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
 Mean0.069
 SD0.368
 Sharpe ratio (Glass type estimate) 0.187
 Sharpe ratio (Hedges UMVUE)0.184
 df55.000
 t0.403
 p0.344
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.722
 Upperbound of 95% confidence interval for Sharpe Ratio1.094
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.724
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)1.092
Statistics related to Sortino ratio
 Sortino ratio0.222
 Upside Potential Ratio1.136
 Upside part of mean0.351
 Downside part of mean-0.283
 Upside SD0.194
 Downside SD0.309
 N nonnegative terms33.000
 N negative terms23.000
Statistics related to linear regression on benchmark
 N of observations56.000
 Mean of predictor0.217
 Mean of criterion0.069
 SD of predictor0.182
 SD of criterion0.368
 Covariance0.001
 r0.015
 b (slope, estimate of beta)0.031
 a (intercept, estimate of alpha)0.062
 Mean Square Error0.138
 DF error54.000
 t(b)0.112
 p(b)0.456
 t(a)0.341
 p(a)0.367
 Lowerbound of 95% confidence interval for beta-0.520
 Upperbound of 95% confidence interval for beta0.581
 Lowerbound of 95% confidence interval for alpha-0.302
 Upperbound of 95% confidence interval for alpha0.426
 Treynor index (mean / b)2.237
 Jensen alpha (a)0.062
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.023
 SD0.474
 Sharpe ratio (Glass type estimate) -0.049
 Sharpe ratio (Hedges UMVUE)-0.048
 df55.000
 t-0.106
 p0.542
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.956
 Upperbound of 95% confidence interval for Sharpe Ratio0.859
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.956
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.859
Statistics related to Sortino ratio
 Sortino ratio-0.053
 Upside Potential Ratio0.766
 Upside part of mean0.333
 Downside part of mean-0.356
 Upside SD0.180
 Downside SD0.434
 N nonnegative terms33.000
 N negative terms23.000
Statistics related to linear regression on benchmark
 N of observations56.000
 Mean of predictor0.198
 Mean of criterion-0.023
 SD of predictor0.178
 SD of criterion0.474
 Covariance-0.002
 r-0.022
 b (slope, estimate of beta)-0.059
 a (intercept, estimate of alpha)-0.012
 Mean Square Error0.229
 DF error54.000
 t(b)-0.162
 p(b)0.564
 t(a)-0.050
 p(a)0.520
 Lowerbound of 95% confidence interval for beta-0.785
 Upperbound of 95% confidence interval for beta0.668
 Lowerbound of 95% confidence interval for alpha-0.479
 Upperbound of 95% confidence interval for alpha0.455
 Treynor index (mean / b)0.396
 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%)0.203
 Expected Shortfall on VaR0.247
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 observations56.000
 Minimum0.429
 Quartile 10.993
 Median1.013
 Quartile 31.038
 Maximum1.250
 Mean of quarter 10.912
 Mean of quarter 21.004
 Mean of quarter 31.024
 Mean of quarter 41.098
 Inter Quartile Range0.044
 Number outliers low3.000
 Percentage of outliers low0.054
 Mean of outliers low0.681
 Number of outliers high5.000
 Percentage of outliers high0.089
 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)0.061
 Expected Shortfall (moments method)NA
 Extreme Value Index (regression method)1.060
 VaR(95%) (regression method)0.057
 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 downs0.076
 Compounded annual return / Expected Shortfall lognormal0.085
ANALYSIS BASED ON DAILY VALUES, FULL HISTORY (Back)
RATIO STATISTICS
Ratio statistics of excess return rates
Statistics related to Sharpe ratio
 Mean0.012
 SD0.261
 Sharpe ratio (Glass type estimate) 0.047
 Sharpe ratio (Hedges UMVUE)0.047
 df1614.000
 t0.102
 p0.499
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.858
 Upperbound of 95% confidence interval for Sharpe Ratio0.951
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.858
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.951
Statistics related to Sortino ratio
 Sortino ratio0.059
 Upside Potential Ratio4.300
 Upside part of mean0.896
 Downside part of mean-0.884
 Upside SD0.157
 Downside SD0.208
 N nonnegative terms522.000
 N negative terms1093.000
Statistics related to linear regression on benchmark
 N of observations1615.000
 Mean of predictor0.210
 Mean of criterion0.012
 SD of predictor0.193
 SD of criterion0.261
 Covariance-0.001
 r-0.022
 b (slope, estimate of beta)-0.029
 a (intercept, estimate of alpha)0.018
 Mean Square Error0.068
 DF error1613.000
 t(b)-0.872
 p(b)0.514
 t(a)0.152
 p(a)0.498
 Lowerbound of 95% confidence interval for beta-0.095
 Upperbound of 95% confidence interval for beta0.037
 Lowerbound of 95% confidence interval for alpha-0.219
 Upperbound of 95% confidence interval for alpha0.255
 Treynor index (mean / b)-0.418
 Jensen alpha (a)0.018
Ratio statistics of excess log return rates
Statistics related to Sharpe ratio
 Mean-0.023
 SD0.270
 Sharpe ratio (Glass type estimate) -0.086
 Sharpe ratio (Hedges UMVUE)-0.086
 df1614.000
 t-0.187
 p0.502
 Lowerbound of 95% confidence interval for Sharpe Ratio-0.991
 Upperbound of 95% confidence interval for Sharpe Ratio0.818
 Lowerbound of 95% CI (Gibbons, Hedeker & Davis approximation)-0.991
 Upperbound of 95% CI (Gibbons, Hedeker & Davis approximation)0.818
Statistics related to Sortino ratio
 Sortino ratio-0.105
 Upside Potential Ratio3.967
 Upside part of mean0.884
 Downside part of mean-0.908
 Upside SD0.153
 Downside SD0.223
 N nonnegative terms522.000
 N negative terms1093.000
Statistics related to linear regression on benchmark
 N of observations1615.000
 Mean of predictor0.191
 Mean of criterion-0.023
 SD of predictor0.193
 SD of criterion0.270
 Covariance-0.001
 r-0.024
 b (slope, estimate of beta)-0.033
 a (intercept, estimate of alpha)-0.017
 Mean Square Error0.073
 DF error1613.000
 t(b)-0.958
 p(b)0.515
 t(a)-0.136
 p(a)0.502
 Lowerbound of 95% confidence interval for beta-0.102
 Upperbound of 95% confidence interval for beta0.035
 Lowerbound of 95% confidence interval for alpha-0.262
 Upperbound of 95% confidence interval for alpha0.228
 Treynor index (mean / b)0.700
 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 observations1615.000
 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 low170.000
 Percentage of outliers low0.105
 Mean of outliers low0.981
 Number of outliers high201.000
 Percentage of outliers high0.124
 Mean of outliers high1.017
Risk estimates for a one-period unit investment (based on Extreme Value Theory)
 Extreme Value Index (moments method)0.829
 VaR(95%) (moments method)0.007
 Expected Shortfall (moments method)0.045
 Extreme Value Index (regression method)0.557
 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 downs0.224
 Compounded annual return / Expected Shortfall lognormal0.704
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 predictor0.236
 Mean of criterion-0.044
 SD of predictor0.179
 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 predictor0.220
 Mean of criterion-0.044
 SD of predictor0.179
 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)-5583158135388585.000
 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)34521443076174016614582223634432.000
 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