### Advanced Statistics: Buzo Principle

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 | 0.075 | ||||

SD | 0.378 | ||||

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

Sharpe ratio (Hedges UMVUE) | 0.196 | ||||

df | 52.000 | ||||

t | 0.417 | ||||

p | 0.339 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -0.736 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.131 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.738 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.129 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.236 | ||||

Upside Potential Ratio | 1.168 | ||||

Upside part of mean | 0.371 | ||||

Downside part of mean | -0.296 | ||||

Upside SD | 0.199 | ||||

Downside SD | 0.318 | ||||

N nonnegative terms | 33.000 | ||||

N negative terms | 20.000 | ||||

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

N of observations | 53.000 | ||||

Mean of predictor | 0.213 | ||||

Mean of criterion | 0.075 | ||||

SD of predictor | 0.185 | ||||

SD of criterion | 0.378 | ||||

Covariance | 0.001 | ||||

r | 0.016 | ||||

b (slope, estimate of beta) | 0.033 | ||||

a (intercept, estimate of alpha) | 0.068 | ||||

Mean Square Error | 0.146 | ||||

DF error | 51.000 | ||||

t(b) | 0.114 | ||||

p(b) | 0.455 | ||||

t(a) | 0.355 | ||||

p(a) | 0.362 | ||||

Lowerbound of 95% confidence interval for beta | -0.542 | ||||

Upperbound of 95% confidence interval for beta | 0.607 | ||||

Lowerbound of 95% confidence interval for alpha | -0.316 | ||||

Upperbound of 95% confidence interval for alpha | 0.452 | ||||

Treynor index (mean / b) | 2.298 | ||||

Jensen alpha (a) | 0.068 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | -0.022 | ||||

SD | 0.488 | ||||

Sharpe ratio (Glass type estimate) | -0.045 | ||||

Sharpe ratio (Hedges UMVUE) | -0.045 | ||||

df | 52.000 | ||||

t | -0.095 | ||||

p | 0.538 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -0.978 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 0.888 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.977 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.888 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | -0.049 | ||||

Upside Potential Ratio | 0.788 | ||||

Upside part of mean | 0.352 | ||||

Downside part of mean | -0.374 | ||||

Upside SD | 0.185 | ||||

Downside SD | 0.447 | ||||

N nonnegative terms | 33.000 | ||||

N negative terms | 20.000 | ||||

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

N of observations | 53.000 | ||||

Mean of predictor | 0.194 | ||||

Mean of criterion | -0.022 | ||||

SD of predictor | 0.181 | ||||

SD of criterion | 0.488 | ||||

Covariance | -0.002 | ||||

r | -0.022 | ||||

b (slope, estimate of beta) | -0.060 | ||||

a (intercept, estimate of alpha) | -0.010 | ||||

Mean Square Error | 0.243 | ||||

DF error | 51.000 | ||||

t(b) | -0.159 | ||||

p(b) | 0.563 | ||||

t(a) | -0.042 | ||||

p(a) | 0.517 | ||||

Lowerbound of 95% confidence interval for beta | -0.818 | ||||

Upperbound of 95% confidence interval for beta | 0.698 | ||||

Lowerbound of 95% confidence interval for alpha | -0.503 | ||||

Upperbound of 95% confidence interval for alpha | 0.483 | ||||

Treynor index (mean / b) | 0.368 | ||||

Jensen alpha (a) | -0.010 | ||||

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

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

VaR(95%) | 0.208 | ||||

Expected Shortfall on VaR | 0.252 | ||||

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

VaR(95%) | 0.044 | ||||

Expected Shortfall on VaR | 0.107 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 53.000 | ||||

