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Advanced Statistics: Easy Does It

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
 Mean0.114
 SD0.984
 Sharpe ratio (Glass type estimate) 0.116
 Sharpe ratio (Hedges UMVUE)0.106
 df9.000
 t0.106
 p0.459
 Lowerbound of 95% confidence interval for Sharpe Ratio-2.035
 Upperbound of 95% confidence interval for Sharpe Ratio2.260
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-2.042
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)2.253
Statistics related to Sortino ratio
 Sortino ratio0.180
 Upside Potential Ratio1.946
 Upside part of mean1.229
 Downside part of mean-1.115
 Upside SD0.688
 Downside SD0.632
 N nonnegative terms3.000
 N negative terms7.000
Statistics related to linear regression on benchmark
 N of observations10.000
 Mean of predictor0.251
 Mean of criterion0.114
 SD of predictor0.131
 SD of criterion0.984
 Covariance-0.056
 r-0.434
 b (slope, estimate of beta)-3.263
 a (intercept, estimate of alpha)0.932
 Mean Square Error0.885
 DF error8.000
 t(b)-1.361
 p(b)0.895
 t(a)0.781
 p(a)0.229
 Lowerbound of 95% confidence interval for beta-8.792
 Upperbound of 95% confidence interval for beta2.265
 Lowerbound of 95% confidence interval for alpha-1.819
 Upperbound of 95% confidence interval for alpha3.683
 Treynor index (mean / b)-0.035
 Jensen alpha (a)0.932
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.345
 SD1.034
 Sharpe ratio (Glass type estimate) -0.334
 Sharpe ratio (Hedges UMVUE)-0.305
 df9.000
 t-0.305
 p0.616
 Lowerbound of 95% confidence interval for Sharpe Ratio-2.477
 Upperbound of 95% confidence interval for Sharpe Ratio1.828
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-2.457
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)1.847
Statistics related to Sortino ratio
 Sortino ratio-0.431
 Upside Potential Ratio1.298
 Upside part of mean1.039
 Downside part of mean-1.384
 Upside SD0.574
 Downside SD0.801
 N nonnegative terms3.000
 N negative terms7.000
Statistics related to linear regression on benchmark
 N of observations10.000
 Mean of predictor0.240
 Mean of criterion-0.345
 SD of predictor0.127
 SD of criterion1.034
 Covariance-0.054
 r-0.410
 b (slope, estimate of beta)-3.336
 a (intercept, estimate of alpha)0.455
 Mean Square Error1.000
 DF error8.000
 t(b)-1.270
 p(b)0.880
 t(a)0.360
 p(a)0.364
 Lowerbound of 95% confidence interval for beta-9.394
 Upperbound of 95% confidence interval for beta2.722
 Lowerbound of 95% confidence interval for alpha-2.459
 Upperbound of 95% confidence interval for alpha3.370
 Treynor index (mean / b)0.103
 Jensen alpha (a)0.455
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.405
 Expected Shortfall on VaR0.472
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.262
 Expected Shortfall on VaR0.481
ORDER STATISTICS
Quartiles of return rates
 Number of observations10.000
 Minimum0.593
 Quartile 10.928
 Median1.000
 Quartile 31.149
 Maximum1.498
 Mean of quarter 10.704
 Mean of quarter 20.992
 Mean of quarter 31.000
 Mean of quarter 41.345
 Inter Quartile Range0.221
 Number outliers low1.000
 Percentage of outliers low0.100
 Mean of outliers low0.593
 Number of outliers high1.000
 Percentage of outliers high0.100
 Mean of outliers high1.498
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)-2112.260
 VaR(95%) (moments method)0.225
 Expected Shortfall (moments method)NA
 Extreme Value Index (regression method)-5.041
 VaR(95%) (regression method)1.005
 Expected Shortfall (regression method)1.006
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations1.000
 Minimum0.515
 Quartile 10.515
 Median0.515
 Quartile 30.515
 Maximum0.515
 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)-0.266
 Compounded annual return (geometric extrapolation)-0.260
 Calmar ratio (compounded annual return / max draw down)-0.505
 Compounded annual return / average of 25% largest draw downsNA
