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Advanced Statistics: Gray Swan

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
 Mean0.744
 SD0.568
 Sharpe ratio (Glass type estimate) 1.310
 Sharpe ratio (Hedges UMVUE)1.248
 df16.000
 t1.559
 p0.318
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.416
 Upperbound of 95% confidence interval for Sharpe Ratio2.998
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.455
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)2.950
Statistics related to Sortino ratio
 Sortino ratio2.158
 Upside Potential Ratio3.522
 Upside part of mean1.214
 Downside part of mean-0.470
 Upside SD0.480
 Downside SD0.345
 N nonnegative terms13.000
 N negative terms4.000
Statistics related to linear regression on benchmark
 N of observations17.000
 Mean of predictor0.162
 Mean of criterion0.744
 SD of predictor0.116
 SD of criterion0.568
 Covariance0.005
 r0.077
 b (slope, estimate of beta)0.377
 a (intercept, estimate of alpha)0.683
 Mean Square Error0.342
 DF error15.000
 t(b)0.299
 p(b)0.451
 t(a)1.283
 p(a)0.303
 Lowerbound of 95% confidence interval for beta-2.310
 Upperbound of 95% confidence interval for beta3.064
 Lowerbound of 95% confidence interval for alpha-0.451
 Upperbound of 95% confidence interval for alpha1.817
 Treynor index (mean / b)1.972
 Jensen alpha (a)0.683
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean0.566
 SD0.591
 Sharpe ratio (Glass type estimate) 0.958
 Sharpe ratio (Hedges UMVUE)0.913
 df16.000
 t1.141
 p0.363
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.735
 Upperbound of 95% confidence interval for Sharpe Ratio2.623
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.764
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)2.589
Statistics related to Sortino ratio
 Sortino ratio1.387
 Upside Potential Ratio2.717
 Upside part of mean1.109
 Downside part of mean-0.543
 Upside SD0.435
 Downside SD0.408
 N nonnegative terms13.000
 N negative terms4.000
Statistics related to linear regression on benchmark
 N of observations17.000
 Mean of predictor0.154
 Mean of criterion0.566
 SD of predictor0.116
 SD of criterion0.591
 Covariance0.002
 r0.035
 b (slope, estimate of beta)0.180
 a (intercept, estimate of alpha)0.539
 Mean Square Error0.372
 DF error15.000
 t(b)0.137
 p(b)0.478
 t(a)0.977
 p(a)0.346
 Lowerbound of 95% confidence interval for beta-2.631
 Upperbound of 95% confidence interval for beta2.991
 Lowerbound of 95% confidence interval for alpha-0.637
 Upperbound of 95% confidence interval for alpha1.714
 Treynor index (mean / b)3.145
 Jensen alpha (a)0.539
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.208
 Expected Shortfall on VaR0.261
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.050
 Expected Shortfall on VaR0.124
ORDER STATISTICS
Quartiles of return rates
 Number of observations17.000
 Minimum0.657
 Quartile 11.013
 Median1.074
 Quartile 31.190
 Maximum1.288
 Mean of quarter 10.872
 Mean of quarter 21.044
 Mean of quarter 31.157
 Mean of quarter 41.238
 Inter Quartile Range0.177
 Number outliers low1.000
 Percentage of outliers low0.059
 Mean of outliers low0.657
 Number of outliers high0.000
 Percentage of outliers high0.000
 Mean of outliers highNA
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)-0.059
 VaR(95%) (regression method)0.116
 Expected Shortfall (regression method)0.183
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations3.000
 Minimum0.006
 Quartile 10.143
 Median0.279
 Quartile 30.311
 Maximum0.343
 Mean of quarter 10.006
 Mean of quarter 20.279
 Mean of quarter 3NA
 Mean of quarter 40.343
 Inter Quartile Range0.169
 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)0.970
 Compounded annual return (geometric extrapolation)0.841
 Calmar ratio (compounded annual return / max draw down)2.450
 Compounded annual return / average of 25% largest draw downs2.450
 Compounded annual return / Expected Shortfall lognormal3.220
ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean0.643
 SD0.421
 Sharpe ratio (Glass type estimate) 1.528
 Sharpe ratio (Hedges UMVUE)1.526
 df489.000
 t1.824
 p0.034
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.117
 Upperbound of 95% confidence interval for Sharpe Ratio3.173
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.119
