### 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 | 0.411 | ||||

SD | 0.763 | ||||

Sharpe ratio (Glass type estimate) | 0.539 | ||||

Sharpe ratio (Hedges UMVUE) | 0.524 | ||||

df | 28.000 | ||||

t | 0.838 | ||||

p | 0.205 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -0.734 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.803 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.744 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.793 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.941 | ||||

Upside Potential Ratio | 2.806 | ||||

Upside part of mean | 1.226 | ||||

Downside part of mean | -0.814 | ||||

Upside SD | 0.620 | ||||

Downside SD | 0.437 | ||||

N nonnegative terms | 17.000 | ||||

N negative terms | 12.000 | ||||

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

N of observations | 29.000 | ||||

Mean of predictor | 0.155 | ||||

Mean of criterion | 0.411 | ||||

SD of predictor | 0.110 | ||||

SD of criterion | 0.763 | ||||

Covariance | 0.011 | ||||

r | 0.132 | ||||

b (slope, estimate of beta) | 0.918 | ||||

a (intercept, estimate of alpha) | 0.269 | ||||

Mean Square Error | 0.593 | ||||

DF error | 27.000 | ||||

t(b) | 0.691 | ||||

p(b) | 0.248 | ||||

t(a) | 0.501 | ||||

p(a) | 0.310 | ||||

Lowerbound of 95% confidence interval for beta | -1.806 | ||||

Upperbound of 95% confidence interval for beta | 3.641 | ||||

Lowerbound of 95% confidence interval for alpha | -0.831 | ||||

Upperbound of 95% confidence interval for alpha | 1.369 | ||||

Treynor index (mean / b) | 0.448 | ||||

Jensen alpha (a) | 0.269 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | 0.141 | ||||

SD | 0.741 | ||||

Sharpe ratio (Glass type estimate) | 0.191 | ||||

Sharpe ratio (Hedges UMVUE) | 0.186 | ||||

df | 28.000 | ||||

t | 0.297 | ||||

p | 0.384 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.073 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.451 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.076 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.447 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.276 | ||||

Upside Potential Ratio | 2.088 | ||||

Upside part of mean | 1.070 | ||||

Downside part of mean | -0.929 | ||||

Upside SD | 0.518 | ||||

Downside SD | 0.513 | ||||

N nonnegative terms | 17.000 | ||||

N negative terms | 12.000 | ||||

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

N of observations | 29.000 | ||||

Mean of predictor | 0.148 | ||||

Mean of criterion | 0.141 | ||||

SD of predictor | 0.108 | ||||

SD of criterion | 0.741 | ||||

Covariance | 0.008 | ||||

r | 0.103 | ||||

b (slope, estimate of beta) | 0.706 | ||||

a (intercept, estimate of alpha) | 0.037 | ||||

Mean Square Error | 0.563 | ||||

DF error | 27.000 | ||||

t(b) | 0.540 | ||||

p(b) | 0.297 | ||||

t(a) | 0.071 | ||||

p(a) | 0.472 | ||||

Lowerbound of 95% confidence interval for beta | -1.977 | ||||

Upperbound of 95% confidence interval for beta | 3.389 | ||||

Lowerbound of 95% confidence interval for alpha | -1.030 | ||||

Upperbound of 95% confidence interval for alpha | 1.104 | ||||

Treynor index (mean / b) | 0.200 | ||||

Jensen alpha (a) | 0.037 | ||||

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

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

VaR(95%) | 0.288 | ||||

Expected Shortfall on VaR | 0.347 | ||||

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 | 29.000 | ||||

Minimum | 0.633 | ||||

Quartile 1 | 0.848 | ||||

Median | 1.015 | ||||

Quartile 3 | 1.190 | ||||

Maximum | 1.667 | ||||

Mean of quarter 1 | 0.786 | ||||

Mean of quarter 2 | 0.975 | ||||

Mean of quarter 3 | 1.113 | ||||

Mean of quarter 4 | 1.313 | ||||

Inter Quartile Range | 0.341 | ||||

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) | 0.258 | ||||

Expected Shortfall (moments method) | 0.414 | ||||

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

VaR(95%) (regression method) | 0.208 | ||||

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) | 0.234 | ||||

Compounded annual return (geometric extrapolation) | 0.204 | ||||

Calmar ratio (compounded annual return / max draw down) | 0.264 | ||||

Compounded annual return / average of 25% largest draw downs | 0.264 | ||||

Compounded annual return / Expected Shortfall lognormal | 0.587 | ||||

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

RATIO STATISTICS | |||||

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

Statistics related to Sharpe ratio | |||||

Mean | 0.339 | ||||

SD | 0.663 | ||||

Sharpe ratio (Glass type estimate) | 0.511 | ||||

Sharpe ratio (Hedges UMVUE) | 0.510 | ||||

df | 834.000 | ||||

t | 0.796 | ||||

p | 0.213 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -0.747 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.769 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.748 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.769 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.895 | ||||

