Research Collection7076/eth... · Abstract This Master Thesis deals with the most important...

Research Collection

Master Thesis

Optimal PortfoliosThe Benefts of Advanced Techniques in Risk Management andPortfolio Optimization

Author(s): Meng, Nicolas

Publication Date: 2013

Permanent Link: https://doi.org/10.3929/ethz-a-009900690

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Swiss Federal Institute of Technology Zurich Seminar forStatistics

Department of Mathematics

Master Thesis Spring 2013

Nicolas Meng, CFA, M.A. HSG

Optimal Portfolios -

The Benefits of Advanced Techniques in Risk

Management and Portfolio Optimization

Submission Date: May 31th 2013

Co-Adviser Dr. Markus KalischAdviser: Prof. Dr. Sara van de Geer

iii

To my family and friends.

iv Abstract

Abstract

This Master Thesis deals with the most important challenges facing practitioners in port-folio and risk management. It embeds a variety of risk- and optimization methodologiesinto a common framework and performs an empirical backtest on a typical sector rotationstrategy in the US market. The objective of this study is to evaluate the impact of wrongassumptions in risk modeling and portfolio optimization, as a recent survey showed thatpractitioners are still using simplified approaches based on wrong assumptions, despiteempirical evidence that contradicts their assumptions.

This thesis embeds a variety of risk and optimization methods into a common frameworkand performs an empirical backtest on a typical sector rotation strategy in the US mar-ket. First, we apply different risk forecast models to the empirical data. Apart from anunconditional model still prominently practiced, a constant conditional correlation (CCC)and dynamic conditional correlation (DCC) model are implemented and the forecastingperformance is evaluated on the risk measures of volatility, VaR, and CVaR. There isclear empirical evidence that the unconditional model performs poorly and lead to severeunderforecasting and clustering of loss during the financial crisis of 2008. The more com-plex DCC model provided the most accurate forecasts, followed by the CCC model. Thisdemonstrates that wrong model assumptions lead to unacceptable results in practice.

Based on forecasts from all risk models, two optimization approaches are tested. Anadapted version of the traditional mean-variance optimization is employed. Additionally,a relatively new method of diversification optimization is implemented and comparedagainst return maximization, subject to a CVaR constraint. Using this comparison, weexamine the effect of estimation error on the expected returns and risk parameters. Asa diversification approach is invariant to the estimates of expected returns, we assumethat it should provide more stability to an optimized portfolio. We were able to confirmthe concerns about estimation error and found that return maximization does not lead tooptimal portfolios out-of-sample. In contrast, the empirical results of the diversification-CVaR strategy are promising. Maximum diversification of independent risk factors leadsto better performance in terms of both, realized risk and returns. In light of these findings,we question the practice of using the traditional method of return maximization, as thecost of ignoring estimation error in the optimization seems to be significant.

Finally, we conclude that the standard approach still followed by a majority of practition-ers does not deliver satisfactory results due to wrong assumptions about the statisticalproperties of the financial markets. We conclude that conditional risk estimates and theproblem field of estimation errors are important aspects that cannot be neglected solelyfor the sake of simplicity.

v

vi CONTENTS

Contents

1 Introduction 11.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Risk Modeling Framework 52.1 Notations, Loss Operators and its Distributions . . . . . . . . . . . . . . . . 52.2 Risk Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Unconditional Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Conditionality in Financial Markets . . . . . . . . . . . . . . . . . . 92.2.3 Conditional Heteroscedasticity in Financial Markets . . . . . . . . . 102.2.4 Constant Conditional Correlations . . . . . . . . . . . . . . . . . . . 152.2.5 Dynamic Conditional Correlations . . . . . . . . . . . . . . . . . . . 16

2.3 Projecting the Portfolio Loss Distribution . . . . . . . . . . . . . . . . . . . 192.3.1 Unconditional Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Constant Conditional Correlations . . . . . . . . . . . . . . . . . . . 202.3.3 Dynamic Conditional Correlations . . . . . . . . . . . . . . . . . . . 212.3.4 Improving Simulation Accuracy by Moment Matching . . . . . . . . 21

2.4 Mapping the Risk Factors into Portfolio Loss . . . . . . . . . . . . . . . . . 22

3 Portfolio Optimization Framework 253.1 General Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.2 Value at Risk (VaR) . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.3 Conditional Value at Risk (CVaR) . . . . . . . . . . . . . . . . . . . 28

3.3 Mean-CVaR Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.1 Minimum Risk for Given Expected Return . . . . . . . . . . . . . . 283.3.2 Maximum Return for Upper Risk Boundary . . . . . . . . . . . . . . 303.3.3 Smooth Approximation Approach . . . . . . . . . . . . . . . . . . . 313.3.4 Reliable Estimation of Expected Returns . . . . . . . . . . . . . . . 32

3.4 Maximally Diversified Portfolios . . . . . . . . . . . . . . . . . . . . . . . . 333.4.1 Risk Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4.2 Diversification Distribution . . . . . . . . . . . . . . . . . . . . . . . 353.4.3 Conditional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Risk Forecast and Optimization Evaluation Framework 394.1 Volatility Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.1 Volatility Forecast Bias . . . . . . . . . . . . . . . . . . . . . . . . . 404.1.2 Q-Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Value at Risk (VaR) Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Conditional Value at Risk (CVaR) Forecasts . . . . . . . . . . . . . . . . . . 424.4 Optimization Strategy Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Backtesting Framework 455.1 Rolling Window Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Methods, Materials and Parameters . . . . . . . . . . . . . . . . . . . . . . 47

5.2.1 Development Environment . . . . . . . . . . . . . . . . . . . . . . . 475.2.2 Raw Asset Prices and Risk Factors . . . . . . . . . . . . . . . . . . . 47

CONTENTS vii

5.2.3 Risk Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 485.2.4 Projection and Mapping to Portfolio Loss . . . . . . . . . . . . . . . 485.2.5 Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2.6 Risk Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6 Backtesting Results 516.1 Volatility Based Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2 Value at Risk (VaR) Based Forecasts . . . . . . . . . . . . . . . . . . . . . . 526.3 Conditional Value at Risk (CVaR) Based Forecasts . . . . . . . . . . . . . . 536.4 Optimization Strategy Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 54

7 Discussion 577.1 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Bibliography 60

A Tables and Figures 65A.1 Summary Statistics of the Underlying Assets . . . . . . . . . . . . . . . . . 65A.2 VaR Evaluation for the Equal Weight Portfolio . . . . . . . . . . . . . . . . 66A.3 VaR Evaluation for the Mean-CVaR Optimized Portfolios . . . . . . . . . . 69A.4 VaR Evaluation for the Diversification-CVaR Optimized Portfolio . . . . . . 71A.5 Realized Returns and Drawdowns . . . . . . . . . . . . . . . . . . . . . . . . 73A.6 Realized Loss vs Risk Forecast for Equal-Weight Portfolios . . . . . . . . . 78A.7 Realized Loss vs Risk Forecast for Optimized Portfolios . . . . . . . . . . . 81A.8 Sample Autocorrelations of GARCH residuals . . . . . . . . . . . . . . . . . 86

viii LIST OF FIGURES

List of Figures

2.1 Sample autocorrelation function of the Consumer Discretionary SPDR ETF(XLY) between the time period of 1999 - 2012. . . . . . . . . . . . . . . . . 10

2.2 Sample autocorrelation function of the squared risk factor changes of theConsumer Discretionary SPDR ETF (XLY) between the time period of 1999- 2012. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Sample autocorrelation function of the squared standardized residuals ofthe Consumer Discretionary SPDR ETF (XLY) between the time period of1999 - 2012. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Conditional analysis of the variance of eigenportfolios λi based upon a bud-get constraint. The letter u indicates the unconstrained eigenportfolio,whereas c refers to the constrained portfolios. The analysis was performedon date 12/28/2001. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Cumulative contribution of the conditional eigenportfolios to total varianceon date 12/28/2001. Note that the first 4 eigenportfolios explain approxi-mately 80% of the dispersion of the underlying assets. . . . . . . . . . . . . 38

5.1 Step-wise approach implemented to estimate a risk model, optimize portfo-lios and evaluate the performance. . . . . . . . . . . . . . . . . . . . . . . . 46

A.1 Performance summary of the mean-CVaR optimized strategy (black) underthe unconditional risk model versus the equal weight benchmark portfolio(red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.2 Performance summary of the mean-CVaR optimized strategy (black) underthe CCC-normal risk model versus the equal weight benchmark portfolio(red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.3 Performance summary of the mean-CVaR optimized strategy (black) underthe CCC-t risk model versus the equal weight benchmark portfolio (red). . 74

A.4 Performance summary of the mean-CVaR optimized strategy (black) underthe DCC-normal risk model versus the equal weight benchmark portfolio(red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

A.5 Performance summary of the mean-CVaR optimized strategy (black) underthe DCC-t risk model versus the equal weight benchmark portfolio (red). . 75

A.6 Performance summary of the diversification-CVaR optimized strategy (black)under the unconditional risk model versus the equal weight benchmark port-folio (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

A.7 Performance summary of the diversification-CVaR optimized strategy (black)under the CCC-normal risk model versus the equal weight benchmark port-folio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

A.8 Performance summary of the diversification-CVaR optimized strategy (black)under the CCC-t risk model versus the equal weight benchmark portfolio(red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

A.9 Performance summary of the diversification-CVaR optimized strategy (black)under the DCC-normal risk model versus the equal weight benchmark port-folio (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

LIST OF FIGURES ix

A.10 Performance summary of the diversification-CVaR optimized strategy (black)under the DCC-t risk model versus the equal weight benchmark portfolio(red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

A.11 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the equalweight strategy under the unconditional model. . . . . . . . . . . . . . . . . 78

A.12 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the equalweight strategy under the CCC-normal model. . . . . . . . . . . . . . . . . 78

A.13 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the equalweight strategy under the CCC-t model. . . . . . . . . . . . . . . . . . . . . 79

A.14 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the equalweight strategy under the DCC-normal model. . . . . . . . . . . . . . . . . 79

A.15 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the equalweight strategy under the DCC-normal model. . . . . . . . . . . . . . . . . 80

A.16 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the mean-CVaR optimized strategy under the unconditional model. . . . . . . . . . . 81

A.17 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the mean-CVaR optimized strategy under the CCC-normal model. . . . . . . . . . . . 81

A.18 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the mean-CVaR optimized strategy under the CCC-t model. . . . . . . . . . . . . . . 82

A.19 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the mean-CVaR optimized strategy under the DCC-normal model. . . . . . . . . . . . 82

A.20 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the mean-CVaR optimized strategy under the DCC-normal model. . . . . . . . . . . . 83

A.21 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the diversification-CVaR optimized strategy under the unconditional model. . . . . . . . . . . 83

A.22 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the diversification-CVaR optimized strategy under the CCC-normal model. . . . . . . . . . . . 84

A.23 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the diversification-CVaR optimized strategy under the CCC-t model. . . . . . . . . . . . . . . 84

A.24 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the diversification-CVaR optimized strategy under the DCC-normal model. . . . . . . . . . . . 85

A.25 Risk forecast (purple=VaR, blue=CVaR) versus realized loss for the diversification-CVaR optimized strategy under the DCC-t model. . . . . . . . . . . . . . . 85

A.26 Sample autocorrelations of GARCH residuals for the asset XLY. . . . . . . 86A.27 Sample autocorrelations of GARCH residuals for the asset XLP. . . . . . . 86A.28 Sample autocorrelations of GARCH residuals for the asset XLE. . . . . . . 87A.29 Sample autocorrelations of GARCH residuals for the asset XLF. . . . . . . 87A.30 Sample autocorrelations of GARCH residuals for the asset XLV. . . . . . . 88A.31 Sample autocorrelations of GARCH residuals for the asset XLI. . . . . . . . 88A.32 Sample autocorrelations of GARCH residuals for the asset XLB. . . . . . . 89A.33 Sample autocorrelations of GARCH residuals for the asset XLK. . . . . . . 89A.34 Sample autocorrelations of GARCH residuals for the asset XLU. . . . . . . 90

x LIST OF TABLES

List of Tables

2.1 Fitting results of a GARCH(1,1) model to all assets. . . . . . . . . . . . . . 132.2 Fitting results of a GARCH(2,2) model to all assets. . . . . . . . . . . . . . 14

6.1 Bias and q-stat for the volatility forecasts for the assets and the equal weightportfolio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2 Results of a statistical test for CVaR forecasting accuracy for the optimizedportfolios. The CVaR forecast bias is denoted E [Lt+1 − CV aRt+1]. . . . . . 53

6.3 Results of a statistical test for CVaR forecasting accuracy, applied to theoptimized portfolios. The CVaR forecast bias is denoted E [Lt+1 − CV aRt+1]. 53

6.4 Performance summary statistics of the mean-CVaR optimization strategyfor a target CVaR of 1.5%. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.5 Performance summary statistics of the diversification-CVaR optimizationstrategy for a target CVaR of 1.5%. . . . . . . . . . . . . . . . . . . . . . . 55

A.1 Daily summary statistics for the Select Sector SPDR ETFs for the timeperiod of 01/01/1999 until 08/31/2012. . . . . . . . . . . . . . . . . . . . . 65

A.2 Expected versus actual violations of VaR for the equal weight portfoliounder the unconditional model. . . . . . . . . . . . . . . . . . . . . . . . . . 66

A.3 Expected versus actual violations of VaR for the equal weight portfoliounder the CCC-normal model. . . . . . . . . . . . . . . . . . . . . . . . . . 66

A.4 Expected versus actual violations of VaR for the equal weight portfoliounder the CCC-t model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

A.5 Expected versus actual violations of VaR for the equal weight portfoliounder the DCC-normal model. . . . . . . . . . . . . . . . . . . . . . . . . . 67

A.6 Expected versus actual violations of VaR for the equal weight portfoliounder the DCC-t model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.7 Expected versus actual violations of VaR for the mean-CVaR optimizedportfolio under the unconditional model. . . . . . . . . . . . . . . . . . . . . 69

A.8 Expected versus actual violations of VaR for the mean-CVaR optimizedportfolio under the CCC-normal model. . . . . . . . . . . . . . . . . . . . . 69

A.9 Expected versus actual violations of VaR for the mean-CVaR optimizedportfolio under the CCC-t model. . . . . . . . . . . . . . . . . . . . . . . . . 70

A.10 Expected versus actual violations of VaR for the mean-CVaR optimizedportfolio under the DCC-normal model. . . . . . . . . . . . . . . . . . . . . 70

A.11 Expected versus actual violations of VaR for the mean-CVaR optimizedportfolio under the DCC-t model. . . . . . . . . . . . . . . . . . . . . . . . . 70

A.12 Expected versus actual violations of VaR for the diversification-CVaR op-timized portfolio under the unconditional model. . . . . . . . . . . . . . . . 71

A.13 Expected versus actual violations of VaR for the diversification-CVaR op-timized portfolio under the CCC-normal model. . . . . . . . . . . . . . . . . 71

A.14 Expected versus actual violations of VaR for the diversification-CVaR op-timized portfolio under the CCC-t model. . . . . . . . . . . . . . . . . . . . 72

A.15 Expected versus actual violations of VaR for the diversification-CVaR op-timized portfolio under the DCC-normal model. . . . . . . . . . . . . . . . . 72

A.16 Expected versus actual violations of VaR for the diversification-CVaR op-timized portfolio under the DCC-t model. . . . . . . . . . . . . . . . . . . . 72

Chapter 1

Introduction

Predicting stock market returns has always been a popular subject among academics andpractitioners, as this particular prediction can immediately be turned into a financial profit.However, the efficient market hypothesis, which is still a fundamental part of modern Eco-nomics, contradicts the natural desire of traders, fund managers, and quantitative analystswho justify their salaries by beating the markets. Bachelier (1900) was the first to modelstock market movements as an unpredictable random walk in his notable thesis, ”Theoryof Speculation” and thus laid the foundations for modern finance. While containing manysimilarities to Bachelier’s work, it was Fama (1965) who developed the efficient markethypothesis, which is widely discussed in Economics and always challenged by academicsand practitioners alike. Generally, the EMH states that it is impossible, based on theinformation available prior to investment, to consistently outperform market returns on arisk adjusted basis (Meng, 2012).

More formally, in the theory of the efficient market hypothesis, asset returns are assumedto follow a martingale difference:

E [Rt | Ft−1] = 0 (1.0.0.1)

The sigma algebra Ft−1 represents the entire knowledge (filtering) at time t − 1. Hence,the EMH states that conditioned on the past, the expectation of asset returns will bealways zero and consequently can be considered a fair game. Although this assumptionis still widely used in Finance, particularly in derivative pricing, the EMH also has beenchallenged by contradictive empirical findings. For example, Lo and MacKinlay (2001)demonstrated that the EMH is not fully valid and cross autocorrelations could be partiallyused to predict future stock returns. However, condensing the available information into asuccessful investment strategy remains a significant task and only a few market participantsare able to consistently outperform a passive benchmark portfolio (Carhart, 1997).

In contrast to asset return prediction, the field of risk management uniquely concernsdispersion of loss over time. An investor is not only concerned about possible profit butalso about the associated potential loss. As investors can be considered a heterogenousgroup, it becomes evident that their risk tolerance is subject to considerable variation.For instance, a pension plan is very sensitive to potential future losses due to its legalobligations, whereas a wealthy private investor may be more risk tolerant. Since thefinancial crisis of 2008 and its severe effect on international wealth, risk management

1

2 Introduction

has become the most important task in portfolio management. While return prediction isdifficult due to the EMH, levels of dispersion as risk forecasts can be statistically modelled.

To put risk management into a statistical framework, we need to define a risk measureas a functional of a loss distribution (McNeil, Frey, and Embrechts, 2005). Apart fromtheoretical properties, it is important that this measure coincides with the investor’s intu-ition about risk. Due to the groundbreaking work of Markowitz (1952), the most popularnotion of risk has been variance as the second distributional moment and thus a measureof deviation from the mean. Although still widely used among financial practitioners, inthis thesis we will see that it can produce dangerous pitfalls. The relatively new measuresof Value at Risk (VaR) and Conditional Value at Risk (CVaR) partially overcome theweaknesses of the variance as a notion of risk.

Once the relevant risk measure has been defined, it is crucial to assign an appropriatedistribution to the random portfolio loss. However, as the portfolio is a combination ofseveral assets, it is necessary to write the portfolio loss as a function of a set of risk factors,whose multivariate distribution should be estimated. It is still very common among assetmanagers to assume that financial returns are iid. realizations over time. Hence, in orderto build a risk model, historical data needs to be fitted to an unconditional distributionalmodel.

While this approach seems straight-forward from a practitioners perspective, the assump-tion of iid. realizations as valid is questionable. Specifically, conditional heteroscedasticitywas widely found in financial markets and theoretically well described in the work ofBollerslev (1986). Intuitively, this effect means that large shocks in financial markets aredependent on the past history and likely followed by further large market movements.Furthermore, Engle and Sheppard (2001) introduced a conditional model for correlationsin financial markets.

While these effects are empirically well established for a variety of assets, there are stilldoubts about the practical relevance to asset managers, as the more complex approaches re-quire statistical modeling knowledge not always available to asset managers. For example,Amenc, Goltz, and Lioui (2011) surveyed risk modeling techniques among practitionersin Europe and found that 60% of the participants still rely on a simple sample-based es-timate of the covariance matrix. In light of this discrepancy between empirical evidenceand standards in practice, a central subject of investigation in this thesis is the effect ofconditional versus unconditional models in financial markets. In the presence of volatilityclustering, it is expected that the unconditional model will underperform its conditionalcounterpart. Furthermore, the same survey indicated that a majority of investors assumenormality in risk modeling. Therefore, we will also compare the investment performanceand risk forecasting accuracy as a result of different distributional assumptions.

Once the risk model has been defined and estimated, using historical data, it is crucial toperform an optimal allocation decision under the investor’s specific risk constraints. Thisimportant task is part of portfolio optimization, whose foundations were laid by the mean-variance framework of Markowitz (1952), in which he proposed optimizing the trade-offbetween expected portfolio return and variance. Rockafellar and Uryasev (2000) extendedthis approach to the CVaR risk measure that we prefer for theoretical properties andintuitiveness. However, as DeMiguel, Garlappi, and Uppal (2009) show in their empiricalstudy, it is questionable if these approaches can deliver acceptable results in the presence ofestimation error in the expected returns. A return maximization approach is theoretically

1.1 Outline 3

appealing if the expected asset returns are known a-priori. However, if a reliable estimationis not possible due to a high level of noise in the data, a maximization algorithm may failand even amplify the present estimation error. In this context, it is surprising that amajority of practitioners still follow this traditional approach, according to Amenc et al.(2011).

As summarized in Qian (2011), the risk parity approach, in contrast to the mean-varianceoptimization, focuses on the maximization of diversification and assumes that risky strate-gies are fairly rewarded in the market equilibrium. The main advantage of this relativelynew method is that expected returns do not need to be estimated and incorporated intothe optimization process. While there are a variety of risk diversification approaches, weimplemented the relatively new method of Meucci (2009a), as it decomposes risk intostatistically independent factors, which greatly facilitates the diversification optimization.Finally, we compare the diversification approach to the traditional mean-CVaR optimiza-tion and expect that due to the reduction of estimation error, the former will produce morestable portfolios that adhere better to risk constraints. In this study, we will also evaluatethe average realized return of both strategies. Whereas the mean-CVaR optimization bydefinition should deliver maximum expected returns, the diversification-CVaR approachis invariant to returns. However, if the realized return of the former does not significantlyexceed the latter, this indicates that the traditional return maximization approaches arenot viable in practice due to estimation error. In this case, a risk parity approach shouldbe the preferred option.

