Consider a payer forward start swap where the swap begins at time Tn and it matures at time TN. Assume that the accrual period is of length δ. In swaps, payments are made inarrears, therefore the first payment occurs at Tn+1 = Tn + δ.
(a) Assuming a notional principal of $1, show that the total value of the payer swap at time t ‰¤ Tn is

where S is the fixed rate (annualized) specified in the contract.
(b) Find the expression for the forward swap rate, Rswap(t), at time t ‰¤ Tn using the results in part (a).Here Rswap(t) is the fixed rate S above which gives IIp(t) = 0.
(c) Also show that the forward swap rate, Rswap(t), can be written as weighted average of simple forward rates, that is

and find wj(t). F(t; Tjˆ’1, Tj) is the simple forward rate for [Tjˆ’1, Tj] prevailing at t.

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j=n+1 j=n+1