### Advanced Statistics: Green Dog

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 | 0.396 | ||||

SD | 0.750 | ||||

Sharpe ratio (Glass type estimate) | 0.528 | ||||

Sharpe ratio (Hedges UMVUE) | 0.514 | ||||

df | 29.000 | ||||

t | 0.835 | ||||

p | 0.205 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -0.723 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.771 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.732 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.761 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.922 | ||||

Upside Potential Ratio | 2.758 | ||||

Upside part of mean | 1.185 | ||||

Downside part of mean | -0.789 | ||||

Upside SD | 0.610 | ||||

Downside SD | 0.429 | ||||

N nonnegative terms | 17.000 | ||||

N negative terms | 13.000 | ||||

Statistics related to linear regression on benchmark | |||||

N of observations | 30.000 | ||||

Mean of predictor | 0.145 | ||||

Mean of criterion | 0.396 | ||||

SD of predictor | 0.109 | ||||

SD of criterion | 0.750 | ||||

Covariance | 0.011 | ||||

r | 0.135 | ||||

b (slope, estimate of beta) | 0.930 | ||||

a (intercept, estimate of alpha) | 0.261 | ||||

Mean Square Error | 0.572 | ||||

DF error | 28.000 | ||||

t(b) | 0.721 | ||||

p(b) | 0.239 | ||||

t(a) | 0.507 | ||||

p(a) | 0.308 | ||||

Lowerbound of 95% confidence interval for beta | -1.714 | ||||

Upperbound of 95% confidence interval for beta | 3.575 | ||||

Lowerbound of 95% confidence interval for alpha | -0.792 | ||||

Upperbound of 95% confidence interval for alpha | 1.313 | ||||

Treynor index (mean / b) | 0.426 | ||||

Jensen alpha (a) | 0.261 | ||||

Ratio statistics of excess log return rates | |||||

Statistics related to Sharpe ratio | |||||

Mean | 0.135 | ||||

SD | 0.728 | ||||

Sharpe ratio (Glass type estimate) | 0.186 | ||||

Sharpe ratio (Hedges UMVUE) | 0.181 | ||||

df | 29.000 | ||||

t | 0.294 | ||||

p | 0.386 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.056 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.425 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.060 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.421 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.268 | ||||

Upside Potential Ratio | 2.053 | ||||

Upside part of mean | 1.035 | ||||

Downside part of mean | -0.900 | ||||

Upside SD | 0.510 | ||||

Downside SD | 0.504 | ||||

N nonnegative terms | 17.000 | ||||

N negative terms | 13.000 | ||||

Statistics related to linear regression on benchmark | |||||

N of observations | 30.000 | ||||

Mean of predictor | 0.138 | ||||

Mean of criterion | 0.135 | ||||

SD of predictor | 0.108 | ||||

SD of criterion | 0.728 | ||||

Covariance | 0.008 | ||||

r | 0.104 | ||||

b (slope, estimate of beta) | 0.705 | ||||

a (intercept, estimate of alpha) | 0.038 | ||||

Mean Square Error | 0.543 | ||||

DF error | 28.000 | ||||

t(b) | 0.555 | ||||

p(b) | 0.292 | ||||

t(a) | 0.075 | ||||

p(a) | 0.470 | ||||

Lowerbound of 95% confidence interval for beta | -1.900 | ||||

Upperbound of 95% confidence interval for beta | 3.310 | ||||

Lowerbound of 95% confidence interval for alpha | -0.983 | ||||

Upperbound of 95% confidence interval for alpha | 1.058 | ||||

Treynor index (mean / b) | 0.192 | ||||

Jensen alpha (a) | 0.038 | ||||

Risk estimates for a one-period unit investment (parametric) | |||||

assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||

VaR(95%) | 0.284 | ||||

Expected Shortfall on VaR | 0.342 | ||||

assuming Pareto losses only (using partial moments from Sortino statistics) | |||||

VaR(95%) | 0.140 | ||||

Expected Shortfall on VaR | 0.267 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 30.000 | ||||

Minimum | 0.633 | ||||

Quartile 1 | 0.849 | ||||

Median | 1.015 | ||||

Quartile 3 | 1.180 | ||||

Maximum | 1.667 | ||||

Mean of quarter 1 | 0.786 | ||||

Mean of quarter 2 | 0.973 | ||||

Mean of quarter 3 | 1.088 | ||||

Mean of quarter 4 | 1.298 | ||||

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 for a one-period unit investment (based on Extreme Value Theory) | |||||

