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 | 0.029 | ||||
SD | 0.306 | ||||
Sharpe ratio (Glass type estimate) | 0.093 | ||||
Sharpe ratio (Hedges UMVUE) | 0.092 | ||||
df | 73.000 | ||||
t | 0.231 | ||||
p | 0.409 | ||||
Lowerbound of 95% confidence interval for Sharpe Ratio | -0.697 | ||||
Upperbound of 95% confidence interval for Sharpe Ratio | 0.882 | ||||
Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.697 | ||||
Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.882 | ||||
Statistics related to Sortino ratio | |||||
Sortino ratio | 0.114 | ||||
Upside Potential Ratio | 1.013 | ||||
Upside part of mean | 0.253 | ||||
Downside part of mean | -0.224 | ||||
Upside SD | 0.175 | ||||
Downside SD | 0.249 | ||||
N nonnegative terms | 27.000 | ||||
N negative terms | 47.000 | ||||
Statistics related to linear regression on benchmark | |||||
N of observations | 74.000 | ||||
Mean of predictor | 0.288 | ||||
Mean of criterion | 0.029 | ||||
SD of predictor | 0.234 | ||||
SD of criterion | 0.306 | ||||
Covariance | 0.001 | ||||
r | 0.008 | ||||
b (slope, estimate of beta) | 0.010 | ||||
a (intercept, estimate of alpha) | 0.026 | ||||
Mean Square Error | 0.095 | ||||
DF error | 72.000 | ||||
t(b) | 0.066 | ||||
p(b) | 0.474 | ||||
t(a) | 0.194 | ||||
p(a) | 0.423 | ||||
Lowerbound of 95% confidence interval for beta | -0.298 | ||||
Upperbound of 95% confidence interval for beta | 0.318 | ||||
Lowerbound of 95% confidence interval for alpha | -0.237 | ||||
Upperbound of 95% confidence interval for alpha | 0.289 | ||||
Treynor index (mean / b) | 2.820 | ||||
Jensen alpha (a) | 0.026 | ||||
Ratio statistics of excess log return rates | |||||
Statistics related to Sharpe ratio | |||||
Mean | -0.032 | ||||
SD | 0.377 | ||||
Sharpe ratio (Glass type estimate) | -0.084 | ||||
Sharpe ratio (Hedges UMVUE) | -0.083 | ||||
df | 73.000 | ||||
t | -0.208 | ||||
p | 0.582 | ||||
Lowerbound of 95% confidence interval for Sharpe Ratio | -0.873 | ||||
Upperbound of 95% confidence interval for Sharpe Ratio | 0.706 | ||||
Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.872 | ||||
Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.707 | ||||
Statistics related to Sortino ratio | |||||
Sortino ratio | -0.093 | ||||
Upside Potential Ratio | 0.704 | ||||
Upside part of mean | 0.238 | ||||
Downside part of mean | -0.270 | ||||
Upside SD | 0.161 | ||||
Downside SD | 0.338 | ||||
N nonnegative terms | 27.000 | ||||
N negative terms | 47.000 | ||||
Statistics related to linear regression on benchmark | |||||
N of observations | 74.000 | ||||
Mean of predictor | 0.259 | ||||
Mean of criterion | -0.032 | ||||
SD of predictor | 0.225 | ||||
SD of criterion | 0.377 | ||||
Covariance | -0.000 | ||||
r | -0.003 | ||||
b (slope, estimate of beta) | -0.005 | ||||
a (intercept, estimate of alpha) | -0.030 | ||||
Mean Square Error | 0.144 | ||||
DF error | 72.000 | ||||
t(b) | -0.024 | ||||
p(b) | 0.509 | ||||
t(a) | -0.188 | ||||
p(a) | 0.574 | ||||
Lowerbound of 95% confidence interval for beta | -0.398 | ||||
Upperbound of 95% confidence interval for beta | 0.389 | ||||
Lowerbound of 95% confidence interval for alpha | -0.351 | ||||
Upperbound of 95% confidence interval for alpha | 0.291 | ||||
Treynor index (mean / b) | 6.715 | ||||
Jensen alpha (a) | -0.030 | ||||
Risk estimates for a one-period unit investment (parametric) | |||||
assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
