### 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 | 0.084 | ||||

SD | 0.307 | ||||

Sharpe ratio (Glass type estimate) | 0.275 | ||||

Sharpe ratio (Hedges UMVUE) | 0.258 | ||||

df | 13.000 | ||||

t | 0.297 | ||||

p | 0.448 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.548 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 2.087 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.559 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.076 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.615 | ||||

Upside Potential Ratio | 2.302 | ||||

Upside part of mean | 0.315 | ||||

Downside part of mean | -0.231 | ||||

Upside SD | 0.263 | ||||

Downside SD | 0.137 | ||||

N nonnegative terms | 3.000 | ||||

N negative terms | 11.000 | ||||

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

N of observations | 14.000 | ||||

Mean of predictor | 0.337 | ||||

Mean of criterion | 0.084 | ||||

SD of predictor | 0.210 | ||||

SD of criterion | 0.307 | ||||

Covariance | 0.003 | ||||

r | 0.051 | ||||

b (slope, estimate of beta) | 0.074 | ||||

a (intercept, estimate of alpha) | 0.059 | ||||

Mean Square Error | 0.102 | ||||

DF error | 12.000 | ||||

t(b) | 0.176 | ||||

p(b) | 0.475 | ||||

t(a) | 0.181 | ||||

p(a) | 0.474 | ||||

Lowerbound of 95% confidence interval for beta | -0.843 | ||||

Upperbound of 95% confidence interval for beta | 0.992 | ||||

Lowerbound of 95% confidence interval for alpha | -0.655 | ||||

Upperbound of 95% confidence interval for alpha | 0.773 | ||||

Treynor index (mean / b) | 1.136 | ||||

Jensen alpha (a) | 0.059 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | 0.044 | ||||

SD | 0.284 | ||||

Sharpe ratio (Glass type estimate) | 0.156 | ||||

Sharpe ratio (Hedges UMVUE) | 0.147 | ||||

df | 13.000 | ||||

t | 0.169 | ||||

p | 0.470 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.662 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.969 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.668 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.963 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.309 | ||||

Upside Potential Ratio | 1.981 | ||||

Upside part of mean | 0.285 | ||||

Downside part of mean | -0.240 | ||||

Upside SD | 0.234 | ||||

Downside SD | 0.144 | ||||

N nonnegative terms | 3.000 | ||||

N negative terms | 11.000 | ||||

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

N of observations | 14.000 | ||||

Mean of predictor | 0.313 | ||||

Mean of criterion | 0.044 | ||||

SD of predictor | 0.198 | ||||

SD of criterion | 0.284 | ||||

Covariance | 0.004 | ||||

r | 0.064 | ||||

b (slope, estimate of beta) | 0.092 | ||||

a (intercept, estimate of alpha) | 0.016 | ||||

Mean Square Error | 0.087 | ||||

DF error | 12.000 | ||||

t(b) | 0.221 | ||||

p(b) | 0.468 | ||||

t(a) | 0.052 | ||||

p(a) | 0.492 | ||||

Lowerbound of 95% confidence interval for beta | -0.812 | ||||

Upperbound of 95% confidence interval for beta | 0.995 | ||||

Lowerbound of 95% confidence interval for alpha | -0.644 | ||||

Upperbound of 95% confidence interval for alpha | 0.675 | ||||

Treynor index (mean / b) | 0.485 | ||||

Jensen alpha (a) | 0.016 | ||||

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

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

VaR(95%) | 0.123 | ||||

Expected Shortfall on VaR | 0.152 | ||||

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

VaR(95%) | 0.058 | ||||

Expected Shortfall on VaR | 0.108 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 14.000 | ||||

Minimum | 0.888 | ||||

Quartile 1 | 1.000 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.000 | ||||

Maximum | 1.276 | ||||

Mean of quarter 1 | 0.943 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.095 | ||||

Inter Quartile Range | 0.000 | ||||

Number outliers low | 3.000 | ||||

Percentage of outliers low | 0.214 | ||||

Mean of outliers low | 0.924 | ||||

Number of outliers high | 3.000 | ||||

Percentage of outliers high | 0.214 | ||||

Mean of outliers high | 1.126 | ||||

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) | -1.090 | ||||

VaR(95%) (regression method) | 0.123 | ||||

Expected Shortfall (regression method) | 0.136 | ||||

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) | 0.093 | ||||

Compounded annual return (geometric extrapolation) | 0.093 | ||||

Calmar ratio (compounded annual return / max draw down) | 0.623 | ||||

Compounded annual return / average of 25% largest draw downs | 0.623 | ||||

Compounded annual return / Expected Shortfall lognormal | 0.607 | ||||

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

RATIO STATISTICS | |||||

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

Statistics related to Sharpe ratio | |||||

Mean | 0.058 | ||||

SD | 0.196 | ||||

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

Sharpe ratio (Hedges UMVUE) | 0.296 | ||||

df | 427.000 | ||||

t | 0.331 | ||||

p | 0.371 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.461 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 2.054 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.461 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 2.053 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.422 | ||||

