Advanced Statistics: High-Probability, Mean-Reversion ETF Options Strategy
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 | -0.340 | ||||
| SD | 0.550 | ||||
| Sharpe ratio (Glass type estimate) | -0.619 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.609 | ||||
| df | 48.000 | ||||
| t | -1.250 | ||||
| p | 0.891 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.593 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.362 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.586 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.369 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.628 | ||||
| Upside Potential Ratio | 0.288 | ||||
| Upside part of mean | 0.156 | ||||
| Downside part of mean | -0.497 | ||||
| Upside SD | 0.111 | ||||
| Downside SD | 0.542 | ||||
| N nonnegative terms | 11.000 | ||||
| N negative terms | 38.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 49.000 | ||||
| Mean of predictor | 0.421 | ||||
| Mean of criterion | -0.340 | ||||
| SD of predictor | 0.274 | ||||
| SD of criterion | 0.550 | ||||
| Covariance | 0.000 | ||||
| r | 0.002 | ||||
| b (slope, estimate of beta) | 0.005 | ||||
| a (intercept, estimate of alpha) | -0.342 | ||||
| Mean Square Error | 0.309 | ||||
| DF error | 47.000 | ||||
| t(b) | 0.015 | ||||
| p(b) | 0.494 | ||||
| t(a) | -1.135 | ||||
| p(a) | 0.869 | ||||
| Lowerbound of 95% confidence interval for beta | -0.584 | ||||
| Upperbound of 95% confidence interval for beta | 0.593 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.949 | ||||
| Upperbound of 95% confidence interval for alpha | 0.264 | ||||
| Treynor index (mean / b) | -75.632 | ||||
| Jensen alpha (a) | -0.342 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.676 | ||||
| SD | 1.006 | ||||
| Sharpe ratio (Glass type estimate) | -0.672 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.661 | ||||
| df | 48.000 | ||||
| t | -1.357 | ||||
| p | 0.909 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.647 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.311 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.640 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.318 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.670 | ||||
| Upside Potential Ratio | 0.149 | ||||
| Upside part of mean | 0.150 | ||||
| Downside part of mean | -0.826 | ||||
| Upside SD | 0.106 | ||||
| Downside SD | 1.009 | ||||
| N nonnegative terms | 11.000 | ||||
| N negative terms | 38.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 49.000 | ||||
| Mean of predictor | 0.380 | ||||
| Mean of criterion | -0.676 | ||||
| SD of predictor | 0.253 | ||||
| SD of criterion | 1.006 | ||||
| Covariance | 0.006 | ||||
| r | 0.022 | ||||
| b (slope, estimate of beta) | 0.088 | ||||
| a (intercept, estimate of alpha) | -0.709 | ||||
| Mean Square Error | 1.033 | ||||
| DF error | 47.000 | ||||
| t(b) | 0.152 | ||||
| p(b) | 0.440 | ||||
| t(a) | -1.292 | ||||
| p(a) | 0.899 | ||||
| Lowerbound of 95% confidence interval for beta | -1.079 | ||||
| Upperbound of 95% confidence interval for beta | 1.256 | ||||
| Lowerbound of 95% confidence interval for alpha | -1.814 | ||||
| Upperbound of 95% confidence interval for alpha | 0.396 | ||||
| Treynor index (mean / b) | -7.645 | ||||
| Jensen alpha (a) | -0.709 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.414 | ||||
| Expected Shortfall on VaR | 0.478 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.123 | ||||
| Expected Shortfall on VaR | 0.270 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 49.000 | ||||
| Minimum | 0.183 | ||||
| Quartile 1 | 1.000 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.000 | ||||
| Maximum | 1.117 | ||||
| Mean of quarter 1 | 0.855 | ||||
| Mean of quarter 2 | 1.000 | ||||
| Mean of quarter 3 | 1.000 | ||||
| Mean of quarter 4 | 1.057 | ||||
| Inter Quartile Range | 0.000 | ||||
| Number outliers low | 4.000 | ||||
| Percentage of outliers low | 0.082 | ||||
| Mean of outliers low | 0.528 | ||||
| Number of outliers high | 11.000 | ||||
| Percentage of outliers high | 0.224 | ||||
| Mean of outliers high | 1.062 | ||||
| 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) | -0.709 | ||||
| VaR(95%) (regression method) | 0.364 | ||||
| Expected Shortfall (regression method) | 0.601 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 2.000 | ||||
| Minimum | 0.174 | ||||
| Quartile 1 | 0.369 | ||||
| Median | 0.563 | ||||
| Quartile 3 | 0.758 | ||||
| Maximum | 0.952 | ||||
| Mean of quarter 1 | 0.174 | ||||
| Mean of quarter 2 | NA | ||||
| Mean of quarter 3 | NA | ||||
| Mean of quarter 4 | 0.952 | ||||
| Inter Quartile Range | 0.389 | ||||
| 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) | -0.226 | ||||
| Compounded annual return (geometric extrapolation) | -0.468 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.492 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.492 | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.980 | ||||
| ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.490 | ||||
| SD | 0.593 | ||||
| Sharpe ratio (Glass type estimate) | -0.827 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.826 | ||||
| df | 1084.000 | ||||
| t | -1.682 | ||||
| p | 0.526 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.790 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.137 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.790 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.138 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -1.126 | ||||
| Upside Potential Ratio | 2.092 | ||||
| Upside part of mean | 0.911 | ||||
