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Another day, another article demonizing CDS

November 28, 2009 in Finance

Bloomberg has a new article up about how the CDS market is starting to crumble – the sort of piece that looks like it’s been sitting on a back burner waiting for an excuse to stoke the flames of derivative fear (thanks, Dubai!).

One of the article’s chief arguments is that “credit-default swaps tied to Thomson SA, the Paris-based owner of film processor Technicolor Inc., paid some holders 30 percent less than those with contracts expiring a day later.” First, however, a technicality – CDS can only expire on four days of the year: the 20th of March, June, September and December (the so-called “roll dates”). Thus, the description of contracts that expire “a day later” is inaccurate. This brings me to a key point: one of the nice things about (most) fixed income is that the terms are… well, fixed. Traders know in advance when a contract will terminate, as well as the quantity (or at least the terms) and timing of any future cashflows. Those definitions extend to procedures in the event of default.

The credit event in Thomson’s case was one of restructuring, the procedures for which were recently updated as part of ISDA’s new “small bang” European protocol. Naturally, in a restructuring event – the debate over whether it should even constitute an event will be left for another post – it may be tough to claim that insurance should pay out. On the one hand, the fixed income product that was being insured just had its terms adjusted (no longer fixed, no longer the same!). On the other hand, the present value of the cashflows should be unchanged which in theory would make investors indifferent (obviously, that’s not the case). Without a cessation of payments, it’s hard to claim that insurance should pay the balance. Therefore, rather than have all insurance contracts pay out uniformly for all referenced bonds, which would fail to capture the odd nature of the restructuring event, traders agreed to set up “buckets” which will each pay out a value deemed fair by market action. The buckets are divided by time to maturity; in Thomson’s case, there were multiple buckets including a 2.5 year bucket, a 5 year bucket, and a 7.5 year bucket. This way, debtholders could more accurately match their insurance claim to the affected bonds.

The crux of Bloomberg’s argument seems to be that a swap maturing on the last roll date of one bucket would pay differently than one maturing on the first roll date of the next bucket (note the semantics – none of this “maturing one day later” language). But under the terms of the protocol, which market participants ratified, that seems appropriate to me. Remember, fixed income means terms are defined in advance. If Thomson had bonds that matured one day before the restructuring was announced, then those bonds would pay out par while bonds maturing the next day would presumably have crashed on the revelation that there isn’t cash to pay them in full (remember, unlike CDS, bonds can and do mature on any day of the year). That actually just happened with the Nakheel December 2009 bonds, which were trading well above par before Dubai’s surprise announcement brought them back to the 70’s overnight. In sum, the fact that some fixed income instruments are treated differently than other is not alarming – maturity and seniority are prime components of the fixed income market and naturally force bonds into differently performing buckets on a daily basis.

So if we can’t fault CDS for the fact that one contract pays out differently than another, maybe we can find something to be upset about because the 2.5 year bucket recovered 30% more than the 5 year bucket (in CDS terms, recall that recovering more means the contract pays less: if a bond recovers its full value, the insurance would pay out nothing at all). But here’s a secret: the disparity arose because of problems in the underlying cash market, not the derivatives market! Okay, it’s not really a secret. Euroweek figured it out well before the auction even took place:

Most of Thomson’s deliverable obligations are thought to be complex private placements and little is known about their documentation. It is possible that none will be deemed eligible for delivery.

CDS payouts aren’t determined by a bunch of traders standing in a room shouting – they are set by the market-clearing price on bonds (“deliverable obligations”) that are submitted by CDS holders in return for insurance payouts. It’s a straightforward system: CDS buyers purchase bonds in the market, then give them to the CDS sellers in return for their par value. The net payment is therefore par less the bonds traded price, or recovery. If there are few bonds available, or little transparency or liquidity about those bonds, then their market price will fluctuate for technical reasons rather than fundamentals. This phenomenon can occur with any traded security: short squeezes are perhaps the most familiar example. That’s exactly what happened with Thomson – so few of the short-dated deliverables were available for public trading that the market clearing price was bid up extremely high. In the next bucket, bonds were more liquid and so reflected recovery more accurately.