To compare the underlying risk models and optimization strategies, it is important to em-ploy a sound evaluation framework. We applied out-of-sample risk forecast methodologieson a univariate asset, as well as a multivariate portfolio level. Furthermore, we assessed theperformance over time to identify periods of risk clustering. In terms of risk measures, weapplied the volatility evaluation techniques, presented in Menchero, Morozov, and Pasqua(2013), as well as the statistical tests for VaR and CVaR, that were first introduced byMcNeil and Frey (2000). Finally, we evaluate the realized risk and return of each strategyagainst the imposed optimization constraints and expect that if the model is accurate, therealized numbers should closely match the investment constraints.

1.1 Outline

The remainder of this thesis is structured as follows. In chapter 2 the theory for allrisk models is elaborated and presented in a consistent framework. Furthermore, the riskprojection to the investment horizon will be outlined and the applied simulation approachwill be introduced. Chapter 3 is based on a set of simulated scenarios and presents theportfolio optimization framework. In particular, the mean-CVaR and diversification-CVaRmethods are outlined in detail. The evaluation methodology is presented in chapter 4 andcomprises several risk measures, such as volatility, VaR and CVaR. The entire empiricalbacktesting framework is outlined in chapter 5. The results are presented in chapter 6 andfinally discussed and interpreted in chapter 7.

4 Introduction

Chapter 2

Risk Modeling Framework

2.1 Notations, Loss Operators and its Distributions

To develop the methods of risk management and portfolio optimization described in thisthesis, we will firstly embed the stochastic nature of financial returns into a commonmathematical framework. Since this thesis focuses on Risk Management and all opti-mization methods deal with risk measures, this thesis uses the notation of McNeil et al.(2005) which reflects the widely accepted industry standards, as first introduced by theRiskMetrics Group.

Since the financial value of any asset over time is subject to a high degree of randomness, itseems natural to model assets as random variables whose domain is a classical probabilityspace. We denote the random forces, which drive asset prices, as the risk factors of therelevant market. In a second step, the asset prices are modeled as a function of therandom risk factors. These functions describe the deterministic relationship between theunderlying randomness of the market and the asset prices.

Vt,i = f(t,Zt)

Zt, a random vector of dimension d, represents the risk factors that determine the randompart of the asset price at time t. Furthermore, we define the multivariate change in riskfactors as Xt := Zt − Zt−1. For some measurable function f : R+ × Rd → R, Vti is thevalue of a financial asset i and is fully determined at time t, once the risk factors Zt areknown. We are now able to define the loss of our one asset portfolio at time t + 1 as thenegative difference of its value between time t+ 1 and t as follows:

Lt+1 = lt(Xt+1) (2.1.0.1)

whereas

lt(x) := − (f(t+ 1,Zt + x)− f(t,Zt)) (2.1.0.2)

lt : R→ R is called the loss operator which maps risk factor changes x into portfolio losses.Its linearized version may be obtained through a first-order approximation of the function

5

6 Risk Modeling Framework

f and is particularly useful if the computation of lt(x) is too complicated:

l∆t x) := −(ft(t,Zt) +

d∑i=1

fzi(t,Zt)xi

)(2.1.0.3)

l∆t (x) expresses the sensitivity to a small risk factor change x at the current risk factorlevel Zt through the partial derivatives fzi .

Since this thesis only encompasses a limited number of equity assets, we directly definethe logarithms of the asset prices as risk factors. In this rather simple case, there is aone-to-one mapping from a risk factor to an asset price.1

Zt,i := lnSt,i

In this specific case, the functional relationship can be easily inverted to St,i = exp(Zt,i)and it can be immediately seen that the stock price St,i is modeled as a combination ofthe random risk factor Zt,i and the deterministic relationship of the exponent. However,if we extend this perspective to the full range of asset types in the financial market, therelationship between asset prices and risk drivers is not that simple. For example, fixedincome security prices usually include a fixed retirement date, and therefore the pricesfollow a time-dependent pattern. However, if we define the underlying risk factor as thelogarithmic price, it is difficult to model it as a random variable due to the non-stationarity.In this case, yield to maturities are chosen as the underlying risk factors, since they exhibita stationary behavior. A good summary of different asset types and proposed risk factorscan be found in Meucci (2009b).

The transition from the single security to a portfolio of assets is not complicated as thevalue is calculated as a weighted sum of the individual asset prices, where λi represents thenumber of shares of an asset i. The portfolio value at time t may be easily calculated asa linear combination of share prices and yields Vt =

∑di=1 λiSt,i. Since we want to model

portfolio losses as a function of risk factors, we substitute St,i with exp(Zt,i) and apply(2.1.0.2) in order to obtain the portfolio loss at time t+ 1 as a function of the random riskfactor changes Xt+1:

Lt+1 = −d∑i=1

λiSt,i (exp(Xt+1,i)− 1) (2.1.0.4)

The linearized version using vector notation and applying (2.1.0.3) is displayed below:

L∆t+1 = −

d∑i=1

λiSt,iXt+1,i = −Vtw′X (2.1.0.5)

As previously stated, risk factors and their changes are modeled as random variables.Hence, the losses Lt and L∆

t , which are dependent on the random risk factor changesXt+1,i are also random and can be fully described by their probability distribution functionFL(l) = P (L ≤ l), which is not currently assigned any parametric model. As extensivelydescribed in Section 3.2, we are able to derive informative statistics S(FL) that we believe

1If dimensionality is very high, the estimation of multivariate risk factor distributions may becomeproblematic and may require modeling asset prices in a factor model, using a few common risk factors,such as interest rates and general market prices. Logarithms instead of share prices are used, followingstandards in Finance. The reason for this standard is that logarithmic returns better fit to distributionsthat are analytically trackable.

2.2 Risk Factor Models 7

capture the investor’s understanding of risk. As the most popular functional of the lossdistribution, many practitioners calculate the second central moment (Variance) whichthey interpret as portfolio risk, since it describes the average deviation from the mean.

The linear loss operator is particularly appealing in this context, since it allows us toexpress the portfolio loss as a linear combination of the random vector X which representsthe change in risk factor levels. Hence, the linearization allows us to more easily calculatestatistics of the portfolio loss distribution such as the first two moments:

E[l∆t (X)

]= −Vtw′µ and Var

(l∆t (X)

)= V 2

t w′Σw (2.1.0.6)

2.2 Risk Factor Models

In Section 2.1, the functional relationship between the portfolio loss Lt and the change inrisk drivers Xt was fully described trough (2.1.0.4). It becomes clear that in order to derivethe loss distribution of a stock portfolio, we need to understand the underlying multivariatedistribution of the portfolio’s risk factor changes Xt. Assigning a distributional modelis not a trivial task since the loss distribution and the consequent investment decisiondepend on these assumptions. For this reason, a substantial part of this thesis deals withthe appropriate selection of distributional models for Xt and its assessment. To be a goodcandidate, a model should be well aligned with empirical data. As historical stock marketdata is readily available, we can calculate a large number of empirical realizations of riskfactor changes through the relationships outlined in Section 2.1 as follows:

xt,i = lnst,i − lnst−1,i (2.2.0.7)

As in this case, the historical risk factor changes are realizations of a random variable, thenotation has changed. We would expect that an accurate risk model is widely in accor-dance with the historical realizations under the assumption of time-stationarity. Amongacademics and financial practitioners, it is safe to assume stock market returns to be sta-tionary over a short window of time.2 Even though not directly relevant for this thesis, itmust be noted that these stationarity assumptions are not true for logarithmic returns inother cases. As outlined in the previous Section, the distribution of bond returns changesover the life of the instrument. A summary of stationarity assumptions for various financialinstruments may be found in Meucci (2009b).

The requirement that a model fits the historical realizations of risk factor changes, leadsus to expect that it also accurately models the loss distribution through the functionalrelationship of (2.1.0.4). Hence, it is important to evaluate the accuracy of risk forecastsunder various distributional models, as outlined in this Chapter 4.

The following sections exhibit a set of different risk models with ascending complexity.There are two main dimensions to identify, the conditional property and the distributionalassumptions of the model’s residuals. We will briefly introduce a simple unconditionalmodel, which is still used among many financial practitioners due to its straight-forward

2Since economic and political conditions change over time, the stationarity assumptions may not applyfor stock returns in the long-term, as they are subject to market regime shifts. However, it is well establishedamong financial practitioners that equity returns may be considered stationary if the estimation windowis not excessively long. The problem of non-stationarity has been considered in the determination of theestimation window in this thesis.


implementation. In a subsequent step, we will introduce conditional models for both,univariate volatility and correlations. Various versions with different distributional as-sumptions will be investigated.

2.2.1 Unconditional Model

Apart from the equal-weight investment strategy, an unconditional normal model serves asan ideal benchmark throughout the risk modeling and portfolio optimization framework.As the number of parameters is fairly low and its estimation is straightforward, it is stillvery popular among unsophisticated investors. Conversely, more complex models mustdeliver a significant increase in risk forecast accuracy and investment performance. Theunconditional model in this thesis adheres to the following assumptions.

Xt := N (µ,Σ) iid. (2.2.1.1)

The model assumes that all logarithmic stock returns are iid. realizations of the multivari-ate normal distribution with mean µ and covariance matrix Σ. An important distinctionto the conditional models in this thesis is the property that realizations are consideredindependent over time. Due to this iid. assumption we may state:

pX(xt | xt−1) = pX(xt)

As in the majority of financial modeling problems, the parameters µ and Σ are unknownand must be estimated from historical data, as described in (2.2.0.7). Based on theseassumptions, we can use the maximum likelihood estimator, which coincides with thesample estimators for mean vector and covariance matrix. The notations follow Mardia,Kent, and Bibby (1979).

µX = n−1X′1 Σ = n−1(X− µX)′(X− µX) (2.2.1.2)

Note that this version of the covariance estimator is biased, which is corrected in thisthesis by applying factor n

n−1 .

A further convenient property of this simple unconditional model is the fact that it is easyto derive the linearized portfolio loss distribution. Revisiting the relationship of (2.4.0.4)and calculating the first two moments yields:

l∆t = −Vtw′x (2.2.1.3)

E[l∆t (X)

]= −Vtw′µ and Var

(l∆t (X)

)= V 2

t w′Σw (2.2.1.4)

As the change of random risk factors Xt follows a multivariate normal distribution, itslinear combination w′Xt is univariate normally distributed. Hence, the linearized portfolioloss exhibits a N

(−Vtw′µ, V 2

t w′Σw)

distribution. Note that the linear loss is merely afirst order approximation to the true loss function, which may lead to inaccuracies in riskestimation, particularly if the time horizon is long. For this reason, we aim at deriving theloss distribution, applying the exact functional relationship as stated in (2.1.0.4), which isdifficult to track analytically, particularly when moving to more complex models. For thisreason, we rely on an enhanced simulation approach for all risk models, which is outlinedin Section 2.3.


2.2.2 Conditionality in Financial Markets

While simplicity and a parsimonious set of parameters speak in favor of the previouslypresented unconditional model, the requirement of iid. realizations may be too strict andlead to inaccuracy in risk forecasts over time. For this reason, it is necessary to empiricallyexamine the underlying time series for the presence of time-conditionality. Indeed, ifpresent, we would expect that the conditional models will significantly outperform theunconditional benchmark model. Following again the notations of McNeil et al. (2005),we may write conditional models as follows:

P [Xt = xt | Ft−1] 6= P [Xt = xt] (2.2.2.1)

Hence, we would like to model the multivariate distribution of random risk factor changes,conditional upon the sigma algebra Ft−1 which represents all available information up topoint in time t− 1. Furthermore, we assume the conditional distribution is different fromthe unconditional case. In Finance, we imply that the history contains information, whichmay be used for making a better informed decision at a certain point in time. However,if this is not the case and the conditional distribution equals the unconditional case, wewould introduce unnecessary estimation error by extending the complexity of the model.Hence, it is crucial to detect time-dependency in empirical data, in order to select anappropriate model. Naturally, the general case of modeling the full dependency of dis-tribution P (Xt) on the past is not feasible in the presence of a limited set of data. Asa consequence, further structure must be introduced to estimate the model in practice.In this context, we focus solely on the first two moments of the distribution P (Xt) ascandidates for time dependency. An intuitive and effective approach for the detection ofconditionality are the autocorrelation plots of the underlying data. For this reason, itbecomes necessary to define the first two moments of the weakly stationary time series.For a more comprehensive and rigorous treatment of time series theory, we suggest thatthe reader refers to Tsay (2002), whose notation we adopt in this Section.

Definition 2.2.2.1. A weakly stationary time series has the first two moments

µ(t) = µ t ∈ Zγ(h) = γ(t− h, t) t, h ∈ Z

With weak stationarity we thus imply that the mean of a random variable Xt remainsconstant over time. Furthermore, the (auto-)covariance γ(h) between different pointsin time t, uniquely depends on the lag h. From the definition of the autocovarianceand the general relationship of covariance and correlation, we can directly derive theautocorrelation function as:

ρ(h) =γ(h)

γ(0)(2.2.2.2)

γ(0) represents the autocovariance at lag 0 and is equivalent to the unconditional varianceof the time series. In the presence of time conditionality of the risk factor changes, we wouldexpect to observe autocorrelations that are significantly different from zero. However, ascan be clearly seen from Figure 2.1, there is no evidence of non-zero autocorellations forthe sample asset XLY and the small exceedances of the confidence bounds beyond the first


0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Figure 2.1: Sample autocorrelation function of the Consumer Discretionary SPDR ETF(XLY) between the time period of 1999 - 2012.

lag, should be regarded as spurious realizations. We performed this visual inspection onall assets and no structural presence of non-zero autocorrelations could be found. Henceit is safe to claim that our assets follow a weak white noise process, which is defined aspossessing zero autocorrelations for all lags h greater than zero. Due to this empiricalresults in the specific case of the underlying assets, we decided that it is not necessary tomodel the time series mean of the assets using a conditional model such as the popularAutoregressive Moving Average Model.

However, we cannot automatically conclude that the risk factor changes follow a strictwhite noise process that is defined to be an iid. sequence, as zero autocorrelations donot necessarily imply stochastic independence. Indeed, plotting the autocorrelations ofthe squared risk factor changes X2

t in Figure 2.2 reveals evidence for higher-order timedependency. The positive autocorrelations suggest that the second moment of the condi-tional distribution P [Xt | Ft−1] is not independent from its past. In finance, this effectis often described as conditional volatility, which can be understood as the conditionalstandard deviation of the process at a certain point in time t. The following Section onconditional heteroscedasticity presents the most popular approaches in Econometrics tomodel conditional volatility.

2.2.3 Conditional Heteroscedasticity in Financial Markets

Although, we deal with 10 assets in this thesis, as a first step we establish a univariateconditional volatility model and then embed it into a multivariate context at a later stage.


0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Figure 2.2: Sample autocorrelation function of the squared risk factor changes of theConsumer Discretionary SPDR ETF (XLY) between the time period of 1999 - 2012.

As highlighted in Section 2.2.2, there is no empirical evidence for conditional means, thus,we did not apply an ARMA model. However, since this model is a defining block in time-series analysis and other models depend upon this, we briefly provide the definition. Thenotations are closely followed from Tsay (2002), who provides a comprehensive introduc-tion to financial time series analysis.

Definition 2.2.3.1. The general ARMA(p,q) model is defined as:

Xt = φ0 +p∑i=1

φiXt−i + Ut −q∑i=1

θiUt−i

whereas Ut is a white-noise process.

It can be seen that the risk factor changes Xt linearly depend on its previous realizationsXt−i. Parameter p hereby determines the the extent of past dependence (lag). Further-more, Xt also depends on the previous white-noise innovations Ut−i as well as its currentrealization Ut. This is a very rich class of models and many time series processes can bedescribed if conditionality in the mean of a time series needs to be modeled. However, asillustrated in Figure 2.1, the empirical autocorrelations of xt are not significantly differentfrom zero and as Tsay (2002) theoretically show, we would expect a exponential decay ofthe theoretical autocorrelation function of an ARMA process.

Nonetheless, as we have evidence for positive autocorrelations in quadratic risk factorchanges x2

t , we would apply the General Autoregressive Conditional Hetroskedacity (GARCH)model which shares many similarities with the ARMA process, outlined in (2.2.3.1).


Definition 2.2.3.2. The GARCH(m,s) model is defined as:

Xt = σtεt, σ2t = α0 +

m∑i=1

αiX2t−i +

s∑j=1

βjσ2t−j

whereas εt is an iid. process with mean 0 and variance 1.

The GARCH models, first proposed by Bollerslev (1986), introduce two principal time de-pendencies. The autoregressive part states that the variance σ2

t depends upon its formerrealizations σ2

t−1 through parameters βj . Furthermore, the model exhibits conditional het-eroscedasticity as the process’ variance depends on previous quadratic risk factor changesX2ti through parameters αi. These two building blocks of the GARCH model were the

result of the older Autoregressive Conditional Hetroskedacity (ARCH) introduced by En-gle (1982) and solely exhibited dependence upon the quadratic risk factor changes. Eventhough the ARCH model was groundbreaking to the discipline of econometrics, it hadsome practical weaknesses, particularly the necessity of incorporating many lag parame-ters of αi. For instance, Tsay (2002) showed that in order to adequately model the S&P500stock market index, one needs to parameterize 9 lags in order to appropriately describethe volatility process. In a multivariate context, this is certainly not feasible anymore andfor this reason, the richer class of GARCH models from (2.2.3.2), has been considered inthis thesis.

Prior to applying the model to empirical observations, it is useful to summarize importantproperties of the model. The results are taken from Tsay (2002) and partially followwithout precise proof. A rigorous theoretical derivation of the topic can be found inBollerslev (1986).

First, it is helpful to reformulate (2.2.3.2) in order to highlight the connection to theARMA process:

Theorem 2.2.3.3. A GARCH process can be written as a process of the squared riskfactor changes X2

t :

X2t = α0 +

max(m,s)∑i=1

(αi + βi)X2t−i + ηt −

s∑j=1

βjηt−j

Proof. i.) Choose ηt = X2t − σ2

t , thus σ2t = X2

t − ηt

ii.) For (i=0, ..., s) plug in σ2t−i = X2

t−i − ηt−i

It is not difficult to see that the new innovations have the same properties as in the ARMAprocess. The zero mean assumption may be checked by plugging in E [ηt] = E

[X2t − σ2

t

]and applying definition of Xt = σtεt . As the random variable εt has mean zero andvariance 1 the result follows immediately. It can also be verified that Cov (ηt, ηt−j) = 0 for

j ≥ 1, as plugging in yields Cov(σ2t ε

2t − σ2

t , σ2t ε

2t−j − σ2

t−j

)and by definition of (2.2.3.2)

εt are iid.

Based upon these results, the connection to the ARMA process in (2.2.3.1) follows imme-diately. We are able to represent the process of de-meaned squared risk factor changesthrough an ARMA model that is equivalent to the GARCH process, defined in (2.2.3.2).


Now it becomes clear that looking at the squared empirical observations was a useful em-pirical method to determine whether a GARCH model is appropriate. The graphic resultsin Figure 2.2 clearly indicate this is the case.

Based upon the similarity of the ARMA and GARCH processes, we may use the theoryon the former model to calculate unconditional moments.

Corollary 2.2.3.4. For instance, a GARCH process’s underlying volatility yields (withoutproof):

E[X2t

]= α0

1−∑max(m,s)

i=1(αi+βi)

Furthermore, in the particular case of a GARCH(1,1) model, which is the most commonlyused in financial applications, the kurtosis is larger than in an unconditional normal model,even if the shocks εt stem from a gaussian process.

Corollary 2.2.3.5. Provided that 2α21−(α1 +β1)2 > 0, the fourth moment of the GARCH

model is:

E[X4t ]

E[X2t ]

2 = 3(1−(α1+β1)2)1−(α1+β1)2−2α2

1> 3

This result appears to be particularly convenient for modeling financial asset returns,which in most cases exhibit fat tails. Whereas the unconditional normal model from Sec-tion 2.2.1 was not able to capture this distributional property, a GARCH(1,1) model maymore accurately fit the non-normality of the unconditional distribution, even if relyingon an analytically simple and trackable gaussian process for the innovations. In a first

Table 2.1: Fitting results of a GARCH(1,1) model to all assets.

α0 t(α0) p α1 t(α1) p β1 t(β1) p

XLY 0.01 3.91 0.00 0.08 9.45 0.00 0.92 116.71 0.00XLP 0.01 3.93 0.00 0.08 8.09 0.00 0.91 89.03 0.00XLE 0.04 3.68 0.00 0.07 8.64 0.00 0.92 93.91 0.00XLF 0.02 4.12 0.00 0.12 11.39 0.00 0.88 83.91 0.00XLV 0.02 4.60 0.00 0.10 9.37 0.00 0.89 79.19 0.00XLI 0.02 3.90 0.00 0.08 9.27 0.00 0.91 97.22 0.00XLB 0.04 4.12 0.00 0.08 8.80 0.00 0.90 84.74 0.00XLK 0.01 3.72 0.00 0.08 8.84 0.00 0.92 107.13 0.00XLU 0.02 4.10 0.00 0.11 9.40 0.00 0.88 68.68 0.00

attempt to model the univariate volatility dynamics, we fitted a GARCH(1,1) model tothe entire dataset for all assets. As the results in Table 2.1 uniformly demonstrate, thereis quite a strong and statistically significant presence of both components in the GARCHmodel. Despite performing multiple testing without proper significance level correction,the strong p values and its presence across all assets makes it safe to assume the presenceof GARCH effects at the lag 1. One may be now tempted to extend the model to furtherlags and in the case of statistical significance, incorporate them into the final risk model.However, in the context of multivariate models, the picture looks different. For example,the addition of a further lag would imply the estimation of an additional 18 parameters.As previously stated however, sparsity and simplicity are important characteristics for risk


0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Figure 2.3: Sample autocorrelation function of the squared standardized residuals of theConsumer Discretionary SPDR ETF (XLY) between the time period of 1999 - 2012.

models and if no clear empirical presence of an effect is observable, we may decide in favorof the simpler model. The results of a GARCH(2,2) fit is shown in Table 2.2. While forsome assets there is empirical evidence for GARCH effects at lag 2, the picture is not uni-form anymore and the significance is not as clear. In this light, we decided to incorporatethe parsimonious GARCH(1,1) into the following multivariate context. Finally, we want

Table 2.2: Fitting results of a GARCH(2,2) model to all assets.

