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pj


Total Posts: 3604
Joined: Jun 2004
 
Posted: 2020-09-14 09:38
Say you have a complicated FX swap in CHF/EUR.
You currency is CHF.
How does one choose the discount curve?


The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

nikol


Total Posts: 1352
Joined: Jun 2005
 
Posted: 2020-09-14 10:22
First, you work out each currency separately:
Each currency discount link to OIS (CHF OIS, EUR OIS)
Each currency payout link to its IBOR (be aware it will disappear)

The same ideas apply about collateral.

After that you involve CHF/EUR cross currency spread to account for exchange between the two legs.

ref papers: Fujii, Takahashi. somewhere there is thesis by Fujii.

UPD. Error fix: base spread to cross currency spread.

... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c)

pj


Total Posts: 3604
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Posted: 2020-09-14 10:32
Thank you, nikol!
Just what i need.
Yes, I have heard that IBOR might disappear.

The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

nikol


Total Posts: 1352
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Posted: 2020-09-14 14:51
U r welcome.
Be aware of my error (too quick) Hammertime
Bad me: not basis spread, but cross-currency spread.

IBOR comes as "3d party" benchmark which is used to fix payout. Hence you will have to change IBOR to the new index in the contract.

Main idea with discount per currency is that it's linked to that currency account and you should not have arbitrage with cash from other products in same currency, e.g. fix-float IRS, to avoid internal arbitrage.

One step further, market based discount linked to OIS will give you market price, but not YOUR fair price, as you very likely have different funding and you have to know how much you are different from the market. That come withing XVA (CVA, FVA, etc) and these sort of things.

... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c)

pj


Total Posts: 3604
Joined: Jun 2004
 
Posted: 2020-09-14 15:03
I was reading too quick and automatically parsed it as cross-currency
spread
Cool

The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

nikol


Total Posts: 1352
Joined: Jun 2005
 
Posted: 2020-09-14 15:35
Double error information transmission problem.
We have proved that!!!

But 2 idiots dont make 1 wiseman. They rather kill him.)))

... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c)

tomgailey
Banned

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Posted: 2020-10-02 11:31
great)

arkestra


Total Posts: 54
Joined: Apr 2007
 
Posted: 2020-12-03 23:49
Your discounting level is set by the applicable collateral for the deal. Assuming this isn't going through a clearing house, the collateral will be specified by the Credit Support Annex (CSA) you have with the counterparty.

For instance, if the CSA states collateralisation is USD OIS, then you should cross-currency-adjust USD OIS into CHF and EUR and discount the flows accordingly.

If the CSA states collateralisation is a choice of a number of levels, i.e. USDOIS/EUROIS/GBPOIS then technically speaking you should take the max of the cross-currency-adjusted rates with an upward adjustment to account for the optionality of being able to substitute collateral. I used to work somewhere with lots of USD/GBP/EUR CSAs and getting the optionality integrated with the risk systems was a pig.

Even more technically speaking you could change your optionality calculation depending on whether the CSA is of the New York variety (where you can only substitute collateral as the mark-to-market changes) or of the London variety (where you can substitute the entire collateral balance from one day to the next).

The construction of cross-currency adjustment curves can be done with FX fwds at the short end but at some point you need to get xccy swaps in there. FX fwds are easy, since you've just got one forward exchange of flows on the same date - so the only discounting sensitivity is second-order from correlation between FX levels and IR levels - in practice a lot of people just set that to zero. Xccy swaps are trickier as you there need to make an assumption about what discounting level market quotes are relevant to. 4 years ago that would have been USD 3m Libor but since then I've ditched interest rates trading for stochastic arb, so don't know what the market quoting default is nowadays. If xccy swaps go through clearing houses nowadays you should use whatever collateralisation they specify. This is in the same way that IR swap market rates should be understood as being quoted with reference to discounting at local currency OIS as that's what LCH etc collateralise at.

The CSA may not be OIS. Might be US govt bonds or anything really. The base discounting curve will vary accordingly. It might be quite hard to define if the collateral isn't something with a decent term structure defined by a liquid market.

