1. 1. JP Omega Ltd. Discussion Document Berlin, January 5, 2015 JP Omega Ltd. JP Omega Ltd. Optimization Simulation "Out-of-sample" Johnson-Omega optimal HFRI Hedge Fund Strategies Portfolio during Lehman Collapse.
2. 2. a) HFRI (Total) indices since January 1990 were chosen as well understood by professionals (various back-tests with investable and shortable assets available upon request). b) Free constraints within 130 30 portfolio framework. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Estimates: Mean equals moving average 6M, Volatility equals standard-deviation 12M. Skewness and Kurtosis measured over entire available history to capture extreme events. Findings: a) Volatility insufficient to measure risk. b) Characteristic and significant skewness & kurtosis patterns (findings from ex-ante and ex-post LTCM crisis again affirmed, Risk 2005 and 2008). Key task: a) Merge of volatility, skewness & kurtosis for a comprehensive risk measure to overcome ambiguities. b) Exclude implicit and explicit impact of "noisy" moments (order 5 and higher) estimates, through their endogenous determination (justified through c2 -tests). i) Correlation (G) matrix ii) Co-Skewness, (3-rank tensors) HFRI Event-Driven iii) Co-Kurtosis, (4-rank tensors) HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Remarks: a) All correlations are substantial, positive and highly significant (>99% C.L.), but do not reflect "correlation to 1" phenomenon in case of market dislocations, due to quick dilution. b) The Variance-CoVariance Matrix S = diag(s1,s5)Gdiag(s1,s5) as it is common practice. c) Co-Skew and Co-Kurtosis account non-discretionarily for "Correlation to 1" phenomenon as they are only weakly diluted over time and memorize rare events over decades. JP Omega Ltd. captures direction of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. captures magnitude of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. Co-moments impact visualized via expected marginal attribution analysis. 0,41 0,67 0,58 0,51 0,41 1 0,70 0,67 1 0,60 0,51 0,56 0,60 0,60 1 1 0,79 0,70 0,56 0,67 0,79 1 0,67 0,60 0,58 "fat tail" to downside underestimation 3. Expected Co-Moments (Co-Variance, Co-Skewness and Co-Kurtosis) estimated on entire available history HFRI Event- Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value moderate attractive upside tail properties overestimation -0,2% 1,2% -0,78 22,9% 8,88 0,8% low -0,4% 2,1% 0,40 2,1% 0,76 2,0% neutral -2,0% 3,9% -0,82 6,8% 4,04 3,7% high "fat tail" to downside underestimation "fat tail" to downside underestimation -0,9% 2,8% 0,16 30,4% 1,35 2,1% high neutral "fat tail" characteristics Traditional, s.t. volatility Advanced, "fat tail" asymmetry (skewness) and magnitude (kurtosis) in rare events. Traditional vs. Advanced Risk Estimation -0,4% 1,6% -1,16 0,4% 4,14 2,3% low I. Universe & Expected Central and Co-Moment estimation prior to Lehman Collapse (ex ante 31-Aug-2008) 1. The Universe 2. Expected Central Moments (Mean, Volatility, Skewness and Excess Kurtosis) Risk Classification (Hedge Fund Strategy "stand-alone") Mean (mn) Volatility (sn) Skewness (skn) p-value Excess Kurtosis (kun) p-value
3. 3. a) HFRI (Total) indices since January 1990 were chosen as well understood by professionals (various back-tests with investable and shortable assets available upon request). b) Free constraints within 130 30 portfolio framework. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Estimates: Mean equals moving average 6M, Volatility equals standard-deviation 12M. Skewness and Kurtosis measured over entire available history to capture extreme events. Findings: a) Volatility insufficient to measure risk. b) Characteristic and significant skewness & kurtosis patterns (findings from ex-ante and ex-post LTCM crisis again affirmed, Risk 2005 and 2008). Key task: a) Merge of volatility, skewness & kurtosis for a comprehensive risk measure to overcome ambiguities. b) Exclude implicit and explicit impact of "noisy" moments (order 5 and higher) estimates, through their endogenous determination (justified through c2 -tests). i) Correlation (G) matrix ii) Co-Skewness, (3-rank tensors) HFRI Event-Driven iii) Co-Kurtosis, (4-rank tensors) HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Remarks: a) All correlations are substantial, positive and highly significant (>99% C.L.), but do not reflect "correlation to 1" phenomenon in case of market dislocations, due to quick dilution. b) The Variance-CoVariance Matrix S = diag(s1,s5)Gdiag(s1,s5) as it is common practice. c) Co-Skew and Co-Kurtosis account non-discretionarily for "Correlation to 1" phenomenon as they are only weakly diluted over time and memorize rare events over decades. JP Omega Ltd. captures direction of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. captures magnitude of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. Co-moments impact visualized via expected marginal attribution analysis. 0,41 0,67 0,58 0,51 0,41 1 0,70 0,67 1 0,60 0,51 0,56 0,60 0,60 1 1 0,79 0,70 0,56 0,67 0,79 1 0,67 0,60 0,58 "fat tail" to downside underestimation 3. Expected Co-Moments (Co-Variance, Co-Skewness and Co-Kurtosis) estimated on entire available history HFRI Event- Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value moderate attractive upside tail properties overestimation -0,2% 1,2% -0,78 22,9% 8,88 0,8% low -0,4% 2,1% 0,40 2,1% 0,76 2,0% neutral -2,0% 3,9% -0,82 6,8% 4,04 3,7% high "fat tail" to downside underestimation "fat tail" to downside underestimation -0,9% 2,8% 0,16 30,4% 1,35 2,1% high neutral "fat tail" characteristics Traditional, s.t. volatility Advanced, "fat tail" asymmetry (skewness) and magnitude (kurtosis) in rare events. Traditional vs. Advanced Risk Estimation -0,4% 1,6% -1,16 0,4% 4,14 2,3% low I. Universe & Expected Central and Co-Moment estimation prior to Lehman Collapse (ex ante 31-Aug-2008) 1. The Universe 2. Expected Central Moments (Mean, Volatility, Skewness and Excess Kurtosis) Risk Classification (Hedge Fund Strategy "stand-alone") Mean (mn) Volatility (sn) Skewness (skn) p-value Excess Kurtosis (kun) p-value
4. 4. a) HFRI (Total) indices since January 1990 were chosen as well understood by professionals (various back-tests with investable and shortable assets available upon request). b) Free constraints within 130 30 portfolio framework. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Estimates: Mean equals moving average 6M, Volatility equals standard-deviation 12M. Skewness and Kurtosis measured over entire available history to capture extreme events. Findings: a) Volatility insufficient to measure risk. b) Characteristic and significant skewness & kurtosis patterns (findings from ex-ante and ex-post LTCM crisis again affirmed, Risk 2005 and 2008). Key task: a) Merge of volatility, skewness & kurtosis for a comprehensive risk measure to overcome ambiguities. b) Exclude implicit and explicit impact of "noisy" moments (order 5 and higher) estimates, through their endogenous determination (justified through c2 -tests). i) Correlation (G) matrix ii) Co-Skewness, (3-rank tensors) HFRI Event-Driven iii) Co-Kurtosis, (4-rank tensors) HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Remarks: a) All correlations are substantial, positive and highly significant (>99% C.L.), but do not reflect "correlation to 1" phenomenon in case of market dislocations, due to quick dilution. b) The Variance-CoVariance Matrix S = diag(s1,s5)Gdiag(s1,s5) as it is common practice. c) Co-Skew and Co-Kurtosis account non-discretionarily for "Correlation to 1" phenomenon as they are only weakly diluted over time and memorize rare events over decades. JP Omega Ltd. captures direction of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. captures magnitude of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. Co-moments impact visualized via expected marginal attribution analysis. 0,41 0,67 0,58 0,51 0,41 1 0,70 0,67 1 0,60 0,51 0,56 0,60 0,60 1 1 0,79 0,70 0,56 0,67 0,79 1 0,67 0,60 0,58 "fat tail" to downside underestimation 3. Expected Co-Moments (Co-Variance, Co-Skewness and Co-Kurtosis) estimated on entire available history HFRI Event- Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value moderate attractive upside tail properties overestimation -0,2% 1,2% -0,78 22,9% 8,88 0,8% low -0,4% 2,1% 0,40 2,1% 0,76 2,0% neutral -2,0% 3,9% -0,82 6,8% 4,04 3,7% high "fat tail" to downside underestimation "fat tail" to downside underestimation -0,9% 2,8% 0,16 30,4% 1,35 2,1% high neutral "fat tail" characteristics Traditional, s.t. volatility Advanced, "fat tail" asymmetry (skewness) and magnitude (kurtosis) in rare events. Traditional vs. Advanced Risk Estimation -0,4% 1,6% -1,16 0,4% 4,14 2,3% low I. Universe & Expected Central and Co-Moment estimation prior to Lehman Collapse (ex ante 31-Aug-2008) 1. The Universe 2. Expected Central Moments (Mean, Volatility, Skewness and Excess Kurtosis) Risk Classification (Hedge Fund Strategy "stand-alone") Mean (mn) Volatility (sn) Skewness (skn) p-value Excess Kurtosis (kun) p-value
