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Advanced Statistics: Buzo Principle

Advanced Statistics module contributed by Jules Ellis, C2 Member.

Jules has written a very helpful guide for these statistics.

MonthlyDaily (all)Daily (last 6 mo.)
ANALYSIS BASED ON MONTHLY VALUES, FULL HISTORY (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean0.073
 SD0.374
 Sharpe ratio (Glass type estimate) 0.194
 Sharpe ratio (Hedges UMVUE)0.192
 df53.000
 t0.412
 p0.341
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.731
 Upperbound of 95% confidence interval for Sharpe Ratio1.118
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.733
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)1.116
Statistics related to Sortino ratio
 Sortino ratio0.231
 Upside Potential Ratio1.157
 Upside part of mean0.364
 Downside part of mean-0.291
 Upside SD0.197
 Downside SD0.315
 N nonnegative terms33.000
 N negative terms21.000
Statistics related to linear regression on benchmark
 N of observations54.000
 Mean of predictor0.206
 Mean of criterion0.073
 SD of predictor0.184
 SD of criterion0.374
 Covariance0.001
 r0.017
 b (slope, estimate of beta)0.034
 a (intercept, estimate of alpha)0.066
 Mean Square Error0.143
 DF error52.000
 t(b)0.121
 p(b)0.452
 t(a)0.350
 p(a)0.364
 Lowerbound of 95% confidence interval for beta-0.533
 Upperbound of 95% confidence interval for beta0.601
 Lowerbound of 95% confidence interval for alpha-0.310
 Upperbound of 95% confidence interval for alpha0.442
 Treynor index (mean / b)2.120
 Jensen alpha (a)0.066
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.022
 SD0.483
 Sharpe ratio (Glass type estimate) -0.047
 Sharpe ratio (Hedges UMVUE)-0.046
 df53.000
 t-0.099
 p0.539
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.970
 Upperbound of 95% confidence interval for Sharpe Ratio0.878
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.970
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.878
Statistics related to Sortino ratio
 Sortino ratio-0.051
 Upside Potential Ratio0.780
 Upside part of mean0.345
 Downside part of mean-0.368
 Upside SD0.183
 Downside SD0.442
 N nonnegative terms33.000
 N negative terms21.000
Statistics related to linear regression on benchmark
 N of observations54.000
 Mean of predictor0.188
 Mean of criterion-0.022
 SD of predictor0.180
 SD of criterion0.483
 Covariance-0.002
 r-0.022
 b (slope, estimate of beta)-0.059
 a (intercept, estimate of alpha)-0.011
 Mean Square Error0.238
 DF error52.000
 t(b)-0.159
 p(b)0.563
 t(a)-0.047
 p(a)0.519
 Lowerbound of 95% confidence interval for beta-0.808
 Upperbound of 95% confidence interval for beta0.689
 Lowerbound of 95% confidence interval for alpha-0.494
 Upperbound of 95% confidence interval for alpha0.471
 Treynor index (mean / b)0.378
 Jensen alpha (a)-0.011
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.207
 Expected Shortfall on VaR0.250
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.045
 Expected Shortfall on VaR0.107
ORDER STATISTICS
Quartiles of return rates
 Number of observations54.000
 Minimum0.429
 Quartile 10.993
 Median1.014
 Quartile 31.040
 Maximum1.250
 Mean of quarter 10.912
 Mean of quarter 21.005
 Mean of quarter 31.025
 Mean of quarter 41.098
 Inter Quartile Range0.047
 Number outliers low2.000
 Percentage of outliers low0.037
 Mean of outliers low0.561
 Number of outliers high5.000
 Percentage of outliers high0.093
 Mean of outliers high1.161
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)1.149
 VaR(95%) (moments method)0.064
 Expected Shortfall (moments method)NA
 Extreme Value Index (regression method)1.060
 VaR(95%) (regression method)0.059
 Expected Shortfall (regression method)NA
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations11.000
 Minimum0.006
 Quartile 10.018
 Median0.040
 Quartile 30.044
 Maximum0.703
 Mean of quarter 10.010
 Mean of quarter 20.030
 Mean of quarter 30.042
 Mean of quarter 40.276
 Inter Quartile Range0.026
 Number outliers low0.000
 Percentage of outliers low0.000
 Mean of outliers lowNA
 Number of outliers high1.000
 Percentage of outliers high0.091
 Mean of outliers high0.703
