### 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 | 0.328 | ||||

SD | 0.654 | ||||

Sharpe ratio (Glass type estimate) | 0.502 | ||||

Sharpe ratio (Hedges UMVUE) | 0.501 | ||||

df | 858.000 | ||||

t | 0.793 | ||||

p | 0.214 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -0.739 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.742 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.739 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.742 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.879 | ||||

Upside Potential Ratio | 7.678 | ||||

Upside part of mean | 2.866 | ||||

Downside part of mean | -2.538 | ||||

Upside SD | 0.537 | ||||

Downside SD | 0.373 | ||||

N nonnegative terms | 371.000 | ||||

N negative terms | 488.000 | ||||

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

N of observations | 859.000 | ||||

Mean of predictor | 0.142 | ||||

Mean of criterion | 0.328 | ||||

SD of predictor | 0.121 | ||||

SD of criterion | 0.654 | ||||

Covariance | 0.009 | ||||

r | 0.110 | ||||

b (slope, estimate of beta) | 0.595 | ||||

a (intercept, estimate of alpha) | 0.244 | ||||

Mean Square Error | 0.423 | ||||

DF error | 857.000 | ||||

t(b) | 3.233 | ||||

p(b) | 0.001 | ||||

t(a) | 0.591 | ||||

p(a) | 0.277 | ||||

Lowerbound of 95% confidence interval for beta | 0.234 | ||||

Upperbound of 95% confidence interval for beta | 0.956 | ||||

Lowerbound of 95% confidence interval for alpha | -0.566 | ||||

Upperbound of 95% confidence interval for alpha | 1.053 | ||||

Treynor index (mean / b) | 0.552 | ||||

Jensen alpha (a) | 0.244 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | 0.135 | ||||

SD | 0.609 | ||||

Sharpe ratio (Glass type estimate) | 0.222 | ||||

Sharpe ratio (Hedges UMVUE) | 0.222 | ||||

df | 858.000 | ||||

t | 0.351 | ||||

p | 0.363 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.018 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.463 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.018 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.462 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.342 | ||||

Upside Potential Ratio | 6.950 | ||||

Upside part of mean | 2.748 | ||||

Downside part of mean | -2.613 | ||||

Upside SD | 0.463 | ||||

Downside SD | 0.395 | ||||

N nonnegative terms | 371.000 | ||||

N negative terms | 488.000 | ||||

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

N of observations | 859.000 | ||||

Mean of predictor | 0.135 | ||||

Mean of criterion | 0.135 | ||||

SD of predictor | 0.121 | ||||

SD of criterion | 0.609 | ||||

Covariance | 0.008 | ||||

r | 0.109 | ||||

b (slope, estimate of beta) | 0.552 | ||||

a (intercept, estimate of alpha) | 0.061 | ||||

Mean Square Error | 0.367 | ||||

DF error | 857.000 | ||||

t(b) | 3.224 | ||||

p(b) | 0.001 | ||||

t(a) | 0.159 | ||||

p(a) | 0.437 | ||||

Lowerbound of 95% confidence interval for beta | 0.216 | ||||

Upperbound of 95% confidence interval for beta | 0.888 | ||||

Lowerbound of 95% confidence interval for alpha | -0.692 | ||||

Upperbound of 95% confidence interval for alpha | 0.815 | ||||

Treynor index (mean / b) | 0.245 | ||||

Jensen alpha (a) | 0.061 | ||||

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 | 0.039 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 859.000 | ||||

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 | 75.000 | ||||

Percentage of outliers low | 0.087 | ||||

Mean of outliers low | 0.945 | ||||

Number of outliers high | 68.000 | ||||

Percentage of outliers high | 0.079 | ||||

Mean of outliers high | 1.061 | ||||

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

Extreme Value Index (moments method) | 0.453 | ||||

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

Expected Shortfall (moments method) | 0.049 | ||||

Extreme Value Index (regression method) | 0.313 | ||||

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) | 0.226 | ||||

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

Calmar ratio (compounded annual return / max draw down) | 0.249 | ||||

Compounded annual return / average of 25% largest draw downs | 0.970 | ||||

Compounded annual return / Expected Shortfall lognormal | 3.021 | ||||

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

RATIO STATISTICS | |||||

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

Statistics related to Sharpe ratio | |||||

Mean | 1.617 | ||||

SD | 1.036 | ||||

Sharpe ratio (Glass type estimate) | 1.561 | ||||

Sharpe ratio (Hedges UMVUE) | 1.554 | ||||

df | 171.000 | ||||

t | 1.104 | ||||

p | 0.447 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.218 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 4.335 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.223 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 4.331 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 6.404 | ||||