5. 5. a) HFRI (Total) indices since January 1990 were chosen as well understood by professionals (various back-tests with investable and shortable assets available upon request). b) Free constraints within 130 30 portfolio framework. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Estimates: Mean equals moving average 6M, Volatility equals standard-deviation 12M. Skewness and Kurtosis measured over entire available history to capture extreme events. Findings: a) Volatility insufficient to measure risk. b) Characteristic and significant skewness & kurtosis patterns (findings from ex-ante and ex-post LTCM crisis again affirmed, Risk 2005 and 2008). Key task: a) Merge of volatility, skewness & kurtosis for a comprehensive risk measure to overcome ambiguities. b) Exclude implicit and explicit impact of "noisy" moments (order 5 and higher) estimates, through their endogenous determination (justified through c2 -tests). i) Correlation (G) matrix ii) Co-Skewness, (3-rank tensors) HFRI Event-Driven iii) Co-Kurtosis, (4-rank tensors) HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Remarks: a) All correlations are substantial, positive and highly significant (>99% C.L.), but do not reflect "correlation to 1" phenomenon in case of market dislocations, due to quick dilution. b) The Variance-CoVariance Matrix S = diag(s1,s5)Gdiag(s1,s5) as it is common practice. c) Co-Skew and Co-Kurtosis account non-discretionarily for "Correlation to 1" phenomenon as they are only weakly diluted over time and memorize rare events over decades. JP Omega Ltd. captures direction of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. captures magnitude of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. Co-moments impact visualized via expected marginal attribution analysis. 0,41 0,67 0,58 0,51 0,41 1 0,70 0,67 1 0,60 0,51 0,56 0,60 0,60 1 1 0,79 0,70 0,56 0,67 0,79 1 0,67 0,60 0,58 "fat tail" to downside underestimation 3. Expected Co-Moments (Co-Variance, Co-Skewness and Co-Kurtosis) estimated on entire available history HFRI Event- Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value moderate attractive upside tail properties overestimation -0,2% 1,2% -0,78 22,9% 8,88 0,8% low -0,4% 2,1% 0,40 2,1% 0,76 2,0% neutral -2,0% 3,9% -0,82 6,8% 4,04 3,7% high "fat tail" to downside underestimation "fat tail" to downside underestimation -0,9% 2,8% 0,16 30,4% 1,35 2,1% high neutral "fat tail" characteristics Traditional, s.t. volatility Advanced, "fat tail" asymmetry (skewness) and magnitude (kurtosis) in rare events. Traditional vs. Advanced Risk Estimation -0,4% 1,6% -1,16 0,4% 4,14 2,3% low I. Universe & Expected Central and Co-Moment estimation prior to Lehman Collapse (ex ante 31-Aug-2008) 1. The Universe 2. Expected Central Moments (Mean, Volatility, Skewness and Excess Kurtosis) Risk Classification (Hedge Fund Strategy "stand-alone") Mean (mn) Volatility (sn) Skewness (skn) p-value Excess Kurtosis (kun) p-value
6. 6. a) HFRI (Total) indices since January 1990 were chosen as well understood by professionals (various back-tests with investable and shortable assets available upon request). b) Free constraints within 130 30 portfolio framework. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Estimates: Mean equals moving average 6M, Volatility equals standard-deviation 12M. Skewness and Kurtosis measured over entire available history to capture extreme events. Findings: a) Volatility insufficient to measure risk. b) Characteristic and significant skewness & kurtosis patterns (findings from ex-ante and ex-post LTCM crisis again affirmed, Risk 2005 and 2008). Key task: a) Merge of volatility, skewness & kurtosis for a comprehensive risk measure to overcome ambiguities. b) Exclude implicit and explicit impact of "noisy" moments (order 5 and higher) estimates, through their endogenous determination (justified through c2 -tests). i) Correlation (G) matrix ii) Co-Skewness, (3-rank tensors) HFRI Event-Driven iii) Co-Kurtosis, (4-rank tensors) HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Remarks: a) All correlations are substantial, positive and highly significant (>99% C.L.), but do not reflect "correlation to 1" phenomenon in case of market dislocations, due to quick dilution. b) The Variance-CoVariance Matrix S = diag(s1,s5)Gdiag(s1,s5) as it is common practice. c) Co-Skew and Co-Kurtosis account non-discretionarily for "Correlation to 1" phenomenon as they are only weakly diluted over time and memorize rare events over decades. JP Omega Ltd. captures direction of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. captures magnitude of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. Co-moments impact visualized via expected marginal attribution analysis. 0,41 0,67 0,58 0,51 0,41 1 0,70 0,67 1 0,60 0,51 0,56 0,60 0,60 1 1 0,79 0,70 0,56 0,67 0,79 1 0,67 0,60 0,58 "fat tail" to downside underestimation 3. Expected Co-Moments (Co-Variance, Co-Skewness and Co-Kurtosis) estimated on entire available history HFRI Event- Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value moderate attractive upside tail properties overestimation -0,2% 1,2% -0,78 22,9% 8,88 0,8% low -0,4% 2,1% 0,40 2,1% 0,76 2,0% neutral -2,0% 3,9% -0,82 6,8% 4,04 3,7% high "fat tail" to downside underestimation "fat tail" to downside underestimation -0,9% 2,8% 0,16 30,4% 1,35 2,1% high neutral "fat tail" characteristics Traditional, s.t. volatility Advanced, "fat tail" asymmetry (skewness) and magnitude (kurtosis) in rare events. Traditional vs. Advanced Risk Estimation -0,4% 1,6% -1,16 0,4% 4,14 2,3% low I. Universe & Expected Central and Co-Moment estimation prior to Lehman Collapse (ex ante 31-Aug-2008) 1. The Universe 2. Expected Central Moments (Mean, Volatility, Skewness and Excess Kurtosis) Risk Classification (Hedge Fund Strategy "stand-alone") Mean (mn) Volatility (sn) Skewness (skn) p-value Excess Kurtosis (kun) p-value
7. 7. a) HFRI (Total) indices since January 1990 were chosen as well understood by professionals (various back-tests with investable and shortable assets available upon request). b) Free constraints within 130 30 portfolio framework. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Estimates: Mean equals moving average 6M, Volatility equals standard-deviation 12M. Skewness and Kurtosis measured over entire available history to capture extreme events. Findings: a) Volatility insufficient to measure risk. b) Characteristic and significant skewness & kurtosis patterns (findings from ex-ante and ex-post LTCM crisis again affirmed, Risk 2005 and 2008). Key task: a) Merge of volatility, skewness & kurtosis for a comprehensive risk measure to overcome ambiguities. b) Exclude implicit and explicit impact of "noisy" moments (order 5 and higher) estimates, through their endogenous determination (justified through c2 -tests). i) Correlation (G) matrix ii) Co-Skewness, (3-rank tensors) HFRI Event-Driven iii) Co-Kurtosis, (4-rank tensors) HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Remarks: a) All correlations are substantial, positive and highly significant (>99% C.L.), but do not reflect "correlation to 1" phenomenon in case of market dislocations, due to quick dilution. b) The Variance-CoVariance Matrix S = diag(s1,s5)Gdiag(s1,s5) as it is common practice. c) Co-Skew and Co-Kurtosis account non-discretionarily for "Correlation to 1" phenomenon as they are only weakly diluted over time and memorize rare events over decades. JP Omega Ltd. captures direction of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. captures magnitude of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. Co-moments impact visualized via expected marginal attribution analysis. 0,41 0,67 0,58 0,51 0,41 1 0,70 0,67 1 0,60 0,51 0,56 0,60 0,60 1 1 0,79 0,70 0,56 0,67 0,79 1 0,67 0,60 0,58 "fat tail" to downside underestimation 3. Expected Co-Moments (Co-Variance, Co-Skewness and Co-Kurtosis) estimated on entire available history HFRI Event- Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value moderate attractive upside tail properties overestimation -0,2% 1,2% -0,78 22,9% 8,88 0,8% low -0,4% 2,1% 0,40 2,1% 0,76 2,0% neutral -2,0% 3,9% -0,82 6,8% 4,04 3,7% high "fat tail" to downside underestimation "fat tail" to downside underestimation -0,9% 2,8% 0,16 30,4% 1,35 2,1% high neutral "fat tail" characteristics Traditional, s.t. volatility Advanced, "fat tail" asymmetry (skewness) and magnitude (kurtosis) in rare events. Traditional vs. Advanced Risk Estimation -0,4% 1,6% -1,16 0,4% 4,14 2,3% low I. Universe & Expected Central and Co-Moment estimation prior to Lehman Collapse (ex ante 31-Aug-2008) 1. The Universe 2. Expected Central Moments (Mean, Volatility, Skewness and Excess Kurtosis) Risk Classification (Hedge Fund Strategy "stand-alone") Mean (mn) Volatility (sn) Skewness (skn) p-value Excess Kurtosis (kun) p-value
