Question

Six-month LIBOR is 3.5%. LIBOR forward rates for the 6- to 12-month period and for the 12- to 18-month period are both 3.7%. Swap rates for 2- and 3-year semiannual pay swaps are 3.6% and 3.8%, respectively. Estimate the LIBOR forward rates for maturities of 18-month to 2 years, 2 to 2.5 years, and 2.5 to 3 years. Assume that the 2.5-year swap rate is the average of the 2- and 3-year swap rates and that OIS zero rates for all maturities are 3.2%. OIS rates are expressed with continuous compounding, all other rates are expressed with semiannual compounding.

Answer 1

Suppose the 18 month to 2 year forward rate is F. The two year swap rate is 3.6% Setting the value of the 2 years swap equal to zero:

(0.035-0.036)e-0.032*0.5+(0.037-0.036)e-0.032*1.0+(0.037-0.036)e-0.032*1.5+(F-0.036)e-0.032*2=0

Which gives F=0.0562. The 18 month to 2 year LIBOR rate is therefore 5.62%

Suppose next that the 2 year to 2.5 year forward rate is F. The 2.5 year swap rate is 3.8% setting the value of the 2.5 year swap equal to zero:

(0.035-0.038)e-0.032*0.5+(0.037-0.038)e-0.032*1.0+(0.037-0.038)e-0.032*1.5+(0.037-0.038)e-0.032*2(F-0.038)e-0.032*2.5=0

Which gives F=0.0592. The 2 to 2.5 year LIBOR rate is therefore 5.92%

Suppose next that the 2.5 year to 3 year forward rate is F. The 2.5 year swap rate is 3.8% setting the value of the 2.5 year swap equal to zero:

(0.035-0.038)e-0.032*0.5+(0.037-0.038)e-0.032*1.0+(0.037-0.038)e-0.032*1.5+(0.037-0.038)e-0.032*2+(0.037-0.038)e-0.032*2.5+(0.037-0.038)e-0.032*3=0

Which gives F=0.0614. The 2 to 2.5 year LIBOR rate is therefore 6.14%