### how to derive yield curve from interest rate swap?

• athos

8 years ago

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 [email protected] 8.5%.

, how could this swap contract help deriving the 5 years' yield rate? I am not a quant. I am still/just learning about IRS recently. My question was **why/how forward rates are used to calculate Interest Rate Swap?** I was told that we need to build the swap curve before we try to value floating leg of an IRS. The main reason being, at the start of the IRS contract we do not have *realistic LIBOR* rates for the entire term and to calculate all the cashflows. Thus we use zero-rate curve derived from yields of defined/liquid securities to build swap curve (bootstrapping). Then use the rates from each tenor in Swap curve to value the cashflows of IRS floating leg. Conceptually above may be correct. And hint me if I am "believing" wrong concept. Market wise, above may be no longer in practice but *improvised* versions.

• 8 years ago

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. Can you please expand a bit on this: "Suppose we know that the 2-year swap rate is 5%, which implies that a bond with a semiannual coupon of 5% per annum sells for par" ? Do you mean a bond representing the fixed leg of the swap? @armensg90 Since the 2-year bond is at par, the fixed coupon payments over the 2 years match the payments in the fixed leg of the 2-year swap exactly. Hence the par rate of the bond is the same as the par swap rate. @wsw Since this is the accepted answer, I'd recommend that you incorporate the feedback from Matt and Phil below. The methodology in this answer is unfortunately quite outdated.

• {{ error }}