Risk-based asset allocation : a forward looking approach

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(2) Risk-based asset allocation: a forward looking approach by Lamukanyani Alson Mantshimuli A minor dissertation submitted to the Faculty of Economic and Financial Sciences in partial fulfillment of the requirements for the degree of Master of Commerce in Financial Economics at the University of Johannesburg Supervisor: Professor John Muteba Mwamba August 16, 2016.

(3) Abstract The portfolio allocation problem is characterised by two factors; risk and expected return. This is mainly explained by the Markowitz (1952) mean-variance framework. The frequency and severity of recent financial crises has led to an increase in calls for improved asset allocation methods in the asset management industry. Asset allocation strategies should protect investor capital and result in higher relative returns in turbulent times. Modern portfolio theory has been heavily criticised (Lee, 2011; Roncalli, 2013) for failure to provide adequate diversification to protect fund managers during crises, hence the emergence of risk-based asset allocation methods that focus on portfolio construction based on risk and diversification. The crises led to poor performance of different portfolios and funds, especially those with high exposure to equities. Risk-based allocation methods try to achieve investors’ goals of safety and higher returns, irrespective of future market behaviour. Six risk-based asset allocation strategies were explored and contrasted; Equally weighted, Risk parity, Most Diversified, Minimum Correlation, Minimum variance and the Minimum CVaR portfolio. This was done in an effort to find the method which performs better when investors have different investment goals. Predicted risk measures were applied as inputs in these risk-based asset allocation methods (i.e. a forward looking approach was taken). The study focused on comparisons of the risk-based asset allocation methods using forwardlooking risk measures in the South African market. The main results of the study include the finding that risk-based asset allocation methods are effective in protecting investors’ capital and achieve higher returns than the market portfolio during crisis periods compared to other periods as expected. It was also found that the Minimum Correlation Portfolio performed better than all other risk-based asset allocation during the crisis period, which means it is the best risk-based asset allocation method to use during crisis periods in the South African market . There has not been a lot of studies on the perfomance of the Minimum Correlation Portfolio, and this result shows the need for a comprehensive study on all risk-based asset allocation methods in different countries/regions to determine which risk-based asset allocation technique is best for different regions. Key Words: Risk-based strategies, Markowitz mean-variance framework, Financial crises, Predictive risk measures, Asset allocation.. ii.

(4) Acknowledgments First of all, I would like to thank God for all the courage and strength when writing this dissertation. I am grateful to my supervisor Professor J.M Mwamba for all the guidance and advice on the structure and the content of the dissertation. I would also like to thank Dr Frank Riedel for his inputs in some sections of the research. A special thanks to Raphael Nkomo who introduced me to the concept of risk-based asset allocation and guided me through the structure of the dissertation. Last but not least I would like to thank my family and friends especially my partner Elizabeth, for all their support and encouragement. Their words of encouragement helped me pull though difficult challenges I faced in completing this work.. iii.

(5) Contents List of Figures. vi. List of Tables. vii. 1 Introduction. 4. 1.1 Research Problem and Question . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Organisation of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 2 Literature Review: Risk-based asset allocation methods 9 2.1 Equally weighted portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Most Diversified Portfolio. 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. 2.3 Minimum Variance Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11. 2.4 Risk-parity portfolios: Equal Risk Contribution and Inverse Volatility Portfolios. ........................................ 2.5 Minimum Correlation Portfolio 2.6 Minimum CVaR Portfolio. .......................... 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. 3 Methodology: Risk Measures and the Construction of Risk-Based Portfolios 3.1. 12. 17. Risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17. 3.1.1. Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. 3.1.2. Semi-variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20. 3.1.3. Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20. 3.1.4. Conditional Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . .. 22. 3.1.5. Modelling and forecasting of risk measures . . . . . . . . . . . . . . .. 22. 3.1.6. Risk allocation and portfolio risk contributions . . . . . . . . . . . . .. 31. 3.2 3.2.1 3.2.2 3.2.3. Construction of Risk-based portfolios . . . . . . . . . . . . . . . . . . . . . .. 34. Equally weighted portfolio . . . . . . . . . . . . . . . . . . . . . . . .. 34. Most Diversified portfolio. ......................... 34. Minimum variance portfolio . . . . . . . . . . . . . . . . . . . . . . .. 37. iv.

(6) 3.2.4. Risk-parity portfolios: Equal Risk Contribution and Inverse Volatility Portfolios. .................................. 38. 3.2.5. Minimum Correlation Portfolio. 3.2.6. Minimum CVaR portfolio. 3.2.7. Summary and comparison of risk-based asset allocation techniques . .. 4 Empirical Analysis. ..................... ......................... 39 42 44. 47. 4.1 Data Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Risk-based Portfolios: Back-test . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Risk-based Portfolios: Predictive Risk Measures Applied . . . . . . . . . . . 4.3.1. Forecasting risk measures . . . . . . . . . . . . . . . . . . . . . . . .. 58. 4.3.2. Portfolio Performance. 63. 5 Conclusions. ........................... 58. 67. 5.1 Main Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.2 Direction of further studies . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69. References. 70. Appendices. 77 Appendix A: Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77. Appendix B: Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 81. List of Figures 1.1 Risk Parity Performance (Source: BlackRock Investment) 3.1 Normal CDF. ........... .................................... 7 41. 4.1 JSE Indices performance Indexed . . . . . . . . . . . . . . . . . . . . . . . .. 49. 4.2 4.3 4.4 4.5 4.6 4.7. 53 54 55 60 61 61. Risk-Based Asset Allocation: Pre-Crisis Back-test Performance ....... Risk-Based Asset Allocation: Crisis Period Back-test Performance . . . . . . Risk-Based Asset Allocation: Post-Crisis Back-test Performance . . . . . . . JSE Sectors Correlations: Pre-Crisis . . . . . . . . . . . . . . . . . . . . . . . JSE Sectors Correlations: Crisis Period . . . . . . . . . . . . . . . . . . . . . JSE Sectors Correlations: Post-Crisis . . . . . . . . . . . . . . . . . . . . . .. v.

(7) 4.8 Risk-based portfolio performance using DCC Garch (1,1) covariance as inputs: Pre-Crisis leading to Crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 64. 4.9 Risk-based portfolio performance using DCC Garch (1,1) covariance as inputs: Crisis leading to Post-Crisis. ............................ 65. 5.1 JSE Indices performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 81. 5.2 JSE Sectors Return Distributions: Pre-Crisis . . . . . . . . . . . . . . . . . . 82 5.3 JSE Sectors Return Distributions: Crisis Period . . . . . . . . . . . . . . . . 83 5.4 JSE Sectors Return Distributions: Post-Crisis . . . . . . . . . . . . . . . . . 84 5.5 JSE Sectors Return Distributions(observations in probability terms) . . . . . 85 5.6 Most Diversified Portfolio Estimated Weights: Pre-Crisis . . . . . . . . . . . 86 5.7 Minimum Variance Portfolio Estimated Weights: Pre-Crisis . . . . . . . . . . 87 5.8 2 Asset Risk Parity Portfolio Estimated Weights: Pre-Crisis ......... 88 5.9 Equal Risk Contribution Risk Parity Portfolio Estimated Weights: Pre-Crisis 89 5.10 Minimum Correlation Portfolio Estimated Weights: Pre-Crisis . . . . . . . . 90 5.11 Minimum CVaR Portfolio Estimated Weights: Pre-Crisis . . . . . . . . . . . 91 5.12 Most Diversified Portfolio Estimated Weights: Crisis Period ......... 92 5.13 Minimum Variance Portfolio Estimated Weights: Crisis Period . . . . . . . . 93 5.14 2 Asset Risk Parity Portfolio Estimated Weights: Crisis Period . . . . . . . . 94 5.15 Equal Risk Contribution Risk Parity Portfolio Estimated Weights: Crisis Period 95 5.16 Minimum Correlation Portfolio Estimated Weights: Crisis Period ...... 96 5.17 Minimum CVaR Portfolio Estimated Weights: Crisis Period ......... 97 5.18 Risk-Based Asset Allocation Pre-Crisis Back-test: 2 Asset Case ....... 98 5.19 Risk-Based Asset Allocation Crisis Period Back-test: 2 Asset Case . . . . . . 99 5.20 Risk-Based Asset Allocation Post-Crisis Back-test: 2 Asset Case . . . . . . . 100. List of Tables 1.1 Average loss contribution for the 60/40 portfolio (Qian, 2005). ......... 6. 3.1. Comparison of risk-based asset allocation strategies . . . . . . . . . . . . . .. 46. 4.1. JSE Sectors Descriptive Statistics : Pre-Crisis . . . . . . . . . . . . . . . . .. 50. 4.2 JSE Sectors Descriptive Statistics : Crisis Period ............... 4.3 JSE Sectors Descriptive Statistics : Post-Crisis . . . . . . . . . . . . . . . . . 4.4 Portfolio Performance Measures: Pre-Crisis . . . . . . . . . . . . . . . . . . . 4.5 Portfolio Performance Measures: Crisis Period . . . . . . . . . . . . . . . . . 4.6 Portfolio Performance Measures: Post-Crisis . . . . . . . . . . . . . . . . . . 4.7 The variance-covariance matrix: Pre-Crisis . . . . . . . . . . . . . . . . . . . 4.8 The variance-covariance matrix: Crisis Period . . . . . . . . . . . . . . . . . 4.9 The variance-covariance matrix: Post-Crisis ................... vi. 50 50 56 57 57 59 59 59.

