(t)Sij

(t)Cij(t)ε ij

The steps involved in wavelet shrinkage technique are as follows:

Step 1: Calculate the wavelet coefficient matrix by applying a wavelet (Symlet) transform to the data.

ωψ

(7) ψ(ε)ψ(C)ψ(S)ω Step 2: Modify the detail coefficients corresponding to wavelet coefficient by applying a global threshold operator 2logNλ , where N is the size of the data to

obtain the estimate of the wavelet coefficients of C. ω

(8) ωω Step 3: Inverse the modified detail coefficients to obtain the denoised coefficients.

(9) )ω(ψC 1 After carrying out several experiments on wavelet

functions for the efficient noise removal from the covariance matrix, symlet 3 wavelet function have been chosen to act as the forward and inverse

transformations and 1 in equations (7) and (9)

respectively, because of high correlation between the original covariance matrix and symlet 3 based denoised covariance matrix. In addition to a wavelet, there is also a need to select number of multiresolution levels. For denoising method, five multiresolution levels are chosen because it is the maximum level after which no improvement in reliability of optimal portfolios is felt.

ψ ψ

6. Reliability of Wavelet filtered Portfolio Sets Following Tola et al., [2005] and Pai and Michel [2008], we also adopted the same measure to compute the reliability of portfolios obtained using wavelet based filters. This section gives a report on the procedure used to find predicted and realized risk required to find the reliability of a portfolio. In addition, it presents the experimental results regarding measurement of reliability of wavelet denoised portfolio sets which is the thematic objective of this paper. Furthermore, these results are compared with those of Markowitz, RMT based filter and K-means cluster analysis. 6.1. Predicted and Realized risk – a brief note Let us consider a time t0, when the optimization is expected to take place. Given a time series of T trading

days for N assets, the optimal weights for the t0-T period are computed by making use of the observed

*ip


6

mean returns and volatilities of the (t0+T) and the covariance matrix of the (t0-T) period. Thus, an efficient frontier is traced for the (t0-T) time horizon by making use of these parameters representing the predicted risk-return curve. Now, the

optimal weights , the mean returns , volatilities

and the covariance matrix of the (t0+T) period are used to compute the risk-return couple and an efficient frontier is traced representing the realized risk-return curve.

T)(ti

0μ

*ip

T)(ti

0σ

T)(ti

0μ

T)(ti

0σ

Figures 2(a) and 2(b) illustrates the predicted and realized risk-return graphs obtained for BSE200 and Nikkei225 data sets. The risk-return graphs of the k-means cluster analysis based model have been compared with those of the RMT based model and Markowitz model, which employs a “noisy” covariance matrix. K-means cluster analysis considers an investable universe with K=20 indicating small portfolio set for the BSE200 data set and K=50 representing large portfolio set for the Nikkei225 data sets. A visual observation of graphs indicates that the deviation between predicted and realized risk-return curves of k-means cluster analysis based model is less when compared to other two models.

Figures 3(a) and 3(b) gives the comparison of the predicted and realized risk-return graphs obtained for wavelet based filter model with k-means cluster analysis based model, which is a better method when compared with earlier models. Furthermore, the graphs shows less deviation between the predicted and realized risk-return curves of wavelet based filter model when compared with k-means cluster based model.

Figure 2(a). Predicted and Realized risk-return graphs for the

Markowitz model, RMT filter based model and Investable universe (K=20) for BSE200 (July 2001 – July 2006) data set

Figure 2(b). Predicted and Realized risk-return graphs for the

Markowitz model, RMT filter based model and Investable universe (K=50) for Nikkei225 (March 2002 – March 2007) data

set

6.2. Measuring the reliability of portfolio sets

For a given expected portfolio return, the corresponding predicted and realized risks, and

respectively could be obtained by a linear

interpolation of the known values reflected on the risk-return curves. The reliability coefficient R is calculated based on predicted and realized risks of a portfolio, measuring the goodness of different filtering methods. If R is small, then the corresponding portfolio set is said to be more reliable.

