Mr A$.$ Gaylord manages a pension fund and believes that the his stock

selection ability is excellent. However, he is worried because the market could go down.

He considers entering an equity swap where each quarter i$,$ up to quarter M$,$ he pays

counterparty B the previous quarters total rate of return $$ on the S&P $500$ index times

some notional principal and receives payments at a fixed rate r on the same principal.

The total rate of return includes dividends. Specifically, $1+=(+$

$)/1,$where $$

and $$ are the values of the index at i and the dividends received from i $1$ to i$,$

respectively. Derive the value of such a swap by the following steps: