Posts from the long tail:

Risk

Professor Risk

December 13, 2009 in Math, Risk

David Spiegelhalter is the Professor of the Public Understanding of Risk at Cambridge University. He has recently produced the following video to encourage better practices in the casual perception of risky behaviors:

I think it’s a brilliant video and would love to have been one of Professor Spegelhalter’s students. I firmly believe that the study of risk and statistics more generally suffers more than anything from a particularly awful and dare I say boring curriculum, not to mention one which many teachers choose to render in terms beyond the grasp of many students. Efforts like this go a long way toward alleviating that obstacle and I applaud the professor for his work.

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Philosophies of risk management, briefly

November 18, 2009 in Finance, Risk

My last post made me think of a common question in risk management: “what is risk?”

A lot of time is spent deciding the various metrics, exposures, values, sensitivities, etc. that are considered “risks.” In the previous post, a simple change of perspective – is risk defined by dollars invested or shares controlled? – resulted in a dramatically different investment decision (granted, it was for a hypothetical insider trade… so it was definitely more illustrative than practical).

In the investment equation, the very definition of risk is a variable, not a constant.

This is probably much more interesting to talk about than write about, given the open-endedness of the question, but I would like to highlight how critical that question is. Before risks can be managed, they must be measured; and before they are measured, they must be identified. It’s very easy, particularly in a time when we are inundated by numbers and statistics, to look for a catch-all metric, or overlook risks that critical thinking would expose. Risk is rarely obvious.

Readers will know I am hardly espousing any sort of dive into complicated models or quantitative nonsense, merely an appeal to reason: every investment decision carries a unique set of risks which need to be identified and defined – from dollars invested, to sensitivities, to position in a larger portfolio, to leverage, and so on. More than merely identifying them, they should be understood – even VaR has its use, remember.

It wouldn’t be right to end this without a HHG2G quote which is almost, but not quite, entirely unrelated. Let’s just say it’s about things we take for granted:

Time is an illusion. Lunchtime doubly so.

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Stock is more efficient than options are more efficient than stock

November 18, 2009 in Finance, Risk

There is a very interesting debate taking place on the profitability of options as opposed to the underlying stock. It originates in this post from Ultimi Barbarorum on options volume following the Palm/3Com announcement, and continues in the comments on Felix Salmon’s coverage of that post.

The crux of the argument is the spike in options volatility immediately preceding the merger announcement, which many took as a clear sign of insider trading. Baruch argues that options are notoriously volatile, so one spike is hardly a smoking gun, and I agree (see also: superstitions regarding trading on option expiration days). However, I echo Felix in noting that it’s very hard, therefore, to draw any conclusion about insider trading whatsoever. Baruch’s second point is that if it were insider trading, it was misguided – the insiders could have made more money and attracted far less attention by trading the underlying stock rather than the options. This is where the debate lies – and I confess up front that my immediate impulse was to say “that can’t be right.” In fact, it could be right, depending on your point of view.

(p.s. hats off to Baruch for introducing his post with “Before the Zero Hedge folks get the pitchforks out, let’s stop and think a bit.”)

Baruch writes:

Had someone concrete knowledge of the 3Com deal, it would be far more efficient to buy the stock. The most important of the “Greeks”, as options dudes call the panoply of statistics surrounding options, is “delta”, the rate of change in the value of the option relative to the value of the shares (it’s a function of volatility, time to expiry, a whole lot of stuff, don’t trouble your head), and this is always less than one. 3Com options buyers made far less money on the takeover by buying options than they would if they had bought the stock.

It will be instructive here to discuss delta – I know some of TGR’s readers are already familiar with concept, and I hope you will excuse this detour.

“Delta” is a mathematical (as opposed to financial!) derivative of the option formula, as described by Baruch – but it is easier to understand as a “hedge ratio.” It tells you how many shares of stock you need to hedge your exposure to an option (I’m going to assume from here on that we are discussing calls). To see why, run back to the math for one second and consider that delta is the amount that the price of the option will rise if the stock price goes up by $1 – ok, now ignore the math. If the option is way in the money and trades at its intrinsic value, then it will gain $1 for every $1 the stock rises – a delta of 1. It the option is at the money, then its as likely about whether it will ultimately pay off at all, and so it only gains $0.50 for every $1 the stock rises – a delta of .5. Thus, if you want to hedge your option exposure, you would short [delta] shares for every option you hold. Delta is always less than 1; no option will gain more than $1 for every $1 the stock price rises.

