Advanced Statistics: Optimum Systems
Advanced Statistics module contributed by Jules Ellis, C2 Member. Jules has written a very helpful guide for these statistics. | |||||
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| ANALYSIS BASED ON MONTHLY VALUES, FULL HISTORY (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.040 | ||||
| SD | 0.170 | ||||
| Sharpe ratio (Glass type estimate) | -0.234 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.231 | ||||
| df | 60.000 | ||||
| t | -0.528 | ||||
| p | 0.700 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.103 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.637 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.101 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.639 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.335 | ||||
| Upside Potential Ratio | 1.212 | ||||
| Upside part of mean | 0.143 | ||||
| Downside part of mean | -0.183 | ||||
| Upside SD | 0.120 | ||||
| Downside SD | 0.118 | ||||
| N nonnegative terms | 12.000 | ||||
| N negative terms | 49.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 61.000 | ||||
| Mean of predictor | 0.315 | ||||
| Mean of criterion | -0.040 | ||||
| SD of predictor | 0.243 | ||||
| SD of criterion | 0.170 | ||||
| Covariance | -0.005 | ||||
| r | -0.133 | ||||
| b (slope, estimate of beta) | -0.093 | ||||
| a (intercept, estimate of alpha) | -0.010 | ||||
| Mean Square Error | 0.029 | ||||
| DF error | 59.000 | ||||
| t(b) | -1.029 | ||||
| p(b) | 0.846 | ||||
| t(a) | -0.131 | ||||
| p(a) | 0.552 | ||||
| Lowerbound of 95% confidence interval for beta | -0.273 | ||||
| Upperbound of 95% confidence interval for beta | 0.087 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.171 | ||||
| Upperbound of 95% confidence interval for alpha | 0.150 | ||||
| Treynor index (mean / b) | 0.428 | ||||
| Jensen alpha (a) | -0.010 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.054 | ||||
| SD | 0.169 | ||||
| Sharpe ratio (Glass type estimate) | -0.318 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.314 | ||||
| df | 60.000 | ||||
| t | -0.716 | ||||
| p | 0.762 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.188 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.555 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.185 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.557 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.430 | ||||
| Upside Potential Ratio | 1.094 | ||||
| Upside part of mean | 0.136 | ||||
| Downside part of mean | -0.190 | ||||
| Upside SD | 0.113 | ||||
| Downside SD | 0.125 | ||||
| N nonnegative terms | 12.000 | ||||
| N negative terms | 49.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 61.000 | ||||
| Mean of predictor | 0.281 | ||||
| Mean of criterion | -0.054 | ||||
| SD of predictor | 0.243 | ||||
| SD of criterion | 0.169 | ||||
| Covariance | -0.006 | ||||
| r | -0.144 | ||||
| b (slope, estimate of beta) | -0.100 | ||||
| a (intercept, estimate of alpha) | -0.026 | ||||
| Mean Square Error | 0.028 | ||||
| DF error | 59.000 | ||||
| t(b) | -1.117 | ||||
| p(b) | 0.866 | ||||
| t(a) | -0.324 | ||||
| p(a) | 0.626 | ||||
| Lowerbound of 95% confidence interval for beta | -0.279 | ||||
| Upperbound of 95% confidence interval for beta | 0.079 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.183 | ||||
| Upperbound of 95% confidence interval for alpha | 0.132 | ||||
| Treynor index (mean / b) | 0.537 | ||||
| Jensen alpha (a) | -0.026 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.081 | ||||
| Expected Shortfall on VaR | 0.099 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.047 | ||||
| Expected Shortfall on VaR | 0.090 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 61.000 | ||||
| Minimum | 0.865 | ||||
| Quartile 1 | 0.999 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.000 | ||||
| Maximum | 1.174 | ||||
| Mean of quarter 1 | 0.953 | ||||
| Mean of quarter 2 | 1.000 | ||||
| Mean of quarter 3 | 1.000 | ||||
| Mean of quarter 4 | 1.052 | ||||
| Inter Quartile Range | 0.001 | ||||
| Number outliers low | 13.000 | ||||
| Percentage of outliers low | 0.213 | ||||
| Mean of outliers low | 0.942 | ||||
| Number of outliers high | 13.000 | ||||
| Percentage of outliers high | 0.213 | ||||