(8) 4.10 GARCH models comparisons: Pre-Crisis . . . . . . . . . . . . . . . . . . . . 4.11 GARCH models comparisons: Crisis Period .................. 4.12 GARCH models comparisons: Post-Crisis . . . . . . . . . . . . . . . . . . . . 4.13 Portfolio Performance Measures: Predictive Risk Measures Applied (PreCrisis leading to Crisis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Portfolio Performance Measures: Predictive Risk Measures Applied (Crisis leading to Post-Crisis) ............................... 62 62 62. 5.1. 77. Risk-based portfolio weights: Pre-Crisis . . . . . . . . . . . . . . . . . . . . .. 5.2 5.3 5.4 5.5. 66 66. Risk-based portfolio weights: Crisis Period . . . . . . . . . . . . . . . . . . . 77 Risk-based portfolio weights: Post-Crisis . . . . . . . . . . . . . . . . . . . . 77 DCC(1,1) GARCH with Multivariate t Distribution Model Results: Pre-Crisis 78 DCC(1,1) GARCH with Multivariate t Distribution Model Results: Crisis Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.6 DCC(1,1) GARCH with Multivariate t Distribution Model Results: Post-Crisis 80. vii.

(9) Chapter 1 Introduction The decision of how to optimally allocate funds in a portfolio is certainly one of the most important decisions in the asset management industry. It is a portfolio manager’s job to meet and exceed investment objectives such as maximising expected returns, beating the market portfolio, performing better than the benchmark and offering capital protection for investors. Investors and portfolio managers use different types of techniques to help them allocate funds in a portfolio optimally. The focus of this research is on the analysis and comparison of one particular type of asset allocation technique ; the risk-based asset allocation. The analysis is performed using predictive risk measures in the South African equity market. The following risk-based asset allocation methods are analysed(examined and applied): the Equally Weighted Portfolio, the Most Diversified Portfolio, the Minimum Variance Portfolio, the Risk Parity Portfolio (Equal Risk Contribution and the Inverse Volatility), the Minimum Correlation Portfolio and the Minimum CVaR portfolio. Interest in risk-based approaches to asset allocation has grown over the past few years, with investors and portfolio managers becoming more interested in portfolio risks and the diversification of those risks. Improving risk management is one of the important themes that have become apparent over the past years in the asset management industry, especially after the 2007/2008 global financial crises. These crises caused investors to question what went wrong with their portfolios, which they believed to have been properly diversified (Lee, 2010). Important literature on this subject includes seminal papers by Lee (2010) and Allen (2010), as well as a book by Roncalli (2013) on risk parity and risk budgeting. The study builds upon this literature, among others, in the analysis of risk-based asset allocation methods. It is important for fund managers to find asset allocation methods that strive towards meeting their investment goals, irrespective of the future performance of financial markets. Riskbased asset allocation methods have gained considerable amounts of attention since the crisis as investors turn their focus on improving risk diversification and try to move away from the traditional diversification of wealth. Risk-based asset allocation approaches are preferred in the asset management realm because asset allocation can be determined by using only risk inputs, without the need to estimate expected returns. Recent studies on riskbased asset allocation methods have found that they perform better in terms of producing risk-adjusted returns of a portfolio rather than the traditional Mean Variance Optimisation (MVO)-based asset allocation methods (Allen, 2010; Peters, 2010; Rapport & Nottebohm, 2012). An empirical study by Romahi and Santiago (2012) highlighted the clear risk reduction from portfolios when using risk-based asset allocation methods, and showed that they are “the point of least regret” since they produce results which are closer to ex post optimal than other asset allocation methods.. 4.

(10) Any academic study on asset allocation and portfolio construction would not be a proper study without mentioning the mean-variance portfolio construction framework developed by Markowitz (1952). The Markowitz framework is the cornerstone of Modern Portfolio Theory and describes how risk averse investors can assemble a portfolio of assets such that portfolio expected returns are maximised for a given level of risk, or portfolio risk is minimised for a given level of return. Modern Portfolio Theory faced a lot of criticism for the lack of diversification (see Roncalli, 2013 and Lee, 2010) since portfolios that were constructed based on the mean-variance framework were found not to be diversified enough to withstand financial crises and protect investors’ capital, hence the emergence of asset allocation approaches such as risk-based asset allocation methods. Techniques such as the risk-based asset allocation methods focus on creating portfolio construction methods based on risk and diversification rather than using expected returns in portfolio construction. The failure of mean-variance based asset allocation methods to protect investor capital, especially during crises periods, highlights the gap between theory and practice in asset allocation or portfolio theory, and shows that there is a need for more discussion on riskbased asset allocation theory in literature. Shortfalls of the mean variance framework such as the challenge of estimating expected returns without large estimation errors, all contribute to the under-performance of MVO-based portfolios. One of the challenges of using expected returns as inputs in asset allocation is that they are usually estimated using historical data. This may result in over-estimating expected returns of a certain asset due to past strong performance of that asset. Risk-based asset allocation methods on the other hand are expected returns insensitive, and hence avoid most of the problems associated with estimating expected returns. The estimation of the expected returns is considered to be the most difficult step in asset allocation as a result of the challenges mentioned above, and for this reason there is a wide range of models that can be used to estimate expected returns, including the Capital Asset Pricing Model. However, economic events such the 2008 financial crisis showed that depending on expected returns in asset allocation does not lead to robust portfolios which are diversified enough to maximise returns and ensure the protection of investor capital. Another challenge in the mean-variance based portfolio is that a portfolio consisting of both fixed income instruments and equities will usually have a higher contribution of risk from equities because they are relatively highly volatile and usually consist of about half of many asset managers’ portfolios. The higher risk contributions from equities result in underdiversified portfolios, and this is where risk-based asset allocation methods are introduced in order to make sure portfolios are sufficiently diversified to withstand potential stock market crises and are able to produce the desired returns. To illustrate this point further, an example from Qian (2005) is analysed. Table 1.1 (in the next page) shows the average loss contribution for the 60% Equity/40% bonds portfolio using monthly returns data from the Russel 1000 Index and the Lehman Aggregate Bond Indices from the years 1983 to 2004. From Table 1.1, it can be seen that a greater percentage of the overall 60/40 portfolio loss came from equities, with equities contributing more than 100% for a portfolio loss of more than 3%. 5.