The key to this delta business is that as long as delta is less than 1, you need to short fewer shares than the amount you control via options in order to hedge your option exposure. Put another way, it takes more options than shares to create the same exposure (on a per-share basis) to the underlying stock. If the stock price moves up, the dollar gain from holding that stock will be greater than the dollar gain from the options, hence the argument that stocks are “far more efficient” than options.

The closing price for COMS on November 10th, the day before the option purchasing frenzy, was 5.41. The $5 November calls cost slightly more than their intrinsic value at $0.55, trading with a delta of 0.72. On November 12th, the stock closed at $7.46, representing a gain of $2.05, whereas the options finished at $2.50, gaining just $1.95. Share for share, the stock outperformed.

However, shares controlled is an poor metric for comparing investments. This is particularly true for options, where you may not know until the day they expire if you actually control those shares or not! Instead, for risk management purposes we think of the number of shares the position is likely to control, given the current state of the world: the probability-weighted number of shares. Unsurprisingly, it’s the same as the number of shares it takes to delta-hedge the position. From this observation, a nice property of delta is revealed: it may be roughly interpreted as the probability of an option finishing in the money.

The important philosophical point here is not to make the mistake of thinking that the number of options you buy is equal to the number of shares you own – that’s only true the day they mature in the money. To set up the same exposure in options as we have with shares, at the time of purchase, we need to buy a few extra options. Specifically, for the November calls with a delta of .72, we need 1/.72 or 1.39 options for every share. Run the numbers and you’ll see that this results in a final profit of $2.71 on the option side, vs $2.05 for the shares. If an insider bought options on a delta-adjusted share basis, he’d find the options more profitable than the stock.

(If you constantly adjust the number of options to correspond to the prevailing delta, you’ll wind up making $2.05 on your options – this process is called dynamic delta-hedging [that's a real aside for this post, because the discontinuity in COMS stock price would make the rebalancing futile].)

So, on the basis of shares-at-maturity, stock yielded a better dollar profit. On the basis of shares-at-trade, options would have been preferable. There’s an argument to be made that, as an insider, you know the options will finish in the money, so shares-at-maturity is the right way to consider it. But there’s a third exposure metric: capital at risk.

You can look at capital at risk as either 1) the maximum loss you could experience OR (if you’re an insider who knows the trade will be profitable) as the opportunity cost of capital. This is very straightforward to explain: those options only cost $0.55; the stock cost $5.41. The percentage gain on the options is 355%; for the stock it’s just 38%. If you consider your exposure in terms of dollars invested, rather than shares controlled, you’d find the options a far better bet: they cost almost 90% less than the stock but return nearly as much per contract! So for every dollar you could put into the stock, you could instead put into options and return 10x as much. Options, from this perspective, are far more effective.

So this all depends on how you look at your risk and exposure. Baruch assumes that his insiders want to control a certain number of shares, and from that perspective they should absolutely have transacted stock instead of options (assuming that, with their perfect knowledge, they skip over the delta-adjust share argument). Personally, I would look at it from a capital at risk perspective – if I’m willing to spend $5.41/share to make $2.05, why not put that to work in options and make $19.18?

It all depends on your perspective – both answers could be correct, given some set of portfolio constraints and different definitions of risk/exposure.

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Haven’t we seen this movie before?

October 14, 2009 in Finance, Risk

In a recent profile of KKR, Breakingviews.com (via the NYTimes) attempted to value the company by taking a look at Blackstone’s operations. I don’t have any comment on the analysis itself, but two excerpts stood out in my mind:

[Blackstone] didn’t do as well collecting performance fees and investment gains because its holdings have been falling in value. But if history is any guide, its investments should rebound.

…

So, although it sounds generous given the last year’s market conditions, it’s not unreasonable to assume those [illiquid assets] might gain 20 percent annually from their current valuations for the next five years.

Does anyone else shudder a little when reading sentences like these?

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The rise of VaR

October 1, 2009 in Finance, Quotes, Risk

Simon Johnson and James Kwak take a look at how VaR got to be so popular in the first place. They make the insightful observation that a bad (or at least an incomplete) model can gain acceptance not only because of its simplicity but, oddly, because of its output as well.

Indeed, VaR succeeded not just because it seemed to capture risk accurately (“losses exceeded only 5% of the time” and so on), but because it provided the answer that financial agents were looking for. Most cynically, its greatest disadvantage – failing to look at what actually happens in crisis times, rather than just defining the crisis itself – turned into its biggest sell point when it came to market adoption. In a bizarre twist, the model was chosen because it gave the right answer; not because it answered the right question.