| Downside part of mean | -1.401 | ||||
| Upside SD | 0.404 | ||||
| Downside SD | 0.435 | ||||
| N nonnegative terms | 77.000 | ||||
| N negative terms | 1008.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 1085.000 | ||||
| Mean of predictor | 0.444 | ||||
| Mean of criterion | -0.490 | ||||
| SD of predictor | 0.325 | ||||
| SD of criterion | 0.593 | ||||
| Covariance | 0.009 | ||||
| r | 0.044 | ||||
| b (slope, estimate of beta) | 0.081 | ||||
| a (intercept, estimate of alpha) | -0.526 | ||||
| Mean Square Error | 0.352 | ||||
| DF error | 1083.000 | ||||
| t(b) | 1.456 | ||||
| p(b) | 0.472 | ||||
| t(a) | -1.800 | ||||
| p(a) | 0.535 | ||||
| Lowerbound of 95% confidence interval for beta | -0.028 | ||||
| Upperbound of 95% confidence interval for beta | 0.190 | ||||
| Lowerbound of 95% confidence interval for alpha | -1.100 | ||||
| Upperbound of 95% confidence interval for alpha | 0.048 | ||||
| Treynor index (mean / b) | -6.074 | ||||
| Jensen alpha (a) | -0.526 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.667 | ||||
| SD | 0.599 | ||||
| Sharpe ratio (Glass type estimate) | -1.114 | ||||
| Sharpe ratio (Hedges UMVUE) | -1.113 | ||||
| df | 1084.000 | ||||
| t | -2.267 | ||||
| p | 0.534 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -2.078 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | -0.150 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -2.078 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.149 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -1.354 | ||||
| Upside Potential Ratio | 1.716 | ||||
| Upside part of mean | 0.846 | ||||
| Downside part of mean | -1.512 | ||||
| Upside SD | 0.342 | ||||
| Downside SD | 0.493 | ||||
| N nonnegative terms | 77.000 | ||||
| N negative terms | 1008.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 1085.000 | ||||
| Mean of predictor | 0.391 | ||||
| Mean of criterion | -0.667 | ||||
| SD of predictor | 0.324 | ||||
| SD of criterion | 0.599 | ||||
| Covariance | 0.009 | ||||
| r | 0.045 | ||||
| b (slope, estimate of beta) | 0.083 | ||||
| a (intercept, estimate of alpha) | -0.699 | ||||
| Mean Square Error | 0.358 | ||||
| DF error | 1083.000 | ||||
| t(b) | 1.471 | ||||
| p(b) | 0.472 | ||||
| t(a) | -2.371 | ||||
| p(a) | 0.546 | ||||
| Lowerbound of 95% confidence interval for beta | -0.028 | ||||
| Upperbound of 95% confidence interval for beta | 0.193 | ||||
| Lowerbound of 95% confidence interval for alpha | -1.278 | ||||
| Upperbound of 95% confidence interval for alpha | -0.121 | ||||
| Treynor index (mean / b) | -8.079 | ||||
| Jensen alpha (a) | -0.699 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.061 | ||||
| Expected Shortfall on VaR | 0.076 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.018 | ||||
| Expected Shortfall on VaR | 0.039 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 1085.000 | ||||
| Minimum | 0.625 | ||||
| Quartile 1 | 1.000 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.000 | ||||
| Maximum | 1.634 | ||||
| Mean of quarter 1 | 0.979 | ||||
| Mean of quarter 2 | 1.000 | ||||
| Mean of quarter 3 | 1.000 | ||||
| Mean of quarter 4 | 1.014 | ||||
| Inter Quartile Range | 0.000 | ||||
| Number outliers low | 96.000 | ||||
| Percentage of outliers low | 0.088 | ||||
| Mean of outliers low | 0.941 | ||||
| Number of outliers high | 78.000 | ||||
| Percentage of outliers high | 0.072 | ||||
| Mean of outliers high | 1.049 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | -0.414 | ||||
| VaR(95%) (moments method) | 0.003 | ||||
| Expected Shortfall (moments method) | 0.005 | ||||
| Extreme Value Index (regression method) | 0.067 | ||||
| VaR(95%) (regression method) | 0.017 | ||||
| Expected Shortfall (regression method) | 0.047 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 14.000 | ||||
| Minimum | 0.002 | ||||
| Quartile 1 | 0.009 | ||||
| Median | 0.046 | ||||
| Quartile 3 | 0.155 | ||||
| Maximum | 0.952 | ||||
| Mean of quarter 1 | 0.005 | ||||
| Mean of quarter 2 | 0.026 | ||||
| Mean of quarter 3 | 0.082 | ||||
| Mean of quarter 4 | 0.410 | ||||
| Inter Quartile Range | 0.146 | ||||
| Number outliers low | 0.000 | ||||
| Percentage of outliers low | 0.000 | ||||
| Mean of outliers low | NA | ||||
| Number of outliers high | 1.000 | ||||
| Percentage of outliers high | 0.071 | ||||
| Mean of outliers high | 0.952 | ||||
| Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.525 | ||||
| VaR(95%) (moments method) | 0.482 | ||||
| Expected Shortfall (moments method) | 1.123 | ||||
| Extreme Value Index (regression method) | 1.737 | ||||
| VaR(95%) (regression method) | 0.765 | ||||
| Expected Shortfall (regression method) | NA | ||||
| COMBINED STATISTICS | |||||
| Annualized return (arithmetic extrapolation) | -0.223 | ||||
| Compounded annual return (geometric extrapolation) | -0.464 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.487 | ||||
| Compounded annual return / average of 25% largest draw downs | -1.131 | ||||
| Compounded annual return / Expected Shortfall lognormal | -6.123 | ||||
| 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 | 1.438 | ||||
| Mean of criterion | -0.044 | ||||
| SD of predictor | 0.386 | ||||
| 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 | 1.360 | ||||
| Mean of criterion | -0.044 | ||||
| SD of predictor | 0.387 | ||||
| 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) | -8598232147497379.000 | ||||
| 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) | -69124280397843500322878459478016.000 | ||||
| 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 | ||||