Advanced Statistics: Ruben Lucas
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 | -0.098 | ||||
| SD | 0.287 | ||||
| Sharpe ratio (Glass type estimate) | -0.343 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.335 | ||||
| df | 31.000 | ||||
| t | -0.560 | ||||
| p | 0.710 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.544 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.863 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.538 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.868 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.442 | ||||
| Upside Potential Ratio | 1.637 | ||||
| Upside part of mean | 0.365 | ||||
| Downside part of mean | -0.463 | ||||
| Upside SD | 0.176 | ||||
| Downside SD | 0.223 | ||||
| N nonnegative terms | 15.000 | ||||
| N negative terms | 17.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 32.000 | ||||
| Mean of predictor | 0.584 | ||||
| Mean of criterion | -0.098 | ||||
| SD of predictor | 0.322 | ||||
| SD of criterion | 0.287 | ||||
| Covariance | 0.046 | ||||
| r | 0.501 | ||||
| b (slope, estimate of beta) | 0.447 | ||||
| a (intercept, estimate of alpha) | -0.359 | ||||
| Mean Square Error | 0.064 | ||||
| DF error | 30.000 | ||||
| t(b) | 3.172 | ||||
| p(b) | 0.002 | ||||
| t(a) | -2.052 | ||||
| p(a) | 0.976 | ||||
| Lowerbound of 95% confidence interval for beta | 0.159 | ||||
| Upperbound of 95% confidence interval for beta | 0.735 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.717 | ||||
| Upperbound of 95% confidence interval for alpha | -0.002 | ||||
| Treynor index (mean / b) | -0.220 | ||||
| Jensen alpha (a) | -0.359 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.140 | ||||
| SD | 0.294 | ||||
| Sharpe ratio (Glass type estimate) | -0.475 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.464 | ||||
| df | 31.000 | ||||
| t | -0.776 | ||||
| p | 0.778 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.678 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.734 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.669 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.742 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.583 | ||||
| Upside Potential Ratio | 1.456 | ||||
| Upside part of mean | 0.349 | ||||
| Downside part of mean | -0.489 | ||||
| Upside SD | 0.167 | ||||
| Downside SD | 0.240 | ||||
| N nonnegative terms | 15.000 | ||||
| N negative terms | 17.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 32.000 | ||||
| Mean of predictor | 0.526 | ||||
| Mean of criterion | -0.140 | ||||
| SD of predictor | 0.285 | ||||
| SD of criterion | 0.294 | ||||
| Covariance | 0.042 | ||||
| r | 0.496 | ||||
| b (slope, estimate of beta) | 0.511 | ||||
| a (intercept, estimate of alpha) | -0.409 | ||||
| Mean Square Error | 0.067 | ||||
| DF error | 30.000 | ||||
| t(b) | 3.131 | ||||
| p(b) | 0.002 | ||||
| t(a) | -2.263 | ||||
| p(a) | 0.984 | ||||
| Lowerbound of 95% confidence interval for beta | 0.178 | ||||
| Upperbound of 95% confidence interval for beta | 0.845 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.778 | ||||
| Upperbound of 95% confidence interval for alpha | -0.040 | ||||
| Treynor index (mean / b) | -0.273 | ||||
| Jensen alpha (a) | -0.409 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.140 | ||||
| Expected Shortfall on VaR | 0.170 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.093 | ||||
| Expected Shortfall on VaR | 0.158 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 32.000 | ||||
| Minimum | 0.790 | ||||
| Quartile 1 | 0.945 | ||||
| Median | 0.987 | ||||
| Quartile 3 | 1.060 | ||||
| Maximum | 1.139 | ||||
| Mean of quarter 1 | 0.891 | ||||
| Mean of quarter 2 | 0.962 | ||||
| Mean of quarter 3 | 1.030 | ||||
| Mean of quarter 4 | 1.098 | ||||
| Inter Quartile Range | 0.114 | ||||
| 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) | -0.754 | ||||
