Wavelet portfolio optimization: Investment horizons, stability in time and rebalancing

• Thesis title in Czech: Wavelet portfolio optimization: Investment horizons, stability in time and rebalancing
• Thesis title in English: Wavelet portfolio optimization: Investment horizons, stability in time and rebalancing
• English key words: wavelets, MODWT, portfolio optimization, fractal market hypothesis
• Academic year of topic announcement: 2013/2014
• Thesis type: diploma thesis
• Thesis language: angličtina
• Department: Institute of Economic Studies (23-IES)
• Supervisor: prof. PhDr. Ladislav Krištoufek, Ph.D.
• Author: hidden - assigned by the advisor
• Date of registration: 29.05.2014
• Date of assignment: 30.05.2014
• Date and time of defence: 22.09.2015 00:00
• Venue of defence: IES
• Date of electronic submission: 31.07.2015
• Date of proceeded defence: 22.09.2015
• Opponents: PhDr. Jiří Kukačka, Ph.D.

Preliminary scope of work in English

Motivation:
Financial trading has experienced a huge increase during the past decades. Both researchers and practitioners have tried to develop many techniques how to beat the market. One of the core tools is Markowitz (1952) mean-variance portfolio selection, which combines several assets based on their covariance structure with the goal to minimize the overall portfolio variance. The output of this procedure is a set of weights of all the assets that should form an efficient market portfolio. Based on the theory of Efficient Market Hypothesis of Fama (1970) no one should be able to consistently earn abnormal returns different from the returns of the efficient portfolio.

One of the theories that might contradict to EMH is Fractal Market Hypothesis of Peters (1994). It states that financial markets are composed of heterogenous agents with different investment horizons that react differently to inflow of newly published information. Kristoufek (2013) concludes that if there is a sufficient number of traders with wide range of investment horizons, demand meets supply and the market works efficiently. However, if there is a dominant investment horizon, the market is not cleared and thus does not work efficiently any more.

The goal of this diploma thesis is to scrutinize how the portfolio selection evolves both over time and at different investment horizons. In order to estimate the covariance structure at various investment horizons a relatively novel approach of Maximum Overlap Discrete Wavelet Transform (MODTW) as described in Percival (2000) will be used. This approach allows us to decompose the overall covariance structure into several increments that are associated with different investment scales. Based on this approach we should be able to determine how the portfolio composition evolves over time and at different scales. Moreover, we will also try to find out how does the effect of rebalancing influences both short-term and long-term traders.

Hypotheses:
Hypothesis #1: The set of assets of mean-variance portfolio differ in time and at different investment horizons.
Hypothesis #2: It is possible to earn abnormal returns in some specific time periods and at some investment horizons.
Hypothesis #3: Short-term traders achieve abnormal returns by using low scales covariance structure and high rebalancing, while long- term traders achieve abnormal returns by using high scales covariance structure and rare rebalancing.
Hypothesis #4: The previous results are robust to different types of mother wavelets.


Methodology:
The first step is to use a rolling window to estimate MODWT coefficients at various scales and in time. Based on this procedure we will obtain the covariance structure of all our assets as it changes in time and at different scales. This approach has already proven useful in volatility estimation of some Central European markets in Dajcman and Kavkler (2014) and in volatility estimation of DJIA in Gallegati (2007).
Subsequently, we will apply the methodology of Markowitz (1952) to the scaled covariance structures to obtain minimum-variance portfolio at various horizons. A similar approach was applied in Fernandez and Lucey (2006) who used Capital Asset Pricing Model to compute betas at various scales.

In order to measure the performance of our portfolios at different scales, we will compute returns of equally weighted portfolio and of sample covariance portfolio. Then, we will use different rebalancing horizons of the portfolio selection to show what is the influence of rebalancing on different investment horizons. Finally, we will repeat all the procedure to show whether our results are robust to different mother wavelets.

One of the possible extensions is the use of Continuous Wavelet Transform (CWT) and Wavelet Coherence (WC). So far it has not appear in the literature with connection to portfolio selection, but many papers use CWT to detect significant areas of wavelets (Kristoufek 2013) and WC to detect significant areas of comovement (Razdan 2003, Rua and Nunes 2009). Therefore, it should be possible to exploit those features in portfolio selection.

Expected Contribution:
Most of the literature that uses wavelet approach to time series analysis is focused on 3D plots showing either univariate wavelet coefficients or coherence coefficients. This is a nice way how to reveal temporarily increased anomalies or correlations. However, most of the research stops at this point and does not reveal further implications. This diploma thesis will go beyond the results of wavelet analysis and will show its implications to portfolio selection at various investment horizons. Moreover, the problem of rebalancing will be also closely scrutinized. To the best knowledge of the author a similar analysis of this extent has not yet been published.

Outline:
1. Literature: I will summarize existing literature on heterogenous agents, various methods of portfolio optimization and the use of wavelets in economics and finance. I will briefly describe the theory behind the Fractal Market Hypothesis in order to motivate for the further use of wavelets.

2. Methodology: I will explain the mechanisms behind portfolio optimization and wavelets estimation.

3. Data: I will describe the data selection and its main features.

4. Results: I will present the results; mainly increments of covariance structure, portfolio composition and portfolio returns.

5. Conclusion: I will summarize both the theoretical and empirical results.

Core Bibliography:
Dajcman S. & A. Kavkler (2014): "Wavelet analysis of stock return energy decomposition and return comovement – a case of some Central European and developed European stock markets," E&M Economics and Management 17(1), pp 104-120.
Fama, E. (1970): "Efficient Capital Markets: A Review of Theory and Empirical Work," Journal of Finance 25(2), pp 383-417.

Fernandez, V. & B. M. Lucey (2007): "Portfolio management under sudden changes in volatility and heterogeneous investment horizons," Physica A: Statistical Mechanics and its Applications 375(2), pp 612-624.

Gallegati, M. (2008): "Wavelet analysis of stock returns and aggregate economic activity," Computational Statistics & Data Analysis, 52(6), pp 3061-3074.

Kristoufek, L. (2013): "Fractal Markets Hypothesis and the Global Financial Crisis: Wavelet Power Evidence," Scientific Report, 3, art. 2857.

Markowitz, H. (1952): "Portfolio Selection," Journal of Finance 7(1), pp. 77-91.

Percival D. B. & A. T. Walden (2000): Wavelet Methods for Time Series Analysis, Cambridge University Press

Peters, E. (1994): Fractal Market analysis – Applying Chaos Theory to Investment and Analysis. John Wiley & Sons.

Razdan, A. (2004): "Wavelet correlation coefficient of ‘strongly correlated’ time series," Physica A: Statistical Mechanics and its Applications 333(15), pp 335-342.

Rua, A. & L. C. Nunes (2009): "International comovement of stock market returns: A wavelet analysis, " Journal of Empirical Finance 16(4), pp 632-639.