2004 ACM Symposium on Applied Computing
The Time Diversification Monitoring of a Stock Portfolio: An Approach Based on the Fractal Dimension Mehmed Kantardzic
[email protected]
Pedram Sadeghian
[email protected]
Chun Shen
[email protected]
Computer Engineering and Computer Science Department University of Louisville Louisville, Kentucky, USA other in one’s portfolio, or collection of assets. By diversifying, the investor aims to reduce the risk of an entire portfolio depreciating in value, if a few of the assets within the portfolio are depreciated. Due to the volatile and unpredictable nature of the stock market, diversifying a stock portfolio rather than concentrating on similar stocks has been recommended as a means of portfolio risk reduction [4, 5].
ABSTRACT
Diversification is a technique used to reduce the risk of investment and is accomplished by including uncorrelated and independent stocks in one’s portfolio. By diversifying, the investor aims to reduce the risk of an entire portfolio depreciating in value, if a few of the assets within the portfolio are depreciated. In the past, the correlation coefficient has been used as a basis for diversification. However, the correlation coefficient is problematic since it can not capture nonlinear dependency, and analyzing pair-by-pair stocks in the portfolio does not always give the best estimation of diversification for the entire portfolio.
The main methodology for analyzing the interdependence of asset diversification is the covariance matrix CV, where the elements of the matrix are products of the two assets’ volatilities and their correlation [3]. The covariance CVij may vary, not because the correlation between the two assets change, but simply because their individual volatilities change. This effect can be corrected by considering the correlation matrix C = [ ij] instead of the covariance matrix. The matrix C may be estimated from time series of asset returns: Cij is estimated by the sample correlation r between assets i and j [3]. The correlation r, also known as the Pearson Product Moment Correlation, is one of the most prevalent statistics used in science and business today. Pearson Product Moment Correlation consists of the covariance divided by the square root of the product of the standard deviations of the two variables.
In this paper we present a simple, but efficient methodology for monitoring portfolio diversification, which can capture most of the nonlinear phenomena in a portfolio. We propose a measurement of portfolio diversification through the fractal dimension parameter. Monitoring this parameter in a time domain represents the basis for automatic detection of significant changes in portfolio diversification. When the fractal dimension is significantly reduced, the algorithm eliminates stocks that are highly correlated and adds new uncorrelated stocks to the portfolio. We tested our method using real historical stock data and obtained significant improvements in the time diversification of selected stock portfolios.
The main problem with correlation coefficient as a measure of diversification is that this parameter represents a linear measure of the dependency of the two time series variables. Measurement of diversification based on linear model can fail because of the nonlinear dependence amongst the real-world assets in a portfolio analyzed through time [9]. There is considerable literature on the changing nature of relations between asset’s time series under different stock market conditions. Volatility is found to be asymmetric and the correlation is high when the aggregate market is down, but when the market is up, the correlation is little different from the normal period [10]. Therefore it is not recommended to extrapolate the performance of the portfolio including its diversification through the period of 15-20 years. There is the concept of time diversification and it explains why diversification is not a static process. First, the distributions are not stationary for long time periods because of technological and structural changes in the economy [7]. Second, as you hold the portfolio, your time horizon shortens. As time passes, the risk
Keywords
Stock Market, Fractal Dimension, Time Diversification, Data Streams, Data Mining
1. INTRODUCTION
Diversification is a popular method used to reduce the risks of investment. Simply put, the idea is to “not put all your eggs in one basket.” The basic methodology of diversification is to include assets that are not highly correlated and dependent to each Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. SAC ’04, March 14-17, 2004, Nicosia, Cyprus. Copyright 2004 ACM 1-58113-812-1/03/04…$5.00.
