Advanced Statistics: Compounding Income Stream
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 | 0.167 | ||||
| SD | 0.579 | ||||
| Sharpe ratio (Glass type estimate) | 0.289 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.285 | ||||
| df | 59.000 | ||||
| t | 0.645 | ||||
| p | 0.261 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.591 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.165 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.593 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.163 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 0.524 | ||||
| Upside Potential Ratio | 2.105 | ||||
| Upside part of mean | 0.671 | ||||
| Downside part of mean | -0.504 | ||||
| Upside SD | 0.480 | ||||
| Downside SD | 0.319 | ||||
| N nonnegative terms | 37.000 | ||||
| N negative terms | 23.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 60.000 | ||||
| Mean of predictor | 0.311 | ||||
| Mean of criterion | 0.167 | ||||
| SD of predictor | 0.237 | ||||
| SD of criterion | 0.579 | ||||
| Covariance | 0.040 | ||||
| r | 0.288 | ||||
| b (slope, estimate of beta) | 0.702 | ||||
| a (intercept, estimate of alpha) | -0.051 | ||||
| Mean Square Error | 0.313 | ||||
| DF error | 58.000 | ||||
| t(b) | 2.290 | ||||
| p(b) | 0.013 | ||||
| t(a) | -0.190 | ||||
| p(a) | 0.575 | ||||
| Lowerbound of 95% confidence interval for beta | 0.088 | ||||
| Upperbound of 95% confidence interval for beta | 1.316 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.587 | ||||
| Upperbound of 95% confidence interval for alpha | 0.485 | ||||
| Treynor index (mean / b) | 0.238 | ||||
| Jensen alpha (a) | -0.051 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 0.020 | ||||
| SD | 0.536 | ||||
| Sharpe ratio (Glass type estimate) | 0.038 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.037 | ||||
| df | 59.000 | ||||
| t | 0.084 | ||||
| p | 0.467 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.839 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.914 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.839 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.914 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 0.055 | ||||
| Upside Potential Ratio | 1.592 | ||||
| Upside part of mean | 0.583 | ||||
| Downside part of mean | -0.563 | ||||
| Upside SD | 0.385 | ||||
| Downside SD | 0.366 | ||||
| N nonnegative terms | 37.000 | ||||
| N negative terms | 23.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 60.000 | ||||
| Mean of predictor | 0.280 | ||||
| Mean of criterion | 0.020 | ||||
| SD of predictor | 0.224 | ||||
| SD of criterion | 0.536 | ||||
| Covariance | 0.033 | ||||
| r | 0.277 | ||||
| b (slope, estimate of beta) | 0.663 | ||||
| a (intercept, estimate of alpha) | -0.166 | ||||
| Mean Square Error | 0.270 | ||||
| DF error | 58.000 | ||||
| t(b) | 2.195 | ||||
| p(b) | 0.016 | ||||
| t(a) | -0.671 | ||||
| p(a) | 0.747 | ||||
| Lowerbound of 95% confidence interval for beta | 0.058 | ||||
| Upperbound of 95% confidence interval for beta | 1.269 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.661 | ||||
| Upperbound of 95% confidence interval for alpha | 0.329 | ||||
| Treynor index (mean / b) | 0.030 | ||||
| Jensen alpha (a) | -0.166 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.223 | ||||
| Expected Shortfall on VaR | 0.271 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.080 | ||||
| Expected Shortfall on VaR | 0.169 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 60.000 | ||||
| Minimum | 0.676 | ||||
| Quartile 1 | 0.966 | ||||
| Median | 1.018 | ||||
| Quartile 3 | 1.049 | ||||
| Maximum | 1.764 | ||||
| Mean of quarter 1 | 0.845 | ||||
| Mean of quarter 2 | 0.998 | ||||
| Mean of quarter 3 | 1.028 | ||||
| Mean of quarter 4 | 1.199 | ||||
| Inter Quartile Range | 0.083 | ||||
| Number outliers low | 6.000 | ||||
| Percentage of outliers low | 0.100 | ||||
| Mean of outliers low | 0.742 | ||||
| Number of outliers high | 5.000 | ||||
| Percentage of outliers high | 0.083 | ||||
| Mean of outliers high | 1.414 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | -0.677 | ||||