8. 8. a) HFRI (Total) indices since January 1990 were chosen as well understood by professionals (various back-tests with investable and shortable assets available upon request). b) Free constraints within 130 30 portfolio framework. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Estimates: Mean equals moving average 6M, Volatility equals standard-deviation 12M. Skewness and Kurtosis measured over entire available history to capture extreme events. Findings: a) Volatility insufficient to measure risk. b) Characteristic and significant skewness & kurtosis patterns (findings from ex-ante and ex-post LTCM crisis again affirmed, Risk 2005 and 2008). Key task: a) Merge of volatility, skewness & kurtosis for a comprehensive risk measure to overcome ambiguities. b) Exclude implicit and explicit impact of "noisy" moments (order 5 and higher) estimates, through their endogenous determination (justified through c2 -tests). i) Correlation (G) matrix ii) Co-Skewness, (3-rank tensors) HFRI Event-Driven iii) Co-Kurtosis, (4-rank tensors) HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Remarks: a) All correlations are substantial, positive and highly significant (>99% C.L.), but do not reflect "correlation to 1" phenomenon in case of market dislocations, due to quick dilution. b) The Variance-CoVariance Matrix S = diag(s1,s5)Gdiag(s1,s5) as it is common practice. c) Co-Skew and Co-Kurtosis account non-discretionarily for "Correlation to 1" phenomenon as they are only weakly diluted over time and memorize rare events over decades. JP Omega Ltd. captures direction of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. captures magnitude of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. Co-moments impact visualized via expected marginal attribution analysis. 0,41 0,67 0,58 0,51 0,41 1 0,70 0,67 1 0,60 0,51 0,56 0,60 0,60 1 1 0,79 0,70 0,56 0,67 0,79 1 0,67 0,60 0,58 "fat tail" to downside underestimation 3. Expected Co-Moments (Co-Variance, Co-Skewness and Co-Kurtosis) estimated on entire available history HFRI Event- Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value moderate attractive upside tail properties overestimation -0,2% 1,2% -0,78 22,9% 8,88 0,8% low -0,4% 2,1% 0,40 2,1% 0,76 2,0% neutral -2,0% 3,9% -0,82 6,8% 4,04 3,7% high "fat tail" to downside underestimation "fat tail" to downside underestimation -0,9% 2,8% 0,16 30,4% 1,35 2,1% high neutral "fat tail" characteristics Traditional, s.t. volatility Advanced, "fat tail" asymmetry (skewness) and magnitude (kurtosis) in rare events. Traditional vs. Advanced Risk Estimation -0,4% 1,6% -1,16 0,4% 4,14 2,3% low I. Universe & Expected Central and Co-Moment estimation prior to Lehman Collapse (ex ante 31-Aug-2008) 1. The Universe 2. Expected Central Moments (Mean, Volatility, Skewness and Excess Kurtosis) Risk Classification (Hedge Fund Strategy "stand-alone") Mean (mn) Volatility (sn) Skewness (skn) p-value Excess Kurtosis (kun) p-value
9. 9. a) HFRI (Total) indices since January 1990 were chosen as well understood by professionals (various back-tests with investable and shortable assets available upon request). b) Free constraints within 130 30 portfolio framework. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Estimates: Mean equals moving average 6M, Volatility equals standard-deviation 12M. Skewness and Kurtosis measured over entire available history to capture extreme events. Findings: a) Volatility insufficient to measure risk. b) Characteristic and significant skewness & kurtosis patterns (findings from ex-ante and ex-post LTCM crisis again affirmed, Risk 2005 and 2008). Key task: a) Merge of volatility, skewness & kurtosis for a comprehensive risk measure to overcome ambiguities. b) Exclude implicit and explicit impact of "noisy" moments (order 5 and higher) estimates, through their endogenous determination (justified through c2 -tests). i) Correlation (G) matrix ii) Co-Skewness, (3-rank tensors) HFRI Event-Driven iii) Co-Kurtosis, (4-rank tensors) HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Remarks: a) All correlations are substantial, positive and highly significant (>99% C.L.), but do not reflect "correlation to 1" phenomenon in case of market dislocations, due to quick dilution. b) The Variance-CoVariance Matrix S = diag(s1,s5)Gdiag(s1,s5) as it is common practice. c) Co-Skew and Co-Kurtosis account non-discretionarily for "Correlation to 1" phenomenon as they are only weakly diluted over time and memorize rare events over decades. JP Omega Ltd. captures direction of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. captures magnitude of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. Co-moments impact visualized via expected marginal attribution analysis. 0,41 0,67 0,58 0,51 0,41 1 0,70 0,67 1 0,60 0,51 0,56 0,60 0,60 1 1 0,79 0,70 0,56 0,67 0,79 1 0,67 0,60 0,58 "fat tail" to downside underestimation 3. Expected Co-Moments (Co-Variance, Co-Skewness and Co-Kurtosis) estimated on entire available history HFRI Event- Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value moderate attractive upside tail properties overestimation -0,2% 1,2% -0,78 22,9% 8,88 0,8% low -0,4% 2,1% 0,40 2,1% 0,76 2,0% neutral -2,0% 3,9% -0,82 6,8% 4,04 3,7% high "fat tail" to downside underestimation "fat tail" to downside underestimation -0,9% 2,8% 0,16 30,4% 1,35 2,1% high neutral "fat tail" characteristics Traditional, s.t. volatility Advanced, "fat tail" asymmetry (skewness) and magnitude (kurtosis) in rare events. Traditional vs. Advanced Risk Estimation -0,4% 1,6% -1,16 0,4% 4,14 2,3% low I. Universe & Expected Central and Co-Moment estimation prior to Lehman Collapse (ex ante 31-Aug-2008) 1. The Universe 2. Expected Central Moments (Mean, Volatility, Skewness and Excess Kurtosis) Risk Classification (Hedge Fund Strategy "stand-alone") Mean (mn) Volatility (sn) Skewness (skn) p-value Excess Kurtosis (kun) p-value
10. 10. a) HFRI (Total) indices since January 1990 were chosen as well understood by professionals (various back-tests with investable and shortable assets available upon request). b) Free constraints within 130 30 portfolio framework. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Estimates: Mean equals moving average 6M, Volatility equals standard-deviation 12M. Skewness and Kurtosis measured over entire available history to capture extreme events. Findings: a) Volatility insufficient to measure risk. b) Characteristic and significant skewness & kurtosis patterns (findings from ex-ante and ex-post LTCM crisis again affirmed, Risk 2005 and 2008). Key task: a) Merge of volatility, skewness & kurtosis for a comprehensive risk measure to overcome ambiguities. b) Exclude implicit and explicit impact of "noisy" moments (order 5 and higher) estimates, through their endogenous determination (justified through c2 -tests). i) Correlation (G) matrix ii) Co-Skewness, (3-rank tensors) HFRI Event-Driven iii) Co-Kurtosis, (4-rank tensors) HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Remarks: a) All correlations are substantial, positive and highly significant (>99% C.L.), but do not reflect "correlation to 1" phenomenon in case of market dislocations, due to quick dilution. b) The Variance-CoVariance Matrix S = diag(s1,s5)Gdiag(s1,s5) as it is common practice. c) Co-Skew and Co-Kurtosis account non-discretionarily for "Correlation to 1" phenomenon as they are only weakly diluted over time and memorize rare events over decades. JP Omega Ltd. captures direction of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. captures magnitude of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. Co-moments impact visualized via expected marginal attribution analysis. 0,41 0,67 0,58 0,51 0,41 1 0,70 0,67 1 0,60 0,51 0,56 0,60 0,60 1 1 0,79 0,70 0,56 0,67 0,79 1 0,67 0,60 0,58 "fat tail" to downside underestimation 3. Expected Co-Moments (Co-Variance, Co-Skewness and Co-Kurtosis) estimated on entire available history HFRI Event- Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value moderate attractive upside tail properties overestimation -0,2% 1,2% -0,78 22,9% 8,88 0,8% low -0,4% 2,1% 0,40 2,1% 0,76 2,0% neutral -2,0% 3,9% -0,82 6,8% 4,04 3,7% high "fat tail" to downside underestimation "fat tail" to downside underestimation -0,9% 2,8% 0,16 30,4% 1,35 2,1% high neutral "fat tail" characteristics Traditional, s.t. volatility Advanced, "fat tail" asymmetry (skewness) and magnitude (kurtosis) in rare events. Traditional vs. Advanced Risk Estimation -0,4% 1,6% -1,16 0,4% 4,14 2,3% low I. Universe & Expected Central and Co-Moment estimation prior to Lehman Collapse (ex ante 31-Aug-2008) 1. The Universe 2. Expected Central Moments (Mean, Volatility, Skewness and Excess Kurtosis) Risk Classification (Hedge Fund Strategy "stand-alone") Mean (mn) Volatility (sn) Skewness (skn) p-value Excess Kurtosis (kun) p-value