Of course, how much you actually need to worry about this is another question. Discounting sensitivity depends on the mismatch between the timing of cashflows in and out. If everything's happening inside a couple of years from today, there's unlikely to be a lot of net discounting sensitivity, unless you have something deliberately constructed to maximise the discounting delta. Goldman Sachs used to be good at sticking that kind of payload in back when everyone was more naive about discounting calculations - I've seen people come radically unstuck from those - if GS offer you a structure with a large discounting sensitivity, walk away slowly & count your fingers afterwards.

The first stage of knowledge is to be able to understand the sensitivity of the trade to the discounting level. This will inform the rest of your approach. I would concentrate on getting a handle on that first if I were you and then you will have some idea about how much you really need to care about all the rest. E.g. only a tiny fraction of market participants need to care about things like USDOIS/EUROIS/GBPOIS substitution optionality, and you will hopefully not be one of them.

Reading all this back reaffirms my decision to leave all this crap firmly in the rear view mirror :D

Down pokey quaint streets in Cambridge / Cycles our distant spastic heritage

pj


Total Posts: 3604
Joined: Jun 2004
 
Posted: 2020-12-04 11:24
Thank you very much, arkestra!
Your post is pure gold.

Before leaving this crap, could you give some references?
Books, articles?

A disclaimer, my firm wants to make its name in this topics
and I was actively resisting to learn about it.
But now, I am yielding to the greater powers.

The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

arkestra


Total Posts: 54
Joined: Apr 2007
 
Posted: 2020-12-09 22:38
Nikol mentioned Fujii and Takashi and I would likewise recommend you print off all their stuff and start ploughing through. I am not aware of any books. Someone even suggested I wrote a book with them on this stuff a while back, which shows you how desperate things are on the information front.

This paper in particular is a good way in: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1440633 (Takeshi, A Note on Construction of Multiple Swap Curves with and without Collateral).

The paper above looks like it has the relevant formulae, so in the below I'll concentrate on the why and the how rather than lots of equations.

0) WHY DOES DISCOUNTING MATTER WHEN BOOTSTRAPPING?

First off, why is discounting a concern when bootstrapping? The central point is that the breakeven IRS rate is going to change depending on your discounting/collateralisation, assuming your yield curve is not flat.

E.g. assuming your projection curve level is rising over time, then pay-fixed swaps will pay cashflow net towards the start (floating rates are low relative to the fixed rate), and receive cashflows at the end (floating rates are high). Bump the discounting level by a constant factor and the received cashflows at the end will have their PV lowered by more than the paid cashflows at the start, because there is more time for discounting to affect their PV. So there is a net negative effect on PV for the payer of fixed rates, meaning that (where forward rates slope upwards) the breakeven rate decreases as discounting levels are bumped up evenly across the term structure.

Hopefully I have that the right way around :D anyway the point is that time asymmetry between when cashflows are paid and received creates discounting sensitivity. So a given set of IRS quotes over some range of maturities will imply different projected Libor forwards depending on your assumed collateralisation.

Same applies to xccy basis swap rates - assuming the forward curves don't have the same shape between the 2 currencies you will again have a mismatch in the timing of cashflows and so an overall PV sensitivity to discounting changes, e.g. if both discounting curves get bumped by the same factor, the breakeven spread will change. Exactly how depends on the nature of the difference between the curves in each currency.

1) WHAT DISCOUNTING LEVEL SHOULD BE USED WHEN BOOTSTRAPPING?

OK so we've established we care about which discounting level we use. What discounting level *should* we use? The answer is that market quotes will have an assumed rate of discounting, which any trader should be aware of.

For normal IRS as of 4 years ago market swap quotes assumed clearing house collateralisation (LCH/SwapClear/etc) and the reference interest rate for this was Local Currency OIS (see caveat below). I imagine that's still the same but that might be worth checking.

For XCCY swaps, it was the case that market swap quotes assumed USD3M libor discounting (or the equivalent level in other currencies) but this may well have changed if people have started clearing these through LCH/etc, so you should definitely check that.

