Advanced Statistics: Mirabilia
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 | 0.069 | ||||
| SD | 0.587 | ||||
| Sharpe ratio (Glass type estimate) | 0.117 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.114 | ||||
| df | 32.000 | ||||
| t | 0.194 | ||||
| p | 0.424 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.066 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.299 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.068 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.297 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 0.163 | ||||
| Upside Potential Ratio | 1.854 | ||||
| Upside part of mean | 0.781 | ||||
| Downside part of mean | -0.713 | ||||
| Upside SD | 0.395 | ||||
| Downside SD | 0.421 | ||||
| N nonnegative terms | 19.000 | ||||
| N negative terms | 14.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 33.000 | ||||
| Mean of predictor | 0.608 | ||||
| Mean of criterion | 0.069 | ||||
| SD of predictor | 0.278 | ||||
| SD of criterion | 0.587 | ||||
| Covariance | -0.023 | ||||
| r | -0.144 | ||||
| b (slope, estimate of beta) | -0.303 | ||||
| a (intercept, estimate of alpha) | 0.253 | ||||
| Mean Square Error | 0.348 | ||||
| DF error | 31.000 | ||||
| t(b) | -0.809 | ||||
| p(b) | 0.788 | ||||
| t(a) | 0.599 | ||||
| p(a) | 0.277 | ||||
| Lowerbound of 95% confidence interval for beta | -1.068 | ||||
| Upperbound of 95% confidence interval for beta | 0.461 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.608 | ||||
| Upperbound of 95% confidence interval for alpha | 1.114 | ||||
| Treynor index (mean / b) | -0.226 | ||||
| Jensen alpha (a) | 0.253 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.124 | ||||
| SD | 0.663 | ||||
| Sharpe ratio (Glass type estimate) | -0.187 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.182 | ||||
| df | 32.000 | ||||
| t | -0.310 | ||||
| p | 0.621 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.368 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.997 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.365 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.000 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.224 | ||||
| Upside Potential Ratio | 1.290 | ||||
| Upside part of mean | 0.712 | ||||
| Downside part of mean | -0.836 | ||||
| Upside SD | 0.350 | ||||
| Downside SD | 0.552 | ||||
| N nonnegative terms | 19.000 | ||||
| N negative terms | 14.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 33.000 | ||||
| Mean of predictor | 0.557 | ||||
| Mean of criterion | -0.124 | ||||
| SD of predictor | 0.261 | ||||
| SD of criterion | 0.663 | ||||
| Covariance | -0.020 | ||||
| r | -0.118 | ||||
| b (slope, estimate of beta) | -0.300 | ||||
| a (intercept, estimate of alpha) | 0.043 | ||||
| Mean Square Error | 0.447 | ||||
| DF error | 31.000 | ||||
| t(b) | -0.661 | ||||
| p(b) | 0.743 | ||||
| t(a) | 0.091 | ||||
| p(a) | 0.464 | ||||
| Lowerbound of 95% confidence interval for beta | -1.225 | ||||
| Upperbound of 95% confidence interval for beta | 0.625 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.928 | ||||
| Upperbound of 95% confidence interval for alpha | 1.014 | ||||
| Treynor index (mean / b) | 0.413 | ||||
| Jensen alpha (a) | 0.043 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.278 | ||||
| Expected Shortfall on VaR | 0.331 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.123 | ||||
| Expected Shortfall on VaR | 0.247 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 33.000 | ||||
| Minimum | 0.458 | ||||
| Quartile 1 | 0.902 | ||||
| Median | 1.019 | ||||
| Quartile 3 | 1.096 | ||||
| Maximum | 1.349 | ||||
| Mean of quarter 1 | 0.815 | ||||
| Mean of quarter 2 | 0.975 | ||||
| Mean of quarter 3 | 1.067 | ||||
| Mean of quarter 4 | 1.204 | ||||
