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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
 Mean0.055
 SD0.359
 Sharpe ratio (Glass type estimate) 0.154
 Sharpe ratio (Hedges UMVUE)0.151
 df50.000
 t0.317
 p0.376
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.798
 Upperbound of 95% confidence interval for Sharpe Ratio1.104
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.800
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)1.102
Statistics related to Sortino ratio
 Sortino ratio0.189
 Upside Potential Ratio1.304
 Upside part of mean0.382
 Downside part of mean-0.326
 Upside SD0.203
 Downside SD0.293
 N nonnegative terms32.000
 N negative terms19.000
Statistics related to linear regression on benchmark
 N of observations51.000
 Mean of predictor0.209
 Mean of criterion0.055
 SD of predictor0.189
 SD of criterion0.359
 Covariance-0.002
 r-0.023
 b (slope, estimate of beta)-0.045
 a (intercept, estimate of alpha)0.065
 Mean Square Error0.132
 DF error49.000
 t(b)-0.165
 p(b)0.565
 t(a)0.349
 p(a)0.364
 Lowerbound of 95% confidence interval for beta-0.591
 Upperbound of 95% confidence interval for beta0.502
 Lowerbound of 95% confidence interval for alpha-0.307
 Upperbound of 95% confidence interval for alpha0.436
 Treynor index (mean / b)-1.234
 Jensen alpha (a)0.065
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.021
 SD0.417
 Sharpe ratio (Glass type estimate) -0.051
 Sharpe ratio (Hedges UMVUE)-0.050
 df50.000
 t-0.105
 p0.542
 Lowerbound of 95% confidence interval for Sharpe Ratio-1.001
 Upperbound of 95% confidence interval for Sharpe Ratio0.900
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-1.001
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.901
Statistics related to Sortino ratio
 Sortino ratio-0.058
 Upside Potential Ratio0.985
 Upside part of mean0.362
 Downside part of mean-0.383
 Upside SD0.188
 Downside SD0.367
 N nonnegative terms32.000
 N negative terms19.000
Statistics related to linear regression on benchmark
 N of observations51.000
 Mean of predictor0.190
 Mean of criterion-0.021
 SD of predictor0.185
 SD of criterion0.417
 Covariance-0.004
 r-0.058
 b (slope, estimate of beta)-0.130
 a (intercept, estimate of alpha)0.003
 Mean Square Error0.176
 DF error49.000
 t(b)-0.403
 p(b)0.656
 t(a)0.016
 p(a)0.494
 Lowerbound of 95% confidence interval for beta-0.777
 Upperbound of 95% confidence interval for beta0.517
 Lowerbound of 95% confidence interval for alpha-0.424
 Upperbound of 95% confidence interval for alpha0.431
 Treynor index (mean / b)0.163
 Jensen alpha (a)0.003
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.181
 Expected Shortfall on VaR0.220
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.049
 Expected Shortfall on VaR0.115
ORDER STATISTICS
Quartiles of return rates
 Number of observations51.000
 Minimum0.550
 Quartile 10.989
 Median1.015
 Quartile 31.044
 Maximum1.250
 Mean of quarter 10.900
 Mean of quarter 21.005
 Mean of quarter 31.027
 Mean of quarter 41.102
 Inter Quartile Range0.055
 Number outliers low3.000
 Percentage of outliers low0.059
 Mean of outliers low0.679
 Number of outliers high5.000
 Percentage of outliers high0.098
 Mean of outliers high1.161
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)1.052
 VaR(95%) (moments method)0.086
 Expected Shortfall (moments method)NA
 Extreme Value Index (regression method)1.129
 VaR(95%) (regression method)0.077
 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
 Maximum0.697
 Mean of quarter 10.010
 Mean of quarter 20.030
 Mean of quarter 30.042
 Mean of quarter 40.274
 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 high0.697
Risk estimates based on draw downs (based on Extreme Value Theory)
 Extreme Value Index (moments method)1.023
 VaR(95%) (moments method)0.246
 Expected Shortfall (moments method)NA
 Extreme Value Index (regression method)3.269
 VaR(95%) (regression method)1.046
 Expected Shortfall (regression method)NA
COMBINED STATISTICS