11. 11. a) HFRI (Total) indices since January 1990 were chosen as well understood by professionals (various back-tests with investable and shortable assets available upon request). b) Free constraints within 130 30 portfolio framework. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Estimates: Mean equals moving average 6M, Volatility equals standard-deviation 12M. Skewness and Kurtosis measured over entire available history to capture extreme events. Findings: a) Volatility insufficient to measure risk. b) Characteristic and significant skewness & kurtosis patterns (findings from ex-ante and ex-post LTCM crisis again affirmed, Risk 2005 and 2008). Key task: a) Merge of volatility, skewness & kurtosis for a comprehensive risk measure to overcome ambiguities. b) Exclude implicit and explicit impact of "noisy" moments (order 5 and higher) estimates, through their endogenous determination (justified through c2 -tests). i) Correlation (G) matrix ii) Co-Skewness, (3-rank tensors) HFRI Event-Driven iii) Co-Kurtosis, (4-rank tensors) HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value Remarks: a) All correlations are substantial, positive and highly significant (>99% C.L.), but do not reflect "correlation to 1" phenomenon in case of market dislocations, due to quick dilution. b) The Variance-CoVariance Matrix S = diag(s1,s5)Gdiag(s1,s5) as it is common practice. c) Co-Skew and Co-Kurtosis account non-discretionarily for "Correlation to 1" phenomenon as they are only weakly diluted over time and memorize rare events over decades. JP Omega Ltd. captures direction of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. captures magnitude of co-movements in case of market dislocations. Long-term memory, due to weak dilution over time. Co-moments impact visualized via expected marginal attribution analysis. 0,41 0,67 0,58 0,51 0,41 1 0,70 0,67 1 0,60 0,51 0,56 0,60 0,60 1 1 0,79 0,70 0,56 0,67 0,79 1 0,67 0,60 0,58 "fat tail" to downside underestimation 3. Expected Co-Moments (Co-Variance, Co-Skewness and Co-Kurtosis) estimated on entire available history HFRI Event- Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value moderate attractive upside tail properties overestimation -0,2% 1,2% -0,78 22,9% 8,88 0,8% low -0,4% 2,1% 0,40 2,1% 0,76 2,0% neutral -2,0% 3,9% -0,82 6,8% 4,04 3,7% high "fat tail" to downside underestimation "fat tail" to downside underestimation -0,9% 2,8% 0,16 30,4% 1,35 2,1% high neutral "fat tail" characteristics Traditional, s.t. volatility Advanced, "fat tail" asymmetry (skewness) and magnitude (kurtosis) in rare events. Traditional vs. Advanced Risk Estimation -0,4% 1,6% -1,16 0,4% 4,14 2,3% low I. Universe & Expected Central and Co-Moment estimation prior to Lehman Collapse (ex ante 31-Aug-2008) 1. The Universe 2. Expected Central Moments (Mean, Volatility, Skewness and Excess Kurtosis) Risk Classification (Hedge Fund Strategy "stand-alone") Mean (mn) Volatility (sn) Skewness (skn) p-value Excess Kurtosis (kun) p-value
12. 12. The density with the derivative g', whereas: Moments of HFRI strategies estimated prior to Lehman Collapse a) Cover entire skewness and kurtosis space and therefore able to account for strategy specific asymmetry and fat tail characteristics. b) No influence by "noise" as no estimation of instable moments of order 5 or higher. c) Contain Normal and Log-Normal distributions as special cases. d) Independent to economically and statistically insignificant moments of order >=5. e) Cover bi-modal, but no economically meaning less multi-modal distributions. f) Much more flexible than Weibull, student-t or two-sided t distributions. g) c2 -tests indicate appropriateness (available upon requrest). JP Omega Ltd. i.e. Johnson distributions are transformed versions of Gauss distributions. The 4 parameters g, d, m and l uniquely depend on 4 moments. 3. Johnson densities of HFRI Strategies (Linear) 4. Conclusion & Features of Johnson distributions: II. Johnson distributions - Basics & Features 1. Johnson distributions cover the entire Skewness and Kurtosis Space 2. Definition of Johnson distributions Bounded Johnson SB Log-Normal SL (negatively skewed) Unbounded Johnson SU Normal SN Log-Normal SL (positively skewed) Event Driven Equity Hedge Emerging Markets Macro RelativeValue 0 3 6 9 12 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 Kurtosis Skewness l m d l m dg x gexj x g 2 2 2 1 )( )(:1ln )(: 1 ln )(:ln : :)( 2 SUJohnsonUnboundedifyy SBJohnsonBoundedif y y SLNormalLogify SNNormalify yg 0 5 10 15 20 25 30 35 40 45 50 -10% -5% 0% 5% 10% Return HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value
13. 13. The density with the derivative g', whereas: Moments of HFRI strategies estimated prior to Lehman Collapse a) Cover entire skewness and kurtosis space and therefore able to account for strategy specific asymmetry and fat tail characteristics. b) No influence by "noise" as no estimation of instable moments of order 5 or higher. c) Contain Normal and Log-Normal distributions as special cases. d) Independent to economically and statistically insignificant moments of order >=5. e) Cover bi-modal, but no economically meaning less multi-modal distributions. f) Much more flexible than Weibull, student-t or two-sided t distributions. g) c2 -tests indicate appropriateness (available upon requrest). JP Omega Ltd. i.e. Johnson distributions are transformed versions of Gauss distributions. The 4 parameters g, d, m and l uniquely depend on 4 moments. 3. Johnson densities of HFRI Strategies (Linear) 4. Conclusion & Features of Johnson distributions: II. Johnson distributions - Basics & Features 1. Johnson distributions cover the entire Skewness and Kurtosis Space 2. Definition of Johnson distributions Bounded Johnson SB Log-Normal SL (negatively skewed) Unbounded Johnson SU Normal SN Log-Normal SL (positively skewed) Event Driven Equity Hedge Emerging Markets Macro RelativeValue 0 3 6 9 12 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 Kurtosis Skewness l m d l m dg x gexj x g 2 2 2 1 )( )(:1ln )(: 1 ln )(:ln : :)( 2 SUJohnsonUnboundedifyy SBJohnsonBoundedif y y SLNormalLogify SNNormalify yg 0 5 10 15 20 25 30 35 40 45 50 -10% -5% 0% 5% 10% Return HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value
14. 14. The density with the derivative g', whereas: Moments of HFRI strategies estimated prior to Lehman Collapse a) Cover entire skewness and kurtosis space and therefore able to account for strategy specific asymmetry and fat tail characteristics. b) No influence by "noise" as no estimation of instable moments of order 5 or higher. c) Contain Normal and Log-Normal distributions as special cases. d) Independent to economically and statistically insignificant moments of order >=5. e) Cover bi-modal, but no economically meaning less multi-modal distributions. f) Much more flexible than Weibull, student-t or two-sided t distributions. g) c2 -tests indicate appropriateness (available upon requrest). JP Omega Ltd. II. Johnson distributions - Basics & Features 1. Johnson distributions cover the entire Skewness and Kurtosis Space 2. Definition of Johnson distributions 3. Johnson densities of HFRI Strategies (Logarithmic) 4. Conclusion & Features of Johnson distributions: i.e. Johnson distributions are transformed versions of Gauss distributions. The 4 parameters g, d, m and l uniquely depend on 4 moments. Bounded Johnson SB Log-Normal SL (negatively skewed) Unbounded Johnson SU Normal SN Log-Normal SL (positively skewed) Event Driven Equity Hedge Emerging Markets Macro RelativeValue 0 3 6 9 12 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 Kurtosis Skewness l m d l m dg x gexj x g 2 2 2 1 )( 0,00001 0,0001 0,001 0,01 0,1 1 10 100 -10% -5% 0% 5% 10% Return HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value )(:1ln )(: 1 ln )(:ln : :)( 2 SUJohnsonUnboundedifyy SBJohnsonBoundedif y y SLNormalLogify SNNormalify yg
15. 15. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error -8,5% 1 0,51 0,46 1,61E-03 617,64 -9,0% 1 0,56 0,35 7,84E-02 10,84 -20,0% 1 0,28 1,82 8,41E-05 11884 -6,8% 1 0,75 0,08 0,06 14,01 -6,6% 2 1,49 0,18 1,52 0,15 -15,8% 0 0,52 0,52 0,01 0,01 -5,1% 2 2,02 0,00 1,17 0,59 -4,3% 5 6,22 0,24 13,54 5,39 -11,5% 2 1,77 0,03 0,59 3,41 -3,4% 6 5,80 0,01 10,45 1,89 -1,9% 30 26,25 0,53 54,91 11,30 -7,3% 4 6,85 1,18 9,79 3,43 -1,7% 10 17,30 3,08 44,77 27,00 0,5% 82 84,33 0,06 101,71 3,82 -3,0% 33 29,45 0,43 57,10 10,17 0,0% 54 50,13 0,30 92,55 16,06 2,9% 117 115,97 0,01 86,30 10,92 1,3% 100 106,45 0,39 118,08 2,77 1,7% 126 110,30 2,24 92,55 12,09 5,3% 49 50,23 0,03 33,51 7,15 5,5% 133 124,08 0,64 87,26 23,98 3,4% 81 95,28 2,14 44,77 29,32 7,6% 10 10,55 0,03 5,93 2,79 9,7% 22 25,59 0,50 22,97 0,04 5,1% 16 15,21 0,04 10,45 2,95 10,0% 1 1,91 0,43 0,48 0,58 14,0% 2 2,61 0,14 2,13 0,01 More 1 0,70 0,13 1,24 0,04 More 1 0,49 0,52 0,02 55,38 More 1 0,39 0,96 0,07 12,61 c2 -stat 8,48 806,49 c2 -stat 2,39 193,21 c2 -stat 6,62 12025 HFRI Macro HFRI Relative Value Historical Strategy Moments used to fit distribution Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error Start-Date 31.12.1989 -4,8% 1 0,35 1,18 3,60E+00 1,88 -7,4% 1 0,32 1,42 3,29E-07 3036144 End-Date 31.10.2014 -3,2% 5 3,65 0,50 15,75 7,34 -5,9% 0 0,37 0,37 0,00 0,00 Mean Volatility -1,7% 14 24,09 4,23 45,97 22,23 -4,4% 2 0,93 1,22 0,05 73,62 HFRI Event-Driven 0,91% 1,93% -1,27 4,08 -0,1% 82 71,59 1,51 79,49 0,08 -2,9% 0 2,72 2,72 2,57 2,57 HFRI Equity Hedge 1,01% 2,60% -0,24 1,93 1,5% 95 92,52 0,07 81,49 2,24 -1,5% 5 9,51 2,14 33,32 24,07 HFRI Emerging Markets 1,01% 4,00% -0,83 3,85 3,1% 59 62,52 0,20 49,54 1,81 0,0% 42 42,17 0,00 116,67 47,79 HFRI Macro 0,92% 2,13% 0,60 1,15 4,7% 22 28,17 1,35 17,84 0,97 1,5% 181 166,48 1,27 112,96 40,98 HFRI Relative Value 0,80% 1,24% -2,11 13,92 6,2% 12 10,21 0,31 3,80 17,66 3,0% 62 72,51 1,52 30,22 33,43 7,8% 7 3,34 3,99 0,48 88,78 4,5% 4 2,77 0,54 2,18 1,53 Critical Levels More 1 1,55 0,19 0,04 24,94 More 1 0,22 2,71 0,04 22,41 Confidence 90,0% 95,0% 97,5% 99,0% 99,9% c2 -stat 13,54 252,83 c2 -stat 13,90 3036475 Cut-Off 12,017 14,067 16,013 18,475 24,322 JP Omega Ltd. Johnson Gauss Johnson Gauss III. Johnson distributions - Fit to HFRI (Total) Indices 1. Calculation of c2 -stats of HFRI (Total) Indices Johnson Gauss Closer investigated in subsequent section. c2 -stat indicates: The hypothesis - "Johnson distributions describe HFRI Indices appropriate" - cannot be rejected at 95% confidence level. For ED, EH and EM even at 90% confidence level. Johnson Gauss Johnson Gauss Skewness for Gauss for Johnson 2. Conclusion: Kurtosis (Excess)