| Inter Quartile Range | 0.194 | ||||
| Number outliers low | 1.000 | ||||
| Percentage of outliers low | 0.030 | ||||
| Mean of outliers low | 0.458 | ||||
| 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) | 0.397 | ||||
| VaR(95%) (moments method) | 0.215 | ||||
| Expected Shortfall (moments method) | 0.381 | ||||
| Extreme Value Index (regression method) | 0.848 | ||||
| VaR(95%) (regression method) | 0.173 | ||||
| Expected Shortfall (regression method) | 0.726 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 1.000 | ||||
| Minimum | 0.641 | ||||
| Quartile 1 | 0.641 | ||||
| Median | 0.641 | ||||
| Quartile 3 | 0.641 | ||||
| Maximum | 0.641 | ||||
| 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) | -0.072 | ||||
| Compounded annual return (geometric extrapolation) | -0.077 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.120 | ||||
| Compounded annual return / average of 25% largest draw downs | NA | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.231 | ||||
| ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 0.083 | ||||
| SD | 0.616 | ||||
| Sharpe ratio (Glass type estimate) | 0.134 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.134 | ||||
| df | 730.000 | ||||
| t | 0.224 | ||||
| p | 0.411 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.039 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.308 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.039 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.307 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 0.174 | ||||
| Upside Potential Ratio | 6.548 | ||||
| Upside part of mean | 3.118 | ||||
| Downside part of mean | -3.035 | ||||
| Upside SD | 0.390 | ||||
| Downside SD | 0.476 | ||||
| N nonnegative terms | 339.000 | ||||
| N negative terms | 392.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 731.000 | ||||
| Mean of predictor | 0.597 | ||||
| Mean of criterion | 0.083 | ||||
| SD of predictor | 0.359 | ||||
| SD of criterion | 0.616 | ||||
| Covariance | -0.016 | ||||
| r | -0.074 | ||||
| b (slope, estimate of beta) | -0.127 | ||||
| a (intercept, estimate of alpha) | 0.158 | ||||
| Mean Square Error | 0.378 | ||||
| DF error | 729.000 | ||||
| t(b) | -1.996 | ||||
| p(b) | 0.977 | ||||
| t(a) | 0.428 | ||||
| p(a) | 0.335 | ||||
| Lowerbound of 95% confidence interval for beta | -0.251 | ||||
| Upperbound of 95% confidence interval for beta | -0.002 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.568 | ||||
| Upperbound of 95% confidence interval for alpha | 0.885 | ||||
| Treynor index (mean / b) | -0.653 | ||||
| Jensen alpha (a) | 0.158 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.123 | ||||
| SD | 0.657 | ||||
| Sharpe ratio (Glass type estimate) | -0.187 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.187 | ||||
| df | 730.000 | ||||
| t | -0.312 | ||||
| p | 0.622 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.360 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.987 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.360 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.987 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.228 | ||||
| Upside Potential Ratio | 5.665 | ||||
| Upside part of mean | 3.045 | ||||
| Downside part of mean | -3.167 | ||||
| Upside SD | 0.377 | ||||
| Downside SD | 0.537 | ||||
| N nonnegative terms | 339.000 | ||||
| N negative terms | 392.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 731.000 | ||||
| Mean of predictor | 0.530 | ||||
| Mean of criterion | -0.123 | ||||
| SD of predictor | 0.365 | ||||
| SD of criterion | 0.657 | ||||
| Covariance | -0.016 | ||||
| r | -0.066 | ||||
| b (slope, estimate of beta) | -0.119 | ||||
| a (intercept, estimate of alpha) | -0.060 | ||||
| Mean Square Error | 0.430 | ||||
| DF error | 729.000 | ||||