| VaR(95%) (moments method) | 0.114 | ||||
| Expected Shortfall (moments method) | 0.126 | ||||
| Extreme Value Index (regression method) | -0.007 | ||||
| VaR(95%) (regression method) | 0.132 | ||||
| Expected Shortfall (regression method) | 0.179 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 2.000 | ||||
| Minimum | 0.057 | ||||
| Quartile 1 | 0.161 | ||||
| Median | 0.265 | ||||
| Quartile 3 | 0.369 | ||||
| Maximum | 0.473 | ||||
| Mean of quarter 1 | 0.057 | ||||
| Mean of quarter 2 | NA | ||||
| Mean of quarter 3 | NA | ||||
| Mean of quarter 4 | 0.473 | ||||
| Inter Quartile Range | 0.208 | ||||
| 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) | -0.085 | ||||
| Compounded annual return (geometric extrapolation) | -0.091 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.193 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.193 | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.537 | ||||
| ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.079 | ||||
| SD | 0.374 | ||||
| Sharpe ratio (Glass type estimate) | -0.212 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.212 | ||||
| df | 719.000 | ||||
| t | -0.351 | ||||
| p | 0.637 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.394 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.970 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.394 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.971 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.298 | ||||
| Upside Potential Ratio | 7.144 | ||||
| Upside part of mean | 1.897 | ||||
| Downside part of mean | -1.977 | ||||
| Upside SD | 0.262 | ||||
| Downside SD | 0.266 | ||||
| N nonnegative terms | 321.000 | ||||
| N negative terms | 399.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 720.000 | ||||
| Mean of predictor | 0.657 | ||||
| Mean of criterion | -0.079 | ||||
| SD of predictor | 0.370 | ||||
| SD of criterion | 0.374 | ||||
| Covariance | 0.071 | ||||
| r | 0.516 | ||||
| b (slope, estimate of beta) | 0.520 | ||||
| a (intercept, estimate of alpha) | -0.421 | ||||
| Mean Square Error | 0.103 | ||||
| DF error | 718.000 | ||||
| t(b) | 16.124 | ||||
| p(b) | 0.000 | ||||
| t(a) | -2.164 | ||||
| p(a) | 0.985 | ||||
| Lowerbound of 95% confidence interval for beta | 0.457 | ||||
| Upperbound of 95% confidence interval for beta | 0.583 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.802 | ||||
| Upperbound of 95% confidence interval for alpha | -0.039 | ||||
| Treynor index (mean / b) | -0.152 | ||||
| Jensen alpha (a) | -0.421 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.149 | ||||
| SD | 0.375 | ||||
| Sharpe ratio (Glass type estimate) | -0.398 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.397 | ||||
| df | 719.000 | ||||
| t | -0.659 | ||||
| p | 0.745 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.580 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.785 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.580 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.785 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.541 | ||||
| Upside Potential Ratio | 6.753 | ||||
| Upside part of mean | 1.864 | ||||
| Downside part of mean | -2.013 | ||||
| Upside SD | 0.254 | ||||
| Downside SD | 0.276 | ||||
| N nonnegative terms | 321.000 | ||||
| N negative terms | 399.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 720.000 | ||||
| Mean of predictor | 0.589 | ||||
| Mean of criterion | -0.149 | ||||
| SD of predictor | 0.364 | ||||
| SD of criterion | 0.375 | ||||
| Covariance | 0.069 | ||||
| r | 0.507 | ||||
| b (slope, estimate of beta) | 0.523 | ||||
| a (intercept, estimate of alpha) | -0.458 | ||||
| Mean Square Error | 0.105 | ||||
| DF error | 718.000 | ||||
| t(b) | 15.772 | ||||
| p(b) | 0.000 | ||||
| t(a) | -2.332 | ||||
| p(a) | 0.990 | ||||