16. 16. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error -8,5% 1 0,51 0,46 1,61E-03 617,64 -9,0% 1 0,56 0,35 7,84E-02 10,84 -20,0% 1 0,28 1,82 8,41E-05 11884 -6,8% 1 0,75 0,08 0,06 14,01 -6,6% 2 1,49 0,18 1,52 0,15 -15,8% 0 0,52 0,52 0,01 0,01 -5,1% 2 2,02 0,00 1,17 0,59 -4,3% 5 6,22 0,24 13,54 5,39 -11,5% 2 1,77 0,03 0,59 3,41 -3,4% 6 5,80 0,01 10,45 1,89 -1,9% 30 26,25 0,53 54,91 11,30 -7,3% 4 6,85 1,18 9,79 3,43 -1,7% 10 17,30 3,08 44,77 27,00 0,5% 82 84,33 0,06 101,71 3,82 -3,0% 33 29,45 0,43 57,10 10,17 0,0% 54 50,13 0,30 92,55 16,06 2,9% 117 115,97 0,01 86,30 10,92 1,3% 100 106,45 0,39 118,08 2,77 1,7% 126 110,30 2,24 92,55 12,09 5,3% 49 50,23 0,03 33,51 7,15 5,5% 133 124,08 0,64 87,26 23,98 3,4% 81 95,28 2,14 44,77 29,32 7,6% 10 10,55 0,03 5,93 2,79 9,7% 22 25,59 0,50 22,97 0,04 5,1% 16 15,21 0,04 10,45 2,95 10,0% 1 1,91 0,43 0,48 0,58 14,0% 2 2,61 0,14 2,13 0,01 More 1 0,70 0,13 1,24 0,04 More 1 0,49 0,52 0,02 55,38 More 1 0,39 0,96 0,07 12,61 c2 -stat 8,48 806,49 c2 -stat 2,39 193,21 c2 -stat 6,62 12025 HFRI Macro HFRI Relative Value Historical Strategy Moments used to fit distribution Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error Start-Date 31.12.1989 -4,8% 1 0,35 1,18 3,60E+00 1,88 -7,4% 1 0,32 1,42 3,29E-07 3036144 End-Date 31.10.2014 -3,2% 5 3,65 0,50 15,75 7,34 -5,9% 0 0,37 0,37 0,00 0,00 Mean Volatility -1,7% 14 24,09 4,23 45,97 22,23 -4,4% 2 0,93 1,22 0,05 73,62 HFRI Event-Driven 0,91% 1,93% -1,27 4,08 -0,1% 82 71,59 1,51 79,49 0,08 -2,9% 0 2,72 2,72 2,57 2,57 HFRI Equity Hedge 1,01% 2,60% -0,24 1,93 1,5% 95 92,52 0,07 81,49 2,24 -1,5% 5 9,51 2,14 33,32 24,07 HFRI Emerging Markets 1,01% 4,00% -0,83 3,85 3,1% 59 62,52 0,20 49,54 1,81 0,0% 42 42,17 0,00 116,67 47,79 HFRI Macro 0,92% 2,13% 0,60 1,15 4,7% 22 28,17 1,35 17,84 0,97 1,5% 181 166,48 1,27 112,96 40,98 HFRI Relative Value 0,80% 1,24% -2,11 13,92 6,2% 12 10,21 0,31 3,80 17,66 3,0% 62 72,51 1,52 30,22 33,43 7,8% 7 3,34 3,99 0,48 88,78 4,5% 4 2,77 0,54 2,18 1,53 Critical Levels More 1 1,55 0,19 0,04 24,94 More 1 0,22 2,71 0,04 22,41 Confidence 90,0% 95,0% 97,5% 99,0% 99,9% c2 -stat 13,54 252,83 c2 -stat 13,90 3036475 Cut-Off 12,017 14,067 16,013 18,475 24,322 JP Omega Ltd. Johnson Gauss Johnson Gauss III. Johnson distributions - Fit to HFRI (Total) Indices 1. Calculation of c2 -stats of HFRI (Total) Indices Johnson Gauss Closer investigated in subsequent section. c2 -stat indicates: The hypothesis - "Johnson distributions describe HFRI Indices appropriate" - cannot be rejected at 95% confidence level. For ED, EH and EM even at 90% confidence level. Johnson Gauss Johnson Gauss Skewness for Gauss for Johnson 2. Conclusion: Kurtosis (Excess)
17. 17. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error -8,5% 1 0,51 0,46 1,61E-03 617,64 -9,0% 1 0,56 0,35 7,84E-02 10,84 -20,0% 1 0,28 1,82 8,41E-05 11884 -6,8% 1 0,75 0,08 0,06 14,01 -6,6% 2 1,49 0,18 1,52 0,15 -15,8% 0 0,52 0,52 0,01 0,01 -5,1% 2 2,02 0,00 1,17 0,59 -4,3% 5 6,22 0,24 13,54 5,39 -11,5% 2 1,77 0,03 0,59 3,41 -3,4% 6 5,80 0,01 10,45 1,89 -1,9% 30 26,25 0,53 54,91 11,30 -7,3% 4 6,85 1,18 9,79 3,43 -1,7% 10 17,30 3,08 44,77 27,00 0,5% 82 84,33 0,06 101,71 3,82 -3,0% 33 29,45 0,43 57,10 10,17 0,0% 54 50,13 0,30 92,55 16,06 2,9% 117 115,97 0,01 86,30 10,92 1,3% 100 106,45 0,39 118,08 2,77 1,7% 126 110,30 2,24 92,55 12,09 5,3% 49 50,23 0,03 33,51 7,15 5,5% 133 124,08 0,64 87,26 23,98 3,4% 81 95,28 2,14 44,77 29,32 7,6% 10 10,55 0,03 5,93 2,79 9,7% 22 25,59 0,50 22,97 0,04 5,1% 16 15,21 0,04 10,45 2,95 10,0% 1 1,91 0,43 0,48 0,58 14,0% 2 2,61 0,14 2,13 0,01 More 1 0,70 0,13 1,24 0,04 More 1 0,49 0,52 0,02 55,38 More 1 0,39 0,96 0,07 12,61 c2 -stat 8,48 806,49 c2 -stat 2,39 193,21 c2 -stat 6,62 12025 HFRI Macro HFRI Relative Value Historical Strategy Moments used to fit distribution Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error Start-Date 31.12.1989 -4,8% 1 0,35 1,18 3,60E+00 1,88 -7,4% 1 0,32 1,42 3,29E-07 3036144 End-Date 31.10.2014 -3,2% 5 3,65 0,50 15,75 7,34 -5,9% 0 0,37 0,37 0,00 0,00 Mean Volatility -1,7% 14 24,09 4,23 45,97 22,23 -4,4% 2 0,93 1,22 0,05 73,62 HFRI Event-Driven 0,91% 1,93% -1,27 4,08 -0,1% 82 71,59 1,51 79,49 0,08 -2,9% 0 2,72 2,72 2,57 2,57 HFRI Equity Hedge 1,01% 2,60% -0,24 1,93 1,5% 95 92,52 0,07 81,49 2,24 -1,5% 5 9,51 2,14 33,32 24,07 HFRI Emerging Markets 1,01% 4,00% -0,83 3,85 3,1% 59 62,52 0,20 49,54 1,81 0,0% 42 42,17 0,00 116,67 47,79 HFRI Macro 0,92% 2,13% 0,60 1,15 4,7% 22 28,17 1,35 17,84 0,97 1,5% 181 166,48 1,27 112,96 40,98 HFRI Relative Value 0,80% 1,24% -2,11 13,92 6,2% 12 10,21 0,31 3,80 17,66 3,0% 62 72,51 1,52 30,22 33,43 7,8% 7 3,34 3,99 0,48 88,78 4,5% 4 2,77 0,54 2,18 1,53 Critical Levels More 1 1,55 0,19 0,04 24,94 More 1 0,22 2,71 0,04 22,41 Confidence 90,0% 95,0% 97,5% 99,0% 99,9% c2 -stat 13,54 252,83 c2 -stat 13,90 3036475 Cut-Off 12,017 14,067 16,013 18,475 24,322 JP Omega Ltd. Johnson Gauss Johnson Gauss III. Johnson distributions - Fit to HFRI (Total) Indices 1. Calculation of c2 -stats of HFRI (Total) Indices Johnson Gauss Closer investigated in subsequent section. c2 -stat indicates: The hypothesis - "Johnson distributions describe HFRI Indices appropriate" - cannot be rejected at 95% confidence level. For ED, EH and EM even at 90% confidence level. Johnson Gauss Johnson Gauss Skewness for Gauss for Johnson 2. Conclusion: Kurtosis (Excess) All exceeded: Gauss to be rejected at high Confidence Levels.
18. 18. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error -8,5% 1 0,51 0,46 1,61E-03 617,64 -9,0% 1 0,56 0,35 7,84E-02 10,84 -20,0% 1 0,28 1,82 8,41E-05 11884 -6,8% 1 0,75 0,08 0,06 14,01 -6,6% 2 1,49 0,18 1,52 0,15 -15,8% 0 0,52 0,52 0,01 0,01 -5,1% 2 2,02 0,00 1,17 0,59 -4,3% 5 6,22 0,24 13,54 5,39 -11,5% 2 1,77 0,03 0,59 3,41 -3,4% 6 5,80 0,01 10,45 1,89 -1,9% 30 26,25 0,53 54,91 11,30 -7,3% 4 6,85 1,18 9,79 3,43 -1,7% 10 17,30 3,08 44,77 27,00 0,5% 82 84,33 0,06 101,71 3,82 -3,0% 33 29,45 0,43 57,10 10,17 0,0% 54 50,13 0,30 92,55 16,06 2,9% 117 115,97 0,01 86,30 10,92 1,3% 100 106,45 0,39 118,08 2,77 1,7% 126 110,30 2,24 92,55 12,09 5,3% 49 50,23 0,03 33,51 7,15 5,5% 133 124,08 0,64 87,26 23,98 3,4% 81 95,28 2,14 44,77 29,32 7,6% 10 10,55 0,03 5,93 2,79 9,7% 22 25,59 0,50 22,97 0,04 5,1% 16 15,21 0,04 10,45 2,95 10,0% 1 1,91 0,43 0,48 0,58 14,0% 2 2,61 0,14 2,13 0,01 More 1 0,70 0,13 1,24 0,04 More 1 0,49 0,52 0,02 55,38 More 1 0,39 0,96 0,07 12,61 c2 -stat 8,48 806,49 c2 -stat 2,39 193,21 c2 -stat 6,62 12025 HFRI Macro HFRI Relative Value Historical Strategy Moments used to fit distribution Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error Start-Date 31.12.1989 -4,8% 1 0,35 1,18 3,60E+00 1,88 -7,4% 1 0,32 1,42 3,29E-07 3036144 End-Date 31.10.2014 -3,2% 5 3,65 0,50 15,75 7,34 -5,9% 0 0,37 0,37 0,00 0,00 Mean Volatility -1,7% 14 24,09 4,23 45,97 22,23 -4,4% 2 0,93 1,22 0,05 73,62 HFRI Event-Driven 0,91% 1,93% -1,27 4,08 -0,1% 82 71,59 1,51 79,49 0,08 -2,9% 0 2,72 2,72 2,57 2,57 HFRI Equity Hedge 1,01% 2,60% -0,24 1,93 1,5% 95 92,52 0,07 81,49 2,24 -1,5% 5 9,51 2,14 33,32 24,07 HFRI Emerging Markets 1,01% 4,00% -0,83 3,85 3,1% 59 62,52 0,20 49,54 1,81 0,0% 42 42,17 0,00 116,67 47,79 HFRI Macro 0,92% 2,13% 0,60 1,15 4,7% 22 28,17 1,35 17,84 0,97 1,5% 181 166,48 1,27 112,96 40,98 HFRI Relative Value 0,80% 1,24% -2,11 13,92 6,2% 12 10,21 0,31 3,80 17,66 3,0% 62 72,51 1,52 30,22 33,43 7,8% 7 3,34 3,99 0,48 88,78 4,5% 4 2,77 0,54 2,18 1,53 Critical Levels More 1 1,55 0,19 0,04 24,94 More 1 0,22 2,71 0,04 22,41 Confidence 90,0% 95,0% 97,5% 99,0% 99,9% c2 -stat 13,54 252,83 c2 -stat 13,90 3036475 Cut-Off 12,017 14,067 16,013 18,475 24,322 JP Omega Ltd. Johnson Gauss Johnson Gauss III. Johnson distributions - Fit to HFRI (Total) Indices 1. Calculation of c2 -stats of HFRI (Total) Indices Johnson Gauss Closer investigated in subsequent section. c2 -stat indicates: The hypothesis - "Johnson distributions describe HFRI Indices appropriate" - cannot be rejected at 95% confidence level. For ED, EH and EM even at 90% confidence level. Johnson Gauss Johnson Gauss Skewness for Gauss for Johnson 2. Conclusion: Kurtosis (Excess) All exceeded: Gauss to be rejected at high Confidence Levels. Not just because of tails: Poor fit even in central bins.