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)3.171
Statistics related to Sortino ratio
 Sortino ratio2.381
 Upside Potential Ratio10.469
 Upside part of mean2.828
 Downside part of mean-2.185
 Upside SD0.324
 Downside SD0.270
 N nonnegative terms230.000
 N negative terms260.000
Statistics related to linear regression on benchmark
 N of observations490.000
 Mean of predictor0.161
 Mean of criterion0.643
 SD of predictor0.121
 SD of criterion0.421
 Covariance0.009
 r0.183
 b (slope, estimate of beta)0.634
 a (intercept, estimate of alpha)0.541
 Mean Square Error0.171
 DF error488.000
 t(b)4.115
 p(b)0.000
 t(a)1.556
 p(a)0.060
 Lowerbound of 95% confidence interval for beta0.331
 Upperbound of 95% confidence interval for beta0.937
 Lowerbound of 95% confidence interval for alpha-0.142
 Upperbound of 95% confidence interval for alpha1.225
 Treynor index (mean / b)1.014
 Jensen alpha (a)0.541
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean0.555
 SD0.419
 Sharpe ratio (Glass type estimate) 1.324
 Sharpe ratio (Hedges UMVUE)1.322
 df489.000
 t1.580
 p0.057
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.321
 Upperbound of 95% confidence interval for Sharpe Ratio2.968
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.322
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)2.966
Statistics related to Sortino ratio
 Sortino ratio1.998
 Upside Potential Ratio10.002
 Upside part of mean2.777
 Downside part of mean-2.222
 Upside SD0.315
 Downside SD0.278
 N nonnegative terms230.000
 N negative terms260.000
Statistics related to linear regression on benchmark
 N of observations490.000
 Mean of predictor0.153
 Mean of criterion0.555
 SD of predictor0.121
 SD of criterion0.419
 Covariance0.009
 r0.182
 b (slope, estimate of beta)0.628
 a (intercept, estimate of alpha)0.458
 Mean Square Error0.170
 DF error488.000
 t(b)4.090
 p(b)0.000
 t(a)1.324
 p(a)0.093
 Lowerbound of 95% confidence interval for beta0.326
 Upperbound of 95% confidence interval for beta0.930
 Lowerbound of 95% confidence interval for alpha-0.222
 Upperbound of 95% confidence interval for alpha1.139
 Treynor index (mean / b)0.883
 Jensen alpha (a)0.458
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.035
 Expected Shortfall on VaR0.044
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.015
 Expected Shortfall on VaR0.031
ORDER STATISTICS
Quartiles of return rates
 Number of observations490.000
 Minimum0.876
 Quartile 10.994
 Median1.000
 Quartile 31.011
 Maximum1.146
 Mean of quarter 10.977
 Mean of quarter 20.998
 Mean of quarter 31.004
 Mean of quarter 41.029
 Inter Quartile Range0.018
 Number outliers low25.000
 Percentage of outliers low0.051
 Mean of outliers low0.950
 Number of outliers high25.000
 Percentage of outliers high0.051
 Mean of outliers high1.059
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)0.151
 VaR(95%) (moments method)0.019
 Expected Shortfall (moments method)0.029
 Extreme Value Index (regression method)0.033
 VaR(95%) (regression method)0.022
 Expected Shortfall (regression method)0.033
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations32.000
 Minimum0.000
 Quartile 10.007
 Median0.026
 Quartile 30.049
 Maximum0.374
 Mean of quarter 10.002
 Mean of quarter 20.018
 Mean of quarter 30.035
 Mean of quarter 40.151
 Inter Quartile Range0.042
 Number outliers low0.000
 Percentage of outliers low0.000
 Mean of outliers lowNA
 Number of outliers high3.000
 Percentage of outliers high0.094
 Mean of outliers high0.289
Risk estimates based on draw downs (based on Extreme Value Theory)
 Extreme Value Index (moments method)0.569
 VaR(95%) (moments method)0.156
 Expected Shortfall (moments method)0.406
 Extreme Value Index (regression method)0.214
 VaR(95%) (regression method)0.135
 Expected Shortfall (regression method)0.216
COMBINED STATISTICS
 Annualized return (arithmetic extrapolation)0.945
 Compounded annual return (geometric extrapolation)0.820
 Calmar ratio (compounded annual return / max draw down)2.194
 Compounded annual return / average of 25% largest draw downs5.436
 Compounded annual return / Expected Shortfall lognormal18.652
ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean0.830
 SD0.429
 Sharpe ratio (Glass type estimate) 1.938
