Advanced Statistics: Bottoms up
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 | -0.240 | ||||
| SD | 0.450 | ||||
| Sharpe ratio (Glass type estimate) | -0.533 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.528 | ||||
| df | 87.000 | ||||
| t | -1.444 | ||||
| p | 0.924 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.260 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.197 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.256 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.200 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.552 | ||||
| Upside Potential Ratio | 0.318 | ||||
| Upside part of mean | 0.138 | ||||
| Downside part of mean | -0.378 | ||||
| Upside SD | 0.128 | ||||
| Downside SD | 0.435 | ||||
| N nonnegative terms | 29.000 | ||||
| N negative terms | 59.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 88.000 | ||||
| Mean of predictor | 0.207 | ||||
| Mean of criterion | -0.240 | ||||
| SD of predictor | 0.349 | ||||
| SD of criterion | 0.450 | ||||
| Covariance | -0.020 | ||||
| r | -0.127 | ||||
| b (slope, estimate of beta) | -0.163 | ||||
| a (intercept, estimate of alpha) | -0.206 | ||||
| Mean Square Error | 0.202 | ||||
| DF error | 86.000 | ||||
| t(b) | -1.183 | ||||
| p(b) | 0.880 | ||||
| t(a) | -1.225 | ||||
| p(a) | 0.888 | ||||
| Lowerbound of 95% confidence interval for beta | -0.438 | ||||
| Upperbound of 95% confidence interval for beta | 0.111 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.541 | ||||
| Upperbound of 95% confidence interval for alpha | 0.128 | ||||
| Treynor index (mean / b) | 1.469 | ||||
| Jensen alpha (a) | -0.206 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -1.349 | ||||
| SD | 3.273 | ||||
| Sharpe ratio (Glass type estimate) | -0.412 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.409 | ||||
| df | 87.000 | ||||
| t | -1.116 | ||||
| p | 0.866 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.137 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.315 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.135 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.318 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.412 | ||||
| Upside Potential Ratio | 0.040 | ||||
| Upside part of mean | 0.131 | ||||
| Downside part of mean | -1.480 | ||||
| Upside SD | 0.116 | ||||
| Downside SD | 3.275 | ||||
| N nonnegative terms | 29.000 | ||||
| N negative terms | 59.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 88.000 | ||||
| Mean of predictor | 0.145 | ||||
| Mean of criterion | -1.349 | ||||
| SD of predictor | 0.349 | ||||
| SD of criterion | 3.273 | ||||
| Covariance | -0.102 | ||||
| r | -0.089 | ||||
| b (slope, estimate of beta) | -0.840 | ||||
| a (intercept, estimate of alpha) | -1.227 | ||||
| Mean Square Error | 10.748 | ||||
| DF error | 86.000 | ||||
| t(b) | -0.833 | ||||
| p(b) | 0.796 | ||||
| t(a) | -1.006 | ||||
| p(a) | 0.841 | ||||
| Lowerbound of 95% confidence interval for beta | -2.844 | ||||
| Upperbound of 95% confidence interval for beta | 1.164 | ||||
| Lowerbound of 95% confidence interval for alpha | -3.651 | ||||
| Upperbound of 95% confidence interval for alpha | 1.197 | ||||
| Treynor index (mean / b) | 1.606 | ||||
| Jensen alpha (a) | -1.227 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.811 | ||||
| Expected Shortfall on VaR | 0.866 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.085 | ||||
| Expected Shortfall on VaR | 0.191 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 88.000 | ||||
| Minimum | 0.000 | ||||
| Quartile 1 | 0.990 | ||||
| Median | 0.999 | ||||
| Quartile 3 | 1.010 | ||||
| Maximum | 1.281 | ||||
| Mean of quarter 1 | 0.888 | ||||
| Mean of quarter 2 | 0.996 | ||||
| Mean of quarter 3 | 1.002 | ||||
| Mean of quarter 4 | 1.049 | ||||
| Inter Quartile Range | 0.020 | ||||
| Number outliers low | 7.000 | ||||
| Percentage of outliers low | 0.080 | ||||