Upside Potential Ratio | 7.788 | ||||

Upside part of mean | 2.949 | ||||

Downside part of mean | -2.610 | ||||

Upside SD | 0.545 | ||||

Downside SD | 0.379 | ||||

N nonnegative terms | 371.000 | ||||

N negative terms | 464.000 | ||||

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

N of observations | 835.000 | ||||

Mean of predictor | 0.155 | ||||

Mean of criterion | 0.339 | ||||

SD of predictor | 0.118 | ||||

SD of criterion | 0.663 | ||||

Covariance | 0.009 | ||||

r | 0.114 | ||||

b (slope, estimate of beta) | 0.638 | ||||

a (intercept, estimate of alpha) | 0.240 | ||||

Mean Square Error | 0.435 | ||||

DF error | 833.000 | ||||

t(b) | 3.301 | ||||

p(b) | 0.001 | ||||

t(a) | 0.566 | ||||

p(a) | 0.286 | ||||

Lowerbound of 95% confidence interval for beta | 0.259 | ||||

Upperbound of 95% confidence interval for beta | 1.018 | ||||

Lowerbound of 95% confidence interval for alpha | -0.593 | ||||

Upperbound of 95% confidence interval for alpha | 1.073 | ||||

Treynor index (mean / b) | 0.531 | ||||

Jensen alpha (a) | 0.240 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | 0.141 | ||||

SD | 0.618 | ||||

Sharpe ratio (Glass type estimate) | 0.227 | ||||

Sharpe ratio (Hedges UMVUE) | 0.227 | ||||

df | 834.000 | ||||

t | 0.354 | ||||

p | 0.362 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.031 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.486 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.031 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.485 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.350 | ||||

Upside Potential Ratio | 7.049 | ||||

Upside part of mean | 2.827 | ||||

Downside part of mean | -2.687 | ||||

Upside SD | 0.469 | ||||

Downside SD | 0.401 | ||||

N nonnegative terms | 371.000 | ||||

N negative terms | 464.000 | ||||

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

N of observations | 835.000 | ||||

Mean of predictor | 0.148 | ||||

Mean of criterion | 0.141 | ||||

SD of predictor | 0.118 | ||||

SD of criterion | 0.618 | ||||

Covariance | 0.008 | ||||

r | 0.113 | ||||

b (slope, estimate of beta) | 0.593 | ||||

a (intercept, estimate of alpha) | 0.053 | ||||

Mean Square Error | 0.377 | ||||

DF error | 833.000 | ||||

t(b) | 3.295 | ||||

p(b) | 0.001 | ||||

t(a) | 0.134 | ||||

p(a) | 0.447 | ||||

Lowerbound of 95% confidence interval for beta | 0.240 | ||||

Upperbound of 95% confidence interval for beta | 0.947 | ||||

Lowerbound of 95% confidence interval for alpha | -0.722 | ||||

Upperbound of 95% confidence interval for alpha | 0.828 | ||||

Treynor index (mean / b) | 0.237 | ||||

Jensen alpha (a) | 0.053 | ||||

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

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

VaR(95%) | 0.053 | ||||

Expected Shortfall on VaR | 0.066 | ||||

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

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

Expected Shortfall on VaR | 0.039 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 835.000 | ||||

Minimum | 0.762 | ||||

Quartile 1 | 0.994 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.009 | ||||

Maximum | 1.591 | ||||

Mean of quarter 1 | 0.971 | ||||

Mean of quarter 2 | 0.999 | ||||

Mean of quarter 3 | 1.003 | ||||

Mean of quarter 4 | 1.031 | ||||

Inter Quartile Range | 0.015 | ||||

Number outliers low | 73.000 | ||||

Percentage of outliers low | 0.087 | ||||

Mean of outliers low | 0.945 | ||||

Number of outliers high | 60.000 | ||||

Percentage of outliers high | 0.072 | ||||

Mean of outliers high | 1.065 | ||||

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

Extreme Value Index (moments method) | 0.402 | ||||

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

Expected Shortfall (moments method) | 0.046 | ||||

Extreme Value Index (regression method) | 0.274 | ||||

VaR(95%) (regression method) | 0.026 | ||||

Expected Shortfall (regression method) | 0.048 | ||||

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) | 0.233 | ||||

Compounded annual return (geometric extrapolation) | 0.203 | ||||

Calmar ratio (compounded annual return / max draw down) | 0.257 | ||||

Compounded annual return / average of 25% largest draw downs | 1.000 | ||||

Compounded annual return / Expected Shortfall lognormal | 3.074 | ||||

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

RATIO STATISTICS | |||||

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

Statistics related to Sharpe ratio | |||||

Mean | 1.284 | ||||

SD | 1.067 | ||||

Sharpe ratio (Glass type estimate) | 1.203 | ||||

Sharpe ratio (Hedges UMVUE) | 1.198 | ||||

df | 171.000 | ||||

t | 0.851 | ||||

p | 0.459 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.573 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 3.976 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.577 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 3.972 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 3.825 | ||||