| Mean of outliers high | 1.060 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 1.231 | ||||
| VaR(95%) (moments method) | 0.015 | ||||
| Expected Shortfall (moments method) | NA | ||||
| Extreme Value Index (regression method) | -0.112 | ||||
| VaR(95%) (regression method) | 0.043 | ||||
| Expected Shortfall (regression method) | 0.064 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 2.000 | ||||
| Minimum | 0.144 | ||||
| Quartile 1 | 0.203 | ||||
| Median | 0.262 | ||||
| Quartile 3 | 0.321 | ||||
| Maximum | 0.380 | ||||
| Mean of quarter 1 | 0.144 | ||||
| Mean of quarter 2 | NA | ||||
| Mean of quarter 3 | NA | ||||
| Mean of quarter 4 | 0.380 | ||||
| Inter Quartile Range | 0.118 | ||||
| Number outliers low | 0.000 | ||||
| Percentage of outliers low | 0.000 | ||||
| Mean of outliers low | NA | ||||
| Number of outliers high | 0.000 | ||||
| Percentage of outliers high | 0.000 | ||||
| Mean of outliers high | NA | ||||
| Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | NA | ||||
| VaR(95%) (moments method) | NA | ||||
| Expected Shortfall (moments method) | NA | ||||
| Extreme Value Index (regression method) | NA | ||||
| VaR(95%) (regression method) | NA | ||||
| Expected Shortfall (regression method) | NA | ||||
| COMBINED STATISTICS | |||||
| Annualized return (arithmetic extrapolation) | -0.009 | ||||
| Compounded annual return (geometric extrapolation) | -0.010 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.025 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.025 | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.096 | ||||
| ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 0.051 | ||||
| SD | 0.467 | ||||
| Sharpe ratio (Glass type estimate) | 0.110 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.110 | ||||
| df | 1352.000 | ||||
| t | 0.249 | ||||
| p | 0.497 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.753 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.972 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.753 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.972 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 0.181 | ||||
| Upside Potential Ratio | 3.694 | ||||
| Upside part of mean | 1.046 | ||||
| Downside part of mean | -0.995 | ||||
| Upside SD | 0.372 | ||||
| Downside SD | 0.283 | ||||
| N nonnegative terms | 254.000 | ||||
| N negative terms | 1099.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 1353.000 | ||||
| Mean of predictor | 0.398 | ||||
| Mean of criterion | 0.051 | ||||
| SD of predictor | 0.382 | ||||
| SD of criterion | 0.467 | ||||
| Covariance | 0.051 | ||||
| r | 0.287 | ||||
| b (slope, estimate of beta) | 0.351 | ||||
| a (intercept, estimate of alpha) | -0.088 | ||||
| Mean Square Error | 0.201 | ||||
| DF error | 1351.000 | ||||
| t(b) | 10.993 | ||||
| p(b) | 0.320 | ||||
| t(a) | -0.447 | ||||
| p(a) | 0.508 | ||||
| Lowerbound of 95% confidence interval for beta | 0.288 | ||||
| Upperbound of 95% confidence interval for beta | 0.413 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.476 | ||||
| Upperbound of 95% confidence interval for alpha | 0.299 | ||||
| Treynor index (mean / b) | 0.146 | ||||
| Jensen alpha (a) | -0.088 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.053 | ||||
| SD | 0.458 | ||||
| Sharpe ratio (Glass type estimate) | -0.117 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.117 | ||||
| df | 1352.000 | ||||
| t | -0.265 | ||||
| p | 0.504 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.979 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.746 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.979 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.746 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.162 | ||||
| Upside Potential Ratio | 3.003 | ||||
| Upside part of mean | 0.990 | ||||
| Downside part of mean | -1.044 | ||||
| Upside SD | 0.318 | ||||
| Downside SD | 0.330 | ||||
| N nonnegative terms | 254.000 | ||||
| N negative terms | 1099.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 1353.000 | ||||
| Mean of predictor | 0.323 | ||||
| Mean of criterion | -0.053 | ||||
| SD of predictor | 0.391 | ||||
| SD of criterion | 0.458 | ||||
| Covariance | 0.055 | ||||
| r | 0.309 | ||||