 Annualized return (arithmetic extrapolation)0.024
 Compounded annual return (geometric extrapolation)0.023
 Calmar ratio (compounded annual return / max draw down)0.033
 Compounded annual return / average of 25% largest draw downs0.084
 Compounded annual return / Expected Shortfall lognormal0.105
ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean0.028
 SD0.312
 Sharpe ratio (Glass type estimate) 0.088
 Sharpe ratio (Hedges UMVUE)0.088
 df1479.000
 t0.183
 p0.497
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.857
 Upperbound of 95% confidence interval for Sharpe Ratio1.033
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.857
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)1.033
Statistics related to Sortino ratio
 Sortino ratio0.122
 Upside Potential Ratio4.590
 Upside part of mean1.040
 Downside part of mean-1.013
 Upside SD0.215
 Downside SD0.227
 N nonnegative terms523.000
 N negative terms957.000
Statistics related to linear regression on benchmark
 N of observations1480.000
 Mean of predictor0.202
 Mean of criterion0.028
 SD of predictor0.196
 SD of criterion0.312
 Covariance-0.002
 r-0.029
 b (slope, estimate of beta)-0.046
 a (intercept, estimate of alpha)0.037
 Mean Square Error0.098
 DF error1478.000
 t(b)-1.109
 p(b)0.514
 t(a)0.244
 p(a)0.497
 Lowerbound of 95% confidence interval for beta-0.127
 Upperbound of 95% confidence interval for beta0.035
 Lowerbound of 95% confidence interval for alpha-0.259
 Upperbound of 95% confidence interval for alpha0.333
 Treynor index (mean / b)-0.600
 Jensen alpha (a)0.037
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.021
 SD0.314
 Sharpe ratio (Glass type estimate) -0.068
 Sharpe ratio (Hedges UMVUE)-0.068
 df1479.000
 t-0.142
 p0.502
 Lowerbound of 95% confidence interval for Sharpe Ratio-1.013
 Upperbound of 95% confidence interval for Sharpe Ratio0.877
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-1.013
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.877
Statistics related to Sortino ratio
 Sortino ratio-0.089
 Upside Potential Ratio4.211
 Upside part of mean1.019
 Downside part of mean-1.041
 Upside SD0.201
 Downside SD0.242
 N nonnegative terms523.000
 N negative terms957.000
Statistics related to linear regression on benchmark
 N of observations1480.000
 Mean of predictor0.183
 Mean of criterion-0.021
 SD of predictor0.196
 SD of criterion0.314
 Covariance-0.002
 r-0.032
 b (slope, estimate of beta)-0.052
 a (intercept, estimate of alpha)-0.012
 Mean Square Error0.099
 DF error1478.000
 t(b)-1.236
 p(b)0.516
 t(a)-0.079
 p(a)0.501
 Lowerbound of 95% confidence interval for beta-0.134
 Upperbound of 95% confidence interval for beta0.030
 Lowerbound of 95% confidence interval for alpha-0.310
 Upperbound of 95% confidence interval for alpha0.286
 Treynor index (mean / b)0.416
 Jensen alpha (a)-0.012
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.028
 Expected Shortfall on VaR0.034
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.008
 Expected Shortfall on VaR0.018
ORDER STATISTICS
Quartiles of return rates
 Number of observations1480.000
 Minimum0.810
 Quartile 10.998
 Median1.000
 Quartile 31.002
 Maximum1.278
 Mean of quarter 10.989
 Mean of quarter 21.000
 Mean of quarter 31.000
 Mean of quarter 41.012
 Inter Quartile Range0.004
 Number outliers low134.000
 Percentage of outliers low0.091
 Mean of outliers low0.976
 Number of outliers high159.000
 Percentage of outliers high0.107
 Mean of outliers high1.022
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)0.841
 VaR(95%) (moments method)0.009
 Expected Shortfall (moments method)0.060
 Extreme Value Index (regression method)0.598
 VaR(95%) (regression method)0.007
 Expected Shortfall (regression method)0.021
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)0.024
 Compounded annual return (geometric extrapolation)0.023
 Calmar ratio (compounded annual return / max draw down)0.032
 Compounded annual return / average of 25% largest draw downs0.245