| Mean of outliers low | 0.687 | ||||
| Number of outliers high | 8.000 | ||||
| Percentage of outliers high | 0.091 | ||||
| Mean of outliers high | 1.100 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 1.299 | ||||
| VaR(95%) (moments method) | 0.081 | ||||
| Expected Shortfall (moments method) | NA | ||||
| Extreme Value Index (regression method) | 1.660 | ||||
| VaR(95%) (regression method) | 0.063 | ||||
| Expected Shortfall (regression method) | NA | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 5.000 | ||||
| Minimum | 0.030 | ||||
| Quartile 1 | 0.043 | ||||
| Median | 0.062 | ||||
| Quartile 3 | 0.314 | ||||
| Maximum | 1.000 | ||||
| Mean of quarter 1 | 0.036 | ||||
| Mean of quarter 2 | 0.062 | ||||
| Mean of quarter 3 | 0.314 | ||||
| Mean of quarter 4 | 1.000 | ||||
| Inter Quartile Range | 0.271 | ||||
| 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 | 0.200 | ||||
| Mean of outliers high | 1.000 | ||||
| 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) | -0.136 | ||||
| Compounded annual return (geometric extrapolation) | -0.729 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.729 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.729 | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.842 | ||||
| ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.157 | ||||
| SD | 0.606 | ||||
| Sharpe ratio (Glass type estimate) | -0.259 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.259 | ||||
| df | 1927.000 | ||||
| t | -0.702 | ||||
| p | 0.510 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.981 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.464 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.981 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.464 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.308 | ||||
| Upside Potential Ratio | 2.289 | ||||
| Upside part of mean | 1.166 | ||||
| Downside part of mean | -1.323 | ||||
| Upside SD | 0.329 | ||||
| Downside SD | 0.509 | ||||
| N nonnegative terms | 698.000 | ||||
| N negative terms | 1230.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 1928.000 | ||||
| Mean of predictor | 0.342 | ||||
| Mean of criterion | -0.157 | ||||
| SD of predictor | 0.576 | ||||
| SD of criterion | 0.606 | ||||
| Covariance | -0.025 | ||||
| r | -0.072 | ||||
| b (slope, estimate of beta) | -0.076 | ||||
| a (intercept, estimate of alpha) | -0.131 | ||||
| Mean Square Error | 0.366 | ||||
| DF error | 1926.000 | ||||
| t(b) | -3.165 | ||||
| p(b) | 0.536 | ||||
| t(a) | -0.587 | ||||
| p(a) | 0.507 | ||||
| Lowerbound of 95% confidence interval for beta | -0.123 | ||||
| Upperbound of 95% confidence interval for beta | -0.029 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.569 | ||||
| Upperbound of 95% confidence interval for alpha | 0.307 | ||||
| Treynor index (mean / b) | 2.073 | ||||
| Jensen alpha (a) | -0.131 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -1.345 | ||||
| SD | 3.295 | ||||
| Sharpe ratio (Glass type estimate) | -0.408 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.408 | ||||
| df | 1927.000 | ||||
| t | -1.107 | ||||
| p | 0.516 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.131 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.315 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.130 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.315 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.410 | ||||
| Upside Potential Ratio | 0.340 | ||||
| Upside part of mean | 1.117 | ||||
| Downside part of mean | -2.462 | ||||
| Upside SD | 0.303 | ||||
| Downside SD | 3.282 | ||||
| N nonnegative terms | 698.000 | ||||
| N negative terms | 1230.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 1928.000 | ||||
| Mean of predictor | 0.177 | ||||
| Mean of criterion | -1.345 | ||||
| SD of predictor | 0.575 | ||||
| SD of criterion | 3.295 | ||||
| Covariance | -0.006 | ||||