| b (slope, estimate of beta) | 0.363 | ||||
| a (intercept, estimate of alpha) | -0.171 | ||||
| Mean Square Error | 0.190 | ||||
| DF error | 1351.000 | ||||
| t(b) | 11.962 | ||||
| p(b) | 0.306 | ||||
| t(a) | -0.888 | ||||
| p(a) | 0.515 | ||||
| Lowerbound of 95% confidence interval for beta | 0.303 | ||||
| Upperbound of 95% confidence interval for beta | 0.422 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.547 | ||||
| Upperbound of 95% confidence interval for alpha | 0.206 | ||||
| Treynor index (mean / b) | -0.147 | ||||
| Jensen alpha (a) | -0.171 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.046 | ||||
| Expected Shortfall on VaR | 0.057 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.012 | ||||
| Expected Shortfall on VaR | 0.026 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 1353.000 | ||||
| Minimum | 0.586 | ||||
| Quartile 1 | 1.000 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.000 | ||||
| Maximum | 1.622 | ||||
| Mean of quarter 1 | 0.985 | ||||
| Mean of quarter 2 | 1.000 | ||||
| Mean of quarter 3 | 1.000 | ||||
| Mean of quarter 4 | 1.016 | ||||
| Inter Quartile Range | 0.000 | ||||
| Number outliers low | 272.000 | ||||
| Percentage of outliers low | 0.201 | ||||
| Mean of outliers low | 0.982 | ||||
| Number of outliers high | 259.000 | ||||
| Percentage of outliers high | 0.191 | ||||
| Mean of outliers high | 1.021 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.483 | ||||
| VaR(95%) (moments method) | 0.005 | ||||
| Expected Shortfall (moments method) | 0.014 | ||||
| Extreme Value Index (regression method) | 0.407 | ||||
| VaR(95%) (regression method) | 0.011 | ||||
| Expected Shortfall (regression method) | 0.027 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 5.000 | ||||
| Minimum | 0.036 | ||||
| Quartile 1 | 0.111 | ||||
| Median | 0.184 | ||||
| Quartile 3 | 0.418 | ||||
| Maximum | 0.481 | ||||
| Mean of quarter 1 | 0.074 | ||||
| Mean of quarter 2 | 0.184 | ||||
| Mean of quarter 3 | 0.418 | ||||
| Mean of quarter 4 | 0.481 | ||||
| Inter Quartile Range | 0.306 | ||||
| Number outliers low | 0.000 | ||||
| Percentage of outliers low | 0.000 | ||||
| Mean of outliers low | NA | ||||
| Number of outliers high | 0.000 | ||||
| Percentage of outliers high | 0.000 | ||||
| Mean of outliers high | NA | ||||
| Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | NA | ||||
| VaR(95%) (moments method) | NA | ||||
| Expected Shortfall (moments method) | NA | ||||
| Extreme Value Index (regression method) | NA | ||||
| VaR(95%) (regression method) | NA | ||||
| Expected Shortfall (regression method) | NA | ||||
| COMBINED STATISTICS | |||||
| Annualized return (arithmetic extrapolation) | -0.009 | ||||
| Compounded annual return (geometric extrapolation) | -0.009 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.020 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.020 | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.165 | ||||
| ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.044 | ||||
| SD | 0.000 | ||||
| Sharpe ratio (Glass type estimate) | NA | ||||
| Sharpe ratio (Hedges UMVUE) | NA | ||||
| df | NA | ||||
| t | NA | ||||
| p | NA | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | NA | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | NA | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | NA | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | NA | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -16.186 | ||||
| Upside Potential Ratio | 0.000 | ||||
| Upside part of mean | 0.000 | ||||
| Downside part of mean | -0.044 | ||||
| Upside SD | 0.000 | ||||
| Downside SD | 0.003 | ||||
| N nonnegative terms | 0.000 | ||||
| N negative terms | 131.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 1.195 | ||||
| Mean of criterion | -0.044 | ||||
| SD of predictor | 0.510 | ||||
| SD of criterion | 0.000 | ||||
| Covariance | 0.000 | ||||
| r | NA | ||||
| b (slope, estimate of beta) | NA | ||||
| a (intercept, estimate of alpha) | NA | ||||
| Mean Square Error | NA | ||||
| DF error | NA | ||||
| t(b) | NA | ||||
| p(b) | NA | ||||
| t(a) | NA | ||||
| p(a) | NA | ||||
| Lowerbound of 95% confidence interval for beta | NA | ||||
| Upperbound of 95% confidence interval for beta | NA | ||||
