Mr. A. Gaylord manages a pension fund and believes that 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 quarter's total rate of return \(r_{i}\) 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, \(1+r_{i}=\left(S_{i}+d_{i}\right) / S_{i-1}\), where \(S_{i}\) and \(d_{i}\) 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:

(a) Let \(V_{i-1}\left(S_{i}+d_{i}\right)\) denote the value at time \(i-1\) of receiving \(S_{i}+d_{i}\) at time \(i\). Argue that \(V_{i-1}\left(S_{i}+d_{i}\right)=S_{i-1}\) and find \(V_{i-1}\left(r_{i}\right)\).

(b) Find \(V_{0}\left(r_{i}\right)\).

(c) Find \(\sum_{i=1}^{M} V_{0}\left(r_{i}\right)\).

(d) Find the value of the swap.