# Swap Fixed/Floating

A \$10 million 1-year semi-annual-pay LIBOR-based interest-rate swap was initiated 90 days ago when LIBOR was 4.8 percent. The fixed rate on the swap is 5 percent, current 90-day LIBOR is 5 percent and 270-day LIBOR is 5.4 percent. The value of the swap to the fixed-rate payer is closest to: A) \$15,633. B) \$12,465. C) -\$7,667. D) \$19,229. Can someone help explain this to me. I am not getting it yet. Specifically how you get the PV of the floating rate side.

I get close to A. Is that correct? The floating payment is always set one period prior to the date it is supposed to be paid. 90 days ago (at initiation) LIBOR was 4.8%. So at 180 days the floating rate payer must pay (.048/2) = .024. Since the floating rate is = to the market rate, the value at 180 days = 1.024. All you have to do then is multiply it by the discount factor of (1/(1+(.05*(90/360)) = about 1.01136.

Yea, A is correct. A) \$15,633. The fixed rate payments are 0.05 × (180/360) × 10,000,000 = 250,000. The present value of the remaining payments are 250,000/(1 + 0.05 × (90/360)) + 10,250,000/(1+ 0.054 × (270/360)) = \$10,097,947. The floating payment in 90 days is 0.048 × (180/360) = 240,000 and the present value is 240,000/(1 + 0.05/4) = \$237,037. The second floating-rate payment combined with 1 at the end of the swap has a present value of 1 on the first payment date. The present value of 1 is 1/(1 + 0.05 × (90/360)) = 0.987654321 so the present value of the second floating rate payment combined with the principal amount is \$9,876,543. The total value is 9,876,543 + 237,037 = \$10,113,580. The value of the swap to the fixed-rate payer is 10,113,580 – 10,097,947 = \$15,633.

Did my explanation help? You can see if you multiply my floating value (1.01136) by the notional value you get close to 10,113,580 if that helps you understand what I did.

What is messing me up is for the fixed rate side we figure out what the fixed payment is at 180 days and then at the end of the year + the \$10million. We then discount both of those back to PV. On the floating side, we figure out the payment at 180 days and then only the \$10million at the end of the year. Then discount both of those back to PV. Why are we not figuring out the floating rate payment for the end of the year? I am sure this is easy but my mind is not making the connection. Or maybe my thoughts are messed up.

agree with nibs. if for some reason you’re doing them out with actual cash flows (i did this 1st to help get me there and then i started working on the discount factors), you with floating just need to get to your next pay date. so just like nibs said, since it was 4.8% when you started, semi-annual pay gets you 2.4% each pay date- 2.4% x 10 mil = 240k take the 240k/1 + (.05 x 90/360) = 237,037.037 but then you have to do the principal- remember floating side just get y’self to the next pay date so 10,000,000/1 + (.05 x 90/360) = 9,876,543.210 add those to get \$\$ on float side of 10,113,580.25 fix side do you have? -wise it’s \$10,097,946.74 so 10,113,580.25 - 10,097,946.74 = \$15,633.50688 fixed guy paid less so he’s the winnahhhhh as they’d say here in boston. nibs- i forgot how to do the discount stuff- can you show me out quick the discount way to get your fixed side so i could get my answer in discounts subtracted and then just x by the notional? BLOU23 taught this to me about 3 weeks ago and i haven’t looked at it since… it’s fuzzy now. i think the discount way in the long run is faster and makes sense no matter what kind of swap you’re doing.

Someone else can chime in on the intuition because I don’t think I can explain it well. The trick is you know all the payments to be made by the fixed rate payer. You right them out on a time line and discount them all back. For the floating side, the “equivalent bond” will always be trading at par since its coupon rate is = to the market rate. Since you know the value at the next payment date one period before, you just discount that expected value one period into the future back to the present.

I get A as well.

If you have all the discount factors, the fixed side is = to the coupon payment multiplied by the sum of the discount factors except the final discount factor which is used for the maturity value (1+.025) in ma example below (if that makes sense). Ex. A two year semi swap with a fixed rate of 5% Discount factors are .99, .98, .97, .96 (all made up) The value of the fixed is = to .025*(.99+.98+.97) + 1.025*(.96) Is that what you were asking?

Ok, I think I am almost there. It’s coming back to me I think. So if we have a 2 year semiannual pay SWAP, for the floating side we would figure out the floating payments for .5, 1, and 1.5 years and then for the final payment it is just the principal amount? All discounted back to PV of course.

The nice thing about valuing the floating side is you don’t have to go out past the next settlement date to value it. You have the value at the next settlement because you know the coupon rate = market rate = par value. All you have to do is discount that value back to the present. The market value at the settlement date is 1 (par) plus the coupon payment .024 (in your example). That means the value of the floating is 1.024 at the next settlement date and that needs to be brought back to the present.

Ok, got it. Thanks, I remembered reading about that the first time around but I could not make the connection while going back through questions. Thanks Niblita and bannisja!!

yes, remembered it thx. just taking the .05(90/360) + 1 and then the reciprocal of that to get the .987654 discount factor and then just like you said you take that x .025 and add it to the 1.025 x the .961076 discount which was the .057(270/360) + 1 and reciprocal. i’m good now. wandering- if this stuff confuses you, post up any more and we’ll get you there. i for the LONGEST time could not do a swap to save my life and one day it just kind of clicked. they still take me a while but i feel better with them. draw a picture- i think slouis said it a while back- it to me is crucial in seeing the whole thing in my head. my cfa booklet (but not exam ticket) is going to be one big timeline drawing if we get a swaps vignette!

and yes wandering- floating guy just get to that next payment!

Today is my SWAP day…so hopefully I have it down by tomorrow.

since I did not get the right answer in first place, i’m going to post my answer too… (well it’s the one elaborated above, but somehow I do need the practice… ) fix: the fixed rate is 5% paid semi-annual on notional of 10Mio. today----------------90days------------------------180days--------------------------360days -----------------------PV??----------------------------250,000-------------------------10,250,000 discount rate--------aa-< 90d=1+0.05*90/360 discount rate--------xx--------------------------------------<270d=1+0.054*270/360 floating - the LIBOR at initiation was 4.8% paid semi-annual… remember it is reset on the first payment day. today----------------90days------------------------180days--------------------------360days -----------------------PV??----------------------------10,240,000----------------not important------ discount rate--------xxx-< 90d=1+0.05*90/360 Value to fixed rate payer… (he receives floating, pays fix) V=10,113,580.25 - 10,097,946.74 = solution A

good work barthezz, that’s the way I do it too. Timelines!

Thanks barthezz, that visual depiction with the timeline really helps.

Anyone got a good example of plain vanilla currency swap? I’m always scared when I see these. A U.S. firm (U.S.) and a foreign firm (F) engage in a plain-vanilla currency swap. The U.S. firm pays fixed in the FC and receives floating in dollars. The fixed rate at initiation and at the end of the swap was 5 percent. The variable rate at the end of year 1 was 4 percent, at the end of year 2 was 6 percent, and at the end of year 3 was 7 percent. At the beginning of the swap, \$2 million was exchanged at an exchange rate of 2 foreign units per \$1. At the end of the swap period the exchange rate was 1.75 foreign units per \$1. At the end of year 3, firm F will pay firm U.S.: A) \$120,000. B) 280,000 foreign units. C) \$140,000. D) 80,000 foreign units.