# Collateralized Interest Rate Swap

I am struggeling with the wording "Collateralized" IRS and try to get an understanding out of it based on an example. Especially what it means that in the multi curve models the expectations are calibrated such that the net present value of a swap equals zero (PV Fixed - PV Floating).

I have the following IRS:

Notional 1 Mio. Euro
Fixed Rate Leg = 2%
Floating Rate Leg (Tenor) = 6M Euribor without spread
Maturity = 2 years
Tenor Floating = 6 Month
Tenor Fixed = 12 Month
Day Count Conventions Fixed = Actual/360
Day Count Conventions Floating = Actual/360
Discount Curve = OIS USD


How would an example look like (with math and with numerical numbers)?

• Well, "collateralized" just means the counterparty has put up enough collateral that you don't have to worry about counterparty risk when you price it. So you can analyse it as a vanilla interest rate swap. Sep 19 '17 at 17:02
• @noob2, Thanks for the answer. But how would the collaterlized curve look like? Which Data should I use? I am looking for the OIS adjusted forward curve. Sep 19 '17 at 17:05
• LIBOR curve for projecting Floating Payments and OIS curve for discounting both Floating and Fixed Payments. Find the fixed rate that will make PV of both streams equal. Sep 19 '17 at 17:07
• @noob2, do you have a mathematical and numerical example? Sep 19 '17 at 18:14

Collateralised means that when the IRS is negatively valued (i.e. a liability) for one of the counterparties then they post collateral to the other respective counterparty (i.e. the asset holder) to protect them against default of the liability owner.

Collateral comes in many forms. The 'gold standard' is cash remunerated at the OIS rate, but it could be corporate bonds or equities or some other weaker form of collateral. For standard swaps cash@OIS is the default collateral type specified in the CSA (credit support annex) and this is why the discount factor curve used for standard IRS is the the OIS curve.

Your (annual) fixed leg is 2%. Suppose you received fixed on EUR100mm and the 6M IBOR rate was 1%, this would imply you make a payment of EUR0.5mm in 6M time. At that point the remaining cashflows on your swap mean it is now an asset of EUR0.5mm, so the counterparty will repay the cashflow, that you just paid to them, back to you. But now the money is in the form of collateral and you will pay the counterparty interest at OIS on the EUR0.5mm.

• thanks a lot for the answer. But how would the collaterlized curve look like? Which Data should I use? I am looking for the OIS adjusted forward curve Sep 21 '17 at 19:11
• You keep using the same term; 'OIS adjusted forward curve'. There is no 'adjusted' in swap terminology so I don't know what you are talking about. The collateralised discount curve (when the collateral is cash@OIS) should be treated as the same as the OIS forecast curve in that currency. When you ask 'what data should I use?' Does this mean how to actually create the curve. If thats the case you must be able to acquire market data such as OIS swap prices for different tenors. Often these are quoted to at least the first 1Y in PARs or fwds. Or take IBOR futures rates and IBOR/OIS basis swaps.
– Attack68
Sep 21 '17 at 21:49
• what I mean is, A crucial starting point for the Multi-Curve framework, is the discount curve. This curve has to be correctly constructed because not only is it used to discount the future cash flows generated by the derivative but it intervenes in the construction of the forward curves as well. What I want to understand is how I can build that curve. Is there an example? Oct 31 '17 at 18:21
• Its called multi-curve framework because multiple curves need to be solved simultaneously, they cannot (and particularly the discount curve cannot) be solved independently in a specific order. Here is an example: solve the OIS curve to 10Y given the 20 prices; 10 IRSs versus 3M IBOR each year {1Y, 2Y,.. 10Y} and 10Y 3M-IBOR/OIS basis swaps each year {1Y, 2Y, ...,10Y}. Observe that every price is dependent on both the 3M-IBOR and OIS discount curves so you cannot solve these curves independently, but you can arrive at a unique solution that solves both at once and gives the answer.
– Attack68
Nov 1 '17 at 20:53
• And if you want to know how to do it you need to use numerical optimisers targeting an objective function which minimses the error in your input prices. The bibliography of the book "pricing and trading interest rate derivatives: a practical guide to swaps" includes an excel spreadsheet to accompany the chapter on curve building. In it the author uses excel's nonlinear GRG solver, which is inefficient but practical for educational purposes.
– Attack68
Nov 1 '17 at 21:03