VaR(95%) | 0.166 | ||||
Expected Shortfall on VaR | 0.202 | ||||
assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
VaR(95%) | 0.049 | ||||
Expected Shortfall on VaR | 0.109 | ||||
ORDER STATISTICS | |||||
Quartiles of return rates | |||||
Number of observations | 74.000 | ||||
Minimum | 0.466 | ||||
Quartile 1 | 1.000 | ||||
Median | 1.000 | ||||
Quartile 3 | 1.025 | ||||
Maximum | 1.268 | ||||
Mean of quarter 1 | 0.936 | ||||
Mean of quarter 2 | 1.000 | ||||
Mean of quarter 3 | 1.007 | ||||
Mean of quarter 4 | 1.081 | ||||
Inter Quartile Range | 0.025 | ||||
Number outliers low | 7.000 | ||||
Percentage of outliers low | 0.095 | ||||
Mean of outliers low | 0.843 | ||||
Number of outliers high | 8.000 | ||||
Percentage of outliers high | 0.108 | ||||
Mean of outliers high | 1.136 | ||||
Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
Extreme Value Index (moments method) | 0.673 | ||||
VaR(95%) (moments method) | 0.009 | ||||
Expected Shortfall (moments method) | 0.035 | ||||
Extreme Value Index (regression method) | 1.107 | ||||
VaR(95%) (regression method) | 0.053 | ||||
Expected Shortfall (regression method) | NA | ||||
DRAW DOWN STATISTICS | |||||
Quartiles of draw downs | |||||
Number of observations | 11.000 | ||||
Minimum | 0.003 | ||||
Quartile 1 | 0.016 | ||||
Median | 0.038 | ||||
Quartile 3 | 0.048 | ||||
Maximum | 0.694 | ||||
Mean of quarter 1 | 0.008 | ||||
Mean of quarter 2 | 0.027 | ||||
Mean of quarter 3 | 0.043 | ||||
Mean of quarter 4 | 0.269 | ||||
Inter Quartile Range | 0.032 | ||||
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 | 0.694 | ||||
Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
Extreme Value Index (moments method) | 1.237 | ||||
VaR(95%) (moments method) | 0.282 | ||||
Expected Shortfall (moments method) | NA | ||||
Extreme Value Index (regression method) | 5.421 | ||||
VaR(95%) (regression method) | 1.915 | ||||
Expected Shortfall (regression method) | NA | ||||
COMBINED STATISTICS | |||||
Annualized return (arithmetic extrapolation) | 0.013 | ||||
Compounded annual return (geometric extrapolation) | 0.013 | ||||
Calmar ratio (compounded annual return / max draw down) | 0.018 | ||||
Compounded annual return / average of 25% largest draw downs | 0.047 | ||||
Compounded annual return / Expected Shortfall lognormal | 0.062 | ||||
ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back) | |||||
RATIO STATISTICS | |||||
Ratio statistics of excess return rates | |||||
Statistics related to Sharpe ratio | |||||
Mean | 0.020 | ||||
SD | 0.320 | ||||
Sharpe ratio (Glass type estimate) | 0.062 | ||||
Sharpe ratio (Hedges UMVUE) | 0.062 | ||||
df | 1636.000 | ||||
t | 0.156 | ||||
p | 0.498 | ||||
Lowerbound of 95% confidence interval for Sharpe Ratio | -0.722 | ||||
Upperbound of 95% confidence interval for Sharpe Ratio | 0.846 | ||||
Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.722 | ||||
Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.846 | ||||
Statistics related to Sortino ratio | |||||
Sortino ratio | 0.088 | ||||
Upside Potential Ratio | 4.073 | ||||
Upside part of mean | 0.920 | ||||
Downside part of mean | -0.900 | ||||
Upside SD | 0.226 | ||||
Downside SD | 0.226 | ||||
N nonnegative terms | 457.000 | ||||
N negative terms | 1180.000 | ||||
Statistics related to linear regression on benchmark | |||||
N of observations | 1637.000 | ||||
Mean of predictor | 0.327 | ||||
Mean of criterion | 0.020 | ||||