It reminds me of a passage from the ever-insightful Hitchhiker’s Guide to the Galaxy:

“I checked it very thoroughly,” said the computer, “and that quite definitely is the answer. I think the problem, to be quite honest with you, is that you’ve never actually known what the question is.”

“But it was the Great Question! The Ultimate Question of Life, the Universe and Everything!” howled Loonquawl.

“Yes,” said Deep Thought with the air of one who suffers fools gladly, “but what actually is it?”

But you don’t come here for HHG2G quotes (or do you?). Here’s the key excerpt from Johnson and Kwak’s analysis:

David Colander made this point about economic models: The sociology of the economics profession gave preference to elegant mathematical models that could describe the world using the smallest number of parameters. “Common sense does not advance one very far within the economics profession,” he says.

A similar point can be made about VAR models. Sure, maybe all the financial professionals who design and work with VAR know about its shortcomings, both mathematical and practical. But nevertheless, using VAR brought concrete benefits to specific actors in the banking world by helping them rationalize bad bets. If common sense would lead a risk manager to crack down on a trader taking large, risky bets, then the trader is better off if the risk manager uses VAR instead.

Not only that, but imagine the situation of the chief risk manager of a bank in, say, 2004. As Andrew Lo has argued, if he tried to reduce his bank’s exposure to structured securities such as collateralized debt obligations, he would be out of a job; VAR gave him a handy tool to rationalize a situation that defied common sense but that made his bosses only too happy. And at the top levels, chief executives and directors who probably did not understand the shortcomings of VAR were biased in its favor because it told them a story they wanted to hear.

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Measuring aggregate risk in CDS markets

September 25, 2009 in Finance, Risk

The ECB recently published this lengthy report (PDF link) on the state of the CDS market, with particular focus on counterparty risk. It is well worth a read for either a cursory overview or more in-depth look at the mechanics and concerns of that market.

Section 3.4 regarding counterparty risk measures was especially interesting to me. Consider the passage on the use of gross outstanding notional as an indicator of risk (emphasis mine):

The notional amount of a credit default swap refers to the nominal amount of protection bought or sold on the underlying bond or loan. Notional amounts are the basis on which cash flow payments are calculated.

The gross notional amount reported by the BIS is the total of the notional amounts of all transactions that have not yet matured, prior to taking into account all offsetting transactions between pairs of counterparties. As outlined above, gross notional amounts thus represent a cumulative total of past transactions. Using gross notional amounts as an indicator of counterparty risk may be misleading, as many trades are concluded with a single counterparty.

Once negotiated, CDSs bind both counterparties until the agreed maturity. Market participants basically have three choices when increasing or reducing their CDS exposures.

First, they can terminate the contract, provided the counterparty agrees to the early termination. Second, they can fi nd a third party to replace them in the contract, provided the counterparty consents to the transfer of obligations (“novation”). As a third option, dealers that want to unwind or hedge their positions can also enter into offsetting transactions, sometimes (though not necessarily) negotiated with the same counterparty as the hedged deal. The third solution is used extensively, and so the number of trades has surged, resulting in an increase in total gross notional amounts. Indeed, this technique, by contrast with the other two, does not eliminate previous deals and instead adds them together. The end result is that external market commentators tend to pay too much attention to the gross market values in relation to other measures of the real economy such as GDP, whereas net notional amounts, where accounted for, may be downplayed or perceived as being very low or moderate in relative terms given the huge gross notional amounts outstanding.

The gross notional amount reported by the BIS is the total of the notional amounts of all transactions that have not yet matured, prior to taking into account all offsetting transactions between pairs of counterparties. As outlined above, gross notional amounts thus represent a cumulative total of past transactions. Using gross notional amounts as an indicator of counterparty risk may be misleading, as many trades are concluded with a single counterparty.

Once negotiated, CDSs bind both counterparties until the agreed maturity. Market participants basically have three choices when increasing or reducing their CDS exposures.

First, they can terminate the contract, provided the counterparty agrees to the early termination. Second, they can fi nd a third party to replace them in the contract, provided the counterparty consents to the transfer of obligations (“novation”). As a third option, dealers that want to unwind or hedge their positions can also enter into offsetting transactions, sometimes (though not necessarily) negotiated with the same counterparty as the hedged deal. The third solution is used extensively, and so the number of trades has surged, resulting in an increase in total gross notional amounts. Indeed, this technique, by contrast with the other two, does not eliminate previous deals and instead adds them together. The end result is that external market commentators tend to pay too much attention to the gross market values in relation to other measures of the real economy such as GDP, whereas net notional amounts, where accounted for, may be downplayed or perceived as being very low or moderate in relative terms given the huge gross notional amounts outstandingNotional amounts are the basis on which cash flow payments are calculated.