pσ

Rσ


7

Figure 3(a). Predicted and Realized risk-return graphs for the k-means clustered based model (K=20) and Wavelet based filter

model for BSE200 (July 2001 – July 2006) data set

Figure 3(b). Predicted and Realized risk-return graphs for the k-means clustered based model (K=50) and Wavelet based filter

model for Nikkei225 (March 2002 – March 2007) data set

For and , the predicted and realized risk

respectively for a given level of expected return, the portfolio reliability coefficient R is given by

pσ Rσ

p

Rp

σ

σσR

(10)

Figures 4(a) and 4(b) compares the reliability of the portfolio set obtained using k-means cluster based model with those of Markowitz and RMT based filter for BSE200 and Nikkei225 data sets respectively. The investable universes of K=20 and K=50 for BSE200 and Nikkei225 data sets respectively are considered for k-means cluster analysis. The graph reports smaller reliability (R) for portfolio sets obtained using k-means cluster based model when compared to the rest.

Figure 4(a). A Comparison of the reliabilities of the k-means clustered portfolio set (K=20) with the Markowitz and RMT

filtered models of the BSE200 data set

Figure 4(b). A Comparison of the reliabilities of the k-means clustered portfolio set (K=50) with the Markowitz and RMT

filtered models of the Nikkei225 data sets

Furthermore, figures 5(a) and 5(b) illustrates the reliability of the portfolio set obtained using wavelet based filter model and the k-means cluster based model. The small value of reliability R reported by wavelet based filtered portfolio sets are indicative of the better reliability of portfolios, compared to others. Hence the optimal portfolios obtained using wavelet based filter model is more reliable.


8

Figure 5(a). A Comparison of the reliabilities of the wavelet based filtering model with the k-means clustered portfolio set

(K=20) for the BSE200 data sets

Figure 5(b). A Comparison of the reliabilities of the wavelet based filtering model with the k-means clustered portfolio set

(K=50) for the Nikkei225 data sets

7. Hybridizing K-means cluster analysis and Wavelet filters for better reliability

Although wavelet based filter model exhibits better reliability when compared to other filtering methods, k-means cluster analysis is also required to eliminate the cardinality constraint from the problem model. This reduces the number of design variables in the problem, thereby making it easy to be solved by any heuristic methods.

Hence wavelet based filter model and k-means cluster analysis can be combined sequentially to form a hybrid model, before the application of any solution strategy. First, the empirical covariance matrix obtained from historical return series is denoised using wavelet based filter. Then the denoised covariance matrix and the expected return are used as input to k-means cluster analysis to group N assets into K clusters. Now, from each cluster, an asset is selected randomly or preferably as discussed in section 4.2.

Figures 6(a) and 6(b) illustrates the predicted and realized risk of the portfolio sets obtained for the hybrid model, which combines wavelet based filter and

k-means cluster analysis with K=20 and K=50 for BSE200 and Nikkei225 data sets respectively. Thus the risk-return graphs obtained for the hybrid model have been compared with that of portfolio sets obtained using wavelet based filter. The observation of this graph shows less deviation between the predicted and realized risk curves obtained for hybrid model when compared to other.

Figure 6(a). Predicted and Realized risk-return graphs for the

wavelet based filter and Hybrid model (Wavelet based filter and k-means cluster with K=20) for BSE200 (July 2001 – July 2006)

data set

Figure 6(b). Predicted and Realized risk-return graphs for the wavelet based filter and Hybrid model (Wavelet based filter and k-means cluster with K=50) for Nikkei225 (March 2002 – March

2007) data set

Figures 7(a) and 7(b) compares the reliability of the

portfolio sets obtained using the hybrid model and wavelet based filter model. The smaller reliability coefficient value exhibited by the hybrid model shows that optimal portfolios obtained using this model is more reliable than the rest.


9

Figure 7(a). A Comparison of the reliabilities of the wavelet

based filtering model and hybrid model (K=20) for BSE200 (July 2001 – July 2006) data sets

Figure 7(b). A Comparison of the reliabilities of the wavelet

based filtering model and hybrid model (K=50) for Nikkei225 (March 2002 – March 2007) data sets

7.1. Measuring the efficiency of expected returns obtained using hybrid model

Furthermore, statistical analysis is carried out to prove that efficient frontiers traced using hybrid model (Sequential combination of Wavelet based filter and k-means cluster analysis) gives better expected return when compared to the recently developed k-means cluster analysis.