Upside Potential Ratio | 4.978 | ||||

Upside part of mean | 0.686 | ||||

Downside part of mean | -0.628 | ||||

Upside SD | 0.139 | ||||

Downside SD | 0.138 | ||||

N nonnegative terms | 54.000 | ||||

N negative terms | 374.000 | ||||

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

N of observations | 428.000 | ||||

Mean of predictor | 0.304 | ||||

Mean of criterion | 0.058 | ||||

SD of predictor | 0.182 | ||||

SD of criterion | 0.196 | ||||

Covariance | -0.000 | ||||

r | -0.005 | ||||

b (slope, estimate of beta) | -0.005 | ||||

a (intercept, estimate of alpha) | 0.060 | ||||

Mean Square Error | 0.039 | ||||

DF error | 426.000 | ||||

t(b) | -0.101 | ||||

p(b) | 0.540 | ||||

t(a) | 0.338 | ||||

p(a) | 0.368 | ||||

Lowerbound of 95% confidence interval for beta | -0.108 | ||||

Upperbound of 95% confidence interval for beta | 0.098 | ||||

Lowerbound of 95% confidence interval for alpha | -0.288 | ||||

Upperbound of 95% confidence interval for alpha | 0.407 | ||||

Treynor index (mean / b) | -11.008 | ||||

Jensen alpha (a) | 0.060 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | 0.039 | ||||

SD | 0.196 | ||||

Sharpe ratio (Glass type estimate) | 0.198 | ||||

Sharpe ratio (Hedges UMVUE) | 0.198 | ||||

df | 427.000 | ||||

t | 0.221 | ||||

p | 0.412 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.559 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.956 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.559 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.955 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.277 | ||||

Upside Potential Ratio | 4.810 | ||||

Upside part of mean | 0.676 | ||||

Downside part of mean | -0.637 | ||||

Upside SD | 0.137 | ||||

Downside SD | 0.141 | ||||

N nonnegative terms | 54.000 | ||||

N negative terms | 374.000 | ||||

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

N of observations | 428.000 | ||||

Mean of predictor | 0.288 | ||||

Mean of criterion | 0.039 | ||||

SD of predictor | 0.181 | ||||

SD of criterion | 0.196 | ||||

Covariance | -0.000 | ||||

r | -0.004 | ||||

b (slope, estimate of beta) | -0.004 | ||||

a (intercept, estimate of alpha) | 0.040 | ||||

Mean Square Error | 0.039 | ||||

DF error | 426.000 | ||||

t(b) | -0.083 | ||||

p(b) | 0.533 | ||||

t(a) | 0.227 | ||||

p(a) | 0.410 | ||||

Lowerbound of 95% confidence interval for beta | -0.107 | ||||

Upperbound of 95% confidence interval for beta | 0.099 | ||||

Lowerbound of 95% confidence interval for alpha | -0.307 | ||||

Upperbound of 95% confidence interval for alpha | 0.388 | ||||

Treynor index (mean / b) | -8.976 | ||||

Jensen alpha (a) | 0.040 | ||||

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

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

VaR(95%) | 0.017 | ||||

Expected Shortfall on VaR | 0.021 | ||||

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

VaR(95%) | 0.006 | ||||

Expected Shortfall on VaR | 0.013 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 428.000 | ||||

Minimum | 0.946 | ||||

Quartile 1 | 1.000 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.000 | ||||

Maximum | 1.071 | ||||

Mean of quarter 1 | 0.993 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.008 | ||||

