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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
 Mean0.075
 SD0.378
 Sharpe ratio (Glass type estimate) 0.198
 Sharpe ratio (Hedges UMVUE)0.196
 df52.000
 t0.417
 p0.339
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.736
 Upperbound of 95% confidence interval for Sharpe Ratio1.131
 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 ratio0.236
 Upside Potential Ratio1.168
 Upside part of mean0.371
 Downside part of mean-0.296
 Upside SD0.199
 Downside SD0.318
 N nonnegative terms33.000
 N negative terms20.000
Statistics related to linear regression on benchmark
 N of observations53.000
 Mean of predictor0.213
 Mean of criterion0.075
 SD of predictor0.185
 SD of criterion0.378
 Covariance0.001
 r0.016
 b (slope, estimate of beta)0.033
 a (intercept, estimate of alpha)0.068
 Mean Square Error0.146
 DF error51.000
 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 beta0.607
 Lowerbound of 95% confidence interval for alpha-0.316
 Upperbound of 95% confidence interval for alpha0.452
 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
 SD0.488
 Sharpe ratio (Glass type estimate) -0.045
 Sharpe ratio (Hedges UMVUE)-0.045
 df52.000
 t-0.095
 p0.538
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.978
 Upperbound of 95% confidence interval for Sharpe Ratio0.888
 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 Ratio0.788
 Upside part of mean0.352
 Downside part of mean-0.374
 Upside SD0.185
 Downside SD0.447
 N nonnegative terms33.000
 N negative terms20.000
Statistics related to linear regression on benchmark
 N of observations53.000
 Mean of predictor0.194
 Mean of criterion-0.022
 SD of predictor0.181
 SD of criterion0.488
 Covariance-0.002
 r-0.022
 b (slope, estimate of beta)-0.060
 a (intercept, estimate of alpha)-0.010
 Mean Square Error0.243
 DF error51.000
 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 beta0.698
 Lowerbound of 95% confidence interval for alpha-0.503
 Upperbound of 95% confidence interval for alpha0.483
 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 VaR0.252
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.044
 Expected Shortfall on VaR0.107
ORDER STATISTICS
Quartiles of return rates
 Number of observations53.000
 Minimum0.429
 Quartile 10.992
 Median1.015
 Quartile 31.041
 Maximum1.250
 Mean of quarter 10.912
 Mean of quarter 21.006
 Mean of quarter 31.027
 Mean of quarter 41.102
 Inter Quartile Range0.049
 Number outliers low2.000
 Percentage of outliers low0.038
 Mean of outliers low0.561
 Number of outliers high5.000
 Percentage of outliers high0.094
 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)0.066
 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 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)0.032
 Compounded annual return / average of 25% largest draw downs0.080
 Compounded annual return / Expected Shortfall lognormal0.088
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
 df1547.000
 t0.117
 p0.498
 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 Ratio4.392
 Upside part of mean0.935
 Downside part of mean-0.920
 Upside SD0.161
 Downside SD0.213
 N nonnegative terms522.000
 N negative terms1026.000
Statistics related to linear regression on benchmark
 N of observations1548.000
 Mean of predictor0.206
 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 error1546.000
 t(b)-0.867
 p(b)0.511
 t(a)0.166
 p(a)0.498
 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)-0.484
 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) -0.081
 Sharpe ratio (Hedges UMVUE)-0.081
 df1547.000
 t-0.173
 p0.503
 Lowerbound of 95% confidence interval for Sharpe Ratio-1.005
 Upperbound of 95% confidence interval for Sharpe Ratio0.843
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-1.005
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.843
Statistics related to Sortino ratio
 Sortino ratio-0.099
 Upside Potential Ratio4.052
 Upside part of mean0.923
 Downside part of mean-0.945
 Upside SD0.156
 Downside SD0.228
 N nonnegative terms522.000
 N negative terms1026.000
Statistics related to linear regression on benchmark
 N of observations1548.000
 Mean of predictor0.188
