Advanced Statistics: ES Formula Test
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 | 0.337 | ||||
| SD | 0.560 | ||||
| Sharpe ratio (Glass type estimate) | 0.603 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.597 | ||||
| df | 79.000 | ||||
| t | 1.557 | ||||
| p | 0.062 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.164 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.366 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.167 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.362 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 1.086 | ||||
| Upside Potential Ratio | 2.122 | ||||
| Upside part of mean | 0.659 | ||||
| Downside part of mean | -0.322 | ||||
| Upside SD | 0.471 | ||||
| Downside SD | 0.311 | ||||
| N nonnegative terms | 18.000 | ||||
| N negative terms | 62.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 80.000 | ||||
| Mean of predictor | 0.209 | ||||
| Mean of criterion | 0.337 | ||||
| SD of predictor | 0.236 | ||||
| SD of criterion | 0.560 | ||||
| Covariance | -0.012 | ||||
| r | -0.091 | ||||
| b (slope, estimate of beta) | -0.216 | ||||
| a (intercept, estimate of alpha) | 0.383 | ||||
| Mean Square Error | 0.314 | ||||
| DF error | 78.000 | ||||
| t(b) | -0.807 | ||||
| p(b) | 0.789 | ||||
| t(a) | 1.706 | ||||
| p(a) | 0.046 | ||||
| Lowerbound of 95% confidence interval for beta | -0.748 | ||||
| Upperbound of 95% confidence interval for beta | 0.317 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.064 | ||||
| Upperbound of 95% confidence interval for alpha | 0.829 | ||||
| Treynor index (mean / b) | -1.563 | ||||
| Jensen alpha (a) | 0.383 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 0.183 | ||||
| SD | 0.557 | ||||
| Sharpe ratio (Glass type estimate) | 0.329 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.326 | ||||
| df | 79.000 | ||||
| t | 0.850 | ||||
| p | 0.199 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.432 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.089 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.435 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.087 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 0.469 | ||||
| Upside Potential Ratio | 1.455 | ||||
| Upside part of mean | 0.569 | ||||
| Downside part of mean | -0.386 | ||||
| Upside SD | 0.395 | ||||
| Downside SD | 0.391 | ||||
| N nonnegative terms | 18.000 | ||||
| N negative terms | 62.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 80.000 | ||||
| Mean of predictor | 0.181 | ||||
| Mean of criterion | 0.183 | ||||
| SD of predictor | 0.223 | ||||
| SD of criterion | 0.557 | ||||
| Covariance | -0.012 | ||||
| r | -0.095 | ||||
| b (slope, estimate of beta) | -0.239 | ||||
| a (intercept, estimate of alpha) | 0.227 | ||||
| Mean Square Error | 0.311 | ||||
| DF error | 78.000 | ||||
| t(b) | -0.846 | ||||
| p(b) | 0.800 | ||||
| t(a) | 1.021 | ||||
| p(a) | 0.155 | ||||
| Lowerbound of 95% confidence interval for beta | -0.800 | ||||
| Upperbound of 95% confidence interval for beta | 0.323 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.215 | ||||
| Upperbound of 95% confidence interval for alpha | 0.669 | ||||
| Treynor index (mean / b) | -0.769 | ||||
| Jensen alpha (a) | 0.227 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.221 | ||||
| Expected Shortfall on VaR | 0.270 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.080 | ||||
| Expected Shortfall on VaR | 0.172 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 80.000 | ||||
| Minimum | 0.553 | ||||
| Quartile 1 | 1.000 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.000 | ||||
| Maximum | 1.624 | ||||
| Mean of quarter 1 | 0.904 | ||||
| Mean of quarter 2 | 1.000 | ||||
| Mean of quarter 3 | 1.000 | ||||
| Mean of quarter 4 | 1.223 | ||||
| Inter Quartile Range | 0.000 | ||||
| Number outliers low | 11.000 | ||||
| Percentage of outliers low | 0.138 | ||||
| Mean of outliers low | 0.826 | ||||
| Number of outliers high | 19.000 | ||||
| Percentage of outliers high | 0.237 | ||||
| Mean of outliers high | 1.235 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 1.291 | ||||
| VaR(95%) (moments method) | 0.003 | ||||
| Expected Shortfall (moments method) | NA | ||||
| Extreme Value Index (regression method) | -0.117 | ||||
| VaR(95%) (regression method) | 0.117 | ||||
| Expected Shortfall (regression method) | 0.215 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 6.000 | ||||
| Minimum | 0.098 | ||||
| Quartile 1 | 0.195 | ||||
| Median | 0.310 | ||||
| Quartile 3 | 0.420 | ||||
| Maximum | 0.448 | ||||
| Mean of quarter 1 | 0.132 | ||||
| Mean of quarter 2 | 0.283 | ||||
| Mean of quarter 3 | 0.336 | ||||
| Mean of quarter 4 | 0.448 | ||||
| Inter Quartile Range | 0.225 | ||||
| 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) | 0.533 | ||||
| Compounded annual return (geometric extrapolation) | 0.255 | ||||
| Calmar ratio (compounded annual return / max draw down) | 0.570 | ||||
| Compounded annual return / average of 25% largest draw downs | 0.571 | ||||
| Compounded annual return / Expected Shortfall lognormal | 0.946 | ||||
| ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 0.436 | ||||
| SD | 0.771 | ||||
| Sharpe ratio (Glass type estimate) | 0.565 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.565 | ||||
| df | 1767.000 | ||||
| t | 1.468 | ||||
| p | 0.478 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.190 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.320 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.190 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.320 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 1.093 | ||||
