Advanced Statistics: faith in research
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 | 1521.988 | ||||
| SD | 3138.424 | ||||
| Sharpe ratio (Glass type estimate) | 0.485 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.478 | ||||
| df | 50.000 | ||||
| t | 1.000 | ||||
| p | 0.161 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.473 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.438 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.478 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.433 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 2099.155 | ||||
| Upside Potential Ratio | 2100.178 | ||||
| Upside part of mean | 1522.730 | ||||
| Downside part of mean | -0.742 | ||||
| Upside SD | 3138.409 | ||||
| Downside SD | 0.725 | ||||
| N nonnegative terms | 9.000 | ||||
| N negative terms | 42.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 51.000 | ||||
| Mean of predictor | 0.382 | ||||
| Mean of criterion | 1521.988 | ||||
| SD of predictor | 0.260 | ||||
| SD of criterion | 3138.424 | ||||
| Covariance | -61.625 | ||||
| r | -0.076 | ||||
| b (slope, estimate of beta) | -914.122 | ||||
| a (intercept, estimate of alpha) | 1870.762 | ||||
| Mean Square Error | 9993239.313 | ||||
| DF error | 49.000 | ||||
| t(b) | -0.531 | ||||
| p(b) | 0.701 | ||||
| t(a) | 1.121 | ||||
| p(a) | 0.134 | ||||
| Lowerbound of 95% confidence interval for beta | -4374.286 | ||||
| Upperbound of 95% confidence interval for beta | 2546.042 | ||||
| Lowerbound of 95% confidence interval for alpha | -1481.637 | ||||
| Upperbound of 95% confidence interval for alpha | 5223.162 | ||||
| Treynor index (mean / b) | -1.665 | ||||
| Jensen alpha (a) | 1870.762 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.455 | ||||
| SD | 6.166 | ||||
| Sharpe ratio (Glass type estimate) | -0.074 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.073 | ||||
| df | 50.000 | ||||
| t | -0.152 | ||||
| p | 0.560 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.024 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.877 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.023 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.878 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.104 | ||||
| Upside Potential Ratio | 0.549 | ||||
| Upside part of mean | 2.400 | ||||
| Downside part of mean | -2.855 | ||||
| Upside SD | 4.263 | ||||
| Downside SD | 4.373 | ||||
| N nonnegative terms | 9.000 | ||||
| N negative terms | 42.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 51.000 | ||||
| Mean of predictor | 0.342 | ||||
| Mean of criterion | -0.455 | ||||
| SD of predictor | 0.257 | ||||
| SD of criterion | 6.166 | ||||
| Covariance | 0.096 | ||||
| r | 0.061 | ||||
| b (slope, estimate of beta) | 1.458 | ||||
| a (intercept, estimate of alpha) | -0.954 | ||||
| Mean Square Error | 38.657 | ||||
| DF error | 49.000 | ||||
| t(b) | 0.426 | ||||
| p(b) | 0.336 | ||||
| t(a) | -0.295 | ||||
| p(a) | 0.615 | ||||
| Lowerbound of 95% confidence interval for beta | -5.424 | ||||
| Upperbound of 95% confidence interval for beta | 8.340 | ||||
| Lowerbound of 95% confidence interval for alpha | -7.457 | ||||
| Upperbound of 95% confidence interval for alpha | 5.549 | ||||
| Treynor index (mean / b) | -0.312 | ||||
| Jensen alpha (a) | -0.954 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.948 | ||||
| Expected Shortfall on VaR | 0.971 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.191 | ||||
| Expected Shortfall on VaR | 0.410 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 51.000 | ||||
| Minimum | 0.000 | ||||
| Quartile 1 | 1.000 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.000 | ||||
| Maximum | 6471.000 | ||||
| Mean of quarter 1 | 0.769 | ||||
| Mean of quarter 2 | 1.000 | ||||
| Mean of quarter 3 | 1.000 | ||||
| Mean of quarter 4 | 498.818 | ||||
| Inter Quartile Range | 0.000 | ||||
| Number outliers low | 6.000 | ||||
| Percentage of outliers low | 0.118 | ||||
| Mean of outliers low | 0.500 | ||||
| Number of outliers high | 11.000 | ||||
| Percentage of outliers high | 0.216 | ||||
| Mean of outliers high | 589.331 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | -52.710 | ||||
| VaR(95%) (moments method) | 0.001 | ||||
| Expected Shortfall (moments method) | 0.001 | ||||
| Extreme Value Index (regression method) | -0.997 | ||||
| VaR(95%) (regression method) | 0.509 | ||||
| Expected Shortfall (regression method) | 0.705 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 2.000 | ||||
| Minimum | 0.701 | ||||
| Quartile 1 | 0.775 | ||||
| Median | 0.850 | ||||
| Quartile 3 | 0.925 | ||||
| Maximum | 1.000 | ||||
| Mean of quarter 1 | 0.701 | ||||
| Mean of quarter 2 | NA | ||||
| Mean of quarter 3 | NA | ||||
| Mean of quarter 4 | 1.000 | ||||
| Inter Quartile Range | 0.150 | ||||
| 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) | -0.194 | ||||
| Compounded annual return (geometric extrapolation) | -0.337 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.337 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.337 | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.347 | ||||
| ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 1498.571 | ||||
| SD | 3114.036 | ||||
| Sharpe ratio (Glass type estimate) | 0.481 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.481 | ||||
| df | 1130.000 | ||||
| t | 1.000 | ||||
| p | 0.485 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.462 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.425 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.463 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.424 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 1903.757 | ||||
