how to derive yield curve from interest rate swap?

According to some textbooks, to derive the yield curve, quote

• overnight to 1 week: rates from interbank money market deposit,


• 1 month to 1 year: LIBOR;


• 1 year to 7 years: Interest Rate Swap;



• 7 years above: government bond.

I'm a bit lost here: how can an IRS rate be used to derive yield curve?

Yield rate is the discount rate, if $yield (5 years) = 4.1 \%$ , it means the NPV of 1 dollar 5 years later is $NPV ( 1 dollar, 5 years) = 1/[(1+4.1\%)^5] = 0.818$.

While interest rate swap is a contract among to legs. Assume a 5 years' IRS contract is

• leg A pays fixed rate to B @ 8.5%, while A receives floating rate @ LIBOR +1.5%


• leg B pays floating rate to A @ LIBOR +1.5%, B receives fixed rate@ 8.5%.

, how could this swap contract help deriving the 5 years' yield rate?

Answer

You should take a look at the example from Hull's book.

Assume that the 6-month, 12-month, 18-month zero rates are 4%, 4.5%, and 4.8%, respectively.

Suppose we know that the 2-year swap rate is 5%, which implies that a 2-year bond with a semiannual coupon of 5% per annum sells for par:

$$2.5 e^{-0.04 \bullet 0.5} + 2.5 e^{-0.045 \bullet 1.0} + 2.5 e^{-0.048 \bullet 1.5} + 102.5 e^{-2 \bullet R} = 100 \; .$$ Solving for $R$ above gives a 2-year zero rate $R$ of 4.953%. We can keep going to compute the 3-year zero rates, etc.