Caveat: what I'm saying about IRS clearing discounting rates is (probably still) true for relatively major currencies, but clearing for some minor currencies may involve USDOIS levels of interest rather than native currency OIS. The clearing houses will have documentation on this.

2) HOW DO I BOOTSTRAP IRS?

So now we know what discounting level to use, we can bootstrap swap curves a la the Takeshi paper. If your currency has liquid outright OIS swap quotes (and maybe also OIS futures at the short end) you can get the OIS level without reference to Libor, then boostrap the Libor forwards in a second step. Otherwise if OIS is expressed as a spread to Libor you'll need to dual-bootstrap OIS and the main Libor frequency for the currency simultaneously, ending up with a discount curve (CCYOIS) and a projection curve (CCYxM).

Now you can figure out projection curves for all other tenors of Libor (1M, 3M, 6M, 12M) by bootstrapping using the Libor basis swap levels for your projected libors (keeping discounting at Local Currency OIS).

4) HOW DO I BOOTSTRAP XCCY?

Next you can get xccy basis between discounting levels by bootstrapping off xccy spread quotes. At the time I was dealing with this, the assumption was collateralisation at USD3M equivalent, meaning that boostrapping allow determination of (the equivalent of) USD3M is in all currencies. In the absence of a clearing house driven level of collateralisation, USD3M has an advantage as a reference discounting level, as it makes the USD side PV (where principal is exchanges) come to zero. Well, very close to zero, but not quite if you pay close attention to your projection start and end dates relative to your payment dates. Did I mention I'm glad I'm not doing this any more?

This may well have changed - e.g. clearing houses were discussing schemes such as taking the OIS level for the first currency to appear in a list going something like (USD, EUR, GBP, JPY, ...) so a EUR/USD swap would be USDOIS and a GBP/EUR swap would be GBPOIS. No idea if this actually went ahead or not. Looked overly complex to me but it was a tricky one - local currency OIS for IRS trades was far simpler to go with. Anyway point being that you will end up with an equivalence of a given level between two currencies, but this may not be USD3M if xccy swaps are routinely clearing nowadays, with this being reflected in the market consensus on discounting for quotes on Reuters etc.

5) HOW DO I TRANSLATE LEVELS BETWEEN CURRENCIES?

The existence of a shared reference level then allows you to translate discounting levels between currencies, assuming that the ratio between levels remains constant. So say you know (USD3M in GBP), (USD3M in USD), (GBPOIS in GBP). And you are collateralised at GBPOIS (e.g. your CSA specified everything is collateralised in GBP and that's the interest rate it specified. Then you would discount GBP flows at GBPOIS, and USD flows at (GBPOIS in USD), which is defined as multiplying discount factors like this:

(GBPOIS in USD) = (GBPOIS in GBP) * (USD3M in USD) / (USD3M in GBP)

6) WHAT WAS MY QUESTION AGAIN?

To get back to your original question - the correct discounting level is the xccy adjusted version of the level implied by your CSA. So if your complicated FX swap was collateralised at USDOIS by your CSA with the counterparty, you would discount the CHF side at (USDOIS in CHF) and the EUR side at (USDOIS in EUR).

7) WHAT ABOUT MULTI-CURRENCY CSAS?

Maybe your CSA says you have a choice: you can post GBP (interest rate GBPOIS) or USD (interest rate USDOIS). Here the correct discounting curve in (say) EUR is to construct a synthetic curve where each daily forward is the max of (GBPOIS in EUR) and (USDOIS in EUR).

You may want to add a correction for the ability to change collateral - this is important if you are running a lot of discounting risk because otherwise you may get sharp discontinuous changes of sensitivity between one day and the next if one curve's forward rates crosses another - GBPOIS and EUROIS certainly used to have this problem, for instance. You might have 10 years where the xccy-adjusted levels are virtually identical. Without an optionality correction then the deltas switch back and forth from day to day. With optionality then if the levels are close, you get risk split between them, which is far more sensible.