The gross notional amount reported by the BIS is the total of the notional amounts of all transactions that have not yet matured, prior to taking into account all offsetting transactions between pairs of counterparties. As outlined above, gross notional amounts thus represent a cumulative total of past transactions. Using gross notional amounts as an indicator of counterparty risk may be misleading, as many trades are concluded with a single counterparty.

Once negotiated, CDSs bind both counterparties until the agreed maturity. Market participants basically have three choices when increasing or reducing their CDS exposures.

First, they can terminate the contract, provided the counterparty agrees to the early termination. Second, they can fi nd a third party to replace them in the contract, provided the counterparty consents to the transfer of obligations (“novation”). As a third option, dealers that want to unwind or hedge their positions can also enter into offsetting transactions, sometimes (though not necessarily) negotiated with the same counterparty as the hedged deal. The third solution is used extensively, and so the number of trades has surged, resulting in an increase in total gross notional amounts. Indeed, this technique, by contrast with the other two, does not eliminate previous deals and instead adds them together. The end result is that external market commentators tend to pay too much attention to the gross market values in relation to other measures of the real economy such as GDP, whereas net notional amounts, where accounted for, may be downplayed or perceived as being very low or moderate in relative terms given the huge gross notional amounts outstanding.

It’s easy to come up with an example which illustrates the problems with gross notionals (the ECB’s “third solution”):

Dealer A sells $1mm of protection to Fund X. The gross notional at this time is $1mm, and the maximum that could be lost (in an extreme case with 0% recovery and the original contract transacted at a zero spread) is also $1mm. Now Dealer B sells $1mm of protection on the same name to Fund Y. The gross notional is $2mm, and so is the maximum loss in the market. But what if Dealer B had sold CDS to Dealer A instead? Then the gross notional would still be $2mm, but only $1mm could be lost, as Dealer A has hedged its position completely. Thus, gross notional has overstated the risk present in the marketplace.

Net notional is a much better measure, but, in line with my parenthetical aside, does not quite capture the risk at hand; it only does so under extreme circumstances. (It also isn’t nearly as dramatic a number, so the media is more loathe to deal with it.)

In my experience, jump to default (JTD) and jump-or-bleed to safety (JTS) measures are instructive methods for evaluating risk. Most commonly, these measures are evaluated with respect to the reference issuer, but they are easily applied to the counterparty as well. However, calculating them in aggregate – at the market level – requires knowledge of the various contracts’ market values, data which is not presently made public (gross and net notional values are available from the DTCC).

Finally, the ECB makes the salient point that any market-wide counterparty risk measure must account for collateralization. There is some ambiguity there, however, because a contract which is fully collateralized on a mark-to-market basis still has considerable counterparty risk in a jump event. Frequently, protection buyers may find that to be wrong-way risk, meaning that the exposure to a counterparty is inversely related to that counterparty’s credit rating. For example, a counterparty defaults, driving credit spreads wider (a profitable event for the protection buyer) but also making other counterparties more likely to default (a very bad thing for the protection buyer).

In failing to find a clear, universal or simple risk metric for this market – which I don’t think is necessarily preferable given the over-reliance and under-comprehension placed on VaR after its wide dissemination – we may find that the best outcome is to strive for transparency in understanding. A strong education in the mechanics and risks of complex markets is an important step forward and a necessary prerequisite for market participants in both direct and regulatory roles.

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Yet more risky testimony

September 11, 2009 in Finance, Risk

Nassim Taleb and Chris Whalen also participated in Wednesday’s House hearing on risk management. The full text of their remarks are available here (Taleb) and here (Whalen).

Taleb’s thoughts are familiar, consisting largely of his well-known opinions on VaR and financial regulation. Whalen, however, provides an excellent quote:

The problem is not with models themselves. The trouble happens when they are (a) improperly constructed and then (b) deliberately misapplied by individuals working in the financial markets.

Yes, models are just the tool! Unfortunately, Whalen followed it up with this misstep:

We take a different view. We don’t actually believe there is such a thing a a “Black Swan.” Our observations tell us that a more likely explanation is that leaders in finance and politics simply made the mistake of, again, believing in what were in fact flawed models and blinded themselves to what should have been plainly calculable innovation risks destined to be unsustainable.

