Advanced Statistics: Monte Carlo FX Statistical Arbitrage - SP
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 | 168.089 | ||||
| SD | 219.927 | ||||
| Sharpe ratio (Glass type estimate) | 0.764 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.750 | ||||
| df | 40.000 | ||||
| t | 1.413 | ||||
| p | 0.083 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.314 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.833 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.323 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.823 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 140.124 | ||||
| Upside Potential Ratio | 141.904 | ||||
| Upside part of mean | 170.224 | ||||
| Downside part of mean | -2.135 | ||||
| Upside SD | 222.579 | ||||
| Downside SD | 1.200 | ||||
| N nonnegative terms | 14.000 | ||||
| N negative terms | 27.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 41.000 | ||||
| Mean of predictor | 0.519 | ||||
| Mean of criterion | 168.089 | ||||
| SD of predictor | 0.253 | ||||
| SD of criterion | 219.927 | ||||
| Covariance | 6.479 | ||||
| r | 0.116 | ||||
| b (slope, estimate of beta) | 101.120 | ||||
| a (intercept, estimate of alpha) | 115.580 | ||||
| Mean Square Error | 48936.132 | ||||
| DF error | 39.000 | ||||
| t(b) | 0.732 | ||||
| p(b) | 0.234 | ||||
| t(a) | 0.828 | ||||
| p(a) | 0.206 | ||||
| Lowerbound of 95% confidence interval for beta | -178.372 | ||||
| Upperbound of 95% confidence interval for beta | 380.611 | ||||
| Lowerbound of 95% confidence interval for alpha | -166.665 | ||||
| Upperbound of 95% confidence interval for alpha | 397.825 | ||||
| Treynor index (mean / b) | 1.662 | ||||
| Jensen alpha (a) | 115.580 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.712 | ||||
| SD | 7.658 | ||||
| Sharpe ratio (Glass type estimate) | -0.093 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.091 | ||||
| df | 40.000 | ||||
| t | -0.172 | ||||
| p | 0.568 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.153 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.968 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.152 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.969 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.125 | ||||
| Upside Potential Ratio | 1.094 | ||||
| Upside part of mean | 6.215 | ||||
| Downside part of mean | -6.926 | ||||
| Upside SD | 5.000 | ||||
| Downside SD | 5.679 | ||||
| N nonnegative terms | 14.000 | ||||
| N negative terms | 27.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 41.000 | ||||
| Mean of predictor | 0.477 | ||||
| Mean of criterion | -0.712 | ||||
| SD of predictor | 0.246 | ||||
| SD of criterion | 7.658 | ||||
| Covariance | 0.831 | ||||
| r | 0.441 | ||||
| b (slope, estimate of beta) | 13.747 | ||||
| a (intercept, estimate of alpha) | -7.275 | ||||
| Mean Square Error | 48.421 | ||||
| DF error | 39.000 | ||||
| t(b) | 3.073 | ||||
| p(b) | 0.002 | ||||
| t(a) | -1.681 | ||||
| p(a) | 0.950 | ||||
| Lowerbound of 95% confidence interval for beta | 4.697 | ||||
| Upperbound of 95% confidence interval for beta | 22.797 | ||||
| Lowerbound of 95% confidence interval for alpha | -16.030 | ||||
| Upperbound of 95% confidence interval for alpha | 1.480 | ||||
| Treynor index (mean / b) | -0.052 | ||||
| Jensen alpha (a) | -7.275 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.975 | ||||
| Expected Shortfall on VaR | 0.987 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.484 | ||||
| Expected Shortfall on VaR | 0.891 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 41.000 | ||||
| Minimum | 0.001 | ||||
| Quartile 1 | 0.804 | ||||
| Median | 0.987 | ||||
| Quartile 3 | 1.086 | ||||
| Maximum | 390.000 | ||||
| Mean of quarter 1 | 0.427 | ||||
| Mean of quarter 2 | 0.912 | ||||
| Mean of quarter 3 | 1.024 | ||||
| Mean of quarter 4 | 59.140 | ||||
| Inter Quartile Range | 0.281 | ||||
| Number outliers low | 6.000 | ||||
| Percentage of outliers low | 0.146 | ||||
| Mean of outliers low | 0.155 | ||||
| Number of outliers high | 7.000 | ||||
| Percentage of outliers high | 0.171 | ||||
| Mean of outliers high | 83.949 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.170 | ||||
| VaR(95%) (moments method) | 0.542 | ||||
| Expected Shortfall (moments method) | 0.843 | ||||
| Extreme Value Index (regression method) | -2.053 | ||||
| VaR(95%) (regression method) | 0.511 | ||||
| Expected Shortfall (regression method) | 0.518 | ||||
| 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) | -0.263 | ||||
| Compounded annual return (geometric extrapolation) | -0.487 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.487 | ||||
| Compounded annual return / average of 25% largest draw downs | NA | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.493 | ||||
| ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 690.614 | ||||
| SD | 408.442 | ||||
| Sharpe ratio (Glass type estimate) | 1.691 | ||||
| Sharpe ratio (Hedges UMVUE) | 1.689 | ||||
| df | 899.000 | ||||
| t | 3.134 | ||||
| p | 0.001 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | 0.630 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 2.751 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.629 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.750 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 247.223 | ||||
| Upside Potential Ratio | 252.075 | ||||
| Upside part of mean | 704.169 | ||||
| Downside part of mean | -13.554 | ||||
| Upside SD | 410.429 | ||||
| Downside SD | 2.793 | ||||
| N nonnegative terms | 337.000 | ||||
| N negative terms | 563.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 900.000 | ||||
| Mean of predictor | 0.556 | ||||
| Mean of criterion | 690.614 | ||||
| SD of predictor | 0.320 | ||||
| SD of criterion | 408.442 | ||||
| Covariance | 4.309 | ||||
| r | 0.033 | ||||
| b (slope, estimate of beta) | 41.997 | ||||
| a (intercept, estimate of alpha) | 667.246 | ||||
| Mean Square Error | 166829.535 | ||||
| DF error | 898.000 | ||||
| t(b) | 0.988 | ||||
| p(b) | 0.162 | ||||
| t(a) | 3.010 | ||||
| p(a) | 0.001 | ||||
| Lowerbound of 95% confidence interval for beta | -41.466 | ||||
| Upperbound of 95% confidence interval for beta | 125.459 | ||||
| Lowerbound of 95% confidence interval for alpha | 232.246 | ||||
| Upperbound of 95% confidence interval for alpha | 1102.246 | ||||
| Treynor index (mean / b) | 16.444 | ||||
| Jensen alpha (a) | 667.246 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.708 | ||||
| SD | 14.325 | ||||
| Sharpe ratio (Glass type estimate) | -0.049 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.049 | ||||
| df | 899.000 | ||||
| t | -0.092 | ||||
| p | 0.536 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -1.107 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.008 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.107 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.008 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.072 | ||||
| Upside Potential Ratio | 3.179 | ||||
| Upside part of mean | 31.163 | ||||
| Downside part of mean | -31.871 | ||||
| Upside SD | 10.435 | ||||
| Downside SD | 9.803 | ||||
| N nonnegative terms | 337.000 | ||||
| N negative terms | 563.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 900.000 | ||||
| Mean of predictor | 0.505 | ||||
| Mean of criterion | -0.708 | ||||
| SD of predictor | 0.320 | ||||
| SD of criterion | 14.325 | ||||
| Covariance | 0.402 | ||||
| r | 0.088 | ||||
| b (slope, estimate of beta) | 3.928 | ||||
| a (intercept, estimate of alpha) | -2.691 | ||||
| Mean Square Error | 203.858 | ||||
| DF error | 898.000 | ||||
| t(b) | 2.640 | ||||
| p(b) | 0.004 | ||||
| t(a) | -0.348 | ||||
| p(a) | 0.636 | ||||
| Lowerbound of 95% confidence interval for beta | 1.008 | ||||
| Upperbound of 95% confidence interval for beta | 6.849 | ||||
| Lowerbound of 95% confidence interval for alpha | -17.881 | ||||
| Upperbound of 95% confidence interval for alpha | 12.500 | ||||
| Treynor index (mean / b) | -0.180 | ||||
| Jensen alpha (a) | -2.691 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.767 | ||||
| Expected Shortfall on VaR | 0.832 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.134 | ||||
| Expected Shortfall on VaR | 0.295 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 900.000 | ||||
| Minimum | 0.003 | ||||
| Quartile 1 | 0.982 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.013 | ||||
| Maximum | 443.000 | ||||
| Mean of quarter 1 | 0.798 | ||||
| Mean of quarter 2 | 0.996 | ||||
| Mean of quarter 3 | 1.003 | ||||
| Mean of quarter 4 | 11.748 | ||||
| Inter Quartile Range | 0.031 | ||||
| Number outliers low | 118.000 | ||||
| Percentage of outliers low | 0.131 | ||||
| Mean of outliers low | 0.647 | ||||
| Number of outliers high | 114.000 | ||||
| Percentage of outliers high | 0.127 | ||||
| Mean of outliers high | 22.186 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 1.213 | ||||
| VaR(95%) (moments method) | 0.165 | ||||
| Expected Shortfall (moments method) | NA | ||||
| Extreme Value Index (regression method) | 0.028 | ||||
| VaR(95%) (regression method) | 0.133 | ||||
| Expected Shortfall (regression method) | 0.209 | ||||
| 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) | -0.261 | ||||
| Compounded annual return (geometric extrapolation) | -0.485 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.485 | ||||
| Compounded annual return / average of 25% largest draw downs | NA | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.583 | ||||
| ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 1571.885 | ||||
| SD | 661.220 | ||||