| Upside Potential Ratio | 4.456 | ||||
| Upside part of mean | 1.775 | ||||
| Downside part of mean | -1.340 | ||||
| Upside SD | 0.660 | ||||
| Downside SD | 0.398 | ||||
| N nonnegative terms | 274.000 | ||||
| N negative terms | 1494.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 1768.000 | ||||
| Mean of predictor | 0.238 | ||||
| Mean of criterion | 0.436 | ||||
| SD of predictor | 0.259 | ||||
| SD of criterion | 0.771 | ||||
| Covariance | -0.009 | ||||
| r | -0.045 | ||||
| b (slope, estimate of beta) | -0.133 | ||||
| a (intercept, estimate of alpha) | 0.467 | ||||
| Mean Square Error | 0.593 | ||||
| DF error | 1766.000 | ||||
| t(b) | -1.877 | ||||
| p(b) | 0.522 | ||||
| t(a) | 1.574 | ||||
| p(a) | 0.481 | ||||
| Lowerbound of 95% confidence interval for beta | -0.272 | ||||
| Upperbound of 95% confidence interval for beta | 0.006 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.115 | ||||
| Upperbound of 95% confidence interval for alpha | 1.049 | ||||
| Treynor index (mean / b) | -3.280 | ||||
| Jensen alpha (a) | 0.467 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 0.181 | ||||
| SD | 0.703 | ||||
| Sharpe ratio (Glass type estimate) | 0.257 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.257 | ||||
| df | 1767.000 | ||||
| t | 0.668 | ||||
| p | 0.490 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.498 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.012 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.498 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.012 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 0.375 | ||||
| Upside Potential Ratio | 3.366 | ||||
| Upside part of mean | 1.622 | ||||
| Downside part of mean | -1.441 | ||||
| Upside SD | 0.512 | ||||
| Downside SD | 0.482 | ||||
| N nonnegative terms | 274.000 | ||||
| N negative terms | 1494.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 1768.000 | ||||
| Mean of predictor | 0.204 | ||||
| Mean of criterion | 0.181 | ||||
| SD of predictor | 0.258 | ||||
| SD of criterion | 0.703 | ||||
| Covariance | -0.008 | ||||
| r | -0.046 | ||||
| b (slope, estimate of beta) | -0.124 | ||||
| a (intercept, estimate of alpha) | 0.206 | ||||
| Mean Square Error | 0.493 | ||||
| DF error | 1766.000 | ||||
| t(b) | -1.918 | ||||
| p(b) | 0.523 | ||||
| t(a) | 0.761 | ||||
| p(a) | 0.491 | ||||
| Lowerbound of 95% confidence interval for beta | -0.251 | ||||
| Upperbound of 95% confidence interval for beta | 0.003 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.325 | ||||
| Upperbound of 95% confidence interval for alpha | 0.737 | ||||
| Treynor index (mean / b) | -1.458 | ||||
| Jensen alpha (a) | 0.206 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.068 | ||||
| Expected Shortfall on VaR | 0.085 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.016 | ||||
| Expected Shortfall on VaR | 0.036 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 1768.000 | ||||
| Minimum | 0.483 | ||||
| Quartile 1 | 1.000 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.000 | ||||
| Maximum | 2.224 | ||||
| Mean of quarter 1 | 0.980 | ||||
| Mean of quarter 2 | 1.000 | ||||
| Mean of quarter 3 | 1.000 | ||||
| Mean of quarter 4 | 1.027 | ||||
| Inter Quartile Range | 0.000 | ||||
| Number outliers low | 221.000 | ||||
| Percentage of outliers low | 0.125 | ||||
| Mean of outliers low | 0.960 | ||||
| Number of outliers high | 280.000 | ||||
| Percentage of outliers high | 0.158 | ||||
| Mean of outliers high | 1.043 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.022 | ||||
| VaR(95%) (moments method) | 0.003 | ||||
| Expected Shortfall (moments method) | 0.006 | ||||
| Extreme Value Index (regression method) | 0.257 | ||||
| VaR(95%) (regression method) | 0.019 | ||||
| Expected Shortfall (regression method) | 0.049 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 25.000 | ||||
| Minimum | 0.003 | ||||
| Quartile 1 | 0.010 | ||||
| Median | 0.095 | ||||
| Quartile 3 | 0.242 | ||||
| Maximum | 0.676 | ||||
| Mean of quarter 1 | 0.005 | ||||
| Mean of quarter 2 | 0.047 | ||||
| Mean of quarter 3 | 0.153 | ||||
| Mean of quarter 4 | 0.431 | ||||
| Inter Quartile Range | 0.232 | ||||
| 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 | 0.040 | ||||
| Mean of outliers high | 0.676 | ||||
| Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | -0.305 | ||||
| VaR(95%) (moments method) | 0.449 | ||||
| Expected Shortfall (moments method) | 0.528 | ||||
| Extreme Value Index (regression method) | 0.059 | ||||
| VaR(95%) (regression method) | 0.513 | ||||
| Expected Shortfall (regression method) | 0.692 | ||||
| COMBINED STATISTICS | |||||
| Annualized return (arithmetic extrapolation) | 0.527 | ||||
| Compounded annual return (geometric extrapolation) | 0.252 | ||||
| Calmar ratio (compounded annual return / max draw down) | 0.373 | ||||
| Compounded annual return / average of 25% largest draw downs | 0.584 | ||||
| Compounded annual return / Expected Shortfall lognormal | 2.967 | ||||
| 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 | 0.971 | ||||
| Mean of criterion | -0.044 | ||||
| SD of predictor | 0.511 | ||||
| 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 | 0.839 | ||||
| Mean of criterion | -0.044 | ||||
| SD of predictor | 0.515 | ||||
| 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) | -8755287692541668.000 | ||||
| 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) | -335129179166404354587875342286848.000 | ||||
| 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 | ||||