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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
 Mean0.014
 SD0.265
 Sharpe ratio (Glass type estimate) 0.052
 Sharpe ratio (Hedges UMVUE)0.052
 df1574.000
 t0.111
 p0.499
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.864
 Upperbound of 95% confidence interval for Sharpe Ratio0.968
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.864
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.968
Statistics related to Sortino ratio
 Sortino ratio0.065
 Upside Potential Ratio4.354
 Upside part of mean0.919
 Downside part of mean-0.905
 Upside SD0.159
 Downside SD0.211
 N nonnegative terms522.000
 N negative terms1053.000
Statistics related to linear regression on benchmark
 N of observations1575.000
 Mean of predictor0.217
 Mean of criterion0.014
 SD of predictor0.194
 SD of criterion0.265
 Covariance-0.001
 r-0.022
 b (slope, estimate of beta)-0.030
 a (intercept, estimate of alpha)0.020
 Mean Square Error0.070
 DF error1573.000
 t(b)-0.871
 p(b)0.514
 t(a)0.163
 p(a)0.497
 Lowerbound of 95% confidence interval for beta-0.098
 Upperbound of 95% confidence interval for beta0.038
 Lowerbound of 95% confidence interval for alpha-0.223
 Upperbound of 95% confidence interval for alpha0.263
 Treynor index (mean / b)-0.456
 Jensen alpha (a)0.020
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.023
 SD0.274
 Sharpe ratio (Glass type estimate) -0.083
 Sharpe ratio (Hedges UMVUE)-0.083
 df1574.000
 t-0.179
 p0.502
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.999
 Upperbound of 95% confidence interval for Sharpe Ratio0.832
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.999
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.833
Statistics related to Sortino ratio
 Sortino ratio-0.101
 Upside Potential Ratio4.017
 Upside part of mean0.907
 Downside part of mean-0.930
 Upside SD0.154
 Downside SD0.226
 N nonnegative terms522.000
 N negative terms1053.000
Statistics related to linear regression on benchmark
 N of observations1575.000
 Mean of predictor0.198
 Mean of criterion-0.023
 SD of predictor0.193
 SD of criterion0.274
 Covariance-0.001
 r-0.024
 b (slope, estimate of beta)-0.034
 a (intercept, estimate of alpha)-0.016
 Mean Square Error0.075
 DF error1573.000
 t(b)-0.956
 p(b)0.515
 t(a)-0.126
 p(a)0.502
 Lowerbound of 95% confidence interval for beta-0.104
 Upperbound of 95% confidence interval for beta0.036
 Lowerbound of 95% confidence interval for alpha-0.267
 Upperbound of 95% confidence interval for alpha0.235
 Treynor index (mean / b)0.670
 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 observations1575.000
 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 low149.000
 Percentage of outliers low0.095
 Mean of outliers low0.979
 Number of outliers high182.000
 Percentage of outliers high0.116
 Mean of outliers high1.018
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)0.855
 VaR(95%) (moments method)0.008
 Expected Shortfall (moments method)0.058
 Extreme Value Index (regression method)0.579
 VaR(95%) (regression method)0.007
 Expected Shortfall (regression method)0.018
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)0.022
 Compounded annual return (geometric extrapolation)0.021
 Calmar ratio (compounded annual return / max draw down)0.030
 Compounded annual return / average of 25% largest draw downs0.230
 Compounded annual return / Expected Shortfall lognormal0.713
ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean-0.171
 SD0.091
 Sharpe ratio (Glass type estimate) -1.877
 Sharpe ratio (Hedges UMVUE)-1.869
 df171.000
 t-1.327
 p0.564
 Lowerbound of 95% confidence interval for Sharpe Ratio-4.653
 Upperbound of 95% confidence interval for Sharpe Ratio0.905
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-4.647
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.910
Statistics related to Sortino ratio
 Sortino ratio-1.978
 Upside Potential Ratio0.480
 Upside part of mean0.041
 Downside part of mean-0.212
 Upside SD0.029
 Downside SD0.086
 N nonnegative terms1.000
 N negative terms171.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.548
 Mean of criterion-0.171
 SD of predictor0.193
 SD of criterion0.091
 Covariance-0.000
 r-0.013
 b (slope, estimate of beta)-0.006
 a (intercept, estimate of alpha)-0.167
 Mean Square Error0.008
 DF error170.000
 t(b)-0.174
 p(b)0.507
 t(a)-1.282
 p(a)0.549
 Lowerbound of 95% confidence interval for beta-0.078
 Upperbound of 95% confidence interval for beta0.065
 Lowerbound of 95% confidence interval for alpha-0.425
 Upperbound of 95% confidence interval for alpha0.090
 Treynor index (mean / b)27.183
 Jensen alpha (a)-0.167
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.175
 SD0.093
 Sharpe ratio (Glass type estimate) -1.886
 Sharpe ratio (Hedges UMVUE)-1.877
 df171.000
 t-1.333
 p0.564
 Lowerbound of 95% confidence interval for Sharpe Ratio-4.662
 Upperbound of 95% confidence interval for Sharpe Ratio0.896
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-4.656
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.901
Statistics related to Sortino ratio
 Sortino ratio-1.980
 Upside Potential Ratio0.464
 Upside part of mean0.041
 Downside part of mean-0.216
 Upside SD0.029
 Downside SD0.088
 N nonnegative terms1.000
 N negative terms171.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.529
 Mean of criterion-0.175
 SD of predictor0.193
 SD of criterion0.093
 Covariance-0.000
 r-0.013
 b (slope, estimate of beta)-0.006
 a (intercept, estimate of alpha)-0.172
 Mean Square Error0.009
 DF error170.000
 t(b)-0.171
 p(b)0.507
 t(a)-1.290
 p(a)0.549
 Lowerbound of 95% confidence interval for beta-0.079
 Upperbound of 95% confidence interval for beta0.067
 Lowerbound of 95% confidence interval for alpha-0.434
 Upperbound of 95% confidence interval for alpha0.091
 Treynor index (mean / b)27.805
 Jensen alpha (a)-0.172
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.009
 Expected Shortfall on VaR0.011
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.002
 Expected Shortfall on VaR0.005
ORDER STATISTICS
Quartiles of return rates
 Number of observations172.000
 Minimum0.949
 Quartile 11.000
 Median1.000
 Quartile 31.000
 Maximum1.021
 Mean of quarter 10.998
 Mean of quarter 21.000
 Mean of quarter 31.000
 Mean of quarter 41.000
 Inter Quartile Range0.000
 Number outliers low2.000
 Percentage of outliers low0.012
 Mean of outliers low0.958
 Number of outliers high1.000
 Percentage of outliers high0.006
 Mean of outliers high1.021
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)-1.904
 VaR(95%) (regression method)-0.492
 Expected Shortfall (regression method)NA
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations1.000
 Minimum0.083
 Quartile 10.083
 Median0.083
 Quartile 30.083
 Maximum0.083
 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)-0.127
 Compounded annual return (geometric extrapolation)-0.123
 Calmar ratio (compounded annual return / max draw down)-1.488
 Compounded annual return / average of 25% largest draw downsNA
 Compounded annual return / Expected Shortfall lognormal-11.400