…

If financial markets and the models used to describe them are limited to those instruments that can be verified objectively, then we no longer need to fear from the ravages of Black Swans or systemic risk. The source of systemic risk in the financial markets is fear born from the complexity of opaque securities for which there is no underlying basis.

…

If we accept that the sudden change in market conditions or the “Black Swan” event that Taleb and other theorists have so elegantly described arises from a breakdown in prudential regulation and basic common sense, and not from some unknowable market mechanism, then we no longer need to fear surprises or systemic risk. We need to simply ensure that all of the financial instruments in our marketplace have an objective basis, including a visible, cash basis market that is visible to all market participants. If investors cannot price a security without reference to subjective models, then the security should be banned from the US markets as a matter of law and regulation.

What a frightening belief! This notion of “objectivity” simply doesn’t exist. In a strict sense, it would mean that all market participants agree on a security’s value. This comes in two varieties: first, the value of the security is known at all future points. This is a failure because no one would ever speculate on such a product.

Second, a standardized model could be used, in which everyone agrees on the level of uncertainty in the market and that it is the right description of reality. Obviously this is impossible because agreeing that a model is right does not make it so – the emperor is still naked. Moreover, to suggest that all we need to do to avoid crash risk is come up with a “correct” model is naive, as we don’t have a universally accepted “correct” model for anything!

Claiming that Black Swans only exist because the prior model didn’t encompass them is all very nice; it is however tautological and unhelpful. In fact, I find it especially interesting that this is part of Whalen’s testimony, because Bookstaber’s testimony at the same hearing includes a near-perfect rebuttal in its appendix, in which he declares that blaming “fat tails” is a straw man argument:

We are not, after all, talking about physics, about timeless and universal laws of the universe when we deal with securities. Weird stuff happens. And the place where the imperfection is most telling is in risk management.

When the risk manager misses the equivalent of a force five hurricane, we ask what is wrong with his methods. By definition, what he missed was a ten or twenty standard deviation event, so we tell him he ignored fat tails. There you have it, you failed because you did not incorporate fat tails. This is tautological. If I miss a large risk – which will occur on occasion even if I am fully competent; that is why they are called risks – I will have failed to account for a fat tailed event. I can tell you that ahead of time. I can tell you now – as can everyone in risk management – that I will miss something. If after the fact you want to castigate me for not incorporating sufficiently fat tailed events, let the flogging begin.

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Bookstaber’s testimony on risk

September 11, 2009 in Finance, Risk

Rick Bookstaber testified to the House on Wednesday regarding risk management; the text of his remarks is available here. It is a must-read.

The bulk of his testimony focuses on VaR: it’s use, misuse and role in the recent crisis. I find his greatest insight in this paragraph:

I remember a cartoon that showed a man sitting behind a desk with a name plate that read ‘Risk Manager’. The man sitting in front of the desk said, “Be careful? That’s all you can tell me, is to be careful?” Stopping with the observation that extreme events can occur in the markets and redrawing the distribution accordingly is about as useful as saying “be careful.” A better approach is to accept the limitations of VaR, and then try to understand the nature of the extreme events, the market crises where VaR fails. If we understand the dynamics of market crisis, we may be able to improve risk management to make it work when it is of the greatest importance.

But stop reading here and check out the testimony in its entirety. It is an excellent example of gentle writing that nonetheless contains a critical message. Also, don’t miss a reprint of Bookstaber’s attack on fat-tails as a straw man argument. I hadn’t seen such a description in print before, but I’m inclined to agree.

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Deconstructing the Gaussian copula, part III

August 11, 2009 in Finance, Math, Risk

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

In the first two parts of this series, I respectively addressed some misperceptions about the Gaussian copula and described its common use in CDO pricing. Part III focuses more on the model components and the intuition driving them.

I am a staunch supporter of a “models are just the tool” viewpoint, an opinion more elaborately and memorably stated by George Box as, “All models are wrong, but some are useful.” With that in mind, what you will find here is not a campaign against the Gaussian copula itself; merely its blind application to certain problems in finance. I find it as difficult to blame this model alone for the 2008 recession as I find it hard to blame the sinking of the Titanic on its hull design (new research actually suggests the rivets were more at fault) – while it certainly contributed to a general sense of invincibility and well-more-than-advisable risk taking, it is naive to think that in the absence of this notorious model, 2008 would have turned out just fine.

As I recently (and strangely, given my campaigns against it) stated about VaR, the Gaussian copula does exactly what it is supposed to do – the error lies in its interpretation and its application in the first place. I join Paul Wilmott in his crusade for less equations and more common sense among quantitative financiers: getting the number is good but explaining it is better.

