Portfolio simulation using Cholesky decomposition - r

Question

I do a MC simulation of a portfolio with 4 assets (Bond, equity, equity, cash market)
I use monthly steps and my simulation horizon is 10 years i.e. 120 steps. My final goal is to compute the yearly expected shortfall, i.e. taking the worst 5% of the Portfolio Returns.
The simulation seems to be ok - at the first glance. However, I have the impression the drift dominates the process over time, so my expected shortfall is even positive for the long end. Also the expected shortfall eventually decreases when I increase the weights for equity. This is also true when I set the expected return for each asset to zero and hence the increased risk should drag the expected shortfall down.
I expect a bug in my code but can't see it. Any advice highly appreciated!
#maturity in years
maturity <- 10
#Using monthly steps
nsteps <- maturity*12
dt <- maturity / nsteps
#number of assets
nAssets = 4
#number of simulations
nTrails = 10000
#expected return p.a. for each asset, stored in vector BM.mu
BM.mu <- rep(NA,nAssets)
BM.mu[1] <- 0.0072
BM.mu[2] <- 0.0365
BM.mu[3] <- 0.04702
BM.mu[4] <- 0.0005
#defining variable size
simulated.Returns <- array(NA, dim = c(nsteps+1, nTrails, nAssets))
cumulative.PortReturns <- matrix(rep(NA,nsteps*nTrails), nrow = nsteps, ncol = nTrails)
ES <- rep(NA, maturity)
#defining my monthly correlation and covariance matrix
corr_matrix <- matrix(c(1.000000000, -0.05081574, -0.07697585, 0.0051,
-0.050815743, 1.00000000, 0.80997805, -0.3540,
-0.076975850, 0.80997805, 1.00000000, -0.3130,
0.005095699, -0.35365332, -0.31278506, 1.0000), nrow = 4, ncol = 4)
cov_matrix <- matrix(c(1.44e-04, -2.20e-05, -3.86e-05, 8.44e-08,
-2.20e-05, 1.30e-03, 1.22e-03, -1.76e-05,
-3.86e-05, 1.22e-03, 1.75e-03, -1.81e-05,
8.44e-08, -1.76e-05, -1.81e-05, 1.90e-06), nrow = 4, ncol = 4)
#defining my portfolio weights
port.weights <- c(0.72, 0.07, 0.07, 0.14)
#performing cholesky decomposition
R <- (chol(corr_matrix))
#generating standard-normal, random variables
x <- array(rnorm(nsteps*nTrails*nAssets), c(nsteps*nTrails,nAssets))
#generating correlated standard-normal, random variables
ep <- x %*% R
#defining the drift
drift <- BM.mu - 0.5 * diag(cov_matrix)
#generating asset paths
temp = array(exp(as.vector(drift %*% t(dt)) + t(ep *sqrt(diag(cov_matrix)))), c(nAssets,nsteps,nTrails))
for(i in 2:nsteps) temp[,i,] = temp[,i,] * temp[,(i-1),]
#changing dimension of the array temp from dim(nAssets, nsteps, nTrails) to dim(nsteps, nAssets, nTrails)
simulated.Returns <- aperm(temp,c(2,1,3))
#computing portfolio returns for each simulation (nTrails). To do this, each step is weighted with "port.weights"
#Since I generate continuous returns, I first transform them into discrete, multiply with weights and then transform back into continuous.
for (z in 1:nTrails) {
for (i in 1:nsteps) cumulative.PortReturns[i,z] = log(1+((exp(simulated.Returns[i,,z]-1)-1) %*% port.weights))
}
#Finally I compute the monthly expected shortfall (5%-level) by taking the average of the 5% worst portfolio yields
#I do steps of 12 as I calculate the ES at the end of each year
z = 0
for (i in seq(12, nsteps, by = 12 )) {
z = z + 1
ES[z] <- mean(sort(cumulative.PortReturns[i,]) [1:(0.05*nTrails)])
}
#plotting a sample of simulated portfolio returns
#library(QRM)
plot(as.timeSeries(cumulative.PortReturns[,1:100]), plot.type = 'single')

Answer 1

From your comments, you have defined BM.mu to be the annual expected return for each asset. However, you are simulating each sample path using monthly rather than annual steps. This then needs to be incorporated in your drift variable by scaling BM.mu to the expected monthly return accordingly:
#defining the drift
drift <- BM.mu/12 - 0.5 * diag(cov_matrix)
Without this, you are computing the drift value using an annual expected return value and a monthly covariance matrix. This is resulting in a larger drift than you expect, which would impact on the results you are seeing.