(11) This study illustrates how risky portfolios such as the 60/40 portfolios which were once thought as diversified but are actually not diversified in terms of risk or loss contributions. The same principle applies to equity in portfolios where diversification is thought of as a mixture of company shares from different sectors or geographic locations with different risk and reward profiles. This example, once again, illustrates the need for a study of methods that better allocate risk across all asset classes and across all assets for improved diversification. Risk-based asset allocation methods, such as the risk-parity portfolio, speak directly to this point, and aim at distributing risk equally across all portfolio assets.. Portfolio Loss 2% 3% 4%. Equities Contribution 95.6% 100.1% 101.9%. Bonds Contribution 4.4% -0,10% -1,90%. Table 1.1: Average loss contribution for the 60/40 portfolio (Qian, 2005). Of all the risk-based asset allocation approaches, risk-parity has gained more popularity since the 2008 financial crisis. Risk parity supports the view that risk contribution in a portfolio should derive equally from all the assets that comprise the portfolio, although this will result in lower weights for risky assets in the portfolio. To achieve a higher return, investors can leverage the portfolio with a short cash position. A backtest study by BlackRock Investment Management (de Martel & Ransenberg, 2014) showed that although risk parity based portfolios do not beat benchmarks entirely, they would have performed better during financial crises, and this could have saved investors millions. Figure 1.1(Source: BlackRock Investment) on the next page shows the performance of the risk-based strategy compared to the S & P 500, as well as cash plus 5% since 1930. As it can be seen from the graph, the strategy performed better during the 1930s great depression, the 1976 debt crisis, and the 2008 sub-prime crisis.. 6.

(12) Figure 1.1: Risk Parity Performance (Source: BlackRock Investment) We discuss risk-based asset allocation methods using a forward-looking approach in order to move away from estimates based on historical data and to bridge the gap between theory and practice. It is important to take a forward-looking approach in the study of risk-based portfolios, or any portfolios for that matter, because a portfolio based on historical inputs will be exposed to large estimation errors (over- and under-estimating the assets’ weight for a particular portfolio due to past performance of those assets, which is not necessarily the same as future performance). Predictive risk measures are used because risk measures such as variance are the main inputs in a risk-based portfolio, which means that they impact the performance of a portfolio more than other factors. The closer the predicted values are to the true values of risk measures, the higher the capacity of the portfolio manager to minimise portfolio risk, because they can select with more certainty on how risky these assets are.. Although other studies have attempted to take a forward-looking or predictive approach to risk-based asset allocation1, no one (to our knowledge) has used predicted risk measures in the construction of risk-based portfolios. The fact that this study is being done on the South African market Johannesburg Stock Exchange) makes the study unique. In summary, the real-world problem for investors and portfolio managers is that portfolios created using methods based on mean-variance-based asset allocation do not perform well during crisis. 1. See Roncalli, (2013) on illustrations of the simulation of risk-based indices. 7.

(13) periods because of their dependence on expected returns. This leads to under-diversification of portfolios and hence the emergence of risk-based asset allocation methods.. 1.1. Research Problem and Question. Risk-based asset allocation methods discussed here aim at improving the minimisation of portfolio risk and the protection of investor capital, especially during financial crises. Practitioners and academics are becoming increasingly interested in finding rigid asset allocation methods that result in better performance of portfolios in turbulent times. Although there has been extensive research (Allen, 2010; Lee, 2010; Roncalli, 2013; among others) on the performance of risk-based asset allocation methods, no research has been done focusing on the performance of these methods in the South African market. Hence the research problem centres on the fact that we do not know how the different risk-based asset allocation methods perform against each other in the South African market. The main contribution of this research is to fill this gap, thereby giving South African portfolio managers knowledge on the application of risk-based methods in the South African market. The research problem makes us question the performance of risk-based asset allocation methods in the South African market, hence the question at hand is: How do risk-based asset allocation methods perform in the South African market?. 1.2. Organisation of study. The dissertation consists of five chapters including this introductory chapter. The dissertation proceeds by discussing current literature on the six main risk-based asset allocation methods. In the methodology chapter, the theoretical principles that the riskbased asset allocation methods build upon are analysed. This is done by discussing different risk measures that can be used in the construction of a risk-based asset allocation portfolios. Methods used to find estimates or forecasts of these risk measures are also examined, because a forward looking approach was taken. The methodology chapter concludes with a discussion on how risk-based asset portfolios can be constructed using mathematical expressions, as this will be of help in the actual construction of portfolios in the empirical section. In Chapter 4, the empirical analysis of the study is presented. In the empirical study, the data used (JSE sector indices) is described, and a backtest for the risk-based portfolios is performed. Estimated portfolio weights of risk-based portfolios using predictive risk measures are found. Finally, the conclusions of the study are presented in Chapter 5 by reviewing the most important take outs of the study, as well as, the important findings of the study.. Chapter 2 8.

(14) Literature Review: Risk-based asset allocation methods The literature review contains an overview of the existing academic literature on the main risk-based asset allocation methods. This is done in order to gain a proper understanding of these methods. The methodology on constructing these portfolios is then discussed. The chapter concludes with a review of the six main methods (Equally Weighted Portfolio, the Most Diversified Portfolio, Minimum Variance Portfolio, the Risk Parity Portfolio (Equal Risk Contribution and the Inverse Volatility), Minimum Correlation Portfolio and the Minimum CVaR Portfolio),as well as, other methods not often discussed in academia or used in practice.. 2.1. Equally weighted portfolio. The equally weighted portfolio or the “1/n” portfolio, where all assets in the portfolio are given the same weight, is considered to be the simplest of risk-based portfolios. DeMiguel, Garlappi and Uppal (2009) did an extensive study on the equally weighted portfolio and compared it to the sample-based mean-variance model and its extensions (extensions aimed at reducing the estimation error of variances and expected returns). They discovered that of the 14 asset allocation methods, including the Bayes-Stein and minimum variance allocation methods, none of them consistently performed better than the equally weighted strategy in terms of the Sharpe ratio. However, this does not mean that the authors encourage the use of the “1/n” asset allocation in investment decisions, but rather to use it as a benchmark for optimized portfolios (which may be less efficient because of estimation errors). It is also better to use the equally weighted portfolio as a benchmark for other portfolios due to the ease of implementation. Other research on the “1/n” asset allocation have not found any conclusive evidence of a consistent out-performance of other sophisticated optimised asset allocation methodologies such as the sample-based mean-variance optimal portfolio, and the minimum-variance portfolio over the equally weighted portfolio (Bloomfield, Leftwich, & Long, 1977; Jorion, 1991; Brown, Hwang & In, 2013). The study by Bloomfield et al. (1977) compares the equally weighted portfolio to different sophisticated asset allocation strategies that are weighed periodically, as well as the trading of the different strategies. They found that sophisticated asset allocation methods which require portfolio weights to be frequently revised, may actually lead to inferior results to the equally weighted portfolio as a buy and hold strategy.This is as a result of the high transaction costs associated with revaluing these portfolios. The presence of estimation errors when using optimal asset allocation methods which depend on expected returns and risk inputs, can also be the cause of poor performance of the optimal asset allocation methods compared to the “1/n” strategy.This is because estimation errors may lead to extreme weights that fluctuate all the time (Brown et al., 2013). 9.