Minimum | 0.429 | ||||

Quartile 1 | 0.992 | ||||

Median | 1.015 | ||||

Quartile 3 | 1.041 | ||||

Maximum | 1.250 | ||||

Mean of quarter 1 | 0.912 | ||||

Mean of quarter 2 | 1.006 | ||||

Mean of quarter 3 | 1.027 | ||||

Mean of quarter 4 | 1.102 | ||||

Inter Quartile Range | 0.049 | ||||

Number outliers low | 2.000 | ||||

Percentage of outliers low | 0.038 | ||||

Mean of outliers low | 0.561 | ||||

Number of outliers high | 5.000 | ||||

Percentage of outliers high | 0.094 | ||||

Mean of outliers high | 1.161 | ||||

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

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

VaR(95%) (moments method) | 0.066 | ||||

Expected Shortfall (moments method) | NA | ||||

Extreme Value Index (regression method) | 1.060 | ||||

VaR(95%) (regression method) | 0.060 | ||||

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

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 11.000 | ||||

Minimum | 0.006 | ||||

Quartile 1 | 0.018 | ||||

Median | 0.040 | ||||

Quartile 3 | 0.044 | ||||

Maximum | 0.703 | ||||

Mean of quarter 1 | 0.010 | ||||

Mean of quarter 2 | 0.030 | ||||

Mean of quarter 3 | 0.042 | ||||

Mean of quarter 4 | 0.276 | ||||

Inter Quartile Range | 0.026 | ||||

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 | 0.091 | ||||

Mean of outliers high | 0.703 | ||||

Risk estimates based on draw downs (based on Extreme Value Theory) | |||||

Extreme Value Index (moments method) | 1.030 | ||||

VaR(95%) (moments method) | 0.248 | ||||

Expected Shortfall (moments method) | NA | ||||

Extreme Value Index (regression method) | 3.283 | ||||

VaR(95%) (regression method) | 1.059 | ||||

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

COMBINED STATISTICS | |||||

Annualized return (arithmetic extrapolation) | 0.023 | ||||

Compounded annual return (geometric extrapolation) | 0.022 | ||||

Calmar ratio (compounded annual return / max draw down) | 0.032 | ||||

Compounded annual return / average of 25% largest draw downs | 0.080 | ||||

Compounded annual return / Expected Shortfall lognormal | 0.088 | ||||

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

RATIO STATISTICS | |||||

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

Statistics related to Sharpe ratio | |||||

Mean | 0.015 | ||||

SD | 0.268 | ||||

Sharpe ratio (Glass type estimate) | 0.057 | ||||

Sharpe ratio (Hedges UMVUE) | 0.057 | ||||

df | 1532.000 | ||||

t | 0.120 | ||||

p | 0.498 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -0.872 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 0.985 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.872 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.985 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.071 | ||||

Upside Potential Ratio | 4.414 | ||||

Upside part of mean | 0.944 | ||||

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

Upside SD | 0.162 | ||||

Downside SD | 0.214 | ||||

N nonnegative terms | 522.000 | ||||

N negative terms | 1011.000 | ||||

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

N of observations | 1533.000 | ||||

Mean of predictor | 0.217 | ||||

Mean of criterion | 0.015 | ||||

SD of predictor | 0.194 | ||||

SD of criterion | 0.268 | ||||

Covariance | -0.001 | ||||

r | -0.022 | ||||

b (slope, estimate of beta) | -0.031 | ||||

a (intercept, estimate of alpha) | 0.022 | ||||

Mean Square Error | 0.072 | ||||

DF error | 1531.000 | ||||

t(b) | -0.870 | ||||

p(b) | 0.514 | ||||

t(a) | 0.172 | ||||

p(a) | 0.497 | ||||

Lowerbound of 95% confidence interval for beta | -0.100 | ||||

Upperbound of 95% confidence interval for beta | 0.039 | ||||

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

Upperbound of 95% confidence interval for alpha | 0.272 | ||||

Treynor index (mean / b) | -0.495 | ||||

Jensen alpha (a) | 0.022 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | -0.022 | ||||

SD | 0.277 | ||||

Sharpe ratio (Glass type estimate) | -0.080 | ||||

Sharpe ratio (Hedges UMVUE) | -0.080 | ||||

df | 1532.000 | ||||

t | -0.169 | ||||

p | 0.502 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -1.009 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 0.848 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -1.009 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.848 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | -0.097 | ||||

