Average Asset Correlation

Article Quant Q&A · Author: AK88

Summary

The document examines two formulas for measuring average correlation among assets in a weighted portfolio. It presents an R example using simulated returns to build a covariance matrix, correlation matrix, and portfolio variance, then compares the two measures. The initial nested loops sum across incorrect index pairs, which includes self-pairs and duplicates terms, producing inconsistent results.

The accepted correction restricts the loops to unique pairs where the second asset follows the first. It also gives vectorized matrix calculations: zeroing the correlation matrix diagonal for the first measure and using the upper triangle for the volatility-weighted pair sum in the second. The example offers a practical implementation lesson, but it does not provide empirical market evidence or discuss estimation uncertainty, missing data, or behavior with degenerate weights or zero volatility.

Key ideas

Average correlation can be computed using portfolio weights and pairwise asset correlations.
The double sums should include each distinct asset pair once, excluding self-pairs.
Portfolio variance and individual variances provide the numerator for a volatility-weighted correlation measure.
Matrix operations can replace nested loops when calculating the weighted sums.

Tags

Statistics Portfolio construction

Full text

# Implementation of total correlation of assets in R


# 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.

## Answer by AK88 (score 1, accepted)

https://quant.stackexchange.com/a/34777

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):

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.

Calculating Average Asset Correlation from Portfolio Weights

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