(15) A study by Kritzman, Page and Turkington (2010) showed that if some naive inputs are used, optimized portfolios usually outperform equally weighted portfolios when longer term samples are relied on for estimating expected returns. The difficulty with the “1/n” portfolio is that it cannot be placed in the mean variance framework since it does not take into account expected returns and risk. Arguments have also been made about the level of diversification of the strategy. Since the portfolio ignores the moments of the return distribution, the degree of diversification will be sensitive to the universe of assets under consideration (Lee, 2011). The risks of the assets in this portfolio are very different,therefore the use of equal weightings strategy will lead to a high concentration of risk, because high risk assets will be allocated the same weight with low risk assets. The equally weighted portfolio is included in the study for comparisons and completeness of the study of risk-based approaches, although it is not technically a risk-based approach because it is not dependent on any risk measures to determine the portfolio.. 2.2. Most Diversified Portfolio. The investment objective of an asset manager can just be to achieve the highest level of diversification in a portfolio (this can be an objective within the risk minimisation goal or vision of an asset manager). A portfolio is said to be diversified if it is not heavily exposed to an individual asset. The Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) introduced the concept of risk which can be diversified away (systematic risk) and risk which cannot be diversified away (unsystematic or idiosyncratic risk). The Most Diversified Portfolio (MDP) only focuses on the idiosyncratic risk. Some of the common diversification measures in asset allocation include the Herfindahl-Hirschman index (HHI), the diversification ratio, entropy, and the percentage of risk explained by the systematic factors in a systematic-plus-idiosyncratic factor model (Meucci, 2009). The Most Diversified Portfolio (MDP) is one of the risk-based asset allocation methods that has recently gained increasing attention. More research has gone into defining a proper diversification measure for a portfolio and in 2008, Choueifaty and Coignard (2008) introduced a measure of portfolio diversification named the diversification ratio. It is defined as the ratio of the weighted average volatilities of assets in a portfolio to the total portfolio volatility. This definition was further expanded by Choueifaty, Froidure, and Reynier (2011) by decomposing the diversification ratio into two parts: the Concentration Ratio and the volatility-weighted average correlation. This was done because portfolios with highly correlated assets or concentrated weights cannot be constructed properly, and would lead to low diversification ratios. In performing a comparison analysis of the portfolio, Choueifaty and Coignard (2008) used the Standard & Poor’s 500 index and the Dow Jones index daily performances to compare the most diversified portfolio with the market cap-weighted benchmark, the minimum variance portfolio and the equal weighted portfolio. The empirical result of the study showed that the MDP has higher Sharpe ratios than most market-cap weighted indices, and results in better returns in the long run. When assets in the portfolio have the same Sharpe Ratios, the Most 10.

(16) Diversified Portfolio has the highest possible Sharpe Ratio. Sapra (2011) also showed that in setting the objective of maximising the diversification ratio, the resulting portfolios were found to have higher ex-ante and ex-post Sharpe ratios compared to the market portfolio using the MSCI US, MSCI Europe, and MSCI Developed World indices.This is why the MDP is also described as the efficient alternative to the market cap-weighted index portfolio. The diversification ratio is not only used in the MDP, but can also be used in comparing how different portfolio strategies are diversified. In a paper on parity strategies and maximum diversification, Schoen (2012) used the diversification ratio to compare the risk parity method with other risk-based asset allocation in terms of its performance with regard to diversification, and found that the risk parity portfolio offers greater diversification compared to the equally weighted portfolio and the 60% Equity/40% bonds portfolio.. 2.3. Minimum Variance Portfolio. In the aftermath of the global financial crises, the Minimum Variance Portfolio (MVP) is one of the risk-based asset allocation methods that gained the most popularity amongst investors, especially those who are risk-averse. The Minimum Variance Portfolio is a portfolio of assets that has the lowest volatility, and can be found in the efficient frontier without using expected returns as inputs. It is sometimes referred to as the Global Minimum Variance Portfolio. Empirical studies have shown that the MVP often yields better out of sample results than the Markowitz-based mean-variance portfolio (Ledoit & Wolf, 2001; Clarke, de Silva & Thorley, 2006; Behr, Guttler & Miebs (2008)). Ledoit and Wolf (2001) used the MVP in a study of improving the estimation of the larger covariance matrix of stocks during portfolio selection using the New York Stock Exchange (NYSE) and the American Stock Exchange (AMEX) stocks returns, and showed an improvement of the MVP-based covariance estimator upon other estimators included in the study. Behr, Guttler and Miebs (2008) examined the worth of using the MVP in asset allocation by analysing its risk-adjusted performance. They used bootstrap methods for statistical inference with regard to performance measures (the Sharpe ratio, the Sortino ratio and Alpha measures) of portfolios compared to the MVP, and provided that constrained minimum-variance portfolios perform better compared to a value weighted benchmark. They also confirmed that the MVP does not outperform naively diversified portfolios such as the Equally Weighted Portfolio in terms of the Sharpe ratio. The good performance of the MVP can be attributed to the low volatility effect which Blitz and van Vliet (2007) describe as a phenomenon where stocks with low volatility earn high risk-adjusted returns. They performed an empirical study in the US, European and Asian markets using stock data from 1985 to 2006, and showed that stocks with historically low volatility had higher risk-adjusted returns in terms of the Sharpe ratios and the CAPM alphas. These results are also consistent with the results from a study by Ang, Hodrick, Xing and Zhang (2006) which showed that stocks with high idiosyncratic risk are less rewarded than stocks with low idiosyncratic risk.. 11.

(17) As with the maximum diversification portfolio and in light of the frequency of the financial crises that the market has experienced over the past few years, one also have to ask the question of whether the minimum variance portfolio is really diversified. The minimum variance portfolio results in equal marginal risk contribution for all the assets in the portfolio, but this does not mean that there are equal contribution of risk from different assets; hence, a minimum variance portfolio will not necessarily guarantee a fully diversified portfolio, even though it has the lowest volatility. Research on the minimum variance portfolio is mostly focused on the estimation of the variances. The covariance can be estimated more precisely than expected returns, and this is the advantage of using the minimum variance approach. Kempf and Memmel (2003) studied the distribution of estimated weights in a minimum variance portfolio and also determined the extent of the estimation risk that an investor faces. Other concerns of the minimum variance portfolio are the sensitivity of the portfolio weight to volatility estimates. This is because of the tendency of the portfolio to allocate more weight to low variance assets when minimising the portfolio variance. Clark et al. (2006) describe cases in which constraints can be applied in the optimisation process to minimise portfolio weight concentration. Concerns about the most diversified portfolio are mainly around the degree of diversification that the portfolio offers as well as the definition and interpretation of diversification. The most diversified portfolio maximises the diversification ratio, which is equivalent to maximising the difference between the volatility of the portfolio and the weighted sum of the volatilities of assets in the portfolio. Meucci (2009) explains the difference between the portfolio volatility and the weighted sum of volatilities of each stock as a differential diversification measure and not an absolute measure of risk in the portfolio. Kirchner and Zuncekl (2011) advocate for the use of a diversification measure which takes into account the probability distribution of the final portfolio value, since this would incorporate assets exposures and characteristics.. 2.4 Risk-parity portfolios: Equal Risk Contribution and Inverse Volatility Portfolios The method that receives the most attention of the all risk-based allocation methods is probably the risk parity or equal risk contribution asset allocation method. It has been studied extensively since the 2008 financial crises, although it may have not been referred to by the same name (see Neurich, 2008; Maillard, Roncalli and Teiletche, 2010; Lee, 2011 for example). The term ”risk-parity” was first introduced by Qian (2005), who described it as an asset allocation method that aims to allocate market risk equally among different asset classes. He also described this method as an ”efficient beta portfolio”, which means it is a well-diversified portfolio and rewards market risk sufficiently. Risk contribution is an important concept in risk-parity. Qian (2005) argues that this is because it is an accurate indicator of loss contribution in a portfolio. Using the portfolio loss of the 60/40 portfolio as shown in Table 1.1 above, portfolio losses of over 2% on average are characterised by having over 96% of these portfolio losses coming from equities. Therefore it is important to have a. 12.

