Advanced Statistics: Rank and Rule I
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 | 1.715 | ||||
| SD | 1.045 | ||||
| Sharpe ratio (Glass type estimate) | 1.641 | ||||
| Sharpe ratio (Hedges UMVUE) | 1.606 | ||||
| df | 36.000 | ||||
| t | 2.881 | ||||
| p | 0.003 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | 0.452 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 2.809 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.430 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.783 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 3.989 | ||||
| Upside Potential Ratio | 5.302 | ||||
| Upside part of mean | 2.279 | ||||
| Downside part of mean | -0.564 | ||||
| Upside SD | 1.060 | ||||
| Downside SD | 0.430 | ||||
| N nonnegative terms | 25.000 | ||||
| N negative terms | 12.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 37.000 | ||||
| Mean of predictor | 0.532 | ||||
| Mean of criterion | 1.715 | ||||
| SD of predictor | 0.296 | ||||
| SD of criterion | 1.045 | ||||
| Covariance | 0.266 | ||||
| r | 0.860 | ||||
| b (slope, estimate of beta) | 3.033 | ||||
| a (intercept, estimate of alpha) | 0.103 | ||||
| Mean Square Error | 0.293 | ||||
| DF error | 35.000 | ||||
| t(b) | 9.953 | ||||
| p(b) | -0.000 | ||||
| t(a) | 0.295 | ||||
| p(a) | 0.385 | ||||
| Lowerbound of 95% confidence interval for beta | 2.415 | ||||
| Upperbound of 95% confidence interval for beta | 3.652 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.605 | ||||
| Upperbound of 95% confidence interval for alpha | 0.810 | ||||
| Treynor index (mean / b) | 0.565 | ||||
| Jensen alpha (a) | 0.103 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 1.162 | ||||
| SD | 0.985 | ||||
| Sharpe ratio (Glass type estimate) | 1.180 | ||||
| Sharpe ratio (Hedges UMVUE) | 1.156 | ||||
| df | 36.000 | ||||
| t | 2.073 | ||||
| p | 0.023 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | 0.024 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 2.321 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.008 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.303 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 1.901 | ||||
| Upside Potential Ratio | 3.059 | ||||
| Upside part of mean | 1.870 | ||||
| Downside part of mean | -0.708 | ||||
| Upside SD | 0.826 | ||||
| Downside SD | 0.611 | ||||
| N nonnegative terms | 25.000 | ||||
| N negative terms | 12.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 37.000 | ||||
| Mean of predictor | 0.480 | ||||
| Mean of criterion | 1.162 | ||||
| SD of predictor | 0.277 | ||||
| SD of criterion | 0.985 | ||||
| Covariance | 0.231 | ||||
| r | 0.846 | ||||
| b (slope, estimate of beta) | 3.007 | ||||
| a (intercept, estimate of alpha) | -0.283 | ||||
| Mean Square Error | 0.284 | ||||
| DF error | 35.000 | ||||
| t(b) | 9.371 | ||||
| p(b) | -0.000 | ||||
| t(a) | -0.830 | ||||
| p(a) | 0.794 | ||||
| Lowerbound of 95% confidence interval for beta | 2.356 | ||||
| Upperbound of 95% confidence interval for beta | 3.659 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.974 | ||||
| Upperbound of 95% confidence interval for alpha | 0.409 | ||||
| Treynor index (mean / b) | 0.386 | ||||
| Jensen alpha (a) | -0.283 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.310 | ||||
| Expected Shortfall on VaR | 0.384 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.078 | ||||
| Expected Shortfall on VaR | 0.180 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 37.000 | ||||
| Minimum | 0.385 | ||||
| Quartile 1 | 0.990 | ||||
| Median | 1.086 | ||||
| Quartile 3 | 1.266 | ||||
| Maximum | 1.988 | ||||
| Mean of quarter 1 | 0.831 | ||||
| Mean of quarter 2 | 1.042 | ||||
| Mean of quarter 3 | 1.194 | ||||
| Mean of quarter 4 | 1.555 | ||||
| Inter Quartile Range | 0.276 | ||||
| Number outliers low | 1.000 | ||||
| Percentage of outliers low | 0.027 | ||||
| Mean of outliers low | 0.385 | ||||
| Number of outliers high | 3.000 | ||||