____________________ This research has been supported with the KSEF-148-502-03-56 grant.
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tolerance of the investor is declining because the need for liquidity increases at the end of the investment horizon. The portfolio is actually becoming more risky with time, not less risky [8]. The portfolio manager should pick the best portfolio based on the time diversification model, but would then have to be ready to revise the portfolio, monthly if necessary. Therefore, time diversification is simply an aggressive utility function for choosing the best portfolio from an efficient frontier. The time diversification approach is valid only if the portfolio manager commits to continuous monitoring and revision of the portfolio. Because of technological changes, structural changes, and the general business cycle, a review of the portfolio strategies should take place every 1-36 months. Even with a 20-30 year investment horizon, the manager has to constantly monitor the portfolio as investor risk tolerance, expectations, probability distributions, and technology changes [8, 9].
fractal dimension [2], as shown in the equation: Dq = ( 1 / (q-1) ) . ( d ( log
q i pi )
/ d ( log r) )
There is a whole family of fractal dimensions (i.e. Hausdorff fractal dimension for q=0). Specifically, the correlation fractal dimension (q = 2) has gained attention in the literature. Although a data set may have a large embedding dimension (= number of features in the data set), it is the intrinsic dimension (= correlation fractal dimension) that reveals information about the dependence of the features within the data set [11, 12]. Changes in the correlation fractal dimension means changes in the distribution of points in the multidimensional data set and that is the parameter we are using in the analysis of portfolio diversification. The correlation fractal dimension is usually calculated by means of the box-counting plot. Let N(r) be i pi2, then the plot of N(r), for different values of r, versus r in a log-log scale is called the box-counting plot. The linear slope of the plot is the estimation of the correlation fractal dimension [2, 12].
The purpose of this paper is to present a simple, but efficient portfolio diversification methodology which is time dependent and can capture most of nonlinear phenomena in a portfolio. We propose a measurement of portfolio diversification through the fractal dimension parameter. Monitoring this parameter in a time domain represents the basis for automatic detection of significant changes in portfolio diversification and appropriate modifications of the portfolio. Basic definitions from fractal theory are introduced in section 2, and preliminary experiments with the fractal dimension in financial time series are explained in section 3. Proposed methodology for time diversification monitoring is given in section 4 with specific algorithms for monitoring, portfolio reduction, and portfolio extension. Experimental results and discussion are given in section 5, while conclusion and further research challenges conclude the paper in section 6.
Correlation fractal dimension as a measure of intrinsic dimensionality has important properties applicable to the analysis of financial time series, and these properties are described and verified through the experiments presented in the next section. While the statistical methods based on a correlation matrix are essentially univariate or bivariate analyses, and they deal with one or two asset at a time—most practical problems in portfolio assessment deal with the management of portfolios containing a large number of assets (typically more than twenty). Therefore, the appropriate analysis requires information on the joint distribution in the data set [3, 6]. With the experiments in section 3, we wanted to show the advantages of multidimensional analysis using correlation fractal dimension compared with traditional correlation matrices.
2. FRACTAL DIMENSION IN FINANCE
Fractals are ‘objects’ that are found in nature (i.e. coastline, fern tips) or are generated through algorithms (i.e. Cantor set, Koch curve). One of the defining characteristics of fractals is self – similarity. In other words, fractals possess symmetry across scale, with each small part of the object replicating the structure of the whole. Fractal geometry is closely linked with the subject of chaotic dynamics [1].
3. EXPERIMENTS WITH FRACTAL DIMENSION OF FINANCIAL TIME SERIES
Recent mathematical techniques are able to demonstrate that financial markets display chaotic behavior. The patterns of investment returns come from spiky, fat tailed distributions, where three and four standard deviation events are common. The high degrees of skewness and kurtosis reported, confirm that the distribution is non-normal. Skewness measures the presence of long-tails – the normal distribution has a skewness coefficient of zero. Non-normality means that the traditional mean/variance models will not be appropriate for financial time series [5]. Conventional statistical techniques lack the power to distinguish random and deterministic components of time series.
1. Some common sense interpretations of stock portfolio diversification (random selection, different groups, large number of stocks, etc.) are not valid in many situations, and an objective approach with adequate measure in portfolio optimization is necessary.
