Advanced Statistics: Khoura
Advanced Statistics module contributed by Jules Ellis, C2 Member. Jules has written a very helpful guide for these statistics. | |||||
| |||||
| ANALYSIS BASED ON MONTHLY VALUES, FULL HISTORY (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 0.929 | ||||
| SD | 1.812 | ||||
| Sharpe ratio (Glass type estimate) | 0.513 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.482 | ||||
| df | 13.000 | ||||
| t | 0.554 | ||||
| p | 0.404 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.322 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 2.328 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.342 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.306 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 1.056 | ||||
| Upside Potential Ratio | 2.621 | ||||
| Upside part of mean | 2.304 | ||||
| Downside part of mean | -1.375 | ||||
| Upside SD | 1.533 | ||||
| Downside SD | 0.879 | ||||
| N nonnegative terms | 8.000 | ||||
| N negative terms | 6.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 14.000 | ||||
| Mean of predictor | 1.001 | ||||
| Mean of criterion | 0.929 | ||||
| SD of predictor | 0.520 | ||||
| SD of criterion | 1.812 | ||||
| Covariance | -0.095 | ||||
| r | -0.101 | ||||
| b (slope, estimate of beta) | -0.351 | ||||
| a (intercept, estimate of alpha) | 1.280 | ||||
| Mean Square Error | 3.522 | ||||
| DF error | 12.000 | ||||
| t(b) | -0.351 | ||||
| p(b) | 0.550 | ||||
| t(a) | 0.638 | ||||
| p(a) | 0.409 | ||||
| Lowerbound of 95% confidence interval for beta | -2.530 | ||||
| Upperbound of 95% confidence interval for beta | 1.828 | ||||
| Lowerbound of 95% confidence interval for alpha | -3.089 | ||||
| Upperbound of 95% confidence interval for alpha | 5.650 | ||||
| Treynor index (mean / b) | -2.646 | ||||
| Jensen alpha (a) | 1.280 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.434 | ||||
| SD | 1.782 | ||||
| Sharpe ratio (Glass type estimate) | -0.244 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.229 | ||||
| df | 13.000 | ||||
| t | -0.263 | ||||
| p | 0.546 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -2.056 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.578 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -2.046 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.587 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.305 | ||||
| Upside Potential Ratio | 1.170 | ||||
| Upside part of mean | 1.668 | ||||
| Downside part of mean | -2.103 | ||||
| Upside SD | 0.965 | ||||
| Downside SD | 1.426 | ||||
| N nonnegative terms | 8.000 | ||||
| N negative terms | 6.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 14.000 | ||||
| Mean of predictor | 0.868 | ||||
| Mean of criterion | -0.434 | ||||
| SD of predictor | 0.421 | ||||
| SD of criterion | 1.782 | ||||
| Covariance | -0.242 | ||||
| r | -0.322 | ||||
| b (slope, estimate of beta) | -1.362 | ||||
| a (intercept, estimate of alpha) | 0.749 | ||||
| Mean Square Error | 3.082 | ||||
| DF error | 12.000 | ||||
| t(b) | -1.179 | ||||
| p(b) | 0.661 | ||||
| t(a) | 0.392 | ||||
| p(a) | 0.444 | ||||
| Lowerbound of 95% confidence interval for beta | -3.880 | ||||
| Upperbound of 95% confidence interval for beta | 1.155 | ||||
| Lowerbound of 95% confidence interval for alpha | -3.413 | ||||
| Upperbound of 95% confidence interval for alpha | 4.911 | ||||
| Treynor index (mean / b) | 0.319 | ||||
| Jensen alpha (a) | 0.749 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.586 | ||||
| Expected Shortfall on VaR | 0.661 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.237 | ||||
| Expected Shortfall on VaR | 0.492 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 14.000 | ||||
| Minimum | 0.317 | ||||
| Quartile 1 | 0.931 | ||||
| Median | 1.047 | ||||
| Quartile 3 | 1.154 | ||||
| Maximum | 2.584 | ||||
| Mean of quarter 1 | 0.622 | ||||
| Mean of quarter 2 | 0.990 | ||||
| Mean of quarter 3 | 1.113 | ||||
| Mean of quarter 4 | 1.584 | ||||
