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
 Mean0.071
 SD0.371
 Sharpe ratio (Glass type estimate) 0.190
 Sharpe ratio (Hedges UMVUE)0.188
 df54.000
 t0.408
 p0.343
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.727
 Upperbound of 95% confidence interval for Sharpe Ratio1.106
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.728
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)1.104
Statistics related to Sortino ratio
 Sortino ratio0.227
 Upside Potential Ratio1.147
 Upside part of mean0.358
 Downside part of mean-0.287
 Upside SD0.196
 Downside SD0.312
 N nonnegative terms33.000
 N negative terms22.000
Statistics related to linear regression on benchmark
 N of observations55.000
 Mean of predictor0.218
 Mean of criterion0.071
 SD of predictor0.184
 SD of criterion0.371
 Covariance0.001
 r0.015
 b (slope, estimate of beta)0.030
 a (intercept, estimate of alpha)0.064
 Mean Square Error0.140
 DF error53.000
 t(b)0.109
 p(b)0.457
 t(a)0.346
 p(a)0.365
 Lowerbound of 95% confidence interval for beta-0.526
 Upperbound of 95% confidence interval for beta0.586
 Lowerbound of 95% confidence interval for alpha-0.307
 Upperbound of 95% confidence interval for alpha0.435
 Treynor index (mean / b)2.342
 Jensen alpha (a)0.064
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.023
 SD0.479
 Sharpe ratio (Glass type estimate) -0.048
 Sharpe ratio (Hedges UMVUE)-0.047
 df54.000
 t-0.102
 p0.541
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.963
 Upperbound of 95% confidence interval for Sharpe Ratio0.868
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.963
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.868
Statistics related to Sortino ratio
 Sortino ratio-0.052
 Upside Potential Ratio0.773
 Upside part of mean0.339
 Downside part of mean-0.362
 Upside SD0.181
 Downside SD0.438
 N nonnegative terms33.000
 N negative terms22.000
Statistics related to linear regression on benchmark
 N of observations55.000
 Mean of predictor0.200
 Mean of criterion-0.023
 SD of predictor0.180
 SD of criterion0.479
 Covariance-0.002
 r-0.022
 b (slope, estimate of beta)-0.059
 a (intercept, estimate of alpha)-0.011
 Mean Square Error0.233
 DF error53.000
 t(b)-0.161
 p(b)0.564
 t(a)-0.047
 p(a)0.519
 Lowerbound of 95% confidence interval for beta-0.793
 Upperbound of 95% confidence interval for beta0.675
 Lowerbound of 95% confidence interval for alpha-0.487
 Upperbound of 95% confidence interval for alpha0.465
 Treynor index (mean / b)0.388
 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%)0.205
 Expected Shortfall on VaR0.249
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 observations55.000
 Minimum0.429
 Quartile 10.993
 Median1.014
 Quartile 31.039
 Maximum1.250
 Mean of quarter 10.912
 Mean of quarter 21.005
 Mean of quarter 31.025
 Mean of quarter 41.098
 Inter Quartile Range0.046
 Number outliers low3.000
 Percentage of outliers low0.055
 Mean of outliers low0.681
 Number of outliers high5.000
 Percentage of outliers high0.091
 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)0.063
 Expected Shortfall (moments method)NA
 Extreme Value Index (regression method)1.060
 VaR(95%) (regression method)0.058
 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)0.022
 Compounded annual return (geometric extrapolation)0.021
 Calmar ratio (compounded annual return / max draw down)0.030
 Compounded annual return / average of 25% largest draw downs0.078
 Compounded annual return / Expected Shortfall lognormal0.086
ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean0.013
 SD0.262
 Sharpe ratio (Glass type estimate) 0.048
 Sharpe ratio (Hedges UMVUE)0.048
 df1603.000
 t0.104
 p0.498
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.859
 Upperbound of 95% confidence interval for Sharpe Ratio0.956
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.859
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.956
Statistics related to Sortino ratio
 Sortino ratio0.060
 Upside Potential Ratio4.315
 Upside part of mean0.902
 Downside part of mean-0.890
 Upside SD0.158
 Downside SD0.209
 N nonnegative terms522.000
 N negative terms1082.000
Statistics related to linear regression on benchmark
 N of observations1604.000
 Mean of predictor0.216
 Mean of criterion0.013
 SD of predictor0.193
 SD of criterion0.262
 Covariance-0.001
 r-0.022
 b (slope, estimate of beta)-0.030
 a (intercept, estimate of alpha)0.019
 Mean Square Error0.069
 DF error1602.000
 t(b)-0.873
 p(b)0.511
 t(a)0.156
 p(a)0.498
 Lowerbound of 95% confidence interval for beta-0.096
 Upperbound of 95% confidence interval for beta0.037
 Lowerbound of 95% confidence interval for alpha-0.220
 Upperbound of 95% confidence interval for alpha0.258
 Treynor index (mean / b)-0.427
 Jensen alpha (a)0.019
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.023
 SD0.271
 Sharpe ratio (Glass type estimate) -0.086
 Sharpe ratio (Hedges UMVUE)-0.086
 df1603.000
 t-0.185
 p0.503
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.993
 Upperbound of 95% confidence interval for Sharpe Ratio0.822
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.993
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.822
Statistics related to Sortino ratio
 Sortino ratio-0.104
 Upside Potential Ratio3.981
 Upside part of mean0.890
 Downside part of mean-0.914
 Upside SD0.153
 Downside SD0.224
 N nonnegative terms522.000