In our preliminary analysis of stock portfolio diversification using the fractal dimension (FD) calculated by [13], we want to verify some hypothesis that are already described in the literature [4, 9]:
2. Analysis of portfolio diversification based on a simple correlation matrix may produce false interpretation of diversification quality. In the first set of experiments, we analyzed the FD factor for different stock pairs. The analysis was performed based on stock time series for the period of 10 years using the closing price as a daily value. Only selected pairs of stocks are given in Table 1.
Fractal analysis offers an alternative to conventional statistical measures of portfolio diversification. The fractal dimension characterizes fractal sets representing multidimensional time series. By embedding the data set in an n-dimensional grid with cell sides of size r, we can compute the frequency with which data points fall into the i-th cell, pi, and compute Dq, the generalized
The results in Table 1 show that the common sense approach in diversifying portfolio by selecting stocks from different industrial groups does not hold in many cases. Sometimes, traditional assumptions will be supported, such as in the case of stocks in experiments 3a and 3b where diversification (= higher FD) is
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achieved by choosing stocks from different industrial groups. Experiments showed that often the time series for the stocks in the same group have significantly higher FD (1a and 2a) than the stocks from different groups (1b and 2b). Therefore, a simplified approach in diversification, which is based only on selection of stocks from different industrial groups, could produce very low quality of diversification. Deeper, more objective analysis is necessary.
the portfolio consisting of stocks from different industrial groups (case 2) does not automatically improve diversification. Stocks in case 4 are from the same group, and still the FD is significantly higher (FD=1.70) compared with the “diversified” case 2 (FD=1.41). This anomaly is not occurring in general. In some cases, such as 3, 5, and 6 common accepted rules (different groups = better diversification of the portfolio) are still valid. But this ambiguity in interpretation of correlations requires integrated n-dimensional analysis of diversification. The fractal dimension is a good candidate for an objective measure.
Table 1. Fractal dimension for 2D portfolios through 10 years
Stocks pairs of same groups 1a 2a
Stock pairs of different groups
Paired Stocks
FD
Honda + Toyota (both auto)
1.53
IBM + AT&T
1.42
2b
Intel (tech.) + Boeing (aviation)
1.16
1.06
3b
Delta (aviation) +
1.57
1b
Dell + HP
FD
Disney (ent.) +
1.08
4. TIME DIVERSIFICATION MONITORING AND CONTROL: TDMC METHODOLOGY
Dell (tech)
(both tech) 3a
Paired Stocks
(both tech)
A common research challenge in a portfolio monitoring method is the simplicity of computation on one hand, and simplicity but expressiveness of time–varying model representation on the other hand [6]. Using the fractal dimension as a time dependent parameter, we can design a simple, incremental way of deciding when to break the monitoring process for the portfolio under consideration and take actions to improve its diversification. The central idea is to modify the standard box-counting algorithm for FD computation in such a way that it will incrementally compute the FD of a changing multidimensional time series defined for a given time window [12]. When the changes in FD are over the given threshold value, the process switches to the algorithm for changing the portfolio (reduction + extension) to improve its diversification through time. When the new portfolio is generated, the monitoring process continues. To describe the proposed methodology we will start with formal definitions of input and outputs of the monitoring process, and then describe the three main algorithms of our methodology:
Disney (ent.)
