Term (days) | Rate |

180 | 9% |

360 | 9.75% |

540 | 10.2% |

720 | 10.5% |

What should the fixed rate be?

**A**:

- Compute the bond discount factor: the PV of a 6-mon LIBOR zero coupon bond for maturity of T days is DF(T) = 1 / [1 + L(0, T) * (T/360)]. think of this discount factor as the value of spot LIBOR deposit that pays $1 T days later.

Term (days) Spot LIBOR Rate Discount Factor 180 6m LIBOR = L(0, 180) = 9% B(0, 180) = 1 / (1+L(0, 180)*(180/360)) = 1 / (1+9%*(180/360)) = 0.9569 360 12m LIBOR = L(0, 360) = 9.75% B(0, 360) = 1 / (1+L(0, 360)*(180/360)) = 1 / (1+9.75%*(360/360)) = 0.9112 540 18m LIBOR = L(0, 540) = 10.2% B(0, 540) = 1 / (1+L(0, 540)*(540/360)) = 1 / (1+10.2%*(540/360)) = 0.8673 720 24m LIBOR = L(0, 720) = 10.5% B(0, 720) = 1 / (1+L(0, 720)*(720/360)) = 1 / (1+10.5%*(360/360)) = 0.8264

- R = (360/T) * (1 - DF(T)) / (DF(1) + DF(2) + ... +DF(N)) = (360/180) * [(1 - 0.8264) / (0.9569 + 0.9112 + 0.8673 + 0.8264)] = 0.0975
- Swap fixed payments would be $20,000,000 x 0.0975 x 180/360 = $975,000

*Bonus Points*

- The interest amount is reset on each coupon reset date and paid one period after, but its value on reset date is par.

Suppose today is day 360, and the LIBOR on that day is l. Looking ahead to day 540, we anticipate receiving 1.0, the final principal payment, plus l*180/540. What is the value of this amount on day 360?We would discount it by the appropriate 6-mon LIBOR: Value on day 360 = (Payment on day 540) * (One-period discount factor) = [1.0 + l * (180/540)] * {1 / [1 + (l * (180/540))]} = 1.0

- R = (360/T) * (1 - DF(T)) / (DF(1) + DF(2) + ... +DF(N))
- The PV of a $1 fixed-rate bond = R * ( DF(1) + DF(2) + ... +DF(N) ) + DF(N)
- Initial swap PV ($1 principal) = PV(fixed) - PV(float) = V(fixed) - $1 = 0

*Category: C++ Quant > Derivatives > Swaps*

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