α0 t(α0) p α1 t(α1) p α2 t(α2) p β1 t(β1) p β2 t(β2) pXLY 0.02 1.98 0.05 0.03 1.51 0.13 0.07 1.70 0.09 0.61 0.85 0.39 0.28 0.42 0.67XLP 0.01 3.15 0.00 0.06 3.59 0.00 0.06 2.20 0.03 0.52 1.73 0.08 0.36 1.30 0.19XLE 0.07 3.64 0.00 0.05 5.29 0.00 0.09 7.74 0.00 0.00 0.00 1.00 0.84 47.18 0.00XLF 0.03 3.99 0.00 0.14 9.86 0.00 0.00 0.00 1.00 0.69 6.53 0.00 0.17 1.80 0.07XLV 0.02 4.54 0.00 0.07 3.35 0.00 0.04 1.90 0.06 0.88 63.57 0.00 0.00 0.03 0.98XLI 0.03 4.01 0.00 0.02 1.51 0.13 0.11 6.08 0.00 0.46 3.04 0.00 0.39 2.79 0.01XLB 0.06 3.82 0.00 0.03 1.74 0.08 0.10 4.67 0.00 0.45 2.02 0.04 0.40 1.99 0.05XLK 0.02 3.58 0.00 0.02 1.12 0.26 0.10 5.27 0.00 0.50 2.30 0.02 0.37 1.82 0.07XLU 0.02 3.96 0.00 0.13 7.95 0.00 0.00 0.00 1.00 0.66 5.67 0.00 0.19 1.81 0.07

to briefly verify that the standardized squared residuals from the model are in accordancewith the iid. assumptions on εt and indeed do not exhibit autocorrelations. This wouldalso be an indication that a GARCH(1,1) model was able to sufficiently capture the effectsof conditional hetroskedacity. While a set of formal statistical tests does exist, becauseof the error inflation of multiple testing and the lack of intuitivity, we decided to visuallyinspect the empirical autocorrelation function and verify that the assumption of zero au-tocorrelation on the squared residuals is not violated. Figure 2.3 shows this as an examplefor the asset XLY and indicates that the positive autocorrelation could be removed. Thisgraph should be compared against the raw squared series of risk factor changes X2

t infigure 2.2, where autocorrelation was dominantly present. The same analysis has been


performed for the entire set of assets and its results are presented in Appendix A.8. Theonly asset which potentially violates the residual assumption is the Financial Sector ETF(XLF). However, the degree of violation is very low and may be the result of pure ran-domness. Since there were no significant autocorrelations present, we can finally considerthe GARCH(1,1) as a valid candidate for our univariate risk model.

2.2.4 Constant Conditional Correlations

Although the univariate GARCH(1,1) model is capable of mimicking the effects of condi-tional heteroscedasticity, this is only helpful in the single asset case. As in most financialapplications however, we deal with a portfolio of different assets, which are driven by amultivariate process of risk factors Xt. Hence, in order to produce accurate portfoliorisk forecasts, we need to accurately model the dependency between individual risk factorchanges. First, we will employ a model with an unconditional dependency structure. Wewill then relax this constraint in the next Section, where we introduce time-conditional de-pendency models. With no loss of generality, we again assume that the risk factor changesXt have mean equal to zero.

Definition 2.2.4.1. We can then define the multivariate generalization of a conditionalvolatility model as follows:

Xt = H1/2t Zt

whereas,

• Zt is a d× 1 vector of risk factor changes,

• H1/2t is the square root of a d× d conditional covariance matrix at time t,

• Zt an iid. d× 1 random vector with E [Zt] = 0 and Cov (Zt) = I.

Similar to (2.2.3.2), the amplitude of the risk factor changes depends upon the volatilityprocess at time t. However, in the multivariate context, σ2

t becomes covariance matrixHt. Indeed it is easy to check, that based upon this definition, the variance of Xt indeed

coincides with Ht as Var(H

1/2t Zt

)= H

1/2′

t IH1/2t .

This representation of multivariate conditional volatility processes is a general class ofmodels, since the time dependent structure of Ht may be defined in several ways. Theclassical decomposition of the covariance matrix into its diagonal variance components andthe correlation matrix, provides us with the flexibility to detach the correlation from thevolatility process. In the first step, we will keep the latter constant whereas the formermatrix is conditionally modeled.

Ht = DtRDt (2.2.4.1)

Dt is a diagonal matrix with non-zero elements h1/211,t, ..., h

1/2dd,t. The missing time subscript

of R here indicates that the correlation matrix is not dependent on time. This decom-position is very convenient as it allows us to describe the univariate volatility processesindependently. For instance, we may directly model the diagonal components as univariateGARCH processes, as defined in (2.2.3.2).


Definition 2.2.4.2. The uncorrelated individual GARCH(m,s) processes can be re-writtenin matrix notation:

diag(Ht) = α0 +m∑i=1

AiXt−i �Xt−i +s∑j=1

Bjht−j

Assuming that there are no volatility spill-over effects from one asset to another, Ai (d×d)and Bj (d× d) are required to be diagonal. Even though this condition could be relaxed,this would imply a quadratic explosion in the number of parameters to be estimated, whichis not feasible for the purpose of this thesis.

As a last step, we shall blend the GARCH volatilities in the diagonals of Dt with theunconditional correlation matrix R, as stated in (2.2.4.2). The main advantage of thismodel consists in the relative sparsity of parameters and simplicity for estimation of theunconditional correlations. However, it may be too restrictive to assume time-constant cor-relations, which may lead to inaccurate risk forecasts and portfolios. A similar estimationis performed on the more general DCC model and is presented in Section 2.2.5.

2.2.5 Dynamic Conditional Correlations

If we assume that time-conditionality is not only present in the diagonal volatility pro-cesses but also in the dependency structure, we need to relax the assumption of constantcorrelations, as implied in Section 2.2.4. While there is a huge variety of methodologiesfor modeling correlations, an important restriction in practice is sparsity in parameters.Therefore, it is not possible to model each correlation pair separately in time as this wouldimply a quadratic explosion in the number of parameters. An important breakthrough wasthe proposition of the dynamic conditional correlation GARCH model (DCC-GARCH) asfirst proposed by Engle (2002).

As the main distinction of the DCC to the CCC model consists in time-varying correlations,the decomposition of covariance matrix Ht reformulates to:

Ht = DtRtDt (2.2.5.1)

The subscript in the correlation matrix now indicates that it is no longer assumed to beconstant over time. While there are a variety of methods to model the time dynamics ofcorrelations, the following theoretical and practical restrictions need to be considered.

• In order to ensure the feasibility of parameter estimation, the complexity of thetime varying process should be limited. Hence, due to the number of pair-wisecorrelations, in practice it is not possible to model the time dynamics individually.

• For obvious reasons, the covariance matrix Ht of the process needs to always bepositive definite and symmetric.

• The absolute value of all elements of Rt must be limited to 1 in order to representa valid correlation matrix.

As the DCC-GARCH model fulfills all criteria, it has emerged among financial practition-ers as a popular solution for modeling time-varying correlations.


Definition 2.2.5.1. According to the proposition of Engle (2002), Rt is modeled as:

Rt = Q∗−1t QtQ

∗−1t

Qt = (1− a− b)Q + aεt−1ε′t−1 + bQt−1

Q: The unconditional covariance matrix of the standardized errors εtQ∗t =

√diag(Qt)

It can be seen that the standardization of Qt by multiplying with Q∗−1t ensures that Rt

is a correlation matrix with its elements bound to absolute value 1. Positive definitenessis guaranteed by restricting parameters a and b to be positive and its sum to be smallerthan 1. Furthermore, DCC yields a sparse model as a and b are the only parameters thatmodel the time dynamics in correlations.

Estimation

Due to the high number of parameters, the estimation of a DCC-GARCH model is nottrivial. While a two stage MLE is the most common estimation procedure, there havebeen proposals for bayesian approaches, leveraging Markov Chain Monte Carlo methods.A recent investigation in this field has been performed by M. Concepcion Ausin (2010).The bayesian approach would be particularly appealing for modeling estimation risk infinancial markets. However, the convergence of MCMC may be problematic and its successhas only been demonstrated in cases with an exemplary low number of assets. For thisreason, we decided to use the traditional MLE approach. In this Section, we briefly derivethe estimation algorithm for multivariate Gaussian distributed innovations. For furtherdistributions and more details, the work of Orskaug (2009) offers an excellent overview.We closely follow their notation and structure in this Section.

We assume that as in (2.2.4.1), the error terms Zt follow a multivariate normal distribution

with E [Zt] = 0 and Cov (Zt) = I. Hence, based on the relationship Xt = H1/2t Zt, the

likelihood function reads:

L(θ) =T∏t=1

1

(2π)d/2 |Ht|1/2exp(−1

2X′tH

−1t Xt) (2.2.5.2)

θ represents the set of model parameters and can be divided into the subsets (φ,ψ).φ = (φ1, ...,φd) contain the parameters from the univariate GARCH processes i = 1, . . . , d,such as φi = (α0i, α1i, β1i). ψ denotes the parameters for the conditional correlations anddecomposes into ψ = (a, b).

Following the standard procedure of MLE, we take logarithms and plug-in the decompo-sition of Ht = DtRtDt:

ln(L(θ)) = −1

2

T∑t=1

(dln(2π) + ln(|DtRtDt|) + X′tD−1t R−1

t D−1t Xt) (2.2.5.3)

= −1

2

T∑t=1

(dln(2π) + 2ln(|Dt|) + ln(|Rt|) + X′tD−1t R−1

t D−1t Xt) (2.2.5.4)


Due to the large number of parameters, simultaneous maximization of the likelihood func-tion does not seem feasible and a two stage procedure is applied instead. First, the param-eters of the univariate GARCH processes are estimated, ignoring the correlation structure.For this reason, we may set the conditional correlations Rt equal to the identity matrix I:

ln(L1(φ)) = −1

2

T∑t=1

(dln(2π) + 2ln(|Dt|) + X′tD−1t ID−1

t Xt) (2.2.5.5)

ln(L1(φ)) = −1

2

T∑t=1

(dln(2π) +n∑i=1

(ln(hit) +X2it

hit)) (2.2.5.6)

= −N∑i=1

(−1

2

T∑t=1

[ln(hit) +X2it

hit] + constant) (2.2.5.7)

(2.2.5.7) equals the sum of likelihood functions of univariate GARCH processes. This isa very convenient formulation for estimation, as we can express the likelihood function asa sum of N univariate processes. Hence, when taking derivate with respect to a process’parameter subset φi, the terms related to all other processes in the sum disappear. Weare then able to maximize the likelihood function of each process i individually. In theestimation algorithm, ht = α0 + α1x

2t−1 + β1ht−1 needs to be evaluated recursively. After

the estimation of parameters φi, the conditional volatilities hii,t can be calculated, usingthe same relationship and the estimated parameters. Based on these volatility estimates,we can calculate the residuals of the first step as εt = D−1

t Xt.

In a second step, the likelihood function is minimized with respect to ψ = (a, b), giventhe already estimated parameters φ and diagonal conditional volatility matrices Dt. Thisyields:

ln(L2(ψ) = −1

2

T∑t=1

(nln(2π) + 2ln(|Dt|) + ln(|Rt|) + X′tD−1t R−1

t D−1t Xt) (2.2.5.8)

ln(L2(ψ) = −1

2

T∑t=1

(nln(2π) + 2ln(|Dt|) + ln(|Rt|) + ε′tR−1t εt) (2.2.5.9)

The constant terms can be ignored in the likelihood maximization. Engle and Sheppard(2001) showed that this quasi-maximum likelihood procedure yields consistent and asymp-totically normal estimators. The same MLE process can be similarly derived for other er-ror distributions. In this thesis, the multivariate normal and student-t distributions wereapplied.

Testing for Dynamic Conditional Correlations

In order to justify the inclusion of a DCC model in the study, it would be advantageousto obtain empirical evidence that suggests a dynamic model for correlations, as it exhibitshigher complexity than the simple CCC model from Section 2.2.4. If in fact, correlationswere assumed to be constant, a dynamic model would unduly inflate the estimation errorand diminish the investment performance of such a model. Hence, we employ a test fordynamic conditional correlations that was proposed by Engle and Sheppard (2001) in their

2.3 Projecting the Portfolio Loss Distribution 19

work about the theoretical properties of DCC. The following testing procedure is proposed:

H0 : Rt = RHa : vechu(Rt) = vechu(R) + β1vech

u(Rt−1) + ...βpvechu(Rt−p)

In this context, vechu refers to the upper diagonal entries of the correlation matrix. Hence,we would like to reject the null hypothesis that the conditional correlation matrix Rt

is equal to its unconditional version R. For this reason, we perform a full parameterestimation of a CCC-GARCH model as specified in (2.2.5.1). We are then able to estimatethe residuals using the correlations and volatility estimates of the model:

εt = R−1/2D−1t xt (2.2.5.10)

Under the null hypothesis, it is expected that these residuals are iid. with covariancematrix I. As a next step we define the following auxiliary variable for our time-seriesregression as Yt = vechu(εtε

′t − I). Note that the dimension of Yt is d(d− 1)/2 for d risk

factors. Under the null hypothesis, Yt is uncorrelated over time. We therefore, performthe following time-series regression:

Yt = α+ β1Yt−1 + ...+ βsYt−s + υ (2.2.5.11)

As all target variables depend on the same scalars βi, the estimation of this linear-modelis quite simple. For each element in Yt, we have T samples from the available history.In order to construct the target variable sample vector in the regression, we simply stackr = d(d−1)/2 times the T samples of each element of the target variable yt. We denote thistarget sample vector u = (y′t1, ...,y

′tr)′, whereas yti represents the historical samples for

element i. For the lagged versions, yt−s we proceed similarly and construct the explanatorymatrix V that will have dimension Tr× (s+ 1). The parameters β then can be estimatedby applying least square regression. Under the null hypothesis of β = 0, the test statistic

t = βV′Vβ′

σ2u

follows an asymptotical distribution of χ2s+1.

For the assets in this study, even at a lag of only s = 1, we were able to reject thenull-hypothesis, obtaining a test statistic of t = 118.71. While the rejection of constantcorrelations does not necessarily imply that the DCC model will outperform in termsof risk forecast and investment performance, it certainly warrants incorporating a DCCmodel in this empirical study.

2.3 Projecting the Portfolio Loss Distribution

After constructing the risk models and estimating the parameters in the previous Sections,it is now important to project the random risk factor changes Xt to the desired investmenthorizon h, which will allow us in a subsequent step to derive the portfolio loss distribution,as outlined in the introductory notes to risk modelling in Section 2.1. It is important tonote that when formally deriving the projection, we assume that the parameters are known.However, in practice, as in this empirical study, the true parameters are replaced by theirestimates, according to the described fitting methods. As a function of these parameters,all subsequent steps, including projection, are also subject to estimation error.

More formally, we would like to obtain the multivariate distribution of risk factor changesat time t+ h:

P [Xt+h | Ft] (2.3.0.12)


The projection is performed conditional upon the sigma algebra Ft which contains all avail-able information up to the point in time t, at which we perform the projection. Generally,we may distinguish between an analytical and a simulation approach. While both meth-ods exhibit advantages and disadvantages, it depends on the practical problem setting tofinally determine the appropriate method. While in simple cases the analytical approachmay yield straight-forward exact solutions to a problem, it may become intractable if therisk model is complex or if in a subsequent step complicated functions of the random riskfactor changes are calculated, for which no closed form solution can be derived. On theother hand, while the simulation approach provides maximum flexibility when applied tocomplex models, it may be computationally expensive. As will become clear later in thisthesis, the functional relationships of our risk statistics will be of a complex nature. Wetherefore decided to employ the simulation approach for all risk models. Due to missingrelevance in this thesis, we do not present analytical derivations of the projection step,but rather focus on the simulation methodology applied in this thesis. Another importantfactor to consider in the projection step is the length of the investment horizon h andthe sampling frequency that was used for the model estimation. Specifically, if they aredifferent, this can present challenges. However, if the daily sampling frequency coincideswith the daily investment horizon, as in this study, it is not necessary to consider thepitfalls of this area.

For all models, the following projection step has been performed:

x∗t+1 = S(M,Ft,u∗) (2.3.0.13)

The above function states that our samples x∗t+1 of simulated risk factor changes are afunction of the applied risk model M to the past information Ft as well as the randomlygenerated innovations u∗ from the applicable distribution. Having provided this generaldefinition, we are able to derive the simulation approach for the projection step moreconcretely to all risk models in this study.

2.3.1 Unconditional Model

As defined in (2.2.1.1), the risk factor changes are iid. realizations of a multivariate normaldistribution with parameters µ and Σ. Due to the iid. property, the projection step of(2.3.0.13) is straight-forward and reads:

x∗t+1 = µ+ Σ1/2u∗ (2.3.1.1)

Note that Σ1/2 may be calculated using cholesky decomposition and u∗ are iid. standardnormal samples of a random generator.

2.3.2 Constant Conditional Correlations

According to the CCC model in (2.2.4.1), the risk factor changes are dependent on theconditional volatilities. Transforming this relationship to point in time t + 1, reads asfollows:

x∗t+1 = H1/2t+1u

∗ (2.3.2.1)

Here, u∗ may be either realizations of N (0, I) or t(0, I, υ), as both versions have beentested in this thesis. Note that in order to perform the above simulation, one must know

2.3 Projecting the Portfolio Loss Distribution 21

the predicted conditional volatility at time t+1. According to the CCC model in (2.2.5.1),this is decomposed as Ht+1 = Dt+1RDt+1. As the correlations are constant, we onlyneed to predict the diagonal of Ht+1, using the time conditional relationship of definition(2.2.4.2).

diag Ht+1 = α0 +m∑i=1

AiXt−i+1 �Xt−i+1 +s∑j=1

Bjht−j+1 (2.3.2.2)

Since the above relationship only depends on data available until point in time t, diag(Ht+1)may be easily calculated and applied to generate x∗t+1 according to (2.3.2.1).

2.3.3 Dynamic Conditional Correlations

The simulation algorithm for CCC models can be easily extended to dynamic conditionalcorrelations. Whereas x∗t+1 are also generated applying (2.3.2.1), the decomposition ofthe conditional covariance matrix changes to:

Ht+1 = Dt+1Rt+1Dt+1 (2.3.3.1)

The predicted diagonal variance matrices Dt+1 can be derived, using the same relationshipas in (2.3.2.2). In addition, the conditional correlation matrices need to be projected. Asthe parameters are assumed to be known or have been estimated, we can merely applydefinition (2.2.5.1) for point in time t+ 1:

Rt+1 = Q∗−1t+1 Qt+1Q

∗−1t+1

Qt+1 = (1− a− b)Q + aεtε′t + bQt

(2.3.3.2)

We see again that the conditional correlation matrix uniquely depends on informationwhich is available at time t. Despite the fact that we only used one step ahead predictionsin this study, it is not difficult to derive the distribution of longer horizons by relying onthe simulation approach, as the simulation step can be performed interactively over thedesired investment horizon h.

2.3.4 Improving Simulation Accuracy by Moment Matching

A strong advantage of the simulation approach for risk factor forecasts lies in the flexibilityof the approach. Using a set of generated scenarios, we can virtually calculate any riskmeasure and use it for portfolio optimization. However, as previously noted, this flexi-bility comes at the cost of accuracy. For instance, if we generate 1000 scenarios of theunconditional model in Section (2.3.1), it is expected that the sample mean and variancewill not exactly match the distribution parameters µ and Σ and thus will be subject toestimation error. As both estimators are consistent, this error will diminish with increas-ing simulation size. However, the scenario optimization approach from Section (3.3.1.1)is computationally expensive and we are unable to process an arbitrarily high number ofscenarios.

Meucci (2009c) presented a method that matches the first two moments of a set of sim-ulated samples to their simulation parameters, provided that the random draws originate


from an elliptical distribution. It is easy to verify this in all of our risk models, as theunderlying error distribution was either a multivariate normal or symmetric student-t dis-tribution. Furthermore, elliptical distributions are closed under linear transformations,

thus H1/2t Zt must be elliptical for the CCC and DCC model. The idea of the approach is

to match the first two theoretical moments with the sample counterparts from the randomsimulations. In this way, the simulated set becomes a more exact approximation to thetrue distribution in further calculations.

Subsequently, we assume that x∗i is a multivariate random draw from an elliptical distri-bution of mean m and variance S. First, we impose mean zero on our random samplesby defining yi = x∗i − n−1∑N

i=1 x∗i . In a next step, we apply a linear transformation suchthat afterwards the sample covariance matrix Sy matches the true covariance matrix Sythat was used in the random generator:

yi = Byi (2.3.4.1)

To satisfy the desired condition, based upon the equivariance of the covariance matrix wecan write the following Riccati equation:

S ≡ BSyB B ≡ B′ (2.3.4.2)

The detailed steps for solving the above Riccati problem are outlined in Meucci (2009c)or more generally in Petkov, Christov, and Konstantinov (1991). After B is calculated,the following linear transformation can be performed to ensure that the first two samplemoments of our simulations match with their theoretical counterparts:

xi = m + Byi (2.3.4.3)

2.4 Mapping the Risk Factors into Portfolio Loss

As stated in the introduction in Section 2.1, our ultimate goal is modeling the futuredistribution of the portfolio loss. As the change in risk factors is logarithmic return, it istempting to map the produced scenarios directly into the portfolio loss, using the linearloss operator from (2.1.0.3):

L∆t+1 = −

d∑i=1

λiSt,iXt+1,i = −Vtw′X (2.4.0.4)

However, it must be noted that this represents a major pitfall in portfolio management ashighlighted by Meucci (2010). Logarithmic returns as an artificial construct, are not equalto what the investor will finally receive and when incorporating as such into a portfoliooptimization, it will result in sub-optimal portfolios. Even though the investment horizonin this thesis is rather short, we decided to use the exact loss operator. Since we areapplying a simulation approach, we do not rely on the analytical convenience of lineartransformations. In order to calculate the portfolio loss, we merely need to revert the initialdefinition of the risk factor changes in (2.2.0.7) and multiply by the portfolio weights. The

2.4 Mapping the Risk Factors into Portfolio Loss 23

exact portfolio loss has already been exposed in (2.1.0.4) and we substitute the randomvariable by the modified (moment matching) simulation draw i:

lt+1,i = −d∑j=1

λjst,i(exp(xjt+1,i)− 1) (2.4.0.5)

In this representation, the index i refers to the particular realization from the multivariatesimulation, whereas j denotes the asset. For the sake of comparison, we would ratherwork with relative portfolio weights w and therefore standardize (2.4.0.5) with the totalportfolio value vt.

lt+1,i = −∑dj=1 λjst,i(exp(x

jt+1,i)− 1)

vt= −

d∑j=1

wt,j(exp(xjt+1,i)− 1) (2.4.0.6)

Hence, after applying the exponential to the randomly generated scenarios, the portfolioloss can be presented as a linear combination of the individual losses and the relativeweights. Particularly in portfolio optimization, this linear relationship will be very helpful.