| Lowerbound of 95% confidence interval for beta | 0.458 | ||||
| Upperbound of 95% confidence interval for beta | 0.588 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.843 | ||||
| Upperbound of 95% confidence interval for alpha | -0.072 | ||||
| Treynor index (mean / b) | -0.285 | ||||
| Jensen alpha (a) | -0.458 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.038 | ||||
| Expected Shortfall on VaR | 0.047 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.018 | ||||
| Expected Shortfall on VaR | 0.037 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 720.000 | ||||
| Minimum | 0.839 | ||||
| Quartile 1 | 0.990 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.009 | ||||
| Maximum | 1.165 | ||||
| Mean of quarter 1 | 0.975 | ||||
| Mean of quarter 2 | 0.996 | ||||
| Mean of quarter 3 | 1.004 | ||||
| Mean of quarter 4 | 1.026 | ||||
| Inter Quartile Range | 0.020 | ||||
| Number outliers low | 21.000 | ||||
| Percentage of outliers low | 0.029 | ||||
| Mean of outliers low | 0.933 | ||||
| Number of outliers high | 27.000 | ||||
| Percentage of outliers high | 0.037 | ||||
| Mean of outliers high | 1.060 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.166 | ||||
| VaR(95%) (moments method) | 0.024 | ||||
| Expected Shortfall (moments method) | 0.036 | ||||
| Extreme Value Index (regression method) | 0.208 | ||||
| VaR(95%) (regression method) | 0.022 | ||||
| Expected Shortfall (regression method) | 0.033 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 13.000 | ||||
| Minimum | 0.000 | ||||
| Quartile 1 | 0.002 | ||||
| Median | 0.013 | ||||
| Quartile 3 | 0.025 | ||||
| Maximum | 0.568 | ||||
| Mean of quarter 1 | 0.001 | ||||
| Mean of quarter 2 | 0.008 | ||||
| Mean of quarter 3 | 0.022 | ||||
| Mean of quarter 4 | 0.252 | ||||
| Inter Quartile Range | 0.023 | ||||
| Number outliers low | 0.000 | ||||
| Percentage of outliers low | 0.000 | ||||
| Mean of outliers low | NA | ||||
| Number of outliers high | 3.000 | ||||
| Percentage of outliers high | 0.231 | ||||
| Mean of outliers high | 0.252 | ||||
| Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.350 | ||||
| VaR(95%) (moments method) | 0.119 | ||||
| Expected Shortfall (moments method) | 0.241 | ||||
| Extreme Value Index (regression method) | 1.346 | ||||
| VaR(95%) (regression method) | 0.473 | ||||
| Expected Shortfall (regression method) | NA | ||||
| COMBINED STATISTICS | |||||
| Annualized return (arithmetic extrapolation) | -0.091 | ||||
| Compounded annual return (geometric extrapolation) | -0.100 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.176 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.397 | ||||
| Compounded annual return / Expected Shortfall lognormal | -2.116 | ||||
| ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 0.041 | ||||
| SD | 0.501 | ||||
| Sharpe ratio (Glass type estimate) | 0.081 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.080 | ||||
| df | 130.000 | ||||
| t | 0.057 | ||||
| p | 0.497 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -2.691 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 2.853 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -2.691 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.852 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 0.118 | ||||
| Upside Potential Ratio | 8.103 | ||||
| Upside part of mean | 2.793 | ||||
| Downside part of mean | -2.753 | ||||
| Upside SD | 0.360 | ||||
| Downside SD | 0.345 | ||||
| N nonnegative terms | 48.000 | ||||
| N negative terms | 83.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 1.159 | ||||
| Mean of criterion | 0.041 | ||||
| SD of predictor | 0.470 | ||||
| SD of criterion | 0.501 | ||||
| Covariance | 0.136 | ||||
| r | 0.579 | ||||
| b (slope, estimate of beta) | 0.617 | ||||
| a (intercept, estimate of alpha) | -0.674 | ||||
| Mean Square Error | 0.168 | ||||