19. 19. HFRI Event-Driven HFRI Equity Hedge HFRI Emerging Markets Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error -8,5% 1 0,51 0,46 1,61E-03 617,64 -9,0% 1 0,56 0,35 7,84E-02 10,84 -20,0% 1 0,28 1,82 8,41E-05 11884 -6,8% 1 0,75 0,08 0,06 14,01 -6,6% 2 1,49 0,18 1,52 0,15 -15,8% 0 0,52 0,52 0,01 0,01 -5,1% 2 2,02 0,00 1,17 0,59 -4,3% 5 6,22 0,24 13,54 5,39 -11,5% 2 1,77 0,03 0,59 3,41 -3,4% 6 5,80 0,01 10,45 1,89 -1,9% 30 26,25 0,53 54,91 11,30 -7,3% 4 6,85 1,18 9,79 3,43 -1,7% 10 17,30 3,08 44,77 27,00 0,5% 82 84,33 0,06 101,71 3,82 -3,0% 33 29,45 0,43 57,10 10,17 0,0% 54 50,13 0,30 92,55 16,06 2,9% 117 115,97 0,01 86,30 10,92 1,3% 100 106,45 0,39 118,08 2,77 1,7% 126 110,30 2,24 92,55 12,09 5,3% 49 50,23 0,03 33,51 7,15 5,5% 133 124,08 0,64 87,26 23,98 3,4% 81 95,28 2,14 44,77 29,32 7,6% 10 10,55 0,03 5,93 2,79 9,7% 22 25,59 0,50 22,97 0,04 5,1% 16 15,21 0,04 10,45 2,95 10,0% 1 1,91 0,43 0,48 0,58 14,0% 2 2,61 0,14 2,13 0,01 More 1 0,70 0,13 1,24 0,04 More 1 0,49 0,52 0,02 55,38 More 1 0,39 0,96 0,07 12,61 c2 -stat 8,48 806,49 c2 -stat 2,39 193,21 c2 -stat 6,62 12025 HFRI Macro HFRI Relative Value Historical Strategy Moments used to fit distribution Bin Frequency Theoretical Error Theoretical Error Bin Frequency Theoretical Error Theoretical Error Start-Date 31.12.1989 -4,8% 1 0,35 1,18 3,60E+00 1,88 -7,4% 1 0,32 1,42 3,29E-07 3036144 End-Date 31.10.2014 -3,2% 5 3,65 0,50 15,75 7,34 -5,9% 0 0,37 0,37 0,00 0,00 Mean Volatility -1,7% 14 24,09 4,23 45,97 22,23 -4,4% 2 0,93 1,22 0,05 73,62 HFRI Event-Driven 0,91% 1,93% -1,27 4,08 -0,1% 82 71,59 1,51 79,49 0,08 -2,9% 0 2,72 2,72 2,57 2,57 HFRI Equity Hedge 1,01% 2,60% -0,24 1,93 1,5% 95 92,52 0,07 81,49 2,24 -1,5% 5 9,51 2,14 33,32 24,07 HFRI Emerging Markets 1,01% 4,00% -0,83 3,85 3,1% 59 62,52 0,20 49,54 1,81 0,0% 42 42,17 0,00 116,67 47,79 HFRI Macro 0,92% 2,13% 0,60 1,15 4,7% 22 28,17 1,35 17,84 0,97 1,5% 181 166,48 1,27 112,96 40,98 HFRI Relative Value 0,80% 1,24% -2,11 13,92 6,2% 12 10,21 0,31 3,80 17,66 3,0% 62 72,51 1,52 30,22 33,43 7,8% 7 3,34 3,99 0,48 88,78 4,5% 4 2,77 0,54 2,18 1,53 Critical Levels More 1 1,55 0,19 0,04 24,94 More 1 0,22 2,71 0,04 22,41 Confidence 90,0% 95,0% 97,5% 99,0% 99,9% c2 -stat 13,54 252,83 c2 -stat 13,90 3036475 Cut-Off 12,017 14,067 16,013 18,475 24,322 JP Omega Ltd. Johnson Gauss Johnson Gauss III. Johnson distributions - Fit to HFRI (Total) Indices 1. Calculation of c2 -stats of HFRI (Total) Indices Johnson Gauss Closer investigated in subsequent section. c2 -stat indicates: The hypothesis - "Johnson distributions describe HFRI Indices appropriate" - cannot be rejected at 95% confidence level. For ED, EH and EM even at 90% confidence level. Johnson Gauss Johnson Gauss Skewness for Gauss for Johnson 2. Conclusion: Kurtosis (Excess)
20. 20. 1. Mean, mn 2. (Co-) Variance, S=(covmn) 3. (Co-) Skewness, co-sklmn 4. (Co-) Kurtosis, co-kuklmn k,l,m and n=1,5 These moments are estimated, fix and not subject to optimization. These weights are subject to optimization. JP Omega Ltd. 90% 95% 97,5% 99% 99,9% 0,1% 1% 2,5% 5% 10% 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.a) Mean, m(w)=wT mm 2.b) Variance, s2 (w)=wT Sw 2.c) Skewness, sk(w) 2.d) Kurtosis (excess), ku(w) VII. Optimization of JP Omega 1.A) Constituent Moments 1.B) Constituent Weights (w) HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega - 5% 95% -4,1%
21. 21. 1. Mean, mn 2. (Co-) Variance, S=(covmn) 3. (Co-) Skewness, co-sklmn 4. (Co-) Kurtosis, co-kuklmn k,l,m and n=1,5 These moments are estimated, fix and not subject to optimization. These weights are subject to optimization. JP Omega Ltd. 90% 95% 97,5% 99% 99,9% 0,1% 1% 2,5% 5% 10% 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.a) Mean, m(w)=wT mm 2.b) Variance, s2 (w)=wT Sw 2.c) Skewness, sk(w) 2.d) Kurtosis (excess), ku(w) VII. Optimization of JP Omega 1.A) Constituent Moments 1.B) Constituent Weights (w) HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega - 5% 95% -4,1%
22. 22. 1. Mean, mn 2. (Co-) Variance, S=(covmn) 3. (Co-) Skewness, co-sklmn 4. (Co-) Kurtosis, co-kuklmn k,l,m and n=1,5 These moments are estimated, fix and not subject to optimization. These weights are subject to optimization. JP Omega Ltd. 90% 95% 97,5% 99% 99,9% 0,1% 1% 2,5% 5% 10% 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.a) Mean, m(w)=wT mm 2.b) Variance, s2 (w)=wT Sw 2.c) Skewness, sk(w) 2.d) Kurtosis (excess), ku(w) VII. Optimization of JP Omega 1.A) Constituent Moments 1.B) Constituent Weights (w) HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega - 5% 95% -4,1%
23. 23. 1. Mean, mn 2. (Co-) Variance, S=(covmn) 3. (Co-) Skewness, co-sklmn 4. (Co-) Kurtosis, co-kuklmn k,l,m and n=1,5 These moments are estimated, fix and not subject to optimization. These weights are subject to optimization. JP Omega Ltd. 90% 95% 97,5% 99% 99,9% 0,1% 1% 2,5% 5% 10% 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.a) Mean, m(w)=wT mm 2.b) Variance, s2 (w)=wT Sw 2.c) Skewness, sk(w) 2.d) Kurtosis (excess), ku(w) VII. Optimization of JP Omega 1.A) Constituent Moments 1.B) Constituent Weights (w) HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega - 5% 95% -4,1%
24. 24. 1. Mean, mn 2. (Co-) Variance, S=(covmn) 3. (Co-) Skewness, co-sklmn 4. (Co-) Kurtosis, co-kuklmn k,l,m and n=1,5 These moments are estimated, fix and not subject to optimization. These weights are subject to optimization. JP Omega Ltd. 90% 95% 97,5% 99% 99,9% 0,1% 1% 2,5% 5% 10% 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.a) Mean, m(w)=wT mm 2.b) Variance, s2 (w)=wT Sw 2.c) Skewness, sk(w) 2.d) Kurtosis (excess), ku(w) VII. Optimization of JP Omega 1.A) Constituent Moments 1.B) Constituent Weights (w) HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega - 5% 95% -4,1%
25. 25. 1. Mean, mn 2. (Co-) Variance, S=(covmn) 3. (Co-) Skewness, co-sklmn 4. (Co-) Kurtosis, co-kuklmn k,l,m and n=1,5 These moments are estimated, fix and not subject to optimization. These weights are subject to optimization. JP Omega Ltd. 90% 95% 97,5% 99% 99,9% 0,1% 1% 2,5% 5% 10% 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.a) Mean, m(w)=wT mm 2.b) Variance, s2 (w)=wT Sw 2.c) Skewness, sk(w) 2.d) Kurtosis (excess), ku(w) VII. Optimization of JP Omega 1.A) Constituent Moments 1.B) Constituent Weights (w) HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega - 5% 95% -4,1%
26. 26. 1. Mean, mn 2. (Co-) Variance, S=(covmn) 3. (Co-) Skewness, co-sklmn 4. (Co-) Kurtosis, co-kuklmn k,l,m and n=1,5 These moments are estimated, fix and not subject to optimization. These weights are subject to optimization. JP Omega Ltd. 90% 95% 97,5% 99% 99,9% 0,1% 1% 2,5% 5% 10% 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.a) Mean, m(w)=wT mm 2.b) Variance, s2 (w)=wT Sw 2.c) Skewness, sk(w) 2.d) Kurtosis (excess), ku(w) VII. Optimization of JP Omega 1.A) Constituent Moments 1.B) Constituent Weights (w) HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega - 5% 95% -4,1%
27. 27. 1. Mean, mn 2. (Co-) Variance, S=(covmn) 3. (Co-) Skewness, co-sklmn 4. (Co-) Kurtosis, co-kuklmn k,l,m and n=1,5 These moments are estimated, fix and not subject to optimization. These weights are subject to optimization. JP Omega Ltd. 90% 95% 97,5% 99% 99,9% 0,1% 1% 2,5% 5% 10% 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.a) Mean, m(w)=wT mm 2.b) Variance, s2 (w)=wT Sw 2.c) Skewness, sk(w) 2.d) Kurtosis (excess), ku(w) VII. Optimization of JP Omega 1.A) Constituent Moments 1.B) Constituent Weights (w) HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega - 5% 95% -4,1% Hurdle=Mean= -0,79%
28. 28. 1. Mean, mn 2. (Co-) Variance, S=(covmn) 3. (Co-) Skewness, co-sklmn 4. (Co-) Kurtosis, co-kuklmn k,l,m and n=1,5 These moments are estimated, fix and not subject to optimization. These weights are subject to optimization. JP Omega Ltd. 90% 95% 97,5% 99% 99,9% 0,1% 1% 2,5% 5% 10% 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.a) Mean, m(w)=wT mm 2.b) Variance, s2 (w)=wT Sw 2.c) Skewness, sk(w) 2.d) Kurtosis (excess), ku(w) VII. Optimization of JP Omega Iteration 0 1.A) Constituent Moments 1.B) Constituent Weights (w) HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega - 5% 95% -4,1% Hurdle=Mean= -0,79%
29. 29. 1. Mean, mn 2. (Co-) Variance, S=(covmn) 3. (Co-) Skewness, co-sklmn 4. (Co-) Kurtosis, co-kuklmn k,l,m and n=1,5 These moments are estimated, fix and not subject to optimization. These weights are subject to optimization. JP Omega Ltd. 90% 95% 97,5% 99% 99,9% 0,1% 1% 2,5% 5% 10% 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.a) Mean, m(w)=wT mm 2.b) Variance, s2 (w)=wT Sw 2.c) Skewness, sk(w) 2.d) Kurtosis (excess), ku(w) VII. Optimization of JP Omega Iteration 0 1.A) Constituent Moments 1.B) Constituent Weights (w) HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Call - 5% 95% -4,1%
30. 30. 1. Mean, mn 2. (Co-) Variance, S=(covmn) 3. (Co-) Skewness, co-sklmn 4. (Co-) Kurtosis, co-kuklmn k,l,m and n=1,5 These moments are estimated, fix and not subject to optimization. These weights are subject to optimization. JP Omega Ltd. 90% 95% 97,5% 99% 99,9% 0,1% 1% 2,5% 5% 10% 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.a) Mean, m(w)=wT mm 2.b) Variance, s2 (w)=wT Sw 2.c) Skewness, sk(w) 2.d) Kurtosis (excess), ku(w) VII. Optimization of JP Omega Iteration 0 1.A) Constituent Moments 1.B) Constituent Weights (w) HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Put 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Call - 5% 95% -4,1%
