# Swaps - Fixed Rate

Hey everybody,

I was hoping somebody can help me with regards to the swap fixed rate. I understand how to calculate the swap rate at initiation, and to value a swap XX days after initiation, but what happens to the swap fixed rate AFTER the floating rate is reset to par on each settlement date? Does the fixed rate stay the same throughout the entire swap period, or does it have to be recalculated using the new libor rates / discount factors?

For example, take a 1 year quarterly swap, with a fixed rate of 5.6% annual (1.4% quarterly), and the floating rate at initation was 5% (1.25% quarterly). Move forward 3 months, right after the first settlement period (where either the pay fixed or receive floating was owed \$\$), and the floating rate is reset (say, to 5.2%). Would I have to recalculate the fixed rate using the new libor rates (which would be (1-Z3)/(Z1+Z2+Z3)), or will it still stay the same from when the swap originated (at 5.6%)?

Thanks

It stays the same. You only recalculate if you want to initiate a new swap.

Fixed rate is always fixed during the swap , as the name implies

I am still not clear about the floating side of payments. I can calculate fixed rate pretty easily, discount rates, which are going to be same for fixed and floating, payments from fixed, but when it comes to floating side of payments I get confused. Is it going to be all payments (coupon and principal ) considering 1 period back interest rate?

For float you PV the next cash flow which is 1 (par value always as float resets at settlement) + Coupon factor based on the LIBOR at last settlement (LIBOR today is what your next coupon will be).

Technically for Fixed, as you mentioned, you PV all cashflows, including 1 + Coupon at maturity and compare it to the Float side which is equivalent ot PV of next cashflow + 1 (par).

That’s where I struggle to understand. Why only one cash flow and par value? Can you elaborate little more on this? Okay so this is case when you are valuing the swap at a specific date. What if you were calculating the payoff for fixed or floating party at a specific date. In this case, you simply net the value expected to be received by the floating, which is LIBOR of last period, and by the fixed (coupon payment), without considering the par value. So if I keep calculating the payoff at each date, the value will be more or lesss than the swap value derived using 1 coupon and 1 par payment from the floating side. I think I don’t understand how does resetting affect the cash flows. One last thing, is this only valid for interest rate swaps? I have seen this in case of equity swaps as well in which you only take the value in considerating at the valuation date, however, payment from fixed guy is expected throughout the period. How about currency swaps? do you take just one payment into account or all?

I’m going to give this a shot -

Market value of the fixed at any point in time is the PV of the cash flows (coupon plus notional) until the end of the swap term.

At each reset on the floating, the value of the floating side is 1 x notional amount. How? Because the coupon payment for the floating resets to whatever the LIBOR rate is per the term of the swap (90, 270, 180, whatever).

For simplicity with \$100 notional, if you know 1 year rate is 4% at reset date, you know will receive \$4 in one year. Whats the present value of that bond? At the end of the first year you would receive the full amount of notional (100) + coupon - using the PV to discount it back to the reset date, its 100. You don’t know anything about the cash flows after the next reset date, so the assumption is to value it in 1 period increments. If the rate changes to 5% 90 days into the year, you know what the present value of 104 is discounted back using the 5% *(90/360) discount factor convention

For a plain vanilla swap, the arbitrage portfolio is a fixed bond and floating bond. We know how to value the fixed bond.

We dont know all the cash flows of the floating bond, but we do know that it will be worth par on the floating rate reset dates. So we can value the floating bond based on what we know it will be worth on the next reset date which is also the date cash flows will be exchanged between fixed and floating. As the people above me said we discount (1 + The known coupon) back to today using the new term structure, and we’ve got our floating rate bond value.

The net value of the fixed and floating rate bonds is the market value of the swap position.

Thanks Kwalew and Going_For_CFA for exellent explanation. However, I did understand these concepts but the only thing I am not quite sure is, say I am fixed payer and my friend Mumu is a floating payer. I sweat and pay on each coupon date and notional principal as well to my friend Mumu but what do I get in return, just the notional principal. How is this hedging if I wanted to cover my a** from rising interest rates. Is it because the floating payments already captured in the value of swap? I still need clarity about the other swaps. I think I have rock size holes in the understanding of swaps.

Imagine you have floating rate exposure (say, you are paying LIBOR + 100bps semi-annually to your freind called Sathish). To hedge against rising rates, you could enter into a swap with Mumu, who will pay you LIBOR + 100bps semi-annually. In exchange, you will pay Mumu fixed rate.

Every six months, Mumu will pay you LIBOR + 100bps, which you will use to pay Sathish. You will pay Mumu a fixed rate, regardless of what LIBOR actually does.

This way, you have mitigated interest rate risk by fixing the interest you pay each period to the fixed rate of the swap.

PS: You and Mumu swap coupons (not notional). There is no exchange of notional, that’s why it’s called a “notional”.

@drakesterling I understand this, and perhaps my doubts are even more reinforced now because you are getting net interest each coupon date and principal at the end but while calculating the value from floating side you only consider one floating payment and the principal.

… because the fixed rate is “fixed” for a number of periods, you have to find its present value by looking at all cash flows. With a floating rate, there is a fresh rate at the end of each period, but as of now, only the previous floating rate is known for sure. At the end of the current period you will receive the already set floating rate plus principal. No other unknowns exist. Future floating rates will have an impact, but as of now the value is based on known rates only.

@Dreary hmm but I haven’t seen any financial problems in which they simply ignore the data if it is not available, most of the time they forecast it and calculate. In this, they simply neglect the value. Interesting! How about equity and currency swaps?

Sgupta - don’t let it get too confusing for you. The fixed rate (and therefore the fixed periodic payments) are set in advance. We’re all clear on that. It may help if you think about the underlying purpose of a swap. If you think about the present value of the floating side one period before the end of the swap, it gets a little clearer.

It is January 1st, and our swap deal ends at the end of this year. The floating rate (1 year LIBOR) has just been reset, and (if we did exhange notional) we would pay each other back at the end of the year + coupon amount. What is the floating coupon paying? 1 year LIBOR on Jan 1st. What coupon am I going to receive at the end of the year? 1 year LIBOR * notional amount. What is the notional amount I am going to receive? Whatever the notional amount is. Total cash flow = notional + coupon.

What is the present value of that cash flow today? Notional * (1 + 1year LIBOR) / (1 + LIBOR). 1 year LIBOR divided by 1 year LIBOR is… 1. So, the present value today is the notional amount. In fact, the present value of the floating side at every reset date is the notional amount.

so, the floating rate side is priced as though you were to receive the notional amount at the end of the period (at the next reset date). Why can we do this? Because, we haven’t exchanged notional amounts. A firm or investor enters into a swap to protect against changes in interest rates. Usually, they have a set amount of money they are trying to hedge - ie the notional. If I am a bank that pays depositors based on LIBOR but lends at a fixed rate, I’m in trouble if LIBOR rises above the fixed rate i receive. How much am I in trouble? By the difference in rates * whatever my exposure is (ie, ive taken in 10mm in deposits and made 10mm in loans) So, the floating side needs to always be calculated based on the notional at every point