The BBB Series – Topic 7: Part 1 – Introducing Interest Rate Swaps

Today's focus is on the mechanics of swaps, mainly interest rate swaps. The objective is an understanding of what's being quoted and why. It'll be intense but the key will be developing an intuition for the mkt.
I must first of all give a shout out to @shortendtrader and @aRishisays for their help in making sure I don’t make too much of an ass of myself.
If you have not followed these two already, please do (though SET’s account is locked).
The global swaps market is YUGE with Interest Rate Derivatives Notional Outstanding of US$466.5 trillion at the end of 2020 according to BIS. In comparison the entire US’ debt outstanding (private and public) stands at US$47T as of end-Dec 2020 according to the BIS.
We’re going to break down a plain vanilla interest rate swap first and foremost. Once familiar with that, in Part 2 we'll introduce and apply the concepts to other types of swaps such as ASW, TRS , Basis Swaps and Cross Currency Basis Swaps.
Full disclosure: I do not trade swaps, I only observe and try to understand the market. If you are reading this and an active participant please point out where I’m wrong or missing nuances.
And for the love of Fabozzi and all that is carry, please read the topic on bootstrapping
So first, let’s lay out the concept of a swap – the name itself implies an exchange of two things. A swap always has two “legs” where one cashflow is being swapped for another.
I want you to keep 3 simple principles in mind:

1. A swap is simply 2 fixed income instruments combined, much like a pair trade is 2 instruments traded vs each other
2. We first analyze the swap from initiation i.e. t=0
3. The mkt aims for value equivalence (aka no free lunch)
Why do swaps exist? Well initially and from a high-level perspective swaps are about comparative advantage which is about relative borrowing power and exploiting that.
Suppose the following:

Company A: can borrow Fixed at 5% or Floating at LIBOR+100bps

Company B: can borrow Fixed at 6% or Floating at LIBOR+175bps
Company A is surely the better credit as its financing is lower overall. But say its asset cashflows are variable and their borrowing tends to be fixed, can they match cashflow types and borrow for cheaper?
If Company A borrows fixed @ 5% then agrees with Company B borrowing floating at 3ML+175 to swap cashflows whereby A pays B LIBOR+85 and B pays A 4.95% fixed – what is the net effect?
Company A now has floating rate debt 15bps cheaper, while receiving 5bps less in fixed: Net -10bps

Company B now has fixed rate debt 105bps cheaper, while receiving 90bps less in floating: Net -15bps

Win-win. Graphically it looks like this:
So it’s about transforming one type of cashflow or liability into another type – and banks are heavy users of swaps. Why?

Because a bank typically funds itself through avenues that are short term and therefore change…i.e. float – LIBOR being a prime example of this.
A bank that holds a fixed income bond might want to swap its asset cashflows into floating rate cashflows so it matches their funding profile.
As the usage of swaps increased it became apparent that swaps markets were inherently capturing mkt expectations of interest rates (e.g. if you worried about rising rates you could turn fixed coupons into floating ones ahead of time using swaps).

With this the swaps market grew
The plain vanilla IRS is the most widely traded type of IRS and refers to a fixed-to-float IRS. Because there are two legs to any swap, the market must have a convention to focus on one side so everyone is referencing the same thing.
There must be a payer and a receiver. If the term ‘payer’ is mentioned, who is the payer? The party that is agreeing to pay the FIXED coupon of the swap.

What is the swap rate that gets quoted? The FIXED coupon rate of the swap.
In the words of @shortendtrader: “The second you hear someone say ‘i want to buy a swap’ then you know you can widen the bid/offer a touch” – paying vs receiving not buying vs selling
So for a vanilla IRS the focus is on the FIXED coupon. As we work through it will make more sense why.