| t(b) | -1.785 | ||||
| p(b) | 0.963 | ||||
| t(a) | -0.152 | ||||
| p(a) | 0.560 | ||||
| Lowerbound of 95% confidence interval for beta | -0.249 | ||||
| Upperbound of 95% confidence interval for beta | 0.012 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.833 | ||||
| Upperbound of 95% confidence interval for alpha | 0.714 | ||||
| Treynor index (mean / b) | 1.034 | ||||
| Jensen alpha (a) | -0.060 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.065 | ||||
| Expected Shortfall on VaR | 0.081 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.027 | ||||
| Expected Shortfall on VaR | 0.058 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 731.000 | ||||
| Minimum | 0.627 | ||||
| Quartile 1 | 0.986 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.018 | ||||
| Maximum | 1.188 | ||||
| Mean of quarter 1 | 0.959 | ||||
| Mean of quarter 2 | 0.995 | ||||
| Mean of quarter 3 | 1.007 | ||||
| Mean of quarter 4 | 1.041 | ||||
| Inter Quartile Range | 0.032 | ||||
| Number outliers low | 26.000 | ||||
| Percentage of outliers low | 0.036 | ||||
| Mean of outliers low | 0.891 | ||||
| Number of outliers high | 21.000 | ||||
| Percentage of outliers high | 0.029 | ||||
| Mean of outliers high | 1.094 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.335 | ||||
| VaR(95%) (moments method) | 0.039 | ||||
| Expected Shortfall (moments method) | 0.070 | ||||
| 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 | 4.000 | ||||
| Minimum | 0.015 | ||||
| Quartile 1 | 0.021 | ||||
| Median | 0.036 | ||||
| Quartile 3 | 0.233 | ||||
| Maximum | 0.787 | ||||
| Mean of quarter 1 | 0.015 | ||||
| Mean of quarter 2 | 0.023 | ||||
| Mean of quarter 3 | 0.048 | ||||
| Mean of quarter 4 | 0.787 | ||||
| Inter Quartile Range | 0.212 | ||||
| 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 | 0.250 | ||||
| Mean of outliers high | 0.787 | ||||
| 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) | -0.071 | ||||
| Compounded annual return (geometric extrapolation) | -0.076 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.096 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.096 | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.938 | ||||
| ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 0.182 | ||||
| SD | 0.511 | ||||
| Sharpe ratio (Glass type estimate) | 0.357 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.355 | ||||
| df | 130.000 | ||||
| t | 0.253 | ||||
| p | 0.489 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -2.415 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 3.129 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -2.417 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 3.127 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 0.502 | ||||
| Upside Potential Ratio | 8.732 | ||||
| Upside part of mean | 3.174 | ||||
| Downside part of mean | -2.992 | ||||
| Upside SD | 0.356 | ||||
| Downside SD | 0.364 | ||||
| N nonnegative terms | 62.000 | ||||
| N negative terms | 69.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 0.832 | ||||
| Mean of criterion | 0.182 | ||||
| SD of predictor | 0.461 | ||||
| SD of criterion | 0.511 | ||||
| Covariance | -0.042 | ||||
| r | -0.179 | ||||
| b (slope, estimate of beta) | -0.198 | ||||
| a (intercept, estimate of alpha) | 0.348 | ||||
| Mean Square Error | 0.254 | ||||
| DF error | 129.000 | ||||
| t(b) | -2.070 | ||||
| p(b) | 0.614 | ||||
| t(a) | 0.484 | ||||
| p(a) | 0.473 | ||||
| Lowerbound of 95% confidence interval for beta | -0.388 | ||||
| Upperbound of 95% confidence interval for beta | -0.009 | ||||
| Lowerbound of 95% confidence interval for alpha | -1.072 | ||||
| Upperbound of 95% confidence interval for alpha | 1.767 | ||||
| Treynor index (mean / b) | -0.919 | ||||