Euroweek described it nicely (again, well before the auction even took place):

…it is very likely that there will be a shortage of deliverable obligations and a scramble to get hold of what is available. The consequent short squeeze will drive up prices and the recovery rate much higher than it would otherwise be — good news for protection sellers but bad news for the buyers. For example, the most likely and liquid deliverable obligation, according to Citigroup analysts, is the June 2012 revolver, which would fall in the 2-1/2 to five year maturity bucket. It has been pushed from a 40% price to 70% in recent days.

But the real difficulties lie in the 0 to 2-1/2 year bucket. Thomson, a French media firm, was a regular member of the main iTraxx Europe Index from series 1 to series 7 and was thus much referenced in index CDOs. There are a lot of single name hedges against the name with maturities between now and 2012, putting particular pressure on the 0 to 2-1/2 year bucket.

I’m still waiting for the article titled “CDS auction goes smoothly despite problems in bond market.”

To Bloomberg’s credit, there is a deserved debate over restructuring events and CDS more generally outside the Bang protocols (and even within them). Moreover, the Thomson example – though I disagree with the author’s specific points – is a good one for demonstrating how settling CDS remains a mystifying and seemingly arbitrary process. There is no doubt that further clarity is needed, for the benefit of all market participants. The rest of the article deals with the lack of transparency into what qualifies as a credit event and murkiness following that declaration. I have to point out that though the arguments there have merit, their very existence demonstrates that CDS by nature doesn’t force companies into default or anything along those lines – otherwise these arguments would be settled by a simple imperative to bankrupt the firm.

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Deconstructing the Gaussian copula, part I

June 5, 2009 in Finance, Math

(Parts II, II and a half, and III of this series are also available.)

Newsweek has a new article about Paul Wilmott called “Revenge of the Nerd” which I really enjoyed, with two caveats.

In its opening the article compares quants to aeronautical engineers who design faulty planes (CDOs). The author observes:

Yet while aeronautical engineers who willfully designed a faulty plane might be on trial for criminal negligence, Wall Street’s math gurus are, for the most part, still employed. Strangely, the banks need quants more than ever right now.

But where is the logical conclusion of the aeronautical metaphor, that other engineers would be needed to fix the planes? In that framework, there’s no contradiction.

But that point is minor. Here’s what really bothered me (my comments are inserted in bold):

In 2000, the CDO market was jump-started by David X. Li, who, while working at JPMorgan, created the Gaussian copula function (no, he didn’t), a formula for determining the correlation between the default rates of different securities (no, it’s not). In theory, the model tells you the odds that, if one CDO goes bad, others will too (no, it doesn’t). The apparent genius of the Gaussian copula is its abstraction (true, but not in the way the author means). Rather than relying on the immense amount of data used to figure the odds that a CDO might default (there is no such data; issuers default, not CDOs), Li appeared to have discovered a law of correlation (no, he didn’t). That is, you didn’t need the data; the correlation was just there. Armed with it, quants could price CDOs much faster, and traders could buy and sell them at record speeds. Gaussian was rocket fuel for the CDO market (“Gaussian” is an adjective, not a noun). The global volume of CDO deals went from $157 billion in 2004 to $520 billion in 2006. As more banks got in on the game, the once large profit margins started to shrink. In order for banks to make the same kind of returns, they had to pack more and more loans into a CDO, essentially making bigger bombs. Li was on his way to a Nobel Prize when the world blew up (no, he wasn’t).

I have no problem with the simplification of difficult topics (in fact, I encourage it). I also have no problem with bashing Gaussian copulas as applied to CDO’s (the argument was featured in my thesis years ago). But I have severe issues and frustrations with poor reporting of false information. Will 99% of this paragraph’s readers realize it’s incorrect or even care? Of course not! But it doesn’t make it all right.

Deep breath. Ready to go deeper?

This paragraph seems to be lifted largely from a recent Wired article (my response to that article would evolve into my “models are just the tool” tirade). Anyway:

“David X. Li…created the Gaussian copula function.” The Gaussian copula is rooted in research from more than 250 years ago. In fact, Gauss – a prodigal mathematician whose influence extends far beyond the bell curve – died in 1855! It’s unclear when the first bivariate extensions were arrived at, but Wikipedia notes that it must have been developed by 1872. The copula itself would not be described until 1959, but almost immediately mathematicians used it to decompose the multivariate normal distribution into a pairing of Gaussian marginals and something the new vocabulary termed a Gaussian copula. All David Li did was pair the copula and CDO pricing for the first time.