| VaR(95%) (moments method) | 0.114 | ||||
| Expected Shortfall (moments method) | 0.131 | ||||
| Extreme Value Index (regression method) | -0.585 | ||||
| VaR(95%) (regression method) | 0.187 | ||||
| Expected Shortfall (regression method) | 0.223 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 4.000 | ||||
| Minimum | 0.002 | ||||
| Quartile 1 | 0.006 | ||||
| Median | 0.015 | ||||
| Quartile 3 | 0.183 | ||||
| Maximum | 0.665 | ||||
| Mean of quarter 1 | 0.002 | ||||
| Mean of quarter 2 | 0.007 | ||||
| Mean of quarter 3 | 0.023 | ||||
| Mean of quarter 4 | 0.665 | ||||
| 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 | 1.000 | ||||
| Percentage of outliers high | 0.250 | ||||
| Mean of outliers high | 0.665 | ||||
| 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) | 0.076 | ||||
| Compounded annual return (geometric extrapolation) | 0.066 | ||||
| Calmar ratio (compounded annual return / max draw down) | 0.100 | ||||
| Compounded annual return / average of 25% largest draw downs | 0.100 | ||||
| Compounded annual return / Expected Shortfall lognormal | 0.245 | ||||
| ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 0.227 | ||||
| SD | 0.677 | ||||
| Sharpe ratio (Glass type estimate) | 0.335 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.334 | ||||
| df | 1322.000 | ||||
| t | 0.752 | ||||
| p | 0.490 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.538 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 1.207 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.538 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.207 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 0.482 | ||||
| Upside Potential Ratio | 5.667 | ||||
| Upside part of mean | 2.666 | ||||
| Downside part of mean | -2.439 | ||||
| Upside SD | 0.487 | ||||
| Downside SD | 0.470 | ||||
| N nonnegative terms | 736.000 | ||||
| N negative terms | 587.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 1323.000 | ||||
| Mean of predictor | 0.331 | ||||
| Mean of criterion | 0.227 | ||||
| SD of predictor | 0.280 | ||||
| SD of criterion | 0.677 | ||||
| Covariance | 0.110 | ||||
| r | 0.579 | ||||
| b (slope, estimate of beta) | 1.403 | ||||
| a (intercept, estimate of alpha) | -0.238 | ||||
| Mean Square Error | 0.305 | ||||
| DF error | 1321.000 | ||||
| t(b) | 25.801 | ||||
| p(b) | 0.153 | ||||
| t(a) | -0.964 | ||||
| p(a) | 0.517 | ||||
| Lowerbound of 95% confidence interval for beta | 1.296 | ||||
| Upperbound of 95% confidence interval for beta | 1.510 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.721 | ||||
| Upperbound of 95% confidence interval for alpha | 0.246 | ||||
| Treynor index (mean / b) | 0.162 | ||||
| Jensen alpha (a) | -0.238 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.015 | ||||
| SD | 0.711 | ||||
| Sharpe ratio (Glass type estimate) | -0.022 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.022 | ||||
| df | 1322.000 | ||||
| t | -0.049 | ||||
| p | 0.501 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -0.894 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 0.851 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.894 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.851 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.028 | ||||
| Upside Potential Ratio | 4.686 | ||||
| Upside part of mean | 2.557 | ||||
| Downside part of mean | -2.573 | ||||
| Upside SD | 0.455 | ||||
| Downside SD | 0.546 | ||||
| N nonnegative terms | 736.000 | ||||
| N negative terms | 587.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 1323.000 | ||||
| Mean of predictor | 0.290 | ||||
| Mean of criterion | -0.015 | ||||
| SD of predictor | 0.290 | ||||
| SD of criterion | 0.711 | ||||
| Covariance | 0.128 | ||||
| r | 0.621 | ||||
| b (slope, estimate of beta) | 1.524 | ||||
| a (intercept, estimate of alpha) | -0.457 | ||||
| Mean Square Error | 0.311 | ||||
| DF error | 1321.000 | ||||
| t(b) | 28.803 | ||||
| p(b) | 0.132 | ||||
| t(a) | -1.839 | ||||
| p(a) | 0.532 | ||||
| Lowerbound of 95% confidence interval for beta | 1.420 | ||||
| Upperbound of 95% confidence interval for beta | 1.628 | ||||