(18) risk-parity framework that allows for all assets or asset classes to contribute the same amount of risk to the portfolio (loss contribution). There has been many studies (Maillard et al., 2010; Chaves et al., 2011; Lee, 2011; Roncalli, 2013)conducted on the comparison of the risk-parity method with other risk-based asset allocation methods, such as the minimum variance portfolio, the most diversified portfolio, and the equally weighted portfolio. Roncalli (2013) describes risk-parity as a heuristic method 1 because the main aim of the risk-parity method is to build a portfolio which guarantees the same risk contribution from each asset without trying to find a solution to the optimisation problem. The heuristic nature of the risk-parity approach as described by Roncalli (2013) will always lead to questions of how the portfolio performs in terms of returns when compared to other risk-based portfolio strategies. A study by Chaves et al. (2011) compares the performance of the risk parity portfolio with other asset allocation methods including equal weighting, minimum variance, mean variance optimisation and the 60/40 equity/bond portfolio. The empirical study using US stocks (S & P 500) and a number of indexes 2 showed that the risk parity approach does not consistently outperform the equal weighted or the 60/40 equity/bond portfolio on a risk-adjusted basis. However, it tends to consistently outperform the optimised strategies such as the minimum variance portfolio and the mean variance optimal portfolio. This may be as a result of the positive characteristics of the risk-parity approach such as a balanced risk-allocation, which provides for better investment protection from risky assets as well as being less volatile in nature. There is often a confusion on what constitutes a risk parity strategy, where both the inverse volatility method and the equal risk contribution being often regarded as the risk-parity approach. Some people consider the equal risk contribution as the risk parity (Lee, 2010) while others considers the inverse volatility strategy as the risk parity approach (Fisher, Maymin, & Maymin, 2012; Galane, 2014). We do not try to define and choose between these two techniques on whether which one is the correct reference to the risk parity approach however we will show that although they are considered to be different strategies, they are essentially the same. The inverse volatility portfolio weights are proven to be a consequence of the equal risk contribution mathematics, hence it does not make sense to separate these two strategies. This is shown in the methodology section which discusses the construction of risk-based asset allocation portfolios. We will start by explaining equal risk contribution and showing how this results in inverse volatility portfolio weights. The Equal Risk Contribution (ERC) approach results in an allocation of assets that ensures an equal contribution of risk from each asset in the portfolio.. Roncalli (2013) defines heuristic asset allocation methods as “experience-based techniques and trial and error methods, which do not correspond to the optimal solution of an optimization problem” 2 BarCap U.S Long Treasury Index, BarCap U.S. Investment Grade Corporate Bond Index, JP Morgan Global Government Bond Index, BarCap U.S. High Yield Corporate Bond Index, S & P 500 Index, MSCI EAFE Index, MSCI Emerging Market Index, Dow Jones UBS Commodity Index, and FTSE NAREIT US Real Estate Index. 1. 13.

(19) As a result of this equal risk contribution, a portfolio allocated to use this approach will have lower volatility asset classes such as fixed income comprising a greater percentage of the portfolio than before (Allen, 2010). An equal risk contribution portfolio might also result in lower expected returns, since less volatile assets tend to have lower expected returns and would show much more weight in the risk parity portfolio. Asset managers looking for higher returns than the risk-parity portfolio will need to leverage in order to create the potential for getting the desired return. Asness, Frazzini, and Pedersen (2012) show that applying leverage in the risk-parity portfolio results in higher expected returns for the portfolio. Leverage can be applied to increase the potential of the risk parity approach in order to earn a higher expected return, thus ensuring that the portfolio is well-constructed having an acceptable risk. Although leverage has its own risks and challenges, most risk parity investors prefer it to taking the risk of a higher concentration on risky assets in pursuit of higher expected returns. This is because leverage is a better risk, and leveraging the best unlevered risk parity portfolio should result in excess returns compared to taking on concentration risk (Asness, 2014).. 2.5. Minimum Correlation Portfolio. The Minimum Correlation Portfolio(MCP) is formed based on the minimum correlation algorithm by Bee, Kapler and Rittenhouse (2012). The algorithm stems from the formula of the portfolio variance or standard deviation, which is a measure of the risk of the portfolio. Since portfolio diversification is not only a factor of the number of assets in a portfolio but also how correlated these assets are, it is important to take a correlation of assets in the portfolio in the process of asset selection. This is why this portfolio is very similar to the MVP, although in this case the correlation of assets was only minimised, which lead to a minimum variance of the portfolio. Bee et al. (2012) developed the minimum correlation algorithm with the aim of minimising or reducing the portfolio variance from the correlation function of the variance. This is an optimisation method since we are trying to get weights of a portfolio which results in minimum correlation of a portfolio. The MCP is also useful because it can be used in conjunction with other risk-based asset allocation methods; for example, O’Toole (2014) used it to solve the maximum value of the diversification ratio in the Maximum Diversification Portfolio. The maximum diversification ratio shows an increase in the potential of the portfolio to be diversified with the lowest possible correlations among portfolio assets, and results in the required portfolio weights for the Maximum Diversification Portfolio. A study by Barber, Bennet and Gvozdeva (2015) compared the performance of the MCP with other asset allocation methods such as the Minimum Variance Portfolio and the Maximum Sharpe Ratio portfolio, across several regions (developed markets and emerging markets) in an effort to propose a framework that relies on modern portfolio theory but also reduces the sensitivity to estimation error. The study did not find any overwhelming evidence that the MCP performs better that other asset allocation methods; for example, it had a Sharpe Ratio of 0.39 while the Equal Risk Contribution Portfolio had a Sharpe ratio of 0.43, and the Minimum Variance Portfolio had a Sharpe ratio of 0.41 over the period January 2000 to July 2014. There is very limited 14.

(20) literature on the Minimum Correlation Portfolio since it is a new concept and there are already other established risk-based asset allocation methods which are related to it, such as the Minimum Variance Portfolio. The construction of the algorithm which forms the portfolio is detailed in the methodology chapter under Construction of risk-based asset allocation portfolios section (Section 3.2).. 2.6. Minimum CVaR Portfolio. Value-at-Risk (VaR) is one of the most commonly used risk measure in financial markets. It can be defined as the amount of capital sufficient to cover losses from a portfolio over a certain time period. Research on the applicability of VaR to portfolio construction has mainly been in the Mean-Variance Optimisation space, with the aim of maximising expected returns. This is subject to the constraint that the expected maximum loss should not exceed Value-atRisk limits (Campbell, Huisman & Koedijk, 2001; Gu, 2013). Conditional Value-at-risk (CVaR) is an extension of VaR that takes into account the shape of the loss distribution in the tails. It allows the portfolio manager to manage the risk of big losses better than VaR because it models the tail end of the distribution and hence it is used to manage probability of large losses in a portfolio. Krokhmal, Palmquist and Uryasev (2001) performed a portfolio optimisation study with CVaR as both the objective (minimising CVaR for a given level of return) and constraint (maximising expected returns for a given limit of CVaR). This Markowitz-based optimisation study resulted in a stable and efficient optimisation algorithm which was useful to consider when we looked at risk-based optimisation, as seen in the next chapter. An alternative to the normal VaR and the CVaR is the Entropic Value-at-risk (EVaR) risk measure, which measures risk by taking into account the risk aversion level of the individual investor, hence resulting in different values for different investors with the same portfolio. EVaR has also been used in portfolio construction and optimal asset allocation. For example, Firouzi and Loung (2014) used EVaR in the optimisation problem and this yielded an explicit formula for the objective function, which means that the optimisation problem can be solved without using numerical approximations. In the Minimum CVaR Portfolio, portfolio weights are found by striking a balance between the return objectives of the portfolio manager or investor, and the allocation of risk (CVaR) across the portfolio. The construction of the portfolio is based on work done by Boudt, Carl and Peterson (2013), which focused on risk contribution portfolios using downside risk budgets such as CVAR rather than portfolio variance as an objective or constraint (minimising risk objective or equal risk contribution constraint), just as most of the above risk-based asset allocation methods did. This method aims at finding portfolio weights that minimise the largest CVaR risk contribution in the portfolio. The empirical analysis of this study used monthly data1 from January 1976 to June 2010, and showed results supporting. Merrill Lynch Domestic Master index, the S & P Goldman Sachs commodity index, the S & P 500 index and the National Association of Real Estate Investment Trusts Index (NAREIT) 1. 15.

