Statistics - Non-Parametric Approaches

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[...] Resampled again and again with replacement of existing with new Imprecise estimates: Sample taken from normal distribution Reason: Dimensionality ignored Large data available: Imprecision less DATA: Emperically and Mathematically derived Minute level data(smallest available unit) a)HS estimate available b)Divide this into bins (smaller parts). For this a)From resampled samples to get decision on width of bin and returns mean is taken New its positioning has to be taken VAR is derived from this mean To make decisions over the return data it can be represented using histogram, kernel etc More accurate than HS Used for any confidence Interval HISTOGRAM VS KERNAL Se rie 1. [...]

[...] In histogram s1 the data trend can be derived by taking bins (each bar) as positioning and drawing straight line joining all points Series Later online straight line is viewed and area under graph is treated as probability density function 3. In case of surrogate density function, the area under the graph is taken such that the line drawn produces smooth curve In non parametric density estimator method, purpose remains to give some guidance on how to get the smooth curve 1. Theoretically Kernal is considered superior method but not practically suitable. [...]

[...] Practically histogram is transparent and easier to check 2. Different types of Kernels exist: Box, Triangular and Epanechinikov kernels, Gaussian one Data is arranged dimensionally as per kernel choice VAR and ES is estimated Irrespective of the methods results are subjected to number of sources of error : errors in mapping approximation etc Natural limit exist over how much data can be fitted to actual CLASSIFICATION: BASED ON WEIGHTS Equally weighted Historical Simulation Age weighted Historical Simulation Volatility weighted Historical Simulation Correlation weighted Historical Simulation EQUALLY WEIGHTED HISTORICAL SIMULATION As all observations are treated equally some events are underestimated while others are overestimated In case some major event is overlooked or no response generated means VAR estimate with more error Potential ghost effect exist: Loss Observations Depiction Reason Small cluster of high loss observations VAR under-estimated As all equally weighted and chances of occurrence more VAR over-estimated As chances of occurrence less. [...]

[...] Statistics - Non-Parametric Approaches PURPOSE To get the confidence interval for Historical simulated VAR and Expected Shortfall weighted VAR for non parametric approaches FEATURES Resolves high dimensionality problems as no constraint over distribution type Weighted average of returns used for estimation No Problem with variancecovariance matrices ( This exist when data is segregated. e.g. Here only one master table exist with all data , so no case of variance -covariance ) EXTENSIONS Many times data analysis is not suitable by non parametric method and there is need to convert it into semi parametric form (e.g. [...]

[...] VAR estimates can still be insufficiently responsive to change in underlying risk Reduced distortion caused by events (occured in past but unlikely to recur Reduced ghost effect: As extreme events less weighted so as time passes new observation grows (old observation are kept - improved efficiency) VOLATILITY WEIGHTED HISTORICAL SIMULATION Historical returns are adjusted to take volatility into consideration Volatility adjusted return ( = (recent forecast of volatility of asset i/estimated volatility based on GARCH & EWMA forecast) * r ADVANTAGES OVER TRADITIONAL & BRW AGE WEIGHTED Volatiltiy changes are taken into consideration in direct form and in natural way Incorporates information from GARCH forecast into HS VAR and ES estimation When volatility is high, in this VAR & Expected shortfall would exceed the maximum loss calculated( obtained) through historical simulation and would be close to real one Superior VAR estimate than BRW one CORRELATION WEIGHTED HISTORICAL SIMULATION More involved than volatility weighted One step ahead of volatility weighted adjustment and along with volatility correlations among various returns considered To get correlation adjusted return choleski decomposition is used: CHOLESKI DECOMPOSITION Historical and Current returns available Decomposed using choleski decomposition into square root matrix and uncorrelated noise process Square root matrix are used to get correlated adjusted return FILTERED HISTORICAL SIMULATION • Semi-parametric method: Combination of Historical simulation + GARCH flexibility • The Current returns has mean + Error component • GARCH equation is used to get updated volatilities STEPS DETAILS 1 Forecast volatilities using GARCH 2 Based on new volatilities get realised returns 3 Divide the realised returns such that it produces set of standardised returns that are independent of each other and identically distributed say returns (r1,r2,r3,r4,r5,r6,r7,r8,r9,r10) All independent and when plotted forms normal distribution 4 Put these returns (say M in count ) into historical simulation 5 Bootstrap it ( sample, resample and simulate) 6 End result is based on simulation logic (model) portfolio return value and corresponding loss 7 Based on stated confidence interval get VAR ADVANTAGES OF FILTERED HISTORICAL SIMULATION Non Parametric benefit of HS with current market effect of volatility (GARCH) Faster (For larger portfolios too) Help to get VAR and ES exceeding max historical loss Correlation structure maintained Autocorrelation and past cross-correlation in asset returns can be accounted for Reverse confidence interval can also be estimated ADVANTAGES AND DISADVANTAGES OF NON-PARAMETRIC METHODS • ADVANTAGES Intuitive and conceptually simple Can accommodate fat tails, skewness and non-normal features Can accommodate any position (e.g. [...]