A set of expected returns corresponding to 10 different risk values on the efficient frontiers for BSE200 data sets given in figure8 is used for analysis.

Figure 8. Efficient frontier traced using K-means cluster

analysis (K=20) and Hybrid model (Wavelet based filter and k-means cluster filter (K=20)) for BSE200 (July 2001 – July 2006)

data sets

Let represents the expected return of the efficient

frontier traced for hybrid model and represents the expected return obtained for the efficient frontier traced using k-means cluster analysis.

hμ

kμ

Paired t-test is the optimal choice of statistical test used to compare the efficiency of expected returns obtained in different models.

The null hypothesis (H0) and alternate hypothesis

(H1) is given by H0: <= and Hhμ kμ 1: h > kμ μ

For assumed 5% level of significance, the results show that H0 is rejected and H1 is accepted. This implies that expected returns obtained using hybrid model is better and efficient when compared to the expected returns obtained using other models. 8. Conclusions

In this paper, the impact of noisy covariance matrix on the portfolio optimization problem and also various denoising models such as RMT based filter, K-means cluster based filter and newly proposed wavelet based model used for denoising covariance matrix, which is the primary input of portfolio optimization problem have been investigated. Finally, experimental studies have been undertaken on the Bombay Stock Exchange (BSE200 index: July 2001 to July 2006) and Tokyo Stock Exchange (Nikkei225 index: March 2002 to March 2007) data sets.The conclusions are, I. The hybrid model, which is the sequential

combination of wavelet based filter and k-means cluster analysis, serves to yield reliable portfolio sets when compared to Markowitz model, RMT based filter, k-means cluster analysis and also simple wavelet based filter models.

II. Inclusion of k-means cluster analysis for the elimination of cardinality constraint promotes


10

dimensionality reduction, i.e., reducing the number of design variables, which could be exploited by any computational approaches, thereby leading to faster convergence.

III. Irrespective of diversification in small or large portfolios, the hybrid model based on wavelet filter and k-means cluster analysis yields portfolio sets in realistic time.

IV. Statistical inference drawn from paired t-test proves that expected returns obtained using hybrid model is better when compared to the expected returns obtained using previous filter models.

Parallel efforts to investigate the application of other heuristic approaches, using wavelet based denoised covariance matrix for the solution of the complex constrained portfolio optimization problem with the inclusion of other constraints modeling investor preferences and Market norms such as class constraint, linear short sales constraint and transaction cost constraints are already in progress. 9. References [1] Alberto Fernandez and Sergio Gomez, “Portfolio

selection using neural networks”, Computers and Operations Research, 34, 2007, pp.1177-1191.

[2] Chopra and W.T.Ziemba,“The effects of errors in means, variances and covariances on optimal portfolio choice”, J. Portfolio Management, 1993, pp. 6-11.

[3] Elton, E.J. and M.J. Gruber, Modern Portfolio Theory

and Investment Analysis, J. Wiley and Sons, New York, 1995

[4] Farrell, Jr., J, Portfolio management Theory and

applications, McGraw Hill, New York, 2nd ed., 1997.

[5] G. Bonanno, G. Caldarelli, F. Lilli, S. Micciche, N. Vandewalle and R.N. Mantegna, Eur. Phys. J. B, 38, 363, 2004.

[6] G. Papp, S.Z. Pafka, M.A. Nowak and I. Kandor, “Random matrix filtering in portfolio optimization”, Acta Physica Polonica B, Vol.36, 9, 2005.

[7] G.A. Vijayalakshmi Pai and Thierry Michel, “Evolutionary optimization of constrained k-means clustered assets for diversification in small portfolios”, IEEE Transactions on Evolutionary Computation, 2008, (accepted).

[8] G.A. Vijayalakshmi Pai and Thierry Michel, “On measuring the reliability of k-means clustered financial portfolio sets for Cardinality Constrained Portfolio optimization”, In Mathematical and computational models, (Eds.) R Nadarajan, R Anitha and C Porkodi., Dec 2007, pp. 313-323.

[9] H. Markowitz,“Portfolio selection”, Journal of finance,

1952.

[10] K.V Mardia, J.T. Kent and J.M. Bibby, Multivariate Analysis, Academic Press, San Diego, 1979.