| Lowerbound of 95% confidence interval for alpha | -0.944 | ||||
| Upperbound of 95% confidence interval for alpha | 0.030 | ||||
| Treynor index (mean / b) | -0.010 | ||||
| Jensen alpha (a) | -0.457 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.070 | ||||
| Expected Shortfall on VaR | 0.087 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.019 | ||||
| Expected Shortfall on VaR | 0.044 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 1323.000 | ||||
| Minimum | 0.474 | ||||
| Quartile 1 | 0.996 | ||||
| Median | 1.001 | ||||
| Quartile 3 | 1.004 | ||||
| Maximum | 1.323 | ||||
| Mean of quarter 1 | 0.964 | ||||
| Mean of quarter 2 | 0.999 | ||||
| Mean of quarter 3 | 1.002 | ||||
| Mean of quarter 4 | 1.039 | ||||
| Inter Quartile Range | 0.008 | ||||
| Number outliers low | 201.000 | ||||
| Percentage of outliers low | 0.152 | ||||
| Mean of outliers low | 0.945 | ||||
| Number of outliers high | 189.000 | ||||
| Percentage of outliers high | 0.143 | ||||
| Mean of outliers high | 1.062 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 0.800 | ||||
| VaR(95%) (moments method) | 0.025 | ||||
| Expected Shortfall (moments method) | 0.142 | ||||
| Extreme Value Index (regression method) | 0.200 | ||||
| VaR(95%) (regression method) | 0.032 | ||||
| Expected Shortfall (regression method) | 0.058 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 57.000 | ||||
| Minimum | 0.000 | ||||
| Quartile 1 | 0.000 | ||||
| Median | 0.004 | ||||
| Quartile 3 | 0.008 | ||||
| Maximum | 0.755 | ||||
| Mean of quarter 1 | 0.000 | ||||
| Mean of quarter 2 | 0.002 | ||||
| Mean of quarter 3 | 0.006 | ||||
| Mean of quarter 4 | 0.078 | ||||
| Inter Quartile Range | 0.007 | ||||
| Number outliers low | 0.000 | ||||
| Percentage of outliers low | 0.000 | ||||
| Mean of outliers low | NA | ||||
| Number of outliers high | 7.000 | ||||
| Percentage of outliers high | 0.123 | ||||
| Mean of outliers high | 0.143 | ||||
| Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | 1.088 | ||||
| VaR(95%) (moments method) | 0.050 | ||||
| Expected Shortfall (moments method) | NA | ||||
| Extreme Value Index (regression method) | 1.210 | ||||
| VaR(95%) (regression method) | 0.033 | ||||
| Expected Shortfall (regression method) | NA | ||||
| COMBINED STATISTICS | |||||
| Annualized return (arithmetic extrapolation) | 0.031 | ||||
| Compounded annual return (geometric extrapolation) | 0.029 | ||||
| Calmar ratio (compounded annual return / max draw down) | 0.039 | ||||
| Compounded annual return / average of 25% largest draw downs | 0.371 | ||||
| Compounded annual return / Expected Shortfall lognormal | 0.336 | ||||
| ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back) | |||||
| RATIO STATISTICS | |||||
| Ratio statistics of excess return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | 0.265 | ||||
| SD | 1.120 | ||||
| Sharpe ratio (Glass type estimate) | 0.237 | ||||
| Sharpe ratio (Hedges UMVUE) | 0.235 | ||||
| df | 130.000 | ||||
| t | 0.167 | ||||
| p | 0.493 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -2.536 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 3.008 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -2.537 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 3.007 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | 0.374 | ||||
| Upside Potential Ratio | 9.591 | ||||
| Upside part of mean | 6.796 | ||||
| Downside part of mean | -6.531 | ||||
| Upside SD | 0.861 | ||||
| Downside SD | 0.709 | ||||
| N nonnegative terms | 60.000 | ||||
| N negative terms | 71.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 1.035 | ||||
| Mean of criterion | 0.265 | ||||
| SD of predictor | 0.448 | ||||
| SD of criterion | 1.120 | ||||
| Covariance | 0.259 | ||||
| r | 0.516 | ||||
| b (slope, estimate of beta) | 1.288 | ||||
| a (intercept, estimate of alpha) | -1.067 | ||||
| Mean Square Error | 0.927 | ||||
| DF error | 129.000 | ||||
| t(b) | 6.838 | ||||
| p(b) | 0.187 | ||||
| t(a) | -0.776 | ||||
| p(a) | 0.543 | ||||
| Lowerbound of 95% confidence interval for beta | 0.915 | ||||
| Upperbound of 95% confidence interval for beta | 1.660 | ||||
