Question: Consider a payer forward start swap where the swap begins at some fixed time Tn in the future and expires at time TM > Tn.

Consider a payer forward start swap where the swap begins at some fixed time Tn in the future and expires at time TM > Tn. We assume the accrual period is of length Î´, measured in years. Since payments are made in-arrears, the first payment occurs at Tn+1 = Tn + Î´ and the final payment at TM+1. First, convince yourself that at time t

Where R is the fixed rate (annualized) specified in the contract and L(Tj, Tj) is the spot LIBOR rate applying to the interval [Tj, Tj+1] = [Tj, Tj + Î´].
LIBOR rates are annualized rates based on simple compounding. This means that one dollar invested at time Tj until time Tj + Î´ at the LIBOR rate L(Tj, Tj) will be worth 1 + Î´L(Tj, Tj) dollars at time Tj + Î´.
Assuming a notional principal of $1, show that

, (19.39) M+1 j=n+1

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