Correlation

The Correlation Term

Correlation changes risk through the variance cross term.

highlighted = computed this step

The cross term

Correlation enters through the cross term. With the same weights and sigmas, rho can be +1, 0, or -1 in this exact-core model.

cross term=2w(1−w)ρσAσB\text{cross term}=2w(1-w)\rho\sigma_A\sigma_B

Same or opposite moves

When rho is +1, the cross term adds variance. When rho is -1, it subtracts variance. When rho is 0, the cross term is zero.

ρ∈{−1,0,+1}\rho\in\{-1,0,+1\}

Restricted rho

Real-world rho is estimated from data with uncertainty and is rarely exactly one of these three values. Here rho is a stated exact input. This is descriptive, not investment advice.

rho is a stated model input here\text{rho is a stated model input here}