### Advanced Statistics: ETF OPTION TIMER

Advanced Statistics module contributed by Jules Ellis, C2 Member. Jules has written a very helpful guide for these statistics. | |||||

| |||||

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

RATIO STATISTICS | |||||

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

Statistics related to Sharpe ratio | |||||

Mean | -0.005 | ||||

SD | 0.095 | ||||

Sharpe ratio (Glass type estimate) | -0.048 | ||||

Sharpe ratio (Hedges UMVUE) | -0.047 | ||||

df | 28.000 | ||||

t | -0.074 | ||||

p | 0.529 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.308 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.213 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.307 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.214 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | -0.072 | ||||

Upside Potential Ratio | 1.470 | ||||

Upside part of mean | 0.092 | ||||

Downside part of mean | -0.097 | ||||

Upside SD | 0.069 | ||||

Downside SD | 0.063 | ||||

N nonnegative terms | 5.000 | ||||

N negative terms | 24.000 | ||||

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

N of observations | 29.000 | ||||

Mean of predictor | 0.148 | ||||

Mean of criterion | -0.005 | ||||

SD of predictor | 0.090 | ||||

SD of criterion | 0.095 | ||||

Covariance | 0.001 | ||||

r | 0.172 | ||||

b (slope, estimate of beta) | 0.181 | ||||

a (intercept, estimate of alpha) | -0.031 | ||||

Mean Square Error | 0.009 | ||||

DF error | 27.000 | ||||

t(b) | 0.908 | ||||

p(b) | 0.186 | ||||

t(a) | -0.462 | ||||

p(a) | 0.676 | ||||

Lowerbound of 95% confidence interval for beta | -0.228 | ||||

Upperbound of 95% confidence interval for beta | 0.591 | ||||

Lowerbound of 95% confidence interval for alpha | -0.171 | ||||

Upperbound of 95% confidence interval for alpha | 0.108 | ||||

Treynor index (mean / b) | -0.025 | ||||

Jensen alpha (a) | -0.031 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | -0.009 | ||||

SD | 0.095 | ||||

Sharpe ratio (Glass type estimate) | -0.094 | ||||

Sharpe ratio (Hedges UMVUE) | -0.091 | ||||

df | 28.000 | ||||

t | -0.145 | ||||

p | 0.557 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.354 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.168 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.352 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.170 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | -0.137 | ||||

Upside Potential Ratio | 1.389 | ||||

Upside part of mean | 0.090 | ||||

Downside part of mean | -0.098 | ||||

Upside SD | 0.067 | ||||

Downside SD | 0.065 | ||||

N nonnegative terms | 5.000 | ||||

N negative terms | 24.000 | ||||

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

N of observations | 29.000 | ||||

Mean of predictor | 0.143 | ||||

Mean of criterion | -0.009 | ||||

SD of predictor | 0.089 | ||||

SD of criterion | 0.095 | ||||

Covariance | 0.002 | ||||

r | 0.181 | ||||

b (slope, estimate of beta) | 0.192 | ||||

a (intercept, estimate of alpha) | -0.036 | ||||

Mean Square Error | 0.009 | ||||

DF error | 27.000 | ||||

t(b) | 0.956 | ||||

p(b) | 0.174 | ||||

t(a) | -0.538 | ||||

p(a) | 0.703 | ||||

Lowerbound of 95% confidence interval for beta | -0.220 | ||||

Upperbound of 95% confidence interval for beta | 0.603 | ||||

Lowerbound of 95% confidence interval for alpha | -0.174 | ||||

Upperbound of 95% confidence interval for alpha | 0.102 | ||||

Treynor index (mean / b) | -0.046 | ||||

Jensen alpha (a) | -0.036 | ||||

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

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

VaR(95%) | 0.045 | ||||

Expected Shortfall on VaR | 0.055 | ||||

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

VaR(95%) | 0.025 | ||||

Expected Shortfall on VaR | 0.048 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 29.000 | ||||

Minimum | 0.930 | ||||

Quartile 1 | 1.000 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.000 | ||||

Maximum | 1.074 | ||||

Mean of quarter 1 | 0.982 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.035 | ||||

Inter Quartile Range | 0.000 | ||||

Number outliers low | 3.000 | ||||

Percentage of outliers low | 0.103 | ||||

Mean of outliers low | 0.951 | ||||

Number of outliers high | 6.000 | ||||

Percentage of outliers high | 0.207 | ||||

Mean of outliers high | 1.040 | ||||

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) | -1.319 | ||||

VaR(95%) (regression method) | 0.065 | ||||

Expected Shortfall (regression method) | 0.079 | ||||

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 2.000 | ||||

Minimum | 0.027 | ||||

Quartile 1 | 0.037 | ||||

Median | 0.048 | ||||

Quartile 3 | 0.059 | ||||

Maximum | 0.070 | ||||

Mean of quarter 1 | 0.027 | ||||

Mean of quarter 2 | NA | ||||

Mean of quarter 3 | NA | ||||

Mean of quarter 4 | 0.070 | ||||

Inter Quartile Range | 0.021 | ||||

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) | 0.037 | ||||

Compounded annual return (geometric extrapolation) | 0.036 | ||||

Calmar ratio (compounded annual return / max draw down) | 0.515 | ||||

Compounded annual return / average of 25% largest draw downs | 0.515 | ||||

Compounded annual return / Expected Shortfall lognormal | 0.646 | ||||

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

RATIO STATISTICS | |||||

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

Statistics related to Sharpe ratio | |||||

Mean | -0.006 | ||||

SD | 0.093 | ||||

Sharpe ratio (Glass type estimate) | -0.060 | ||||

Sharpe ratio (Hedges UMVUE) | -0.060 | ||||

df | 854.000 | ||||

t | -0.094 | ||||

p | 0.537 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.303 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.184 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.303 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.184 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | -0.083 | ||||