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)0.031
 Compounded annual return / average of 25% largest draw downs0.079
 Compounded annual return / Expected Shortfall lognormal0.087
ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean0.015
 SD0.267
 Sharpe ratio (Glass type estimate) 0.055
 Sharpe ratio (Hedges UMVUE)0.055
 df1548.000
 t0.117
 p0.499
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.869
 Upperbound of 95% confidence interval for Sharpe Ratio0.979
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.869
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.979
Statistics related to Sortino ratio
 Sortino ratio0.069
 Upside Potential Ratio4.391
 Upside part of mean0.934
 Downside part of mean-0.920
 Upside SD0.161
 Downside SD0.213
 N nonnegative terms522.000
 N negative terms1027.000
Statistics related to linear regression on benchmark
 N of observations1549.000
 Mean of predictor0.208
 Mean of criterion0.015
 SD of predictor0.194
 SD of criterion0.267
 Covariance-0.001
 r-0.022
 b (slope, estimate of beta)-0.030
 a (intercept, estimate of alpha)0.021
 Mean Square Error0.071
 DF error1547.000
 t(b)-0.867
 p(b)0.514
 t(a)0.166
 p(a)0.497
 Lowerbound of 95% confidence interval for beta-0.099
 Upperbound of 95% confidence interval for beta0.038
 Lowerbound of 95% confidence interval for alpha-0.226
 Upperbound of 95% confidence interval for alpha0.268
 Treynor index (mean / b)-0.483
 Jensen alpha (a)0.021
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.022
 SD0.276
 Sharpe ratio (Glass type estimate) -0.082
 Sharpe ratio (Hedges UMVUE)-0.081
 df1548.000
 t-0.173
 p0.502
 Lowerbound of 95% confidence interval for Sharpe Ratio-1.005
 Upperbound of 95% confidence interval for Sharpe Ratio0.842
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-1.005
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.842
Statistics related to Sortino ratio
 Sortino ratio-0.099
 Upside Potential Ratio4.051
 Upside part of mean0.922
 Downside part of mean-0.944
 Upside SD0.156
 Downside SD0.228
 N nonnegative terms522.000
 N negative terms1027.000
Statistics related to linear regression on benchmark
 N of observations1549.000
 Mean of predictor0.189
 Mean of criterion-0.022
 SD of predictor0.194
 SD of criterion0.276
 Covariance-0.001
 r-0.024
 b (slope, estimate of beta)-0.035
 a (intercept, estimate of alpha)-0.016
 Mean Square Error0.076
 DF error1547.000
 t(b)-0.953
 p(b)0.515
 t(a)-0.123
 p(a)0.502
 Lowerbound of 95% confidence interval for beta-0.105
 Upperbound of 95% confidence interval for beta0.036
 Lowerbound of 95% confidence interval for alpha-0.271
 Upperbound of 95% confidence interval for alpha0.239
 Treynor index (mean / b)0.652
 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%)0.024
 Expected Shortfall on VaR0.030
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.007
 Expected Shortfall on VaR0.016
ORDER STATISTICS
Quartiles of return rates
 Number of observations1549.000
 Minimum0.810
 Quartile 10.999
 Median1.000
 Quartile 31.002
 Maximum1.158
 Mean of quarter 10.990
 Mean of quarter 21.000
 Mean of quarter 31.000
 Mean of quarter 41.011
 Inter Quartile Range0.003
 Number outliers low145.000
 Percentage of outliers low0.094
 Mean of outliers low0.978
 Number of outliers high172.000
 Percentage of outliers high0.111
 Mean of outliers high1.019
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)0.850
 VaR(95%) (moments method)0.008
 Expected Shortfall (moments method)0.057
 Extreme Value Index (regression method)0.575
 VaR(95%) (regression method)0.007
 Expected Shortfall (regression method)0.019
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations71.000
 Minimum0.000
 Quartile 10.003
 Median0.008
 Quartile 30.023
 Maximum0.703
 Mean of quarter 10.001
 Mean of quarter 20.005
 Mean of quarter 30.016
 Mean of quarter 40.093
 Inter Quartile Range0.020
 Number outliers low0.000
 Percentage of outliers low0.000
 Mean of outliers lowNA
 Number of outliers high11.000
 Percentage of outliers high0.155
 Mean of outliers high0.134
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 downs0.234