| r | -0.003 | ||||
| b (slope, estimate of beta) | -0.017 | ||||
| a (intercept, estimate of alpha) | -1.342 | ||||
| Mean Square Error | 10.865 | ||||
| DF error | 1926.000 | ||||
| t(b) | -0.129 | ||||
| p(b) | 0.501 | ||||
| t(a) | -1.104 | ||||
| p(a) | 0.513 | ||||
| Lowerbound of 95% confidence interval for beta | -0.273 | ||||
| Upperbound of 95% confidence interval for beta | 0.239 | ||||
| Lowerbound of 95% confidence interval for alpha | -3.725 | ||||
| Upperbound of 95% confidence interval for alpha | 1.042 | ||||
| Treynor index (mean / b) | 79.779 | ||||
| Jensen alpha (a) | -1.342 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.288 | ||||
| Expected Shortfall on VaR | 0.344 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.013 | ||||
| Expected Shortfall on VaR | 0.030 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 1928.000 | ||||
| Minimum | 0.000 | ||||
| Quartile 1 | 0.999 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.001 | ||||
| Maximum | 1.334 | ||||
| Mean of quarter 1 | 0.981 | ||||
| Mean of quarter 2 | 1.000 | ||||
| Mean of quarter 3 | 1.000 | ||||
| Mean of quarter 4 | 1.018 | ||||
| Inter Quartile Range | 0.002 | ||||
| Number outliers low | 272.000 | ||||
| Percentage of outliers low | 0.141 | ||||
| Mean of outliers low | 0.967 | ||||
| Number of outliers high | 287.000 | ||||
| Percentage of outliers high | 0.149 | ||||
| Mean of outliers high | 1.028 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 1.339 | ||||
| VaR(95%) (moments method) | 0.012 | ||||
| Expected Shortfall (moments method) | NA | ||||
| Extreme Value Index (regression method) | 0.941 | ||||
| VaR(95%) (regression method) | 0.010 | ||||
| Expected Shortfall (regression method) | 0.197 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 15.000 | ||||
| Minimum | 0.001 | ||||
| Quartile 1 | 0.042 | ||||
| Median | 0.093 | ||||
| Quartile 3 | 0.168 | ||||
| Maximum | 1.000 | ||||
| Mean of quarter 1 | 0.022 | ||||
| Mean of quarter 2 | 0.072 | ||||
| Mean of quarter 3 | 0.128 | ||||
| Mean of quarter 4 | 0.459 | ||||
| Inter Quartile Range | 0.126 | ||||
| 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 | 0.133 | ||||
| Mean of outliers high | 0.706 | ||||
| Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | -0.024 | ||||
| VaR(95%) (moments method) | 0.414 | ||||
| Expected Shortfall (moments method) | 0.581 | ||||
| Extreme Value Index (regression method) | 0.844 | ||||
| VaR(95%) (regression method) | 0.795 | ||||
| Expected Shortfall (regression method) | 5.308 | ||||
| COMBINED STATISTICS | |||||
| Annualized return (arithmetic extrapolation) | -0.136 | ||||
| Compounded annual return (geometric extrapolation) | -0.728 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.728 | ||||
| Compounded annual return / average of 25% largest draw downs | -1.585 | ||||
| Compounded annual return / Expected Shortfall lognormal | -2.112 | ||||
| ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -3.434 | ||||
| SD | 1.571 | ||||
| Sharpe ratio (Glass type estimate) | -2.185 | ||||
| Sharpe ratio (Hedges UMVUE) | -2.172 | ||||
| df | 130.000 | ||||
| t | -1.545 | ||||
| p | 0.567 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -4.965 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.604 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -4.957 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.612 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -2.174 | ||||
| Upside Potential Ratio | 0.022 | ||||
| Upside part of mean | 0.035 | ||||
| Downside part of mean | -3.469 | ||||
| Upside SD | 0.017 | ||||
| Downside SD | 1.580 | ||||
| N nonnegative terms | 13.000 | ||||
| N negative terms | 118.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 2.075 | ||||
| Mean of criterion | -3.434 | ||||
| SD of predictor | 0.795 | ||||
| SD of criterion | 1.571 | ||||
| Covariance | 0.048 | ||||
| r | 0.039 | ||||