Gaussian Dependance

Copulas are nothing more than descriptions of how two or more random variables relate to each other. To be more specific, copulas refer to the co-behaviors of uniform random variables only; but any distribution may be transformed to the uniform case via its CDF, and that is the appeal of copula models: they describe dependance without concern of the marginal distributions. The Gaussian copula, we may conclude, doesn’t necessarily have anything to do with normal distributions as we typically think of them (i.e. in the “normal distributions are useless in finance” sense)! Rather, it describes the sort of dependance that arises when a bunch of normally-distributed variables are correlated with each other.

Gaussian dependance isn’t easy to describe like a Gaussian distribution is. For the latter case, just think of a bell curve. The former is more difficult to identify, so here’s a picture of a two uniform random variables with a Gaussian dependance structure (click to zoom):

A first observation is that the dependence is regular (meaning even) and smooth. It lacks any significant clustering. More importantly, it lacks a property called tail dependence. Tail dependance is the probability of observing extreme observations in all random variables at once. Strictly speaking, it measures the probability of observing joint tail events. As you move further out in the tail, that probability converges to 1 in the limit for structures exhibiting tail dependence. It is extremely surprising and counter-intuitive to learn that the Gaussian copula lacks tail dependence. In plain English, this means that tail events in the Gaussian copula are asymptotically independent of each other – and that is the chief problem with using Gaussian dependence in finance.

In finance, extreme events co-occur all the time, as recent memory bears witness. If risk management is the process of ascertaining, measuring, and avoiding those situations, then doesn’t it seem a little odd to use a model which is explicitly unable to account for them? Tail dependence is a necessary condition for a dependence model in finance. The Student t copula exhibits it and is only marginally more difficult to implement than a Gaussian copula; but simplicity is king and there was obviously a decision made at some point that tail events didn’t require consideration, anyway. It brings to mind my favorite VaR metaphor as an airbag that always works, except in a crash.

Correlation

Another element of the Gaussian model which does not carry well to finance is the idea that linear correlation is a sufficient statistic for the dependence distribution. Consider these two plots, each of which shows two variables that, by construction, have a correlation of 0.7 to each other. First, a Gaussian dependence structure (this looks different than the above plot because the former was the copula itself, as indicated by the uniform marginals, whereas this is a full copula-derived multivariate distribution):

Next, a dependance structure exhibiting lower tail dependance (this is from a Clayton copula and is a stylized depiction of behaviors more characteristic of finance). You can plainly see the impact of the tail dependance, in contrast to the Gaussian plot above:

The two distributions are very obviously different, and yet if you merely measured their correlation you’d describe them in exactly the same way. Correlation alone is insufficient to describe more complex dependence structures such as those observed in finance. And yet, it is the only descriptive statistic of a multivariate Gaussian distribution.

Financial covariates tend to resemble the second plot – when a large negative event occurs in one, it more than likely will occur in the other. This, by the way, accounts for some of the skewness in financial distributions – it is possible to have two perfectly normal distributions whose combination is nonetheless skewed if the dependence structure exhibits tail effects like this.

Again, we have a call for clarity: it is imperative for the underlying dynamic of any model to resemble the behaviors of the system in question.

The Single Correlation Factor

In a CDO pricing framework based on the Gaussian copula, not only is correlation the sole determinant of the dependence structure, it is assumed to be the same for every name in the basket. This has caused much alarm. Certainly, using more factors would provide a more accurate model – allowing different industries to have different correlations, for example. Unfortunately, this comes at the cost of model accuracy.

It is very important, where possible, for a model to have no more than one unobserved input for every output. Think of a Black-Scholes option: future volatility can not be known, so we plug in whatever value gets the model to spit out the current market price of the option (a “the market is always right” approach). If there were two volatilities (say, a short term value and a long term value), we would be unable to create a consistent model, for there would likely be an infinite number of volatility pairs that would satisfy the market price. For every additional parameter, we need one more output metric to match. If we could match an option’s price and also it’s delta, just for arguments sake, then there is probably a unique combination of two volatilities for that output space.

This is why using multiple correlations is problematic not just from a fitting standpoint, but from a model integrity standpoint – if you take the thousands of necessary pairwise correlations and estimate just a handful of them incorrectly, the model could deliver completely spurious results.

(For a very concrete example of this, consider pricing a mezzanine or senior CDO tranche, which requires two correlation inputs. Without knowledge of the corresponding equity tranche price – and consequently the attachment point correlation – this becomes a very difficult puzzle indeed).