Inter Quartile Range | 0.000 | ||||

Number outliers low | 43.000 | ||||

Percentage of outliers low | 0.100 | ||||

Mean of outliers low | 0.983 | ||||

Number of outliers high | 54.000 | ||||

Percentage of outliers high | 0.126 | ||||

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) | 0.001 | ||||

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

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

VaR(95%) (regression method) | 0.005 | ||||

Expected Shortfall (regression method) | 0.015 | ||||

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) | 0.087 | ||||

Compounded annual return (geometric extrapolation) | 0.087 | ||||

Calmar ratio (compounded annual return / max draw down) | 0.341 | ||||

Compounded annual return / average of 25% largest draw downs | 0.341 | ||||

Compounded annual return / Expected Shortfall lognormal | 4.027 | ||||

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

RATIO STATISTICS | |||||

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

Statistics related to Sharpe ratio | |||||

Mean | -0.044 | ||||

SD | 0.000 | ||||

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

Sharpe ratio (Hedges UMVUE) | NA | ||||

df | NA | ||||

t | NA | ||||

p | NA | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | NA | ||||

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

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

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

Statistics related to Sortino ratio | |||||

Sortino ratio | -18.547 | ||||

Upside Potential Ratio | 0.000 | ||||

Upside part of mean | 0.000 | ||||

Downside part of mean | -0.044 | ||||

Upside SD | 0.000 | ||||

Downside SD | 0.002 | ||||

N nonnegative terms | 0.000 | ||||

N negative terms | 172.000 | ||||

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

N of observations | 172.000 | ||||

Mean of predictor | 0.195 | ||||

Mean of criterion | -0.044 | ||||

SD of predictor | 0.172 | ||||

SD of criterion | 0.000 | ||||

Covariance | 0.000 | ||||

r | NA | ||||

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

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

Mean Square Error | NA | ||||

DF error | NA | ||||

t(b) | NA | ||||

p(b) | NA | ||||

t(a) | NA | ||||

p(a) | NA | ||||

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

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

Lowerbound of 95% confidence interval for alpha | NA | ||||

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

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

Jensen alpha (a) | NA | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | -0.044 | ||||

SD | 0.000 | ||||

Sharpe ratio (Glass type estimate) | -7936535306446244.000 | ||||

Sharpe ratio (Hedges UMVUE) | -7901674975671223.000 | ||||

df | 171.000 | ||||

t | -5611977934314593.000 | ||||

p | 1.000 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | NA | ||||

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

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -8739115425704417.000 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -7064234525638030.000 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | -18.547 | ||||

Upside Potential Ratio | 0.000 | ||||

Upside part of mean | 0.000 | ||||

Downside part of mean | -0.044 | ||||

Upside SD | 0.000 | ||||

Downside SD | 0.002 | ||||

N nonnegative terms | 0.000 | ||||

N negative terms | 172.000 | ||||

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

N of observations | 172.000 | ||||

Mean of predictor | 0.180 | ||||

Mean of criterion | -0.044 | ||||

SD of predictor | 0.172 | ||||

SD of criterion | 0.000 | ||||

Covariance | -0.000 | ||||

r | -0.000 | ||||

b (slope, estimate of beta) | -0.000 | ||||

a (intercept, estimate of alpha) | -0.044 | ||||

Mean Square Error | 0.000 | ||||

DF error | 170.000 | ||||

t(b) | -0.000 | ||||

p(b) | 0.500 | ||||

t(a) | -5586620237671944.000 | ||||

p(a) | 1.000 | ||||

Lowerbound of 95% confidence interval for beta | -0.000 | ||||

Upperbound of 95% confidence interval for beta | 0.000 | ||||

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

Upperbound of 95% confidence interval for alpha | -0.044 | ||||

Treynor index (mean / b) | 23977339601330866766702413086720.000 | ||||

Jensen alpha (a) | -0.044 | ||||

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

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

VaR(95%) | 0.000 | ||||

Expected Shortfall on VaR | 0.000 | ||||

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

VaR(95%) | 0.000 | ||||

Expected Shortfall on VaR | 0.000 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 172.000 | ||||

Minimum | 1.000 | ||||

Quartile 1 | 1.000 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.000 | ||||

Maximum | 1.000 | ||||

Mean of quarter 1 | 1.000 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.000 | ||||

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 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) | NA | ||||

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

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

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 0.000 | ||||

Minimum | NA | ||||

Quartile 1 | NA | ||||

Median | NA | ||||

Quartile 3 | NA | ||||

Maximum | NA | ||||

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 | NA | ||||

Mean of outliers low | NA | ||||

Number of outliers high | 0.000 | ||||

Percentage of outliers high | NA | ||||

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) | 0.000 | ||||

Compounded annual return (geometric extrapolation) | 0.000 | ||||

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

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

Compounded annual return / Expected Shortfall lognormal | 0.000 |