(21) that minimum CVaR portfolios have low risk, high diversification as well as low portfolio turnover. Another study on the Minimum CVaR Portfolio is by Rockafellar and Uryasev (2000) on the optimisation of Conditional Value-at-Risk, with a particular focus on calculating the VaR of a portfolio and optimising CVaR. At the same time it has the capability of being combined with mathematical optimisation techniques such as linear programming and nonsmooth programming in order to optimise portfolios with a large number of assets. As an illustration of this approach to portfolio optimisation and asset selection, Rockafellar and Uryasev (2000), showed that a portfolio CVaR can be minimised efficiently by using linear programming and nonsmooth programming. It also lowers the level of VaR in a portfolio. It can also be used with scenario-based methods to optimise portfolios, leading to easier ways of evaluating derivatives such as options and futures. This is a different approach to the one explained by Boudt et al. (2013), but they all aim at minimising the portfolio CVaR.. 16.

(22) Chapter 3 Methodology: Risk Measures and the Construction of Risk-Based Portfolios This chapter looks at the tools needed to construct risk-based portfolios and how to actually allocate assets using these methods. In order to construct a risk-based asset allocation portfolio, it is important to first discuss different risk measures that can be used in the construction of these portfolios. This is done in the next section.. 3.1. Risk measures. This section discusses the following risk measures; variance, semi-variance, Value-at-risk (VaR) and Conditional Value-at-risk (CVaR). The use of these risk measures depends on preference and the ability of different users to compute them (Galane, 2014). The discussion on risk measures starts by first discussing some properties that make risk measures coherent. This is then followed by a discussion on the most popular and easy ways to calculate risk measures, such as the variance and semi-variance methods as well as the valueat-risk. It is important to have a discussion on risk measures used in financial markets because the focus of the discussion is on risk-based asset allocation methods which are based on risk measures. This means it is important to find appropriate risk measures to use in the asset allocation process. In discussing risk measures, it is valuable to first discuss properties that make different risk measures desirable to use in portfolio management. A risk measure is important because it enables one to understand and characterise investor risk preferences in portfolio selection. A popular word in describing a good risk measure is “coherent”. Following a paper by Artzner (1999) on the application of coherent risk measures to capital requirements in insurance, one can now describe properties that risk measures should satisfy in order to be coherent. Let R(p) be a risk measure of a certain portfolio p, axioms that make the risk measure coherent are now discussed; i. Sub-additivity The risk of two portfolios is less than or equal the risk of two portfolios added together. R(p1 + p2) ≤ R(p1) + R(p2), where p1 and p2 represents portfolio 1 and 2 respectively. This property illustrate the idea that pooling risks from different assets in a portfolio lead to diversification. 17.

(23) ii. Positive Homogeneity R(λp) = λR(p) if λ ≥ 0, where λ is the degree of homogeneity or the leverage factor. If λ = 0 , then no leverage is applied and the risk of the portfolio does not change. The axiom means that leveraging and de-leveraging of the portfolio increases the risk measure in the same magnitude. The axiom has been criticized for ignoring the concentration of risk and has been associated with increasing liquidity problems in a portfolio (Haugh, 2010). iii. Monotonicity If p1 ≺ p2, then R(p1) ≥ R(p2), where ≺ shows preference (the object on the right is preferred to the one on the left) while ≥ shows greater than (the object on the left is greater than the one on the right). If portfolio 2 is always preferred to portfolio 1 i.e has better returns under different conditions, then the risk measure R(p1) should be higher than risk measure R(p2). iv. Translation Invariance R(p + c) = R(P) - c for c ∈ R Adding some cash amount c to the portfolio will reduce the portfolio risk by the same amount of cash added. A risk measure which satisfy sub-additivity, translation invariance, monotonicity, and positive homogeneity is called a coherent risk measure. The criticisms of the sub-additivity and the positive homogeneity properties led to the development of a fifth risk measure called convexity. Convexity ensures that diversification of assets in a portfolio does not increase the total risk of the portfolio and it can be defined as follows; R(λp1 + (1 − λ)p2) ≤ λR(p1) + (1 − λ)R(p2). Four different risk measures that can be used in risk-based asset allocation methods (variance, semi-variance, value-at-risk and conditional value-at-risk) are now analysed. These risk measures do not always satisfy the coherent and convexity properties of a good risk measure e.g. for Value-at-Risk, the sub-additivity axiom does not hold in general. Furthermore, the standard deviation of the portfolio is not coherent because it violates the translation invariance axiom, since adding cash to portfolio will not necessarily decrease the risk of a 18.

(24) portfolio. Although these risk measures do not always satisfy the coherent and convexity risk measures, we do consider them and use them as coherent for the purpose of portfolio management because some of the axioms are not well adapted to portfolio management.. 3.1.1. Variance. Variance is one of the most popular risk measure in the investment industry. For a given random variable such as asset returns R with n data points or prices given, the mean can be estimated as follows;. ,. (3.1). and the variance as;. .. (3.2). Some of the important basic properties of the variance include the following; For given constant k,l and returns of asset i, Ri and asset j, Rj , i. Var(Ri) ≥ 0. ii. Var(Ri=k) ⇐⇒ Var(Ri)= 0 , i.e if the variance of some data points is zero, then all variables are the same and the variance of a constant is zero. iii. Var(Ri+ k)= Var(Ri). iv. Var(kRi) = k2Var(Ri). v. Var(kRi± lRj) = k2Var(Ri) + l2Var(Rj) ± 2klCov(Ri,Rj), where Cov(Ri,Rj) is the covariance between the returns of asset i and asset j. It is from the above variance properties that one can define the variance of the portfolio with n assets; .. 3.1.2. Semi-variance. The semi-variance concept is similar to the variance concept, but instead of calculating it by using the whole set of observations, only observations below a certain benchmark are needed. This is done to specifically measure the downside risk of the portfolio. While variance measures the volatility of an asset, semi-variance measures the negative volatility or fluctuation of the asset. From the definition of variance given in Section 3.1.1, the semivariance of a stock price with n number of prices over a certain period of time can be defined as follows;. 19.

(25) .. (3.3). In this case, the mean is used as the benchmark of downside variance which means only stock prices that result in returns below the expected return are taken. Some researchers have explored the use of semi-variance instead of variance in mean-variance optimisation (Hongan and Warren, 1974; Estrada, 2007) so it is important for us to consider using it in the risk-based portfolio construction framework as well. Following work by Estrada (2007), semi-variance can be defined as follows; Var(R) = E[min(R − b,0)]2,. (3.4). which can be written as ,. (3.5). where b is any benchmark that an investor chooses in their definition of downside risk. The square root of the semi variance will give us the semi-deviation which is an important measure of downside risk. To find a portfolio semi-variance, it is important to first get the semi-covariance between two assets, and this is challenging to define. The semi-covariance between assets 1 and 2 is given by the following equations; Covar(R) = σ12 = E[min(R1 − b,0) × min(R2 − b,0)],. (3.6). equivalently, n. Covar(. .. (3.7). i=1. Although the semi-variance makes much more sense to use as a risk measure, it has its own challenges, including the fact that the variance-covariance matrix is endogenous and not exogenous as in the original mean-variance portfolio. This means it is not an input into the model but rather forms part of the mean-variance model, and this makes it difficult to calculate the portfolio variance (Estrada, 2007).. 3.1.3. Value-at-Risk. Value-at-Risk (VaR) can be defined as the maximum potential loss in the value of an asset or portfolio over a specified period of time with a given confidence interval. This means VaR is determined by two parameters, the confidence or probability level and the time period. VaR is formally defined as follows; VaR of a portfolio with a value X at a given probability level p ∈ (0;1) at a time t + 1 is given by; VaRp(X) = min{m : P(mRt + X ≤ 0) ≤ p}. (3.8). where Rt is the return of a risk free asset at time t . This means VaR of a position X at a time t+1 can be explained as the smallest amount m that if added to the position at a time t and 20.

