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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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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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VaR at risk

July 31, 2009 in Finance, Risk

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 should stop using it.

Some, including regulators who base capital reserve requirements on this metric, may call VaR a “measure of market risk” and “predictor of future losses.” But it is neither of those things. Its forecast of how much an investor can lose from a trading position is entirely calculated from historical data. It’s a mathematical tool that simply reflects what happened to a portfolio of assets during a certain past period. (The person supplying the data to the model can essentially select any dates.)

I’m certainly not one to stick up for VaR as a risk measure by any means, but sometimes you have to stand up for the underdog (or, in this case, the recently toppled dictator). In truth, VaR has a very concrete, well defined use: it measures the edge of a distribution’s tail. How one makes use of that information is another story.

How one calculates it is up for even more debate. VaR is not “entirely calculated from historical data,” though it is quite informed by it. A proper risk model of any sort will involve some form of forward-looking simulation (absent closed form expressions) and account for the firm’s best intuition about the future. So let’s not discount the measure on an incorrect assumption regarding it’s origin.

A well-specified VaR model can do a surprisingly good job of defining the distribution’s tail. From a risk managment perspective, I believe this is insufficient information on which to act. However, I will make this generalization: as VaR increases, the edge of the tail moves farther into negative territory, and “risk” (however you define that measure) has increased as well. Note there is no absolute statement here – I don’t care if VaR is $1 or $1mm. But if that number doubles, you would be wise to conclude that the portfolio’s risk has increased as well.

Another thought – as long as we only care about relative moves in VaR, and not absolute levels, the distinction between a normal and fat tailed distribution becomes less important – yes, a high-kurtosis distribution could conceivably have a lower VaR but longer tail, but if I don’t think the distribution changes significantly over time then my relative-moves insight should hold true.

Caveat: I’m merely suggesting that VaR has a well-defined meaning and a plausible use for that measure; I’m not arguing in favor of it over any other analytic. VaR is but one tool among many.

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Deconstructing the Gaussian copula, part II

July 9, 2009 in Finance, Math, Risk

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

Recently, I addressed a great deal of misinformation regarding the Gaussian copula and it’s role in the 2008 crisis. I would like to try and follow that up with a succinct description of the copula and its use in CDO pricing. (This may seem a defense of the math behind the process, but you know I’m just setting it up for a fall.)

Introduction

David Li’s contribution to quantitative finance was the rapidly-standardized “single factor Gaussian copula” CDO pricing framework. The real crux of the problem was the “single factor” part – not the Gaussian copula itself (though we won’t pull any punches here). In an extraordinarily broad sense, a copula is a mathematical function that describes how two or more random variables interact. “Correlation” is a simple way of describing the copula, which should give the function some intuitive grounding. But let’s back up a second and figure out why we even need a copula in the first place.

Aside: Why Copulas?

If you try to model the behavior of many random variables, you need a multivariate distribution. The most mathematically friendly distributions are from the Gaussian family, including the familiar bell (or normal) curve. This is why such models are prevalent in all manners of statistics. For most purposes, the model is not only easy to work with but asymptotically correct (which is a nice feature, to put it mildly). However, there are some areas where the model choice is more for pragmatic reasons than justified ones – finance being prime among them. Indeed, financial distributions do not behave normally, but only recently have tools been developed that can describe them – and even there large joint distributions are daunting.

So, it is unsurprising that the Gaussian copula arose as a natural choice for modeling the joint distribution inherent to CDOs – which are essentially just collections of many intercorrelated credits.

But I’m getting ahead of myself. (This is much easier to discuss than to write about, I think, because you can guage your audience’s comfort which each boldfaced section before moving on. I hope, brave reader, that you are still there.) Lets talk about CDOs.

CDOs

A CDO is nothing more than a collection of various bonds, all held together in a basket. The principal risk of a CDO is default: the chance that one or more of the bonds will not survive to maturity. To isolate this risk, it is instructive to think of the CDO as a basket of sold CDS contracts, rather than a basket of purchased bonds (and indeed, “synthetic CDOs” are nothing more than CDS portfolios and have rapidly gained market share from bond portfolios). Thus, the buyer of a CDO needs to draw two conclusions regarding the basket:

Will any of the credits default?
When will all of those defaults occur?

The first point is obvious; the second gets at the heart of the problem. Both the timing and the correlation of defaults matter. If the CDO basket is comprised disproportionately of financial companies, then default by one may imply a greater likelihood of default for the others; a more diversified basket may not exhibit such dependencies.

This issue is compounded by the introduction of tranches – a staple of the CDO industry. Again, it is helpful to consider a CDO as a basket of sold CDS. The most junior (or “equity”) tranche has, by definition, sold insurance on the first few issuers to default – say, the first 3. The next tranche does not experience a loss until the 4th issuer defaults. The key here is that when a portfolio is tranched, investors have not sold CDS on specific issuers by name, but rather by time of default. They can not know ahead of time which issuers they are effectively responsible or on the hook for.

Bathtub Correlation

To understand why tranching compounds the correlation problem, think of the CDO as a rectangular bathtub interspaced with mines that represent each issuer’s default. The CDO investors are aboard a boat on one side of the bathtub, and need to cross to the other side. If the boat hits a mine, that issuer defaults, and the explosion of the mine will damage the boat. The equity tranche has an extremely thin hull and will sink quickly; the senior tranche has a thick hull and can withstand many blasts without taking damage. Finally, the boat moves across the bathtub via geometric brownian motion – which is to say, randomly.

In a low-correlation world, the mines are dispersed uniform randomly across the bathtub; hitting one mine does not imply or necessitate hitting any other. With high correlation, the mines cluster somewhere in the water; hitting one mine makes it relatively certain that another will be hit.