Here’s a typical example of a vanilla IRS:
1.The fixed coupon
2.The floating index (notice you can add a spread to it)
3.DC Stripping
4.The fixed leg NPV
5.The floating leg NPV
6.The swap’s premium
At the top the Solver is set to Premium, meaning we will solve Fixed Coupon for a set premium.
Recall the 1st of 3 principles: a swap is made up of 2 instruments. Essentially a vanilla IRS is swapping a fixed coupon bond for a floating one.
a) Leg 1 is a 5y fixed coupon bond paying 0.872421% coupon semi-annually
b) Leg 2 is a 5y floating rate note paying 3M LIBOR every qtr
SWAP PARTS 1 & 2: The fixed and floating coupon
A) The Fixed Coupon leg (like a fixed coupon bond) has a known series of cashflows paid semi-annually, which is $43,621.05 every 6 months for 5 years.

10MM x (0.0872421/2) = 43,621.05
B) The Floating Leg (like a FRN) has an UNKOWN series of cashflows paid quarterly tied to LIBOR. In bootstrapping we learned that we can strip par swaps curves then imply forward zero-coupon rates for any tenor. This is how we generate the future implied 3M LIBOR rates in a swap.
Note: BBG’s swap functions are in a bit of a mess right now – FWCV does not incorporate dual-curve stripping, FWCM has just started to and ICVS displays both. Make sure everything is matched when trying to tie values from each screen (bbg has a paper on this found in help pages).
On the 6) Cashflow tab you can see the swap cashflows – starting from today (Principle #2) what are the fixed coupons every 6 months and floating rates every 3 months?
By now we’re familiar with discounting bond coupons (cashflows) to arrive at Price (NPV).

But what do we use for discounting swaps? That’s where the bootstrapping topic comes in again.
Using spot rates (stripped curve to derive zero rates) we can discount the fixed leg’s cashflows using the swaps curve. Prior to the GFC, IRS discounting used the LIBOR swaps curve which was…made up of LIBOR swaps across different tenors.
So there was recursiveness built in where swaps were ‘self-discounting’. They thought it was basically risk-free. Oops.
The convention now is to discount future cashflows using a true risk-free curve, it was OIS based on Fed Funds but since Oct 2020 has shifted to SOFR curve discounting. You imply fwd cashflows using the same swap curve and discount using a different curve.

More details later on.

We have constant cashflows for fixed and implied forward cashflows for floating and we also have the SOFR spot curve (zero rates) to discount the cashflows.
Recall from the bootstrapping topic that we can take a spot rate and convert it into a discount factor for easy discounting. The screenshot shows the DFs and how we get a NPV for each leg. The sum of the PVs equal the NPV of the leg.

This is the crux of the swap valuation and Principle #3: No Free Lunch. The idea is that at initiation (t=0), both the payer and receiver enter a swap where both sides are currently equal.
If it wasn’t there is hardly any incentive to enter the agreement. Value equivalence is when both NPVs are equal to one another.
The difference in the NPVs is the premium, which logically means the premium should be 0 at inception.
The premium is expressed as a percent of par – so a premium of 1 is 1% of par notional – on $10MM is $100,000.
IMPORTANT: The premium on SWPM is always expressed in the 1st Leg’s perspective of NPV with respect to paying/receiving. By default Leg 1 in an IRS on SWPM will be the fixed leg as per convention.
Hence if you are receiving fixed, the premium is the upfront for the PAYER. So a +1.00 premium in this screenshot means the person receiving the fixed leg has an NPV $100k higher than the NPV of the counterparty’s floating leg.
To be fair – the fixed PAYER (i.e. floating receiver) should be paid $100k up front. This is to make both parties economically equal at inception.
Let’s look at premium another way: Because there is an existing swap curve which determines the floating rate resets, and there is an SOFR curve to discount cashflows, neither of them can be changed by trader of a swap – so what can be negotiated?
The trader can only negotiate the fixed coupon or a spread over the floating rate. For the par swaps curve we assume 0 spread on floating, leaving only the fixed coupon.
At 2% fixed, the NPVs of the two legs net out to be 5.57% in favour of the fixed coupon – i.e. the fixed leg NPV is too high vs the floating leg NPV given where swaps are trading.
KEY TAKEAWAY – the swap rate you see quoted on a daily basis is the FIXED coupon rate of the swap required to ensure the premium of a swap is 0 – i.e. the swap is trading at par.

