### Advanced Statistics: Bravo Tango

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 | 0.119 | ||||

SD | 0.349 | ||||

Sharpe ratio (Glass type estimate) | 0.342 | ||||

Sharpe ratio (Hedges UMVUE) | 0.315 | ||||

df | 10.000 | ||||

t | 0.327 | ||||

p | 0.375 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.719 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 2.386 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.737 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.367 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.772 | ||||

Upside Potential Ratio | 2.599 | ||||

Upside part of mean | 0.401 | ||||

Downside part of mean | -0.282 | ||||

Upside SD | 0.297 | ||||

Downside SD | 0.154 | ||||

N nonnegative terms | 3.000 | ||||

N negative terms | 8.000 | ||||

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

N of observations | 11.000 | ||||

Mean of predictor | 0.394 | ||||

Mean of criterion | 0.119 | ||||

SD of predictor | 0.236 | ||||

SD of criterion | 0.349 | ||||

Covariance | 0.003 | ||||

r | 0.041 | ||||

b (slope, estimate of beta) | 0.061 | ||||

a (intercept, estimate of alpha) | 0.095 | ||||

Mean Square Error | 0.135 | ||||

DF error | 9.000 | ||||

t(b) | 0.124 | ||||

p(b) | 0.452 | ||||

t(a) | 0.221 | ||||

p(a) | 0.415 | ||||

Lowerbound of 95% confidence interval for beta | -1.053 | ||||

Upperbound of 95% confidence interval for beta | 1.175 | ||||

Lowerbound of 95% confidence interval for alpha | -0.878 | ||||

Upperbound of 95% confidence interval for alpha | 1.068 | ||||

Treynor index (mean / b) | 1.955 | ||||

Jensen alpha (a) | 0.095 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | 0.069 | ||||

SD | 0.324 | ||||

Sharpe ratio (Glass type estimate) | 0.212 | ||||

Sharpe ratio (Hedges UMVUE) | 0.195 | ||||

df | 10.000 | ||||

t | 0.203 | ||||

p | 0.422 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.843 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 2.256 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.854 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.244 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.424 | ||||

Upside Potential Ratio | 2.236 | ||||

Upside part of mean | 0.362 | ||||

Downside part of mean | -0.294 | ||||

Upside SD | 0.264 | ||||

Downside SD | 0.162 | ||||

N nonnegative terms | 3.000 | ||||

N negative terms | 8.000 | ||||

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

N of observations | 11.000 | ||||

Mean of predictor | 0.363 | ||||

Mean of criterion | 0.069 | ||||

SD of predictor | 0.222 | ||||

SD of criterion | 0.324 | ||||

Covariance | 0.004 | ||||

r | 0.057 | ||||

b (slope, estimate of beta) | 0.084 | ||||

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

Mean Square Error | 0.116 | ||||

DF error | 9.000 | ||||

t(b) | 0.172 | ||||

p(b) | 0.434 | ||||

t(a) | 0.096 | ||||

p(a) | 0.463 | ||||

Lowerbound of 95% confidence interval for beta | -1.014 | ||||

Upperbound of 95% confidence interval for beta | 1.181 | ||||

Lowerbound of 95% confidence interval for alpha | -0.861 | ||||

Upperbound of 95% confidence interval for alpha | 0.937 | ||||

Treynor index (mean / b) | 0.822 | ||||

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%) | 0.138 | ||||

Expected Shortfall on VaR | 0.170 | ||||

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

VaR(95%) | 0.068 | ||||

Expected Shortfall on VaR | 0.121 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 11.000 | ||||

Minimum | 0.888 | ||||

Quartile 1 | 0.980 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.009 | ||||

Maximum | 1.276 | ||||

Mean of quarter 1 | 0.924 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.126 | ||||