 Compounded annual return / Expected Shortfall lognormal-0.551
ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean0.419
 SD1.214
 Sharpe ratio (Glass type estimate) 0.345
 Sharpe ratio (Hedges UMVUE)0.344
 df295.000
 t0.320
 p0.375
 Lowerbound of 95% confidence interval for Sharpe Ratio-1.768
 Upperbound of 95% confidence interval for Sharpe Ratio2.458
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-1.769
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)2.457
Statistics related to Sortino ratio
 Sortino ratio0.502
 Upside Potential Ratio6.839
 Upside part of mean5.712
 Downside part of mean-5.293
 Upside SD0.879
 Downside SD0.835
 N nonnegative terms70.000
 N negative terms226.000
Statistics related to linear regression on benchmark
 N of observations296.000
 Mean of predictor0.306
 Mean of criterion0.419
 SD of predictor0.203
 SD of criterion1.214
 Covariance0.000
 r0.002
 b (slope, estimate of beta)0.010
 a (intercept, estimate of alpha)0.416
 Mean Square Error1.480
 DF error294.000
 t(b)0.028
 p(b)0.489
 t(a)0.316
 p(a)0.376
 Lowerbound of 95% confidence interval for beta-0.676
 Upperbound of 95% confidence interval for beta0.696
 Lowerbound of 95% confidence interval for alpha-2.174
 Upperbound of 95% confidence interval for alpha3.006
 Treynor index (mean / b)42.498
 Jensen alpha (a)0.416
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.336
 SD1.244
 Sharpe ratio (Glass type estimate) -0.270
 Sharpe ratio (Hedges UMVUE)-0.269
 df295.000
 t-0.250
 p0.599
 Lowerbound of 95% confidence interval for Sharpe Ratio-2.382
 Upperbound of 95% confidence interval for Sharpe Ratio1.844
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-2.382
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)1.844
Statistics related to Sortino ratio
 Sortino ratio-0.355
 Upside Potential Ratio5.677
 Upside part of mean5.368
 Downside part of mean-5.703
 Upside SD0.806
 Downside SD0.946
 N nonnegative terms70.000
 N negative terms226.000
Statistics related to linear regression on benchmark
 N of observations296.000
 Mean of predictor0.286
 Mean of criterion-0.336
 SD of predictor0.202
 SD of criterion1.244
 Covariance0.002
 r0.006
 b (slope, estimate of beta)0.038
 a (intercept, estimate of alpha)-0.347
 Mean Square Error1.554
 DF error294.000
 t(b)0.107
 p(b)0.457
 t(a)-0.257
 p(a)0.601
 Lowerbound of 95% confidence interval for beta-0.667
 Upperbound of 95% confidence interval for beta0.744
 Lowerbound of 95% confidence interval for alpha-2.999
 Upperbound of 95% confidence interval for alpha2.306
 Treynor index (mean / b)-8.724
 Jensen alpha (a)-0.347
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.105
 Expected Shortfall on VaR0.130
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.045
 Expected Shortfall on VaR0.096
ORDER STATISTICS
Quartiles of return rates
 Number of observations296.000
 Minimum0.664
 Quartile 10.994
 Median1.000
 Quartile 31.000
 Maximum1.373
 Mean of quarter 10.939
 Mean of quarter 21.000
 Mean of quarter 31.000
 Mean of quarter 41.067
 Inter Quartile Range0.006
 Number outliers low64.000
 Percentage of outliers low0.216
 Mean of outliers low0.931
 Number of outliers high61.000
 Percentage of outliers high0.206
 Mean of outliers high1.080
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)0.328
 VaR(95%) (moments method)0.031
 Expected Shortfall (moments method)0.060
 Extreme Value Index (regression method)0.280
 VaR(95%) (regression method)0.059
 Expected Shortfall (regression method)0.117
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations1.000
 Minimum0.682
 Quartile 10.682
 Median0.682
 Quartile 30.682
 Maximum0.682
 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)-0.258
 Compounded annual return (geometric extrapolation)-0.253
 Calmar ratio (compounded annual return / max draw down)-0.371
 Compounded annual return / average of 25% largest draw downsNA
