# swaps question

A \$10 million 1-year semi-annual-pay LIBOR-based interest-rate swap was initiated 90 days ago when LIBOR was 4.8%. The fixed rate on the swap is 5%, current 90-day LIBOR is 5% and 270-day LIBOR is 5.4%. The value of the swap to the fixed-rate payer is closest to: A) \$19,229. B) \$15,633. C) \$12,465. ____________________________________________ im having a hard time grasping the time-lines in these and figured if i saw it from someone elses prespective might make things clearer - please show your work.

Ninety days ago, LIBOR was 4.8% Fixed Rate was 5% Semi-Annual swap - so payments are 0.025 and 1.025. Current 90 5% --> 1/(1+0.05*90/360) = 0.98765 * 0.025 = 0.02469 Current 270 5.4% -> 1/(1+0.054*270/360)=0.96108 * 1.025 = 0.98510 Fixed side of swap is now worth = 0.02469 + 0.98510 = 1.00979 Floating side: 90 days ago was a 180 day LIBOR -> 1/(1+0.048*180/360) 0.97656 Now 90 days LIBOR = 0.98765 calculated above. So Floating is worth 0.98765 * 1.024 = 1.01135 You are in a Pay Fixed, Receive Floating -> so -1.00979 + 1.01135 = 0.00156 per \$ of investment. You invested 10 Million => 15636 Ans B

came up with 15,637.03…closest to B ?

Here is how I do it. It takes an extra minute, but this method was much simplier than the ones presented in the books in my opinion. This will look like a long explanation, but it is actually pretty simple when you understand it and you can do it pretty quick. The key is to remember that you value a fixed rate bond like you normally would. Then, you independently value a floating security. Keep in mind the floating rate security resets to par at each payment. First the fixed rate bond: The interest rate is 5.0% semi-annually. With a notional principal of \$10MM, that means two payments of \$250,000. The notional principal of \$10MM also is returned on the second payment. So I actually right out on the page: \$250,000 + \$10,250,000 To calculate the value of the bond, you must discount each of these back. Since this is a 360 day agreement with semiannual payments, from initation payments will come at day 180 and day 360. It says you are 90 days into the agreement, which means the payments are now 90 and 270 days away. The 90 day rate was 5.0%. That is an annual rate, so you must divide by 4 (4 = 90/360) to get how much to discount over the 90 days = 1.25%. You do the same thing with the 270 rate 5.4% = 5.4%*(270/360) = 4.05%. Now I add those underneath my cashflows: \$250,000/(1.0125) + \$10,250,000/(1.0405) = \$10,097,947.74 That’s the value of the fixed rate security. Now the floater: Again I first figure out the cashflows but remember, it resets to par at each payment. Think of it as the bond matures and returns the principal at each payment, then issues a new security at par if that makes sense. So they give you the initial rate of 4.8%. That is semi-annual, so the first coupon is \$240,000. You have the \$10MM notional principal returned with this though, so the actual cashflow is \$10,240,000 in 180 days. I write down on the paper: \$10,240,000 Now 90 days later, the rate to discount it at is the same as the 90 day rate for the fixed security (5%/4 = 1.25%). \$10,240,000 / (1.0125) = \$10,113,580.25 Now subtract the two: \$10,113,580.25 - \$10,097,947.74 = \$15,632.51 When I understood this method, the swap section went from being hard for me, to a very very easy chapter.

Job +1, solid explanation.

Thanks all, job71188 - that was actually pretty straightfwd & makes a lot of sense- appreciate u taking the time

After re-reading Jobs posting, do you know why we value the floating rate assuming it resets to par. I understand the effects, but only in a currency swap would this make intuitive sense. Any ideas?

When the floating leg resets, the next coupon rate is set to the current market rate. Like any bond who’s coupon equals the market rate, it trades at par. As soon as the rates start to change, this no longer holds, but at the reset point, the floating rate bond is equal to par + PV(next coupon), and next coupon is obviously calculated using the rate we just reset to. It’s worth remembering that the float + fixed bond combination is just a way to synthesise the same cashflows as the IRS, and the ability to price the floating bond at the reset point gives us a way to value the floating leg, even though we don’t know the actual future cash flows.

Great explanation Jobs - thanks! So why is there not a second cash flow from the floating bond? At the 270-day mark.