(26) invested in a risk free asset will make sure that the probability of a non-positive value at a time t+1 is less than or equal to the probability value p. Furthermore, it can be shown that there exist the smallest amount m in equation (3.8); m : P(mRt + X ≤ 0) ≤ p p. p Rt. (3.9) p p. This can be written as a right continuous and increasing cumulative distribution function F(m), which implies {m : F(m)) ≥ 1−p} = [m0,∞) for some element m0. This proves that the exist a small element m0 which is the Value-at-Risk. Alternatively, one can define VaR at the confidence level p ∈ (0,1) as the smallest number m such that the probability that a certain loss L exceeds m is not larger than 1−p (McNeil, Frey & Embrechts, 2005). Formally,. VaRp = inf {m ∈ R : P(L > m) ≤ 1 − p} (3.10) = inf{m ∈ R : FL(m) ≥ p}. This description of VaR1 means it can be defined as the quantile of the loss distribution. If a portfolio has a VaR of R1 000 000 over a week with a 95% confidence interval, there is only a 5% chance that the value of the portfolio will drop by more than R1 000 000 over any week. VaR is a predictive risk measure because investors are able to predict the loss on a portfolio for a specified fixed period ahead. As mentioned when discussing properties of risk measures, VaR is not a coherent risk measure because the sub-additivity axiom does not always hold 2 . The next subsection discusses the Conditional Value-at-Risk (CVaR) which is an alternative and an improvement. Infimum of a subset S of a partially ordered set T is the greatest element of T that is less than or equal to all elements of S. 2 VaR of a portfolio of with two assets can be greater than the sum of the two individual asset VaR’s 1. 21.

(27) on VaR because it is coherent and always satisfies the sub-additivity axiom, as shown by Rockafellar and Uryasev (2000).. 3.1.4. Conditional Value-at-Risk. Conditional Value-at-Risk (CVaR) or expected shortfall as it is also known, is an extension of VaR that takes into account the shape of the loss distribution in the tails. The words Conditional Value-at-Risk (CVaR) and Expected Shortfall are used interchangeably in this case. It calculates the likelihood of a certain loss exceeding VaR since relying only on VaR for risk estimation exposes the portfolio manager to major losses in the tail end of the loss distribution which are typically not accessed by VaR. Following the definition of VaR by McNeil et al (2005), given a loss L with E(L) < ∞ and loss distribution FL the expected shortfall at confidence level p ∈ (0,1) is defined as:. (3.11) where qu(FL)du is the quantile function of FL. It can be shown how this is an extension of VaR by; (3.12) CVaR tries to look further into the tail of the loss distribution by averaging Value-at-Risk over all levels where u ≥ p instead of using a certain confidence level p. CVaR was designed to take into account the tail risk that VaR cannot, and measures the risk of extreme losses. The main advantage of CVaR over VaR is that it quantifies tail risk and can capture the probability of great loss for a portfolio with an asymmetric risk profile such as a portfolio with options (Kidd, 2012).. 3.1.5. Modelling and forecasting of risk measures. This paper uses predictive risk measures in the application of risk-based asset allocation methods and this means good forecasting models for the covariance matrix are needed. Predictive risk measures are used in the application of risk-based asset allocation methods and this means that good forecasting models are needed for the covariance matrix. The importance of using the forward looking/predictive approach in this case is that one can avoid problems caused by relying too much on historical returns. Relying too much on historical data in portfolio allocation can result in over-estimation of weights due to a strong past performance of an asset which does not necessarily mean it will have the same performance in the future. This is also because it bridges the gap between theoretical backtesting of portfolio construction techniques and the practical use of these techniques. A portfolio manager is more interested in what will happen to the value of his portfolio in the future rather than the historical performance of individual stocks or asset allocation techniques, hence our effort to forecast risk measures to find estimates with minimum bias. The mean-variance optimisation needed the estimation of expected returns and variance, 22.

(28) while risk-based asset allocation methods need the estimation of risk inputs such as variances, covariance and VaR. This means that the research will focus on forecasting the risk measures under consideration for portfolio construction, and decide which risk measures are more appropriate for the risk-based approaches. In forecasting risk measures, different volatility models such as multivariate Garch models and the Hayishi-Yoshida covariance estimator as well the maximum likelihood estimation of the covariance are explored.. Maximum likelihood covariance estimation In estimating the portfolio covariance matrix using the maximum likelihood method, the normal distribution assumption for returns of assets in the portfolio is made i.e. it is assumed that a vector of returns R is a k ×1 Gaussian vector with R ∼ N(µ,Σ). Given k assets in a portfolio, the k ×k positive definite covariance matrix Σ can be found using the probability distribution function of returns, R; .. (3.13). Suppose there is a sample of observed historical returns for each asset in the portfolio (R1,...,Rn) and one wants to estimate the covariance matrix Σ. (2 normalises the probability density function to make sure it integrates to 1. The first step will be to find the likelihood function which is given by; .. (3.14). The maximum likelihood estimate for the mean vector µ is given by the sample mean R¯ which is simply the average of n returns hence Equation 3.14 can be substituted. The log-likelihood is given by;. .. (3.15). The concentrated likelihood can be described using the trace property, tr(AT B)= tr(BAT ) as follows;. tr(Σ−1(Ri − R¯)(Ri − R¯)T )). (3.16). tr(Σ−1S) where S is the k × k positive definite matrix (assuming there exist a subset of data of k linearly independent observations) which is called the scatter matrix. S is given by; n. 23.

(29) S = X(Ri − R¯)(Ri − R¯)T . i=1. It can be deduced that the maximum likelihood estimator of Σ satisfies the first order condition;. and it follows that the estimator of the covariance matrix is the empirical matrix given by. . This estimator of the covariance matrix Σ is simple and easy to calculate because all that isˆ required is a sample of historical returns of all the assets in a portfolio. It is not the most sophisticated estimator of the covariance matrix and its heavy reliance on historical data can result in biased and inefficient variances and covariance values and this calls for better methods of estimating covariances.. Multivariate GARCH Models The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model of Bollerslev (1986) is used to forecast and model market risk in financial markets. It is used to model financial time series which exhibit volatility clustering over a period of time as well as a leptokurtosis (distribution with excess kurtosis). There have been many variations and extensions of the GARCH model because the standard GARCH model does not take into account leveraged effects (the negative relationship between an asset return and changes in volatility), and it does not account for the non-negativity constraints of the variance. Variations and extensions of the original GARCH model include the GLR-GARCH, the Integrated GARCH, the EGarch, the Asymmetric power ARCH (APARCH), and GARCH-M models. Risk-based portfolio construction requires an estimation of the covariance to determine the portfolio weights, hence the need to use multivariate GARCH models which help in modelling volatility in a portfolio made up of multiple assets. Some of the challenges in multivariate volatility modelling include making sure that the forecast covariance is positive definite, and making sure that the number of parameters to be estimated is limited, since a larger number of parameters complicates the process and delays the estimation of the covariance’s. The multivariate GARCH volatility model includes the following: BEKK (Baba, Engle, Kroner & Kraft, 1995), the constant conditional correlation (CCC) (Bollerslev, 1990), dynamic conditional correlation (DCC) (Engle & Sheppard, 2001), and vector volatility models. To illustrate multivariate volatility models, it is assumed that one models asset returns in a portfolio simultaneously using a vector autoregressive (VAR) model which is an extension of 24.

(30) univariate autoregressive model. Vector of returns (rt = (r1t,r2t,...,rnt)) of n assets at time t can be modelled using the VAR model as follows; rt = µt + at. 25. (3.17).