That is why the swap curve is called a par swaps curve.
PnL and Risk of a Swap

Once the swap has been initiated (at par), swap rates no longer stay the same, they move around just like any other yield. This results in the floating leg NPV of the swap leg changing as the implied forward rates change while…
…the fixed coupon leg cashflows do not change (note that the discount curve changes but because both legs share mutual discounting that effect is neutralized).
If rates were to rise then the floating leg would adjust accordingly while the fixed coupon leg would not – resulting in a loss for the fixed receiver and a gain for the fixed payer.

In other words, the receiver is net long duration and the payer is net short duration.
Recall from our coverage of duration on fixed vs floating rate notes that a floater’s duration is very low while a fixed bond’s is objectively not – this carries over into a swap which is basically a two-asset portfolio of (long/short) floater+fixed.
Special Consideration of the risk profile surrounding Fixings:
@shortendtrader brought up a very interesting aspect here particularly for short end swaps; for something like a 1 year swap vs 3ML there are effectively 4 fixings where the floating leg is effectively a strip of FRAs
Now LIBOR fixing happens at 11:55am London at which point the swap’s cashflows will adjust. This means that at 11am the swap has 100% risk but only 75% of the risk at 11:55am. Such simple intuition but blew my mind.
This wraps up the basic nature of how a vanilla interest rate swap works but the principles translate over to other swaps that you encounter and that we cover in Part 2. Remember the 3 principles for structuring a swap and the 6 basic parts of a swap to take note of.
Before we end this part however, I want to cover some changes and developments that have taken place in the swaps market over the years that have resulted in some added intricacies to swap construction and valuation.
As swaps are OTC derivatives, they are essentially a private agreement between two parties. The agreement that covers this agreement is governed by what is known as an ISDA Master Agreement that includes some standardized elements of a swaps trade...
...including counterparty default scenarios. Not everyone can get an ISDA which means not everyone can trade swaps bilaterally (between two parties)
The Dodd-Frank Act mandated that all eligible swaps (which was a lot) have to be centrally cleared by a clearing house.
This massively shifted the counterparty credit risk over from a bilateral OTC world (with optional initial margins) to a cleared world where margins are mandatory.
The largest swaps clearing house is London Clearing House (LCH) which handles the majority of the world’s cleared vanilla IRS. How it works is you enter a bilateral swap with someone under your ISDA Master Agreement, then you send the swap trade to LCH and once they accept it…
…the original swap gets replaced by two swaps with LCH sitting in the middle and conducting the swap between both parties.
The other part of Dodd-Frank’s impact on swaps was the requirement for eligible swaps to be executed on a Swaps Execution Facility (SEF). A SEF is an electronic platform where you can buy/sell swaps with other market participants much like a formal exchange but for swaps.

Once it became clear that interbank risk was and could be significant in the aftermath of the GFC, discounting using LIBOR curves was no longer appropriate.
Post-GFC the market demanded collateral for swap transactions, which is covered under a Credit Support Annexes within an ISDA Master Agreement.
CSAs were valued using OIS, which as I mentioned in Topic 2, is closer to risk-free. Furthermore, after Dodd-Frank the amount of swaps that had to be cleared increased dramatically and thus, the amount of mandatory margins increased dramatically.
Once the market shifted to collateralized swap transactions, the need to move away from self-discounting was evident.
This introduced the concept of dual-curve stripping. Using the swap curve (which is a ‘risky’ curve) we can imply its forward cashflows, but then to value the swap we have to discount using a different risk-free curve. This is a core aspect of swap valuations.
The key takeaway here is that self-discounting vs dual-curve discounting is a matter of uncollateralized vs collateralized swaps.

