[Returnanalytics-commits] r2068 - pkg/PerformanceAnalytics/sandbox/Meucci/demo

[noreply at r-forge.r-project.org]

Author: mkshah
Date: 2012-06-25 00:48:40 +0200 (Mon, 25 Jun 2012)
New Revision: 2068

Added:
   pkg/PerformanceAnalytics/sandbox/Meucci/demo/InvariantProjection.R
Log:
Created a new demo file for InvariantProjection and checked the results with Meucci's Matlab Code

Added: pkg/PerformanceAnalytics/sandbox/Meucci/demo/InvariantProjection.R
===================================================================
--- pkg/PerformanceAnalytics/sandbox/Meucci/demo/InvariantProjection.R	                        (rev 0)
+++ pkg/PerformanceAnalytics/sandbox/Meucci/demo/InvariantProjection.R	2012-06-24 22:48:40 UTC (rev 2068)
@@ -0,0 +1,78 @@
+#' Annualization and Projection algorithm for invariant
+#'
+#' Project summary statistics to arbitrary horizons under i.i.d. assumption
+#' SYMMYS - Last version of article and code available at http://symmys.com/node/136
+#' Project summary statistics to arbitrary horizons under i.i.d. assumption
+#' see Meucci, A. (2010) "Annualization and General Projection of Skewness, Kurtosis and All Summary Statistics"
+#' GARP Risk Professional, August, pp. 52-54
+#'
+#' @param    N    
+#' @param    K    
+#' @param    X    a numeric vector consisting of a generic (additive) invariant the 
+#'                  follows the general linear and square-root rules for projecting means and volatility
+#'
+#' @return   Ga   a numeric vector with the first 'N' order statistics projected to the horizon 'K'
+#' @export
+#' @author Ram Ahluwalia \email{rahluwalia@@gmail.com}
+#' @examples
+#'           X = GenerateLogNormalDistribution( J = 100000 , a = 01 , m = .2 , s = .4 ) # X = a + exp( m + s * Z ) # generate log-normal distribution
+#'           moments = ProjectInvariant( N = 6 , K = 251 , X )
+
+N = 6   # a numeric with the number of the first N stadardized summary statistics to project
+K = 100 # a numeric with an arbitrary projection horizon
+
+J = 100000  # a numeric with the number of scenarios
+a = -1      # a numeric with the location shift parameter. Mean of distribution will be exp(a)
+m = 0.2     # log of the mean of the distribution
+s = 0.4     # log of the standard deviation of the distribution
+ 
+X = GenerateLogNormalDistribution(J, a, m, s)
+# TODO: Expectations on outputs
+# Ga[1] should equal K*mean(X)
+# Ga[2] should equal sqrt(K)*std(X)
+  
+# show distribution of the invariant. Invariance test: The three distributions should be very similar
+hist( X , 50 , freq = FALSE , main = "Distribution of Invariant" , xlab = "X" )                          # chart 1: distribution of invariant
+hist( X[ 1 : length( X ) / 2 ] , 50 , freq = FALSE , main = "Distribution (1st Half of Pop.)" , xlab = "X" )   # chart 2: distribution of invariant (1st-half of population)
+hist( X[ ( length( X ) / 2 ) : length( X ) ] , 50 , freq = FALSE , main = "Distribution  (2nd Half of Pop.)" , xlab = "X" ) # chart 3: distribution of invariant (2nd-half of population)
+  
+# To compute the standardized summary statistics of Y we need to introduce
+# three sets of players, defined as follows for a generic random variable X: the
+# central moments (15), the non-central moments (16), and the cumulants (17) for each order n
+  
+# step 0: compute single-period standardized statistics (mean, volatility, skew, kurtosis, etc.) step 1: compute central moments
+stats = SummStats( X , N ) # returns ga (standardized statistics), and mu (the central moments)
+  
+# step 2: From the central moments of step 1, we compute the non-central moments. To do so we start
+# with the first non-central moment and apply recursively an identity (formula 20)
+  
+# step 3 of the projection process: From the non-central moments of X-t, we compute the cumulants of X-t.
+# This process follows from the Taylor approximations for any small z and ln(1+x)~x for any small x,
+# and from the definition of the first cumulant in (17). The we apply recursively the identity
+# in formula (21). See Kendall and Stuart (1969)
+mu_ = Central2Raw( stats$mu )
+  
+# step 4: Transform cumulants of X-t into the cumulants of the annualization/projetion Y = X1 + X2 + X3...
+ka = Raw2Cumul( mu_ ) # compute single-period cumulants
+  
+# now compute multi-period cumulants
+# Since X-t is an invariant, all the X-t's are i.i.d. therefore the projected cumulants = k * Ka
+# See also Duc and Schordereret (2008)
+Ka = K * ka
+  
+# step 5: compute multi-period non-central moments
+Mu_ = Cumul2Raw( Ka ) # Transforms cumulants of Y-t into raw moments of Y-t
+  
+# step 6: compute multi-period central moments
+Mu = Raw2Central( Mu_ )
+  
+# step 7: compute multi-period projected standardized statistics of Y-t
+Ga = Mu
+Ga[2] = sqrt( Mu[2] )
+  
+for ( n in 3:N )
+{
+  Ga[ n ] = Mu[ n ] / ( Ga[ 2 ] ^ n )
+}
+  
+print( Ga ) # TODO: add colnames - mean, sd, skew, kurtosis, ...
\ No newline at end of file