Greeks discount the value of LDI strategies

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I enjoyed writing that as the title to this piece and hopefully the choice of title will become clear as you read through. It was fun trying to think up a colourful title to a piece about the dull topic of swap discounting and how the "greeks", more commonly found when describing options, have made an appearance into the discussion on swap discounting.



Other titles that made the shortlist but didn't win through in the end included:

·  "Cross gamma risk hits swap mark-to-market". Discarded for being too geeky.

·  "Who is gamma and why is she cross with LDI?" Discarded for potentially being construed as gender biased.

·  "What do you get when you cross "gamma" with "LDI"? Discarded for being too cheesy.

In the last few months I have heard a few people mention cross-gamma risk and how this is impacting swaps transacted as part of an LDI strategy. And so I thought what better way to spend a Saturday night than with a glass of white wine, a bowl of popcorn and writing about why your swaps have cross-gamma risk. Oh, and no harm having the snooker on in the background since it's looking like the Rocket is in fine form at the Crucible.

(1) Simple folk like me need simple explanations....especially after the first glass of wine.
Many of you will be familiar with the concepts of duration and convexity and how they apply to bonds. Duration (or more specifically modified duration), we are told, is a handy measure for calculating by how much the value of your bond will increase in response to a 1% fall in yields. Of course, we are then told that, unfortunately, this handy rule of thumb works best for small changes in yield and that for larger changes in yield the actual change in the value of your bond will be different to that implied by the bond's duration. This is because it turns out that duration, far from being a single static number, is actually a dynamic number and one that will be different depending on the level of prevailing yields. And so, to our first rule of thumb we must add a second rule of thumb:
- as yields increase, duration decreases
- as yields decrease, duration increases
[In technical terms, this relationship means that bonds are convex, and, more specifically, positively convex.]

(2) I swear they do this just to confuse me.....mind you, not too difficult a task after the second glass of wine.

So, just when I thought I had all that mastered, along come the market practitioners and confuse me by susbtituing the terms duration and convexity with terminology often used in option markets. And so, duration and convexity, become interchangeable with delta and gamma respectively! Recall that the delta of an option (or derivative) is change in the value of the option with respect to a change in the value of the underlying asset on which the option is based. Gamma is then the change in the delta of the option with respect to a change in the value of the underlying asset.

And so the "delta" (duration) of a bond or a swap becomes the change in the value of the bond/swap with respect to a change in interest rates. The "gamma" (convexity) of the bond or swap is then the change in the delta (duration) with respect to a change in interest rates. Delta is also generally referred to as "PV01" - the change in the present value of a swap or bond for a one basis point change in yield.

 [As an aside, you may ask if this is just a loose use of terminology or is this more widespread. Certainly, Flavell ("Swaps and other Derivatives") does indeed make use of "delta" and "gamma" when referring to bond and swap portfolios and their risk management as part of a trading book and so I would be inclined to accept that their use is reasonably widespread and generally well understood and accepted even if some may argue it is technically incorrect].

(3) And then along comes cross gamma...boy this aint going to be easy, especially after the third glass of wine.

So by now I'm just about managing to keep all of this straight in my head, and keep my head vertical after my 3rd glass of di vino bianco (white wine), when along comes cross gamma. [Btw, I am practising my Italian in advance of my trip to Amalfi in a few weeks. So far I know how to ask for a glass of wine and beer - sorted, I say, although the kids may not think so especially seeing as how we plan to hire a car and word has it that the roads are quite treacherous.].

In the world of options, cross gamma arises when the delta of your option (or derivative) changes in response to changes in, not one, but two (or more) underlying assets. Cross gamma is then the change in the option delta with respect to a change in the value of the second asset.

Not so fast, I hear you say, an interest rate swap is surely just based on a single interest rate and so how on earth does it acquire cross gamma risk.

Well, it turns out that the plain old interest rate swap has become a little more complex than we had all previously thought. In particular, when valuing the interest rate swap, it is possible for the future floating rate payments on the interest rate swap to be forecast or estimated using one interest rate and for these same future payments to be discounted back to the present time using a completely different interest rate. When this happens, then your plain old interest rate swap acquires "cross gamma risk". Cross gamma refers to the change in the delta of the swap in response to a change in the (different) interest rate being used to discount your swap.

(4) Going back to basics ...better hold off on the next round, this may require a measure of sobriety.

Many LDI strategies hold interest rate swaps executed against 3-month Sterling LIBOR - so receive a fixed rate of interest and pay floating-rate 3-month Sterling LIBOR.

