Andras Farkas

2012-Oct-13 11:21 UTC

### hep on arithmetic covariance conversion to log-covariance

Dear All, is there a function in R that would help me convert a covariance matrix built based on arithmetic returns to a covariance matrix from log-returns? As an example of the means and covariance from arithmetic: mu <-c(0.094,0.006,1.337,1.046,0.263) sigma <-matrix(c(0.0037,-0.0001,-0.0370,-0.0136,0.0026,-0.0001,0.0001,0.0008,-0.0015,-0.0011,-0.0370,0.0008,1.0466,0.7208,-0.0455,-0.0136,-0.0015,0.7208,1.1717,0.0346,0.0026,-0.0011,-0.0455,0.0346,0.0348),byrow=TRUE,ncol=5) which I would like to conver to a covariance matrix from a log return. My solution probably should be similar to aprevious post at http://stackoverflow.com/questions/7663690/log-covariance-to-arithmetic-covariance-matrix-function, but the other way around? I would greatly apreciate the help, thanks, Andras [[alternative HTML version deleted]]

Berend Hasselman

2012-Oct-13 17:05 UTC

### [R] hep on arithmetic covariance conversion to log-covariance

On 13-10-2012, at 13:21, Andras Farkas wrote:> Dear All, > > is there a function in R that would help me convert a covariance matrix built based on arithmetic returns to a covariance matrix from log-returns? > > As an example of the means and covariance from arithmetic: > > mu <-c(0.094,0.006,1.337,1.046,0.263) > sigma <-matrix(c(0.0037,-0.0001,-0.0370,-0.0136,0.0026,-0.0001,0.0001,0.0008,-0.0015,-0.0011,-0.0370,0.0008,1.0466,0.7208,-0.0455,-0.0136,-0.0015,0.7208,1.1717,0.0346,0.0026,-0.0011,-0.0455,0.0346,0.0348),byrow=TRUE,ncol=5) > > which I would like to conver to a covariance matrix from a log return. My solution probably should be similar to aprevious post at http://stackoverflow.com/questions/7663690/log-covariance-to-arithmetic-covariance-matrix-function, but the other way around? > I would greatly apreciate the help,An answer can be found here: http://stats.stackexchange.com/questions/18607/function-to-convert-arithmetic-to-log-based-covariance-matrix (Found by googling on the words: arithmetic covariance to log covariance). R code: # Convert arithmetic returns to log return3 logreturn <- function(am,asigma) { M <- 1/(1+am) S <- log( diag(M) %*% asigma %*% diag(M) + 1 ) mu <- log(1+am) - diag(S)/2 list(mean=mu, vcov=S) } # Convert log returns to arithmetic returns linreturn <- function(mu,Sigma) { m <- exp(mu+diag(Sigma)/2)-1 x1 <- outer(mu,mu,"+") x2 <- outer(diag(Sigma),diag(Sigma),"+")/2 S <- exp(x1+x2)*(exp(Sigma)-1) list(mean=m,vcov=S) } mu <- c(0.094,0.006,1.337,1.046,0.263) sigma <- matrix(c(0.0037,-0.0001,-0.0370,-0.0136,0.0026,-0.0001,0.0001,0.0008,-0.0015, -0.0011,-0.0370,0.0008,1.0466,0.7208,-0.0455,-0.0136,-0.0015,0.7208, 1.1717,0.0346,0.0026,-0.0011,-0.0455,0.0346,0.0348),byrow=TRUE,ncol=5) # Run both functions z <- logreturn(mu, sigma) z linreturn(z$mean,z$vcov) Result is: # > z <- logreturn(mu, sigma) # > z # $mean # [1] 0.088297346 0.005932669 0.761207033 0.592495120 0.222699189 # # $vcov # [,1] [,2] [,3] [,4] [,5] # [1,] 3.086716e-03 -9.086663e-05 -0.0145776440 -0.0060945085 0.0018799417 # [2,] -9.086663e-05 9.880583e-05 0.0003402197 -0.0007290309 -0.0008661227 # [3,] -1.457764e-02 3.402197e-04 0.1753220459 0.1404118270 -0.0155352556 # [4,] -6.094509e-03 -7.290309e-04 0.1404118270 0.2467830943 0.0133007368 # [5,] 1.879942e-03 -8.661227e-04 -0.0155352556 0.0133007368 0.0215813095 # # > linreturn(z$mean,z$vcov) # $mean # [1] 0.094 0.006 1.337 1.046 0.263 # # $vcov # [,1] [,2] [,3] [,4] [,5] # [1,] 0.0037 -0.0001 -0.0370 -0.0136 0.0026 # [2,] -0.0001 0.0001 0.0008 -0.0015 -0.0011 # [3,] -0.0370 0.0008 1.0466 0.7208 -0.0455 # [4,] -0.0136 -0.0015 0.7208 1.1717 0.0346 # [5,] 0.0026 -0.0011 -0.0455 0.0346 0.0348 Berend