| Lowerbound of 95% confidence interval for alpha | NA | ||||
| Upperbound of 95% confidence interval for alpha | NA | ||||
| Treynor index (mean / b) | NA | ||||
| Jensen alpha (a) | NA | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.044 | ||||
| SD | 0.000 | ||||
| Sharpe ratio (Glass type estimate) | -12492933348697024.000 | ||||
| Sharpe ratio (Hedges UMVUE) | -12420719861132300.000 | ||||
| df | 130.000 | ||||
| t | -8833837887775228.000 | ||||
| p | 1.000 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | NA | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | NA | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -13930479873814932.000 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -10910959848449668.000 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -16.186 | ||||
| Upside Potential Ratio | 0.000 | ||||
| Upside part of mean | 0.000 | ||||
| Downside part of mean | -0.044 | ||||
| Upside SD | 0.000 | ||||
| Downside SD | 0.003 | ||||
| N nonnegative terms | 0.000 | ||||
| N negative terms | 131.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 1.062 | ||||
| Mean of criterion | -0.044 | ||||
| SD of predictor | 0.514 | ||||
| SD of criterion | 0.000 | ||||
| Covariance | 0.000 | ||||
| r | 0.000 | ||||
| b (slope, estimate of beta) | 0.000 | ||||
| a (intercept, estimate of alpha) | -0.044 | ||||
| Mean Square Error | 0.000 | ||||
| DF error | 129.000 | ||||
| t(b) | 0.000 | ||||
| p(b) | 0.500 | ||||
| t(a) | -8728434330972082.000 | ||||
| p(a) | 1.000 | ||||
| Lowerbound of 95% confidence interval for beta | -0.000 | ||||
| Upperbound of 95% confidence interval for beta | 0.000 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.044 | ||||
| Upperbound of 95% confidence interval for alpha | -0.044 | ||||
| Treynor index (mean / b) | -959367932494573002413540772937728.000 | ||||
| Jensen alpha (a) | -0.044 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.000 | ||||
| Expected Shortfall on VaR | 0.000 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.000 | ||||
| Expected Shortfall on VaR | 0.000 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 131.000 | ||||
| Minimum | 1.000 | ||||
| Quartile 1 | 1.000 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.000 | ||||
| Maximum | 1.000 | ||||
| Mean of quarter 1 | 1.000 | ||||
| Mean of quarter 2 | 1.000 | ||||
| Mean of quarter 3 | 1.000 | ||||
| Mean of quarter 4 | 1.000 | ||||
| Inter Quartile Range | 0.000 | ||||
| Number outliers low | 0.000 | ||||
| Percentage of outliers low | 0.000 | ||||
| Mean of outliers low | NA | ||||
| Number of outliers high | 0.000 | ||||
| Percentage of outliers high | 0.000 | ||||
| Mean of outliers high | NA | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | NA | ||||
| VaR(95%) (moments method) | NA | ||||
| Expected Shortfall (moments method) | NA | ||||
| Extreme Value Index (regression method) | NA | ||||
| VaR(95%) (regression method) | NA | ||||
| Expected Shortfall (regression method) | NA | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 0.000 | ||||
| Minimum | NA | ||||
| Quartile 1 | NA | ||||
| Median | NA | ||||
| Quartile 3 | NA | ||||
| Maximum | NA | ||||
| Mean of quarter 1 | NA | ||||
| Mean of quarter 2 | NA | ||||
| Mean of quarter 3 | NA | ||||
| Mean of quarter 4 | NA | ||||
| Inter Quartile Range | 0.000 | ||||
| Number outliers low | 0.000 | ||||
| Percentage of outliers low | NA | ||||
| Mean of outliers low | NA | ||||
| Number of outliers high | 0.000 | ||||
| Percentage of outliers high | NA | ||||
| Mean of outliers high | NA | ||||
| Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | NA | ||||
| VaR(95%) (moments method) | NA | ||||
| Expected Shortfall (moments method) | NA | ||||
| Extreme Value Index (regression method) | NA | ||||
| VaR(95%) (regression method) | NA | ||||
| Expected Shortfall (regression method) | NA | ||||
| COMBINED STATISTICS | |||||
| Annualized return (arithmetic extrapolation) | 0.000 | ||||
| Compounded annual return (geometric extrapolation) | 0.000 | ||||
| Calmar ratio (compounded annual return / max draw down) | NA | ||||
| Compounded annual return / average of 25% largest draw downs | NA | ||||
| Compounded annual return / Expected Shortfall lognormal | 0.000 | ||||