Second set of preliminary experiments analyzed the quality of correlation factor (and corresponding matrix) as a parameter for estimation of stock-portfolio diversification. Our hypothesis was that the analysis of pair-by-pair stocks in the portfolio does not always give the best estimation of diversification quality. Selected results of this analysis are given in Table 2. Table 2 . Comparison of 2D and portfolio correlation FD Case
Portfolio
Correlation (FD)
Correlation (FD)
of pairs
of portfolio
1
GE+Gap+Halliburton
1.47, 1.50, 1.41
1.72
2
AT&T + Halliburton + Boeing
1.46, 1.48, 1.37
1.41
3
IBM + Dell + HP
0.98, 1.06, 1.30
1.28
4
Honda+Ford+ Toyota
1.47, 1.53, 1.46
1.70
5
Intel + AT&T + Toyota
1.29, 1.32, 1.45
1.63
6
HMA + Honda + HP
1.50, 1.46, 1.44
1.58
Input: The data set $={a1, a2,... an} composed of n stocks (portfolio), and N= {b1, b2, …, bm} set of stockscandidates for the portfolio Output: Time-varying portfolio C = {c1, c2, …, ck} defined and changed in the monitoring process where components stocks ci ∈ $ or ci ∈ N. Algorithm for Portfolio Diversification Monitoring (PDM): For the selected time window ∇T, and selected time interval ∇t for monitoring, the monitoring process starts at initial time value t0 . After ∇T period of time it is possible to compute for the first time the FD value for the portfolio. The computation is then repeated after each ∇t until changes in FD are above the threshold value. In that case this algorithm will switch to the second process, the Portfolio Reduction/Extension (PRE) Algorithm.
If we compare the results for cases 1 and 2 in Table 2, we can see that: 1. Correlations for pairs of stocks are very similar for these two cases. So, based on traditional correlation matrix analysis, one could conclude that these two portfolios have the same level of diversification. But, that is not the case! 2. Correlation for 3D portfolio measured by FD is significantly different by showing better quality of diversification for case 1. In other words, 2D correlations doesn’t give all the necessary information about diversification contained in n-dimensional portfolio.
Algorithm for Portfolio Reduction/Extension (PRE): a) Recursive Reduction Phase: Based on the FD concept, we developed an algorithm named Portfolio Reduction/Extension. The algorithm’s input is the data set $={a1, a2,... an}, composed by n stock time series identified by an index ranging from 1 to n, which are defined for the time-
Also, one conclusion we made using Table 1, now is again supported in Table 2. Comparing the cases 2 and 4 we can see that
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window ∇T. In the first step, we calculate the FD of the entire data set $. Then in the step 2 we calculate the FD of the partial data set consisting of n-1 stocks, eliminating systematically oneby-one stock from portfolio. The stock whose absence makes the smallest decrease in the FD parameter is a stock most correlated with others, and therefore it is eliminated from the portfolio. This process continues until the dependant stocks are eliminated.
portfolio, the afl stock is eliminated form the portfolio since its absence makes the smallest decrease in the FD parameter. The process is again repeated with the new 4-D portfolio. This time it is the ge stock that is eliminated form the portfolio. The process is again repeated with the new 3-D portfolio. This time the elimination of any of the three remaining stocks, would reduce the FD more than the threshold (set at 10% for this experiment). Therefore, the algorithm will keep stocks bls, cl, dol as they are sufficiently independent of one another. Figure 1 summarizes how the FD decreases as the number of stocks in the portfolio decreases.
b) Recursive Extension (Selection) Phase: Consider the data set $={a1, a2,..., an} resulted through the reduction phase of the PRE algorithm. It is composed only of the most uncorrelated stocks. If a new stock bp from the set of stock candidates N is added to this data set, the total FD will increase at most one unit if bp is completely uncorrelated with all the other attributes in the data set. On the other hand, if bp is highly correlated with the existing attributes, the FD will increase by a value of almost zero. In a third case, the total FD will increase by an amount between zero and one if the new attribute bp is partially, or ‘fractally’, correlated with the attributes that already exist in the data set [11]. In the iterative process we are selecting stocks for portfolio extension, which will maximally increase the total FD. When the new uncorrelated stocks have been added to the portfolio, the TDMC methodology continues with the PDM algorithm.