Extreme Value Index (moments method) | 0.414 | ||||

VaR(95%) (moments method) | 0.255 | ||||

Expected Shortfall (moments method) | 0.409 | ||||

Extreme Value Index (regression method) | 1.149 | ||||

VaR(95%) (regression method) | 0.206 | ||||

Expected Shortfall (regression method) | NA | ||||

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 3.000 | ||||

Minimum | 0.006 | ||||

Quartile 1 | 0.143 | ||||

Median | 0.279 | ||||

Quartile 3 | 0.525 | ||||

Maximum | 0.771 | ||||

Mean of quarter 1 | 0.006 | ||||

Mean of quarter 2 | 0.279 | ||||

Mean of quarter 3 | NA | ||||

Mean of quarter 4 | 0.771 | ||||

Inter Quartile Range | 0.382 | ||||

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) | 0.226 | ||||

Compounded annual return (geometric extrapolation) | 0.196 | ||||

Calmar ratio (compounded annual return / max draw down) | 0.255 | ||||

Compounded annual return / average of 25% largest draw downs | 0.255 | ||||

Compounded annual return / Expected Shortfall lognormal | 0.573 | ||||

ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back) | |||||

RATIO STATISTICS | |||||

Ratio statistics of excess return rates | |||||

Statistics related to Sharpe ratio | |||||

Mean | 0.324 | ||||

SD | 0.650 | ||||

Sharpe ratio (Glass type estimate) | 0.498 | ||||

Sharpe ratio (Hedges UMVUE) | 0.497 | ||||

df | 869.000 | ||||

t | 0.792 | ||||

p | 0.214 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -0.735 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.730 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.735 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.730 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.872 | ||||

Upside Potential Ratio | 7.630 | ||||

Upside part of mean | 2.830 | ||||

Downside part of mean | -2.507 | ||||

Upside SD | 0.533 | ||||

Downside SD | 0.371 | ||||

N nonnegative terms | 371.000 | ||||

N negative terms | 499.000 | ||||

Statistics related to linear regression on benchmark | |||||

N of observations | 870.000 | ||||

Mean of predictor | 0.156 | ||||

Mean of criterion | 0.324 | ||||

SD of predictor | 0.121 | ||||

SD of criterion | 0.650 | ||||

Covariance | 0.009 | ||||

r | 0.109 | ||||

b (slope, estimate of beta) | 0.587 | ||||

a (intercept, estimate of alpha) | 0.232 | ||||

Mean Square Error | 0.418 | ||||

DF error | 868.000 | ||||

t(b) | 3.230 | ||||

p(b) | 0.001 | ||||

t(a) | 0.569 | ||||

p(a) | 0.285 | ||||

Lowerbound of 95% confidence interval for beta | 0.230 | ||||

Upperbound of 95% confidence interval for beta | 0.944 | ||||

Lowerbound of 95% confidence interval for alpha | -0.568 | ||||

Upperbound of 95% confidence interval for alpha | 1.032 | ||||

Treynor index (mean / b) | 0.551 | ||||

Jensen alpha (a) | 0.232 | ||||

Ratio statistics of excess log return rates | |||||

Statistics related to Sharpe ratio | |||||

Mean | 0.133 | ||||

SD | 0.605 | ||||

Sharpe ratio (Glass type estimate) | 0.220 | ||||

Sharpe ratio (Hedges UMVUE) | 0.220 | ||||

df | 869.000 | ||||

t | 0.350 | ||||

p | 0.363 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.013 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.452 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.013 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.452 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.339 | ||||

Upside Potential Ratio | 6.906 | ||||

Upside part of mean | 2.714 | ||||

Downside part of mean | -2.580 | ||||

Upside SD | 0.460 | ||||

Downside SD | 0.393 | ||||

N nonnegative terms | 371.000 | ||||

N negative terms | 499.000 | ||||

Statistics related to linear regression on benchmark | |||||

N of observations | 870.000 | ||||

Mean of predictor | 0.149 | ||||

Mean of criterion | 0.133 | ||||

SD of predictor | 0.121 | ||||

SD of criterion | 0.605 | ||||

Covariance | 0.008 | ||||

r | 0.109 | ||||

b (slope, estimate of beta) | 0.546 | ||||

a (intercept, estimate of alpha) | 0.052 | ||||

Mean Square Error | 0.362 | ||||

DF error | 868.000 | ||||

t(b) | 3.224 | ||||

p(b) | 0.001 | ||||

t(a) | 0.137 | ||||

p(a) | 0.446 | ||||

Lowerbound of 95% confidence interval for beta | 0.213 | ||||

Upperbound of 95% confidence interval for beta | 0.878 | ||||

Lowerbound of 95% confidence interval for alpha | -0.693 | ||||

Upperbound of 95% confidence interval for alpha | 0.796 | ||||

Treynor index (mean / b) | 0.244 | ||||

Jensen alpha (a) | 0.052 | ||||

Risk estimates for a one-period unit investment (parametric) | |||||

assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||

VaR(95%) | 0.052 | ||||

Expected Shortfall on VaR | 0.065 | ||||

assuming Pareto losses only (using partial moments from Sortino statistics) | |||||