However, in my mind this is one of the more minor problems. That’s not to say it isn’t an issue, but I’d much rather have a single-parameter tail-dependent model than a multi factor Gaussian one. Why? Because it’s more important to me that a model captures downside risk in some regard than that it captures the distribution’s central dynamics more faithfully.

Correlation (again)

We’ve discussed why correlation is insufficient to describe the CDO dynamics, and also why a single-factor model may lack fidelity. But in some ways, the entire discussion is slightly off base. Correlation (as I’ve alluded before) is an implied measure – it is whatever plug gets the model to output the “right” price.

There is a raging debate about how similar correlation is to Black-Scholes volatility, but I think for the purposes of this exercise we can highlight their similarities (though I will not necessarily agree with that under more rigorous terms). both are plug values; both have intuitively “correct” ranges but can not be directly measured or observed; both are the single unobserved input in the most simple pricing models of their respective derivatives.

Because of this, a lot of our reasoning on the problems with correlation goes backwards, since we begin with the premise that correlation is arbitrary and/or unmeasurable, and therefore conclude that a correlation-based model must fail. However, in practice we actually start with a tranche price, and work out the implied correlation value from that price. So I don’t really care if my correlation comes out to 60% or 70% because I’m not going to read too much into that figure – it’s just a parameter that will keep my model ticking consistently with the market, all else equal.

“But wait,” you say, “that’s the dumbest thing I’ve ever heard!” What if the market price is arbitrarily high and implies a correlation greater than 1 (or just 1, since the input is bounded)? Then that’s great, you get the price right in that instant, but the second you try to measure any sort of risk or even price it the next day, you’ll fail because a correlation of 1 doesn’t reflect reality at all. Moreover, take this to its logical conclusion: why not have a model whose sole input and output is just the price. In this scenario, you would see a tranche trading at 20, and set your “model” to 20 (the implied price). Tomorrow, your model still says 20 – so when the actual tranche trades for 19, you need to adjust your “model parameter” (i.e. price) down. Obviously, a ridiculous situation and it speaks to the critical need for any model to balance a reasonable representation (even if a simple one) of reality with an acceptable range of input parameters.

To reiterate, this is why I would prefer a simplistic one-factor tail dependent model to a multifactor Gaussian one.

Other Copula Models

All of this must raise the question, why are we stuck using the Gaussian copula?

And like so much else, the answer is: because its easy.

As mentioned, the Student t copula exhibits tail dependence and is only slightly harder to build than the Gaussian variety. So why not use it? The dark secret (unless you read part II) is that single factor Gaussian copula models are really just massive simplifications of copula-derived mathematics. The engine itself relies on arithmetic and an integral – nothing that would suggest a copula model on the surface. It is the mathematically friendly properties of the Gaussian distribution that make this possible (though frankly, it seems to me a t implementation shouldn’t be much farther off). More obscure copulas, like those in the Archimedean family, don’t necessarily follow “real world” behaviors in high dimensions, as it pertains to finance.

Moreover, like all problems of this ilk, CDOs suffer from a massive curse of dimensionality. In such situations, familiarity is key – in fact, it is sometimes the only hope of finding answers in the massive cosmos of sparse data.

Finally, Gaussian copulas have a nice property – they are easy to explain (keep in mind, lately such explanations aren’t much at all). In particular, the error rates are easy to quantify – we can be 99.975% sure of an outcome. Knowing a concrete chance of failure, even if that probability is completely bogus, makes the model easy to accept. More complicated copula structures, by contrast, are harder to work with (read: make it harder for risk managers to promise certain error rates within certain error bounds).

Finally, more complicated does not necessarily mean better. Even after all I’ve written, a pinch of common sense applied to a single factor Gaussian model might do more wonders than a more advanced model in the hands of a naive user.

Here endeth the lesson.

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Deconstructing the Gaussian copula, part II and a half

August 11, 2009 in Finance, Math, Risk

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

One of the comments to part II of this series asked about recovery assumptions, observing that a 40% static recovery assumption would cause a 60-100% tranche to always be worth par, since it would never take damages. That’s correct and raises an issue I will dance around in part three: recoveries constitute another unobserved input to the copula model.

When issuers default, they do not drop completely out of the picture – there is a some recovery level assumed, on the premise that in bankruptcy there will be enough assets to cover some portion of the debt owed. It could be anywhere from 0% to 100%. If the issuer recovers 100%, the CDO takes no damage (as there is no need to pay out insurance).