[11] Kevin Terhaar, R. Staub and Brian Singer, “Appropriate policy allocation for alternative investments”, Journal of Portfolio Management, 2003 pp.101-110.

[12] L. Giada and M. Marsili, Phys.Rev. E 63, 061101, 2001.

[13] L. Kullmann, J. Kertesz and K. Kaski, Phys. Rev. E 66, 026125, 2002.

[14] L. Laloux, P. Cizeau, J-P. Bouchaud and M. Potters, “Noise Dressing of Financial Correlation Matrices”, Phys. Rev. Lett. 83, 1999, pp.1467-1470.

[15] L.Laurent, P.Cizeau, M.Potters and J-P. Bouchaud, “Random Matrix Theory and financial correlations”, Mathematical models and methods in Applied Sciences, World Scientific Publishing Company, 2007.

[16] M. Bernaschi, L. Grilli and D. Vergni, Physica A, 308, 381, 2002.

[17] M.L. Metha,“Random Matrices”,Academic Press, NY, 1990.

[18] Maringer Dietmar and H. Kellerer, “Optimization of

cardinality constrained portfolios with a hybrid local search algorithm”,OR Spectrum,25(4),2003,pp.481-495.

[19] Maringer Dietmar, Portfolio management with heuristic optimization, Springer, 2005.

[20] N. Basalto, R. Bellotti, F. De Carlo, P. Facchi and S. Pascazio, Physica A ,345, 196, 2005.

[21] N.C. Suganya and G.A. Vijayalakshmi Pai, “Evolution

based Hopfield Neural Network with wavelet based filter for complex - constrained portfolio optimization”, Dynamics of Continuous, Discrete and Impulsive Systems (Series B), Special issue on Optimization and Dynamics in Finance, 2008, (under revision).

[22] N.C. Suganya, G.A. Vijayalakshmi Pai and Thierry

Michel, “Evolution based Hopfield Neural Network for Constrained Optimization of Financial Portfolios”, Proceedings of IEEE Conference on AI Tools in Engineering, March 2008.

[23] O. Ledoit and M. Wolf, “Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection”, Journal of Empirical Finance, 2003.


11

[24] P. Frost and J. Savarino, “An Empirical Bayes Approach to Efficient Portfolio Selection”, Journal of Financial and Quantitative Analysis, 21, 293, 1986.

[25] P. Jorion, “Bayes-Stein Estimation for portfolio analysis”, Journal of Financial and Quantitative Analysis 21, 1986, pp. 279-292.

[26] P. Winker, Optimization heuristics in Econometrics: Applications of Threshold Accepting, John Wiley & Sons, 2001.

[27] R.N. Mantegna and H.E. Stanley, “An Introduction to

Econophysice: Correlations and Complexity in Finance”, Cambridge University Press, Cambridge, 2000, pp.120-128.

[28] S. Galluccio, J-P. Bouchaud and M. Potters, Physica A 259, 449, 1998.

[29] S. Maslov, Physica A 301, 397, 2001.

[30] Snarska Malgorzata and Jakub Krzych, “Automatic trading agent.RMT based Portfolio theory and Portfolio selection”, Acta Physica Polonica B, 37 (11), 2006.

[31] T.J. Chang, N. Meade, J.B. Beasley and Y.M. Sharaiha, “Heuristics for cardinality constrained portfolio optimization”, Computers and Operations Research, 27, 2000, pp.1271-1302.

[32] Tola Vincenzo, Fabrizio Lillo, Mauro Gallegati and Rosario N. Mantegna, “Cluster analysis for portfolio optimization”, arXiv: physics/0507006, VI, July 2005.

[33] V. Plerou, P. Gopikrishnan, B. Rosenow, L.A.N Amaral and H.E. Stanley, “Universal and Non-Universal Properties of Cross-Correlations in Financial Time Series”, Phys. Rev. Lett. 83, 1999, pp.1471-1474.

[34] Y. Meyer, “Wavelets: Algorithms and Applications”,

Society for Industrial and Applied Mathematics, Philadelphia, 101-105, 1993, pp.13-31.


12

wavelet based noise dressing for reliable cardinality ...mirlabs.org/nagpur/paper01.pdfwavelet based...

Documents