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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
 SD0.266
 Sharpe ratio (Glass type estimate) 0.053
 Sharpe ratio (Hedges UMVUE)0.053
 df1560.000
 t0.114
 p0.499
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.867
 Upperbound of 95% confidence interval for Sharpe Ratio0.974
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.867
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.973
Statistics related to Sortino ratio
 Sortino ratio0.067
 Upside Potential Ratio4.374
 Upside part of mean0.927
 Downside part of mean-0.913
 Upside SD0.160
 Downside SD0.212
 N nonnegative terms522.000
 N negative terms1039.000
Statistics related to linear regression on benchmark
 N of observations1561.000
 Mean of predictor0.218
 Mean of criterion0.014
 SD of predictor0.194
 SD of criterion0.266
 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 error1559.000
 t(b)-0.871
 p(b)0.514
 t(a)0.166
 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-0.224
 Upperbound of 95% confidence interval for alpha0.266
 Treynor index (mean / b)-0.469
 Jensen alpha (a)0.021
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.023
 SD0.275
 Sharpe ratio (Glass type estimate) -0.082
 Sharpe ratio (Hedges UMVUE)-0.082
 df1560.000
 t-0.176
 p0.502
 Lowerbound of 95% confidence interval for Sharpe Ratio-1.003
 Upperbound of 95% confidence interval for Sharpe Ratio0.838
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-1.002
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.838
Statistics related to Sortino ratio
 Sortino ratio-0.100
 Upside Potential Ratio4.035
 Upside part of mean0.915
 Downside part of mean-0.938
 Upside SD0.155
 Downside SD0.227
 N nonnegative terms522.000
 N negative terms1039.000
Statistics related to linear regression on benchmark
 N of observations1561.000
 Mean of predictor0.200
 Mean of criterion-0.023
 SD of predictor0.193
 SD of criterion0.275
 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 error1559.000
 t(b)-0.956
 p(b)0.515
 t(a)-0.122
 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-0.269
 Upperbound of 95% confidence interval for alpha0.238
 Treynor index (mean / b)0.658
 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 observations1561.000
 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 low148.000
 Percentage of outliers low0.095
 Mean of outliers low0.979
 Number of outliers high177.000
 Percentage of outliers high0.113
 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)0.056
 Extreme Value Index (regression method)0.572
 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)0.022
 Calmar ratio (compounded annual return / max draw down)0.031
 Compounded annual return / average of 25% largest draw downs0.232
 Compounded annual return / Expected Shortfall lognormal0.716
ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean-1.082
 SD0.551
 Sharpe ratio (Glass type estimate) -1.965
 Sharpe ratio (Hedges UMVUE)-1.956
 df171.000
 t-1.389
 p0.567
 Lowerbound of 95% confidence interval for Sharpe Ratio-4.742
 Upperbound of 95% confidence interval for Sharpe Ratio0.817
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-4.736
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.823
Statistics related to Sortino ratio
 Sortino ratio-2.215
 Upside Potential Ratio1.395
 Upside part of mean0.682
 Downside part of mean-1.764
 Upside SD0.257
 Downside SD0.489
 N nonnegative terms6.000
 N negative terms166.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.596
 Mean of criterion-1.082
 SD of predictor0.186
 SD of criterion0.551
 Covariance-0.001
 r-0.014
 b (slope, estimate of beta)-0.040
 a (intercept, estimate of alpha)-1.058
 Mean Square Error0.305
 DF error170.000
 t(b)-0.176
 p(b)0.507
 t(a)-1.335
 p(a)0.551
 Lowerbound of 95% confidence interval for beta-0.488
 Upperbound of 95% confidence interval for beta0.408
 Lowerbound of 95% confidence interval for alpha-2.623
 Upperbound of 95% confidence interval for alpha0.507
 Treynor index (mean / b)27.044
 Jensen alpha (a)-1.058
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-1.246
 SD0.583
 Sharpe ratio (Glass type estimate) -2.137
 Sharpe ratio (Hedges UMVUE)-2.128
 df171.000
 t-1.511
 p0.573
 Lowerbound of 95% confidence interval for Sharpe Ratio-4.915
 Upperbound of 95% confidence interval for Sharpe Ratio0.647
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-4.909
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.653
Statistics related to Sortino ratio
 Sortino ratio-2.338
 Upside Potential Ratio1.221
 Upside part of mean0.651
 Downside part of mean-1.898
 Upside SD0.242
 Downside SD0.533
 N nonnegative terms6.000
 N negative terms166.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.578
 Mean of criterion-1.246
 SD of predictor0.186
 SD of criterion0.583
 Covariance-0.002
 r-0.015
 b (slope, estimate of beta)-0.048
 a (intercept, estimate of alpha)-1.218
 Mean Square Error0.342
 DF error170.000
 t(b)-0.201
 p(b)0.508
 t(a)-1.453
 p(a)0.555
 Lowerbound of 95% confidence interval for beta-0.524
 Upperbound of 95% confidence interval for beta0.427
 Lowerbound of 95% confidence interval for alpha-2.874
 Upperbound of 95% confidence interval for alpha0.437
 Treynor index (mean / b)25.767
 Jensen alpha (a)-1.218
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.054
 Expected Shortfall on VaR0.066
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.017
 Expected Shortfall on VaR0.039
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 10.980
 Mean of quarter 21.000
 Mean of quarter 31.000
 Mean of quarter 41.008
 Inter Quartile Range0.000
 Number outliers low9.000
 Percentage of outliers low0.052
 Mean of outliers low0.904
 Number of outliers high6.000
 Percentage of outliers high0.035
 Mean of outliers high1.057
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)-2.196
 VaR(95%) (regression method)0.032
 Expected Shortfall (regression method)0.090
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations1.000
 Minimum0.556
 Quartile 10.556
 Median0.556
 Quartile 30.556
 Maximum0.556
 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)-0.904
 Compounded annual return (geometric extrapolation)-0.700
 Calmar ratio (compounded annual return / max draw down)-1.259
 Compounded annual return / average of 25% largest draw downsNA
 Compounded annual return / Expected Shortfall lognormal-10.578