Upside Potential Ratio | 4.072 | ||||

Upside part of mean | 0.932 | ||||

Downside part of mean | -0.954 | ||||

Upside SD | 0.157 | ||||

Downside SD | 0.229 | ||||

N nonnegative terms | 522.000 | ||||

N negative terms | 1011.000 | ||||

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

N of observations | 1533.000 | ||||

Mean of predictor | 0.198 | ||||

Mean of criterion | -0.022 | ||||

SD of predictor | 0.193 | ||||

SD of criterion | 0.277 | ||||

Covariance | -0.001 | ||||

r | -0.024 | ||||

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

a (intercept, estimate of alpha) | -0.015 | ||||

Mean Square Error | 0.077 | ||||

DF error | 1531.000 | ||||

t(b) | -0.956 | ||||

p(b) | 0.516 | ||||

t(a) | -0.116 | ||||

p(a) | 0.502 | ||||

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

Upperbound of 95% confidence interval for beta | 0.037 | ||||

Lowerbound of 95% confidence interval for alpha | -0.273 | ||||

Upperbound of 95% confidence interval for alpha | 0.243 | ||||

Treynor index (mean / b) | 0.635 | ||||

Jensen alpha (a) | -0.015 | ||||

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

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

VaR(95%) | 0.024 | ||||

Expected Shortfall on VaR | 0.030 | ||||

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 | 1533.000 | ||||

Minimum | 0.810 | ||||

Quartile 1 | 0.999 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.002 | ||||

Maximum | 1.158 | ||||

Mean of quarter 1 | 0.990 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.011 | ||||

Inter Quartile Range | 0.003 | ||||

Number outliers low | 138.000 | ||||

Percentage of outliers low | 0.090 | ||||

Mean of outliers low | 0.978 | ||||

Number of outliers high | 166.000 | ||||

Percentage of outliers high | 0.108 | ||||

Mean of outliers high | 1.019 | ||||

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

Extreme Value Index (moments method) | 0.847 | ||||

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

Expected Shortfall (moments method) | 0.058 | ||||

Extreme Value Index (regression method) | 0.580 | ||||

VaR(95%) (regression method) | 0.007 | ||||

Expected Shortfall (regression method) | 0.019 | ||||

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 71.000 | ||||

Minimum | 0.000 | ||||

Quartile 1 | 0.003 | ||||

Median | 0.008 | ||||

Quartile 3 | 0.023 | ||||

Maximum | 0.703 | ||||

Mean of quarter 1 | 0.001 | ||||

Mean of quarter 2 | 0.005 | ||||

Mean of quarter 3 | 0.016 | ||||

Mean of quarter 4 | 0.093 | ||||

Inter Quartile Range | 0.020 | ||||

Number outliers low | 0.000 | ||||

Percentage of outliers low | 0.000 | ||||

Mean of outliers low | NA | ||||

Number of outliers high | 11.000 | ||||

Percentage of outliers high | 0.155 | ||||

Mean of outliers high | 0.134 | ||||

Risk estimates based on draw downs (based on Extreme Value Theory) | |||||

Extreme Value Index (moments method) | 0.721 | ||||

VaR(95%) (moments method) | 0.089 | ||||

Expected Shortfall (moments method) | 0.338 | ||||

Extreme Value Index (regression method) | 0.541 | ||||

VaR(95%) (regression method) | 0.064 | ||||

Expected Shortfall (regression method) | 0.146 | ||||

COMBINED STATISTICS | |||||

Annualized return (arithmetic extrapolation) | 0.023 | ||||

Compounded annual return (geometric extrapolation) | 0.022 | ||||

Calmar ratio (compounded annual return / max draw down) | 0.031 | ||||

Compounded annual return / average of 25% largest draw downs | 0.236 | ||||

Compounded annual return / Expected Shortfall lognormal | 0.723 | ||||

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

RATIO STATISTICS | |||||

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

Statistics related to Sharpe ratio | |||||

Mean | -1.897 | ||||

SD | 0.642 | ||||

Sharpe ratio (Glass type estimate) | -2.953 | ||||

Sharpe ratio (Hedges UMVUE) | -2.940 | ||||

df | 171.000 | ||||

t | -2.088 | ||||

p | 0.600 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -5.738 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | -0.160 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -5.729 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.151 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | -3.265 | ||||