| Percentage of outliers high | 0.081 | ||||
| Mean of outliers high | 1.791 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | -4.433 | ||||
| VaR(95%) (moments method) | 0.042 | ||||
| Expected Shortfall (moments method) | 0.042 | ||||
| Extreme Value Index (regression method) | 0.401 | ||||
| VaR(95%) (regression method) | 0.225 | ||||
| Expected Shortfall (regression method) | 0.520 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 7.000 | ||||
| Minimum | 0.064 | ||||
| Quartile 1 | 0.095 | ||||
| Median | 0.196 | ||||
| Quartile 3 | 0.252 | ||||
| Maximum | 0.658 | ||||
| Mean of quarter 1 | 0.066 | ||||
| Mean of quarter 2 | 0.160 | ||||
| Mean of quarter 3 | 0.229 | ||||
| Mean of quarter 4 | 0.467 | ||||
| Inter Quartile Range | 0.157 | ||||
| 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 | 0.143 | ||||
| Mean of outliers high | 0.658 | ||||
| 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) | 13.049 | ||||
| Compounded annual return (geometric extrapolation) | 2.341 | ||||
| Calmar ratio (compounded annual return / max draw down) | 3.556 | ||||
| Compounded annual return / average of 25% largest draw downs | 5.012 | ||||
| Compounded annual return / Expected Shortfall lognormal | 6.099 | ||||
| ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 2.002 | ||||
| SD | 1.184 | ||||
| Sharpe ratio (Glass type estimate) | 1.691 | ||||
| Sharpe ratio (Hedges UMVUE) | 1.689 | ||||
| df | 819.000 | ||||
| t | 2.991 | ||||
| p | 0.001 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | 0.579 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 2.801 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.578 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.800 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 2.584 | ||||
| Upside Potential Ratio | 8.715 | ||||
| Upside part of mean | 6.753 | ||||
| Downside part of mean | -4.751 | ||||
| Upside SD | 0.903 | ||||
| Downside SD | 0.775 | ||||
| N nonnegative terms | 461.000 | ||||
| N negative terms | 359.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 820.000 | ||||
| Mean of predictor | 0.566 | ||||
| Mean of criterion | 2.002 | ||||
| SD of predictor | 0.336 | ||||
| SD of criterion | 1.184 | ||||
| Covariance | 0.302 | ||||
| r | 0.758 | ||||
| b (slope, estimate of beta) | 2.669 | ||||
| a (intercept, estimate of alpha) | 0.490 | ||||
| Mean Square Error | 0.596 | ||||
| DF error | 818.000 | ||||
| t(b) | 33.265 | ||||
| p(b) | 0.000 | ||||
| t(a) | 1.116 | ||||
| p(a) | 0.132 | ||||
| Lowerbound of 95% confidence interval for beta | 2.512 | ||||
| Upperbound of 95% confidence interval for beta | 2.827 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.372 | ||||
| Upperbound of 95% confidence interval for alpha | 1.351 | ||||
| Treynor index (mean / b) | 0.750 | ||||
| Jensen alpha (a) | 0.490 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 1.285 | ||||
| SD | 1.207 | ||||
| Sharpe ratio (Glass type estimate) | 1.064 | ||||
| Sharpe ratio (Hedges UMVUE) | 1.063 | ||||
| df | 819.000 | ||||
| t | 1.883 | ||||
| p | 0.030 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.045 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 2.173 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.046 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.172 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 1.443 | ||||
| Upside Potential Ratio | 7.183 | ||||
| Upside part of mean | 6.395 | ||||
| Downside part of mean | -5.110 | ||||
| Upside SD | 0.818 | ||||
| Downside SD | 0.890 | ||||
| N nonnegative terms | 461.000 | ||||
| N negative terms | 359.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 820.000 | ||||
| Mean of predictor | 0.508 | ||||
| Mean of criterion | 1.285 | ||||
| SD of predictor | 0.342 | ||||
| SD of criterion | 1.207 | ||||
| Covariance | 0.318 | ||||
| r | 0.771 | ||||
| b (slope, estimate of beta) | 2.719 | ||||
| a (intercept, estimate of alpha) | -0.096 | ||||