Inter Quartile Range | 0.029 | ||||

Number outliers low | 2.000 | ||||

Percentage of outliers low | 0.182 | ||||

Mean of outliers low | 0.906 | ||||

Number of outliers high | 2.000 | ||||

Percentage of outliers high | 0.182 | ||||

Mean of outliers high | 1.180 | ||||

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

Extreme Value Index (moments method) | -7.493 | ||||

VaR(95%) (moments method) | 0.079 | ||||

Expected Shortfall (moments method) | 0.079 | ||||

Extreme Value Index (regression method) | -0.960 | ||||

VaR(95%) (regression method) | 0.126 | ||||

Expected Shortfall (regression method) | 0.137 | ||||

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 2.000 | ||||

Minimum | 0.076 | ||||

Quartile 1 | 0.094 | ||||

Median | 0.112 | ||||

Quartile 3 | 0.130 | ||||

Maximum | 0.148 | ||||

Mean of quarter 1 | 0.076 | ||||

Mean of quarter 2 | NA | ||||

Mean of quarter 3 | NA | ||||

Mean of quarter 4 | 0.148 | ||||

Inter Quartile Range | 0.036 | ||||

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) | 0.119 | ||||

Compounded annual return (geometric extrapolation) | 0.119 | ||||

Calmar ratio (compounded annual return / max draw down) | 0.803 | ||||

Compounded annual return / average of 25% largest draw downs | 0.803 | ||||

Compounded annual return / Expected Shortfall lognormal | 0.700 | ||||

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

RATIO STATISTICS | |||||

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

Statistics related to Sharpe ratio | |||||

Mean | 0.093 | ||||

SD | 0.227 | ||||

Sharpe ratio (Glass type estimate) | 0.409 | ||||

Sharpe ratio (Hedges UMVUE) | 0.409 | ||||

df | 318.000 | ||||

t | 0.394 | ||||

p | 0.347 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.626 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 2.445 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.627 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.444 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.583 | ||||

Upside Potential Ratio | 5.767 | ||||

Upside part of mean | 0.920 | ||||

Downside part of mean | -0.827 | ||||

Upside SD | 0.162 | ||||

Downside SD | 0.160 | ||||

N nonnegative terms | 54.000 | ||||

N negative terms | 265.000 | ||||

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

N of observations | 319.000 | ||||

Mean of predictor | 0.379 | ||||

Mean of criterion | 0.093 | ||||

SD of predictor | 0.180 | ||||

SD of criterion | 0.227 | ||||

Covariance | -0.000 | ||||

r | -0.006 | ||||

b (slope, estimate of beta) | -0.008 | ||||

a (intercept, estimate of alpha) | 0.096 | ||||

Mean Square Error | 0.052 | ||||

DF error | 317.000 | ||||

t(b) | -0.115 | ||||

p(b) | 0.546 | ||||

t(a) | 0.404 | ||||

p(a) | 0.343 | ||||

Lowerbound of 95% confidence interval for beta | -0.148 | ||||

Upperbound of 95% confidence interval for beta | 0.131 | ||||

Lowerbound of 95% confidence interval for alpha | -0.372 | ||||

Upperbound of 95% confidence interval for alpha | 0.564 | ||||

Treynor index (mean / b) | -11.465 | ||||

Jensen alpha (a) | 0.096 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | 0.067 | ||||

SD | 0.228 | ||||

Sharpe ratio (Glass type estimate) | 0.296 | ||||

Sharpe ratio (Hedges UMVUE) | 0.295 | ||||

df | 318.000 | ||||

t | 0.285 | ||||

p | 0.388 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.740 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 2.331 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.740 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.331 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.413 | ||||