8) WHAT ABOUT WEIRD COLLATERAL?

Then there are trickier collateral types - govt bonds of various types. Less liquid special stuff. You may end up putting risk onto proxy curves for that

9) ARE THERE HIDDEN ASSUMPTIONS IN THE ABOVE?

Yep, two of them, that (a) FRA rates are not sensitive to the overall level of discounting rates, and that (b) FX fwd rates are not sensitive to the overall level of discounting rates.

(a) means that you can use the same projected xM Libor curves at any discounting level

(b) means that you can use the kind of relationship in (5) to translate any discounting level between currencies, given one shared reference discounting level

Life becomes very complicated without these assumptions, and they do make sense on a first order basis because neither FRAs nor FX fwds have cash going one way at one forward date, the other way at a later forward date, so there is no net discounting sensitivity. But if you think there is correlation with movements in discounting levels in either or both, then there is still a potential correction you might want to make - or, at least, have some kind of grasp of the magnitude of - this is the same kind of idea as the one driving Futures/FRA convexity adjustments.

10) WHAT ABOUT SUBSTITUTABILITY?

There are 2 flavours of Credit Support Annex: New York and UK law. They have a difference in terms of how ownership of the collateral is treated - see here for more detail: https://jollycontrarian.com/index.php?title=Credit_Support_Annex . I remember sitting through a day's worth of legal debate on this (the things I do to pay the rent). Upshot is that NY CSAs allow free substitution of collateral types from one day to the next, while UK CSAs only let you swap between permitted collateral types as the mark-to-market shifts. At least that was the consensus view 4 years ago - it's probably the same now.

Technically speaking this may influence how you model the price impact of optionality for multi-collateral CSAs. E.g. say GBPOIS is cheapest to deliver (highest xccy-adjusted interest rate), but then EUROIS becomes cheapest. The collateral payer would like to stop posting GBP and start posting EUR. If it's a NY CSA they can do it straight away. If it's a UK CSA they can only do it piecemeal, which may take a *very* long time to completely substitute, dependning on the characteristics of the deal's MTM. So this means a straightforward optionality calc on daily forwards doesn't capture the true exposure.

Practically speaking, this is a pig to deal with systematically. How much difference this makes depends on how you model shifts in the portfolio MTM. Do you *really* care that much? *Probably* not. This is the kind of issue where you want to maybe to a first-order sense check of whether there's a systematic issue, and then take a view on your stance. For instance, you might apply a correction for some subclass of counterparties who can't substitute and have a really static MTM profile, meaning they are effectively stuck on one collateral level for the duration. Or you might realise that any major counterparty has sufficiently volatile MTM that NY vs UK makes naff-all difference and you can forget about it. The right thing to do here depends on the nature of the portfolio MTMs so there is no definite answer I can give other than "check it our it you're worried".

11) LOOKS LIKE A LOT OF STUFF TO WORRY ABOUT. WHAT SHOULD I DO FIRST?

Depends on your motivation/angle. If your firm is going to be a market participant, then the first step is being able to know what your discounting risk is - at the very least, have the discounting sensitivity properly broken out from your Libor/etc projection levels. That way if (sophisticated player X) offer you some fancy combination of equity options that produce a deterministic set of cashflows, have a way of checking the discounting sensitivities before you consider accepting it. Then don't accept it, walk away, and count your fingers to make sure they're all still there. Yes, I do have a real-world story in mind there.

Also you may discover that your discounting risk in many areas is so small you don't have to worry about more abstruse stuff. E.g. if you have little risk against troublesome combinations such as GBPOIS+EUROIS then perhaps pricing in collateral optionality is not going to be a massive priority for you even if you do have some multi-collateral stuff as it only shifts things by a few b.p. - its main use is to smooth transitions between the forward curves. Some places had pretty much all their CSAs in USDOIS and dear God did I envy them.

And if you are providing services to people taking risk in this stuff, then I guess it's about finding our what their exposure is, for the same kind of reason.