It is easy to see that after the application of the exponential, the distribution is no longerelliptical and analytical tractability becomes difficult. Nevertheless, in the simulationapproach, the distribution of portfolio loss L can be approximated by a sufficiently highnumber of scenarios. Moreover, further complex functionals of the loss distribution, whichwill follow in chapter 3.2, can be numerically calculated without any difficulties.

Chapter 3

Portfolio Optimization Framework

3.1 General Portfolio Optimization

The foundations of portfolio optimization were established in the middle of the last cen-tury by the work of Markowitz (1952). His optimization model, which is also known asmean-variance optimization, has been popular in Finance for decades, despite its stronglimitations. Nevertheless, a variety of extensions and modifications have been proposedin literature. Instead of historically aligning them, we first present portfolio optimizationas a general framework in which the most interesting candidates for this thesis can beembedded and illustrated. Finally, we discuss the strengths and limitations of these can-didates, and take a position on two promising optimization approaches, which are testedin the empirical Section.

In general terms, portfolio optimization is defined as finding the optimal trade-off betweenthe expected portfolio return µ and a risk quantity S(FL), where S is supposed to be somerisk statistic that acts on the probability distribution function of the portfolio loss Lt asoutlined in Section 2.1.

The general optimization problem of an investor in its simplest form then reads:

arg minw

S(FL(w))

Subject to : w′µ = r(3.1.0.1)

A rational investor thus aims to minimize the risk of the portfolio, given a certain expectedreturn target that is equivalent to the negative expected loss, as outlined in (2.1.0.6).1

Depending on the investor’s perspective, this relationship can be easily inverted suchthat the expected portfolio return is maximized for a certain upper risk boundary. Theoptimization problem then reformulates to:

arg maxw

w′µ

Subject to : S(FL(w)) ≤ s(3.1.0.2)

The reformulation of (3.1.0.2) is particularly desirable, if the investor puts substantialemphasis on risk, as opposed to a fixed return target. As previously stated, in this thesiswe decided to employ this approach due to the focus on risk control.

1In order to prevent confusion about the switching terminology, we emphasize the obvious equivalenceof return and loss as Rt = −Lt.

25

26 Portfolio Optimization Framework

Unfortunately, not every feasible solution of the above optimization problem may be in-vestable in practice, due to liquidity or investor’s policy constraints. For example, short-selling (negative weights) of securities may be restricted and investment exposure canbe limited to the available funds. In this case, it is necessary to impose the followingrestrictions to the optimization problem:

wi ≥ 0; w′1 = 1 (3.1.0.3)

While the first constraint ensures that the asset weights remain above 0 and thus, short-selling is not possible, the second restriction ensures that all available funds are fullyinvested.

3.2 Risk Measures

In this general optimization framework, the risk statistic S, which depends on the portfolioloss distribution FL, has deliberately not taken any specific functional form yet. However,we will see that its choice is very important, as it finally determines the optimal solution.From a practical perspective, one would prefer a risk measure S that intuitively provides anunderstanding of the extent of investment risks in the underlying probability distributionfunction of the portfolio loss Lt.

A wide variety of risk measures have been proposed in literature and practice. This thesispresents the most important notions of risk, which are commonly used in the context ofportfolio optimization. Furthermore, the theoretical properties of risk measures are brieflysummarized and put into the context of portfolio optimization.

3.2.1 Variance

Variance as a measure of dispersion from the mean has been a popular measure of risk inFinance and due to the success of Markowitz’s portfolio theory, it is still dominant amongfinancial practitioners. It is defined as the second central moment from the portfolio lossdistribution as exposed in (2.1.0.6) and measures the deviation from the expected returnand thus may be understood as a notion of uncertainty, respectively risk. Markowitz (1952)used variance as a risk measure in a portfolio optimization framework. His methodologybecame popular under the name of mean-variance optimization. More concretely, thegeneral problem of (3.1.0.1) can be presented as:

arg minw

w′Σw


Since w is quadratic in the above stated optimization problem, it may be efficiently solvedthrough quadratic programming. Further constraints, as displayed in (3.1.0.3), may beadded to the problem.

Despite its simplicity and analytical tractability, there are various objections to varianceas a risk measure in portfolio optimization. If we assume that the risk factor changesXt follow a multivariate normal distribution, it may be justified to apply mean-varianceoptimization, since the normal distribution is entirely defined through the first two mo-ments. However, in other cases, mean and variance do not disclose the entire structure of

3.2 Risk Measures 27

a probability distribution and it can be even more misleading in the presence of asymme-try. Furthermore, variance also captures the positive deviation from the expected return,which is counterintuitive. As risk should be understood as the potential for experiencing ashortfall, due to its symmetry assumptions, variance equally weights positive and negativedeviations from the expectation (McNeil et al., 2005).

3.2.2 Value at Risk (VaR)

VaR has become the most important risk measure in Finance during the last decades. Eventhough the concept of VaR previously existed among experts, the growing popularitywas triggered by the market crash in 1987 and the public distribution of the new VaRmethodology by the J.P. Morgan’s RiskMetrics Group. Furthermore, VaR now plays animportant role in the Basel II framework (Holton, 2002).

Following the notations of McNeil et al. (2005), the α− V aR is defined as the α-quantileof the loss distribution FL:

V aRα = inf{l ∈ R : FL(l) ≥ α} = inf{l ∈ R : P (L > l) ≤ 1− α} (3.2.2.1)

Hence, VaR is a quantile-based statistic on the loss distribution over a predefined time-frame and it may be practically understood that the probability of the loss, exceedingthreshold V aRα, is lower or equal to 1 − α. In the case of a continuous and strictlyincreasing loss distribution function FL, the V aRα may be easily calculated using the in-verse function and V aRα(L) = F−1

L (α). Despite its popularity, conceptual simplicity, andintuitiveness, there are shortcomings that are relevant for this thesis. One problem froma risk perspective, consists in the fact that VaR merely provides us with a threshold thatshould not be exceeded at a certain confidence level. However, it does not provide us withan estimate of the severity of the loss in this case. Depending on the upper tail of the lossdistribution, this may lead to significant underestimation of the true risk by an investor.A more theoretically derived shortcoming is that VaR violates the subadditivity propertyand therefore is not considered a coherent risk measure.2 (Artzner, Delbaen, Eber, andHeath, 1999). Unfortunately, as Frey and McNeil (2002) were able to show through anillustrative example, the violation of the subadditivtiy requirement3 may lead to highlyconcentrated and risky portfolios in a mean-VaR optimization framework, when return ismaximized for a given level of risk. Following the criticism of VaR w.r.t. coherence, wethus skip the details about possible optimization algorithms and focus on an extension ofVaR that overcomes the undesirable properties of this risk measure. For more theoreticalinsight about the axioms of coherence, we refer to the work of Frey and McNeil (2002), whonot only treat this topic with mathematical rigor, but also empirically show its importancein quantitative risk- and portfolio management.

2According to standard literature, a risk measure is defined to be coherent if it fulfills the axioms oftranslation invariance, subadditivity, positive homogeneity, and monotonicity.

3Subadditivity in this context implies that the aggregated portfolio loss of various assets is lower orequal to the sum of the individual losses. This makes sense, since one would attribute this mathematicalproperty to diversification benefits in practice.


3.2.3 Conditional Value at Risk (CVaR)

Conditional Value at Risk, also known as Expected Shortfall (ES), is closely related to thedefinition of VaR and is defined as follows:

CV aRα =1

1− α

∫ 1

αV aRu(L)du (3.2.3.1)

The integral may be understood as an averaging over all VaRs in the tail, exceedingα. Alternatively, McNeil et al. (2005) show that in the case of continuous FL, CVaRcan be interpreted as the conditional expectation of the loss in the event that the V aRαwas exceeded. The involvement of the entire tail in such a risk measure represents astrong advantage to VaR, as it provides an estimate of the severity of the loss if a certainconfidence level is exceeded. Moreover, Artzner et al. (1999) have shown that CVaR isa coherent risk measure as it overcomes the subadditivity violation of VaR. Lastly, asopposed to Variance, CVaR only focuses on the upper tail of the loss distribution anddoes not rely on symmetry assumptions. Due to this combination of favorable theoreticaland practical properties, we decided to apply CVaR as the common risk measure in ourportfolio optimization framework and empirical analysis.

Owing its importance in this thesis, presented without proof is the following Lemma fromAcerbi and Tasche (2002) which states that the plug-in estimator, applying the empiricalorder statistics, is a consistent estimator for CVaR.

Lemma 3.2.3.1.

limn→∞

∑ni=nα L(i)

n(1− α)= CV aRα a.s.

where L(i) are the i’th order statistics and iid. random variables with df FL.

Hence, if n becomes sufficiently large, we may use lemma (3.2.3.1) as an estimator for CVaRin practice. Owing to the fact that in this thesis, loss scenarios are produced through MonteCarlo simulation based upon a parametric model, the number of scenarios n is only limitedby the computational complexity of the algorithm and we may accurately approximate thetrue CVaR. Monte Carlo simulation and the above stated lemma are particularly helpful,when the integral of (3.2.3.1) is difficult or impossible to calculate for complicated lossdistributions.

3.3 Mean-CVaR Optimization

3.3.1 Minimum Risk for Given Expected Return

According to the discussion in Section 3.2, we decided to apply CVaR as a common riskmeasure in this study due to its theoretical properties as well as its appealing intuitivitywith regard to risk management. As will be shown, portfolio optimization w.r.t. CVaR canbe either reduced to a linear programming problem or a non-linear smooth approximation.Changing the risk measure, the general optimization problem of (3.1.0.1) in its simplest

3.3 Mean-CVaR Optimization 29

form translates into:

arg minw

CV aRα(w)


Since the calculation of CVaR as a function of w and the loss distribution FL is oftendifficult to derive and would need to be individually calculated for different parametricmodels, a scenario-based non-parametric approach is followed instead. If a sufficient highnumber of scenarios can be obtained we may apply Lemma (3.2.3.1) and argue that wecan approximate the true CVaR arbitrarily close through scenarios.

Rockafellar and Uryasev (2000) firstly introduced the scenario-based CVaR optimizationapproach and presented the following auxiliary function for the expression of VaR andCVaR, assuming that the density function pL(x) exists for the loss distribution functionFL(x).4

Fα(w, γ) = γ +1

1− α

∫x∈Rp

[l(x,w)− γ]+pL(x)dx (3.3.1.2)

In the above expression, l(x,w) is again denoted the general loss operator, mapping riskfactor changes x and relative asset weights w into portfolio losses.

The following lemma, applying auxiliary function Fα(w, γ) is needed for the solution ofthe optimization problem (3.2.3.1). The detailed non-trivial proof may be found in Rock-afellar and Uryasev (2000):

Lemma 3.3.1.1. Using the auxiliary function of (3.3.1.2), VaR and CVaR can be ex-pressed as follows:

CV aRα(w) = minγ∈R

Fα(w, γ)

V aRα(w) = arg minγ∈R

Fα(w, γ)

minw

CV aRα(w) = minγ,w

Fα(w, γ)

(3.3.1.3)

Furthermore, Fα(w, γ) is a convex function of γ and continuously differentiable.

According to lemma (3.3.1.1), Fα(w, γ) needs to be minimized over the domain of both,the portfolio weights w and auxiliary variable γ. As mentioned previously, in many cases,particularly in complicated models, the analytical calculation of the integral of (3.3.1.2) isnot desired. In this case and provided that risk factor scenarios are not sparse, pl(x) maybe replaced with the empirical counterpart, yielding:

Fα(w, γ) = γ +1

(1− α)N

N∑i=1

[l(xi,w)− γ]+ (3.3.1.4)

4Note that for the sake of notational simplicity, we skipped the subscript t in this context and presentthe algorithm in a general manner. In practice, the optimization step is performed at any point in time t,using the conditional loss distribution based upon the sigma algebra Ft−1.


As can be seen in this general representation, the loss operator l maps the multivariaterisk factor change scenarios x into portfolio losses, applying the relative asset weights w.Hence, it now should become clearer how random risk factor scenario generation in Section2.3 is connected to the portfolio optimization through its loss operator.

Finally, Fα shall be minimized with respect to w and γ. Conveniently, (3.3.1.4) is trans-formed into a linear optimization problem, introducing auxiliary variables ui and linearconstraints:

minu,w,γ

γ +1

(1− α)N

N∑i=1

ui s.t.

ui ≥ 0

ui ≥ l(xi,w)− γ

(3.3.1.5)

The linear constraints shall ensure equivalency to (3.3.1.4). This is guaranteed, sincethe constraints enforce ui to be equal or greater than l(xi,w) − γ. Moreover, the factthat we are minimizing, implies that ui attains its lower bound from the feasible setui ≥ l(xi,w)− γ. Hence the optimal solution will ensure that equality is reached, namelyui = l(xi,w)− γ. Finally, the constraint ui ≥ 0 enforces the positivity requirement of ui.

Eventually, we may substitute the general loss operator l(xi,w) through a linear combina-tion of the individual asset losses with portfolio weights w, which equals the standardizedversion of portfolio loss, as presented in (2.4.0.6). This yields a linear optimization prob-lem w.r.t portfolio weights w, which can be conveniently solved using linear programmingalgorithms.

3.3.2 Maximum Return for Upper Risk Boundary

As the focus in this thesis is on risk management, we would like to create portfolios thatadhere to a predefined risk budget and accordingly assess their empirical appropriateness.For this purpose, however, the portfolio optimization algorithms must follow the generalform of (3.1.0.2). Hence, there is a need to maximize the expected return, satisfying anupper CVaR boundary:

arg minw

−w′µ

Subject to : CV aRα(w) ≤ c(3.3.2.1)

Krokhmal, Palmquist, and Uryasev (2002) have shown that this problem can also be solvedthrough plugging-in the auxiliary function Fα(w, γ):

arg minw,γ

−w′µ

Subject to : Fα(w, γ) ≤ c(3.3.2.2)

If Fα(w, γ) is substituted through its scenario-based counterpart Fα(w, γ) and ui is intro-duced as an auxiliary variable in analogy to problem (3.3.1.5), the optimization problem

3.3 Mean-CVaR Optimization 31

may be expressed as:

arg minu,w,γ

−w′µ s.t.

γ +1

(1− α)N

N∑i=1

ui ≤ c

ui ≥ 0

ui ≥ L(xi,w)− γ

(3.3.2.3)

If L(xi,w) is linear in w we again have a linear optimization problem as in the CVaRminimization problem of (3.3.1.5).

3.3.3 Smooth Approximation Approach

Although the linear formulation of a CVaR optimization problem is theoretically appeal-ing and should be the approach of first preference in this study, severe difficulties wereexperienced in regard to computational complexity and memory limits. Indeed, the linearconstraint matrix explodes quadratically with increasing scenario numbers N . However,due to the dimensionality of the problem and precision requirements, we rely on a suffi-ciently high number of scenarios, which puts us in a dilemma in terms of accuracy andcomputational feasibility.

In the recent literature, there have been some propositions of more efficient optimizationalgorithms by Bardou, Frikha, et al. (2009) and Alexander, Coleman, and Li (2006).However, their approaches are far from being trivial: they include stochastic optimizationand thus are not considered for this framework. Instead, we aim to improve the efficiency ofthe optimization process through the use of a gradient-descent based approach. Certainly,this may move us away from the desirable global solution, which is guaranteed under thelinear formulation, however it may help to considerably reduce computational costs and,thus, make the scenario approach more viable in practice. Furthermore, it is not difficult tosee that the smooth approach is computationally less expensive than the linear approach,as the constraint matrix of order N2, containing the auxiliary variables, is no longerneeded. The central element of this modified approach again is the result of Rockafellarand Uryasev (2000) which is stated in lemma (3.3.1.1). Ghalanos (2013) proposes a simpleapproximation to the non-smooth function in the brackets, namely:

max(x, 0) ≈ smax(x, 0) =(√x2 + ε+ x)

2(3.3.3.1)

This allows us to reformulate the CVaR representation of (3.3.1.4) with a smooth coun-terpart that can be dealt through a gradient-descent based optimization algorithm:

Fα(w, γ) = γ +1

(1− α)N

N∑i=1

smax(L(xi,w)− γ, 0) (3.3.3.2)

This smooth representation of CVaR finally may be plugged into the optimization problemsof (3.3.1.1) (risk minimization) and (3.3.2.1) (return maximization). As the smooth version


is no longer linear, this optimization problem is not guaranteed to be convex and as aconsequence, an optimization algorithm may be stuck in a local optimum.

In order to ensure that the smooth approach produces acceptable results, a comparisonwas drawn against the linear version on a few examples, in which the differences weresufficiently small for all practical purposes. Owing to the lack of representativity, we donot state these exemplatory comparison results here, but rather refer to the more compre-hensive simulation study of Ghalanos (2013) , who found that the difference in results isnegligible in practice. Due to the substantial improvement of computation time, we solelyemployed the smooth non-linear approach of (3.3.3.2) in all subsequent optimization prob-lems.

Finally, it is important to discuss the appropriate choice of optimization algorithm whichobviously depends on the concrete problem formulation. If CVaR minimization is desired,the non linear but smooth function (3.3.3.2) can be minimized easily, using a Newton-likeapproach. However, if as performed in this study, return maximization is desired instead,the non-linear term is now located in the optimization constraint, which needs particularattention.

For this reason, we applied the methods of sequential quadratic programming (SQP),which are capable of dealing with non-linearity in constraints. A concise introduction toSQP may be found in Boggs and Tolle (1995). In this study, for all optimization problems,the open source implementation of the well known NLopt framework was applied, whichprovides an interface to SQP.

3.3.4 Reliable Estimation of Expected Returns

The mean-CVaR framework, which can be considered an extension to the Markowitz ap-proach, appears to be appealing theoretically, as it balances the trade-off between expectedportfolio return and risk. Although this representation is intuitive, it is based upon theassumption that either the expected returns of the assets are known, or at least can bereliably estimated. Whilst it is out of question that the knowledge of true parameters canbe obtained, there are serious concerns regarding the accuracy of an estimator for expectedreturns. If the estimated expected returns largely deviate from the true parameters, theoptimization will move portfolio weights into the wrong direction and as it lies in thenature of optimization, the estimation error can be even amplified.

Considering the finding that serial correlations in our asset returns are not present, a firstsimple estimator for the asset’s expected returns would be the sample estimator:

µx = n−11′R (3.3.4.1)

R is a n × d matrix, which contains the observed asset returns over the observationperiod n. The classical argumentation from a statistical perspective could be that withincreasing number of historical data, the estimation error will become negligible due tothe asymptotic behavior of the estimator. However, the estimation period can not beindefinitely extended, even in the presence of abundant data. The reason for this limitationlies in the non-stationarity of financial markets over longer time periods. Whereas thestationarity assumption over short time may be approximately true, it does not apply

3.4 Maximally Diversified Portfolios 33

long-term, as economic regimes change and provide different market conditions.5 Hence, inpractice, sample data for return estimation is rather limited and the estimation error thusplays an important role. Unfortunately, it is well established in the literature to portfoliooptimization that noise in the sample mean estimator leads to extreme optimized portfolioswhich do not perform well out of sample. DeMiguel et al. (2009) contributed an extensivestudy concerning the effect of noise on portfolio optimization and accordingly establishedthat in order to estimate the expected returns with acceptable accuracy for optimization,one would require a minimum of 3000 datapoints in the presence of 25 assets. Whereasthere is a variety of approaches, mostly using bayesian methods, we implemented a simpleeconometric model in order to alleviate the difficulty to rank assets’ expected returns.Sharpe (1970) in their groundbreaking work to Finance, firstly explained expected returnsof assets as a function of a systematic market factor:

E(Ri) = Rf + βi(E(Rm)−Rf ) (3.3.4.2)

Rf is a risk-free asset (mostly proxied by US Treasury Bills) and E(Rm) denoted theexpected return of the entire market. Hence, in this model, we assume that the differencesin expected returns are explained fully through the asset exposures βi to the marketfactor. In this study, exposures have been estimated, using p independent least-squareregressions with the assets returns as target variables and the S&P500 market returns asan explanatory variable. Finally, we can consider the following proxy of expected returnsas an input to the optimization algorithm:

ECAPM (Ri) = Rf + βi(E(Rm)−Rf ) (3.3.4.3)

Since, Rf and Rm are constants which apply to all assets, it is easy to check thatarg maxw w′µ is not dependent on them. Hence, using the CAPM model for the expectedasset returns, we may reformulate (3.3.2.2):

minw,γ−w′β

Subject to : Fα(w, γ) ≤ c(3.3.4.4)

We see the main advantage in this approach in the fact that the betas are more stableover time, as they they represent the market structure. Furthermore, it relies on soundeconomic theory, where asset returns are driven by a common underlying systematic part,opposed to the naive sample mean estimator. However, it needs to be proven in theempirical part of this thesis that this method can produce acceptable results.

