Interest Rate Swaps - calculating fixed rate for amortizing notional

Hello everyone,

Perhaps I need a refresher but how do we determine a correct fixed interest rate for an interest rate swap which has amortizing notional? Say amortization interest is paid every quarter, amortization takes place from 100 to 0 in 2 equal installments of 50 at month 6 and month 12. I understand we need to weight it but what is the correct weight?

The fixed rate is calculated so that the present value of the floating rate leg equals the present value of the fixed rate leg. If the notional is amortizing, then the longer maturity interest payments will be on paid on a smaller principal, so the present value factors will be correspondingly smaller.

Consider this example: a 2-year, annual pay, fixed-for-floating interest rate swap in which 40% of the notional is paid on the first settlement date, and the remainder on the second settlement date. (I chose 40% so that it’s obvious when I’m talking about the 60% remaining principal; if I had 50% paid, then 50% is remaining, and it’s less obvious that the 0.5 means the remaining balance, not the paid balance). Current spot rates are:

  • 1-year: s1 = 3%
  • 2-year: s2 = 5%

The 1-year forward rate starting one year from today is:

_1f_1 = \frac{1.05^2}{1.03} - 1 = 0.070388 = 7.0388\%

The present value of the floating leg (assuming an initial principal of 1) is:

\frac{3\%}{1.03} + \frac{0.6\times7.0388\%}{1.05^2} = 6.7433\%

The present value of the fixed leg (where SFR is the swap fixed rate) is:

\frac{SFR}{1.03} + \frac{0.6\times SFR}{1.05^2} = 0.970874\times SFR + 0.544218\times SFR = 1.515091\times SFR

Therefore,

1.515091\times SFR = 6.7433\%
SFR = \frac{6.7433\%}{1.515091} = 4.4507\%

Compare that with SFR = 4.9508\% when the notional doesn’t amortize.

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Thanks for the reply. Just to test my understanding, if we had 3 year floating swap, annual payments with 30% of principal repaid on 1st annual settlement, 50% on 2nd and 20% on 3rd settlement and with the following spot rates 3%, 5% and 6%. Then PV (fixed) = SFR / 1.03 + 0.7 x SFR / 1.05^2 + 0.2 x SFR / 1.06^3 = PV (Float), correct?

That looks right to me.

I’ll work out the fixed rate later today (I’m off to the dentist in a little while, yippee!) and post it. You should do the same (well, except for the dentist part).

I get 4.9218% as the swap fixed rate.