5. EXPERIMENTAL DISCUSSION
RESULTS
Extension Pha se
2
2.2
1.9
2 .1
1.8
2
FD
FD
Re duction Phase
1.7 1.6
1.9 1.8
1.5 5
4
3
1.7
2
3
Figure 1. Reduction Phase
4
5
Number of Stocks
Number of Stocks
Figure 2. Extension Phase
The next stage of the PRE algorithm consists of systematically adding new stocks from a pool of available stocks to build a new diversified 5-D portfolio. In this example, the pool of available stocks consists of Ford, Boeing, and Hewlett Packard. Therefore, N={f, ba, hpq}.
AND
After calculating the FD of all the possible 4-D portfolio, it is the f stock that is added, since its presence results in the greatest amount of diversification (highest FD). In the next iteration, the hpq stock is added, since it results in the more diversified 5-D portfolio. Figure 2 summarizes how the FD increases as uncorrelated stocks are added to the portfolio through the Extension Phase of the PRE algorithm. At the end of this process, our new diversified portfolio consists of BellSouth, ColgatePalmolive, Dole Food Company, Ford, and Hewlett Packard, which has a higher level of diversification based on the FD parameter (FD=2.07).
In order to evaluate the efficacy of the TDMC methodology, we conducted several experiments with different real world data sets. We will explain one experiment in detail in order to clarify all phases of the TDMC methodology. In this simple example, we monitor for ten years (1990-2000) a portfolio consisting of 5 stocks. At the beginning, the 5 stock portfolio includes AFLAC, BellSouth, Colgate-Palmolive, Dole Food Company, and General Electric. Formally, $={afl, bls, cl, dol, ge}. The monitoring process starts in 1990 and the FD is calculated for the window of four years. Therefore, the first calculation is made in 1994 and a calculation is made every subsequent year. We set a threshold of 10% below the maximum calculated FD as a trigger to stop the PDM phase and pass control to the PRE algorithm.
After improving diversification with the PRE algorithm, the maximum FD is set to the new calculated FD and the monitoring process via the PDM algorithm continues. In this example, the PRE algorithm is not triggered anymore before reaching the end of our 10 year analysis (i.e. The FD is within the threshold value). Figure 3 is a graphical representation of the FD values of the original and diversified portfolios through time.
During the first half of the monitoring process, the FD increases and decrease slightly, but the reported FD is within the boundaries of our threshold. Then in 1998, an FD of 1.97 (stocks are becoming more dependant) is reported which is outside of the 10% threshold of the maximum FD (= 2.25). This decrease in the FD, activates the PRE algorithm which will aim to rid the portfolio of dependant stocks and add new stocks to improve the diversification of the portfolio.
T im e Diversification and Monitoring 2.3 2.2 2.1 FD
In the reduction phase, the PRE algorithm systematically eliminates one stock at a time from the portfolio. It looks at the FD of all the possible combinations of n-1 stocks, and gets rid of the stock whose absence causes the least decrease in FD from the FD of the original n stocks. The process is repeated for the new n=n-1 stock until the calculated FD is significantly lower than the original FD value.
FD of original portfolio FD of new portfolio
2 1.9 1.8 1.7 1.6 1994 1995 1996 1997 1998 1999 2000 Year of Analysis
Figure 3. Graph demonstrating improvement in diversification
In this example, the FD of the 5-D portfolio is 1.97. After calculating the FD of all the possible combinations of 4-D
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As a measure of improvement in the diversification of portfolios, we used the following formula: Improvement = ( ( New FD -
the correlation coefficients is problematic since it can not capture nonlinear dependency, and analyzing pair-by-pair stocks in the portfolio does not always give the best estimation of diversification for the entire portfolio.
Old FD) / ( Old FD) ) X 100%
representing the approximation of the area between the two FD curves of the original and the improved portfolio. In the example demonstrated above, there is a 7.45 % improvement. Table 3 lists the results of two other representative experiments using the TDMC methodology. The table shows the stocks in the original portfolio, the stocks in the final diversified portfolio at the end of the 10 year period of analysis, the improvement in diversification as measured by the formula above, and the number of times the PRE algorithm was triggered during the period of analysis. Figure 4 and Figure 5 show the values of the FD for the original and the new improved portfolios through time for the two experiments.