| Upside Potential Ratio | 1906.436 | ||||
| Upside part of mean | 1500.680 | ||||
| Downside part of mean | -2.109 | ||||
| Upside SD | 3114.036 | ||||
| Downside SD | 0.787 | ||||
| N nonnegative terms | 139.000 | ||||
| N negative terms | 992.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 1131.000 | ||||
| Mean of predictor | 0.424 | ||||
| Mean of criterion | 1498.571 | ||||
| SD of predictor | 0.352 | ||||
| SD of criterion | 3114.036 | ||||
| Covariance | -30.563 | ||||
| r | -0.028 | ||||
| b (slope, estimate of beta) | -246.409 | ||||
| a (intercept, estimate of alpha) | 1603.079 | ||||
| Mean Square Error | 9698274.640 | ||||
| DF error | 1129.000 | ||||
| t(b) | -0.937 | ||||
| p(b) | 0.518 | ||||
| t(a) | 1.067 | ||||
| p(a) | 0.480 | ||||
| Lowerbound of 95% confidence interval for beta | -762.530 | ||||
| Upperbound of 95% confidence interval for beta | 269.713 | ||||
| Lowerbound of 95% confidence interval for alpha | -1345.958 | ||||
| Upperbound of 95% confidence interval for alpha | 4552.117 | ||||
| Treynor index (mean / b) | -6.082 | ||||
| Jensen alpha (a) | 1603.079 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.448 | ||||
| SD | 6.078 | ||||
| Sharpe ratio (Glass type estimate) | -0.074 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.074 | ||||
| df | 1130.000 | ||||
| t | -0.153 | ||||
| p | 0.502 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.017 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.870 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.017 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.870 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.103 | ||||
| Upside Potential Ratio | 0.869 | ||||
| Upside part of mean | 3.768 | ||||
| Downside part of mean | -4.216 | ||||
| Upside SD | 4.254 | ||||
| Downside SD | 4.337 | ||||
| N nonnegative terms | 139.000 | ||||
| N negative terms | 992.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 1131.000 | ||||
| Mean of predictor | 0.359 | ||||
| Mean of criterion | -0.448 | ||||
| SD of predictor | 0.365 | ||||
| SD of criterion | 6.078 | ||||
| Covariance | 0.056 | ||||
| r | 0.025 | ||||
| b (slope, estimate of beta) | 0.421 | ||||
| a (intercept, estimate of alpha) | -0.599 | ||||
| Mean Square Error | 36.955 | ||||
| DF error | 1129.000 | ||||
| t(b) | 0.850 | ||||
| p(b) | 0.484 | ||||
| t(a) | -0.204 | ||||
| p(a) | 0.504 | ||||
| Lowerbound of 95% confidence interval for beta | -0.550 | ||||
| Upperbound of 95% confidence interval for beta | 1.392 | ||||
| Lowerbound of 95% confidence interval for alpha | -6.351 | ||||
| Upperbound of 95% confidence interval for alpha | 5.152 | ||||
| Treynor index (mean / b) | -1.065 | ||||
| Jensen alpha (a) | -0.599 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.462 | ||||
| Expected Shortfall on VaR | 0.536 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.026 | ||||
| Expected Shortfall on VaR | 0.058 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 1131.000 | ||||
| Minimum | 0.000 | ||||
| Quartile 1 | 1.000 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.000 | ||||
| Maximum | 6471.000 | ||||
| Mean of quarter 1 | 0.968 | ||||
| Mean of quarter 2 | 1.000 | ||||
| Mean of quarter 3 | 1.000 | ||||
| Mean of quarter 4 | 23.891 | ||||
| Inter Quartile Range | 0.000 | ||||
| Number outliers low | 111.000 | ||||
| Percentage of outliers low | 0.098 | ||||
| Mean of outliers low | 0.919 | ||||
| Number of outliers high | 140.000 | ||||
| Percentage of outliers high | 0.124 | ||||
| Mean of outliers high | 47.273 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.578 | ||||
| VaR(95%) (moments method) | 0.003 | ||||
| Expected Shortfall (moments method) | 0.012 | ||||
| Extreme Value Index (regression method) | 0.494 | ||||
| VaR(95%) (regression method) | 0.020 | ||||
| Expected Shortfall (regression method) | 0.091 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 9.000 | ||||
| Minimum | 0.028 | ||||
| Quartile 1 | 0.088 | ||||
| Median | 0.114 | ||||
| Quartile 3 | 0.305 | ||||
| Maximum | 1.000 | ||||
| Mean of quarter 1 | 0.049 | ||||
| Mean of quarter 2 | 0.102 | ||||
| Mean of quarter 3 | 0.262 | ||||
| Mean of quarter 4 | 0.901 | ||||
| Inter Quartile Range | 0.217 | ||||
| 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 | 0.222 | ||||
| Mean of outliers high | 0.901 | ||||
| Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | -45.817 | ||||
| VaR(95%) (moments method) | 0.641 | ||||
| Expected Shortfall (moments method) | NA | ||||
| Extreme Value Index (regression method) | -2.323 | ||||
| VaR(95%) (regression method) | 1.384 | ||||
| Expected Shortfall (regression method) | 1.396 | ||||
| COMBINED STATISTICS | |||||
| Annualized return (arithmetic extrapolation) | -0.191 | ||||
| Compounded annual return (geometric extrapolation) | -0.332 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.332 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.369 | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.621 | ||||
| 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 | 1.290 | ||||
| Mean of criterion | -0.044 | ||||
| SD of predictor | 0.353 | ||||
| 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 | 1.226 | ||||
| Mean of criterion | -0.044 | ||||
| SD of predictor | 0.351 | ||||
| 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) | -8600894255006553.000 | ||||
| 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) | -237042615051116557317652323762176.000 | ||||
| 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 | ||||