12) INTEREST RATES ARE LOW AT THE MOMENT - SHOULD I CARE ABOUT ANY OF THIS?

Well... if you don't at least account for the risk against varying discounting and projection levels, you're likely to end up selling very cheap liquidity options to your counterparties. That's the big thing to avoid. There are absolutely people out there looking for unsophisticated players so they can build that kind of optionality up.

But I would absolutely recommend a sense of proportion in terms of how much effort is spent on this stuff. I was in a place for some time where all of this really mattered. There was a lot of money to be made - or lost. But I remember dealing with one particular counterparty at the time and they struck me as really clued up. They didn't have a big sensitivity to discounting at that time, but they knew why, knew this was contingent, knew when to re-examine their stance. So they could get on with their daily lives not worrying about it.

I would absolutely recommend getting a stance together on whether you should care about this stuff at all, and if not, when you might start caring, before diving into any of this.



All a bit of a brain dump, but it covers the main steps. Good luck...

Down pokey quaint streets in Cambridge / Cycles our distant spastic heritage

nikol


Total Posts: 1352
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Posted: 2020-12-10 09:28
@arkestra

Thank you for the articles. Apparently, those you mentioned describe the settings in most clear way.

I would add the discussion about discount based on basket of collateral, as source of funding. Most simple formulation,to my taste, comes in presentation of Piterbarg at Barclay's, 2010.

https://eusp.org/images/ec_dep/Lecture_Posters/CollatFund_Slides_02.pdf

It's quite simple.

... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c)

arkestra


Total Posts: 54
Joined: Apr 2007
 
Posted: 2020-12-10 11:41
Yes that's got a good overview of multi-collateral. I see he discusses substitution from P22. Don't know why he doesn't link that discussion to the UK/NY CSA distinction, which I am pretty sure is the driver for whether substitutability applies.

Down pokey quaint streets in Cambridge / Cycles our distant spastic heritage

nikol


Total Posts: 1352
Joined: Jun 2005
 
Posted: 2020-12-10 12:36
> Don't know why he doesn't link that discussion to the UK/NY CSA distinction

I think this is too specific. His presentation has a purpose of showing the idea.

By the way, talking about discounting, we know that it is expectation over cash flows coming to the Funding Account. If not smoothed with expectation operator, something like r(t) = Et[Cash Interest], it would be very jumpy. Hence the negative rates is a pure artifact of this definition and reflects the fact that the Funding Account actually receives cash for funding rather than pays for it. This phenomena is linked to the fact that there are no many possibilities to invest safely.
IR from r(t) = Et[Cash Interest] is a tradeable product with many trading agents working hard on elimination of arbitrage (which is there). From this picture, I am wondering what is the proxy of variance, Vt[Cash Interest]? Liquidity risk premium?

... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c)

pj


Total Posts: 3604
Joined: Jun 2004
 
Posted: 2020-12-10 13:35
Thank you very much, arkestra!
I need to grok the info and
I owe you Beer

The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

arkestra


Total Posts: 54
Joined: Apr 2007
 
Posted: 2020-12-10 21:50
> I think this is too specific. His presentation has a purpose of showing the idea.

The maths is fine and it's a very useful paper that presents a lot of interesting stuff.

But just saying sticky collateral is "more realistic" on Slide 23 is potentially misleading. Pointing that out doesn't invalidate all the good stuff elsewhere in the paper (of which there is lots). It might sound nit-picky - hell, maybe it is kind of nit-picky - but I have seen this distinction cause a non-trivial difference in real life, otherwise I wouldn't have bothered to mention it.

(I think it's perhaps just a reflection of Piterbarg being based in Barcap, which was perhaps predominantly trading using a UK incorporated entity, using lots of UK Law CSAs.)

> From this picture, I am wondering what is the proxy of variance, Vt[Cash Interest]? Liquidity risk premium?

I'd think of liquidity risk as being expressed by the spreads between rates of different tenors, e.g. OIS, 1M, 3M, ... up to 2007/8 these spreads were effectively treated as constant by many market participants. You'd get people trading 3M/6M swaps in reasonable size, going out many years, at a tiny spread. Their systems wouldn't even show the risk. Then the spreads blew out from 1 few bps flat to dozens with a term structure.

