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PORTFOLIO OPTIMIZATION WITH HEDGE FUNDS:
Conditional Value At Risk And Conditional Draw-Down At Risk
For Portfolio Optimization With Alternative Investments
Stephan J¨ohri
Supervisor: PD Dr. Diethelm W¨urtz
Professor: Dr. Kai Nagel
March 16, 2004
Abstract
The aim of this Master’s Thesis is to describe and assess different ways to optimize a portfolio. Special attention is paid to the influence of hedge funds since their returns exhibit special statistical properties.
In the first part of this thesis modern portfolio theory is considered. The Markowitz approach is described and analyzed. It assumes that the assets are identically independently distributed according to the Normal law. CAPM and APT are briefly reviewed.
In the second part we go beyond Markowitz and show that asset returns are in reality not normally distributed, but have fat tails and asymmetries. This is especially true for the returns of hedge funds. These facts justify further investigations for alternative portfolio optimization techniques. We describe and discuss therefore alternative methods that can be found in literature.
They use risk measures different than the standard deviation like Value at Risk or Draw-Down and their derivations Conditional Value at Risk and Conditional Draw-Down at Risk. Based on these methods, the respective optimization problems are formulated and implemented. In the third part we describe the numerical implementation and the used data.
Finally the weight allocations and efficient frontiers that summarize the results of these optimization problems are calculated, analyzed and compared. We focus on the question how optimal portfolios with and without hedge funds are constructed according to the different optimization methods, how useful these methods are in practice and how the results differ. The results are derived by analytical work and simulations on historical and artificial data.
Contents
I Modern Portfolio Theory 7
1 Markowitz Model 7
1.1 Risk Return Framework And Utility Function . . . . . . . . . . . . . . . . . . . . 7
1.2 Selecting Optimal Portfolios: The Efficient Frontier . . . . . . . . . . . . . . . . . 14
2 Capital Asset Pricing Model (CAPM) 27
2.1 Standard Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Arbitrage Pricing Theory (APT) . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
II Beyond Markowitz 34
3 Stylized Facts Of Asset Returns 34
3.1 Distribution Form Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Dependencies Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Results Of Statistical Tests Applied To Market Data . . . . . . . . . . . . . . . . 42
4 Portfolio Construction With Non Normal Asset Returns 48
4.1 Introduction To Risk In General . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Variance As Risk Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5 Value At Risk Measures 52
5.1 Value At Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 Conditional Value At Risk, Expected Shortfall And Tail Conditional Expectation 54
5.3 Mean-Conditional Value At Risk Efficient Portfolios . . . . . . . . . . . . . . . . 58
6 Draw-Down Measures 60
6.1 Draw-Down And Time Under-The-Water . . . . . . . . . . . . . . . . . . . . . . 60
6.2 Conditional Draw-Down At Risk And Conditional Time Under-The-Water At Risk 61
6.3 Mean-Conditional Draw-Down At Risk Efficient Portfolios . . . . . . . . . . . . . 65
7 Comparison Of The Risk Measures 67
III Optimization With Alternative Investments 68
8 Numerical Implementation 68
9 Used Data 69
9.1 Normal Vs. Logarithmic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9.2 Empirical Vs. Simulated Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
10 Evaluation Of The Portfolios 72
10.1 Evaluation With Historical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
10.2 Evaluation With Simulated Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Summary and Outlook 82
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