SD of predictor | 0.319 | ||||
SD of criterion | 0.320 | ||||
Covariance | 0.009 | ||||
r | 0.092 | ||||
b (slope, estimate of beta) | 0.092 | ||||
a (intercept, estimate of alpha) | -0.010 | ||||
Mean Square Error | 0.102 | ||||
DF error | 1635.000 | ||||
t(b) | 3.730 | ||||
p(b) | 0.442 | ||||
t(a) | -0.080 | ||||
p(a) | 0.501 | ||||
Lowerbound of 95% confidence interval for beta | 0.044 | ||||
Upperbound of 95% confidence interval for beta | 0.141 | ||||
Lowerbound of 95% confidence interval for alpha | -0.261 | ||||
Upperbound of 95% confidence interval for alpha | 0.240 | ||||
Treynor index (mean / b) | 0.216 | ||||
Jensen alpha (a) | -0.010 | ||||
Ratio statistics of excess log return rates | |||||
Statistics related to Sharpe ratio | |||||
Mean | -0.032 | ||||
SD | 0.322 | ||||
Sharpe ratio (Glass type estimate) | -0.098 | ||||
Sharpe ratio (Hedges UMVUE) | -0.098 | ||||
df | 1636.000 | ||||
t | -0.245 | ||||
p | 0.503 | ||||
Lowerbound of 95% confidence interval for Sharpe Ratio | -0.882 | ||||
Upperbound of 95% confidence interval for Sharpe Ratio | 0.686 | ||||
Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -0.882 | ||||
Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | 0.686 | ||||
Statistics related to Sortino ratio | |||||
Sortino ratio | -0.131 | ||||
Upside Potential Ratio | 3.714 | ||||
Upside part of mean | 0.896 | ||||
Downside part of mean | -0.928 | ||||
Upside SD | 0.214 | ||||
Downside SD | 0.241 | ||||
N nonnegative terms | 457.000 | ||||
N negative terms | 1180.000 | ||||
Statistics related to linear regression on benchmark | |||||
N of observations | 1637.000 | ||||
Mean of predictor | 0.276 | ||||
Mean of criterion | -0.032 | ||||
SD of predictor | 0.318 | ||||
SD of criterion | 0.322 | ||||
Covariance | 0.009 | ||||
r | 0.087 | ||||
b (slope, estimate of beta) | 0.088 | ||||
a (intercept, estimate of alpha) | -0.056 | ||||
Mean Square Error | 0.103 | ||||
DF error | 1635.000 | ||||
t(b) | 3.539 | ||||
p(b) | 0.445 | ||||
t(a) | -0.436 | ||||
p(a) | 0.507 | ||||
Lowerbound of 95% confidence interval for beta | 0.039 | ||||
Upperbound of 95% confidence interval for beta | 0.137 | ||||
Lowerbound of 95% confidence interval for alpha | -0.309 | ||||
Upperbound of 95% confidence interval for alpha | 0.196 | ||||
Treynor index (mean / b) | -0.358 | ||||
Jensen alpha (a) | -0.056 | ||||
Risk estimates for a one-period unit investment (parametric) | |||||
assuming log normal returns and losses (using central moments from Sharpe statistics) | |||||
VaR(95%) | 0.032 | ||||
Expected Shortfall on VaR | 0.040 | ||||
assuming Pareto losses only (using partial moments from Sortino statistics) | |||||
VaR(95%) | 0.010 | ||||
Expected Shortfall on VaR | 0.022 | ||||
ORDER STATISTICS | |||||
Quartiles of return rates | |||||
Number of observations | 1637.000 | ||||
Minimum | 0.802 | ||||
Quartile 1 | 1.000 | ||||
Median | 1.000 | ||||
Quartile 3 | 1.001 | ||||
Maximum | 1.246 | ||||
Mean of quarter 1 | 0.987 | ||||
Mean of quarter 2 | 1.000 | ||||
Mean of quarter 3 | 1.000 | ||||
Mean of quarter 4 | 1.014 | ||||
Inter Quartile Range | 0.001 | ||||
Number outliers low | 286.000 | ||||
Percentage of outliers low | 0.175 | ||||
Mean of outliers low | 0.982 | ||||
Number of outliers high | 312.000 | ||||
Percentage of outliers high | 0.191 | ||||
Mean of outliers high | 1.018 | ||||
Risk estimates for a one-period unit investment (based on Extreme Value Theory) | |||||
