Aug 19

Variance of correlated variables

by David Harper, CFA, FRM, CIPM

FRM |

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Learning objective

  • Define, calculate and interpret the variance of correlated variables

If two variables (X and Y) are correlated, the variance of their sum is given by…

correlated_vars_sum

…and the variance of their difference is given by:

correlated_vars_diff

Note in both cases: we can substitute the covariance (X,Y) with the product of: correlation (X,Y) * standard deviation (X) * standard deviation (Y)

For example

  • If standard deviation (X) = 2, standard deviation(Y) = 2, and correlation (X,Y) = 0.5, then variance (X + Y) = 12.0 and variance (X-Y) = 4.0
  • If standard deviation (X) = 4, standard deviation(Y) = 3, and correlation (X,Y) = -0.5, then variance (X + Y) = 13.0 and variance (X-Y) = 37.0
  • If standard deviation (X) = 5, standard deviation(Y) = 5, and correlation (X,Y) = 1.0, then variance (X + Y) = 100.0 and variance (X-Y) = 0.0. Notice this is perfect correlation.

Tips for FRM candidate

  • This formula re-appears as two-asset portfolio variance in the Investment discipline
  • The benefits of diversification are summarized mathematically (under mean-variance) in the fact that under imperfect correlation (correlation < 1.0), standard deviation (X,Y) < standard deviation (X) + standard deviation (Y)
  • But under perfect correlation, variance (X,Y) = variance (X) + variance(Y) +2*standard deviation(X)*standard deviation(Y). This equals [variance(x) + variance(Y)]^2, so that standard deviation (X,Y) = standard deviation (X) + standard deviation (Y), if correlation = 1.0.

Computed examples

This is an EditGrid calculator: you can change values without impacting the source!

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