Upside Potential Ratio | 10.967 | ||||

Upside part of mean | 2.768 | ||||

Downside part of mean | -1.152 | ||||

Upside SD | 1.005 | ||||

Downside SD | 0.252 | ||||

N nonnegative terms | 48.000 | ||||

N negative terms | 124.000 | ||||

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

N of observations | 172.000 | ||||

Mean of predictor | 0.135 | ||||

Mean of criterion | 1.617 | ||||

SD of predictor | 0.132 | ||||

SD of criterion | 1.036 | ||||

Covariance | 0.017 | ||||

r | 0.121 | ||||

b (slope, estimate of beta) | 0.947 | ||||

a (intercept, estimate of alpha) | 1.489 | ||||

Mean Square Error | 1.063 | ||||

DF error | 170.000 | ||||

t(b) | 1.589 | ||||

p(b) | 0.440 | ||||

t(a) | 1.020 | ||||

p(a) | 0.461 | ||||

Lowerbound of 95% confidence interval for beta | -0.229 | ||||

Upperbound of 95% confidence interval for beta | 2.123 | ||||

Lowerbound of 95% confidence interval for alpha | -1.394 | ||||

Upperbound of 95% confidence interval for alpha | 4.372 | ||||

Treynor index (mean / b) | 1.708 | ||||

Jensen alpha (a) | 1.489 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | 1.201 | ||||

SD | 0.856 | ||||

Sharpe ratio (Glass type estimate) | 1.403 | ||||

Sharpe ratio (Hedges UMVUE) | 1.397 | ||||

df | 171.000 | ||||

t | 0.992 | ||||

p | 0.452 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.374 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 4.177 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.379 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 4.173 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 4.562 | ||||

Upside Potential Ratio | 9.064 | ||||

Upside part of mean | 2.387 | ||||

Downside part of mean | -1.185 | ||||

Upside SD | 0.815 | ||||

Downside SD | 0.263 | ||||

N nonnegative terms | 48.000 | ||||

N negative terms | 124.000 | ||||

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

N of observations | 172.000 | ||||

Mean of predictor | 0.126 | ||||

Mean of criterion | 1.201 | ||||

SD of predictor | 0.133 | ||||

SD of criterion | 0.856 | ||||

Covariance | 0.014 | ||||

r | 0.120 | ||||

b (slope, estimate of beta) | 0.777 | ||||

a (intercept, estimate of alpha) | 1.104 | ||||

Mean Square Error | 0.727 | ||||

DF error | 170.000 | ||||

t(b) | 1.582 | ||||

p(b) | 0.440 | ||||

t(a) | 0.914 | ||||

p(a) | 0.465 | ||||

Lowerbound of 95% confidence interval for beta | -0.193 | ||||

Upperbound of 95% confidence interval for beta | 1.747 | ||||

Lowerbound of 95% confidence interval for alpha | -1.279 | ||||

Upperbound of 95% confidence interval for alpha | 3.486 | ||||

Treynor index (mean / b) | 1.546 | ||||

Jensen alpha (a) | 1.104 | ||||

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 | 0.088 | ||||

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

VaR(95%) | 0.009 | ||||

Expected Shortfall on VaR | 0.021 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 172.000 | ||||

Minimum | 0.888 | ||||

Quartile 1 | 1.000 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.001 | ||||

Maximum | 1.591 | ||||

Mean of quarter 1 | 0.987 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.032 | ||||

Inter Quartile Range | 0.001 | ||||

Number outliers low | 22.000 | ||||

Percentage of outliers low | 0.128 | ||||

Mean of outliers low | 0.975 | ||||

Number of outliers high | 36.000 | ||||

Percentage of outliers high | 0.209 | ||||

Mean of outliers high | 1.038 | ||||

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

Extreme Value Index (moments method) | -0.006 | ||||

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

Expected Shortfall (moments method) | 0.002 | ||||

Extreme Value Index (regression method) | 0.347 | ||||

VaR(95%) (regression method) | 0.011 | ||||

Expected Shortfall (regression method) | 0.031 | ||||

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 4.000 | ||||

Minimum | 0.026 | ||||

Quartile 1 | 0.044 | ||||

Median | 0.063 | ||||

Quartile 3 | 0.118 | ||||

Maximum | 0.242 | ||||

Mean of quarter 1 | 0.026 | ||||

Mean of quarter 2 | 0.050 | ||||

Mean of quarter 3 | 0.076 | ||||

Mean of quarter 4 | 0.242 | ||||

Inter Quartile Range | 0.074 | ||||

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 | 0.242 | ||||

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) | 1.728 | ||||

Compounded annual return (geometric extrapolation) | 2.474 | ||||

Calmar ratio (compounded annual return / max draw down) | 10.208 | ||||

Compounded annual return / average of 25% largest draw downs | 10.208 | ||||

Compounded annual return / Expected Shortfall lognormal | 28.277 |