31. 31. 1. Mean, mn a) Constituent's (Co-) Moments (1A) and Weights 2. (Co-) Variance, S=(covmn) (1B) uniquely determine portfolio moments (2). 3. (Co-) Skewness, co-sklmn b) Portfolio moments (2) uniquely determine 4. (Co-) Kurtosis, co-kuklmn Johnson density (3) , which is able to capture k,l,m and n=1,5 any "asymmetry" and "fat tail" properties. c) JP Formulas uniquely determine call (blue) These moments are estimated, fix and put (red) prices and its Omega ratio (black). and not subject to optimization. These weights are subject to optimization. d) Johnson Omega ratio (3') is optimized w.r.t. portfolio weights w. e) Johnson distribution (3) provides minimum expected returns (4) at each confidence level. a) mean increasing b) volatility decreasing c) skewness ("asymmetry") becomes positive d) kurtosis ("fat tail" magnitude) decreased -> optimized portfolio with reduced risk and maintained upside potential by bending and shifting return distribution. Omega(w) is equivalent to using closed form representations for call and put prices. JP Omega Ltd. VII. Optimization of JP Omega Iteration 0 HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 1.A) Constituent Moments 1.B) Constituent Weights (w) 0,73% 0,73% Call(mean(winit),mean(w),var(w),skew(w),kurt(w)) Put(mean(winit),mean(w),var(w),skew(w),kurt(w))w.r.t. w =1,00 Initial: Optimized: =1 Omega(wopt)= 0,73% 0,73% Omega(winit)= 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.d) Kurtosis (excess), ku(w)2.c) Skewness, sk(w)2.b) Variance, s2 (w)=wT Sw2.a) Mean, m(w)=wT mm w.r.t. w via ratio: The logic: 3') Optimization setup: max max Improvement through optimization: 0,1% 1% 2,5% 5% 10% 90% 95% 97,5% 99% 99,9% 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Put 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Call 1,00 1,00 0 5 10 15 Omega Initial Optimized - 5% 95% -4,1%
32. 32. 1. Mean, mn a) Constituent's (Co-) Moments (1A) and Weights 2. (Co-) Variance, S=(covmn) (1B) uniquely determine portfolio moments (2). 3. (Co-) Skewness, co-sklmn b) Portfolio moments (2) uniquely determine 4. (Co-) Kurtosis, co-kuklmn Johnson density (3) , which is able to capture k,l,m and n=1,5 any "asymmetry" and "fat tail" properties. c) JP Formulas uniquely determine call (blue) These moments are estimated, fix and put (red) prices and its Omega ratio (black). and not subject to optimization. These weights are subject to optimization. d) Johnson Omega ratio (3') is optimized w.r.t. portfolio weights w. e) Johnson distribution (3) provides minimum expected returns (4) at each confidence level. a) mean increasing b) volatility decreasing c) skewness ("asymmetry") becomes positive d) kurtosis ("fat tail" magnitude) decreased -> optimized portfolio with reduced risk and maintained upside potential by bending and shifting return distribution. Omega(w) is equivalent to using closed form representations for call and put prices. JP Omega Ltd. VII. Optimization of JP Omega Iteration 0 HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 1.A) Constituent Moments 1.B) Constituent Weights (w) 0,73% 0,73% Call(mean(winit),mean(w),var(w),skew(w),kurt(w)) Put(mean(winit),mean(w),var(w),skew(w),kurt(w))w.r.t. w =1,00 Initial: Optimized: =1 Omega(wopt)= 0,73% 0,73% Omega(winit)= 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.d) Kurtosis (excess), ku(w)2.c) Skewness, sk(w)2.b) Variance, s2 (w)=wT Sw2.a) Mean, m(w)=wT mm w.r.t. w via ratio: The logic: 3') Optimization setup: max max Improvement through optimization: 0,1% 1% 2,5% 5% 10% 90% 95% 97,5% 99% 99,9% 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Put 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Call 1,00 1,00 0 5 10 15 Omega Initial Optimized - 5% 95% -4,1%
33. 33. 1. Mean, mn a) Constituent's (Co-) Moments (1A) and Weights 2. (Co-) Variance, S=(covmn) (1B) uniquely determine portfolio moments (2). 3. (Co-) Skewness, co-sklmn b) Portfolio moments (2) uniquely determine 4. (Co-) Kurtosis, co-kuklmn Johnson density (3) , which is able to capture k,l,m and n=1,5 any "asymmetry" and "fat tail" properties. c) JP Formulas uniquely determine call (blue) These moments are estimated, fix and put (red) prices and its Omega ratio (black). and not subject to optimization. These weights are subject to optimization. d) Johnson Omega ratio (3') is optimized w.r.t. portfolio weights w. e) Johnson distribution (3) provides minimum expected returns (4) at each confidence level. a) mean increasing b) volatility decreasing c) skewness ("asymmetry") becomes positive d) kurtosis ("fat tail" magnitude) decreased -> optimized portfolio with reduced risk and maintained upside potential by bending and shifting return distribution. Omega(w) is equivalent to using closed form representations for call and put prices. JP Omega Ltd. VII. Optimization of JP Omega Iteration 0 HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 1.A) Constituent Moments 1.B) Constituent Weights (w) 0,73% 0,73% Call(mean(winit),mean(w),var(w),skew(w),kurt(w)) Put(mean(winit),mean(w),var(w),skew(w),kurt(w))w.r.t. w =1,00 Initial: Optimized: =1 Omega(wopt)= 0,73% 0,73% Omega(winit)= 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.d) Kurtosis (excess), ku(w)2.c) Skewness, sk(w)2.b) Variance, s2 (w)=wT Sw2.a) Mean, m(w)=wT mm w.r.t. w via ratio: The logic: 3') Optimization setup: max max Improvement through optimization: 0,1% 1% 2,5% 5% 10% 90% 95% 97,5% 99% 99,9% 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Put 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Call 1,00 1,00 0 5 10 15 Omega Initial Optimized - 5% 95% -4,1%
34. 34. 1. Mean, mn a) Constituent's (Co-) Moments (1A) and Weights 2. (Co-) Variance, S=(covmn) (1B) uniquely determine portfolio moments (2). 3. (Co-) Skewness, co-sklmn b) Portfolio moments (2) uniquely determine 4. (Co-) Kurtosis, co-kuklmn Johnson density (3) , which is able to capture k,l,m and n=1,5 any "asymmetry" and "fat tail" properties. c) JP Formulas uniquely determine call (blue) These moments are estimated, fix and put (red) prices and its Omega ratio (black). and not subject to optimization. These weights are subject to optimization. d) Johnson Omega ratio (3') is optimized w.r.t. portfolio weights w. e) Johnson distribution (3) provides minimum expected returns (4) at each confidence level. a) mean increasing b) volatility decreasing c) skewness ("asymmetry") becomes positive d) kurtosis ("fat tail" magnitude) decreased -> optimized portfolio with reduced risk and maintained upside potential by bending and shifting return distribution. Omega(w) is equivalent to using closed form representations for call and put prices. JP Omega Ltd. VII. Optimization of JP Omega Iteration 0 HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 1.A) Constituent Moments 1.B) Constituent Weights (w) 0,73% 0,73% Call(mean(winit),mean(w),var(w),skew(w),kurt(w)) Put(mean(winit),mean(w),var(w),skew(w),kurt(w))w.r.t. w =1,00 Initial: Optimized: =1 Omega(wopt)= 0,73% 0,73% Omega(winit)= 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.d) Kurtosis (excess), ku(w)2.c) Skewness, sk(w)2.b) Variance, s2 (w)=wT Sw2.a) Mean, m(w)=wT mm w.r.t. w via ratio: The logic: 3') Optimization setup: max max Improvement through optimization: 0,1% 1% 2,5% 5% 10% 90% 95% 97,5% 99% 99,9% 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Put 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Call 1,00 1,00 0 5 10 15 Omega Initial Optimized - 5% 95% -4,1%
35. 35. 1. Mean, mn a) Constituent's (Co-) Moments (1A) and Weights 2. (Co-) Variance, S=(covmn) (1B) uniquely determine portfolio moments (2). 3. (Co-) Skewness, co-sklmn b) Portfolio moments (2) uniquely determine 4. (Co-) Kurtosis, co-kuklmn Johnson density (3) , which is able to capture k,l,m and n=1,5 any "asymmetry" and "fat tail" properties. c) JP Formulas uniquely determine call (blue) These moments are estimated, fix and put (red) prices and its Omega ratio (black). and not subject to optimization. These weights are subject to optimization. d) Johnson Omega ratio (3') is optimized w.r.t. portfolio weights w. e) Johnson distribution (3) provides minimum expected returns (4) at each confidence level. a) mean increasing b) volatility decreasing c) skewness ("asymmetry") becomes positive d) kurtosis ("fat tail" magnitude) decreased -> optimized portfolio with reduced risk and maintained upside potential by bending and shifting return distribution. Omega(w) is equivalent to using closed form representations for call and put prices. JP Omega Ltd. VII. Optimization of JP Omega Iteration 0 HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 1.A) Constituent Moments 1.B) Constituent Weights (w) 0,73% 0,73% Call(mean(winit),mean(w),var(w),skew(w),kurt(w)) Put(mean(winit),mean(w),var(w),skew(w),kurt(w))w.r.t. w =1,00 Initial: Optimized: =1 Omega(wopt)= 0,73% 0,73% Omega(winit)= 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.d) Kurtosis (excess), ku(w)2.c) Skewness, sk(w)2.b) Variance, s2 (w)=wT Sw2.a) Mean, m(w)=wT mm w.r.t. w via ratio: The logic: 3') Optimization setup: max max Improvement through optimization: 0,1% 1% 2,5% 5% 10% 90% 95% 97,5% 99% 99,9% 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Put 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Call 1,00 1,00 0 5 10 15 Omega Initial Optimized - 5% 95% -4,1%
36. 36. 1. Mean, mn a) Constituent's (Co-) Moments (1A) and Weights 2. (Co-) Variance, S=(covmn) (1B) uniquely determine portfolio moments (2). 3. (Co-) Skewness, co-sklmn b) Portfolio moments (2) uniquely determine 4. (Co-) Kurtosis, co-kuklmn Johnson density (3) , which is able to capture k,l,m and n=1,5 any "asymmetry" and "fat tail" properties. c) JP Formulas uniquely determine call (blue) These moments are estimated, fix and put (red) prices and its Omega ratio (black). and not subject to optimization. These weights are subject to optimization. d) Johnson Omega ratio (3') is optimized w.r.t. portfolio weights w. e) Johnson distribution (3) provides minimum expected returns (4) at each confidence level. a) mean increasing b) volatility decreasing c) skewness ("asymmetry") becomes positive d) kurtosis ("fat tail" magnitude) decreased -> optimized portfolio with reduced risk and maintained upside potential by bending and shifting return distribution. Omega(w) is equivalent to using closed form representations for call and put prices. JP Omega Ltd. VII. Optimization of JP Omega Iteration 0 HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 1.A) Constituent Moments 1.B) Constituent Weights (w) 0,73% 0,73% Call(mean(winit),mean(w),var(w),skew(w),kurt(w)) Put(mean(winit),mean(w),var(w),skew(w),kurt(w))w.r.t. w =1,00 Initial: Optimized: =1 Omega(wopt)= 0,73% 0,73% Omega(winit)= 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.d) Kurtosis (excess), ku(w)2.c) Skewness, sk(w)2.b) Variance, s2 (w)=wT Sw2.a) Mean, m(w)=wT mm w.r.t. w via ratio: The logic: 3') Optimization setup: max max Improvement through optimization: 0,1% 1% 2,5% 5% 10% 90% 95% 97,5% 99% 99,9% 0% 100% Prob.tobebelowreturn -0,79% -0,79%-1,0% -0,5% 0,0% 0,5% 1,97% 1,97% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,06 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 20% 20% 20% -30% 0% 30% 60% 90% Initial (Equi) JP Omega 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Put 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Call 1,00 1,00 0 5 10 15 Omega Initial Optimized - 5% 95% -4,1%