In this paper we presented a novel technique to monitor and maintain a high level of diversification in a stock portfolio. Our TDMC methodology uses the correlation fractal dimension of the portfolio as measure of its diversification. Experiments show that our methodology is able to improve the diversification of a portfolio by about 10%.
7. REFERENCE
[1] Addison, P. Fractals and Chaos: An Illustrated Course. Institute of Physics Publishing, Bristol, 1997.
Figure
[2] Barbara, D., and Chen, P. “Tracking Clusters in Evolving Data Sets,' 'in Proceedings of FLAIRS' 2001, Special Track on Knowledge Discovery and Data Mining , Key West, FL, 2001.
once
4
[3] Cont, R. “Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues,” Quantitative Finance, vol.1, 2001, pp. 223-236.
twice
5
Table 3. Results of two representative experiments Original Portfolio
New Portfolio
Improve
cl, dell, dol, f, ge
cl, f, ge, t, toy
12.89 %
pfe, s, t, tgt, toy
ba, hmc, ibm, msf ,t
9.65 %
PRE triggered
[4] Gravity Investment, Model Portfolio and Diversification Technology. http:/www.gravityinvestments.com/diversification.htm [5] Hedge Fund Research. “Cyber Finance – Investing on The Edge of Chaos”, January 2000.
cl, dell, dol, f, ge / cl, f, ge, t, toy
http://www.hedgeresearch.com/HRGenericResearch.htm
2.1
[6] Mulligan, F. "A Fractal Analysis of Foreign Exchange Markets," International Advances in Economic Research, vol. 6(1), 2000, pp. 33-49.
2
FD
1.9 FD of original portfolio FD of new portfolio
1.8 1.7
[7] Peters, E. Chaos and Order in the Capital Markets. John Wiley & Sons, New York, 1991.
1.6 1.5 1994
1995
1996
1997
1998
1999
2000
[8] Peters, E. Fractal Market Analysis: Applying Chaos Theory to Investment and Economics. John Wiley & Sons, New York, 1994.
Year of Analysis
Figure 4. 12.89% improvement in diversification
[9] Sancetta, A., and Satchell, S. "Changing Correlation and Portfolio Diversification Failure in the Presence of Large Market Losses," Cambridge Working Papers in Economics 0319, Department of Applied Economics, University of Cambridge, 2003.
pfe, s, t, tgt, toy / ba, hmc, ibm, msft, t 2.2
FD
2.1 2
[10] Silvapulle, P., and Granger, C. “Large Returns, Conditional Correlation and Portfolio Diversification: A Value at Risk Approach,” Quantitative Finance, vol. 1, 2001, pp. 542-551.
FD of original portfolio FD of new portfolio
1.9 1.8
[11] Sousa, E., Traina, C., Traina, A., and Faloutsos, C., “How to Use Fractal Dimension to Find Correlations Between Attributes,” in First Workshop on Fractals and Self-similarity in Data Mining: Issues and Approaches (in conjunction with 8th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining), Edmonton, Alberta, Canada, 2002, pp.26-30.
1.7 1994
1995
1996
1997
1998
1999
2000
Year of Analysis
Figure 5. 9.65% improvement in diversification
[12] Traina, C., Traina, A., Wu, L., and Faloutsos, C. “Fast Feature Selection Using Fractal Dimension,” in XV Brazilian Database Symposium, João Pessoa - PA - Brazil, 2000, pp.158171.
6. CONCLUSION Diversification is a technique used to reduce the risk of investment and is accomplished by including uncorrelated and independent stocks in one’s portfolio. In the past, the correlation coefficient has been used as a basis for diversification. However,
[13] Wu, L., and Faloutsos, C. FracDim, Perl Package, 2001. http://www.andrew.cmu.edu/$\sim$lw2j/downloads.html
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