| Mean Square Error | 0.593 | ||||
| DF error | 818.000 | ||||
| t(b) | 34.576 | ||||
| p(b) | 0.000 | ||||
| t(a) | -0.221 | ||||
| p(a) | 0.587 | ||||
| Lowerbound of 95% confidence interval for beta | 2.565 | ||||
| Upperbound of 95% confidence interval for beta | 2.873 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.954 | ||||
| Upperbound of 95% confidence interval for alpha | 0.761 | ||||
| Treynor index (mean / b) | 0.473 | ||||
| Jensen alpha (a) | -0.096 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.111 | ||||
| Expected Shortfall on VaR | 0.138 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.037 | ||||
| Expected Shortfall on VaR | 0.082 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 820.000 | ||||
| Minimum | 0.547 | ||||
| Quartile 1 | 0.985 | ||||
| Median | 1.005 | ||||
| Quartile 3 | 1.034 | ||||
| Maximum | 1.562 | ||||
| Mean of quarter 1 | 0.932 | ||||
| Mean of quarter 2 | 0.996 | ||||
| Mean of quarter 3 | 1.017 | ||||
| Mean of quarter 4 | 1.086 | ||||
| Inter Quartile Range | 0.049 | ||||
| Number outliers low | 52.000 | ||||
| Percentage of outliers low | 0.063 | ||||
| Mean of outliers low | 0.844 | ||||
| Number of outliers high | 42.000 | ||||
| Percentage of outliers high | 0.051 | ||||
| Mean of outliers high | 1.190 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.576 | ||||
| VaR(95%) (moments method) | 0.062 | ||||
| Expected Shortfall (moments method) | 0.167 | ||||
| Extreme Value Index (regression method) | 0.340 | ||||
| VaR(95%) (regression method) | 0.060 | ||||
| Expected Shortfall (regression method) | 0.115 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 48.000 | ||||
| Minimum | 0.000 | ||||
| Quartile 1 | 0.017 | ||||
| Median | 0.063 | ||||
| Quartile 3 | 0.194 | ||||
| Maximum | 0.730 | ||||
| Mean of quarter 1 | 0.010 | ||||
| Mean of quarter 2 | 0.032 | ||||
| Mean of quarter 3 | 0.115 | ||||
| Mean of quarter 4 | 0.409 | ||||
| Inter Quartile Range | 0.177 | ||||
| Number outliers low | 0.000 | ||||
| Percentage of outliers low | 0.000 | ||||
| Mean of outliers low | NA | ||||
| Number of outliers high | 4.000 | ||||
| Percentage of outliers high | 0.083 | ||||
| Mean of outliers high | 0.586 | ||||
| Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | -1.071 | ||||
| VaR(95%) (moments method) | 0.419 | ||||
| Expected Shortfall (moments method) | 0.446 | ||||
| Extreme Value Index (regression method) | -0.315 | ||||
| VaR(95%) (regression method) | 0.497 | ||||
| Expected Shortfall (regression method) | 0.606 | ||||
| COMBINED STATISTICS | |||||
| Annualized return (arithmetic extrapolation) | 20.138 | ||||
| Compounded annual return (geometric extrapolation) | 2.777 | ||||
| Calmar ratio (compounded annual return / max draw down) | 3.803 | ||||
| Compounded annual return / average of 25% largest draw downs | 6.793 | ||||
| Compounded annual return / Expected Shortfall lognormal | 20.116 | ||||
| ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 2.881 | ||||
| SD | 1.673 | ||||
| Sharpe ratio (Glass type estimate) | 1.722 | ||||
| Sharpe ratio (Hedges UMVUE) | 1.712 | ||||
| df | 130.000 | ||||
| t | 1.218 | ||||
| p | 0.447 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.061 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 4.499 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.067 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 4.492 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 2.657 | ||||
| Upside Potential Ratio | 10.643 | ||||
| Upside part of mean | 11.542 | ||||
| Downside part of mean | -8.661 | ||||
| Upside SD | 1.278 | ||||
| Downside SD | 1.084 | ||||
| N nonnegative terms | 78.000 | ||||
| N negative terms | 53.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 1.057 | ||||
| Mean of criterion | 2.881 | ||||
| SD of predictor | 0.505 | ||||
| SD of criterion | 1.673 | ||||
| Covariance | 0.742 | ||||
| r | 0.878 | ||||
| b (slope, estimate of beta) | 2.909 | ||||