3.4 Maximally Diversified Portfolios

In Section 3.3.4 we elaborated a method to deal with the challenges related to the estima-tion of expected returns in the context of portfolio optimization. It is worth noting that theestimation error is present not only in the expected returns but that also the risk modelsin Chapter 2 are affected. This often leads to risk underforecasts, as the estimates are as-sumed to be equal to the true parameters in the optimization, when in fact they are subjectto variation. As estimation and market risk add up and the traditional approach fails to

5For instance, the attacks on the world trade center on 9/11 implied a significant and enduring economicimpact on the airline industry. As a consequence, the assumption that the return’s probability distributionremains the same post-event is not realistic.


account for the former, it is expected that the realized risk will exceed the boundary in theoptimization constraint. Specifically, Lim, Shanthikumar, and Vahn (2011) have shown ina simulation study that a mean-CVaR framework is often bound to fail due to estimationerror and diminishes the favorable theoretical properties of this risk measure. In litera-ture and practice, there have been various proposals to incorporate estimation risk intoa portfolio optimization framework. The most prominent proposal in this direction camefrom Black and Litterman (1991), who have incorporated a bayesian approach towardsparameter estimation. Most recently, M. Concepcion Ausin (2010) proposed a dynamiccopula-GARCH model with bayesian parameter estimation, using MCMC-methods and asimulation approach to derive confidence bounds of optimal portfolios. While theoreticallyappealing, it is important to note that the study was performed on a very small set ofrisk factors and that the implementation of MCMC is non-trivial. Moreover, simulatingscenarios from a set of different models and deriving a distribution of optimal portfolios iscomputationally expensive.

In contrast to the bayesian proposals from statistical estimation theory, the risk parityapproach is relatively new and represents a more solution-oriented method to overcomethe challenges of estimation error in portfolio optimization. The intuition behind this isthat risk should be constrained to be equally distributed among risk factors or assets,thus leading to less extreme optimization results. It is also assumed that investors arefairly rewarded for their exposure to different sources of risk. Under these assumptions,it becomes crucial in portfolio management to diversify a portfolio among different riskfactors, as opposed to return maximization. In this way, portfolios are likely to be moreresilient to inaccuracies in risk estimates as the model parameters are not passed on tounconstrained optimization. Furthermore, by focusing on the equal allocation to riskfactors, rather than the maximization of portfolio return, we do not need to determineexpected returns as outlined in Section 3.3.4.

Qian (2011) provides a basic introduction to the field of diversification and risk parity. Themost simplistic approach of diversification consists in allocating equal funds to each assetin a portfolio. However, this method is bound to fail if risk is not uniformly distributedamong the underlying assets, which is rarely the case. A more sophisticated investor wouldrather account for this and enforce that all assets contribute the same amount to portfoliorisk in total. This could still be problematic, if risk is clustered among a few assets, sharinghigh correlations. Hence, it is more reasonable to allocate risk to a set of underlying riskdrivers, which are able to explain most of the total market variation.

Ideally, we would like to decompose the asset returns into stochastically independentfactors. In a second step, we would be able to maximally diversify the portfolio’s exposuretowards these factors. Due to the orthogonal design, we do not need to account forcorrelations, which greatly simplifies the allocation decision. These properties provided astrong motivation to implement the recent proposal from Meucci (2009a). Apart from theoutlined theoretical properties, it is computationally inexpensive to implement and thussuperior to the rather complex bayesian frameworks. While the approach was originallypresented in the context of classical mean-variance optimization, we firstly embed it intoa scenario-based CVaR optimization framework.


3.4.1 Risk Decomposition

Firstly, it is important to mention that the approach uniquely concerns the covariancematrix and thus risk is simplified to the second moment of the distribution. The dynamicmodels in this thesis, however, produce non-normal conditional distributions for the sim-ulated scenarios. In this light, the diversification method represents a minor inconsistencyto the selected risk measures, which we are accepting in exchange for practical feasibility.

The first step of the approach consists in decomposing the asset returns’ covariance matrixΣ. In practice, this matrix can be estimated, using the sample estimator. As we appliedthe method of moment matching to the simulated risk factor changes, as outlined inSection 2.3.4, no error due to simulation is introduced, as the second central moment ofthe scenario’s distribution is matched perfectly to the conditional covariance matrix in therisk model.

Meucci (2009a) proposes the application of the spectral theorem in order to change thebase of the asset returns such that they are uncorrelated.

E′ΣE ≡ Λ (3.4.1.1)

Λ = diag(λ1, . . . , λd) contains the ordered eigenvalues, corresponding to the eigenvectors inE. In the context of Finance, it is convenient to interpret the eigenvectors as uncorrelatedportfolios. It’s returns R can then be expressed, using the original asset returns R:

R = E′R (3.4.1.2)

It is not difficult to check that the covariance matrix of these eigenportfolios equals thediagonal matrix of eigenvalues Λ. Similarly, we can express a portfolio of assets withweights w as a linear combination of eigenportfolios w = E′w. Through the applicationof the spectral decomposition theorem, we are able to decompose the portfolio variancew′Σw into independent factors which equal the eigenportfolios. Using the new base, theportfolio variance can be simply calculated as the weighted sum of the eigenportfolio’svariances.

Var (Rw) =d∑i=1

w2i λi (3.4.1.3)

3.4.2 Diversification Distribution

Due to the rotation of the original portfolio applying matrix E′, the portfolio’s risk isdecomposed into uncorrelated risk sources. A risk parity approach can now be appliedto the optimization process, attributing equal contribution to each risk factor. For thispurpose, Meucci (2009a) define the following risk concentration curve:

pi ≡w2i λi

Var (Rw)(3.4.2.1)

By normalizing the variances of the individual eigenportfolios, it is guaranteed that∑Ni=1 pn

sums up to 1 and thus can be considered a diversification distribution of risk factors,sharing the properties of a probability distribution. This allows us to measure the level ofdiversification by defining a functional of the entire distribution. The author proposes the


entropy as an ideal measure of diversification, as it is maximized for a uniform probabilitydistribution, which would intuitively represent a well diversified portfolio:

NEnt ≡ exp(−d∑i=1

pilnpi) (3.4.2.2)

The maximum diversification criteria now can be implemented into the optimization frame-work. Namely, the maximum return target is replaced by the entropy of the diversificationdistribution.

arg maxw

NEnt


In contrast to the optimization representation of (3.3.2.1), we entirely disregard portfolioreturns and target portfolio stability, by maximizing its diversification. Obviously, thisdoes not imply that the investor is indifferent about returns, but it is assumed that inequilibrium, taking risks is fairly rewarded. If this assumption is true, full emphasis canbe placed on robustly satisfying the CVaR risk constraint. In the presence of estimationerror, a well diversified portfolio is a better candidate to meeting the risk constraint outof sample rather than a portfolio which is heavily concentrated on a few risk sources dueto noise in empirical data and its amplification through the optimization process.

In order to fit in with our risk model and optimization framework, the following algorithmwas applied to determine the maximum diversification portfolio which satisfies an upperrisk (CVaR) boundary.

i.) Use scenarios from the projection algorithm, outlined in Section 2.3 and estimatethe covariance matrix Σ.

ii.) Perform the spectral decomposition of the estimated covariance matrix Σ = E′ΛE.

iii.) Perform iteratively for each step of the SQP optimization algorithm:

(a) Calculate the rotated portfolio weights w = E−1w.

(b) Determine the diversification distribution of the rotated portfolio weights:

pi =w2i λi

Var(Rw).

(c) Evaluate the optimization problem of (3.4.2.3).

3.4.3 Conditional Analysis

The unconditional approach of decomposing risk into d uncorrelated parts seems to beappropriate if the optimization is fully unconstrained and all eigenportfolios are stable anddo not mirror noise, introduced through estimation error of covariance matrix Σ. However,in fact, most optimization formulation contain the budget constraint of

∑di=1wi = 1, hence

it is not possible to freely combine the eigenfactorportfolios such that NEnt is minimized.Furthermore, it is probable that the eigenportfolios with the smallest variance λi arerather unstable and driven by estimation error at each estimation step of the applied riskmodel. However, diversifying the portfolio with respect to these noisy eigenportfolios is notexpected to provide satisfactory results. The author, thus, proposes a modified eigenvectordecomposition, which is calculated, conditional upon a set of k equality constraints. More


concretely, in the optimization process, we would like to restrict the rebalancing directions,by imposing the following constraints:

A∆w ≡ 0 (3.4.3.1)

A is a k × d matrix whose row contain a constraint. If we restrict the rebalancing di-rections such that the budget constraint

∑di=1wi = 1 is not violated by shifting between

eigenportfolios, the feasible reallocations of weights must satisfy 1′∆w = 0. Hence, we setthe first row of A equal to the vector 1′. Furthermore, if we consider the last eigenvectorsej for j = l, . . . , d to be statistically insignificant, we would like to constrain the rebal-ancing directions such that reallocations on the space, spanned by these eigenvectors arenot possible. In this case, we would read e′j∆w = 0 and the rows of A contain the lastinsignificant eigenvectors. Based on the intuition of constrained reallocation directions,we may now define the d− k eigenvectors, which satisfy the k reallocation constraints.

Definition 3.4.3.1. For i = k + 1, . . . , d the conditional eigenportfolios are defined as:

ei ≡ arg maxe′e=1

e′Σe, such that

Ae = 0

e′Σej = 0, for all existing j

(3.4.3.2)

Due to the imposed constraints, the d−k conditional eigenvectors span the space of feasiblereallocations. This subspace can be easily complemented to the full reallocation space byrecursively calculating the remaining n = 1, . . . , k eigenvectors:

ei ≡ arg maxe′e=1

e′Σe, such that

e′Σej = 0, for all existing j(3.4.3.3)

The optimization algorithm then works similarly as firstly exposed in Section 3.4.2. How-ever, the conditional diversification distribution is now limited to the feasible reallocationdirection and reads as follows:

pi | A ≡w2i λi∑d

m=k+1 w2mλm

, i = k + 1, . . . d (3.4.3.4)

Figure 3.1 exemplarily shows the eigenvalues of a conditional decomposition of the co-variance matrix into eigenportfolios at a specific date. It can be seen that due to theconstraints, only the last 8 eigenportfolios are relevant for the diversification analysis.While the budget constraint is already defined by construction of the optimization prob-lem, it is more difficult to determine how many eigenvectors should be regarded as statis-tically insignificant and consequently disregarded for the definition of the diversificationdistribution. We employ a heuristic approach which is often used in the related PrincipalComponent Analysis (PCA). In Figure 3.2, we plot the cumulative contribution of the firsti conditional eigenportfolios to their total variance

∑dm=k+1 w

2mλm. It is visible that the

first 4 eigenportfolios already explain 80% of the total variance. The marginal increaseof the remaining factors seems to be relatively small, thus, we restrict the diversificationdistribution to the first 4 factors in this study. It is worth mentioning that this heuristicargument cannot provide complete satisfaction, and it would be favorable to have a soundtheoretical underlying to determine the optimal number of factors for the diversificationdistribution. However, for the illustration of the diversification approach, the heuristiccriteria turns out to produce satisfactory results and we leave it up to future research todevelop criteria that exhibit a better theoretical underpinning.


u1 c1 c2 c3 c4 c5 c6 c7 c8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Figure 3.1: Conditional analysis of the variance of eigenportfolios λi based upon a budgetconstraint. The letter u indicates the unconstrained eigenportfolio, whereas c refers to theconstrained portfolios. The analysis was performed on date 12/28/2001.

c1 c2 c3 c4 c5 c6 c7 c8

0.0

0.2

0.4

0.6

0.8

1.0

Figure 3.2: Cumulative contribution of the conditional eigenportfolios to total varianceon date 12/28/2001. Note that the first 4 eigenportfolios explain approximately 80% ofthe dispersion of the underlying assets.

Chapter 4

Risk Forecast and OptimizationEvaluation Framework

As in this study, a variety of risk models and optimization approaches are tested, a soundcommon evaluation methodology becomes crucial. Since the emphasis has been clearly puton risk, we subsequently elaborate the methodology to assess the risk forecast accuracy ofall models, presented in Chapter 2.

It is instructive to validate risk forecasts on three dimensions. Firstly, forecast accuracy canbe assessed either on a single asset (univariate) or portfolio (multivariate) level. The formerfocuses on the univariate forecast ability, whereas the latter also includes the accuracy ofthe dependency forecasts. Secondly, risk forecasts should be assessed over time, as a simpleaverage statistic may be problematic due to cancelling effects. Thirdly, we may measurerisk forecast performance by considering different risk measures, as presented in Section3.2. Consequently, we will commence with volatility forecast validation and in the nextstep VaR and CVaR forecasts from the simulated scenarios are tested. It is held that thiscombination of approaches, which includes all of these perspectives, finally will provide aconclusive insight on the models’ performance.

4.1 Volatility Forecasts

As extensively elaborated in Section 2.2.5, not only the unconditional but also the condi-tional models directly provide us with the covariance matrix forecast Ht+1. It is importantto note this matrix belongs to the risk factor changes Xt, which in this case are denotedthe logarithmic returns. However, the investor is exposed to linear returns and the riskforecast ability must be evaluated on these. For this reason, it is of advantage to firstlyproject the risk factor returns, as outlined in Section 2.3. The applied method of momentmatching ensures that no error is introduced for the first two moments, due to the sim-ulation process. Finally we map the scenarios into portfolio losses as outlined in Section2.4. As a result of this process, we will obtain n portfolio loss (negative returns) scenar-ios which we represent by the sample vector l∗t+1. For the sake of comparability, we willsubsequently state losses as a relative amount of the initial value.1 Based on the scenario

1The conversion can be simply performed by standardizing losses by the security price at time t: St,i.

39

40 Risk Forecast and Optimization Evaluation Framework

set, we may estimate the portfolio loss variance as introduced in Section 3.2.1:

σ2t+1 = (n− 1)−1

n∑i=1

(l∗i,t+1 − µl)2 (4.1.0.1)

As highlighted previously, it is more common in finance to state dispersion in units ofstandard deviation which is often referred to as volatility. Furthermore, we reiterate thatdue to the equivalence ri,t = −li,t, returns and losses can be exchanged in the definitionof the simulated volatility forecast in (4.1.0.1).

4.1.1 Volatility Forecast Bias

Unfortunately, it is impossible to observe the true conditional volatility at a certain pointin time, as only one observed portfolio loss is available. For this reason we may solely relyon the noisy proxy of squared portfolio losses L2

t . This proxy is unbiased as E(L2t ) = σ2

t

under the martingale assumption of E(L) = 02. Menchero et al. (2013) then define thepopular volatility forecast bias, using the imperfect proxy of squared losses.

Firstly, we combine today’s volatility forecast with tomorrow’s loss and calculate out-of-sample z-scores:

Zt =Lt+1

σt+1(4.1.1.1)

Under the assumption that losses are serially uncorrelated3, it may argued that if thevolatility forecasts are unbiased, Zt are true z-scores with V ar(Z2

t ) = 1 and thus, thevolatility bias statistic can be defined as follows:

b =

√√√√ 1

T − 1

T∑t=1

(zt − µz)2 (4.1.1.2)

Hence it becomes clear that underforecasting results in a bias above one, whereas theopposite is true for risk underforecasts.

4.1.2 Q-Statistic

Whilst the bias statistic provides a general insight whether risk is over- or underforecastedon average over a time period, it does not provide us with an adequate estimate of theaccumulated inaccuracy over, as periods of over- and underforecast may cancel out. Forthis reason, it is not recommended to look at bias statistics in an isolated manner. Ideally,a loss function should penalize forecasting inaccuracy and sum up the losses over time.

In the field of decision theory, there exists a huge selection of possible loss function. Patton(2011), however, in their notable theoretical work show, that the Q-like loss function hasunique properties in this context. Firstly, the authors define a loss function as robust, ifit correctly ranks the forecast accuracy of different models, even if the true volatility isreplaced by an unbiased but noisy proxy (e.g. squared realized losses). Furthermore, it isshown that the q statistic is the unique loss function that solely depends on standardized

2This assumption is mostly accepted for equities within empirical finance.3This assumption has been verified in Section 2.2.2.

4.2 Value at Risk (VaR) Forecasts 41

z-scores and therefore enables the comparison of forecasting accuracy on portfolios withdifferent volatility levels. The Q-statistic on z-scores, according to Patton (2011) is definedas follows:

Qt = Z2t − log(Z2

t ) (4.1.2.1)

Menchero et al. (2013) provides the intuition behind the Q-statistic by attributing the firstterm Z2

t to underforecasting penalization, whereas −log(Z2t ) contributes to Q-loss in the

presence of overforecasting. Furthermore, it is no longer possible that the effects cancelout over time. Finally, for the model comparison, the average of the estimated Q-statisticsare calculated over time:

Qt = T−1T∑t=1

Qt (4.1.2.2)

4.2 Value at Risk (VaR) Forecasts

As outlined in Section 3.2.2, VaR has become the most popular measure of risk, despite itstheoretical deficiencies. For this reason, we would like to assess the model’s risk forecastingaccuracy, applying the VaR measure to the portfolio loss distribution. In the context ofour scenario simulation framework, the VaR forecast can be obtained by calculating thesample quantile function q, whereas l∗t+1 is a vector of N simulated scenarios for theforecast horizon t+ 1:

ˆV aRt+1 = q1−α(l∗t+1) (4.2.0.3)

Since VaR, as a quantile-based risk measure, is defined by the upper tail of the lossdistribution, we can only rely on a few set of extreme historical realizations in order toassess the forecast accuracy, in contrast to the assessment of volatility forecasts.

McNeil and Frey (2000) propose a simple VaR exceedance test, which we implemented inthis study. More concretely, as VaR can be interpreted as the threshold which may beexceeded with probability 1− α, it is instructive to examine the out of sample violationsof our VaR forecasts:

It+1 = ILt+1> ˆV aRt(4.2.0.4)

I is the indicator function which takes value 1 if the VaR forecast is violated. Finally wemay use the entire history of VaR forecasts and realized losses in order to calculate thetotal number of exceedance.

NV iol =T∑t=1

ILt+1> ˆV aRt(4.2.0.5)

It is intuitive in the case of accurate forecasts that the estimated number of violations overtime should come close to the theoretical value (1 − α)T . More formally, under the as-sumption of correct forecasts, the number of violations will be iid. binomially distributed(NV iol ∼ B(T, 1 − α)), since a single indicator variable follows a Bernoulli distribution.We are then able to construct the following two-sided binominal test:

Definition 4.2.0.1. Under the null hypothesis of correct VaR forecasts, the total numberof violations should match the theoretical value. Hence, we define the test for VaR ex-ceedance as:


H0: NV iol = (1− α)T

HA: NV iol 6= (1− α)T

In the case that the number of VaR forecast violations is beyond the critical values ata confidence level of 95%, we regard this as empirical evidence that the true VaR doesnot coincide with the model’s estimates. It is also insightful to perform this analysis overdifferent non-overlapping windows, to recognize periods of inaccurate forecasts. Certainly,this window cannot be defined to be arbitrarily small, as only (1 − α)T violations areexpected. In this study, we present violations with a per year resolution, as this providesat least 12.5 expected violations at VaR parameter α = 95%.

4.3 Conditional Value at Risk (CVaR) Forecasts

The conditional forecast for CVaR at time t+1 can be similarly obtained, using the samplecounterpart (plug-in estimator) of the CVaR definition. As Lemma (3.2.3.1) states, thisis a consistent estimator for the theoretical CVaR. Hence, the estimated CVaR at theinvestment horizon is calculated as:

ˆCV aRt+1 =

∑ni=nα l

∗(i)

n(1− α)(4.3.0.6)

l∗(i) represents the i’th order statistic of the simulated scenarios at the horizon t + 1.

Again, McNeil and Frey (2000) present a simple statistical test to asses the CVaR forecastaccuracy. If the CVaR forecast is accurate we expect it to equal the true VaR-conditionalexpected loss:

E[Lt+1 − ˆCV aRt+1

]ILt+1> ˆV aRt+1

= 0 (4.3.0.7)

This motivates the calculation of the following risk exceedance average as a test statistic:

R = T−1T∑t=1

(Lt − ˆCV aRt)ILt+1> ˆV aRt(4.3.0.8)

Definition 4.3.0.2. Finally we would like to assess whether the average CVaR forecastwas accurate by performing the statistical test:

H0: R = 0

HA: R 6= 0

As it is difficult to assign a certain class of parametric distributions to R, a non-parametricbootstrap test was performed, according to Efron and Tibshirani (1994). If the null hy-pothesis was rejected at a significance level of 5%, this was interpreted as evidence ofinaccurate CVaR forecasts. Since this test by construction can only include a few samplesto estimate the mean exceedance, it has been performed over the full historical sample inorder to ensure that the test has sufficient power.

4.4 Optimization Strategy Evaluation 43

4.4 Optimization Strategy Evaluation

Whereas the former Sections of this chapter elaborated the evaluation methodology forrisk forecasts, it is also important to perform an ex-post assessment on the performance ofthe optimization strategies. As five different risk models and two optimization strategieswere subject to this thesis, a total of 10 strategies need to be compared.

A central emphasis in this assessment will be put on the realized risk of the strategieswhich should be compared with the risk constraint in the optimization. Fortunately it isnot complicated to estimate the empirical CVaR of the realized portfolio loss time series,as we may merely apply the estimator, presented in lemma (3.2.3.1). If the risk model wasaccurate it is expected that the realized CVaR should not significantly violate the CVaRconstraint of the optimization. A substantial deviation from the target risk boundary incontrast, indicates that the combination of risk model and optimization strategy failedto meet the investor’s risk constraints. Similarly, the realized volatility and VaR can becalculated, however, as these measures did not form part of the optimization constraint,they cannot be assessed against a target value.