VaR(95%) | 0.018 | ||||

Expected Shortfall on VaR | 0.038 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 870.000 | ||||

Minimum | 0.762 | ||||

Quartile 1 | 0.995 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.009 | ||||

Maximum | 1.591 | ||||

Mean of quarter 1 | 0.972 | ||||

Mean of quarter 2 | 0.999 | ||||

Mean of quarter 3 | 1.003 | ||||

Mean of quarter 4 | 1.031 | ||||

Inter Quartile Range | 0.014 | ||||

Number outliers low | 78.000 | ||||

Percentage of outliers low | 0.090 | ||||

Mean of outliers low | 0.946 | ||||

Number of outliers high | 68.000 | ||||

Percentage of outliers high | 0.078 | ||||

Mean of outliers high | 1.061 | ||||

Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||

Extreme Value Index (moments method) | 0.471 | ||||

VaR(95%) (moments method) | 0.022 | ||||

Expected Shortfall (moments method) | 0.050 | ||||

Extreme Value Index (regression method) | 0.310 | ||||

VaR(95%) (regression method) | 0.025 | ||||

Expected Shortfall (regression method) | 0.047 | ||||

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 32.000 | ||||

Minimum | 0.000 | ||||

Quartile 1 | 0.007 | ||||

Median | 0.026 | ||||

Quartile 3 | 0.049 | ||||

Maximum | 0.788 | ||||

Mean of quarter 1 | 0.002 | ||||

Mean of quarter 2 | 0.018 | ||||

Mean of quarter 3 | 0.035 | ||||

Mean of quarter 4 | 0.203 | ||||

Inter Quartile Range | 0.042 | ||||

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 | 0.094 | ||||

Mean of outliers high | 0.427 | ||||

Risk estimates based on draw downs (based on Extreme Value Theory) | |||||

Extreme Value Index (moments method) | 0.868 | ||||

VaR(95%) (moments method) | 0.204 | ||||

Expected Shortfall (moments method) | 1.609 | ||||

Extreme Value Index (regression method) | 1.426 | ||||

VaR(95%) (regression method) | 0.192 | ||||

Expected Shortfall (regression method) | NA | ||||

COMBINED STATISTICS | |||||

Annualized return (arithmetic extrapolation) | 0.223 | ||||

Compounded annual return (geometric extrapolation) | 0.194 | ||||

Calmar ratio (compounded annual return / max draw down) | 0.246 | ||||

Compounded annual return / average of 25% largest draw downs | 0.957 | ||||

Compounded annual return / Expected Shortfall lognormal | 2.997 | ||||

ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back) | |||||

RATIO STATISTICS | |||||

Ratio statistics of excess return rates | |||||

Statistics related to Sharpe ratio | |||||

Mean | 1.446 | ||||

SD | 1.018 | ||||

Sharpe ratio (Glass type estimate) | 1.421 | ||||

Sharpe ratio (Hedges UMVUE) | 1.415 | ||||

df | 171.000 | ||||

t | 1.005 | ||||

p | 0.451 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.356 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 4.195 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.361 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 4.191 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 6.223 | ||||