 Compounded annual return / Expected Shortfall lognormal0.719
ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean-1.481
 SD0.577
 Sharpe ratio (Glass type estimate) -2.569
 Sharpe ratio (Hedges UMVUE)-2.557
 df171.000
 t-1.816
 p0.587
 Lowerbound of 95% confidence interval for Sharpe Ratio-5.350
 Upperbound of 95% confidence interval for Sharpe Ratio0.220
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-5.342
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.228
Statistics related to Sortino ratio
 Sortino ratio-2.853
 Upside Potential Ratio1.426
 Upside part of mean0.740
 Downside part of mean-2.222
 Upside SD0.260
 Downside SD0.519
 N nonnegative terms9.000
 N negative terms163.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.475
 Mean of criterion-1.481
 SD of predictor0.185
 SD of criterion0.577
 Covariance0.001
 r0.011
 b (slope, estimate of beta)0.036
 a (intercept, estimate of alpha)-1.498
 Mean Square Error0.335
 DF error170.000
 t(b)0.149
 p(b)0.494
 t(a)-1.814
 p(a)0.569
 Lowerbound of 95% confidence interval for beta-0.437
 Upperbound of 95% confidence interval for beta0.508
 Lowerbound of 95% confidence interval for alpha-3.129
 Upperbound of 95% confidence interval for alpha0.132
 Treynor index (mean / b)-41.437
 Jensen alpha (a)-1.498
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-1.663
 SD0.610
 Sharpe ratio (Glass type estimate) -2.726
 Sharpe ratio (Hedges UMVUE)-2.714
 df171.000
 t-1.928
 p0.593
 Lowerbound of 95% confidence interval for Sharpe Ratio-5.509
 Upperbound of 95% confidence interval for Sharpe Ratio0.064
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-5.501
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.072
Statistics related to Sortino ratio
 Sortino ratio-2.949
 Upside Potential Ratio1.257
 Upside part of mean0.709
 Downside part of mean-2.372
 Upside SD0.245
 Downside SD0.564
 N nonnegative terms9.000
 N negative terms163.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.458
 Mean of criterion-1.663
 SD of predictor0.184
 SD of criterion0.610
 Covariance0.001
 r0.009
 b (slope, estimate of beta)0.031
 a (intercept, estimate of alpha)-1.677
 Mean Square Error0.374
 DF error170.000
 t(b)0.122
 p(b)0.495
 t(a)-1.921
 p(a)0.573
 Lowerbound of 95% confidence interval for beta-0.470
 Upperbound of 95% confidence interval for beta0.531
 Lowerbound of 95% confidence interval for alpha-3.400
 Upperbound of 95% confidence interval for alpha0.046
 Treynor index (mean / b)-53.940
 Jensen alpha (a)-1.677
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.057
 Expected Shortfall on VaR0.070
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.022
 Expected Shortfall on VaR0.048
ORDER STATISTICS
Quartiles of return rates
 Number of observations172.000
 Minimum0.810
 Quartile 11.000
 Median1.000
 Quartile 31.000
 Maximum1.158
 Mean of quarter 10.975
 Mean of quarter 21.000
 Mean of quarter 31.000
 Mean of quarter 41.009
 Inter Quartile Range0.000
 Number outliers low17.000
 Percentage of outliers low0.099
 Mean of outliers low0.936
 Number of outliers high9.000
 Percentage of outliers high0.052
 Mean of outliers high1.041
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)0.150
 VaR(95%) (moments method)0.003
 Expected Shortfall (moments method)0.007
 Extreme Value Index (regression method)-0.904
 VaR(95%) (regression method)0.039
 Expected Shortfall (regression method)0.061
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations1.000
 Minimum0.564
 Quartile 10.564
 Median0.564
 Quartile 30.564
 Maximum0.564
 Mean of quarter 1NA
 Mean of quarter 2NA
 Mean of quarter 3NA
 Mean of quarter 4NA
 Inter Quartile Range0.000
 Number outliers low0.000
 Percentage of outliers low0.000
 Mean of outliers lowNA
 Number of outliers high0.000
 Percentage of outliers high0.000
 Mean of outliers highNA
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)-1.110
 Compounded annual return (geometric extrapolation)-0.802
 Calmar ratio (compounded annual return / max draw down)-1.422
 Compounded annual return / average of 25% largest draw downsNA
 Compounded annual return / Expected Shortfall lognormal-11.453