The resets are 180 days ( semi-annual as given in the problem ) , not Quarterly . So there is only 180 and 360 days , no 270 days cash flow

But we’re now 90 days out from t0 - so the cash flows are now 90 days and 270 days out. Job is discounting the first one by 90 days rather than the 180.

Schweser lists the two things that confuses people about swaps the most: 1) When valuing a swap, why do you include notional principal when no notional principal changes hands 2) How do you value a floating rate bond when future cash flows are unknown I’ll address both of these and hopefully it will clear this up. 1) Try to think of a swap in this manner. Person A issues a fixed rate paying bond with principal of \$1,000,000 (the notional principal) at 5.0% (the swap rate). Person B buys this bond. With the \$1,000,000, Person A buys a floating rate paying bond from Person B. So the reason they say no notional principal is exchanged, is Person A would be getting \$1,000,000 from Person B only to turn around and send it back to him when he buys the bond. And at the end of the swap, each person would send the other person \$1,000,000. When you net these, \$0 changes hands. To value a swap though, you need to value a fixed rate bond and then value a floating rate bond. While at the end, the net notional principal that exchanges is \$0, think to yourself that actually what is happening is each party pays each other the notional principal (which happens to be the par value of each bond). That will allow you to value the bonds when you consider that each party will have to return the par value of the bond they issued to the other party at expiration. 2. Really the only thing that you need to remember about floating rate bonds is THEY RESET TO PAR AT EACH COUPON PAYMENT. The only time they deviate from par is inbetween coupon payments/reset dates. This was covered in Level 1 briefly. So while I may not know what the cashflows are going to be past the next coupon date, I don’t need to know them. We only have to worry about the cashflows that take place up until the reset date. Look at the problem I did in my first post on how to handle floating rate bonds. You assume that at each payment, the entire bond is due (coupon payment + principal) then a new bond is issued at the new rate (which happens to be issued at par). I forgot to mention that the method I use also works for any type of currency swap. To value the foreign bond, just multiply the notional amount by the beginning exchange rate then follow the same method. When you have the value of the foreign bond in foreign currency, multiply it by the current exchange rate to bring back to domestic currency.

Right , w.r.t the original date the second rest is at 360 , w.r.t the current date the second reset is at 270 . Job is clearly showing that for fixed payment he is discounting the first at 90 days and the second at 270 days . For the floating you don’t need to do the 270 , because we do not know the rate yet. We’ll only know the rate 90 days later. The floating rate is established at each reset date and determines the next cash flow .

Got it. Thank you both for your help! I wasn’t associating “reset at par” with \$0 value.

I’m sorry… I’m still having trouble with this. I understand that the floating rate note resets to par at the reset dates. I also understand that you will not know the cash flow for the second floating payment until the coupon rate is set (in 90 days in this case). What I don’t understand is why we don’t care about this other future cash flow. In 90 days, the floating rate note will pay the amount shown above. At the same time it will reset to \$1. 270 days from now, there will be another floating rate payment, and the \$1 principal is returned. How does this floating rate payment not have a present value which appears in our calculations of the value of the floating rate note??? e.g. look at it from the perspective of someone holding the floating rate note. If I don’t sell the bond, I know that I’m going to get \$xxx in 90 days, and then \$xxx + principal in 270 days. Regardless of how you think of the bond paying back then reissuing at par, how does this reduce the value of that second cash flow to zero?

Job - You are a lifesaver on this topic. I was totally confused but once I went through your example, I was able to apply it to the rest of the chapter. Thanks for the in depth explanation.

Job great post. Made my life much easier.

From job71188 above: ------------------------- Now the floater: Again I first figure out the cashflows but remember, it resets to par at each payment. Think of it as the bond matures and returns the principal at each payment, then issues a new security at par if that makes sense. So they give you the initial rate of 4.8%. That is semi-annual, so the first coupon is \$240,000. You have the \$10MM notional principal returned with this though, so the actual cashflow is \$10,240,000 in 180 days. I write down on the paper: \$10,240,000 ------------------------------ How do you value a floater if you are at the payment day, say at the first 180 days of the swap? In the above, we took the next payment based on previous 180-LIBOR rate, added principle to it, and discounted to today (day 90). If we are already at day 180, what is the floating value? …saw that and the answer to it at EOC, and thought I add it here, might get you if not careful.

at the payment day - coupon has reset… so it is just the face value (1\$) isn’t it?

It should be 10 mil since it reset at the payment day (day 180)