But wait, there’s more! You see Fed Funds is an unsecured overnight rate whereas LIBOR’s successor SOFR is a Secured Overnight Financing Rate.
You all know about LIBOR’s demise and come December 2021 LIBOR will be no more with SOFR taking its place. This leaves quite a headache for the market and this poor idiot who decided to write a topic on swaps with decade-old information in his head!
In October 2020, London Clearing House (LCH) shifted over from FedFunds (OIS) to SOFR discounting, marking a transition for the swaps market from OIS (EFFR) to SOFR discounting.…
This is why now on SWPM you will notice the discount curve is the Curve 490 – SOFR, which used to default to curve 42 – OIS curve as late as last year.
I’m paraphrasing here so I hope I don’t butcher Rishi’s work. As mentioned before, the GFC brought to light the issues with assuming interbank credit risk was nil.
Rishi beautifully terms as the industry shifting from “money at any risk” to “money with manageable risk”.
The mkt then moved to make adjustments to ascertain the true cost of doing business in swaps by accounting for counterparty default, funding of position, margin, etc. These adjustment became know as Cross-Valuation Adjustments of XVA
#1 Credit Valuation Adjustment, which the CSA that is in the ISDA Master Agreement tries to address by defining and recording the collateral offered by the swap counterparties. A counterparty can default, meaning the payoff will not be realized
This risk is incorporated into the valuation by augmenting the payoff to take account of default. For instance, one can hedge for default risk via Credit Default Swaps, locking in the cost. Thus, we get Credit Valuation Adjustment (or CVA).
#2 - Clear recognition of the vulnerability of banks during the crisis means their default risk is also to be taken into account. Hence, we get Debt Valuation Adjustment (or DVA).
#3 Since derivs pricing is about estimating the cost of setting up a replica portfolio, the cost of funding should be an essential part of the price. As mentioned already, GFC proved interbank credit risk real and significant which led to short-term funding drying up completely
As a result, funding spreads for banks (like Libor-OIS), which had always been an afterthought, widened ~350bps. This is when FVA became a widely accepted part of the business.
FVA, or Funding Value Adjustment is basically about incorporating the cost at which the bank can fund collateral, when necessary, as a spread over the risk-free rate, since it would be compensated for the collateral at that rate
#4 The GFC led to regulators requiring more derivatives to be cleared on exchanges or for the positions that are not exchange cleared, to post initial margin to an independent third party, both leasing to Margin Valuation Adjustment (or MVA).
#5 Regulatory efforts since ’08 to reduce interbank credit risk in times of stress come in the form of tweaks to risk-weighted assets (RWA) & higher cap requirements as a % of RWA.
KVA or Capital Value Adjustment is essentially the cost of raising money from shareholders whereas FVA is the cost of raising money from lenders.
All of this has had a drastic impact on the derivatives business, leading to either innovation like the advent of mark-to-market cross-currency swaps, or a decline in business for other derivatives.
While you could theoretically hedge CVA via CDS, the reality is that CDS is not always readily tradable on the counterpart at hand.
Unhhedged, CVA risk is warehoused on the balance sheet, and a part of the profit is set aside to account for realized losses from counterparty default.
The same for other XVA components as well, which is why most banks have an XVA desk set up to either manage/hedge or calc adjustments in derivative valuations for these risks
Please thank Rishi for the insights on XVA. If he's ok i'll link his full note without my bastardisation of it later.
That’s it for the vanilla IRS, hopefully as understandable as possible!

In the following part we will use the basic principles we covered here and delve into the basics of other commonly mentioned swaps such as ASW, TRS and XCCY.
Bonus question: Today is September 10th 2021. I want to value a par swap ($1MM notional) that started on Sep 3rd when 3ML was 0.1155 and the fixed coupon was 0.887147%. Rates have not moved at all across the curve.
Is the premium still 0? Is the NPV of the swap still 0? Assume ACT/360 daycounts for both legs. I don’t need exact numbers, just an answer based on intuition.
As mentioned, here is a link to @aRishisays' full note on XVAs:

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