Question

A financial institution trades swaps where 12 month LIBOR is exchanged for a fixed rate of interest. Payments are made once a year. The one-year swap rate (i.e., the rate that would be exchanged for 12 month LIBOR in a new one-year swap) is 6 percent. Similarly the two-year swap rate is 6.5 percent.

1. Use this swap data to calculate the one and two year LIBOR zero rates, expressing the rates with continuous compounding.
2. What is the value of an existing swap with a notional principal of $10 million that has two years to go and is such that financial institution pays 7 percent and receives 12 month LIBOR?   Payments are made once a year.
3. What is the value of a forward rate agreement where a rate of 8 percent will be received on a principal of $1 million for the period between one year and two years?

            Note: All rates given in this question are expressed with annual compounding. (Steps plz)

Answer 1

Part (a)

Swap rate = (1 - PV Factor corresponding to the last cash flow date of the swap) / (Sum of all the PV factors)

One year LIBOR zero rate, z1 = one year swap rate = 6%

Let the two year LIBOR zero rate = z2

Hence, 2 year swap rate = [1 - (1 + z2)^-2] / [(1 + z1)^-1 + (1 + z2)^-2]

hence, 6.5% = [1 - (1 + z2)^-2] / [(1 + 6%)^-1 + (1 + z2)^-2] = [1 - (1 + z2)^-2] / [0.9434 + (1 + z2)^-2]

Hence, 6.5% x [0.9434 + (1 + z2)^-2] = [1 - (1 + z2)^-2]

hence, (1 + z2)^-2 = (1 - 6.5% x 0.9434) / (1 + 6.5%) = 0.8814

Hence, the two year LIBOR zero rate, z2 = (1 / 0.8814)^1/2 - 1 = 6.52%

Part (b)

12 months LIBOR rate today = F01 = z1 = 6%

12 months LIBOR, 1 year from now = forward rate = F11 = (1 + z2)² / (1 + z1) - 1 = (1 + 6.52%)² / (1 + 6%) - 1 = 7.04%

PV of all the payments by the financial institution = Notional x 7% / (1 + z1) + Notional x 7% / (1 + z2)² = 10,000,000 x 7%/(1 + 6%) + 10,000,000 x 7%/(1 + 6.52%)² = 1,277,349.63

PV of all the payments received by the financial institution = Notional x F01 / (1 + z1) + Notional x F11 / (1 + z2)² = 10,000,000 x 6%/(1 + 6%) + 10,000,000 x 7.04%/(1 + 6.52%)² = 1,186,110.37

Hence, the value of an existing swap = PV of receipts - PV pf payment = 1,186,110.37 - 1,277,349.63 = - 91,239.26

Part (c)

Value of the FRA = P x (R - F11) / (1 + z2)² = 1,000,000 x (8% - 7.04%) / (1 + 6.52%)² = $ 8,503.85