README.md

In Reckziegel/MomentMatching: Simulations with Exact Means and Covariances

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
#> [1] 0.4293144
max(abs(S - S_sample)) / max(abs(S)) # Sample Dispersion Error
#> [1] 0.1227554

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
#> [1] 0
max(abs(S - S_meucci)) / max(abs(S)) # Dispersion-Matching Error
#> [1] 2.539216e-15

Reckziegel/MomentMatching documentation built on Oct. 15, 2020, 12:04 a.m.