Upside Potential Ratio | 9.599 | ||||

Upside part of mean | 3.222 | ||||

Downside part of mean | -1.938 | ||||

Upside SD | 1.012 | ||||

Downside SD | 0.336 | ||||

N nonnegative terms | 58.000 | ||||

N negative terms | 114.000 | ||||

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

N of observations | 172.000 | ||||

Mean of predictor | 0.212 | ||||

Mean of criterion | 1.284 | ||||

SD of predictor | 0.115 | ||||

SD of criterion | 1.067 | ||||

Covariance | 0.018 | ||||

r | 0.145 | ||||

b (slope, estimate of beta) | 1.349 | ||||

a (intercept, estimate of alpha) | 0.998 | ||||

Mean Square Error | 1.122 | ||||

DF error | 170.000 | ||||

t(b) | 1.911 | ||||

p(b) | 0.427 | ||||

t(a) | 0.663 | ||||

p(a) | 0.475 | ||||

Lowerbound of 95% confidence interval for beta | -0.044 | ||||

Upperbound of 95% confidence interval for beta | 2.741 | ||||

Lowerbound of 95% confidence interval for alpha | -1.974 | ||||

Upperbound of 95% confidence interval for alpha | 3.969 | ||||

Treynor index (mean / b) | 0.952 | ||||

Jensen alpha (a) | 0.998 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | 0.836 | ||||

SD | 0.896 | ||||

Sharpe ratio (Glass type estimate) | 0.933 | ||||

Sharpe ratio (Hedges UMVUE) | 0.929 | ||||

df | 171.000 | ||||

t | 0.660 | ||||

p | 0.468 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.842 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 3.706 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.844 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 3.703 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 2.391 | ||||

Upside Potential Ratio | 8.103 | ||||

Upside part of mean | 2.834 | ||||

Downside part of mean | -1.998 | ||||

Upside SD | 0.823 | ||||

Downside SD | 0.350 | ||||

N nonnegative terms | 58.000 | ||||

N negative terms | 114.000 | ||||

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

N of observations | 172.000 | ||||

Mean of predictor | 0.206 | ||||

Mean of criterion | 0.836 | ||||

SD of predictor | 0.115 | ||||

SD of criterion | 0.896 | ||||

Covariance | 0.015 | ||||

r | 0.145 | ||||

b (slope, estimate of beta) | 1.132 | ||||

a (intercept, estimate of alpha) | 0.603 | ||||

Mean Square Error | 0.790 | ||||

DF error | 170.000 | ||||

t(b) | 1.914 | ||||

p(b) | 0.427 | ||||

t(a) | 0.478 | ||||

p(a) | 0.482 | ||||

Lowerbound of 95% confidence interval for beta | -0.035 | ||||

Upperbound of 95% confidence interval for beta | 2.299 | ||||

Lowerbound of 95% confidence interval for alpha | -1.890 | ||||

Upperbound of 95% confidence interval for alpha | 3.096 | ||||

Treynor index (mean / b) | 0.739 | ||||

Jensen alpha (a) | 0.603 | ||||

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

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

VaR(95%) | 0.074 | ||||

Expected Shortfall on VaR | 0.092 | ||||

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

VaR(95%) | 0.015 | ||||

Expected Shortfall on VaR | 0.033 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 172.000 | ||||

Minimum | 0.888 | ||||

Quartile 1 | 1.000 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.002 | ||||

Maximum | 1.591 | ||||

Mean of quarter 1 | 0.978 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.037 | ||||

Inter Quartile Range | 0.002 | ||||

Number outliers low | 29.000 | ||||

Percentage of outliers low | 0.169 | ||||

Mean of outliers low | 0.967 | ||||

Number of outliers high | 32.000 | ||||

Percentage of outliers high | 0.186 | ||||

Mean of outliers high | 1.049 | ||||

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

Extreme Value Index (moments method) | -0.520 | ||||

VaR(95%) (moments method) | 0.002 | ||||

Expected Shortfall (moments method) | 0.003 | ||||

Extreme Value Index (regression method) | 0.192 | ||||

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

Expected Shortfall (regression method) | 0.043 | ||||

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 2.000 | ||||

Minimum | 0.242 | ||||

Quartile 1 | 0.251 | ||||

Median | 0.260 | ||||

Quartile 3 | 0.269 | ||||

Maximum | 0.278 | ||||

Mean of quarter 1 | 0.242 | ||||

Mean of quarter 2 | NA | ||||

Mean of quarter 3 | NA | ||||

Mean of quarter 4 | 0.278 | ||||

Inter Quartile Range | 0.018 | ||||

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) | 1.105 | ||||

Compounded annual return (geometric extrapolation) | 1.411 | ||||

Calmar ratio (compounded annual return / max draw down) | 5.069 | ||||

Compounded annual return / average of 25% largest draw downs | 5.069 | ||||

Compounded annual return / Expected Shortfall lognormal | 15.259 |