| Lowerbound of 95% confidence interval for alpha | -3.789 | ||||
| Upperbound of 95% confidence interval for alpha | 1.654 | ||||
| Treynor index (mean / b) | 0.206 | ||||
| Jensen alpha (a) | -1.067 | ||||
| Ratio statistics of excess log return rates | |||||
| Statistics related to Sharpe ratio | |||||
| Mean | -0.339 | ||||
| SD | 1.097 | ||||
| Sharpe ratio (Glass type estimate) | -0.309 | ||||
| Sharpe ratio (Hedges UMVUE) | -0.308 | ||||
| df | 130.000 | ||||
| t | -0.219 | ||||
| p | 0.510 | ||||
| Lowerbound of 95% confidence interval for Sharpe Ratio | -3.081 | ||||
| Upperbound of 95% confidence interval for Sharpe Ratio | 2.463 | ||||
| Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -3.080 | ||||
| Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.464 | ||||
| Statistics related to Sortino ratio | |||||
| Sortino ratio | -0.456 | ||||
| Upside Potential Ratio | 8.669 | ||||
| Upside part of mean | 6.459 | ||||
| Downside part of mean | -6.799 | ||||
| Upside SD | 0.800 | ||||
| Downside SD | 0.745 | ||||
| N nonnegative terms | 60.000 | ||||
| N negative terms | 71.000 | ||||
| Statistics related to linear regression on benchmark | |||||
| N of observations | 131.000 | ||||
| Mean of predictor | 0.934 | ||||
| Mean of criterion | -0.339 | ||||
| SD of predictor | 0.446 | ||||
| SD of criterion | 1.097 | ||||
| Covariance | 0.256 | ||||
| r | 0.523 | ||||
| b (slope, estimate of beta) | 1.285 | ||||
| a (intercept, estimate of alpha) | -1.539 | ||||
| Mean Square Error | 0.882 | ||||
| DF error | 129.000 | ||||
| t(b) | 6.962 | ||||
| p(b) | 0.183 | ||||
| t(a) | -1.149 | ||||
| p(a) | 0.564 | ||||
| Lowerbound of 95% confidence interval for beta | 0.920 | ||||
| Upperbound of 95% confidence interval for beta | 1.650 | ||||
| Lowerbound of 95% confidence interval for alpha | -4.190 | ||||
| Upperbound of 95% confidence interval for alpha | 1.111 | ||||
| Treynor index (mean / b) | -0.264 | ||||
| Jensen alpha (a) | -1.539 | ||||
| Risk estimates for a one-period unit investment (parametric) | |||||
| assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
| VaR(95%) | 0.107 | ||||
| Expected Shortfall on VaR | 0.131 | ||||
| assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
| VaR(95%) | 0.061 | ||||
| Expected Shortfall on VaR | 0.107 | ||||
| ORDER STATISTICS | |||||
| Quartiles of return rates | |||||
| Number of observations | 131.000 | ||||
| Minimum | 0.840 | ||||
| Quartile 1 | 0.958 | ||||
| Median | 0.996 | ||||
| Quartile 3 | 1.036 | ||||
| Maximum | 1.262 | ||||
| Mean of quarter 1 | 0.921 | ||||
| Mean of quarter 2 | 0.981 | ||||
| Mean of quarter 3 | 1.014 | ||||
| Mean of quarter 4 | 1.089 | ||||
| Inter Quartile Range | 0.078 | ||||
| Number outliers low | 1.000 | ||||
| Percentage of outliers low | 0.008 | ||||
| Mean of outliers low | 0.840 | ||||
| Number of outliers high | 3.000 | ||||
| Percentage of outliers high | 0.023 | ||||
| Mean of outliers high | 1.238 | ||||
| Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
| Extreme Value Index (moments method) | -0.426 | ||||
| VaR(95%) (moments method) | 0.083 | ||||
| Expected Shortfall (moments method) | 0.096 | ||||
| Extreme Value Index (regression method) | -0.258 | ||||
| VaR(95%) (regression method) | 0.078 | ||||
| Expected Shortfall (regression method) | 0.092 | ||||
| DRAW DOWN STATISTICS | |||||
| Quartiles of draw downs | |||||
| Number of observations | 2.000 | ||||
| Minimum | 0.184 | ||||
| Quartile 1 | 0.270 | ||||
| Median | 0.357 | ||||
| Quartile 3 | 0.444 | ||||
| Maximum | 0.530 | ||||
| Mean of quarter 1 | 0.184 | ||||
| Mean of quarter 2 | NA | ||||
| Mean of quarter 3 | NA | ||||
| Mean of quarter 4 | 0.530 | ||||
| Inter Quartile Range | 0.173 | ||||
| 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) | -0.275 | ||||
| Compounded annual return (geometric extrapolation) | -0.256 | ||||
| Calmar ratio (compounded annual return / max draw down) | -0.482 | ||||
| Compounded annual return / average of 25% largest draw downs | -0.482 | ||||
| Compounded annual return / Expected Shortfall lognormal | -1.947 | ||||