Aside from the risk evaluation, it is important to compare the realized portfolio returnswhich should be calculated on an excess basis (net of risk free rate).4 In this context it is ofparticular interest to analyze whether or not the mean-CVaR version is able to outperformthe diversification-CVaR strategy, as the latter is invariant to expected returns and solelyfocuses on diversification. If this is not the case, we may doubt whether it is useful tooptimize the expected portfolio return in the presence of substantial estimation error.

Further insight concerning an investment strategy may be obtained by comparing therealized return to realized risk. Indeed, a rational investor would determine his returnexpectations depending on the exposure of the strategy towards risk. The most prominentmeasure in this context constitutes the Sharpe ratio which was introduced by Sharpe(1970) and features in every asset manager’s performance report. It is estimated as theratio between realized excess average return (rP − rf ) and volatility σr.

Sr :=rP − rfσr

(4.4.0.9)

Despite the popularity of this performance measure, there are several drawdowns asso-ciated with the symmetry implication of the standard deviation as a risk measure. Forinstance, a low volatility strategy with high probability mass in the upper-tail may delivera high sharpe-ratio over a long period of time, as the measure does not account for thepresence of higher moments.5 However, the potential of extreme losses should matter themost and exactly this aspect of risk is disregarded by applying the sharpe ratio. Hence,it may be dangerous to compare the performance of investment strategies solely on theground of estimated sharpe ratios.

Nonetheless, the idea of risk adjusted performance measures can be generalized to VaRand CVaR. Dowd (2000) adjusted the sharpe ratio to measure excess returns with respect

4It is standard in Finance to evaluate returns on an excess basis. As the risk free rate is guaranteedand may vary over time, excess returns should be interpreted as the implied reward for a risky investmentstrategy.

5The theoretical deficiencies of the volatility/variance as a risk measure are outlined in more detail inSection 3.2.1.


to the realized VaR. It is denoted the excess return on value at risk (EVaR) and is definedas follows:

ˆEV aR :=rP − rf

ˆV aR(4.4.0.10)

As derived in Section 3.2.3, CVaR is considered the risk measure with the most attractivetheoretical properties. Hence, the most appropriate approach to compare the performanceof strategies appears to be the ratio between excess returns and CVaR as a risk measure. Itwas firstly introduced as the conditional sharpe ratio (CSR) by Agarwal and Naik (2004)and is defined:

ˆCSR :=rP − rf

ˆCV aR(4.4.0.11)

It is important to assert that all estimates in this Section should be considered one observedrealization of a strategy over the relevant time period. Hence, if differences among themare small, they may also be considered the effect of pure chance. Obviously, it would be ofadvantage to derive the distribution of these measures in order to perform statistical testson their significance. However, this task is far from trivial as an analytical derivation doesnot seem to be feasible. If we were to follow a simulation approach, a further econometricmodel needs to be fitted to the realized portfolio returns in order to replicate the returnseries of a strategy. In a next step, model-based bootstrap scenarios could be generatedto mimic the statistical properties of the strategies. Due to the complexity, we leave theimplementation of such an approach to future research.

Chapter 5

Backtesting Framework

This thesis is subject to an empirical comparison of a variety of different risk modelsand optimization approaches. In order to ensure comparability between them, it wascrucial to elaborate a common framework in which they are embedded and evaluated. Astep-wise approach hereby guarantees transparency at each modelling level. For instance,we evaluate risk forecast accuracy directly after the projection step before the effects ofoptimization may play a role. Furthermore, it also helps to verify the algorithm at eachmodelling step in order to prevent major pitfalls in risk modelling.

Meucci (2011) as well as his bootcamp in New York (Meucci, 2013) greatly inspired us inthe construction of this framework. We adapted the general framework for the underlyingconcrete case and enriched it with the results of further research which are outlined inchapter 2 and 3. Figure 5.1 summarizes the framework in which the entire study hasbeen embedded. Due to the modular structure, it is also important that the programcode is flexibly written in order to allow to process various models and uniformly evaluatethem. For this reason, all model parameters were separated from the process itself, inorder to ensure a maximum of flexibility in the simulation study. Furthermore, we usedan object-oriented approach to enhance the reusability of program code. Finally, the dataderived at each step was saved for all models in order to ensure quality and comparabilityat all steps. The subsequent Section will briefly describe the concrete process which wasperformed at each level, according to Figure 5.1. In this context, we will also disclose therelevant parameters, which have been applied to the risk models, optimization algorithmand evaluation process. The underlying theory is covered fully in chapters 2 and 3 towhich we refer whenever appropriate.

5.1 Rolling Window Approach

For the sake of clarity, we reiterate that in the field of financial time series, the point in timet plays a crucial role. As we would like to simulate a true investment process in practice,it is crucial to respect the ordering of a time series. Hence, under no circumstances, futuredata should be incorporated into the estimation or optimization process, as this would leadto a forward-looking bias. In order to prevent this effect, all steps in Figure 5.1 followeda rolling window approach. At point in time t, only information up to this moment isavailable to the algorithm. In a subsequent step, the index is incremented to t + 1 and

45

46 Backtesting Framework

Raw Asset PricesSt,i

Risk Factor ChangesZt,i := lnSt,i

Xt,i = lnSt,i − lnSt−1,i

Risk Model Estimation• Unconditional Normal Model• CCC-GARCH (Normal)• CCC-GARCH (Student-t)• DCC-GARCH (Normal)• DCC-GARCH (Student-t)

Projection and Mapping to Portfolio Lossi.) Projection: x∗t+1 = S(M,Ft,u∗)ii.) Moment Matching: xi = m + Byiiii.) Mapping: ljt+1,i = −

∑dj=1 λjst,i(exp(x

jt+1,i)− 1)

Portfolio Optimization• Mean-CVaR Optimization:

arg minw

−w′µ


• Diversification-CVaR Optimization:

arg maxw

NEnt


Risk Model Evaluation• Volatility based evaluation• VaR based evaluation• CVaR based evaluation• Optimization Strategy Evaluation

Figure 5.1: Step-wise approach implemented to estimate a risk model, optimize portfoliosand evaluate the performance.

5.2 Methods, Materials and Parameters 47

the entire estimation and evaluation process is repeated. Future information may be usedsolely for out-of-sample evaluation but not for the estimation, projection or optimizationprocess.

5.2 Methods, Materials and Parameters

5.2.1 Development Environment

The entire code for this empirical study has been developed in R 2.15.2. Whereas wecreated the main framework, comprising all steps in the study, existing packages havebeen used partially where appropriate. The library RMGARCH has been employed for theMLE parameter estimation of the DCC and CCC models. Furthermore, a set of functionshas been adopted from PerformanceAnalytics for reporting purposes. The package nloptrserved as an interface to the sequential quadratic programming solver. Finally, the librarySNOW was used to enhance the estimation speed through multi-processing.

5.2.2 Raw Asset Prices and Risk Factors

We decided to perform the risk modelling and optimization backtests on the US marketsectors and a one month treasury bill which replicates the risk free asset. The main reasonfor this decision was the popular underlying active investment case of rotating allocationsbetween market sectors.

Whereas in reality, an investor may also consider exogenous economic data, we fully focusedon the aspects of risk management in the portfolio allocation decision. A convenientimplication of the sector rotation case is that the number of underlying assets is limited to10, thus, the estimation of model parameters is feasible and no dimensionality reductionis necessary. Moreover, in the US market, there has been a set of liquid exchange-tradedfunds (Select Sector SPDR) which replicate the market sectors. The availability of realinvestable securities is obviously an advantage, compared to a theoretically calculatedindex, which is not based on real transactions.

Table A.1 in Appendix A exposes the summary statistics of the Select Sector SPDRETF daily returns. It is worth noting that the kurtosis is higher than under the normalassumption for all assets. Particularly, the Financial Sector (XLF) exhibits fat tails. Thisis a further indication that a risk and optimization framework should not be based onnormality assumptions.

The one month treasury bill as a proxy for the risk free rate has been included into theanalysis, as from an investor’s perspective it is reasonable to preserve capital during highrisk periods and additionally borrow funds during low risk periods. The treasury bill isfully backed by the US government and as a short term instrument, it is virtually notexposed to interest rate risk. It is also assumed that the borrow rate equals the treasuryinterest rate which is not exactly true in practice but does not impair the conclusionsabout the risk forecast ability of the underlying model.

We retrieved raw price data for all assets for the period of 01/05/1999 - 31/08/2012, usingthe freely available datasource YAHOO Finance. In a further step, the prices were adjustedfor dividends. The treasury bill rates were retrieved directly from the U.S. Department of


Treasury. Based on the raw price data, the risk factor changes were calculated accordingto the theory, presented in Section 2.1.

5.2.3 Risk Model Estimation

The theory on the specific risk models may be found in Section 2.2. The conditional modelshave been implemented, using both multivariate normal and student-t distribution for therandom innovations.

It is difficult to determine the appropriate estimation window for the risk models. Whereas,a long window would reduce the estimation error, it could be subject to non-stationarity,as economic conditions are changing. We set this parameter to 3 year worth of dailydata, as we believe that approximately 750 days will provide sufficient data for estimationpurpose and will mostly fall within one economic cycle. In order to ensure comparability,we decided to apply the same length to all models equally.

5.2.4 Projection and Mapping to Portfolio Loss

Theoretically, the investment horizon could be arbitrarily defined. However, a multi-periodinvestment horizon is more difficult to project and evaluate. The simulation approachmay partially alleviate the increased theoretical complexity of multi- horizon forecasts,however at the cost of computational resources. Furthermore, a longer forecast horizon,would reduce the available data for the statistical tests, which are presented in chapter4. This is particularly harmful for the assessment of extreme risk forecasts such as CVaRand VaR, since only few tail observations are available for evaluation. For this reason, weapply a uniform forecast horizon of one day to all backtest runs. The number of simulatedscenarios has been set to 10’000 as this number is sufficiently high, in order to estimatetail risk measures and still computationally feasible at the same time.1 The methodologyof matching empirical to theoretical moments has been applied equally to all risk models.The theory on the projection step is presented in Section 2.3.

5.2.5 Portfolio Optimization

Two different approaches with respect to optimization have been implemented in thisthesis. Firstly, the mean-CVaR method maximizes expected portfolio returns, subjectto a CVaR constraint. The theory on the estimation of the expected asset returns isoutlined in Section 3.3.4. In the regressions, a weighted least squared estimation has beenperformed, using standardized exponential weights with a half-life of 64. The exponentialdecay ensures that more recent observations receive higher weight in the regression andthe estimated betas become more adaptive. Furthermore, the input data was winsorizedat the levels of 0.05 and 0.95 to prevent jumps due to return outliers. These practicallywell established methods belong to the standard approach of the risk modelling marketleader MSCI and can be exemplarily found in Menchero et al. (2013).

1With these parameter settings, the estimation and optimization of all models in this thesis wereperformed in approximately. 3 days, using an Apple MacBook Pro Core i5, 2.4. The SNOW package hasbeen used to parallelize computations, where no sequential calculations were required.

5.2 Methods, Materials and Parameters 49

The second approach in this thesis consists of diversification maximization, subject to aCVaR constraint. It is intuitive that if portfolios are optimally diversified among differentindependent risk factors, they are more resilient to estimation errors in the underlyingrisk models and thus represent more robust portfolios. The theory and more motivationalbackground on the exact implementation is provided in Section 3.4. It must be noted thatan advantage of the diversification approach is that expected asset returns do not need tobe estimated.

An important parameter at this stage is the CVaR risk constraint c in the optimization.A rather low threshold could lead to the result that the optimized portfolio is investedmainly in the risk free asset, whereas the contrary would lead to high concentration in arisky asset and excessive borrowing. Both extreme results would not represent a commoncase in Asset Management and thus would not be ideal for illustrative purpose in thisstudy. For this reason, we set the CVaR constraint at 1.5% which is approximately. 30%below the realized CVaR of a market portfolio. This makes sense as the objective of riskmanagement is to limit risk exposure to a desired threshold, which mostly is below themarket average.

For both optimization methods we assumed a traditional long-only mutual fund setup, inwhich no assets can be short-sold. As a consequence, the elements of the weights vector wis restricted to be positive in the optimization. The only exception represents the exposureto the risk free asset which is allowed to be negative in the case of borrowing funds duringtimes of low market risk. Furthermore, a budget constraint is imposed requiring that theportfolio weights sum up to 1. These linear constraints may be easily added to the originaloptimization problem, defined in (3.3.2.1) and (3.4.2.3).

w{i;i≥2} ≥ 0

d∑i=1

wi = 1

i = 1 is denoted the risk free asset.

(5.2.5.1)

5.2.6 Risk Model Evaluation

As presented in more detail in Chapter 4, we performed a forecast accuracy evaluationon the risk measures of volatility, VaR and CVaR which are defined in Section 3.2. Theevaluation was performed for all models on a single asset level (univariate) and a portfoliolevel (multivariate). Risk evaluation on the optimized portfolio is of particular interestas estimation error could be amplified due to the optimization algorithm. Furthermore,portfolio risk forecasts are evaluated for the naive equal weighted portfolio. Whereas itwould have been desirable to assess the risk models on a larger set of relevant portfolios,this analysis has been limited by the expensive nature of such data. However, as thisstudy examines mainly the combination of complex risk models and portfolio optimizationmethods, we regard the evaluation performed on the optimized portfolio as sufficient forpractical purposes in this context.

Chapter 6

Backtesting Results

6.1 Volatility Based Forecasts

Table 6.1: Bias and q-stat for the volatility forecasts for the assets and the equal weightportfolio.

Unconditional CCC-normal CCC-t DCC-normal DCC-tbias q-stat bias q-stat bias q-stat bias q-stat bias q-stat

equal 1.098 2.977 1.054 2.496 1.086 2.495 1.005 2.489 1.011 2.479XLY 1.050 2.753 0.983 2.285 0.988 2.278 0.983 2.285 0.988 2.278XLP 0.999 2.529 0.989 2.215 0.988 2.202 0.989 2.215 0.988 2.202XLE 1.050 2.625 0.989 2.288 0.998 2.279 0.989 2.288 0.998 2.279XLF 1.156 3.131 0.990 2.362 0.994 2.346 0.989 2.362 0.994 2.346XLV 1.019 2.724 0.972 2.345 0.969 2.334 0.972 2.345 0.969 2.334XLI 1.076 2.755 0.983 2.355 0.988 2.342 0.982 2.355 0.988 2.342XLB 1.046 2.622 0.976 2.321 0.978 2.312 0.976 2.321 0.978 2.312XLK 0.999 2.722 0.997 2.271 0.996 2.258 0.997 2.271 0.996 2.258XLU 1.038 2.783 0.984 2.290 0.989 2.273 0.984 2.290 0.989 2.273

Table 6.1 displays the bias statistics and q-stats of the single assets and equal-weightportfolio. Note that the unconditional model exhibits a substantial volatility underforecastbias for most assets of this study. For instance, the model underforecasts the volatility ofthe financial sector (XLF) by roughly 15% and the equal weight portfolio by almost 10%.

The CCC-normal and CCC-t models deliver fairly unbiased results for the univariate assetforecasts. However, they exhibit a considerable underforecast bias for the equal-weightportfolio. The results for the DCC-normal and DCC-t appear to be almost unbiased forboth the asset and portfolio level with a slight tendency to overforecast volatility for singleassets.

In terms of the cumulative Q-loss over time, the unconditional model clearly underperformsall conditional models, for both the asset and portfolio level. On an asset level, the DCCand CCC models deliver exactly the same forecasting performance. This is as expected,since the univariate forecasts are invariant to the applied correlation model. On a portfoliolevel, the DCC models consistently outperform their CCC counterparts by 1-2 basis points.

51

52 Backtesting Results

Additionally, applying student-t innovations for the model reduces the cumulative q-lossby another basis point.

The bias and Q-statistics for the optimized portfolios are incorporated into the 6.4 and6.5 model performance summary tables. Similar to the results of single assets and equalweight portfolios, the unconditional model performs poorly. For the mean-CVaR strategyit underforecasts the realized volatility by approximately 15% and accumulates a Q-loss of3.0345 which indicates a considerable underperformance with respect to all other models.However, note that the bias and Q-loss of the unconditional model is substantially reducedin the presence of a diversification-CVaR strategy. The same situation consistently appliesto all other risk models. Hence, the volatility forecasts appear to be more accurate onmean-diversification portfolios, than on their traditional mean-CVaR counterparts. Underboth optimization strategies, the Q-loss was minimized for the DCC-t risk model andall conditional models overforecast volatility in the presence of a diversification-CVaRoptimization strategy.

6.2 Value at Risk (VaR) Based Forecasts

The Value at Risk evaluation results for the equal weight portfolio under all risk modelsare presented in appendix A.2. We compared expected versus realized violations andperformed a statistical test as described in Section 4.2. The tables exhibit the results forthe aggregated time span as well as a per year resolution. The corresponding Figures inappendix A.6 plot the VaR and CVaR risk forecasts, as well as the realized loss over time.Those losses that exceeded VaR forecasts are marked in red.

Note that the actual VaR violations of the unconditional model and the DCC-normal modelcome closest to the expected number of violations (134) for the entire period. However, theunconditional version clearly exhibits substantial concentration of VaR exceedance overtime. For instance, in the middle of the financial crisis of 2008, there were 50 actual VaRviolations, while only 13 would have been expected. Figure A.11 illustrates the unequaldistribution over time. While there were virtually no violations during 2003-2006, therealized losses began exceeding forecasts in 2007. After the financial crisis in 2009, mostobservations are located below the boundaries, now suggesting a risk overforecast bias.Therefore, the null-hypothesis of accurate risk forecasts can clearly be rejected for mostyears of the evaluation period.

In contrast, the DCC-normal model seems to deliver the most accurate VaR forecastover time for the equal weighted portfolio, as only 20 violations could be observed duringthe most severe crisis year of 2008. Generally, the CCC models also outperformed theunconditional model with respect to risk clustering. However on an aggregated level, VaRviolations greatly exceeded the theoretical expectation (eg. 192 vs. 134 for the CCC-tmodel). While, this is not a substantial difference for the DCC models (eg. 158 vs. 134for the DCC-normal model), it should be noted that the statistical test of accurate riskforecasts still rejects the null-hypothesis at the significance level of 5% for all risk models.

The risk forecast evaluation results for the optimized strategies are presented in the sameformat in Appendix A.3 and A.4. The corresponding Figures are located in AppendixA.7. For mean-CVaR optimized portfolios, the observed VaR violations exhibit a patternsimilar to that of the equal weight portfolio. In all cases the number of actual violationssignificantly exceeded the theoretical number and consequently, the null-hypothesis was

6.3 Conditional Value at Risk (CVaR) Based Forecasts 53

rejected for all model candidates. Furthermore, we observe severe VaR violation clusteringfor the unconditional model during the financial crisis.

However, the results look different when assessing VaR forecasts on the diversification-CVaR strategy. The number of actual violations decreased considerably for all risk modelsand the null-hypothesis is only rejected for the CCC-t model. However, the clusteringeffect around the financial crisis can still be observed in all models and is most prominentin the unconditional version. Finally, the DCC-t model exhibits the smallest differenceof actual violations to the theoretical expectation (142 vs 134). In general, DCC modelssubstantially outperform their CCC counterparts by approximately 7% with respect to thenumber of VaR violations. There is a small but noticeable improvement, when applying tinstead of normal innovations to the risk model.

6.3 Conditional Value at Risk (CVaR) Based Forecasts

Table 6.2: Results of a statistical test for CVaR forecasting accuracy for the optimizedportfolios. The CVaR forecast bias is denoted E [Lt+1 − CV aRt+1].

Bias CF Lower CF Upper P-Value

Unconditional 0.4490 0.2728 0.6036 0.0000CCC-normal 0.2003 0.1118 0.2828 0.0000CCC-t 0.0750 -0.0111 0.1558 0.0814DCC-normal 0.1509 0.0625 0.2350 0.0012DCC-t 0.0802 -0.0071 0.1613 0.0658

The results of the statistical test for CVaR risk forecast evaluation are presented in Table6.2 for the equal weight portfolio. On average, the realized loss exceeded the CVaR forecastfor all models, when the corresponding VaR threshold was violated,. This bias is mostevident in the unconditional model (0.4490), followed by the CCC-normal (0.2003) andDCC-normal (0.1509) models. These versions clearly reject the null-hypothesis of accurateCVaR forecasts. For the CCC-t and DCC-t candidates however, the null-hypothesis cannotbe rejected at the significance level, which suggests that no evidence of inaccurate CVaRforecast could be found for these models.

Table 6.3: Results of a statistical test for CVaR forecasting accuracy, applied to theoptimized portfolios. The CVaR forecast bias is denoted E [Lt+1 − CV aRt+1].

Model & Strategy Bias CF Lower CF Upper P-Value

Unconditional mean-CVaR 0.4637 0.3308 0.5912 0.0000Unconditional diversification-CVaR 0.3304 0.2126 0.4381 0.0000

CCC-normal mean-CVaR 0.1756 0.1052 0.2397 0.0000CCC-normal diversifaction-CVaR 0.1248 0.0467 0.1960 0.0032

CCC-t mean-CVaR 0.1234 0.0545 0.1845 0.0008CCC-t diversification-CVaR 0.0341 -0.0347 0.0969 0.3138

DCC-normal mean-CVaR 0.1517 0.0784 0.2171 0.0000DCC-normal diversification-CVaR 0.1157 0.0471 0.1786 0.0026

DCC-t mean-CVaR 0.1190 0.0433 0.1843 0.0022DCC-t diversification-CVaR 0.0787 0.0070 0.1413 0.0350


The results for the optimized strategies are outlined in Table 6.3 and sorted by modelcategory. It is immediately clear that the diversification-CVaR strategy improves theforecast bias, compared to the mean-CVaR counterpart. Furthermore, other tendencies areobserved. T-innovations produce substantially better results than normal innovations forall strategies and models. However, the differences between a DCC versus CCC model areinconsistent. While the DCC versions performs well paired with the mean-CVaR strategy,this picture is partially inverted for the diversification-CVaR strategy. Surprisingly, thebest candidate in this case is the CCC-t model, which is almost unbiased and thus is theonly model that does not reject the null hypothesis of accurate CVaR risk forecasts. Forall other models, this hypothesis is rejected at the significance level.

