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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
 df1551.000
 t0.116
 p0.498
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.868
 Upperbound of 95% confidence interval for Sharpe Ratio0.977
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.868
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.977
Statistics related to Sortino ratio
 Sortino ratio0.068
 Upside Potential Ratio4.387
 Upside part of mean0.933
 Downside part of mean-0.918
 Upside SD0.161
 Downside SD0.213
 N nonnegative terms522.000
 N negative terms1030.000
Statistics related to linear regression on benchmark
 N of observations1552.000
 Mean of predictor0.215
 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 error1550.000
 t(b)-0.868
 p(b)0.511
 t(a)0.167
 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)-0.480
 Jensen alpha (a)0.021
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.023
 SD0.276
 Sharpe ratio (Glass type estimate) -0.082
 Sharpe ratio (Hedges UMVUE)-0.082
 df1551.000
 t-0.174
 p0.503
 Lowerbound of 95% confidence interval for Sharpe Ratio-1.004
 Upperbound of 95% confidence interval for Sharpe Ratio0.841
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-1.004
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.841
Statistics related to Sortino ratio
 Sortino ratio-0.099
 Upside Potential Ratio4.047
 Upside part of mean0.920
 Downside part of mean-0.943
 Upside SD0.156
 Downside SD0.227
 N nonnegative terms522.000
 N negative terms1030.000
Statistics related to linear regression on benchmark
 N of observations1552.000
 Mean of predictor0.196
 Mean of criterion-0.023
 SD of predictor0.194
 SD of criterion0.276
 Covariance-0.001
 r-0.024
 b (slope, estimate of beta)-0.034
 a (intercept, estimate of alpha)-0.016
 Mean Square Error0.076
 DF error1550.000
 t(b)-0.953
 p(b)0.512
 t(a)-0.121
 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)0.654
 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 observations1552.000
 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 low147.000
 Percentage of outliers low0.095
 Mean of outliers low0.979
 Number of outliers high173.000
 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 downs0.233
 Compounded annual return / Expected Shortfall lognormal0.718
ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean-1.393
 SD0.574
 Sharpe ratio (Glass type estimate) -2.428
 Sharpe ratio (Hedges UMVUE)-2.418
 df171.000
 t-1.717
 p0.583
 Lowerbound of 95% confidence interval for Sharpe Ratio-5.209
 Upperbound of 95% confidence interval for Sharpe Ratio0.359
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-5.201
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.366
Statistics related to Sortino ratio
 Sortino ratio-2.704
 Upside Potential Ratio1.432
 Upside part of mean0.738
 Downside part of mean-2.131
 Upside SD0.260
 Downside SD0.515
 N nonnegative terms8.000
 N negative terms164.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.544
 Mean of criterion-1.393
 SD of predictor0.187
 SD of criterion0.574
 Covariance0.001
 r0.013
 b (slope, estimate of beta)0.039
 a (intercept, estimate of alpha)-1.415
 Mean Square Error0.331
 DF error170.000
 t(b)0.166
 p(b)0.494
 t(a)-1.717
 p(a)0.565
 Lowerbound of 95% confidence interval for beta-0.426
 Upperbound of 95% confidence interval for beta0.504
 Lowerbound of 95% confidence interval for alpha-3.041
 Upperbound of 95% confidence interval for alpha0.212
 Treynor index (mean / b)-35.630
 Jensen alpha (a)-1.415
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-1.573
 SD0.607
 Sharpe ratio (Glass type estimate) -2.591
 Sharpe ratio (Hedges UMVUE)-2.579
 df171.000
 t-1.832
 p0.588
 Lowerbound of 95% confidence interval for Sharpe Ratio-5.373
 Upperbound of 95% confidence interval for Sharpe Ratio0.198
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-5.365
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.206
Statistics related to Sortino ratio
 Sortino ratio-2.808
 Upside Potential Ratio1.261
 Upside part of mean0.706
 Downside part of mean-2.279
 Upside SD0.245
 Downside SD0.560
 N nonnegative terms8.000
 N negative terms164.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.527
 Mean of criterion-1.573
 SD of predictor0.186
 SD of criterion0.607
 Covariance0.001
 r0.011
 b (slope, estimate of beta)0.035
 a (intercept, estimate of alpha)-1.591
 Mean Square Error0.371
 DF error170.000
 t(b)0.141
 p(b)0.495
 t(a)-1.827
 p(a)0.569
 Lowerbound of 95% confidence interval for beta-0.458
 Upperbound of 95% confidence interval for beta0.528
 Lowerbound of 95% confidence interval for alpha-3.310
 Upperbound of 95% confidence interval for alpha0.128
 Treynor index (mean / b)-44.523
 Jensen alpha (a)-1.591
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 VaR0.069
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.021
 Expected Shortfall on VaR0.046
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 10.976
 Mean of quarter 21.000
 Mean of quarter 31.000
 Mean of quarter 41.009
 Inter Quartile Range0.000
 Number outliers low16.000
 Percentage of outliers low0.093
 Mean of outliers low0.935
 Number of outliers high8.000
 Percentage of outliers high0.047
 Mean of outliers high1.046
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)0.418
 VaR(95%) (moments method)0.003
 Expected Shortfall (moments method)0.010
 Extreme Value Index (regression method)-0.942
 VaR(95%) (regression method)0.038
 Expected Shortfall (regression method)0.061
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations2.000
 Minimum0.150
 Quartile 10.252
 Median0.353
 Quartile 30.454
 Maximum0.556
 Mean of quarter 10.150
 Mean of quarter 2NA
 Mean of quarter 3NA
 Mean of quarter 40.556
 Inter Quartile Range0.203
 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)-1.069
 Compounded annual return (geometric extrapolation)-0.783
 Calmar ratio (compounded annual return / max draw down)-1.409
 Compounded annual return / average of 25% largest draw downs-1.409
 Compounded annual return / Expected Shortfall lognormal-11.273