| Inter Quartile Range | 0.223 | ||||
| Number outliers low | 2.000 | ||||
| Percentage of outliers low | 0.143 | ||||
| Mean of outliers low | 0.342 | ||||
| Number of outliers high | 1.000 | ||||
| Percentage of outliers high | 0.071 | ||||
| Mean of outliers high | 2.584 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.299 | ||||
| VaR(95%) (moments method) | 0.300 | ||||
| Expected Shortfall (moments method) | 0.568 | ||||
| Extreme Value Index (regression method) | -1.092 | ||||
| VaR(95%) (regression method) | 0.400 | ||||
| Expected Shortfall (regression method) | 0.433 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 4.000 | ||||
| Minimum | 0.032 | ||||
| Quartile 1 | 0.094 | ||||
| Median | 0.388 | ||||
| Quartile 3 | 0.668 | ||||
| Maximum | 0.683 | ||||
| Mean of quarter 1 | 0.032 | ||||
| Mean of quarter 2 | 0.115 | ||||
| Mean of quarter 3 | 0.662 | ||||
| Mean of quarter 4 | 0.683 | ||||
| Inter Quartile Range | 0.574 | ||||
| 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) | -0.314 | ||||
| Compounded annual return (geometric extrapolation) | -0.323 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.473 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.473 | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.489 | ||||
| ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 1.436 | ||||
| SD | 1.815 | ||||
| Sharpe ratio (Glass type estimate) | 0.791 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.789 | ||||
| df | 317.000 | ||||
| t | 0.871 | ||||
| p | 0.192 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.990 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 2.571 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.991 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.569 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 1.265 | ||||
| Upside Potential Ratio | 5.577 | ||||
| Upside part of mean | 6.331 | ||||
| Downside part of mean | -4.895 | ||||
| Upside SD | 1.416 | ||||
| Downside SD | 1.135 | ||||
| N nonnegative terms | 171.000 | ||||
| N negative terms | 147.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 318.000 | ||||
| Mean of predictor | 1.348 | ||||
| Mean of criterion | 1.436 | ||||
| SD of predictor | 0.424 | ||||
| SD of criterion | 1.815 | ||||
| Covariance | 0.219 | ||||
| r | 0.284 | ||||
| b (slope, estimate of beta) | 1.217 | ||||
| a (intercept, estimate of alpha) | -0.204 | ||||
| Mean Square Error | 3.039 | ||||
| DF error | 316.000 | ||||
| t(b) | 5.269 | ||||
| p(b) | 0.000 | ||||
| t(a) | -0.127 | ||||
| p(a) | 0.550 | ||||
| Lowerbound of 95% confidence interval for beta | 0.762 | ||||
| Upperbound of 95% confidence interval for beta | 1.671 | ||||
| Lowerbound of 95% confidence interval for alpha | -3.377 | ||||
| Upperbound of 95% confidence interval for alpha | 2.969 | ||||
| Treynor index (mean / b) | 1.180 | ||||
| Jensen alpha (a) | -0.204 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.406 | ||||
| SD | 2.077 | ||||
| Sharpe ratio (Glass type estimate) | -0.196 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.195 | ||||
| df | 317.000 | ||||
| t | -0.215 | ||||
| p | 0.585 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.975 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.584 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.974 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.584 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.231 | ||||
| Upside Potential Ratio | 3.192 | ||||
| Upside part of mean | 5.624 | ||||
| Downside part of mean | -6.030 | ||||
| Upside SD | 1.095 | ||||
| Downside SD | 1.762 | ||||
| N nonnegative terms | 171.000 | ||||
| N negative terms | 147.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 318.000 | ||||
| Mean of predictor | 1.258 | ||||
| Mean of criterion | -0.406 | ||||
| SD of predictor | 0.414 | ||||
| SD of criterion | 2.077 | ||||
| Covariance | 0.262 | ||||
| r | 0.304 | ||||
| b (slope, estimate of beta) | 1.529 | ||||
