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Hull Ex. 7.3: Valuing a swap in terms of FRAs

ankit4685

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Hi @David Harper CFA FRM,

request your help to understand the below concept, while doing SWAP valuation to calculate floating rate, we divide floating rate by 2. I'm not sure why we doing this . To my understanding is given floating rate is already semiannual ( highlighted in red in the below screenshot )

d hull_ex_7.3.png
 

David Harper CFA FRM

David Harper CFA FRM
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#2
Hi @ankit4685 But the 2.9% is per annum, it is not itself a six-month rate. This is a point that I am constantly making in my videos: interest rate inputs are given in per annum (aka, annualized) terms unless explicitly specified (but there are almost no exceptions really). This is true throughout John Hull's text, which is the authoritative basis. Here is the setup for this valuation and this setup is typical (emphasis mine):
On a notional principal of $100 million, a swap is arrangement is made so as to receive 6-month LIBOR and pay 3% per annum (with semiannual compounding). The swap has a remaining life of 1.25 years. The LIBOR rates with continuous compounding for 3-month, 9-month, and 15- month maturities are 2.8%, 3.2%, and 3.4%, respectively. The 6- month LIBOR rate at the last payment date was 2.9% (with semiannual compounding). These assumptions are shown below.
It is really important to distinguish between compound frequency and the per annum (default) convention. Here are typical illustrate assumptions:
  • The interest rate is 3.0% per annum with semi-annual compound frequency (fully explicit).
    • But this can be written as: "The interest rate is 3.0% with semi-annual compound frequency."
  • The interest rate is 2.5% per annum with annual compound frequency (fully explicit)
    • But this can be written as: "The interest rate is 3.0% compounded annually."
  • "The interest rate is 6.0% compounded monthly" should be interpreted as "The interest rate is 6.0% per annum with monthly compound frequency" ... now, we could instead write here "The one-month interest rate is 0.50%" but we are not leading with the primary convention need to careful ...
So in the above valuation the fully explicated assumption is "The 6- month LIBOR rate at the last payment date was 2.9% per annum (with semiannual compounding) ... " but it's actually not necessary to add the "per annum," although it's helpful.

This is closely related to, or the same as, our tendency (default) to quote a bond yield in its so-called bond-equivalent basis. If a 10-year semi-annual pay bond has a coupon rate of 3.0% and has a price of $109.0, then:
  1. unless explicitly otherwise specified, this is an assumption of a "coupon rate of 3.0% per annum", and
  2. The yield is given by =rate(10*2, 3%*100/2, -109, 100) = 1.0% * 2 = 2.0%; i.e., "the bond's yield to maturity is 2.0% per annum with semi-annual compound frequency" but that's actually the best interpretation of the abbreviated"the bond yield is 2.0%" !
I hope that's helpful!

P.S. Here is a specific video on interest rate swap valuation and notice I stress this "per annum" assumption at about 3:00:

 
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