Upside Potential Ratio | 1.911 | ||||

Upside part of mean | 1.110 | ||||

Downside part of mean | -3.007 | ||||

Upside SD | 0.288 | ||||

Downside SD | 0.581 | ||||

N nonnegative terms | 14.000 | ||||

N negative terms | 158.000 | ||||

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

N of observations | 172.000 | ||||

Mean of predictor | 0.596 | ||||

Mean of criterion | -1.897 | ||||

SD of predictor | 0.174 | ||||

SD of criterion | 0.642 | ||||

Covariance | -0.004 | ||||

r | -0.034 | ||||

b (slope, estimate of beta) | -0.124 | ||||

a (intercept, estimate of alpha) | -1.823 | ||||

Mean Square Error | 0.414 | ||||

DF error | 170.000 | ||||

t(b) | -0.439 | ||||

p(b) | 0.517 | ||||

t(a) | -1.968 | ||||

p(a) | 0.575 | ||||

Lowerbound of 95% confidence interval for beta | -0.684 | ||||

Upperbound of 95% confidence interval for beta | 0.435 | ||||

Lowerbound of 95% confidence interval for alpha | -3.650 | ||||

Upperbound of 95% confidence interval for alpha | 0.005 | ||||

Treynor index (mean / b) | 15.257 | ||||

Jensen alpha (a) | -1.823 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | -2.122 | ||||

SD | 0.677 | ||||

Sharpe ratio (Glass type estimate) | -3.134 | ||||

Sharpe ratio (Hedges UMVUE) | -3.120 | ||||

df | 171.000 | ||||

t | -2.216 | ||||

p | 0.606 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -5.921 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | -0.338 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -5.911 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.329 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | -3.379 | ||||

Upside Potential Ratio | 1.706 | ||||

Upside part of mean | 1.071 | ||||

Downside part of mean | -3.193 | ||||

Upside SD | 0.273 | ||||

Downside SD | 0.628 | ||||

N nonnegative terms | 14.000 | ||||

N negative terms | 158.000 | ||||

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

N of observations | 172.000 | ||||

Mean of predictor | 0.580 | ||||

Mean of criterion | -2.122 | ||||

SD of predictor | 0.173 | ||||

SD of criterion | 0.677 | ||||

Covariance | -0.004 | ||||

r | -0.035 | ||||

b (slope, estimate of beta) | -0.137 | ||||

a (intercept, estimate of alpha) | -2.043 | ||||

Mean Square Error | 0.461 | ||||

DF error | 170.000 | ||||

t(b) | -0.457 | ||||

p(b) | 0.518 | ||||

t(a) | -2.094 | ||||

p(a) | 0.579 | ||||

Lowerbound of 95% confidence interval for beta | -0.727 | ||||

Upperbound of 95% confidence interval for beta | 0.454 | ||||

Lowerbound of 95% confidence interval for alpha | -3.968 | ||||

Upperbound of 95% confidence interval for alpha | -0.117 | ||||

Treynor index (mean / b) | 15.527 | ||||

Jensen alpha (a) | -2.043 | ||||

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

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

VaR(95%) | 0.064 | ||||

Expected Shortfall on VaR | 0.078 | ||||

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

VaR(95%) | 0.029 | ||||

Expected Shortfall on VaR | 0.062 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 172.000 | ||||

Minimum | 0.810 | ||||

Quartile 1 | 1.000 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.000 | ||||

Maximum | 1.158 | ||||

Mean of quarter 1 | 0.966 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.013 | ||||

Inter Quartile Range | 0.000 | ||||

Number outliers low | 26.000 | ||||

Percentage of outliers low | 0.151 | ||||

Mean of outliers low | 0.943 | ||||

Number of outliers high | 14.000 | ||||

Percentage of outliers high | 0.081 | ||||

Mean of outliers high | 1.040 | ||||

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

Extreme Value Index (moments method) | -0.722 | ||||

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

Expected Shortfall (moments method) | 0.006 | ||||

Extreme Value Index (regression method) | -0.506 | ||||

VaR(95%) (regression method) | 0.040 | ||||

Expected Shortfall (regression method) | 0.058 | ||||

DRAW DOWN STATISTICS | |||||

Quartiles of draw downs | |||||

Number of observations | 2.000 | ||||

Minimum | 0.001 | ||||

Quartile 1 | 0.166 | ||||

Median | 0.331 | ||||

Quartile 3 | 0.496 | ||||

Maximum | 0.662 | ||||

Mean of quarter 1 | 0.001 | ||||

Mean of quarter 2 | NA | ||||

Mean of quarter 3 | NA | ||||

Mean of quarter 4 | 0.662 | ||||

Inter Quartile Range | 0.330 | ||||

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) | -1.292 | ||||

Compounded annual return (geometric extrapolation) | -0.875 | ||||

Calmar ratio (compounded annual return / max draw down) | -1.322 | ||||

Compounded annual return / average of 25% largest draw downs | -1.322 | ||||

Compounded annual return / Expected Shortfall lognormal | -11.192 |