Upside Potential Ratio | 5.572 | ||||

Upside part of mean | 0.907 | ||||

Downside part of mean | -0.840 | ||||

Upside SD | 0.158 | ||||

Downside SD | 0.163 | ||||

N nonnegative terms | 54.000 | ||||

N negative terms | 265.000 | ||||

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

N of observations | 319.000 | ||||

Mean of predictor | 0.362 | ||||

Mean of criterion | 0.067 | ||||

SD of predictor | 0.180 | ||||

SD of criterion | 0.228 | ||||

Covariance | -0.000 | ||||

r | -0.005 | ||||

b (slope, estimate of beta) | -0.007 | ||||

a (intercept, estimate of alpha) | 0.070 | ||||

Mean Square Error | 0.052 | ||||

DF error | 317.000 | ||||

t(b) | -0.094 | ||||

p(b) | 0.537 | ||||

t(a) | 0.293 | ||||

p(a) | 0.385 | ||||

Lowerbound of 95% confidence interval for beta | -0.146 | ||||

Upperbound of 95% confidence interval for beta | 0.133 | ||||

Lowerbound of 95% confidence interval for alpha | -0.399 | ||||

Upperbound of 95% confidence interval for alpha | 0.538 | ||||

Treynor index (mean / b) | -10.081 | ||||

Jensen alpha (a) | 0.070 | ||||

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

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

VaR(95%) | 0.020 | ||||

Expected Shortfall on VaR | 0.025 | ||||

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

VaR(95%) | 0.007 | ||||

Expected Shortfall on VaR | 0.016 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 319.000 | ||||

Minimum | 0.946 | ||||

Quartile 1 | 1.000 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.000 | ||||

Maximum | 1.071 | ||||

Mean of quarter 1 | 0.991 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.011 | ||||

Inter Quartile Range | 0.000 | ||||

Number outliers low | 43.000 | ||||

Percentage of outliers low | 0.135 | ||||

Mean of outliers low | 0.983 | ||||

Number of outliers high | 54.000 | ||||

Percentage of outliers high | 0.169 | ||||

Mean of outliers high | 1.016 | ||||

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

Extreme Value Index (moments method) | -0.843 | ||||

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

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

Extreme Value Index (regression method) | 0.136 | ||||

VaR(95%) (regression method) | 0.008 | ||||

Expected Shortfall (regression method) | 0.017 | ||||

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 5.000 | ||||

Minimum | 0.001 | ||||

Quartile 1 | 0.002 | ||||

Median | 0.014 | ||||

Quartile 3 | 0.143 | ||||

Maximum | 0.254 | ||||

Mean of quarter 1 | 0.002 | ||||

Mean of quarter 2 | 0.014 | ||||

Mean of quarter 3 | 0.143 | ||||

Mean of quarter 4 | 0.254 | ||||

Inter Quartile Range | 0.141 | ||||

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) | 0.117 | ||||

Compounded annual return (geometric extrapolation) | 0.118 | ||||

Calmar ratio (compounded annual return / max draw down) | 0.464 | ||||

Compounded annual return / average of 25% largest draw downs | 0.464 | ||||

Compounded annual return / Expected Shortfall lognormal | 4.752 | ||||

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

RATIO STATISTICS | |||||

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

Statistics related to Sharpe ratio | |||||

Mean | -0.231 | ||||

SD | 0.108 | ||||

Sharpe ratio (Glass type estimate) | -2.129 | ||||

Sharpe ratio (Hedges UMVUE) | -2.119 | ||||

df | 171.000 | ||||

t | -1.505 | ||||

p | 0.573 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -4.907 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 0.655 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -4.900 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.662 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | -2.397 | ||||