(4.1) Estimating/projecting the cashflows on these swaps
The future fixed payments are obviously known since these were fixed at execution of the swap, say 4% p.a.. At any point during the lifetime of the swap, the projected future floating rate (LIBOR) payments are unknown since future 3-month Sterling LIBOR fixings are unknown. These future, unknown payments can be estimated using our estimates of future 3-month LIBOR fixings (as implied by a 3-month "LIBOR interest rate curve"). This is the first of the "interest rates" we encounter.

(4.2) Valuing our swap cashflows

To value the swap we now need to discount both streams of cashflows (the fixed cashflows and the floating cashflows) back to the present time. The discount rate used depends on the type of collateral backing this swap⁽¹⁾ (A statement which assumes a massive amount of knowledge but see the additional note at the bottom of this piece for a more detailed explanation of why this is the case).

(4.3) Choosing a discount curve
If, the collateral is "Sterling cash and gilts only", then the swap is likely to be discounted using an interest rate curve derived from SONIA interest rates (a "SONIA interest rate curve") - and this is the second interest rate curve.

(4.4) Impact of choice of discount curve

So, cross gamma risk arises. Cross-gamma risk refers to the fact that the delta of our (3-month LIBOR) interest rate swap will change due to changes in the SONIA interest rate curve i.e. a second, and different, interest rate curve to the one on which the future floating rate payments are based (LIBOR). There are some important observations that need to be highlighted at this point even if they may appear obvious at first reading:

a)    Swap values now depend on two interest rate curves

These two interest rate curves are LIBOR (for projecting future cashflows) and SONIA (for discounting those cashflows).  

A corollary of this is that the swap value therefore also depends on the difference between these two interest rate curves. 

A word of caution. One common mistake made by many, including less experienced practitioners, is to look at the difference between spot 3-month LIBOR and spot SONIA rates and conclude that it is this difference that influences our swap valuation. But, because LDI strategies typically execute long-dated interest rate swaps, the difference that matters is found by instead looking at the difference in fixed rates on LIBOR swaps and SONIA swaps at maturities similar to those of the swaps we hold.  All else equal, the larger the difference in these fixed rates, the larger the impact from SONIA discounting on the value of our swap. This difference is referred to as the LIBOR-SONIA basis. LIBOR-SONIA "basis swaps" are a separately traded instrument and market and connect the LIBOR swap market with the SONIA swap market. So assuming all discounting was done using the SONIA curve because we had a “cash and gilts” CSA then we could write:

• Fixed rate on a 30-year 3-month LIBOR swap = Fixed rate on a 30-year SONIA swap plus market level on a 30-year LIBOR-SONIA basis swap.

b)    LIBOR swaps with SONIA discounting have an embedded LIBOR-SONIA basis swap

It follows from what we said in a) above that, long-dated LIBOR swaps discounted at SONIA, embed a long-dated LIBOR-SONIA basis swap. This means that whenever a LIBOR swap is transacted on a "cash and gilts" CSA then the scheme is also asking the bank to transact a LIBOR-SONIA basis swap. The same applies to swap unwinds and recouponing transactions. We return to this later.

c)     The SONIA curve can rise or fall even if there is no change in the LIBOR curve 

Apologies if this is stating the obvious.

Intuitively, we should expect that a SONIA interest rate curve representing the cost of unsecured overnight borrowing should correlate well with a (3-month) LIBOR interest rate curve representing the cost of unsecured, 3-month borrowing. So that movements in one should mimic movements in the other.

Of course, we now know that during a credit (or more aptly, a liquidity) squeeze, the ensuing panic can create massive uncertainty, even over a short period like 3-months and so 3-month unsecured borrowing rates (LIBOR) can become unhinged from their overnight counterpart, SONIA. The extent of this "unhinging" or dislocation may be largest at shorter-maturities but can often be seen in longer maturities too, i.e. the “LIBOR-SONIA basis” can be large at both short and longer-dated maturities. When this unhinging occurs at longer maturities it will impact the value of LIBOR swaps discounted at SONIA, if these swaps have a non-zero mark-to-market.

d)    The delta of our swap changes 

Moving to SONIA discounting changes the delta of our swaps. The impact has parallels with the impact that convexity has on duration, so that all else being equal then:

-       lower SONIA rates (relative to LIBOR rates) increases the delta of our swap and swap values become even more sensitive to changes in interest rates and

-       higher SONIA rates (relative to LIBOR rates) decreases the delta of our swap and swap values become less sensitive to changes in interest rates.