6.4 Optimization Strategy Evaluation

The results of the performed backtests for all risk model approaches are presented inTable 6.4 for the mean-CVaR optimization, and in Table 6.5 for the diversification-CVaRapproach. Furthermore, the cumulative returns are plotted on a logarithmic scale againstan equal weight benchmark in Appendix A.5.

With respect to realized CVaR, the diversification-CVaR clearly outperformed the mean-CVaR strategy, as the empirical CVaR is closer to the optimization constraint of 1.5%.However, note that in all cases the constraint was exceeded. On the dimension of riskmodels, the DCC-t appears to most closely match the imposed CVaR constraints underboth optimization strategies. Furthermore, the application of t-innovations leads to con-sistently better results in all cases. Finally, the unconditional model greatly exceeds therisk boundary by approximately 30%.

From a return perspective, we observe that the diversification-CVaR approach consistentlyoutperforms its mean-CVaR counterpart by an average of almost 2% over all risk models.

The effect of risk models on realized returns is not uniform. Specifically the modelswith t-innovations produce slightly better results for the mean-CVaR portfolios. However,this effect is inverted when applying a diversification-CVaR approach. Furthermore, acomparison of CCC to DCC models yields mixed results which may not yield a definitiveconclusion. The unconditional model again appears as the worst performer, in terms ofrealized average returns too.

Finally, we compare the model candidates with respect to risk adjusted returns. Specifi-cally, we focus on the conditional Sharpe ratio due to its favorable theoretical properties.Again, the diversification-CVaR portfolios clearly outperform mean-CVaR optimized port-folios and t-innovations improve results on mean-CVaR portfolios, whereas this effect isinverted for the diversification-CVaR approach. There is no observed pattern with respectto distribution of innovations.

Not surprisingly, the unconditional model yields the worst results, as the conditional sharperatio is a function of return and CVaR. Finally, it is interesting to note that the realizedexcess kurtosis is substantially higher for the unconditional model compared to all condi-tional versions.

Table 6.4 also exhibits the realized statistics for a equal weight strategy. Note that thispassive benchmark yields better risk adjusted returns than all mean-CVaR optimized

6.4 Optimization Strategy Evaluation 55

strategies except for the CCC-t risk model. However, it is clearly outperformed by thediversification-CVaR portfolios using a conditional risk model.

Table 6.4: Performance summary statistics of the mean-CVaR optimization strategy fora target CVaR of 1.5%.

Equal Uncond. CCC-n. CCC-t DCC-n. DCC-t

Return (Ann.) % 7.3135 2.6986 5.1958 6.0345 4.8398 5.2946Excess Return (Ann.) % 4.9693 -0.2741 2.2230 3.0618 1.8671 2.3219Sd (Ann.) % 20.1759 13.8231 12.8541 12.6157 12.2369 12.0109Sharpe Ratio 0.2463 -0.0198 0.1729 0.2427 0.1526 0.1933Emp. VaR % 1.9254 1.3188 1.3712 1.3399 1.3180 1.2885Emp. CVaR % 3.0210 2.1217 1.8296 1.7947 1.7442 1.7062EVaR 0.0102 -0.0008 0.0064 0.0091 0.0056 0.0072Cond. Sharpe Ratio 0.0065 -0.0005 0.0048 0.0068 0.0042 0.0054Max Drawdown 0.5696 0.4298 0.2784 0.2612 0.2733 0.2605Skewness 0.0097 -0.0957 -0.2871 -0.2942 -0.2969 -0.2964Kurtosis 7.9257 6.5344 1.1334 1.0920 1.1875 1.1795Bias 1.1520 1.0031 1.0320 0.9518 0.9602Q-stat 3.0345 2.2990 2.3020 2.3132 2.2805

Table 6.5: Performance summary statistics of the diversification-CVaR optimizationstrategy for a target CVaR of 1.5%.

Equal Uncond. CCC-n. CCC-t DCC-n. DCC-t

Mean (Ann.) 7.3135 4.4124 7.4112 6.6514 7.4025 6.0468Excess Return (Ann.) % 4.9693 1.4397 4.4384 3.6787 4.4298 3.0740Sd (Ann.) % 20.1759 12.3727 12.0210 11.6162 11.5411 11.2795Sharpe Ratio 0.2463 0.1164 0.3692 0.3167 0.3838 0.2725Emp. VaR % 1.9254 1.2349 1.2593 1.2176 1.2309 1.2033Emp. CVaR % 3.0210 1.9020 1.6768 1.6082 1.6480 1.6016EVaR 0.0102 0.0046 0.0140 0.0120 0.0143 0.0101Cond. Sharpe Ratio 0.0065 0.0030 0.0105 0.0091 0.0107 0.0076Max Drawdown 0.5696 0.4989 0.2893 0.2960 0.3050 0.2456Skewness 0.0097 -0.3680 -0.3123 -0.2515 -0.3010 -0.3152Kurtosis 7.9257 4.9123 1.1591 0.9357 0.9506 0.9394Bias 1.0391 0.9441 0.9609 0.9076 0.9136Q-Stat 2.6885 2.2834 2.2923 2.2969 2.2605

Chapter 7

Discussion

As previously noted in the introduction, the unconditional estimation of risk from pastdata is still a prominent approach among practitioners in the financial industry as itdoes not rely on a sophisticated mathematical framework and is computationally straight-forward to implement. However, as the evaluation of the backtest clearly showed, thesesimplifications lead to unacceptable results.

First, the unconditional model generally underestimates risk, which is particularly dan-gerous during a financial crisis and may lead to a substantial drawdown in portfolio value.Furthermore, it was shown that VaR violations are highly clustered, indicating that an op-timized portfolio will be overexposed to risk during financial meltdowns and underexposedat stable periods. This clustering is undesirable among investors not only from a risk butalso from a return perspective, as we found empirical evidence of diminished realized port-folio returns. This effect is intuitive, as high risk periods mostly coincide with negativereturns. In terms of CVaR, the empirical results clearly showed that the unconditionalmodel violated the imposed constraint, which leads us to conclude that an effective riskcontrol is not possible.

It is not surprising that the unconditional model does not provide satisfactory results.As there was clear evidence of non-normality and conditional heteroscedasticity in theunderlying asset returns, the unconditional model violates both assumptions. However,it is important to realize that these simplifications do not only exhibit a marginal effecton investment results, as mostly argued among practitioners. Hence, it should not beassumed that returns are iid. realizations and directly estimate distributional momentson them.

Whereas we have clearly demonstrated that conditionality needs to be incorporated inthe univariate volatility estimates, it remains to clarify if correlations also need to bedynamically modeled. The volatility forecast results showed that the DCC models are ableto deliver better results in terms of bias and Q-stat than their CCC counterparts, whichspeaks in favor of the former. Furthermore, modeling dynamic correlations leads to betterrisk forecasts in terms of VaR and CVaR. These results are not surprising as the statisticaltest of Section 2.2.5 already suggested the presence of conditional correlations. However, itis worth noting that the realized CVaR and return of the backtested optimization strategiesdo not clearly confirm this initial finding. Since no intuitive reason for this discrepancycan be found we may be tempted to attribute it to the randomness in the experiment.Finally, we consider the evidence of the comprehensive tests with respect to risk forecast

57

58 Discussion

accuracy as sufficient to conclude that the incorporation of conditional correlations into arisk-optimization model is suggested.

Furthermore, we found consistent evidence that applying the student t-distribution toinnovations leads to better results than the normal counterpart. Whereas these findingsare not clearly visible for volatility as a risk measure, the innovation distribution playsa crucial role for the coherent tail risk measure CVaR, where the forecast accuracy wassubstantially better for all t models. Furthermore, it was found that the realized CVaR ofthe optimized strategies is lower among all applied models. Since financial returns possesslarger tails than a normal distribution, this result is expected as the t-models are ableto better model the tails of the portfolio distribution. We must admit that this patterncould not be confirmed for the VaR-based assessment. However, this does not change ourconclusion as VaR was not the central risk measure to our thesis and exhibits theoreticaldeficiencies. For example, it does not provide any information about the extent of a loss, ifthe VaR level is exceeded. Relying solely on the count of violations and ignoring the extent,does not seem the most appropriate method to assess the effect of applying t-innovations.As a result of mostly positive evidence, we conclude that a student-t innovation should bepreferred in the case of the underlying assets in this thesis. However, to generalize thisresult, the framework should be applied to a much larger asset base.

The relatively new diversification approach of Meucci (2009a), which has never beentested in a CVaR-constraint-based framework, delivered very promising results. On alllevels, the risk forecast accuracy greatly increased when the risk model was applied toa diversification-CVaR compared to a mean-CVaR strategy. Furthermore, the realizedCVaR was substantially closer to the optimization constraint. We emphasize that thisis very important from an investor’s perspective, since the CVaR constraint is regardedas an upper tolerance boundary that should not be exceeded over a long time-frame. Awell-diversified and stable portfolio helps particularly during a severe financial crisis, asa substantial risk constraint violation can rapidly lead to a severe drawdown in portfoliovalue and an asset manager going out of business.

In light of these clear results, we conclude that optimizing diversification to independentrisk factors is a viable approach to tackling the curse of estimation error that often leadsto poorly optimized portfolios. Surprisingly, better diversified portfolios not only lead tosubstantial improvements in terms of realized risk and forecast accuracy, but also resultin consistently higher excess returns that investors may particularly welcome. Certainly,this also raises doubts about the effectiveness of estimating expected asset returns inthe mean-CVaR approach. Specifically, if the expected returns are known, the returnmaximization approach should logically deliver the highest performance. However, if arisk parity approach that is fully invariant to returns can produce better results, it isreasonable to ask whether the estimator of returns is too noisy to serve as a reliable inputfor portfolio optimization.

Additionally, the weakness of a simple mean-CVaR optimization strategy has been shownby the comparison to the naive equal weight strategy. If an optimized portfolio cannotoutperform a passive benchmark in terms of risk adjusted returns, the benefit of the entirerisk modeling and optimization process is questionable. Note, however, that a passivebenchmark cannot consider a risk constraint. Admittedly, the expected returns estimationwas simplified in this thesis and in several cases it can be improved, using exogenouseconomic data. However, we conclude that if an investor does not possess a proven abilityto reliably estimate expected returns, he should favor a risk parity approach and assume

7.1 Future Research 59

that exposure to risk will be rewarded over a sufficiently long investment period.

In this thesis, enhancements to the naive risk modeling and optimization process werepresented. It was clearly shown that volatilities are required to be conditionally modeledin order to provide acceptable results. Furthermore, incorporating conditional correlationsand student-t innovations lead to improved results. Finally, a maximum diversificationapproach greatly enhances the stability of portfolios and increases realized returns. Inlight of these findings, we see substantial potential for asset managers, who historicallyrelied on simplified market assumptions, to improve the quality and performance of theirportfolios.

Finally, we showed that a flexible scenario-based framework can embed and evaluate a va-riety of different model candidates and overcome the analytical intractability of complexrisk and optimization methods. In light of increasingly faster IT-infrastructure, computa-tional costs are relatively low and therefore justify higher model complexity in the questfor modeling true market conditions.

7.1 Future Research

While clear empirical evidence supports our thesis that wrong assumptions in risk andportfolio management can lead to a potentially harmful result, this work may be extendedon the main topics of risk modeling, portfolio optimization and risk forecast evaluation.

We have tested a variety of conditional models, including dynamic dependence structure.As mentioned, there have been further propositions such as dynamic copula models whichaccount for a richer dependence structure. Furthermore, it would be of interest to studythe effect of conditionality in higher moments, such as skewness and kurtosis. While wewould expect such a model to be theoretically more accurate, it remains questionable ifthe additional estimation error would outweigh the theoretical advantage due to the highercomplexity.

The diversification approach delivered promising results and future research into this di-rection is warranted. It would be particularly interesting to blend the entropy criteria withthe expected returns in the optimization process. This would allow to create more stableportfolios while still accounting for the investor’s prior market view. Based on the confi-dence of his forecast, the weight of both criteria in the optimization can be determined.

Finally, it would be helpful to calculate confidence intervals of the realized portfolio statis-tics in Table 6.4 and perform statistical testing. Since the realized time series itself canbe path-dependent, a simple non-parametric bootstrap is not warranted. In this context,it would be beneficial to derive a sound evaluation framework which accounts for differenttime series properties of the realized portfolio returns.

60 Discussion

Bibliography

Acerbi, C. and D. Tasche (2002, July). On the coherence of expected shortfall. Journal ofBanking & Finance 26 (7), 1487–1503.

Agarwal, V. and N. Y. Naik (2004). Risks and portfolio decisions involving hedge funds.Review of Financial Studies 17 (1), 63–98.

Alexander, S., T. F. Coleman, and Y. Li (2006). Minimizing CVaR and VaR for a portfolioof derivatives. Journal of Banking & Finance 30 (2), 583–605.

Amenc, N., F. Goltz, and A. Lioui (2011). Practitioner portfolio construction and perfor-mance measurement: Evidence from Europe. Financial Analysts Journal 67 (3).

Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath (1999). Coherent measures of risk.Mathematical Finance 9 (3), 203–228.

Bachelier, L. (1900). Theorie de la speculation. Gauthier-Villars, Imprimeur-Libraire.

Bardou, O., N. Frikha, et al. (2009). Computing VaR and CVaR using stochastic approx-imation and adaptive unconstrained importance sampling. Monte Carlo Methods andApplications 15 (3), 173–210.

Black, F. and R. B. Litterman (1991). Asset allocation: Combining investor views withmarket equilibrium. The Journal of Fixed Income 1 (2), 7–18.

Boggs, P. T. and J. W. Tolle (1995). Sequential quadratic programming. Acta numer-ica 4 (1), 1–51.

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journalof Econometrics 31 (3), 307–327.

Carhart, M. M. (1997). On persistence in mutual fund performance. The Journal offinance 52 (1), 57–82.

DeMiguel, V., L. Garlappi, and R. Uppal (2009). Optimal versus naive diversification: Howinefficient is the 1/n portfolio strategy? Review of Financial Studies 22 (5), 1915–1953.

Dowd, K. (2000). Adjusting for risk: An improved sharpe ratio. International Review ofEconomics & Finance 9 (3), 209–222.

Efron, B. and R. J. Tibshirani (1994). An Introduction to the Bootstrap. Chapman andHall/CRC.

Engle, R. (2002). Dynamic conditional correlation: A simple class of multivariate gen-eralized autoregressive conditional heteroskedasticity models. Journal of Business andEconomic Statistics 20, 339–350.

61

62 BIBLIOGRAPHY

Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of thevariance of United Kingdom inflation. Econometrica 50 (4), 987–1007.

Engle, R. F. and K. Sheppard (2001). Theoretical and empirical properties of dynamicconditional correlation multivariate GARCH. Technical report.

Fama, E. (1965). Random walks in stock market prices. Financial Analysts Journal 21 (5),55–59.

Frey, R. and A. J. McNeil (2002, July). VaR and expected shortfall in portfolios ofdependent credit risks: Conceptual and practical insights. Journal of Banking & Fi-nance 26 (7), 1317–1334.

Ghalanos, A. (2013, April). Portfolio Optimization in parma (Version 1.5-0).

Holton, G. A. (2002, August). History of value-at-risk: 1922-1998. Working paper.

Krokhmal, P., J. Palmquist, and S. Uryasev (2002). Portfolio optimization with conditionalvalue-at-risk objective and constraints. Journal of Risk 4, 43–68.

Lim, A. E., J. G. Shanthikumar, and G.-Y. Vahn (2011). Conditional value-at-risk inportfolio optimization: Coherent but fragile. Operations Research Letters 39 (3), 163 –171.

Lo, A. W. and A. MacKinlay (2001). A Non-Random Walk Down Wall Street. PrincetonUniversity Press.

M. Concepcion Ausin, H. F. L. (2010). Time-varying joint distribution through copulas.Computational Statistics and Data Analysis 54.

Mardia, K., J. T. Kent, and J. M. Bibby (1979). Multivariate Analysis. Academic Press.

Markowitz, H. (1952). Portfolio selection. The Journal of Finance 7 (1), pp. 77–91.

McNeil, A. J. and R. Frey (2000, November). Estimation of tail-related risk measures forheteroscedastic financial time series: an extreme value approach. Journal of EmpiricalFinance 7 (3-4), 271–300.

McNeil, A. J., R. Frey, and P. Embrechts (2005). Quantitative Risk Management. Prince-ton University Press.

Menchero, J., A. Morozov, and A. Pasqua (2013). Predicting risk at short horizons.

Meng, N. (2012). Creating predictable portfolios using canonical correlation analysis(CCA). Technical report.

Meucci, A. (2009a, May). Managing diversification. Available at SSRN:http://ssrn.com/abstract=1358533, 74–79.

Meucci, A. (2009b). Risk and asset allocation. Springer.

Meucci, A. (2009c). Simulations with exact means and covariances. Available at SSRN1415699 .

Meucci, A. (2010). Quant nugget 2: Linear vs. compounded returns–common pitfalls inportfolio management. GARP Risk Professional , 49–51.

Meucci, A. (2011). ”The prayer” ten-step checklist for advanced risk and portfolio man-agement.

BIBLIOGRAPHY 63

Meucci, A. (2013, May). Advanced risk and portfolio management bootcamp.

Orskaug, E. (2009, June). Multivariate DCC-GARCH model - with various error distri-butions. Technical report, Norsk Regnesentral.

Patton, A. J. (2011). Volatility forecast comparison using imperfect volatility proxies.Journal of Econometrics 160 (1), 246–256.

Petkov, P. H., N. D. Christov, and M. M. Konstantinov (1991). Computational methodsfor linear control systems. Prentice Hall International (UK) Ltd.

Qian, E. (2011). Risk parity and diversification. Journal of Investing 20 (1), 119.

Rockafellar, R. and S. Uryasev (2000). Optimization of conditional value-at-risk. Journalof risk 2, 21–42.

Sharpe, W. F. (1970). Portfolio theory and capital markets, Volume 217. McGraw-HillNew York.

Tsay, R. S. (2002). Analysis of Financial Time Series. John Wiley & Sons, Inc.

64 BIBLIOGRAPHY

Appendix A

Tables and Figures

A.1 Summary Statistics of the Underlying Assets

Table A.1: Daily summary statistics for the Select Sector SPDR ETFs for the timeperiod of 01/01/1999 until 08/31/2012.

XLY XLP XLE XLF XLV XLI XLB XLK XLU

Observations 3439 3439 3439 3439 3439 3439 3439 3439 3439NAs 0 0 0 0 0 0 0 0 0Minimum % -12.36 -6.21 -15.60 -19.36 -10.29 -9.88 -13.25 -9.05 -8.91Quartile 1 % -0.75 -0.51 -0.94 -0.87 -0.58 -0.71 -0.86 -0.88 -0.59Median % 0.04 0.04 0.08 0.00 0.03 0.07 0.05 0.09 0.07Arithmetic Mean % 0.02 0.02 0.04 -0.01 0.02 0.02 0.02 0.00 0.02Geometric Mean % 0.01 0.01 0.02 -0.03 0.01 0.01 0.01 -0.02 0.01Quartile 3 % 0.82 0.56 1.09 0.85 0.65 0.77 0.95 0.87 0.70Maximum % 9.33 6.66 15.25 27.26 11.38 10.17 13.15 14.93 11.40Variance % 0.02 0.01 0.04 0.05 0.01 0.02 0.03 0.03 0.02Stdev % 1.57 1.03 1.88 2.23 1.22 1.47 1.70 1.83 1.29Skewness -0.19 -0.09 -0.42 0.32 -0.08 -0.21 -0.09 0.27 0.13Kurtosis 4.61 4.08 8.35 17.47 8.28 4.78 4.68 4.77 8.18

65

66 Tables and Figures

A.2 VaR Evaluation for the Equal Weight Portfolio

Table A.2: Expected versus actual violations of VaR for the equal weight portfolio underthe unconditional model.

Trading Days Exp. Violations Violations CF Lower CF Upper P-Value

2001 1 0 0 0.0000 0.9750 1.00002002 252 13 29 19.7668 40.5916 0.00002003 252 13 3 0.6204 8.6674 0.00212004 252 13 0 0.0000 3.6620 0.00002005 252 13 3 0.6204 8.6674 0.00212006 251 13 12 6.2610 20.5877 1.00002007 251 13 33 23.1568 45.0941 0.00002008 253 13 50 38.0479 63.7830 0.00002009 252 13 14 7.7369 23.0471 0.66332010 252 13 5 1.6312 11.5146 0.02842011 252 13 9 4.1476 16.8114 0.38402012 169 8 2 0.2428 7.1139 0.0195

Total 2689 134 160 136.7573 185.8606 0.0268

Table A.3: Expected versus actual violations of VaR for the equal weight portfolio underthe CCC-normal model.


2001 1 0 0 0.0000 0.9750 1.00002002 252 13 24 15.6184 34.8741 0.00332003 252 13 8 3.4785 15.5211 0.24472004 252 13 12 6.2608 20.5891 1.00002005 252 13 13 6.9931 21.8233 0.88462006 251 13 14 7.7373 23.0453 0.66242007 251 13 24 15.6194 34.8708 0.00322008 253 13 21 13.1850 31.3928 0.02862009 252 13 18 10.8057 27.8562 0.14512010 252 13 16 9.2543 25.4674 0.31032011 252 13 14 7.7369 23.0471 0.66332012 169 8 7 2.8413 14.1076 0.7259

Total 2689 134 171 146.9868 197.6048 0.0017

A.2 VaR Evaluation for the Equal Weight Portfolio 67

Table A.4: Expected versus actual violations of VaR for the equal weight portfolio underthe CCC-t model.


