 # Swaps and Present Value

Ok, i’m having some trouble understanding the logic behind some of the present value calculations in swaps. Maybe you guys can help me out. In a swap the Z discount factors are 1/(1+Rn) And if we have 90 day libor(where 90 day libor is 0.03), we should do the following calculation 1/{1+[.30*(90/360)]} My question is, if we’re taking the Present value, why is the calculation [1+{.30*(90/360)}], when you’re getting the present value of something shouldn’t the denominator have [1.03^(90/360)]

you are coming up with the present value factor here rather than the present value itself. so yes, if you were coming up with the present value itself you would take the ^(90/360), you are only getting the factor, thus annualizing the yield. L(90) = 3% L(180) = 4% B(90) = 1/(1+.03(90/360)) = .99256 B(180) = 1/(1+.04(180/360)) = .98039 Thus to Fixed Rate ( not annualized) is 1-.98039/ (.99256+.98039) = .00994

i was gonna start a new topic for swaps but i guess a continuation on this topic would do. I want to simplify the SWAP valuation topic as much as possible so here goes! Value = the difference in PV of the Fixed and Floating payments --> thus value to fixed payer = PV(Fixed Payments) - PV(Floating Payments) How to get PV Fixed Payments: 1. Get the Zs: each Z = discount factor =  / [1 + (Spot Rate x (x/360))] 2. Calculate the fixed rate payments: Cn = 1+(last Z) / Sum of all Zs - Question: the Cn here is the fixed rate rate quarterly? i.e. is it always quarterly or does it depend on the Q? 3. Get the PV of the Fixed Payer (this might be the hardest part, but still ok i guess): = [Cn x (Z factor for period 1) + Cn x (Z factor for period 2) … Cn x (Z factor for period n)] + [1 x (Z factor for period n) 4. Get the PV of the Floating payer = (1 + Deannualized First Floating Rate) x (Z Factor for First Period) - I don’t really understand what happened here so a little explanation would be much appreciated! 5. Value to Fixed = [ PV(Fixed) - PV(Floating)] x NP - If (+) then Floating must pay fixed - If (-) then Fixed must pay floating Please correct me if any of the above is not accurate, and don’t forget to answer those questions!!!

"Value = the difference in PV of the Fixed and Floating payments --> thus value to fixed payer = PV(Fixed Payments) - PV(Floating Payments) " This is backwards… Value to fixed payer = PV (Floating payments) - PV (Fixed Payments)

Really? I thought your value would be the excess value over your opposite party!!! Damn this is confusing… What’s the intuition?

fpersic Wrote: ------------------------------------------------------- > Ok, i’m having some trouble understanding the > logic behind some of the present value > calculations in swaps. Maybe you guys can help me > out. > > In a swap the Z discount factors are > > 1/(1+Rn) > > And if we have 90 day libor(where 90 day libor is > 0.03), we should do the following calculation > > 1/{1+[.30*(90/360)]} > > My question is, if we’re taking the Present value, > why is the calculation [1+{.30*(90/360)}], when > you’re getting the present value of something > shouldn’t the denominator have > > [1.03^(90/360)] opposed to multiply .03 by 1/4 > > any help would be much appreciated This is because the swap is based on LIBOR and it is an add on rate.

Usif Wrote: ------------------------------------------------------- > i was gonna start a new topic for swaps but i > guess a continuation on this topic would do. > > I want to simplify the SWAP valuation topic as > much as possible so here goes! > > Value = the difference in PV of the Fixed and > Floating payments > --> thus value to fixed payer = PV(Fixed Payments) > - PV(Floating Payments) > > How to get PV Fixed Payments: > 1. Get the Zs: each Z = discount factor =  / [> 1 + (Spot Rate x (x/360))] > > 2. Calculate the fixed rate payments: Cn = 1+(last > Z) / Sum of all Zs > - Question: the Cn here is the fixed rate rate > quarterly? i.e. is it always quarterly or does it > depend on the Q? NO it is not always quarterly. It depends on the rates used to calculate Z. For example if you’re using L(90), L(180), L(270) etc - you would end up with a quarterly rate. However, if you’re using L(180), L(360), then you’re gonna get a semi annual rate… > > 3. Get the PV of the Fixed Payer (this might be > the hardest part, but still ok i guess): > = [Cn x (Z factor for period 1) + Cn x (Z factor > for period 2) … Cn x (Z factor for period n)] + > [1 x (Z factor for period n) > > 4. Get the PV of the Floating payer > = (1 + Deannualized First Floating Rate) x (Z > Factor for First Period) > - I don’t really understand what happened here so > a little explanation would be much appreciated! Basically what they are saying is that the value of the Floating Rate is equal to the sum of all the payments remaining in the future. There are two parts to this - the coupon payments and the total principal. 1. For floating rates you know the next coupon payment from the previous period… 2. For the ending principal payment you just use 1. The reason for this is that a floating rate not resets its value on every payment date - based on the new term strcture. Hence on the next payment date - the value of the floating rate must be 1. > > 5. Value to Fixed = [ PV(Fixed) - PV(Floating)] x > NP > - If (+) then Floating must pay fixed > - If (-) then Fixed must pay floating > > Please correct me if any of the above is not > accurate, and don’t forget to answer those > questions!!! not sure that was such a good explanation… hope it helps!

Usif Wrote: ------------------------------------------------------- > Really? I thought your value would be the excess > value over your opposite party!!! Damn this is > confusing… > > What’s the intuition? yeah because… the PV of floating payments are what you, as a fixed payer, are going to receive… you pay the fixed and rec. the floating, so the value is the difference between the two…

mumukada + chadtap… thank you.

value of swaption at expiration: is this pretty much calculating the difference between the fixed payment (periodic) between the option and the contract, and then multiplying it by the pv factors for the life of the option? (cash settlement)?