(Again, remember what we said earlier about drawing a distinction between differences in spot LIBOR-SONIA rates versus differences in long-term LIBOR and SONIA rates or the long-term LIBOR-SONIA basis. It is the latter that matters for LDI clients who typically hold long-dated interest rate swaps against 3-month LIBOR).

(5) Show me the money. Oh, and a final glass for the road.

Putting it all together then the financial impact on the pension fund can be summarised as follows:

1. Increased delta (or PV01) on migration to “cash and gilts” CSAs

In a world where the SONIA interest rate curve lies below the LIBOR curve (as it does presently), then cross gamma risk means that a LIBOR swap discounted at SONIA has a higher delta. As yields fall, this delta increases even further. 

For clients migrating to SONIA CSA’s from LIBOR CSA’s, the higher deltas arising on the switch to SONIA discounting means that the scheme is effectively adding PV01 (perhaps inadvertently). Schemes and their advisers should take account of this change in PV01 and adjust their hedges appropriately. 

Of course swaps may be unwound to reduce the unwanted PV01/delta but caution should be exercised (see point 3 below). 

2. Delta (or PV01) charges for migrating to “cash and gilts” CSAs

A bank’s trading desk will see these higher deltas as a risk to be hedged and hence an additional transaction for which the end-user (the pension scheme) must be charged. On large swap portfolios the additional PV01 is not insignificant and the charges not trivial. Caveat emptor.

3. Opportunity costs from inadvertent positions in the LIBOR-SONIA basis

The value (or mark-to-market) of LIBOR swaps discounted at SONIA changes as the LIBOR-SONIA basis changes.

For LDI strategies, pension funds are typically receiving fixed rates on long-dated (LIBOR) interest rate swaps so that:

i) If the swaps are in-the-money to the pension scheme then the in-the-moneyness of these LIBOR swaps will increase when the long-dated LIBOR-SONIA basis widens and decrease when it tightens.

ii) If the swaps are out-of-the-money to the pension scheme then the out-of-the-moneyness of these LIBOR swaps will increase when the long-dated LIBOR-SONIA basis widens and decrease when it tightens. Looked at from the pension scheme's perspective, for out-of-the-money swaps, the pension scheme is hoping that the LIBOR-SONIA basis tightens because that will mean the out-of-the-moneyness of the swaps will decrease resulting in a smaller out-of-the-money position.

iii) If the swaps have no mark-to-market then the changes in the LIBOR-SONIA basis do not affect the value of the swap.

Choosing to unwind or recoupon existing swaps crystallises the LIBOR-SONIA basis at levels prevailing at the time of the unwind. The long-dated LIBOR-SONIA basis is prone to distortion especially in times of crisis.  Even though the impact is most felt at shorter-maturities, the impact at longer maturities is significant and should not be underestimated.

In much the same way that schemes are alive to other distortions - for example with regard to swap spreads - schemes and their advisers should be alive to the possibility of distortions in the LIBOR-SONIA basis.

If a pension scheme's swaps are in-the-money and the basis is unusually narrow at the time of the unwind or recoupon then, arguably the scheme could be losing out from potential future gains should the basis mean revert to its historically wider level. For example, at various times in the last 12 months the forward LIBOR-SONIA basis has been very narrow due to a lack of liquidity in the forward basis swap market. Clients recouponing in-the-money swaps or unwinding their positions would have been locking into this narrow basis and therefore losing out on the potential to benefit from a subsequent widening out of this forward LIBOR-SONIA basis. 

I am not for a moment suggesting running unwanted interest rate risk simply because the basis is unfavourable but rather suggesting that were the basis considered to be unfavourable then perhaps there are other ways to reduce interest rate risk and these should be considered first - for example targeting unwinds at maturities where the basis is considered more attractive or shortening the duration of any bonds held either physically or on repo . 

4. Basis swap charges on unwinds and recoupons 

Each time a LIBOR swap (on SONIA discounting) is executed or restructured then a charge will be made by the bank for the corresponding basis swap position being (inadvertently?) carried out. These charges will be higher if the bank is not well positioned to for the risk it is being asked to take on.

In summary, cross gamma risk is alive and prevalent in your LDI strategy. Make sure you understand the consequences of transactions executed and work to minimise slippage in your LDI implementation. All done and just in time too, the wine's almost up and its the final frame of this session. Tweet

  

Notes:

(1) It is not my purpose here to go into the nuances of what an appropriate interest rate curve is to use for the purposes of discounting. This is covered by many others in a form and depth far better than I ever could. Here is a link to one of the better, albeit more technical, descriptions out there.
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