# Equal weight correlation swap payoff derivation

I am aware that the payoff for a equal weighted correlation swap is;

$$(\rho_K-\rho)*\text{Notional}$$ where $$\rho = \frac{2} {n(n-1)}\sum_{i < j} \rho_{ij}$$ I am wondering how I can derive this $$\rho$$ term.

• What do you mean by derive? It is an average of the (non-diagonal) entries in the correlation matrix. A n by n matrix has $n(n-1)/2$ elements in its upper (or lower) triangle, so to find the average we add these elements and then divide by this number. For example a 3 by 3 matrix has 3*2/2 = 3 elements above the main diagonal. The average of these elements is $(1/3)*(\rho_{12}+\rho_{13}+\rho_{23})$ – Alex C Jun 5 '19 at 2:19
• Thanks Alex that clarifies it. The textbook I was using there was no summation and I was wondering if the coefficient was somehow capturing all of the pairwise correlations if they were assumed to be equal. Do you know are the volatilities used realised or implied for the correlation matrix? – J19 Jun 5 '19 at 12:38
• AFAIK the payoff at maturity is based on the realized correlations during the period until the swap matures. – Alex C Jun 5 '19 at 17:12