 N negative terms1082.000
Statistics related to linear regression on benchmark
 N of observations1604.000
 Mean of predictor0.197
 Mean of criterion-0.023
 SD of predictor0.193
 SD of criterion0.271
 Covariance-0.001
 r-0.024
 b (slope, estimate of beta)-0.034
 a (intercept, estimate of alpha)-0.017
 Mean Square Error0.074
 DF error1602.000
 t(b)-0.959
 p(b)0.512
 t(a)-0.132
 p(a)0.502
 Lowerbound of 95% confidence interval for beta-0.103
 Upperbound of 95% confidence interval for beta0.035
 Lowerbound of 95% confidence interval for alpha-0.263
 Upperbound of 95% confidence interval for alpha0.230
 Treynor index (mean / b)0.689
 Jensen alpha (a)-0.017
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 observations1604.000
 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 41.010
 Inter Quartile Range0.003
 Number outliers low163.000
 Percentage of outliers low0.102
 Mean of outliers low0.980
 Number of outliers high197.000
 Percentage of outliers high0.123
 Mean of outliers high1.017
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)0.845
 VaR(95%) (moments method)0.007
 Expected Shortfall (moments method)0.052
 Extreme Value Index (regression method)0.567
 VaR(95%) (regression method)0.006
 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)0.021
 Calmar ratio (compounded annual return / max draw down)0.030
 Compounded annual return / average of 25% largest draw downs0.226
 Compounded annual return / Expected Shortfall lognormal0.706
ANALYSIS BASED ON DAILY VALUES, LAST 6 MONTHS (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean-0.044
 SD0.000
 Sharpe ratio (Glass type estimate) NA
 Sharpe ratio (Hedges UMVUE)NA
 dfNA
 tNA
 pNA
 Lowerbound of 95% confidence interval for Sharpe RatioNA
 Upperbound of 95% confidence interval for Sharpe RatioNA
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)NA
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)NA
Statistics related to Sortino ratio
 Sortino ratio-18.547
 Upside Potential Ratio0.000
 Upside part of mean0.000
 Downside part of mean-0.044
 Upside SD0.000
 Downside SD0.002
 N nonnegative terms0.000
 N negative terms172.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.402
 Mean of criterion-0.044
 SD of predictor0.177
 SD of criterion0.000
 Covariance0.000
 rNA
 b (slope, estimate of beta)NA
 a (intercept, estimate of alpha)NA
 Mean Square ErrorNA
 DF errorNA
 t(b)NA
 p(b)NA
 t(a)NA
 p(a)NA
 Lowerbound of 95% confidence interval for betaNA
 Upperbound of 95% confidence interval for betaNA
 Lowerbound of 95% confidence interval for alphaNA
 Upperbound of 95% confidence interval for alphaNA
 Treynor index (mean / b)NA
 Jensen alpha (a)NA
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.044
 SD0.000
 Sharpe ratio (Glass type estimate) -7936535306446244.000
 Sharpe ratio (Hedges UMVUE)-7901674975671223.000
 df171.000
 t-5611977934314593.000
 p1.000
 Lowerbound of 95% confidence interval for Sharpe RatioNA
 Upperbound of 95% confidence interval for Sharpe RatioNA
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-8739115425704417.000
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)-7064234525638030.000
Statistics related to Sortino ratio
 Sortino ratio-18.547
 Upside Potential Ratio0.000
 Upside part of mean0.000
 Downside part of mean-0.044
 Upside SD0.000
 Downside SD0.002
 N nonnegative terms0.000
 N negative terms172.000
Statistics related to linear regression on benchmark
 N of observations172.000
 Mean of predictor0.386
 Mean of criterion-0.044
 SD of predictor0.176
 SD of criterion0.000
 Covariance-0.000
 r-0.000
 b (slope, estimate of beta)-0.000
 a (intercept, estimate of alpha)-0.044
 Mean Square Error0.000
 DF error170.000
 t(b)-0.000
 p(b)0.500
 t(a)-5556831020541374.000
 p(a)1.000
 Lowerbound of 95% confidence interval for beta-0.000
 Upperbound of 95% confidence interval for beta0.000
 Lowerbound of 95% confidence interval for alpha-0.044
 Upperbound of 95% confidence interval for alpha-0.044
 Treynor index (mean / b)13322455915738054491807091261440.000
 Jensen alpha (a)-0.044
Risk estimates for a one-period unit investment (parametric)
assuming log normal returns and losses (using central moments from Sharpe statistics)
 VaR(95%)0.000
 Expected Shortfall on VaR0.000
assuming Pareto losses only (using partial moments from Sortino statistics)
 VaR(95%)0.000
 Expected Shortfall on VaR0.000
ORDER STATISTICS
Quartiles of return rates
 Number of observations172.000
 Minimum1.000
 Quartile 11.000
 Median1.000
 Quartile 31.000
 Maximum1.000
 Mean of quarter 11.000
 Mean of quarter 21.000
 Mean of quarter 31.000
 Mean of quarter 41.000
 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 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)NA
 VaR(95%) (regression method)NA
 Expected Shortfall (regression method)NA
DRAW DOWN STATISTICS
Quartiles of draw downs
 Number of observations0.000
 MinimumNA
 Quartile 1NA
 MedianNA
 Quartile 3NA
 MaximumNA
 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 lowNA
 Mean of outliers lowNA
 Number of outliers high0.000
 Percentage of outliers highNA
 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)0.000
 Compounded annual return (geometric extrapolation)0.000
 Calmar ratio (compounded annual return / max draw down)NA
 Compounded annual return / average of 25% largest draw downsNA
 Compounded annual return / Expected Shortfall lognormal0.000