“…a formula for determining the correlation between the default rates of different securities.” Copulas describe the dependence structure of random variables. Correlation is a way of condensing the information contained in the copula down to a single number. The sentence as written suggests that the copula is used to measure correlation when in fact it is the other way around. In fact, you can not even create a Gaussian copula until after you decide what correlation to use.

“In theory, the model tells you the odds that, if one CDO goes bad, others will too.” I assume the author meant to write “if one issuer goes bad” rather than “if one CDO goes bad”, because the Gaussian copula as applied to CDO’s describes the issuers within the CDO, not CDOs to each other. In this framework, the sentence is correct: copulas describe the dependence structure, which essentially means “how one issuer relates to other issuers.” In this case, the thing being measured is default probability.

“The apparent genius of the Gaussian copula is its abstraction.” This is a true statement as it stands: the brilliance of the copula function is that it abstracts the dependence structure from the marginal distributions, meaning the dependence of, for example, two dice numbered 1-6 has the same copula as the behavior of two dice numbered 2-7. Before the development of the copula, the 1-6 dice would have a completely different function than the 2-7 dice, because one would have to account for the marginal differences while defining their dependence.

However, the abstraction the author is referring to is that “you don’t need the data, you only need the correlation” (see below). The Gaussian copula as Li implemented it boils all correlation down to a single number, enabling such an abstraction.

“Rather than relying on the immense amount of data used to figure the odds that a CDO might default…” The available data consists of CDS spreads and bond z-spreads, which may be used to imply a default probability for each issuer. However, to figure out if a CDO will default, one must evaluate the probability of multiple firms defaulting within a given time frame. This is the correlation parameter. Thus, the data alone does not tell you about the likelihood of CDO default.

The default probabilities extracted from historical data are not independent, and so can not simply be added (or multiplied, to be more precise) together. Moreover, the correlation which may be measured in CDS is the correlation of changes in default probability, and the jump to correlations of actual bankruptcy events is much more difficult, not in the least because there are relatively few historical defaults, compared to the number of issuers.

This isn’t to say that the data can’t be used – in fact the data must be used – but the key realization is that without a model (I struggle to think of one capable of handling such data that isn’t a copula), the data yields no worthwhile insights. Merely having the data is not enough to price a CDO.

“Li appeared to have discovered a law of correlation.” As I’ve mentioned, Li did not “discover” anything. He merely applied an existing model to a new dataset.

“You didn’t need the data; the correlation was just there.” Of course you need the data – the correlation is meaningless without the default probabilities extracted from the data. What the author presumably means is that your correlation number does not have to represent the “true” level of correlation observed in your data (which, as I’ve stated, is a nearly impossible thing to observe in the first place).

But having said that, this is probably the one thing the author has correct. After some futile efforts, researchers stopped measuring correlation and started holding a finger in the air to determine the “right” level. Similar to implied volatility in option pricing, correlation was unobservable and the “right” correlation was whatever level made the model price come out the same as the market price. Unfortunately, in a space where traders became so dependent on their models, the chain was circular: markets were informed solely by correlation-based models, which were themselves calibrated to the market.

The critique is not limited to the use of a Gaussian copula, however.

“Gaussian was rocket fuel for the CDO market.” Another true statement, but one which reveals the author’s unfamiliarity: “Gaussian” is an adjective used to describe a type of model. It’s a person’s name. This is like saying “Newtonian revolutionized the world of physics” when you want to talk about a model of gravitational acceleration or “Darwinian turned the study of biology upside down.”

“Li was on his way to a Nobel Prize when the world blew up.” No, he most emphatically was not. This is a repeat of a one of Felix’s statements from the Wired article. Even if the model had been perfectly accurate, do today’s financial journalists think pricing a financial derivative is worthy of a Nobel prize? Black/Scholes/Merton didn’t win a Nobel prize for their option pricing model, they won it for the research they did into the economics of asset pricing. The option model was just a nice benefit on the side.