| a (intercept, estimate of alpha) | -2.330 | ||||
| Mean Square Error | 3.928 | ||||
| DF error | 316.000 | ||||
| t(b) | 5.681 | ||||
| p(b) | 0.000 | ||||
| t(a) | -1.273 | ||||
| p(a) | 0.898 | ||||
| Lowerbound of 95% confidence interval for beta | 0.999 | ||||
| Upperbound of 95% confidence interval for beta | 2.058 | ||||
| Lowerbound of 95% confidence interval for alpha | -5.931 | ||||
| Upperbound of 95% confidence interval for alpha | 1.272 | ||||
| Treynor index (mean / b) | -0.266 | ||||
| Jensen alpha (a) | -2.330 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.192 | ||||
| Expected Shortfall on VaR | 0.233 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.039 | ||||
| Expected Shortfall on VaR | 0.091 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 318.000 | ||||
| Minimum | 0.240 | ||||
| Quartile 1 | 0.985 | ||||
| Median | 1.002 | ||||
| Quartile 3 | 1.019 | ||||
| Maximum | 2.105 | ||||
| Mean of quarter 1 | 0.932 | ||||
| Mean of quarter 2 | 0.995 | ||||
| Mean of quarter 3 | 1.010 | ||||
| Mean of quarter 4 | 1.087 | ||||
| Inter Quartile Range | 0.034 | ||||
| Number outliers low | 13.000 | ||||
| Percentage of outliers low | 0.041 | ||||
| Mean of outliers low | 0.739 | ||||
| Number of outliers high | 21.000 | ||||
| Percentage of outliers high | 0.066 | ||||
| Mean of outliers high | 1.237 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.823 | ||||
| VaR(95%) (moments method) | 0.063 | ||||
| Expected Shortfall (moments method) | 0.367 | ||||
| Extreme Value Index (regression method) | 0.874 | ||||
| VaR(95%) (regression method) | 0.047 | ||||
| Expected Shortfall (regression method) | 0.345 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 14.000 | ||||
| Minimum | 0.011 | ||||
| Quartile 1 | 0.030 | ||||
| Median | 0.058 | ||||
| Quartile 3 | 0.143 | ||||
| Maximum | 0.917 | ||||
| Mean of quarter 1 | 0.021 | ||||
| Mean of quarter 2 | 0.037 | ||||
| Mean of quarter 3 | 0.083 | ||||
| Mean of quarter 4 | 0.478 | ||||
| Inter Quartile Range | 0.114 | ||||
| Number outliers low | 0.000 | ||||
| Percentage of outliers low | 0.000 | ||||
| Mean of outliers low | NA | ||||
| Number of outliers high | 2.000 | ||||
| Percentage of outliers high | 0.143 | ||||
| Mean of outliers high | 0.793 | ||||
| Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.415 | ||||
| VaR(95%) (moments method) | 0.499 | ||||
| Expected Shortfall (moments method) | 1.039 | ||||
| Extreme Value Index (regression method) | 0.964 | ||||
| VaR(95%) (regression method) | 0.628 | ||||
| Expected Shortfall (regression method) | 16.663 | ||||
| COMBINED STATISTICS | |||||
| Annualized return (arithmetic extrapolation) | -0.293 | ||||
| Compounded annual return (geometric extrapolation) | -0.304 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.331 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.636 | ||||
| Compounded annual return / Expected Shortfall lognormal | -1.305 | ||||
| ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 2.310 | ||||
| SD | 2.787 | ||||
| Sharpe ratio (Glass type estimate) | 0.829 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.824 | ||||
| df | 130.000 | ||||
| t | 0.586 | ||||
| p | 0.474 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.946 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 3.601 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.949 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 3.598 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 1.327 | ||||
| Upside Potential Ratio | 6.406 | ||||
| Upside part of mean | 11.153 | ||||
| Downside part of mean | -8.844 | ||||
| Upside SD | 2.167 | ||||
| Downside SD | 1.741 | ||||
| N nonnegative terms | 67.000 | ||||
| N negative terms | 64.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 2.497 | ||||
| Mean of criterion | 2.310 | ||||
| SD of predictor | 0.607 | ||||
| SD of criterion | 2.787 | ||||
| Covariance | 0.504 | ||||
| r | 0.298 | ||||