(31) where with ) with Ft−1 the sigma field1generated by the past information at time t−1. Ht = V ar(at|Ft−1) is a conditional covariance matrix2which takes the following format; h11,t . Ht =. .. .. h12,t h22,t . . .. . . . . .. . . . . .. . h1n,t . . . . . . .. (3.18). hnn,t hn1,t Vech Model The VECH volatility model is a natural multivariate extension of the standard GARCH model which is univariate and provides a full treatment of the covariance matrix Ht meaning one can estimate all parameters in the model even if there is a large number of parameters. The estimation of parameters of the VECH model (VECH-GARCH(1,1))is done using the maximum likelihood methods as in the univariate case although it can be difficult for the optimisation algorithm to converge when there are large parameters involved. To illustrate the VECH presentation, 2 variables case for simplicity are presented;. (3.19) The variance of one of the assets in the portfolio, say asset 1 can be written as follows ;. This variance equation shows that the variance of one asset in a portfolio does not only depend on its own previous lags and residual terms but also depends on the previous variance lags and residual terms of other assets within the portfolio. This is important as it allows the modelled variance to capture the dependence of assets within a portfolio. The estimation of parameters, β’s and α’s, is done using the maximum likelihood process. The number of parameters to be estimated depends on the number of assets in the portfolio. In a portfolio with 2 assets like this one, there are 3 equations like Equation 3.20 with 21 1. Sigma field Ft−1 on a set X is a collection of non empty subsets of X such that the following holds; i. X is. in Ft−1 ii. If a subset A is in Ft−1, then so is the complement of A iii. If An is a sequence of elements of Ft−1, then the union of the An’s is is Ft−1. 2. In the multivariate model, the denotation for the variance is ht instead of σt2. hijt is the covariance between. asset i and asset j at time t and is always equal to hjit.

(32) parameters to be estimated and in a portfolio with 10 assets , there will be 55 equations similar to 3.20 with 4025 parameters to estimate.. 25 Diagonal Vech In order to limit the number of parameters to be estimated while modelling multivariate volatility, the diagonal VECH mode can be used. Each element of the covariance matrix depends only on the past values of itself and the past values of the residual term. In a portfolio of 2 assets , the diagonal VECH reduces the number of parameters to be estimated from 21 to only 9 parameters although this comes at the expense of ignoring important information such as covariances between different assets in a portfolio. The diagonal VECH representation is as follows;. (3.21) The variance of asset 1 can be written as follows ; (3.22) BEKK Model The BEKK model was proposed by Engle et al. (1995) to impose positive definiteness restrictions and allows for dependence if conditional variances of one variable on lagged values of another variable. The number of parameter to be estimated is less than that of the VECH model but more than the diagonal VECH. The BEKK model provides a different representation from the VECH model. For a 2 assets portfolio, the BEKK model covariance Ht is as follows;. (3.23) 27.

(33) The BEKK model is a special case of the VECH because a BEKK representation can be found by multiply both side of the variance and the residual matrices by the coefficients matrices in the VECH representation to get the BEKK model representation. CCC Model The constant conditional correlation (CCC) representation was proposed by Bollerslev(1990) and assumes that the conditional correlation matrix is constant overtime. The CCC model is preferred over other multivariate volatility models because of its simplicity in computation although the assumption of constant correlations can be too restrictive. The model uses a different approach to VECH and BEKK in the sense that it decomposes the covariance conditional variances and conditional correlations which are assumed to be equal. One of the disadvantages of the CCC model is that it is not able to capture interactions between different assets over time in the portfolio because the correlations are constant. It can be shown that the covariance matrix Ht of a 2 assets portfolio under the constant conditional correlation assumption as follows;. (3.24) DCC Model The dynamic conditional correlation (DCC) extends the constant conditional correlation by introducing dynamics to the conditional correlations so that the conditional correlation matrix is no longer constant. It is one of the most important multivariate GARCH model used because it does not make the constant correlation assumption and its easy to use in practice. The DCC model is used in the empirical studies hence its important to have a clear definition and understanding of this model. Engle and Sheppard (2001) and Engle (2002) introduced the idea of dynamic conditional correlation model as a new class of multivariate GARCH models with the ability to estimate large time varying variance covariance matrices. The return of an asset at time t in a univariate case is given by ; rt = µt + at, where at is a residual term(mean-corrected return) which is equal to case,. . In a multivariate. where Ht is an n × n matrix of conditional variances of at given by ; Ht = DtRtDt. (3.25). where Dt is an n × n diagonal matrix of conditional standard deviations of at and Rt is the conditional correlation matrix of at with the same dimensions as can be found by 1 the Cholesky decomposition of Ht. The independent and identically distributed error, t is now an n × 1 vector errors of with properties; ]=0. and 28. (3.26).

(34) The conditional standard deviations matrix, Dt is the same as in the CCC model which means it can written as follows;. (3.27) 1The. covariance matrix matrix is symmetric positive definite hence it can be written as;. . √ which implies. 1/2. Rt D t = H t. but the conditional correlation matrix, Rt is time varying, 1 ρ21,t Rt =. ρ31,t .. ρ12,t ρ13,t 1 ρ23,t ρ32,t 1 . . . ρn2,t. . . . . . . . . . ρn(n−1),t. ρ1n,t. .. ρ2n,t. (3.28). ρ(n−1)n,t 1. ρn1,t It is important to note that, Rt has to be positive definite to ensure that the covariance matrix Ht is positive definite. It is known that Dt is positive definite because all the diagonal entries are greater than 0. Other than the fact that Rt has to be positive definite, the elements of Rt need to be between 0 and 1 since they are correlations between two assets. It is important that these two requirements are guaranteed to be true at all times in the DCC-GARCH model. This can be achieved by decomposing Rt into; ,. (3.29). where, ,. (3.30) where S is the covariance of the standardized errors, ], whereas ω and γ are ∗−1 scalars. Q is a diagonal matrix with the inverse of diagonal elements of Qt;. 29.

(35) (3.31) To make sure that Qt is positive definite and ensure positive variances, some conditions have to be imposed on the scalars ω and γ; ω ≥ 0,. γ≥0. and. ω+γ>1. . In addition to these constraints, one also needs to ensure that the starting value of Qt (when t = 0) is positive definite. Although only a DCC(1,1)-GARCH model will be looked at in this case, the correlation structure results can be extended to the general DCC(M,N)-GARCH model; .. (3.32). The elements of Ht = DtRtDt can simply be written as ; [Ht ]ij = phithjtρij (3.33) where ρij is the correlation between asset i and asset j at a particular time t and ρii = 1. The values of hit can be found using a univariate GARCH model of different orders although the simple GARCH(1,1) model if often adequate (Orskaug,2009); Q. P. hit = αi0 + Xαiqa2i,t−q + Xβiphi,t−p q=1. (3.34). p=1. The next step is to show how one can estimate parameters of a DCC GARCH model for normally distributed (Multivariate Gaussian) standard errors, t. The joint distribution of the standardized errors is given by:. (3.35) using the results of Equation 3.26. The likelihood of the residual term using linear transformation of variables and is ;. can be found. , where θ represents the parameters of the model which can be written as θ = (φ,ψ) =. 30. (3.36).

(36) (φ1,φ2,...φn,ψ),where φi = (α0i,α1i,...,β1i,...,βpi) are the parameters of a univariate GARCH model asset i ∈ (1 : n) and ψ = (ω,γ) are the parameters of the correlation structure as given in Equation 3.30. The log-likelihood of the above likelihood can be written as;. (3.37) Ht = DtRtDt can be substituted in the above equation to get;. (3.38) In order to use the above log-likelihood to estimate the DCC GARCH parameters, a 2 stage estimation process is used since it is difficult to find estimates at one go. The first stage or step gives the estimate of the first group of parameters φi = (α0i,α1i,...,β1i,...,βpi). For a detailed explanation of the steps and the estimation procedure, see Engle and Sheppard (2001). In the first step of the estimation process, Rt is replaced with the identity matrix In in Equation 3.38 which results in the log-likelihood;. (3.39) The result of Equation 3.39 is a quasi-likelihood since it is similar to the log-likelihood function but does not correspond to to any probability distributions. The second step of the estimation process is the estimation of the parameters of the correlation structure, ψ = (ω,γ). In this step, quasi-likelihood in Equation 3.38 is used;. 31.

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