| b (slope, estimate of beta) | 0.076 | ||||
| a (intercept, estimate of alpha) | -3.592 | ||||
| Mean Square Error | 2.485 | ||||
| DF error | 129.000 | ||||
| t(b) | 0.440 | ||||
| p(b) | 0.475 | ||||
| t(a) | -1.591 | ||||
| p(a) | 0.588 | ||||
| Lowerbound of 95% confidence interval for beta | -0.268 | ||||
| Upperbound of 95% confidence interval for beta | 0.421 | ||||
| Lowerbound of 95% confidence interval for alpha | -8.061 | ||||
| Upperbound of 95% confidence interval for alpha | 0.876 | ||||
| Treynor index (mean / b) | -44.892 | ||||
| Jensen alpha (a) | -3.592 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -19.438 | ||||
| SD | 12.514 | ||||
| Sharpe ratio (Glass type estimate) | -1.553 | ||||
| Sharpe ratio (Hedges UMVUE) | -1.544 | ||||
| df | 130.000 | ||||
| t | -1.098 | ||||
| p | 0.548 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -4.329 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.228 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -4.322 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.234 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -1.552 | ||||
| Upside Potential Ratio | 0.003 | ||||
| Upside part of mean | 0.035 | ||||
| Downside part of mean | -19.473 | ||||
| Upside SD | 0.017 | ||||
| Downside SD | 12.524 | ||||
| N nonnegative terms | 13.000 | ||||
| N negative terms | 118.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 1.746 | ||||
| Mean of criterion | -19.438 | ||||
| SD of predictor | 0.815 | ||||
| SD of criterion | 12.514 | ||||
| Covariance | 0.377 | ||||
| r | 0.037 | ||||
| b (slope, estimate of beta) | 0.568 | ||||
| a (intercept, estimate of alpha) | -20.429 | ||||
| Mean Square Error | 157.601 | ||||
| DF error | 129.000 | ||||
| t(b) | 0.420 | ||||
| p(b) | 0.476 | ||||
| t(a) | -1.141 | ||||
| p(a) | 0.564 | ||||
| Lowerbound of 95% confidence interval for beta | -2.107 | ||||
| Upperbound of 95% confidence interval for beta | 3.243 | ||||
| Lowerbound of 95% confidence interval for alpha | -55.865 | ||||
| Upperbound of 95% confidence interval for alpha | 15.006 | ||||
| Treynor index (mean / b) | -34.226 | ||||
| Jensen alpha (a) | -20.429 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.740 | ||||
| Expected Shortfall on VaR | 0.805 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.043 | ||||
| Expected Shortfall on VaR | 0.098 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 131.000 | ||||
| Minimum | 0.000 | ||||
| Quartile 1 | 1.000 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.000 | ||||
| Maximum | 1.012 | ||||
| Mean of quarter 1 | 0.948 | ||||
| Mean of quarter 2 | 1.000 | ||||
| Mean of quarter 3 | 1.000 | ||||
| Mean of quarter 4 | 1.001 | ||||
| Inter Quartile Range | 0.000 | ||||
| Number outliers low | 21.000 | ||||
| Percentage of outliers low | 0.160 | ||||
| Mean of outliers low | 0.919 | ||||
| Number of outliers high | 9.000 | ||||
| Percentage of outliers high | 0.069 | ||||
| Mean of outliers high | 1.002 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 2.371 | ||||
| VaR(95%) (moments method) | 0.003 | ||||
| Expected Shortfall (moments method) | NA | ||||
| Extreme Value Index (regression method) | 3.879 | ||||
| VaR(95%) (regression method) | 0.001 | ||||
| Expected Shortfall (regression method) | NA | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 1.000 | ||||
| Minimum | 1.000 | ||||
| Quartile 1 | 1.000 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.000 | ||||
| Maximum | 1.000 | ||||
| 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 | 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) | -2.000 | ||||
| Compounded annual return (geometric extrapolation) | -1.000 | ||||
| Calmar ratio (compounded annual return / max draw down) | -1.000 | ||||
| Compounded annual return / average of 25% largest draw downs | NA | ||||
| Compounded annual return / Expected Shortfall lognormal | -1.243 | ||||