| b (slope, estimate of beta) | 1.366 | ||||
| a (intercept, estimate of alpha) | -1.102 | ||||
| Mean Square Error | 7.131 | ||||
| DF error | 129.000 | ||||
| t(b) | 3.543 | ||||
| p(b) | 0.313 | ||||
| t(a) | -0.283 | ||||
| p(a) | 0.516 | ||||
| Lowerbound of 95% confidence interval for beta | 0.603 | ||||
| Upperbound of 95% confidence interval for beta | 2.129 | ||||
| Lowerbound of 95% confidence interval for alpha | -8.813 | ||||
| Upperbound of 95% confidence interval for alpha | 6.609 | ||||
| Treynor index (mean / b) | 1.691 | ||||
| Jensen alpha (a) | -1.102 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -2.030 | ||||
| SD | 3.201 | ||||
| Sharpe ratio (Glass type estimate) | -0.634 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.631 | ||||
| df | 130.000 | ||||
| t | -0.449 | ||||
| p | 0.520 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -3.406 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 2.140 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -3.404 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.142 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.745 | ||||
| Upside Potential Ratio | 3.492 | ||||
| Upside part of mean | 9.520 | ||||
| Downside part of mean | -11.550 | ||||
| Upside SD | 1.658 | ||||
| Downside SD | 2.727 | ||||
| N nonnegative terms | 67.000 | ||||
| N negative terms | 64.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 2.309 | ||||
| Mean of criterion | -2.030 | ||||
| SD of predictor | 0.592 | ||||
| SD of criterion | 3.201 | ||||
| Covariance | 0.627 | ||||
| r | 0.331 | ||||
| b (slope, estimate of beta) | 1.791 | ||||
| a (intercept, estimate of alpha) | -6.166 | ||||
| Mean Square Error | 9.192 | ||||
| DF error | 129.000 | ||||
| t(b) | 3.985 | ||||
| p(b) | 0.293 | ||||
| t(a) | -1.398 | ||||
| p(a) | 0.578 | ||||
| Lowerbound of 95% confidence interval for beta | 0.902 | ||||
| Upperbound of 95% confidence interval for beta | 2.680 | ||||
| Lowerbound of 95% confidence interval for alpha | -14.894 | ||||
| Upperbound of 95% confidence interval for alpha | 2.562 | ||||
| Treynor index (mean / b) | -1.134 | ||||
| Jensen alpha (a) | -6.166 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.283 | ||||
| Expected Shortfall on VaR | 0.338 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.074 | ||||
| Expected Shortfall on VaR | 0.167 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 131.000 | ||||
| Minimum | 0.240 | ||||
| Quartile 1 | 0.978 | ||||
| Median | 1.001 | ||||
| Quartile 3 | 1.024 | ||||
| Maximum | 2.105 | ||||
| Mean of quarter 1 | 0.875 | ||||
| Mean of quarter 2 | 0.991 | ||||
| Mean of quarter 3 | 1.012 | ||||
| Mean of quarter 4 | 1.158 | ||||
| Inter Quartile Range | 0.047 | ||||
| Number outliers low | 10.000 | ||||
| Percentage of outliers low | 0.076 | ||||
| Mean of outliers low | 0.686 | ||||
| Number of outliers high | 16.000 | ||||
| Percentage of outliers high | 0.122 | ||||
| Mean of outliers high | 1.285 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.987 | ||||
| VaR(95%) (moments method) | 0.121 | ||||
| Expected Shortfall (moments method) | 9.758 | ||||
| Extreme Value Index (regression method) | 1.011 | ||||
| VaR(95%) (regression method) | 0.097 | ||||
| Expected Shortfall (regression method) | NA | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 3.000 | ||||
| Minimum | 0.113 | ||||
| Quartile 1 | 0.383 | ||||
| Median | 0.653 | ||||
| Quartile 3 | 0.785 | ||||
| Maximum | 0.917 | ||||
| Mean of quarter 1 | 0.113 | ||||
| Mean of quarter 2 | 0.653 | ||||
| Mean of quarter 3 | NA | ||||
| Mean of quarter 4 | 0.917 | ||||
| Inter Quartile Range | 0.402 | ||||
| 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) | -1.259 | ||||
| Compounded annual return (geometric extrapolation) | -0.863 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.941 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.941 | ||||
| Compounded annual return / Expected Shortfall lognormal | -2.550 | ||||