| DF error | 129.000 | ||||
| t(b) | 8.059 | ||||
| p(b) | 0.153 | ||||
| t(a) | -1.150 | ||||
| p(a) | 0.564 | ||||
| Lowerbound of 95% confidence interval for beta | 0.465 | ||||
| Upperbound of 95% confidence interval for beta | 0.768 | ||||
| Lowerbound of 95% confidence interval for alpha | -1.834 | ||||
| Upperbound of 95% confidence interval for alpha | 0.486 | ||||
| Treynor index (mean / b) | 0.066 | ||||
| Jensen alpha (a) | -0.674 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.084 | ||||
| SD | 0.502 | ||||
| Sharpe ratio (Glass type estimate) | -0.168 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.167 | ||||
| df | 130.000 | ||||
| t | -0.119 | ||||
| p | 0.505 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -2.939 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 2.604 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -2.939 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.605 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.235 | ||||
| Upside Potential Ratio | 7.624 | ||||
| Upside part of mean | 2.731 | ||||
| Downside part of mean | -2.815 | ||||
| Upside SD | 0.349 | ||||
| Downside SD | 0.358 | ||||
| N nonnegative terms | 48.000 | ||||
| N negative terms | 83.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 1.046 | ||||
| Mean of criterion | -0.084 | ||||
| SD of predictor | 0.473 | ||||
| SD of criterion | 0.502 | ||||
| Covariance | 0.139 | ||||
| r | 0.587 | ||||
| b (slope, estimate of beta) | 0.623 | ||||
| a (intercept, estimate of alpha) | -0.736 | ||||
| Mean Square Error | 0.166 | ||||
| DF error | 129.000 | ||||
| t(b) | 8.238 | ||||
| p(b) | 0.149 | ||||
| t(a) | -1.264 | ||||
| p(a) | 0.570 | ||||
| Lowerbound of 95% confidence interval for beta | 0.473 | ||||
| Upperbound of 95% confidence interval for beta | 0.773 | ||||
| Lowerbound of 95% confidence interval for alpha | -1.888 | ||||
| Upperbound of 95% confidence interval for alpha | 0.416 | ||||
| Treynor index (mean / b) | -0.135 | ||||
| Jensen alpha (a) | -0.736 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.050 | ||||
| Expected Shortfall on VaR | 0.062 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.028 | ||||
| Expected Shortfall on VaR | 0.053 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 131.000 | ||||
| Minimum | 0.862 | ||||
| Quartile 1 | 0.983 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.014 | ||||
| Maximum | 1.129 | ||||
| Mean of quarter 1 | 0.965 | ||||
| Mean of quarter 2 | 0.994 | ||||
| Mean of quarter 3 | 1.004 | ||||
| Mean of quarter 4 | 1.038 | ||||
| Inter Quartile Range | 0.032 | ||||
| Number outliers low | 1.000 | ||||
| Percentage of outliers low | 0.008 | ||||
| Mean of outliers low | 0.862 | ||||
| Number of outliers high | 4.000 | ||||
| Percentage of outliers high | 0.031 | ||||
| Mean of outliers high | 1.081 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.256 | ||||
| VaR(95%) (moments method) | 0.038 | ||||
| Expected Shortfall (moments method) | 0.059 | ||||
| Extreme Value Index (regression method) | 0.170 | ||||
| VaR(95%) (regression method) | 0.031 | ||||
| Expected Shortfall (regression method) | 0.043 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 2.000 | ||||
| Minimum | 0.037 | ||||
| Quartile 1 | 0.119 | ||||
| Median | 0.201 | ||||
| Quartile 3 | 0.283 | ||||
| Maximum | 0.365 | ||||
| Mean of quarter 1 | 0.037 | ||||
| Mean of quarter 2 | NA | ||||
| Mean of quarter 3 | NA | ||||
| Mean of quarter 4 | 0.365 | ||||
| Inter Quartile Range | 0.164 | ||||
| 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) | -0.040 | ||||
| Compounded annual return (geometric extrapolation) | -0.039 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.108 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.108 | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.633 | ||||