 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 error1546.000
 t(b)-0.953
 p(b)0.512
 t(a)-0.123
 p(a)0.502
 Lowerbound of 95% confidence interval for beta-0.106
 Upperbound of 95% confidence interval for beta0.037
 Lowerbound of 95% confidence interval for alpha-0.272
 Upperbound of 95% confidence interval for alpha0.240
 Treynor index (mean / b)0.651
 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 observations1548.000
 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 low144.000
 Percentage of outliers low0.093
 Mean of outliers low0.978
 Number of outliers high170.000
 Percentage of outliers high0.110
 Mean of outliers high1.019
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)0.848
 VaR(95%) (moments method)0.008
 Expected Shortfall (moments method)0.057
 Extreme Value Index (regression method)0.577
 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-1.516
 SD0.577
 Sharpe ratio (Glass type estimate) -2.627
 Sharpe ratio (Hedges UMVUE)-2.616
 df171.000
 t-1.858
 p0.589
 Lowerbound of 95% confidence interval for Sharpe Ratio-5.409
 Upperbound of 95% confidence interval for Sharpe Ratio0.162
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-5.401
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.170
Statistics related to Sortino ratio
 Sortino ratio-2.916
 Upside Potential Ratio1.424
 Upside part of mean0.740
 Downside part of mean-2.256
 Upside SD0.260
 Downside SD0.520
 N nonnegative terms9.000
 N negative terms163.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.473
 Mean of criterion-1.516
 SD of predictor0.184
 SD of criterion0.577
 Covariance0.001
 r0.010
 b (slope, estimate of beta)0.031
 a (intercept, estimate of alpha)-1.531
 Mean Square Error0.335
 DF error170.000
 t(b)0.129
 p(b)0.495
 t(a)-1.853
 p(a)0.570
 Lowerbound of 95% confidence interval for beta-0.443
 Upperbound of 95% confidence interval for beta0.505
 Lowerbound of 95% confidence interval for alpha-3.162
 Upperbound of 95% confidence interval for alpha0.100
 Treynor index (mean / b)-48.766
 Jensen alpha (a)-1.531
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-1.698
 SD0.610
 Sharpe ratio (Glass type estimate) -2.782
 Sharpe ratio (Hedges UMVUE)-2.770
 df171.000
 t-1.967
 p0.594
 Lowerbound of 95% confidence interval for Sharpe Ratio-5.566
 Upperbound of 95% confidence interval for Sharpe Ratio0.009
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-5.558
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.017
Statistics related to Sortino ratio
 Sortino ratio-3.007
 Upside Potential Ratio1.256
 Upside part of mean0.709
 Downside part of mean-2.407
 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 predictor0.455
 Mean of criterion-1.698
 SD of predictor0.184
 SD of criterion0.610
 Covariance0.001
 r0.008
 b (slope, estimate of beta)0.026
 a (intercept, estimate of alpha)-1.709
 Mean Square Error0.374
 DF error170.000
 t(b)0.102
 p(b)0.496
 t(a)-1.958
 p(a)0.574
 Lowerbound of 95% confidence interval for beta-0.476
 Upperbound of 95% confidence interval for beta0.528
 Lowerbound of 95% confidence interval for alpha-3.433
 Upperbound of 95% confidence interval for alpha0.014
 Treynor index (mean / b)-65.269
 Jensen alpha (a)-1.709
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 10.974
 Mean of quarter 21.000
 Mean of quarter 31.000
 Mean of quarter 41.009
 Inter Quartile Range0.000
 Number outliers low18.000
 Percentage of outliers low0.105
 Mean of outliers low0.938
 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)0.033
 VaR(95%) (moments method)0.003
 Expected Shortfall (moments method)0.006
 Extreme Value Index (regression method)-0.796
 VaR(95%) (regression method)0.037
 Expected Shortfall (regression method)0.058
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations1.000
 Minimum0.571
 Quartile 10.571
 Median0.571
 Quartile 30.571
 Maximum0.571
 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)-1.125
 Compounded annual return (geometric extrapolation)-0.809
 Calmar ratio (compounded annual return / max draw down)-1.415
 Compounded annual return / average of 25% largest draw downsNA
 Compounded annual return / Expected Shortfall lognormal-11.530