| Jensen alpha (a) | 0.348 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 0.052 | ||||
| SD | 0.513 | ||||
| Sharpe ratio (Glass type estimate) | 0.102 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.102 | ||||
| df | 130.000 | ||||
| t | 0.072 | ||||
| p | 0.497 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -2.670 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 2.874 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -2.670 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.873 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 0.140 | ||||
| Upside Potential Ratio | 8.318 | ||||
| Upside part of mean | 3.112 | ||||
| Downside part of mean | -3.060 | ||||
| Upside SD | 0.348 | ||||
| Downside SD | 0.374 | ||||
| N nonnegative terms | 62.000 | ||||
| N negative terms | 69.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 0.726 | ||||
| Mean of criterion | 0.052 | ||||
| SD of predictor | 0.461 | ||||
| SD of criterion | 0.513 | ||||
| Covariance | -0.042 | ||||
| r | -0.180 | ||||
| b (slope, estimate of beta) | -0.200 | ||||
| a (intercept, estimate of alpha) | 0.197 | ||||
| Mean Square Error | 0.256 | ||||
| DF error | 129.000 | ||||
| t(b) | -2.073 | ||||
| p(b) | 0.614 | ||||
| t(a) | 0.274 | ||||
| p(a) | 0.485 | ||||
| Lowerbound of 95% confidence interval for beta | -0.390 | ||||
| Upperbound of 95% confidence interval for beta | -0.009 | ||||
| Lowerbound of 95% confidence interval for alpha | -1.226 | ||||
| Upperbound of 95% confidence interval for alpha | 1.621 | ||||
| Treynor index (mean / b) | -0.262 | ||||
| Jensen alpha (a) | 0.197 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.051 | ||||
| Expected Shortfall on VaR | 0.063 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.027 | ||||
| Expected Shortfall on VaR | 0.052 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 131.000 | ||||
| Minimum | 0.902 | ||||
| Quartile 1 | 0.982 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.019 | ||||
| Maximum | 1.073 | ||||
| Mean of quarter 1 | 0.960 | ||||
| Mean of quarter 2 | 0.995 | ||||
| Mean of quarter 3 | 1.008 | ||||
| Mean of quarter 4 | 1.040 | ||||
| Inter Quartile Range | 0.037 | ||||
| Number outliers low | 2.000 | ||||
| Percentage of outliers low | 0.015 | ||||
| Mean of outliers low | 0.913 | ||||
| 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) | -0.192 | ||||
| VaR(95%) (moments method) | 0.039 | ||||
| Expected Shortfall (moments method) | 0.049 | ||||
| Extreme Value Index (regression method) | -0.059 | ||||
| VaR(95%) (regression method) | 0.042 | ||||
| Expected Shortfall (regression method) | 0.056 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 9.000 | ||||
| Minimum | 0.008 | ||||
| Quartile 1 | 0.021 | ||||
| Median | 0.035 | ||||
| Quartile 3 | 0.042 | ||||
| Maximum | 0.353 | ||||
| Mean of quarter 1 | 0.014 | ||||
| Mean of quarter 2 | 0.033 | ||||
| Mean of quarter 3 | 0.041 | ||||
| Mean of quarter 4 | 0.202 | ||||
| Inter Quartile Range | 0.021 | ||||
| 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 | 0.111 | ||||
| Mean of outliers high | 0.353 | ||||
| Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.939 | ||||
| VaR(95%) (moments method) | 0.201 | ||||
| Expected Shortfall (moments method) | 3.446 | ||||
| Extreme Value Index (regression method) | 4.051 | ||||
| VaR(95%) (regression method) | 1.276 | ||||
| Expected Shortfall (regression method) | NA | ||||
| COMBINED STATISTICS | |||||
| Annualized return (arithmetic extrapolation) | 0.099 | ||||
| Compounded annual return (geometric extrapolation) | 0.101 | ||||
| Calmar ratio (compounded annual return / max draw down) | 0.287 | ||||
| Compounded annual return / average of 25% largest draw downs | 0.501 | ||||
| Compounded annual return / Expected Shortfall lognormal | 1.607 | ||||