# Implementation of total correlation of assets in R

I am trying to implement the (average) total correlation of assets, which is discussed here and here in R. Specifically, I am looking at $\rho_{av(1)}$ and $\rho_{av(2)}$:

$$\rho_{av(1)} = \frac{2 \sum_{i=1}^N \sum_{j>i}^N w_i w_j \rho_{i, j}}{1 - \sum_{i=1}^N w_i^2}$$

$$\rho_{av(2)} = \frac{\sigma^2 - \sum_{i=1}^N w_i^2\sigma_i^2}{2 \sum_{i=1}^N \sum_{j>i}^N w_i w_j \sigma_i \sigma_j}$$

So far, I have the following code:

set.seed(123)
a = rnorm(100, 0, 0.02)
b = rnorm(100, 0, 0.03)
c = rnorm(100, 0, 0.04)
dt = cbind(a, b, c)
wg = runif(ncol(dt))
wg = wg/sum(wg)
wg = round(wg, digits = 3)

options(scipen = 999)
cv = cov(dt)
VARs = diag(cv)
cr = cor(dt)
length(wg)

sum_1 = 0
for (i in 1 : length(wg) ) {
for (j in 2 : length(wg) ) {
sum_1 = sum_1 + 2 * wg[i] * wg[j] * cr[i, j]
}}
rho_1 = sum_1/(1 - sum(wg^2))

sum_2 = 0
for (i in 1 : length(wg) ) {
for (j in 2 : length(wg) ) {
sum_2 = sum_2 + 2 * wg[i] * wg[j] * sqrt(VARs[i]) * sqrt(VARs[j])
}}

PV = t(wg) %*% cv %*% wg
ss = sum(wg^2 * VARs)
rho_2 = (PV - ss)/sum_2

rhos = c(rho_1, rho_2)
rhos


And the results are:

[1]  1.388473356 -0.004104948


I expected these two to be close to each other. I think there might be an error in my code. I would appreciate if somebody could verify the code.

Thanks.

• the inner of each of your for loops should be for(j in i+1 : length(wg)), you are double counting the total variance, and since one is a numerator, an the other a denominator, it's going to push the two father away from each other. I would expect that to only account for a factor of 4 difference though, which is much less than what you have. Additionally, the first number is clearly wrong -> average correlation > 1? i don't think so. – will Jun 19 '17 at 10:06
• in that case my correlation matrix ref gets out of bounds ... Exactly, it should be between [-1; 1]. – AK88 Jun 19 '17 at 10:08
• try for(j in min(i+1, length(wg)) : length(wg))... – will Jun 19 '17 at 10:09
• got closer: 0.10906278; -0.01044804. I think there is still something wrong ... – AK88 Jun 19 '17 at 10:12

Here is the corrected original code (thanks @will):

 sum_1 = 0
for (i in 1 : (length(wg) - 1) ) {
for (j in  min(i+1, length(wg)) : length(wg) ) {
sum_1 = sum_1 + 2 * wg[i] * wg[j] * cr[i, j]
}}
rho_11 = sum_1/(1 - sum(wg^2))

sum_2 = 0
for (i in 1 : (length(wg) - 1) ) {
for (j in  min(i+1, length(wg)) : length(wg) ) {
sum_2 = sum_2 + 2 * wg[i] * wg[j] * sqrt(VARs[i]) * sqrt(VARs[j])
}}

PV = t(wg) %*% cv %*% wg
ss = sum(wg^2 * VARs)
rho_22 = (PV - ss)/sum_2


A vectorized version:

  # double summation in numerator (including multiplier 2)
diag(cr) <- 0
double_sum.1 <- c(crossprod(wg, cr %*% wg))

# single summation in denominator
single_sum.1 <- c(crossprod(wg))
rho_1 = double_sum.1/(1 - single_sum.1)

# single summation in numerator
single_sum.2 <- c(crossprod(wg * sqrt(diag(cv))))
# double summation in denominator
double_sum.2 <- sum(tcrossprod(wg * sqrt(diag(cv)))[upper.tri(diag(length(wg)))])
PV = t(wg) %*% cv %*% wg
rho_2 = (PV - single_sum.2)/ (2 * double_sum.2)


And the comparison of the two methods (initially they were way off):