Extreme Value Index (moments method) | 1.007 | ||||
VaR(95%) (moments method) | 0.008 | ||||
Expected Shortfall (moments method) | NA | ||||
Extreme Value Index (regression method) | 0.538 | ||||
VaR(95%) (regression method) | 0.009 | ||||
Expected Shortfall (regression method) | 0.027 | ||||
DRAW DOWN STATISTICS | |||||
Quartiles of draw downs | |||||
Number of observations | 63.000 | ||||
Minimum | 0.000 | ||||
Quartile 1 | 0.007 | ||||
Median | 0.019 | ||||
Quartile 3 | 0.045 | ||||
Maximum | 0.703 | ||||
Mean of quarter 1 | 0.003 | ||||
Mean of quarter 2 | 0.014 | ||||
Mean of quarter 3 | 0.030 | ||||
Mean of quarter 4 | 0.129 | ||||
Inter Quartile Range | 0.038 | ||||
Number outliers low | 0.000 | ||||
Percentage of outliers low | 0.000 | ||||
Mean of outliers low | NA | ||||
Number of outliers high | 7.000 | ||||
Percentage of outliers high | 0.111 | ||||
Mean of outliers high | 0.215 | ||||
Risk estimates based on draw downs (based on Extreme Value Theory) | |||||
Extreme Value Index (moments method) | 0.539 | ||||
VaR(95%) (moments method) | 0.132 | ||||
Expected Shortfall (moments method) | 0.312 | ||||
Extreme Value Index (regression method) | 0.459 | ||||
VaR(95%) (regression method) | 0.124 | ||||
Expected Shortfall (regression method) | 0.252 | ||||
COMBINED STATISTICS | |||||
Annualized return (arithmetic extrapolation) | 0.013 | ||||
Compounded annual return (geometric extrapolation) | 0.012 | ||||
Calmar ratio (compounded annual return / max draw down) | 0.018 | ||||
Compounded annual return / average of 25% largest draw downs | 0.096 | ||||
Compounded annual return / Expected Shortfall lognormal | 0.308 | ||||
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 | -16.186 | ||||
Upside Potential Ratio | 0.000 | ||||
Upside part of mean | 0.000 | ||||
Downside part of mean | -0.044 | ||||
Upside SD | 0.000 | ||||
Downside SD | 0.003 | ||||
N nonnegative terms | 0.000 | ||||
N negative terms | 131.000 | ||||
Statistics related to linear regression on benchmark | |||||
N of observations | 131.000 | ||||
Mean of predictor | 0.401 | ||||
Mean of criterion | -0.044 | ||||
SD of predictor | 0.414 | ||||
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) | -12492933348697024.000 | ||||
Sharpe ratio (Hedges UMVUE) | -12420719861132300.000 | ||||
df | 130.000 | ||||
t | -8833837887775228.000 | ||||
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) | -13930479873814932.000 | ||||
Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation) | -10910959848449668.000 | ||||
Statistics related to Sortino ratio | |||||
Sortino ratio | -16.186 | ||||
Upside Potential Ratio | 0.000 | ||||
Upside part of mean | 0.000 | ||||
Downside part of mean | -0.044 | ||||
Upside SD | 0.000 | ||||
Downside SD | 0.003 | ||||
N nonnegative terms | 0.000 | ||||
N negative terms | 131.000 | ||||
Statistics related to linear regression on benchmark | |||||
N of observations | 131.000 | ||||
Mean of predictor | 0.315 | ||||
Mean of criterion | -0.044 | ||||
SD of predictor | 0.416 | ||||
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 | 129.000 | ||||
t(b) | 0.000 | ||||
p(b) | 0.500 | ||||
t(a) | -8790133036550921.000 | ||||
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) | -96750048411422887631716045815808.000 | ||||
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 | 131.000 | ||||
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 |