| Sharpe ratio (Glass type estimate) | 2.377 | ||||
| Sharpe ratio (Hedges UMVUE) | 2.364 | ||||
| df | 130.000 | ||||
| t | 1.681 | ||||
| p | 0.427 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.414 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 5.160 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.423 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 5.150 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 511.452 | ||||
| Upside Potential Ratio | 516.030 | ||||
| Upside part of mean | 1585.954 | ||||
| Downside part of mean | -14.069 | ||||
| Upside SD | 665.804 | ||||
| Downside SD | 3.073 | ||||
| N nonnegative terms | 15.000 | ||||
| N negative terms | 116.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 0.994 | ||||
| Mean of criterion | 1571.885 | ||||
| SD of predictor | 0.478 | ||||
| SD of criterion | 661.220 | ||||
| Covariance | -25.117 | ||||
| r | -0.080 | ||||
| b (slope, estimate of beta) | -110.048 | ||||
| a (intercept, estimate of alpha) | 1681.262 | ||||
| Mean Square Error | 437815.173 | ||||
| DF error | 129.000 | ||||
| t(b) | -0.906 | ||||
| p(b) | 0.551 | ||||
| t(a) | 1.782 | ||||
| p(a) | 0.402 | ||||
| Lowerbound of 95% confidence interval for beta | -350.386 | ||||
| Upperbound of 95% confidence interval for beta | 130.291 | ||||
| Lowerbound of 95% confidence interval for alpha | -185.491 | ||||
| Upperbound of 95% confidence interval for alpha | 3548.015 | ||||
| Treynor index (mean / b) | -14.284 | ||||
| Jensen alpha (a) | 1681.262 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.277 | ||||
| SD | 18.203 | ||||
| Sharpe ratio (Glass type estimate) | -0.015 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.015 | ||||
| df | 130.000 | ||||
| t | -0.011 | ||||
| p | 0.500 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -2.787 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 2.757 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -2.787 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.757 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.023 | ||||
| Upside Potential Ratio | 3.273 | ||||
| Upside part of mean | 38.848 | ||||
| Downside part of mean | -39.124 | ||||
| Upside SD | 13.710 | ||||
| Downside SD | 11.868 | ||||
| N nonnegative terms | 15.000 | ||||
| N negative terms | 116.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 0.879 | ||||
| Mean of criterion | -0.277 | ||||
| SD of predictor | 0.479 | ||||
| SD of criterion | 18.203 | ||||
| Covariance | 0.007 | ||||
| r | 0.001 | ||||
| b (slope, estimate of beta) | 0.032 | ||||
| a (intercept, estimate of alpha) | -0.305 | ||||
| Mean Square Error | 333.922 | ||||
| DF error | 129.000 | ||||
| t(b) | 0.010 | ||||
| p(b) | 0.499 | ||||
| t(a) | -0.012 | ||||
| p(a) | 0.501 | ||||
| Lowerbound of 95% confidence interval for beta | -6.593 | ||||
| Upperbound of 95% confidence interval for beta | 6.658 | ||||
| Lowerbound of 95% confidence interval for alpha | -51.766 | ||||
| Upperbound of 95% confidence interval for alpha | 51.156 | ||||
| Treynor index (mean / b) | -8.584 | ||||
| Jensen alpha (a) | -0.305 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.843 | ||||
| Expected Shortfall on VaR | 0.894 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.173 | ||||
| Expected Shortfall on VaR | 0.373 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 131.000 | ||||
| Minimum | 0.005 | ||||
| Quartile 1 | 1.000 | ||||
| Median | 1.000 | ||||
| Quartile 3 | 1.000 | ||||
| Maximum | 350.000 | ||||
| Mean of quarter 1 | 0.787 | ||||
| Mean of quarter 2 | 1.000 | ||||
| Mean of quarter 3 | 1.000 | ||||
| Mean of quarter 4 | 25.030 | ||||
| Inter Quartile Range | 0.000 | ||||
| Number outliers low | 17.000 | ||||
| Percentage of outliers low | 0.130 | ||||
| Mean of outliers low | 0.587 | ||||
| Number of outliers high | 15.000 | ||||
| Percentage of outliers high | 0.115 | ||||
| Mean of outliers high | 53.865 | ||||
| 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) | -1.077 | ||||
| VaR(95%) (regression method) | 0.295 | ||||
| Expected Shortfall (regression method) | 0.374 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 2.000 | ||||
| Minimum | 0.375 | ||||
| Quartile 1 | 0.531 | ||||
| Median | 0.687 | ||||
| Quartile 3 | 0.843 | ||||
| Maximum | 0.999 | ||||
| Mean of quarter 1 | 0.375 | ||||
| Mean of quarter 2 | NA | ||||
| Mean of quarter 3 | NA | ||||
| Mean of quarter 4 | 0.999 | ||||
| Inter Quartile Range | 0.312 | ||||
| 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) | -0.220 | ||||
| Compounded annual return (geometric extrapolation) | -0.207 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.208 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.208 | ||||
| Compounded annual return / Expected Shortfall lognormal | -0.232 | ||||