2001 1 0 0 0.0000 0.9750 1.00002002 252 13 25 16.4393 36.0259 0.00122003 252 13 10 4.8360 18.0848 0.56252004 252 13 13 6.9931 21.8233 0.88462005 252 13 14 7.7369 23.0471 0.66332006 251 13 14 7.7373 23.0453 0.66242007 251 13 26 17.2661 37.1697 0.00072008 253 13 23 14.8013 33.7208 0.00562009 252 13 18 10.8057 27.8562 0.14512010 252 13 17 10.0261 26.6654 0.19302011 252 13 23 14.8023 33.7177 0.00552012 169 8 8 3.4908 15.4027 1.0000

Total 2689 134 191 165.6593 218.8849 0.0000

Table A.5: Expected versus actual violations of VaR for the equal weight portfolio underthe DCC-normal model.


2001 1 0 0 0.0000 0.9750 1.00002002 252 13 21 13.1857 31.3899 0.02042003 252 13 5 1.6312 11.5146 0.02842004 252 13 13 6.9931 21.8233 0.88462005 252 13 13 6.9931 21.8233 0.88462006 251 13 12 6.2610 20.5877 1.00002007 251 13 23 14.8032 33.7145 0.00532008 253 13 20 12.3853 30.2206 0.04212009 252 13 18 10.8057 27.8562 0.14512010 252 13 15 8.4910 24.2615 0.46822011 252 13 11 5.5412 19.3434 0.77242012 169 8 7 2.8413 14.1076 0.7259

Total 2689 134 158 134.9008 183.7220 0.0416


Table A.6: Expected versus actual violations of VaR for the equal weight portfolio underthe DCC-t model.


2001 1 0 0 0.0000 0.9750 1.00002002 252 13 21 13.1857 31.3899 0.02042003 252 13 7 2.8324 14.2110 0.11222004 252 13 13 6.9931 21.8233 0.88462005 252 13 13 6.9931 21.8233 0.88462006 251 13 13 6.9934 21.8217 0.88432007 251 13 24 15.6194 34.8708 0.00322008 253 13 20 12.3853 30.2206 0.04212009 252 13 18 10.8057 27.8562 0.14512010 252 13 15 8.4910 24.2615 0.46822011 252 13 16 9.2543 25.4674 0.31032012 169 8 7 2.8413 14.1076 0.7259

Total 2689 134 167 143.2635 193.3377 0.0053

A.3 VaR Evaluation for the Mean-CVaR Optimized Portfolios 69

A.3 VaR Evaluation for the Mean-CVaR Optimized Port-folios

Table A.7: Expected versus actual violations of VaR for the mean-CVaR optimizedportfolio under the unconditional model.


2002 252 13 25 16.4393 36.0259 0.00122003 252 13 3 0.6204 8.6674 0.00212004 252 13 0 0.0000 3.6620 0.00002005 252 13 8 3.4785 15.5211 0.24472006 251 13 15 8.4913 24.2596 0.46692007 251 13 31 21.4556 42.8474 0.00002008 253 13 57 44.3591 71.3006 0.00002009 252 13 16 9.2543 25.4674 0.31032010 252 13 0 0.0000 3.6620 0.00002011 252 13 7 2.8324 14.2110 0.11222012 169 8 2 0.2428 7.1139 0.0195

Total 2688 134 164 140.4738 190.1344 0.0102

Table A.8: Expected versus actual violations of VaR for the mean-CVaR optimizedportfolio under the CCC-normal model.


2002 252 13 24 15.6184 34.8741 0.00332003 252 13 10 4.8360 18.0848 0.56252004 252 13 17 10.0261 26.6654 0.19302005 252 13 17 10.0261 26.6654 0.19302006 251 13 13 6.9934 21.8217 0.88432007 251 13 26 17.2661 37.1697 0.00072008 253 13 23 14.8013 33.7208 0.00562009 252 13 17 10.0261 26.6654 0.19302010 252 13 15 8.4910 24.2615 0.46822011 252 13 16 9.2543 25.4674 0.31032012 169 8 8 3.4908 15.4027 1.0000

Total 2688 134 186 160.9832 213.5726 0.0000


Table A.9: Expected versus actual violations of VaR for the mean-CVaR optimizedportfolio under the CCC-t model.


2002 252 13 26 17.2649 37.1734 0.00072003 252 13 11 5.5412 19.3434 0.77242004 252 13 14 7.7369 23.0471 0.66332005 252 13 17 10.0261 26.6654 0.19302006 251 13 12 6.2610 20.5877 1.00002007 251 13 25 16.4404 36.0224 0.00122008 253 13 25 16.4383 36.0294 0.00132009 252 13 19 11.5925 29.0402 0.08012010 252 13 16 9.2543 25.4674 0.31032011 252 13 21 13.1857 31.3899 0.02042012 169 8 8 3.4908 15.4027 1.0000

Total 2688 134 194 168.4680 222.0690 0.0000

Table A.10: Expected versus actual violations of VaR for the mean-CVaR optimizedportfolio under the DCC-normal model.


2002 252 13 17 10.0261 26.6654 0.19302003 252 13 7 2.8324 14.2110 0.11222004 252 13 15 8.4910 24.2615 0.46822005 252 13 17 10.0261 26.6654 0.19302006 251 13 11 5.5414 19.3421 0.77232007 251 13 23 14.8032 33.7145 0.00532008 253 13 25 16.4383 36.0294 0.00132009 252 13 17 10.0261 26.6654 0.19302010 252 13 13 6.9931 21.8233 0.88462011 252 13 13 6.9931 21.8233 0.88462012 169 8 7 2.8413 14.1076 0.7259

Total 2688 134 165 141.4035 191.2023 0.0089

Table A.11: Expected versus actual violations of VaR for the mean-CVaR optimizedportfolio under the DCC-t model.


2002 252 13 19 11.5925 29.0402 0.08012003 252 13 9 4.1476 16.8114 0.38402004 252 13 15 8.4910 24.2615 0.46822005 252 13 14 7.7369 23.0471 0.66332006 251 13 11 5.5414 19.3421 0.77232007 251 13 22 13.9921 32.5534 0.01252008 253 13 22 13.9904 32.5594 0.01302009 252 13 17 10.0261 26.6654 0.19302010 252 13 13 6.9931 21.8233 0.88462011 252 13 15 8.4910 24.2615 0.46822012 169 8 7 2.8413 14.1076 0.7259

Total 2688 134 164 140.4738 190.1344 0.0102

A.4 VaR Evaluation for the Diversification-CVaR Optimized Portfolio 71

A.4 VaR Evaluation for the Diversification-CVaR OptimizedPortfolio

Table A.12: Expected versus actual violations of VaR for the diversification-CVaR opti-mized portfolio under the unconditional model.


2002 252 13 18 10.8057 27.8562 0.14512003 252 13 3 0.6204 8.6674 0.00212004 252 13 0 0.0000 3.6620 0.00002005 252 13 4 1.0940 10.1154 0.00852006 251 13 13 6.9934 21.8217 0.88432007 251 13 32 22.3045 43.9723 0.00002008 253 13 49 37.1537 62.7018 0.00002009 252 13 18 10.8057 27.8562 0.14512010 252 13 6 2.2142 12.8773 0.05882011 252 13 4 1.0940 10.1154 0.00852012 169 8 3 0.6212 8.6188 0.0513

Total 2688 134 150 127.4858 175.1562 0.1699

Table A.13: Expected versus actual violations of VaR for the diversification-CVaR opti-mized portfolio under the CCC-normal model.


2002 252 13 19 11.5925 29.0402 0.08012003 252 13 4 1.0940 10.1154 0.00852004 252 13 12 6.2608 20.5891 1.00002005 252 13 13 6.9931 21.8233 0.88462006 251 13 11 5.5414 19.3421 0.77232007 251 13 23 14.8032 33.7145 0.00532008 253 13 22 13.9904 32.5594 0.01302009 252 13 14 7.7369 23.0471 0.66332010 252 13 11 5.5412 19.3434 0.77242011 252 13 17 10.0261 26.6654 0.19302012 169 8 6 2.2204 12.7883 0.4815

Total 2688 134 152 129.3379 177.2993 0.1213


Table A.14: Expected versus actual violations of VaR for the diversification-CVaR opti-mized portfolio under the CCC-t model.


2002 252 13 17 10.0261 26.6654 0.19302003 252 13 7 2.8324 14.2110 0.11222004 252 13 14 7.7369 23.0471 0.66332005 252 13 13 6.9931 21.8233 0.88462006 251 13 12 6.2610 20.5877 1.00002007 251 13 21 13.1865 31.3871 0.01992008 253 13 26 17.2637 37.1771 0.00072009 252 13 14 7.7369 23.0471 0.66332010 252 13 15 8.4910 24.2615 0.46822011 252 13 17 10.0261 26.6654 0.19302012 169 8 6 2.2204 12.7883 0.4815

Total 2688 134 162 138.6152 187.9979 0.0168

Table A.15: Expected versus actual violations of VaR for the diversification-CVaR opti-mized portfolio under the DCC-normal model.


2002 252 13 14 7.7369 23.0471 0.66332003 252 13 3 0.6204 8.6674 0.00212004 252 13 11 5.5412 19.3434 0.77242005 252 13 12 6.2608 20.5891 1.00002006 251 13 15 8.4913 24.2596 0.46692007 251 13 25 16.4404 36.0224 0.00122008 253 13 19 11.5919 29.0427 0.08092009 252 13 15 8.4910 24.2615 0.46822010 252 13 12 6.2608 20.5891 1.00002011 252 13 12 6.2608 20.5891 1.00002012 169 8 7 2.8413 14.1076 0.7259

Total 2688 134 145 122.8607 169.7936 0.3525

Table A.16: Expected versus actual violations of VaR for the diversification-CVaR opti-mized portfolio under the DCC-t model.


2002 252 13 13 6.9931 21.8233 0.88462003 252 13 5 1.6312 11.5146 0.02842004 252 13 12 6.2608 20.5891 1.00002005 252 13 10 4.8360 18.0848 0.56252006 251 13 14 7.7373 23.0453 0.66242007 251 13 23 14.8032 33.7145 0.00532008 253 13 18 10.8052 27.8585 0.14592009 252 13 14 7.7369 23.0471 0.66332010 252 13 11 5.5412 19.3434 0.77242011 252 13 15 8.4910 24.2615 0.46822012 169 8 7 2.8413 14.1076 0.7259

Total 2688 134 142 120.0893 166.5723 0.5066

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Date

−0.

5−

0.4

−0.

3−

0.2

−0.

10.

0

Dra

wdo

wn

Figure A.1: Performance summary of the mean-CVaR opti-mized strategy (black) under the unconditional risk model ver-sus the equal weight benchmark portfolio (red).

−0.

20.

00.

20.

40.

60.

8

Cum

ulat

ive

Ret

urn

−0.

04−

0.02

0.00

0.02

Dai

ly R

etur

n2001−12−31 2003−01−02 2004−01−02 2005−01−03 2006−01−03 2007−07−02 2008−07−01 2009−07−01 2010−07−01 2011−07−01 2012−07−02

Date

−0.

5−

0.4

−0.

3−

0.2

−0.

10.

0

Dra

wdo

wn

Figure A.2: Performance summary of the mean-CVaR opti-mized strategy (black) under the CCC-normal risk model versusthe equal weight benchmark portfolio (red).

74

Table

san

dF

igure

s−

0.2

0.0

0.2

0.4

0.6

0.8

Cum

ulat

ive

Ret

urn

−0.

04−

0.02

0.00

0.02

Dai

ly R

etur

n

2001−12−31 2003−01−02 2004−01−02 2005−01−03 2006−01−03 2007−07−02 2008−07−01 2009−07−01 2010−07−01 2011−07−01 2012−07−02

Date

−0.

5−

0.4

−0.

3−

0.2

−0.

10.

0

Dra

wdo

wn

Figure A.3: Performance summary of the mean-CVaR opti-mized strategy (black) under the CCC-t risk model versus theequal weight benchmark portfolio (red).

−0.

20.

00.

20.

40.

60.

8

Cum

ulat

ive

Ret

urn

−0.

04−

0.02

0.00

0.02

Dai

ly R

etur

n

2001−12−31 2003−01−02 2004−01−02 2005−01−03 2006−01−03 2007−07−02 2008−07−01 2009−07−01 2010−07−01 2011−07−01 2012−07−02

Date

−0.

5−

0.4

−0.

3−

0.2

−0.

10.

0

Dra

wdo

wn

Figure A.4: Performance summary of the mean-CVaR opti-mized strategy (black) under the DCC-normal risk model versusthe equal weight benchmark portfolio (red).

A.4

VaR

Evalu

atio

nfo

rth

eD

iversifi

catio

n-C

VaR

Optim

ized

Portfo

lio75

−0.

20.

00.

20.

40.

60.

8

Cum

ulat

ive

Ret

urn

−0.

04−

0.02

0.00

0.02

Dai

ly R

etur

n

2001−12−31 2003−01−02 2004−01−02 2005−01−03 2006−01−03 2007−07−02 2008−07−01 2009−07−01 2010−07−01 2011−07−01 2012−07−02

Date

−0.

5−

0.4

−0.

3−

0.2

−0.

10.

0

Dra

wdo

wn

Figure A.5: Performance summary of the mean-CVaR opti-mized strategy (black) under the DCC-t risk model versus theequal weight benchmark portfolio (red).

−0.

20.

00.

20.

40.

60.

8

Cum

ulat

ive

Ret

urn

−0.

06−

0.02

0.02

0.04

Dai

ly R

etur

n

2001−12−31 2003−01−02 2004−01−02 2005−01−03 2006−01−03 2007−07−02 2008−07−01 2009−07−01 2010−07−01 2011−07−01 2012−07−02

Date

−0.

5−

0.4

−0.

3−

0.2

−0.

10.

0

Dra

wdo

wn

Figure A.6: Performance summary of the diversification-CVaR optimized strategy (black) under the unconditional riskmodel versus the equal weight benchmark portfolio (red).

76

Table

san

dF

igure

s−

0.2

0.0

0.2

0.4

0.6

0.8

1.0

Cum

ulat

ive

Ret

urn

−0.

04−

0.02

0.00

0.02

Dai

ly R

etur

n

2001−12−31 2003−01−02 2004−01−02 2005−01−03 2006−01−03 2007−07−02 2008−07−01 2009−07−01 2010−07−01 2011−07−01 2012−07−02

Date

−0.

5−

0.4

−0.

3−

0.2

−0.

10.

0

Dra

wdo

wn

Figure A.7: Performance summary of the diversification-CVaR optimized strategy (black) under the CCC-normal riskmodel versus the equal weight benchmark portfolio.

−0.

20.

00.

20.

40.

60.

8

Cum

ulat

ive

Ret

urn

−0.

04−

0.02

0.00

0.02

Dai

ly R

etur

n

2001−12−31 2003−01−02 2004−01−02 2005−01−03 2006−01−03 2007−07−02 2008−07−01 2009−07−01 2010−07−01 2011−07−01 2012−07−02

Date

−0.

5−

0.4

−0.

3−

0.2

−0.

10.

0

Dra

wdo

wn

Figure A.8: Performance summary of the diversification-CVaR optimized strategy (black) under the CCC-t risk modelversus the equal weight benchmark portfolio (red).

A.4

VaR

Evalu

atio

nfo

rth

eD

iversifi

catio

n-C

VaR

Optim

ized

Portfo

lio77

0.0

0.5

1.0

Cum

ulat

ive

Ret

urn

−0.

04−

0.02

0.00

0.02

Dai

ly R

etur

n

2001−12−31 2003−01−02 2004−01−02 2005−01−03 2006−01−03 2007−07−02 2008−07−01 2009−07−01 2010−07−01 2011−07−01 2012−07−02

Date

−0.

5−

0.4

−0.

3−

0.2

−0.

10.

0

Dra

wdo

wn

Figure A.9: Performance summary of the diversification-CVaR optimized strategy (black) under the DCC-normal riskmodel versus the equal weight benchmark portfolio (red).

−0.

20.

00.

20.

40.

60.

8

Cum

ulat

ive

Ret

urn

−0.

04−

0.02

0.00

0.02

Dai

ly R

etur

n

2001−12−31 2003−01−02 2004−01−02 2005−01−03 2006−01−03 2007−07−02 2008−07−01 2009−07−01 2010−07−01 2011−07−01 2012−07−02

Date

−0.

5−

0.4

−0.

3−

0.2

−0.

10.

0

Dra

wdo

wn

Figure A.10: Performance summary of the diversification-CVaR optimized strategy (black) under the DCC-t risk modelversus the equal weight benchmark portfolio (red).

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san

dF

igure

s

A.6 Realized Loss vs Risk Forecast for Equal-Weight Portfolios

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

02

46

8

Loss

%

Figure A.11: Risk forecast (purple=VaR, blue=CVaR) versusrealized loss for the equal weight strategy under the uncondi-tional model.

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

02

46

8

Loss

%

Figure A.12: Risk forecast (purple=VaR, blue=CVaR) ver-sus realized loss for the equal weight strategy under the CCC-normal model.

A.6

Realiz

ed

Loss

vs

Risk

Fore

cast

for

Equal-W

eig

ht

Portfo

lios

79

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

02

46

8

Loss

%

Figure A.13: Risk forecast (purple=VaR, blue=CVaR) ver-sus realized loss for the equal weight strategy under the CCC-tmodel.

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

02

46

8

Loss

%

Figure A.14: Risk forecast (purple=VaR, blue=CVaR) ver-sus realized loss for the equal weight strategy under the DCC-normal model.

80

Table

san

dF

igure

s

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

02

46

8

Loss

%

Figure A.15: Risk forecast (purple=VaR, blue=CVaR) ver-sus realized loss for the equal weight strategy under the DCC-normal model.

A.6

Realiz

ed

Loss

vs

Risk

Fore

cast

for

Equal-W

eig

ht

Portfo

lios

81

A.7 Realized Loss vs Risk Forecast for Optimized Portfolios

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

01

23

45

6

Loss

%

Figure A.16: Risk forecast (purple=VaR, blue=CVaR) versusrealized loss for the mean-CVaR optimized strategy under theunconditional model.

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

01

23

45

Loss

%

Figure A.17: Risk forecast (purple=VaR, blue=CVaR) versusrealized loss for the mean-CVaR optimized strategy under theCCC-normal model.

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san

dF

igure

s

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

01

23

45

Loss

%

Figure A.18: Risk forecast (purple=VaR, blue=CVaR) versusrealized loss for the mean-CVaR optimized strategy under theCCC-t model.

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

01

23

45

Loss

%

Figure A.19: Risk forecast (purple=VaR, blue=CVaR) versusrealized loss for the mean-CVaR optimized strategy under theDCC-normal model.

A.6

Realiz

ed

Loss

vs

Risk

Fore

cast

for

Equal-W

eig

ht

Portfo

lios

83

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

01

23

45

Loss

%

Figure A.20: Risk forecast (purple=VaR, blue=CVaR) versusrealized loss for the mean-CVaR optimized strategy under theDCC-normal model.

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

01

23

45

Loss

%

Figure A.21: Risk forecast (purple=VaR, blue=CVaR) versusrealized loss for the diversification-CVaR optimized strategy un-der the unconditional model.

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Table

san

dF

igure

s

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

01

23

45

Loss

%

Figure A.22: Risk forecast (purple=VaR, blue=CVaR) versusrealized loss for the diversification-CVaR optimized strategy un-der the CCC-normal model.

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

01

23

4

Loss

%

Figure A.23: Risk forecast (purple=VaR, blue=CVaR) versusrealized loss for the diversification-CVaR optimized strategy un-der the CCC-t model.

A.6

Realiz

ed

Loss

vs

Risk

Fore

cast

for

Equal-W

eig

ht

Portfo

lios

85

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

01

23

4

Loss

%

Figure A.24: Risk forecast (purple=VaR, blue=CVaR) versusrealized loss for the diversification-CVaR optimized strategy un-der the DCC-normal model.

Jan 022002

Jul 012003

Jan 032005

Jul 032006

Jan 022008

Jul 012009

Jan 032011

Jul 022012

01

23

4

Loss

%

Figure A.25: Risk forecast (purple=VaR, blue=CVaR) versusrealized loss for the diversification-CVaR optimized strategy un-der the DCC-t model.

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Table

san

dF

igure

s

A.8 Sample Autocorrelations of GARCH residuals

0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

XLY

Figure A.26: Sample autocorrelations of GARCH residualsfor the asset XLY.

0 5 10 15 20 25 30 350.

00.

20.

40.

60.

81.

0

Lag

AC

F

XLP

Figure A.27: Sample autocorrelations of GARCH residualsfor the asset XLP.

A.6

Realiz

ed

Loss

vs

Risk

Fore

cast

for

Equal-W

eig

ht

Portfo

lios

87

0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

XLE

Figure A.28: Sample autocorrelations of GARCH residualsfor the asset XLE.

0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

LagA

CF

XLF

Figure A.29: Sample autocorrelations of GARCH residualsfor the asset XLF.

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Table

san

dF

igure

s

0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

XLV

Figure A.30: Sample autocorrelations of GARCH residualsfor the asset XLV.

0 5 10 15 20 25 30 350.

00.

20.

40.

60.

81.

0

Lag

AC

F

XLI

Figure A.31: Sample autocorrelations of GARCH residualsfor the asset XLI.

A.6

Realiz

ed

Loss

vs

Risk

Fore

cast

for

Equal-W

eig

ht

Portfo

lios

89

0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

XLB

Figure A.32: Sample autocorrelations of GARCH residualsfor the asset XLB.

0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

LagA

CF

XLK

Figure A.33: Sample autocorrelations of GARCH residualsfor the asset XLK.

90

Table

san

dF

igure

s

0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

XLU

Figure A.34: Sample autocorrelations of GARCH residualsfor the asset XLU.

Research Collection7076/eth... · Abstract This Master Thesis deals with the most important...

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