Upside Potential Ratio | 4.146 | ||||

Upside part of mean | 0.275 | ||||

Downside part of mean | -0.281 | ||||

Upside SD | 0.065 | ||||

Downside SD | 0.066 | ||||

N nonnegative terms | 110.000 | ||||

N negative terms | 745.000 | ||||

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

N of observations | 855.000 | ||||

Mean of predictor | 0.145 | ||||

Mean of criterion | -0.006 | ||||

SD of predictor | 0.126 | ||||

SD of criterion | 0.093 | ||||

Covariance | 0.001 | ||||

r | 0.099 | ||||

b (slope, estimate of beta) | 0.073 | ||||

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

Mean Square Error | 0.009 | ||||

DF error | 853.000 | ||||

t(b) | 2.909 | ||||

p(b) | 0.002 | ||||

t(a) | -0.275 | ||||

p(a) | 0.608 | ||||

Lowerbound of 95% confidence interval for beta | 0.024 | ||||

Upperbound of 95% confidence interval for beta | 0.123 | ||||

Lowerbound of 95% confidence interval for alpha | -0.131 | ||||

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

Treynor index (mean / b) | -0.076 | ||||

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

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

Statistics related to Sharpe ratio | |||||

Mean | -0.010 | ||||

SD | 0.093 | ||||

Sharpe ratio (Glass type estimate) | -0.106 | ||||

Sharpe ratio (Hedges UMVUE) | -0.106 | ||||

df | 854.000 | ||||

t | -0.167 | ||||

p | 0.566 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.349 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.137 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.349 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.137 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | -0.146 | ||||

Upside Potential Ratio | 4.064 | ||||

Upside part of mean | 0.273 | ||||

Downside part of mean | -0.283 | ||||

Upside SD | 0.064 | ||||

Downside SD | 0.067 | ||||

N nonnegative terms | 110.000 | ||||

N negative terms | 745.000 | ||||

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

N of observations | 855.000 | ||||

Mean of predictor | 0.137 | ||||

Mean of criterion | -0.010 | ||||

SD of predictor | 0.125 | ||||

SD of criterion | 0.093 | ||||

Covariance | 0.001 | ||||

r | 0.099 | ||||

b (slope, estimate of beta) | 0.073 | ||||

a (intercept, estimate of alpha) | -0.020 | ||||

Mean Square Error | 0.009 | ||||

DF error | 853.000 | ||||

t(b) | 2.908 | ||||

p(b) | 0.002 | ||||

t(a) | -0.338 | ||||

p(a) | 0.632 | ||||

Lowerbound of 95% confidence interval for beta | 0.024 | ||||

Upperbound of 95% confidence interval for beta | 0.123 | ||||

Lowerbound of 95% confidence interval for alpha | -0.135 | ||||

Upperbound of 95% confidence interval for alpha | 0.095 | ||||

Treynor index (mean / b) | -0.134 | ||||

Jensen alpha (a) | -0.020 | ||||

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

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

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

Expected Shortfall on VaR | 0.010 | ||||

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

VaR(95%) | 0.003 | ||||

Expected Shortfall on VaR | 0.006 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 855.000 | ||||

Minimum | 0.963 | ||||

Quartile 1 | 1.000 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.000 | ||||

Maximum | 1.037 | ||||

Mean of quarter 1 | 0.997 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.003 | ||||

Inter Quartile Range | 0.000 | ||||

Number outliers low | 70.000 | ||||

Percentage of outliers low | 0.082 | ||||

Mean of outliers low | 0.991 | ||||

Number of outliers high | 111.000 | ||||

Percentage of outliers high | 0.130 | ||||

Mean of outliers high | 1.006 | ||||

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

Extreme Value Index (moments method) | 0.169 | ||||

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

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

Extreme Value Index (regression method) | 0.070 | ||||

VaR(95%) (regression method) | 0.003 | ||||

Expected Shortfall (regression method) | 0.008 | ||||

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 9.000 | ||||

Minimum | 0.000 | ||||

Quartile 1 | 0.002 | ||||

Median | 0.004 | ||||

Quartile 3 | 0.083 | ||||

Maximum | 0.132 | ||||

Mean of quarter 1 | 0.001 | ||||

Mean of quarter 2 | 0.004 | ||||

Mean of quarter 3 | 0.046 | ||||

Mean of quarter 4 | 0.116 | ||||

Inter Quartile Range | 0.081 | ||||

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) | -2.125 | ||||

VaR(95%) (moments method) | 0.121 | ||||

Expected Shortfall (moments method) | 0.122 | ||||

Extreme Value Index (regression method) | -0.179 | ||||

VaR(95%) (regression method) | 0.141 | ||||

Expected Shortfall (regression method) | 0.165 | ||||

COMBINED STATISTICS | |||||

Annualized return (arithmetic extrapolation) | 0.036 | ||||

Compounded annual return (geometric extrapolation) | 0.035 | ||||

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

Compounded annual return / average of 25% largest draw downs | 0.299 | ||||

Compounded annual return / Expected Shortfall lognormal | 3.381 | ||||

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 | 0.075 | ||||

Mean of criterion | -0.044 | ||||

SD of predictor | 0.110 | ||||

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 | 0.069 | ||||

Mean of criterion | -0.044 | ||||

SD of predictor | 0.111 | ||||

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) | -5592398559240949.000 | ||||

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) | -11873362712668898085609297412096.000 | ||||

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 |