I am really bad a swaps right now. Can someone explain the “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” part to me in the qbank answer? Thank you! 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) $12,465. B) -$7,667. C) $19,229. D) $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.
I thought swaps were a pain in the ass until I figured out I needed to treat them like one of those easy LI quant PV problems. I draw a timeline and figure who gets paid what where, discount the fixed and the variable back to the present and then total them up with whatever signs are required depending on if they want you to be the fixed or the float. I think the answer to your question is that once the floating side makes the first payment its coupon resets to the current LIBOR. Since that coupon = the current mkt int rate… the value of the floating side is the same as the value of a floating bond immediately after the reset, par. With that the easiest way to value the floating side is to value it per unit of the notional amount. The 1.024 is 1 (the value @ t=reset per unit of notional… think par) + .024 (the coupon recieved @ t=reset per unit of notional as determined by LIBOR @ t=0) The rest you got. The discount rate is the current 90 day rate of 0.05. So in this here we take the 1.024 and divide it by 1+ (0.05(90/360) = 1.0114 then multiply by 10M to get 10,113,580. they use the pv factors and that just saves time cause you need to discount the fixed side too. So instead of dividing 1.024 and by 1+ (0.05(90/360) like the easy LI PV q’s they are calc-ing the pv factor as 1 / 1+ (0.05(90/360) = .9877. That goofs me every so often because if you round it off too soon sometimes you don’t get very close to one of the choices and that annoys me.