37. 37. 1. Mean, mn a) Constituent's (Co-) Moments (1A) and Weights 2. (Co-) Variance, S=(covmn) (1B) uniquely determine portfolio moments (2). 3. (Co-) Skewness, co-sklmn b) Portfolio moments (2) uniquely determine 4. (Co-) Kurtosis, co-kuklmn Johnson density (3) , which is able to capture k,l,m and n=1,5 any "asymmetry" and "fat tail" properties. c) JP Formulas uniquely determine call (blue) These moments are estimated, fix and put (red) prices and its Omega ratio (black). and not subject to optimization. These weights are subject to optimization. d) Johnson Omega ratio (3') is optimized w.r.t. portfolio weights w. e) Johnson distribution (3) provides minimum expected returns (4) at each confidence level. a) mean increasing b) volatility decreasing c) skewness ("asymmetry") becomes positive d) kurtosis ("fat tail" magnitude) decreased -> optimized portfolio with reduced risk and maintained upside potential by bending and shifting return distribution. Omega(w) is equivalent to using closed form representations for call and put prices. JP Omega Ltd. VII. Optimization of JP Omega Iteration 1 HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 1.A) Constituent Moments 1.B) Constituent Weights (w) 0,73% 0,73% Call(mean(winit),mean(w),var(w),skew(w),kurt(w)) Put(mean(winit),mean(w),var(w),skew(w),kurt(w))w.r.t. w =1,03 Initial: Optimized: =1 Omega(wopt)= 0,73% 0,71% Omega(winit)= 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.d) Kurtosis (excess), ku(w)2.c) Skewness, sk(w)2.b) Variance, s2 (w)=wT Sw2.a) Mean, m(w)=wT mm w.r.t. w via ratio: The logic: 3') Optimization setup: max max Improvement through optimization: 0,1% 1% 2,5% 5% 10% 90% 95% 97,5% 99% 99,9% 0% 100% Prob.tobebelowreturn -0,79% -0,77%-1,0% -0,5% 0,0% 0,5% 1,97% 1,94% 0% 1% 2% -0,65 -0,65-1,0 -0,5 0,0 0,5 3,06 3,05 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,3% -6,4% -5,1% -4,1% -3,1% 1,4% 2,1% 2,7% 3,6% 6,0% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 20% 19% 20% 21% -30% 0% 30% 60% 90% Initial (Equi) JP Omega 0,73% 0,71% 0,0% 0,4% 0,8% 1,2% Put 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Call 1,00 1,03 0 5 10 15 Omega Initial Optimized - 5% 95% -4,1%
38. 38. 1. Mean, mn a) Constituent's (Co-) Moments (1A) and Weights 2. (Co-) Variance, S=(covmn) (1B) uniquely determine portfolio moments (2). 3. (Co-) Skewness, co-sklmn b) Portfolio moments (2) uniquely determine 4. (Co-) Kurtosis, co-kuklmn Johnson density (3) , which is able to capture k,l,m and n=1,5 any "asymmetry" and "fat tail" properties. c) JP Formulas uniquely determine call (blue) These moments are estimated, fix and put (red) prices and its Omega ratio (black). and not subject to optimization. These weights are subject to optimization. d) Johnson Omega ratio (3') is optimized w.r.t. portfolio weights w. e) Johnson distribution (3) provides minimum expected returns (4) at each confidence level. a) mean increasing b) volatility decreasing c) skewness ("asymmetry") becomes positive d) kurtosis ("fat tail" magnitude) decreased -> optimized portfolio with reduced risk and maintained upside potential by bending and shifting return distribution. Omega(w) is equivalent to using closed form representations for call and put prices. JP Omega Ltd. VII. Optimization of JP Omega Iteration 2 HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 1.A) Constituent Moments 1.B) Constituent Weights (w) 0,73% 0,73% Call(mean(winit),mean(w),var(w),skew(w),kurt(w)) Put(mean(winit),mean(w),var(w),skew(w),kurt(w))w.r.t. w =1,06 Initial: Optimized: =1 Omega(wopt)= 0,73% 0,69% Omega(winit)= 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.d) Kurtosis (excess), ku(w)2.c) Skewness, sk(w)2.b) Variance, s2 (w)=wT Sw2.a) Mean, m(w)=wT mm w.r.t. w via ratio: The logic: 3') Optimization setup: max max Improvement through optimization: 0,1% 1% 2,5% 5% 10% 90% 95% 97,5% 99% 99,9% 0% 100% Prob.tobebelowreturn -0,79% -0,75%-1,0% -0,5% 0,0% 0,5% 1,97% 1,91% 0% 1% 2% -0,65 -0,64-1,0 -0,5 0,0 0,5 3,06 3,03 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,1% -6,3% -5,0% -4,0% -3,0% 1,4% 2,1% 2,7% 3,6% 5,9% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 19% 18% 21% 22% -30% 0% 30% 60% 90% Initial (Equi) JP Omega 0,73% 0,69% 0,0% 0,4% 0,8% 1,2% Put 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Call 1,00 1,06 0 5 10 15 Omega Initial Optimized - 5% 95% -4,1%
39. 39. 1. Mean, mn a) Constituent's (Co-) Moments (1A) and Weights 2. (Co-) Variance, S=(covmn) (1B) uniquely determine portfolio moments (2). 3. (Co-) Skewness, co-sklmn b) Portfolio moments (2) uniquely determine 4. (Co-) Kurtosis, co-kuklmn Johnson density (3) , which is able to capture k,l,m and n=1,5 any "asymmetry" and "fat tail" properties. c) JP Formulas uniquely determine call (blue) These moments are estimated, fix and put (red) prices and its Omega ratio (black). and not subject to optimization. These weights are subject to optimization. d) Johnson Omega ratio (3') is optimized w.r.t. portfolio weights w. e) Johnson distribution (3) provides minimum expected returns (4) at each confidence level. a) mean increasing b) volatility decreasing c) skewness ("asymmetry") becomes positive d) kurtosis ("fat tail" magnitude) decreased -> optimized portfolio with reduced risk and maintained upside potential by bending and shifting return distribution. Omega(w) is equivalent to using closed form representations for call and put prices. JP Omega Ltd. VII. Optimization of JP Omega Iteration 3 HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 1.A) Constituent Moments 1.B) Constituent Weights (w) 0,73% 0,73% Call(mean(winit),mean(w),var(w),skew(w),kurt(w)) Put(mean(winit),mean(w),var(w),skew(w),kurt(w))w.r.t. w =1,09 Initial: Optimized: =1 Omega(wopt)= 0,73% 0,67% Omega(winit)= 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.d) Kurtosis (excess), ku(w)2.c) Skewness, sk(w)2.b) Variance, s2 (w)=wT Sw2.a) Mean, m(w)=wT mm w.r.t. w via ratio: The logic: 3') Optimization setup: max max Improvement through optimization: 0,1% 1% 2,5% 5% 10% 90% 95% 97,5% 99% 99,9% 0% 100% Prob.tobebelowreturn -0,79% -0,73%-1,0% -0,5% 0,0% 0,5% 1,97% 1,88% 0% 1% 2% -0,65 -0,64-1,0 -0,5 0,0 0,5 3,06 3,02 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -10,0% -6,2% -4,9% -3,9% -3,0% 1,4% 2,0% 2,7% 3,5% 5,8% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 19% 17% 21% 23% -30% 0% 30% 60% 90% Initial (Equi) JP Omega 0,73% 0,67% 0,0% 0,4% 0,8% 1,2% Put 0,73% 0,73% 0,0% 0,4% 0,8% 1,2% Call 1,00 1,09 0 5 10 15 Omega Initial Optimized - 5% 95% -4,1%
40. 40. 1. Mean, mn a) Constituent's (Co-) Moments (1A) and Weights 2. (Co-) Variance, S=(covmn) (1B) uniquely determine portfolio moments (2). 3. (Co-) Skewness, co-sklmn b) Portfolio moments (2) uniquely determine 4. (Co-) Kurtosis, co-kuklmn Johnson density (3) , which is able to capture k,l,m and n=1,5 any "asymmetry" and "fat tail" properties. c) JP Formulas uniquely determine call (blue) These moments are estimated, fix and put (red) prices and its Omega ratio (black). and not subject to optimization. These weights are subject to optimization. d) Johnson Omega ratio (3') is optimized w.r.t. portfolio weights w. e) Johnson distribution (3) provides minimum expected returns (4) at each confidence level. a) mean increasing b) volatility decreasing c) skewness ("asymmetry") becomes positive d) kurtosis ("fat tail" magnitude) decreased -> optimized portfolio with reduced risk and maintained upside potential by bending and shifting return distribution. Omega(w) is equivalent to using closed form representations for call and put prices. JP Omega Ltd. VII. Optimization of JP Omega Iteration 4 HFRI Event Driven HFRI Equity Hedge HFRI Emerging Markets HFRI Macro HFRI Relative Value 1.A) Constituent Moments 1.B) Constituent Weights (w) 0,73% 0,73% Call(mean(winit),mean(w),var(w),skew(w),kurt(w)) Put(mean(winit),mean(w),var(w),skew(w),kurt(w))w.r.t. w =1,13 Initial: Optimized: =1 Omega(wopt)= 0,74% 0,65% Omega(winit)= 3) Cumulative Johnson Return Density (depending on portfolio moments and thus on constituents weights) 4) Minimum Expected Return at given C.L. (depending on portfolio moments and thus on constituents weights) 2) Portfolio Moments (depending on constituents weights) 2.d) Kurtosis (excess), ku(w)2.c) Skewness, sk(w)2.b) Variance, s2 (w)=wT Sw2.a) Mean, m(w)=wT mm w.r.t. w via ratio: The logic: 3') Optimization setup: max max Improvement through optimization: 0,1% 1% 2,5% 5% 10% 90% 95% 97,5% 99% 99,9% 0% 100% Prob.tobebelowreturn -0,79% -0,70%-1,0% -0,5% 0,0% 0,5% 1,97% 1,85% 0% 1% 2% -0,65 -0,63-1,0 -0,5 0,0 0,5 3,06 3,00 0 2 4 -10,5% -6,5% -5,1% -4,1% -3,2% 1,4% 2,1% 2,8% 3,6% 6,1% -9,8% -6,1% -4,8% -3,9% -2,9% 1,4% 2,0% 2,6% 3,5% 5,8% -15% -10% -5% 0% 5% 10% Min.exp.return@CL 20% 20% 20% 20% 20%20% 18% 16% 21% 24% -30% 0% 30% 60% 90% Initial (Equi) JP Omega 0,73% 0,65% 0,0% 0,4% 0,8% 1,2% Put 0,73% 0,74% 0,0% 0,4% 0,8% 1,2% Call 1,00 1,13 0 5 10 15 Omega Initial Optimized - 5% 95% -4,1%
41. 41. 1. Mean, mn a) Constituent's (Co-) Moments (1A) and Weights 2. (Co-) Variance, S=(covmn) (1B) uniquely determine portfolio moments (2). 3. (Co-) Skewness, co-sklmn b) Portfolio moments (2) uniquely determine 4. (Co-) Kurtosis, co-kuklmn Johnson density (3) , which is able to capture k,l,m and n=1,5 any "asymmetry" and "fat tail" properties. c) JP Formulas uniquely determine call (blue) These moments are estimated, fix and put (red) prices and its Omega ratio (black). and not subject to optimization. These weights are subject to optimizatio