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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
 SD0.268
 Sharpe ratio (Glass type estimate) 0.057
 Sharpe ratio (Hedges UMVUE)0.057
 df1532.000
 t0.120
 p0.498
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.872
 Upperbound of 95% confidence interval for Sharpe Ratio0.985
 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 ratio0.071
 Upside Potential Ratio4.414
 Upside part of mean0.944
 Downside part of mean-0.929
 Upside SD0.162
 Downside SD0.214
 N nonnegative terms522.000
 N negative terms1011.000
Statistics related to linear regression on benchmark
 N of observations1533.000
 Mean of predictor0.217
 Mean of criterion0.015
 SD of predictor0.194
 SD of criterion0.268
 Covariance-0.001
 r-0.022
 b (slope, estimate of beta)-0.031
 a (intercept, estimate of alpha)0.022
 Mean Square Error0.072
 DF error1531.000
 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 beta0.039
 Lowerbound of 95% confidence interval for alpha-0.228
 Upperbound of 95% confidence interval for alpha0.272
 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
 SD0.277
 Sharpe ratio (Glass type estimate) -0.080
 Sharpe ratio (Hedges UMVUE)-0.080
 df1532.000
 t-0.169
 p0.502
 Lowerbound of 95% confidence interval for Sharpe Ratio-1.009
 Upperbound of 95% confidence interval for Sharpe Ratio0.848
 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 Ratio4.072
 Upside part of mean0.932
 Downside part of mean-0.954
 Upside SD0.157
 Downside SD0.229
 N nonnegative terms522.000
 N negative terms1011.000
Statistics related to linear regression on benchmark
 N of observations1533.000
 Mean of predictor0.198
 Mean of criterion-0.022
 SD of predictor0.193
 SD of criterion0.277
 Covariance-0.001
 r-0.024
 b (slope, estimate of beta)-0.035
 a (intercept, estimate of alpha)-0.015
 Mean Square Error0.077
 DF error1531.000
 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 beta0.037
 Lowerbound of 95% confidence interval for alpha-0.273
 Upperbound of 95% confidence interval for alpha0.243
 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 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 observations1533.000
 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 low138.000
 Percentage of outliers low0.090
 Mean of outliers low0.978
 Number of outliers high166.000
 Percentage of outliers high0.108
 Mean of outliers high1.019
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 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 downs0.236
 Compounded annual return / Expected Shortfall lognormal0.723
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
 SD0.642
 Sharpe ratio (Glass type estimate) -2.953
 Sharpe ratio (Hedges UMVUE)-2.940
 df171.000
 t-2.088
 p0.600
 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 Ratio1.911
 Upside part of mean1.110
 Downside part of mean-3.007
 Upside SD0.288
 Downside SD0.581
 N nonnegative terms14.000
 N negative terms158.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.596
 Mean of criterion-1.897
 SD of predictor0.174
 SD of criterion0.642
 Covariance-0.004
 r-0.034
 b (slope, estimate of beta)-0.124
 a (intercept, estimate of alpha)-1.823
 Mean Square Error0.414
 DF error170.000
 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 beta0.435
 Lowerbound of 95% confidence interval for alpha-3.650
 Upperbound of 95% confidence interval for alpha0.005
 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
 SD0.677
 Sharpe ratio (Glass type estimate) -3.134
 Sharpe ratio (Hedges UMVUE)-3.120
 df171.000
 t-2.216
 p0.606
 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 Ratio1.706
 Upside part of mean1.071
 Downside part of mean-3.193
 Upside SD0.273
 Downside SD0.628
 N nonnegative terms14.000
 N negative terms158.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.580
 Mean of criterion-2.122
 SD of predictor0.173
 SD of criterion0.677
 Covariance-0.004
 r-0.035
 b (slope, estimate of beta)-0.137
 a (intercept, estimate of alpha)-2.043
 Mean Square Error0.461
 DF error170.000
 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 beta0.454
 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 VaR0.078
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.029
 Expected Shortfall on VaR0.062
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 10.966
 Mean of quarter 21.000
 Mean of quarter 31.000
 Mean of quarter 41.013
 Inter Quartile Range0.000
 Number outliers low26.000
 Percentage of outliers low0.151
 Mean of outliers low0.943
 Number of outliers high14.000
 Percentage of outliers high0.081
 Mean of outliers high1.040
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 observations2.000
 Minimum0.001
 Quartile 10.166
 Median0.331
 Quartile 30.496
 Maximum0.662
 Mean of quarter 10.001
 Mean of quarter 2NA
 Mean of quarter 3NA
 Mean of quarter 40.662
 Inter Quartile Range0.330
 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)-1.292
 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