 Sharpe ratio (Hedges UMVUE)1.929
 df171.000
 t1.370
 p0.434
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.845
 Upperbound of 95% confidence interval for Sharpe Ratio4.714
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.850
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)4.708
Statistics related to Sortino ratio
 Sortino ratio2.781
 Upside Potential Ratio10.255
 Upside part of mean3.062
 Downside part of mean-2.232
 Upside SD0.309
 Downside SD0.299
 N nonnegative terms87.000
 N negative terms85.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.369
 Mean of criterion0.830
 SD of predictor0.116
 SD of criterion0.429
 Covariance-0.001
 r-0.015
 b (slope, estimate of beta)-0.054
 a (intercept, estimate of alpha)0.850
 Mean Square Error0.185
 DF error170.000
 t(b)-0.189
 p(b)0.507
 t(a)1.379
 p(a)0.447
 Lowerbound of 95% confidence interval for beta-0.614
 Upperbound of 95% confidence interval for beta0.507
 Lowerbound of 95% confidence interval for alpha-0.367
 Upperbound of 95% confidence interval for alpha2.067
 Treynor index (mean / b)-15.456
 Jensen alpha (a)0.850
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean0.737
 SD0.432
 Sharpe ratio (Glass type estimate) 1.706
 Sharpe ratio (Hedges UMVUE)1.698
 df171.000
 t1.206
 p0.442
 Lowerbound of 95% confidence interval for Sharpe Ratio-1.074
 Upperbound of 95% confidence interval for Sharpe Ratio4.481
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-1.079
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)4.476
Statistics related to Sortino ratio
 Sortino ratio2.383
 Upside Potential Ratio9.747
 Upside part of mean3.015
 Downside part of mean-2.278
 Upside SD0.303
 Downside SD0.309
 N nonnegative terms87.000
 N negative terms85.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.362
 Mean of criterion0.737
 SD of predictor0.116
 SD of criterion0.432
 Covariance-0.001
 r-0.017
 b (slope, estimate of beta)-0.063
 a (intercept, estimate of alpha)0.760
 Mean Square Error0.188
 DF error170.000
 t(b)-0.218
 p(b)0.508
 t(a)1.222
 p(a)0.453
 Lowerbound of 95% confidence interval for beta-0.628
 Upperbound of 95% confidence interval for beta0.503
 Lowerbound of 95% confidence interval for alpha-0.467
 Upperbound of 95% confidence interval for alpha1.987
 Treynor index (mean / b)-11.785
 Jensen alpha (a)0.760
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.036
 Expected Shortfall on VaR0.045
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.015
 Expected Shortfall on VaR0.031
ORDER STATISTICS
Quartiles of return rates
 Number of observations172.000
 Minimum0.876
 Quartile 10.995
 Median1.000
 Quartile 31.013
 Maximum1.072
 Mean of quarter 10.975
 Mean of quarter 20.999
 Mean of quarter 31.007
 Mean of quarter 41.029
 Inter Quartile Range0.018
 Number outliers low10.000
 Percentage of outliers low0.058
 Mean of outliers low0.947
 Number of outliers high9.000
 Percentage of outliers high0.052
 Mean of outliers high1.052
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)-0.041
 VaR(95%) (moments method)0.016
 Expected Shortfall (moments method)0.023
 Extreme Value Index (regression method)-0.219
 VaR(95%) (regression method)0.026
 Expected Shortfall (regression method)0.034
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations17.000
 Minimum0.000
 Quartile 10.006
 Median0.024
 Quartile 30.049
 Maximum0.374
 Mean of quarter 10.002
 Mean of quarter 20.014
 Mean of quarter 30.041
 Mean of quarter 40.160
 Inter Quartile Range0.044
 Number outliers low0.000
 Percentage of outliers low0.000
 Mean of outliers lowNA
 Number of outliers high2.000
 Percentage of outliers high0.118
 Mean of outliers high0.246
Risk estimates based on draw downs (based on Extreme Value Theory)
 Extreme Value Index (moments method)0.453
 VaR(95%) (moments method)0.152
 Expected Shortfall (moments method)0.320
 Extreme Value Index (regression method)1.122
 VaR(95%) (regression method)0.223
 Expected Shortfall (regression method)NA
COMBINED STATISTICS
 Annualized return (arithmetic extrapolation)0.956
 Compounded annual return (geometric extrapolation)1.184
 Calmar ratio (compounded annual return / max draw down)3.168
 Compounded annual return / average of 25% largest draw downs7.407
 Compounded annual return / Expected Shortfall lognormal26.400