A fundamental issue with this paragraph, on top of all these highlights, is that not once does it explain the actual problem. If you read the paragraph, and I asked you why did they blow up, could you tell me? I’m sure you’d say something about the correlation not being reflective of the data. And I’d respond, well then why didn’t we just start using the data, or start using the right correlation?

I’ll try to answer these questions soon in part II.

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Negative swap spreads

June 2, 2009 in Finance

Felix Salmon writes about the negative swap spread – a fascinating turn of events. Or at least, it was when the swap spread went negative almost a year ago.

The swap spread is the extra amount that an interest rate swap yields over a similar Treasury bond. Typically, a swap yields a few basis points more to compensate investors for the extra risk that comes from dealing with a bank instead of with the US government. Shortly after the Lehman default, however, a funny thing happened: the 30 year swap spread became negative. Effectively, it was cheaper to deal with a bank than with the government. Since then, the spread has wavered between 0 and -40 basis points.

What inspired Felix’s post was the large moves in the swap spread over the last few days. He speculates that it is on account of GM-related hedging (and counter-hedging) activity. I’m not convinced that is the case. Though the swap spread moved a massive amount Monday afternoon (rising 15 bps), it also rose 15 bps last Wednesday (May 27) and fell 15 the following day. On Wednesday, the move was tentatively attributed to mortgage-related hedging as Treasury yields moved higher. On Thursday, there was no good explanation. Today, it’s the GM bankruptcy. Forgive me for being skeptical.

Felix’s final point, however, is the one that really surprised me:

The market in interest-rate swaps is enormous — orders of magnitude greater than the market in credit default swaps — and, like most markets, it’s done some pretty crazy things over the past year, with long-dated swap spreads going negative for most of that time. Because there aren’t any systemic implications of things like negative long-dated swap spreads, and because the swaps market is a zero-sum game where for every winner there’s an equal and opposite loser, policymakers and bloggers and pundits haven’t paid much attention to it. That’s fine, they don’t need to. But it’s really important for fixed-income traders, which is why the likes of Jansen spend a lot of time looking at it.

The implication, I believe is that interest rate swaps are different from “something else” (read: CDS) because they lack are a “zero-sum game” and lack “systemic implications.”

But CDS are a zero sum game! In fact, I can’t think of a financial asset that isn’t a zero-sum game. If you follow the trail, it appears that Felix may be referencing either someone who commented on an article of his over at Seeking Alpha, or a different Seeking Alpha article – and you know how I feel about Seeking Alpha – which essentially argue that CDS are not zero sum because they have negative externalities (like requiring bailouts). But, if I may be cynical for a second, the bailout is the only thing preventing CDS from being zero-sum! Zero-sum means no dollars are invented or disappear; every one transfers one-for-one among involved parties. It does NOT mean that dollars are self contained. Saying CDS are not zero sum because they caused the meltdown of 2008 is like saying Russian bonds are not zero sum because they led to LTCM’s bailout in 1998. Every dollar made comes from someone else’s pocket. That’s what zero sum means. Nothing more. Nothing less. The concept that any modern financial contract (including a Ponzi scheme!) is not zero sum is odd. The economic process itself is not zero sum, because wealth can be created (or destroyed), but derivatives thereof (contracts, if you will) are zero sum because every dollar made comes, effectively, from the counterparty. End of rant.

And to the other point, that interest rate swaps lack systemic implications – newly bankrupt Jefferson County, Alabama, begs to differ.

Felix also refers to negative convexity in his post, following it quickly with “don’t ask” and a link to an incomprehensible article. I wrote on the topic a couple months ago, though I can’t promise to be much more clear.

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FRN’s & negative duration

April 16, 2009 in Finance, Math

Floating rate notes (FRN’s) can exhibit a curious property called negative duration.

An FRN is a bond whose coupon is not fixed, but rather changes each year (or each reset period, to be specific). For example, the coupon might be quoted as L+100, meaning 100 bps above than Libor. A familiar if infamous example is an adjustable rate mortgage.