The most simple way to address recovery assumptions is to take them as exogenous static inputs. They don’t necessarily have to be 40% (widely accepted as the long-run average recovery rate). Recovery assumptions could be based on an analyst’s estimate, implied from CDS, taken from a market quote (recoveries are traded quantities), spit out of a random number generator, etc. It doesn’t matter in an exogenously-specified static model.

The second way of dealing with them is to model them as stochastic quantities, which may or may not be correlated with default intensities. A simple (and hardly stochastic) method would be to lower recoveries as more issuers default, on the assumption that in such a scenario the business environment must be very bad. This would have the effect of compounding losses – not only are issuers defaulting, but the surviving issuer recoveries are falling over time. Alternatively, you could get very fancy and let the recoveries dance as random variables – incorporating a gamma process, for example. However, that’s a little beyond the scope of my series (for the moment).

There is generally enough information about recoveries available through research and market information that an exogenous model provides satisfactory results. One question we must ask is whether it is “bad” that 40% recoveries would cause a senior tranche to never take damages. Honestly, I think it isn’t because that’s a scenario that could easily be reflected in reality: if it really happened, then the tranche wouldn’t take damages. It might trade down, however, as people fear recoveries would drop – and that is where the model specification comes into play. It is the role of the model caretaker, not the model itself, to make sure its input parameters and dynamics are reflective of reality.

There’s nothing wrong with creating a risk free bond – and to take this example to extreme, the 99-100% tranche is most likely risk-free under any recovery assumptions. The only problem is to expect to get paid for holding it. Moreover, even a 60-100% tranche with 40% recoveries could trade under par depending on interest rates and (in some cases) prepayment probabilities. But that’s another chapter.

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VaR at risk

July 31, 2009

In a piece called “The Risk Mirage,” BusinessWeek assails its peers for falling for VaR-based evaluations of Goldman’s risk levels:
[A] VaR-based analysis of any firm’s riskiness is useless. VaR lies. Big time. As a predictor of risk, it’s an impostor. It should be consigned to the dustbin. Firms should stop reporting it. Analysts and regulators [...]

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Deconstructing the Gaussian copula, part II

July 9, 2009

A math-free introduction to CDO pricing.

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Interviewing Myron Scholes

May 16, 2009

Speaking of LTCM (and of this Sunday’s Times Magazine, for that matter), here’s an interview that’s going to run with Myron Scholes, who comes off like a bad comedian. The questions are poor and the answers arguably worse. Let’s take a look, shall we?
The second question: “You’re known as the intellectual father of the credit [...]

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Aggravation (but keep reading)

May 16, 2009

I happen to like this article by Niall Ferguson for the Times Magazine, in particular this bit:
Human beings are as good at devising ex post facto explanations for big disasters as they are bad at anticipating those disasters. It is indeed impressive how rapidly the economists who failed to predict this crisis — or predicted [...]

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Monte Carlo: house of cards?

May 8, 2009

The WSJ recently ran apiece on Monte Carlo risk management:
Here is how a typical Monte Carlo retirement-planning tool might work: The user enters information about his age, earnings, assets, retirement-plan contributions, investment mix and other details. The calculator crunches the numbers on hundreds or thousands of potential market scenarios, guided by assumptions about inflation, volatility [...]

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How to lose your money without really trying

April 27, 2009

An author describes a lose-lose strategy.

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A casualty of chance

April 20, 2009

I discovered this Atlantic article (“Why I Fired My Broker“) on MB’s blog. I came to enjoy it in the end, but while I was reading it I was struck by how representative it is of contemporary financial journalism. This is the new cookie cutter article: naive reporter is encouraged by rich capitalists to invest, [...]

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Thoughts on risk management

April 20, 2009

Naked capitalism put out an open call for thoughts on the state of risk management on trading desks. The comments are well worth a curious read (how many times have you said that about a blog post?).
It is interesting that when you get enough academics and practitioners shouting in a room, risk management becomes something [...]

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Models are just the tool

April 17, 2009

I’m a big fan of Emanuel Derman. His memoir My Life as a Quant tells the story of a young physics Ph.D. who stumbled into finance and eventually became the head of Goldman Sach’s Quantitative Risk Strategies group. He currently oversees the financial engineering program at Columbia University and is the CRO of Prisma Capital.
Today he [...]

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Signs of the apocalypse

February 24, 2009

Wired has published an article attacking the Gaussian copula: Recipe for Disaster: The Formula That Killed Wall Street.
It’s a very typical “hate the game, not the player” article which finds fault with a tool rather than the people who use it. Not that I completely disagree with the critique – but imagine my surprise [...]