 Compounded annual return / Expected Shortfall lognormal-1.948
ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean2.410
 SD1.155
 Sharpe ratio (Glass type estimate) 2.087
 Sharpe ratio (Hedges UMVUE)2.078
 df171.000
 t1.476
 p0.429
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.696
 Upperbound of 95% confidence interval for Sharpe Ratio4.865
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.702
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)4.859
Statistics related to Sortino ratio
 Sortino ratio3.507
 Upside Potential Ratio8.124
 Upside part of mean5.583
 Downside part of mean-3.173
 Upside SD0.933
 Downside SD0.687
 N nonnegative terms30.000
 N negative terms142.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.268
 Mean of criterion2.410
 SD of predictor0.188
 SD of criterion1.155
 Covariance-0.017
 r-0.079
 b (slope, estimate of beta)-0.484
 a (intercept, estimate of alpha)2.540
 Mean Square Error1.333
 DF error170.000
 t(b)-1.032
 p(b)0.539
 t(a)1.551
 p(a)0.441
 Lowerbound of 95% confidence interval for beta-1.410
 Upperbound of 95% confidence interval for beta0.442
 Lowerbound of 95% confidence interval for alpha-0.692
 Upperbound of 95% confidence interval for alpha5.773
 Treynor index (mean / b)-4.980
 Jensen alpha (a)2.540
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean1.743
 SD1.159
 Sharpe ratio (Glass type estimate) 1.504
 Sharpe ratio (Hedges UMVUE)1.498
 df171.000
 t1.064
 p0.448
 Lowerbound of 95% confidence interval for Sharpe Ratio-1.274
 Upperbound of 95% confidence interval for Sharpe Ratio4.279
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-1.279
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)4.274
Statistics related to Sortino ratio
 Sortino ratio2.203
 Upside Potential Ratio6.572
 Upside part of mean5.200
 Downside part of mean-3.457
 Upside SD0.847
 Downside SD0.791
 N nonnegative terms30.000
 N negative terms142.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.251
 Mean of criterion1.743
 SD of predictor0.187
 SD of criterion1.159
 Covariance-0.015
 r-0.070
 b (slope, estimate of beta)-0.431
 a (intercept, estimate of alpha)1.851
 Mean Square Error1.344
 DF error170.000
 t(b)-0.912
 p(b)0.535
 t(a)1.126
 p(a)0.457
 Lowerbound of 95% confidence interval for beta-1.365
 Upperbound of 95% confidence interval for beta0.502
 Lowerbound of 95% confidence interval for alpha-1.394
 Upperbound of 95% confidence interval for alpha5.096
 Treynor index (mean / b)-4.043
 Jensen alpha (a)1.851
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.093
 Expected Shortfall on VaR0.116
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.028
 Expected Shortfall on VaR0.063
ORDER STATISTICS
Quartiles of return rates
 Number of observations172.000
 Minimum0.664
 Quartile 11.000
 Median1.000
 Quartile 31.000
 Maximum1.373
 Mean of quarter 10.964
 Mean of quarter 21.000
 Mean of quarter 31.000
 Mean of quarter 41.065
 Inter Quartile Range0.000
 Number outliers low25.000
 Percentage of outliers low0.145
 Mean of outliers low0.937
 Number of outliers high30.000
 Percentage of outliers high0.174
 Mean of outliers high1.093
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)0.454
 VaR(95%) (regression method)0.038
 Expected Shortfall (regression method)0.119
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations4.000
 Minimum0.110
 Quartile 10.111
 Median0.248
 Quartile 30.393
 Maximum0.417
 Mean of quarter 10.110
 Mean of quarter 20.112
 Mean of quarter 30.385
 Mean of quarter 40.417
 Inter Quartile Range0.282
 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)2.888
 Compounded annual return (geometric extrapolation)4.973
 Calmar ratio (compounded annual return / max draw down)11.926
 Compounded annual return / average of 25% largest draw downs11.926
 Compounded annual return / Expected Shortfall lognormal42.788