In Reckziegel/MomentMatching: Simulations with Exact Means and Covariances

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.path = "man/figures/README-",
  out.width = "100%"
)

MomentMatching

MomentMatching ports the MATLAB package Simulations with Exact Means and Covariances into R.

Installation

You can install the development version of MomentMatching from GitHub with:

# install.packages("devtools")
devtools::install_github("Reckziegel/MomentMatching")

Example

library(MomentMatching)

Assume that there are 20 risk-factors with a total of 200 realizations from each risk-exposure.

N <- 20 # number of risk factors
J <- 200 # number of realizations/scenarios

Randomly generate a location parameter M and a scatter matrix S that embeds the dispersion around the "true" parameter M and S.

set.seed(1234)

# ...vector of expected values M...
M <- matrix(runif(N), ncol = N) - 0.5

# ...covariance matrix S
A <- matrix(runif(N * N), ncol = N) - 0.5
S <- A %*% t(A)

Now, if we impose the additional assumption that the data generating process (DGP) is normally distributed and try to get some data from the parameters at hand, we are left with:

# generate sample of size J from multivariate normal N~(M,S)
X <- MASS::mvrnorm(n = J, mu = M, Sigma = S)

But this is just a blurred shadow of the "true" parameters generated above, which can be seen by computing the absolute error of the location and dispersion samples:

# Sample Moments
M_sample <- colMeans(X)
S_sample <- stats::cov(X)

max(abs(M - M_sample)) / max(abs(M)) # Sample Location Error
max(abs(S - S_sample)) / max(abs(S)) # Sample Dispersion Error

To minimize this error, Meucci (2009) suggests an affine transformation that greatly helps to alleviate this problem:

# exact match between sample and population moments
X_ <- MvnRnd(M, S, J)

Now, we can compute the moments of the simulated series X_, as we did with X, to convince ourselves that Moment-Matching mechanism, indeed, offers a simulation that better tracks the original data:

# Moments Matching
M_meucci <- colMeans(X_)
S_meucci <- (nrow(X_) - 1) / nrow(X_) * stats::cov(X_)

# Moment-Matching Errors
max(abs(M - M_meucci)) / max(abs(M)) # Location-Matching Error
max(abs(S - S_meucci)) / max(abs(S)) # Dispersion-Matching Error