Upside Potential Ratio | 1.326 | ||||

Upside part of mean | 0.128 | ||||

Downside part of mean | -0.359 | ||||

Upside SD | 0.051 | ||||

Downside SD | 0.096 | ||||

N nonnegative terms | 6.000 | ||||

N negative terms | 166.000 | ||||

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

N of observations | 172.000 | ||||

Mean of predictor | 0.729 | ||||

Mean of criterion | -0.231 | ||||

SD of predictor | 0.208 | ||||

SD of criterion | 0.108 | ||||

Covariance | -0.002 | ||||

r | -0.067 | ||||

b (slope, estimate of beta) | -0.035 | ||||

a (intercept, estimate of alpha) | -0.205 | ||||

Mean Square Error | 0.012 | ||||

DF error | 170.000 | ||||

t(b) | -0.880 | ||||

p(b) | 0.534 | ||||

t(a) | -1.314 | ||||

p(a) | 0.550 | ||||

Lowerbound of 95% confidence interval for beta | -0.114 | ||||

Upperbound of 95% confidence interval for beta | 0.044 | ||||

Lowerbound of 95% confidence interval for alpha | -0.514 | ||||

Upperbound of 95% confidence interval for alpha | 0.103 | ||||

Treynor index (mean / b) | 6.570 | ||||

Jensen alpha (a) | -0.205 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | -0.237 | ||||

SD | 0.110 | ||||

Sharpe ratio (Glass type estimate) | -2.159 | ||||

Sharpe ratio (Hedges UMVUE) | -2.150 | ||||

df | 171.000 | ||||

t | -1.527 | ||||

p | 0.574 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -4.938 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 0.625 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -4.931 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.631 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | -2.415 | ||||

Upside Potential Ratio | 1.290 | ||||

Upside part of mean | 0.126 | ||||

Downside part of mean | -0.363 | ||||

Upside SD | 0.050 | ||||

Downside SD | 0.098 | ||||

N nonnegative terms | 6.000 | ||||

N negative terms | 166.000 | ||||

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

N of observations | 172.000 | ||||

Mean of predictor | 0.707 | ||||

Mean of criterion | -0.237 | ||||

SD of predictor | 0.207 | ||||

SD of criterion | 0.110 | ||||

Covariance | -0.002 | ||||

r | -0.068 | ||||

b (slope, estimate of beta) | -0.036 | ||||

a (intercept, estimate of alpha) | -0.212 | ||||

Mean Square Error | 0.012 | ||||

DF error | 170.000 | ||||

t(b) | -0.885 | ||||

p(b) | 0.534 | ||||

t(a) | -1.340 | ||||

p(a) | 0.551 | ||||

Lowerbound of 95% confidence interval for beta | -0.116 | ||||

Upperbound of 95% confidence interval for beta | 0.044 | ||||

Lowerbound of 95% confidence interval for alpha | -0.523 | ||||

Upperbound of 95% confidence interval for alpha | 0.100 | ||||

Treynor index (mean / b) | 6.609 | ||||

Jensen alpha (a) | -0.212 | ||||

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

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

VaR(95%) | 0.010 | ||||

Expected Shortfall on VaR | 0.013 | ||||

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

VaR(95%) | 0.004 | ||||

Expected Shortfall on VaR | 0.008 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 172.000 | ||||

Minimum | 0.958 | ||||

Quartile 1 | 1.000 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.000 | ||||

Maximum | 1.029 | ||||

Mean of quarter 1 | 0.996 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.002 | ||||

Inter Quartile Range | 0.000 | ||||

Number outliers low | 10.000 | ||||

Percentage of outliers low | 0.058 | ||||

Mean of outliers low | 0.984 | ||||

Number of outliers high | 6.000 | ||||

Percentage of outliers high | 0.035 | ||||

Mean of outliers high | 1.011 | ||||

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) | -0.013 | ||||

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

Expected Shortfall (regression method) | 0.012 | ||||

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 1.000 | ||||

Minimum | 0.143 | ||||

Quartile 1 | 0.143 | ||||

Median | 0.143 | ||||

Quartile 3 | 0.143 | ||||

Maximum | 0.143 | ||||

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) | -0.184 | ||||

Compounded annual return (geometric extrapolation) | -0.175 | ||||

Calmar ratio (compounded annual return / max draw down) | -1.228 | ||||

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

Compounded annual return / Expected Shortfall lognormal | -13.700 |