Down pokey quaint streets in Cambridge / Cycles our distant spastic heritage

tbretagn


Total Posts: 297
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Posted: 2020-12-11 09:16
@arkestra You'd get people trading 3M/6M swaps in reasonable size, going out many years, at a tiny spread. Their systems wouldn't even show the risk. Then the spreads blew out from 1 few bps flat to dozens with a term structure.

I know a bank desk who pre-GFC had spotted the mounting risk and had built a gigantic position in that (mostly against his own bank). His risk systems blended all the instruments. His only constraints was in the amount of futures he could have. So he kept it all OTC.
I think he made eur 1bn in 2008.

Et meme si ce n'est pas vrai, il faut croire en l'histoire ancienne

NeroTulip


Total Posts: 1080
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Posted: 2020-12-11 14:14
Aaaah, back in the good old days of late 2007, we (bank inflation trading desk) noticed that most market participants were pricing inflation linked bonds asset swaps with libor discounting (instead of OIS, depending on CSA). On the longest maturities, the resulting spread was off by 10-20bps, and you could get 2-way prices 2bps wide in the interbank market. A series of "where's your bid?" and "yours!" ensued. We then bought back the bonds and traded inflation swaps, IRS and basis swaps to hedge out all the risks (except some small residual second order risks we had to sit on). As close to a true arb as I have ever seen. The most surprising thing was that it lasted for *years* and several hundred million PnL. The hardest thing was to keep quiet about it, and just pretend we liked selling asset swaps.

And we were not the sharpest in the market, rumour was that GS made a billion on that in 2008 (possibly what you were alluding to @tbretagn).

Sorry for the threadjack, that brought back some good memories, these were the days!


"Earth: some bacteria and basic life forms, no sign of intelligent life" (Message from a type III civilization probe sent to the solar system circa 2016)

nikol


Total Posts: 1352
Joined: Jun 2005
 
Posted: 2020-12-11 16:12
When multi-curve concept started to get publicity (~2008), I looked back into all available articles I had at hand. Everyone was advocating to use LIBOR for the discount as "market wide used methodology". Which was the strongest argument for quants in other banks to use the LIBOR to discount cash in swaps. However, years earlier (~2000), at FO we used funding curve as discounting to arbitrage equity futures and options. So, I always had the feeling that FO is the way ahead of any public knowledge.

PS. CSA became hot topic in 2008, when everyone started to check if it is signed :))

... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c)

silverside


Total Posts: 1440
Joined: Jun 2004
 
Posted: 2020-12-11 17:53
I remember working for one of the banks mentioned back in the late noughties - who were slightly ahead of the game in implementing funding-cost discounting and made a *lot* of money from it (it took the best part of a decade to work its way across the market into smaller buy-side places :) )

in terms of references, the two presenters that spring to mind are
Piterbarg (Barcap) in relation to CSA / optionality
Christian Fries - more generally on XVA (wrong-way and right-way risk)

btw, should this really be in basics ?





nikol


Total Posts: 1352
Joined: Jun 2005
 
Posted: 2020-12-11 18:46
Why not. Basics seems to be the only place free from hordes of ai-riding invaders.
Besides, this multi-curve discounting is 10+ years old.

... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c)

arkestra


Total Posts: 54
Joined: Apr 2007
 
Posted: 2020-12-13 00:26
> The hardest thing was to keep quiet about it

Yes we had stuff going on too. Things I can't really talk about even now, to be fair to those of my erstwhile workmates still working in the sector, that might potentially have value now: which certainly had value 4 years ago, before I did a runner to the buy side.

> these were the days!

Totally - one problem we had was not taking too much advantage of the less sophisticated (banking) counterparties in situations where it was arguably inappropriate. If that's your main problem you're in a good place.

Down pokey quaint streets in Cambridge / Cycles our distant spastic heritage
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