Fixed-coupon bonds bear interest rate risk, because higher interest rates make their coupons and principal worth less due to heavier discounting. Conversely, lower rates make the bond appreciate. Since FRN coupons reset to higher interest rates, FRN’s have no interest rate risk (actually, they have a little bit of risk, since coupons are set based on Libor at the beginning of the coupon period, but not paid until the end of the period). In the absence of credit risk, they should always trade at or above par.

For a fixed coupon bond, the following relationship exists (this should be a familiar graph to anyone whose taken a basic finance course):

This, of course, is an illustration of bond duration. As the discount rate increases, the bond’s price falls. In an extreme case, if rates went to infinity the bond would go to zero, as any value would be instantly discounted away. If the bond lies on the blue line at the point where the red line is tangent, then the bond’s duration, however you choose to define it (and there are many ways), is essentially the slope of the red line in the graph, or the slope of the price/yield curve at the point of tangency. In other words, it represents the first derivative of the bond’s price with respect to the discount rate, or the amount the bond’s price drops as rates increase. Strictly speaking, duration is a positive number, so it is actually -1 times that slope, and indeed almost all bonds exhibit this property with a negatively-sloped line, indicating positive duration.

FRN’s are a little different, however. Since they bear no interest rate risk, a plot like the one above for an FRN would display a horizontal blue line across from 100.

But what would it look like if the FRN were not trading at par? This situation arises when there are concerns about the issuing company’s credit quality – if the issuer were to go bankrupt tomorrow, then it doesn’t matter that the bond has a nice coupon feature; it will be worthless. Supposing we have such an FRN, trading at a discount to par. If rates went to infinity, it would actually gain value, since the next coupon would be worth an immense amount. Thus, as illustrated in this extreme case, discount FRN’s exhibit negative duration – they increase in value when rates go up.

Negative duration is a very weird thing. Most corporate finance classes teach duration as time (in years) when 50% of a bond’s cashflows have been received (it just so happens that with some math you can show that this happens to be it’s rate sensitivity as well). To say that duration is negative is to imply, by that definition, that 50% of the cashflows were received in the past! I’m not going to worry about that paradox right now, however, except to leave it as evidence for how strange the concept of negative duration is.

But let’s plow ahead and investigate it nonetheless. First, some definitions. For a given FRN, Let L be the Libor rate, C be the coupon spread (i.e. if the coupon is L+100 then C = 100) and N be the number of payment periods (i.e. N = 5 for an annual-pay 5 year bond). Additionally, define S as the additional spread which, when added to Libor, is the rate at which the bond’s cashflows are discounted (i.e. if S = 100 the cashflows are discounted at L+100) . When S < C, the bond will trade at a premium (it’s coupons more than make up for the discounting); when S > C the FRN will trade at a discount (the discounting overwhelms the coupon payments); and when S = C the bond will trade at par (the coupons offset the loss of value to discounting).

For those following along at home, what follows is based on a simple DCF model and is quite easy to implement in Excel. Let N = 5, L = 1%, and C = 300 bps. If S = 400 bps, then the FRN (as expected) will trade at a discounted price of 95.67. Raise L to 1.5% and sure enough, the bond will rise to 95.73: negative duration! But why? Start back with a par FRN. Like the discounted bond, the par FRN also has N = 5, L = 1%, and C = 300, but by necessity has S_p = 300. Now the only difference between these two bonds is that in each period, the cashflows of the discount FRN are discounted by an extra 100 bps. I can represent that difference as an N-period annuity with a 100 bp “coupon” discounted at the rate L+S . This is because both bonds have the same nominal cashflows; the only difference is that one bond’s cashflows are being discounted more heavily. The annuity represents that discounted difference, in this case 100 bps. Now – and this is the important part – note that if I create a portfolio in which I purchase the par FRN and sell the annuity, I have recreated the cashflows of the discount FRN.

The par FRN, as demonstrated, has a duration close to zero. The annuity has positive duration, like most bonds, since raising L reduces the value of the coupon stream. But because I’ve shorted the annuity, my portfolio winds up with a negative duration. Et voila: all it takes is a little security deconstruction to show why discount FRN’s exhibit negative duration.

Please hand in your homework as you leave.

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