Upside Potential Ratio | 10.085 | ||||

Upside part of mean | 2.344 | ||||

Downside part of mean | -0.897 | ||||

Upside SD | 0.991 | ||||

Downside SD | 0.232 | ||||

N nonnegative terms | 43.000 | ||||

N negative terms | 129.000 | ||||

Statistics related to linear regression on benchmark | |||||

N of observations | 172.000 | ||||

Mean of predictor | 0.245 | ||||

Mean of criterion | 1.446 | ||||

SD of predictor | 0.133 | ||||

SD of criterion | 1.018 | ||||

Covariance | 0.018 | ||||

r | 0.132 | ||||

b (slope, estimate of beta) | 1.005 | ||||

a (intercept, estimate of alpha) | 1.200 | ||||

Mean Square Error | 1.023 | ||||

DF error | 170.000 | ||||

t(b) | 1.734 | ||||

p(b) | 0.434 | ||||

t(a) | 0.834 | ||||

p(a) | 0.468 | ||||

Lowerbound of 95% confidence interval for beta | -0.139 | ||||

Upperbound of 95% confidence interval for beta | 2.150 | ||||

Lowerbound of 95% confidence interval for alpha | -1.638 | ||||

Upperbound of 95% confidence interval for alpha | 4.038 | ||||

Treynor index (mean / b) | 1.439 | ||||

Jensen alpha (a) | 1.200 | ||||

Ratio statistics of excess log return rates | |||||

Statistics related to Sharpe ratio | |||||

Mean | 1.050 | ||||

SD | 0.835 | ||||

Sharpe ratio (Glass type estimate) | 1.257 | ||||

Sharpe ratio (Hedges UMVUE) | 1.252 | ||||

df | 171.000 | ||||

t | 0.889 | ||||

p | 0.457 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.520 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 4.030 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.523 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 4.027 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 4.314 | ||||

Upside Potential Ratio | 8.119 | ||||

Upside part of mean | 1.976 | ||||

Downside part of mean | -0.926 | ||||

Upside SD | 0.798 | ||||

Downside SD | 0.243 | ||||

N nonnegative terms | 43.000 | ||||

N negative terms | 129.000 | ||||

Statistics related to linear regression on benchmark | |||||

N of observations | 172.000 | ||||

Mean of predictor | 0.236 | ||||

Mean of criterion | 1.050 | ||||

SD of predictor | 0.134 | ||||

SD of criterion | 0.835 | ||||

Covariance | 0.015 | ||||

r | 0.134 | ||||

b (slope, estimate of beta) | 0.839 | ||||

a (intercept, estimate of alpha) | 0.851 | ||||

Mean Square Error | 0.689 | ||||

DF error | 170.000 | ||||

t(b) | 1.769 | ||||

p(b) | 0.433 | ||||

t(a) | 0.722 | ||||

p(a) | 0.472 | ||||

Lowerbound of 95% confidence interval for beta | -0.097 | ||||

Upperbound of 95% confidence interval for beta | 1.776 | ||||

Lowerbound of 95% confidence interval for alpha | -1.476 | ||||

Upperbound of 95% confidence interval for alpha | 3.179 | ||||

Treynor index (mean / b) | 1.251 | ||||

Jensen alpha (a) | 0.851 | ||||

Risk estimates for a one-period unit investment (parametric) | |||||

assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||

VaR(95%) | 0.069 | ||||

Expected Shortfall on VaR | 0.086 | ||||

assuming Pareto losses only (using partial moments from Sortino statistics) | |||||

VaR(95%) | 0.008 | ||||

Expected Shortfall on VaR | 0.017 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 172.000 | ||||

Minimum | 0.888 | ||||

Quartile 1 | 1.000 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.000 | ||||

Maximum | 1.591 | ||||

Mean of quarter 1 | 0.990 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.027 | ||||

Inter Quartile Range | 0.000 | ||||

Number outliers low | 20.000 | ||||

Percentage of outliers low | 0.116 | ||||

Mean of outliers low | 0.978 | ||||

Number of outliers high | 43.000 | ||||

Percentage of outliers high | 0.250 | ||||

Mean of outliers high | 1.027 | ||||

Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||

Extreme Value Index (moments method) | 0.518 | ||||

VaR(95%) (moments method) | 0.001 | ||||

Expected Shortfall (moments method) | 0.004 | ||||

Extreme Value Index (regression method) | 0.599 | ||||

VaR(95%) (regression method) | 0.006 | ||||

Expected Shortfall (regression method) | 0.030 | ||||

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 2.000 | ||||

Minimum | 0.026 | ||||

Quartile 1 | 0.080 | ||||

Median | 0.134 | ||||

Quartile 3 | 0.188 | ||||

Maximum | 0.242 | ||||

Mean of quarter 1 | 0.026 | ||||

Mean of quarter 2 | NA | ||||

Mean of quarter 3 | NA | ||||

Mean of quarter 4 | 0.242 | ||||

Inter Quartile Range | 0.108 | ||||

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) | 1.456 | ||||

Compounded annual return (geometric extrapolation) | 1.986 | ||||

Calmar ratio (compounded annual return / max draw down) | 8.192 | ||||

Compounded annual return / average of 25% largest draw downs | 8.192 | ||||

Compounded annual return / Expected Shortfall lognormal | 23.152 |