 Compounded annual return / Expected Shortfall lognormal0.662
ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean-2.072
 SD0.788
 Sharpe ratio (Glass type estimate) -2.630
 Sharpe ratio (Hedges UMVUE)-2.619
 df171.000
 t-1.860
 p0.589
 Lowerbound of 95% confidence interval for Sharpe Ratio-5.412
 Upperbound of 95% confidence interval for Sharpe Ratio0.159
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-5.405
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.167
Statistics related to Sortino ratio
 Sortino ratio-3.356
 Upside Potential Ratio2.873
 Upside part of mean1.774
 Downside part of mean-3.846
 Upside SD0.498
 Downside SD0.617
 N nonnegative terms26.000
 N negative terms146.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.509
 Mean of criterion-2.072
 SD of predictor0.175
 SD of criterion0.788
 Covariance-0.013
 r-0.093
 b (slope, estimate of beta)-0.418
 a (intercept, estimate of alpha)-1.859
 Mean Square Error0.619
 DF error170.000
 t(b)-1.218
 p(b)0.546
 t(a)-1.651
 p(a)0.563
 Lowerbound of 95% confidence interval for beta-1.097
 Upperbound of 95% confidence interval for beta0.260
 Lowerbound of 95% confidence interval for alpha-4.083
 Upperbound of 95% confidence interval for alpha0.364
 Treynor index (mean / b)4.953
 Jensen alpha (a)-1.859
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-2.391
 SD0.795
 Sharpe ratio (Glass type estimate) -3.007
 Sharpe ratio (Hedges UMVUE)-2.994
 df171.000
 t-2.127
 p0.602
 Lowerbound of 95% confidence interval for Sharpe Ratio-5.793
 Upperbound of 95% confidence interval for Sharpe Ratio-0.213
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-5.784
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.204
Statistics related to Sortino ratio
 Sortino ratio-3.597
 Upside Potential Ratio2.506
 Upside part of mean1.665
 Downside part of mean-4.056
 Upside SD0.451
 Downside SD0.665
 N nonnegative terms26.000
 N negative terms146.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.494
 Mean of criterion-2.391
 SD of predictor0.175
 SD of criterion0.795
 Covariance-0.014
 r-0.098
 b (slope, estimate of beta)-0.447
 a (intercept, estimate of alpha)-2.171
 Mean Square Error0.630
 DF error170.000
 t(b)-1.286
 p(b)0.549
 t(a)-1.912
 p(a)0.573
 Lowerbound of 95% confidence interval for beta-1.132
 Upperbound of 95% confidence interval for beta0.239
 Lowerbound of 95% confidence interval for alpha-4.411
 Upperbound of 95% confidence interval for alpha0.070
 Treynor index (mean / b)5.354
 Jensen alpha (a)-2.171
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.075
 Expected Shortfall on VaR0.091
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.035
 Expected Shortfall on VaR0.074
ORDER STATISTICS
Quartiles of return rates
 Number of observations172.000
 Minimum0.810
 Quartile 10.999
 Median1.000
 Quartile 31.000
 Maximum1.278
 Mean of quarter 10.956
 Mean of quarter 21.000
 Mean of quarter 31.000
 Mean of quarter 41.021
 Inter Quartile Range0.001
 Number outliers low40.000
 Percentage of outliers low0.233
 Mean of outliers low0.953
 Number of outliers high23.000
 Percentage of outliers high0.134
 Mean of outliers high1.039
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)0.304
 VaR(95%) (moments method)0.010
 Expected Shortfall (moments method)0.019
 Extreme Value Index (regression method)-0.120
 VaR(95%) (regression method)0.039
 Expected Shortfall (regression method)0.058
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations3.000
 Minimum0.002
 Quartile 10.009
 Median0.016
 Quartile 30.359
 Maximum0.703
 Mean of quarter 10.002
 Mean of quarter 20.016
 Mean of quarter 3NA
 Mean of quarter 40.703
 Inter Quartile Range0.351
 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)-1.381
 Compounded annual return (geometric extrapolation)-0.904
 Calmar ratio (compounded annual return / max draw down)-1.286
 Compounded annual return / average of 25% largest draw downs-1.286
 Compounded annual return / Expected Shortfall lognormal-9.955