### 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 | 0.059 | ||||

SD | 0.352 | ||||

Sharpe ratio (Glass type estimate) | 0.169 | ||||

Sharpe ratio (Hedges UMVUE) | 0.167 | ||||

df | 60.000 | ||||

t | 0.380 | ||||

p | 0.353 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -0.702 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 1.038 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.703 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 1.036 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.200 | ||||

Upside Potential Ratio | 1.089 | ||||

Upside part of mean | 0.322 | ||||

Downside part of mean | -0.263 | ||||

Upside SD | 0.186 | ||||

Downside SD | 0.296 | ||||

N nonnegative terms | 33.000 | ||||

N negative terms | 28.000 | ||||

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

N of observations | 61.000 | ||||

Mean of predictor | 0.193 | ||||

Mean of criterion | 0.059 | ||||

SD of predictor | 0.180 | ||||

SD of criterion | 0.352 | ||||

Covariance | 0.001 | ||||

r | 0.018 | ||||

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

a (intercept, estimate of alpha) | 0.053 | ||||

Mean Square Error | 0.126 | ||||

DF error | 59.000 | ||||

t(b) | 0.139 | ||||

p(b) | 0.445 | ||||

t(a) | 0.319 | ||||

p(a) | 0.376 | ||||

Lowerbound of 95% confidence interval for beta | -0.475 | ||||

Upperbound of 95% confidence interval for beta | 0.546 | ||||

Lowerbound of 95% confidence interval for alpha | -0.277 | ||||

Upperbound of 95% confidence interval for alpha | 0.383 | ||||

Treynor index (mean / b) | 1.678 | ||||

Jensen alpha (a) | 0.053 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | -0.025 | ||||

SD | 0.454 | ||||

Sharpe ratio (Glass type estimate) | -0.055 | ||||

Sharpe ratio (Hedges UMVUE) | -0.054 | ||||

df | 60.000 | ||||

t | -0.124 | ||||

p | 0.549 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -0.924 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 0.815 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.924 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.815 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | -0.060 | ||||

Upside Potential Ratio | 0.734 | ||||

Upside part of mean | 0.306 | ||||

Downside part of mean | -0.331 | ||||

Upside SD | 0.172 | ||||

Downside SD | 0.416 | ||||

N nonnegative terms | 33.000 | ||||

N negative terms | 28.000 | ||||

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

N of observations | 61.000 | ||||

Mean of predictor | 0.175 | ||||

Mean of criterion | -0.025 | ||||

SD of predictor | 0.176 | ||||

SD of criterion | 0.454 | ||||

Covariance | -0.002 | ||||

r | -0.021 | ||||

b (slope, estimate of beta) | -0.054 | ||||

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

Mean Square Error | 0.210 | ||||

DF error | 59.000 | ||||

t(b) | -0.161 | ||||

p(b) | 0.564 | ||||

t(a) | -0.073 | ||||

p(a) | 0.529 | ||||

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

Upperbound of 95% confidence interval for beta | 0.619 | ||||

Lowerbound of 95% confidence interval for alpha | -0.439 | ||||

Upperbound of 95% confidence interval for alpha | 0.408 | ||||

Treynor index (mean / b) | 0.461 | ||||

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%) | 0.196 | ||||

Expected Shortfall on VaR | 0.238 | ||||

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

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

Expected Shortfall on VaR | 0.107 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 61.000 | ||||

Minimum | 0.429 | ||||

Quartile 1 | 1.000 | ||||

Median | 1.012 | ||||

Quartile 3 | 1.034 | ||||

Maximum | 1.250 | ||||

Mean of quarter 1 | 0.923 | ||||

Mean of quarter 2 | 1.002 | ||||

Mean of quarter 3 | 1.022 | ||||

Mean of quarter 4 | 1.094 | ||||

Inter Quartile Range | 0.034 | ||||

Number outliers low | 3.000 | ||||

Percentage of outliers low | 0.049 | ||||

Mean of outliers low | 0.681 | ||||

Number of outliers high | 7.000 | ||||

Percentage of outliers high | 0.115 | ||||

Mean of outliers high | 1.140 | ||||

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) | 0.787 | ||||

VaR(95%) (regression method) | 0.053 | ||||

Expected Shortfall (regression method) | 0.323 | ||||

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) | 0.020 | ||||

Compounded annual return (geometric extrapolation) | 0.019 | ||||

Calmar ratio (compounded annual return / max draw down) | 0.027 | ||||

Compounded annual return / average of 25% largest draw downs | 0.070 | ||||

Compounded annual return / Expected Shortfall lognormal | 0.081 | ||||

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

RATIO STATISTICS | |||||

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

Statistics related to Sharpe ratio | |||||

Mean | 0.008 | ||||

SD | 0.250 | ||||

Sharpe ratio (Glass type estimate) | 0.031 | ||||

Sharpe ratio (Hedges UMVUE) | 0.031 | ||||

df | 1757.000 | ||||

t | 0.069 | ||||

p | 0.499 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -0.836 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 0.898 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.836 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.898 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | 0.038 | ||||

Upside Potential Ratio | 4.122 | ||||

Upside part of mean | 0.823 | ||||

Downside part of mean | -0.816 | ||||

Upside SD | 0.151 | ||||

Downside SD | 0.200 | ||||

N nonnegative terms | 522.000 | ||||

N negative terms | 1236.000 | ||||

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

N of observations | 1758.000 | ||||

Mean of predictor | 0.202 | ||||

Mean of criterion | 0.008 | ||||

SD of predictor | 0.200 | ||||

SD of criterion | 0.250 | ||||

Covariance | -0.001 | ||||

r | -0.020 | ||||

b (slope, estimate of beta) | -0.025 | ||||

a (intercept, estimate of alpha) | 0.013 | ||||

Mean Square Error | 0.063 | ||||

DF error | 1756.000 | ||||

t(b) | -0.840 | ||||

p(b) | 0.510 | ||||

t(a) | 0.115 | ||||

p(a) | 0.499 | ||||

Lowerbound of 95% confidence interval for beta | -0.084 | ||||

Upperbound of 95% confidence interval for beta | 0.033 | ||||

Lowerbound of 95% confidence interval for alpha | -0.205 | ||||

Upperbound of 95% confidence interval for alpha | 0.230 | ||||

Treynor index (mean / b) | -0.306 | ||||

Jensen alpha (a) | 0.013 | ||||

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

Statistics related to Sharpe ratio | |||||

Mean | -0.025 | ||||

SD | 0.259 | ||||

Sharpe ratio (Glass type estimate) | -0.097 | ||||

Sharpe ratio (Hedges UMVUE) | -0.097 | ||||

df | 1757.000 | ||||

t | -0.219 | ||||

p | 0.503 | ||||

Lowerbound of 95% confidence interval for Sharpe Ratio | -0.964 | ||||

Upperbound of 95% confidence interval for Sharpe Ratio | 0.770 | ||||

Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.964 | ||||

Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.770 | ||||

Statistics related to Sortino ratio | |||||

Sortino ratio | -0.117 | ||||

Upside Potential Ratio | 3.802 | ||||

Upside part of mean | 0.812 | ||||

Downside part of mean | -0.837 | ||||

Upside SD | 0.146 | ||||

Downside SD | 0.214 | ||||

N nonnegative terms | 522.000 | ||||

N negative terms | 1236.000 | ||||

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

N of observations | 1758.000 | ||||

Mean of predictor | 0.182 | ||||

Mean of criterion | -0.025 | ||||

SD of predictor | 0.200 | ||||

SD of criterion | 0.259 | ||||

Covariance | -0.001 | ||||

r | -0.022 | ||||

b (slope, estimate of beta) | -0.029 | ||||

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

Mean Square Error | 0.067 | ||||

DF error | 1756.000 | ||||

t(b) | -0.924 | ||||

p(b) | 0.511 | ||||

t(a) | -0.173 | ||||

p(a) | 0.502 | ||||

Lowerbound of 95% confidence interval for beta | -0.089 | ||||

Upperbound of 95% confidence interval for beta | 0.032 | ||||

Lowerbound of 95% confidence interval for alpha | -0.245 | ||||

Upperbound of 95% confidence interval for alpha | 0.205 | ||||

Treynor index (mean / b) | 0.877 | ||||

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%) | 0.023 | ||||

Expected Shortfall on VaR | 0.028 | ||||

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

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

Expected Shortfall on VaR | 0.015 | ||||

ORDER STATISTICS | |||||

Quartiles of return rates | |||||

Number of observations | 1758.000 | ||||

Minimum | 0.810 | ||||

Quartile 1 | 0.999 | ||||

Median | 1.000 | ||||

Quartile 3 | 1.001 | ||||

Maximum | 1.158 | ||||

Mean of quarter 1 | 0.991 | ||||

Mean of quarter 2 | 1.000 | ||||

Mean of quarter 3 | 1.000 | ||||

Mean of quarter 4 | 1.010 | ||||

Inter Quartile Range | 0.002 | ||||

Number outliers low | 242.000 | ||||

Percentage of outliers low | 0.138 | ||||

Mean of outliers low | 0.985 | ||||

Number of outliers high | 273.000 | ||||

Percentage of outliers high | 0.155 | ||||

Mean of outliers high | 1.014 | ||||

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

Extreme Value Index (moments method) | 0.817 | ||||

VaR(95%) (moments method) | 0.006 | ||||

Expected Shortfall (moments method) | 0.036 | ||||

Extreme Value Index (regression method) | 0.547 | ||||

VaR(95%) (regression method) | 0.006 | ||||

Expected Shortfall (regression method) | 0.016 | ||||

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) | 0.020 | ||||

Compounded annual return (geometric extrapolation) | 0.019 | ||||

Calmar ratio (compounded annual return / max draw down) | 0.027 | ||||

Compounded annual return / average of 25% largest draw downs | 0.206 | ||||

Compounded annual return / Expected Shortfall lognormal | 0.673 | ||||

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 | 0.132 | ||||

Mean of criterion | -0.044 | ||||

SD of predictor | 0.252 | ||||

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 | 0.100 | ||||

Mean of criterion | -0.044 | ||||

SD of predictor | 0.252 | ||||

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) | -5594252855485242.000 | ||||

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) | 77077945923260818615174478954496.000 | ||||

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 |