Value of a plain vanilla interest rate swap based on 2 simultaneous bond positions

[Vince Loh]

Hi,

Could someone please help me understand why do we only need to calculate the PV of one floating rate payment cash flow (chapter 7 - Swap)? We receive coupon payment for floating rate every 6 months (similar to Fixed) - shouldn't those payment stream be taken into account?

Thanks in advance

Vince

 

[David Harper CFA FRM]

Hi @VinceL

The coupons are taken into account, but they represent future cash flows. A key fact about the floating rate note (i.e., the floating rate bond leg of the swap) is that it prices exactly to par immediately after every coupon if we assume the rate used for the floating coupon is the same rate used to discount cash flows (this is not necessarily the case, but if further clarifications are not provided, it is assumption). I will use the simplest possible example I can think of:

Consider a floating note, that just paid a coupon, with only two annual payments remaining when the one year spot rate is 3.0% and the one year forward rate, F(1,2), is 5.0%; i.e., the two-year spot rate is ~4.0%. The expected cash flows are:

• $3.0 in one year; i.e., 3% measured now but paid in arrears
• $104.00 in two years; i.e., 4% is the expected one year spot in one year

If the same rates are used to discount to present value (PV), the bond must price exactly to par:

• 3/1.03 = $2.91
  
• 105/(1.03*1.05) = 105/(1.04^2) = $97.08 such that the PV = $2.91 + 97.08 = $100.00

If we change to semi-annual coupons and compound frequency, it's won't matter. However, it's only exactly par at the instant after the coupon pays, then the bond trades above or below based on the changing market rate. But this is why we only need to discount the next cash flow: the par value does "impound" the future expected floating cash flows but discounts them back to one value. I hope that explains!