| a (intercept, estimate of alpha) | -0.195 | ||||
| Mean Square Error | 0.644 | ||||
| DF error | 129.000 | ||||
| t(b) | 20.881 | ||||
| p(b) | 0.025 | ||||
| t(a) | -0.170 | ||||
| p(a) | 0.510 | ||||
| Lowerbound of 95% confidence interval for beta | 2.633 | ||||
| Upperbound of 95% confidence interval for beta | 3.185 | ||||
| Lowerbound of 95% confidence interval for alpha | -2.459 | ||||
| Upperbound of 95% confidence interval for alpha | 2.069 | ||||
| Treynor index (mean / b) | 0.990 | ||||
| Jensen alpha (a) | -0.195 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 1.485 | ||||
| SD | 1.678 | ||||
| Sharpe ratio (Glass type estimate) | 0.885 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.880 | ||||
| df | 130.000 | ||||
| t | 0.626 | ||||
| p | 0.473 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.891 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 3.657 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.894 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 3.654 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 1.242 | ||||
| Upside Potential Ratio | 9.044 | ||||
| Upside part of mean | 10.813 | ||||
| Downside part of mean | -9.328 | ||||
| Upside SD | 1.171 | ||||
| Downside SD | 1.196 | ||||
| N nonnegative terms | 78.000 | ||||
| N negative terms | 53.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 0.927 | ||||
| Mean of criterion | 1.485 | ||||
| SD of predictor | 0.511 | ||||
| SD of criterion | 1.678 | ||||
| Covariance | 0.760 | ||||
| r | 0.887 | ||||
| b (slope, estimate of beta) | 2.916 | ||||
| a (intercept, estimate of alpha) | -1.218 | ||||
| Mean Square Error | 0.603 | ||||
| DF error | 129.000 | ||||
| t(b) | 21.865 | ||||
| p(b) | 0.022 | ||||
| t(a) | -1.103 | ||||
| p(a) | 0.561 | ||||
| Lowerbound of 95% confidence interval for beta | 2.652 | ||||
| Upperbound of 95% confidence interval for beta | 3.180 | ||||
| Lowerbound of 95% confidence interval for alpha | -3.405 | ||||
| Upperbound of 95% confidence interval for alpha | 0.968 | ||||
| Treynor index (mean / b) | 0.509 | ||||
| Jensen alpha (a) | -1.218 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.152 | ||||
| Expected Shortfall on VaR | 0.187 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.066 | ||||
| Expected Shortfall on VaR | 0.134 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 131.000 | ||||
| Minimum | 0.696 | ||||
| Quartile 1 | 0.957 | ||||
| Median | 1.011 | ||||
| Quartile 3 | 1.071 | ||||
| Maximum | 1.319 | ||||
| Mean of quarter 1 | 0.881 | ||||
| Mean of quarter 2 | 0.990 | ||||
| Mean of quarter 3 | 1.036 | ||||
| Mean of quarter 4 | 1.138 | ||||
| Inter Quartile Range | 0.114 | ||||
| Number outliers low | 3.000 | ||||
| Percentage of outliers low | 0.023 | ||||
| Mean of outliers low | 0.746 | ||||
| Number of outliers high | 3.000 | ||||
| Percentage of outliers high | 0.023 | ||||
| Mean of outliers high | 1.288 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | -0.641 | ||||
| VaR(95%) (moments method) | 0.110 | ||||
| Expected Shortfall (moments method) | 0.124 | ||||
| Extreme Value Index (regression method) | -0.139 | ||||
| VaR(95%) (regression method) | 0.128 | ||||
| Expected Shortfall (regression method) | 0.169 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 5.000 | ||||
| Minimum | 0.008 | ||||
| Quartile 1 | 0.149 | ||||
| Median | 0.296 | ||||
| Quartile 3 | 0.480 | ||||
| Maximum | 0.730 | ||||
| Mean of quarter 1 | 0.078 | ||||
| Mean of quarter 2 | 0.296 | ||||
| Mean of quarter 3 | 0.480 | ||||
| Mean of quarter 4 | 0.730 | ||||
| Inter Quartile Range | 0.332 | ||||
| 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) | 2.296 | ||||
| Compounded annual return (geometric extrapolation) | 3.613 | ||||
| Calmar ratio (compounded annual return / max draw down) | 4.948 | ||||
| Compounded annual return / average of 25% largest draw downs | 4.948 | ||||
| Compounded annual return / Expected Shortfall lognormal | 19.289 | ||||