As a consequence, equity investors prefer high correlation. They are indifferent to hitting just a few mines or many, as they are wiped out in both situations. Therefore, they prefer the mines to be clustered, as this leaves more clear paths across the bathtub. In contrast, senior investors prefer low correlation – they can withstand glancing off a few mines, but hitting a cluster would wipe them out.

From this intuitive example, it should be clear that not only the timing of the defaults, but also their expected clustering (i.e. correlation) is important when valuing a CDO tranche.

Correlation in the Guassian Copula

Let us first draw the connection I’ve sketched out already: CDOs are composed of many issuers that may interact with each other; and a multivarite normal distribution is a common method of describing such behavior. So far, so good.

Like any Gaussian multivariate model, the Gaussian copula takes as parameters the correlation of every pair of variables under consideration. (In other words, to make the model work, you need to “explain” to it how every issuer interacts with every other issuer – these are the parameters.) Thus, the number of parameters increases with the square of the number of variables being considered – specifically, there are $\frac{N(N-1)}{2}$ parameters. If you had a CDO of 100 names, you would need to compute 4,950 parameters to describe their behavior! It doesn’t take a statistical degree to appreciate the flimsiness of a model which relies on such assumptions – it’s just too many to estimate reliably. Clearly, the traditional model simply won’t do.

Enter David Li, whose principal contribution to this field is to boil 4,950 parameters down to just one.

Shocking! Dastardly! The decision that caused the 2008 crisis! Well, not really. Though I am full of doubts about the validity of the Gaussian copula for this task in the first place, I do not think that the compression of its parameter space is the chief culprit by any means.

What Li was suggesting amounted to this: instead of modeling the intricate inter-corporate correlation structure, in which financials are highly correlated to each other but bear little semblance to utilities, which themselves are very similar, he said why not just model everything at the average correlation of the CDO names? Actually, he just said that one correlation level will be enough to describe the CDO price – he did not say it was the average (I just added that to make the notion more tolerable at first glance). He didn’t care if you chose a higher or lower correlation than any pair in the whole CDO exhibited; his claim was that there was some single number that would get the model to output a price that matched the market.

Before we get up in arms about this let’s remember that most financial instruments are priced this way. One or more variables of the equation are left free to change, such that for some level the model will output the “correct” (or market-observed) price. With options, this is called volatility; with swaps this is the fixed rate; with bonds this is the yield – I particularly like the last example because most people assume this is limited to derivatives. It’s not, “real” securities exhibit this problem too — for stocks, it’s called a P/E ratio.

So, we’ve boiled correlation down to one parameter which can take any value, but forces all issuers to have the same correlation to each other AND (this is a much more important caveat) exhibit a Gaussian dependance structure.

Now What? This Is Getting Boring.

Ok, let’s price a CDO.

If I have CDS prices for all the issuers in my CDO, I can back out the probability of each issuer defaulting. (That’s a whole other lecture, but please take my word that if we have the price of default insurance, we can calculate the probability of default. Otherwise I’ll go on for another 2000 words…) This answers my first question: will defaults occur? Combine that with a correlation number and I can answer the second question: when will all the defaults occur? So now I can price the CDO, right? Unfortunately, no.

The default probabilities backed out of the CDS data are conditional default probabilities, meaning they have the market’s 4,950 correlation factors baked into them. Company A may be doing fine, but it’s very correlated to company B which is not so healthy. The result is that company A’s CDS will exhibit a relatively high default probability even though that’s more B’s fault than A’s.

In statistics, we like to deal with independent or unconditional probabilities, because the math becomes dramatically easier. So the conditional probabilities extracted from the CDS are not so useful, and must be transformed into independent probabilities. To achieve this goal, we do something that I think is very clever:

We set up a model in which defaults are driven by a shared “market factor” and an idiosyncratic factor, similar to a regression with one dependent variable and an error term, hence the name “single factor model.” Now, I know I just said there are two factors, but one is specific to each individual issuer, so it doesn’t count as one of the model factors — if this troubles you, chalk it up to statistical nuance. Anyway, the two drivers are weighted by a correlation term; as correlation increases the market factor dominates, and as it decreases the idiosyncratic factor dominates.

Now, suppose for a moment we knew the value of the [random] market factor. In this case, default would be driven solely by the idiosyncratic factor (since the market factor is fixed, and we have chosen it such that all names either are – or are not – in default). The idiosyncratic factor is, by definition, independent across all issuers. Therefore, we have artificially created a scenario in which defaults are independent for each issuer by conditioning the market factor on a certain level. More specifically, we have generated a set of conditionally-independent default probabilities. Now, repeat the process for every issuer and every market factor level. The result is a complete picture of how every issuer behaves in every possible situation. From this, the unconditionally independent probabilities can be extracted.

(If that isn’t quite clear, suffice to say there’s a bit of math behind it. Interestingly, the math is surprisingly simple, but with the exception of the number of factors in a Gaussian model I have promised not to write out any equations in this post, so in the absence of symbols I hope you will accept my reasoning.)

So now, we have the probability of every issuer independently defaulting at any given time – with that information, it is relatively straightforward to figure out the expected loss on the portfolio. In fact, it’s mainly arithmetic at this point: the value of the portfolio is just the probability-weighted average payoff of all the issuers.

And that’s really it – that’s how the Gaussian copula is used to price a CDO, or a collection of sold CDS on many issuers. We calculate the default probabilities from the CDS, then we use the Gaussian copula to tell us how they relate to each other. You’ll notice that I never actually mentioned the copula when discussing the probability model – that’s because you don’t really need it. It happens that the copula math simplifies nicely into something that is almost, but not quite, entirely unlike a copula (hey! a Douglas Adams reference!). However, the copula-based approach is more informative, even if copula-specific math per se doesn’t enter the picture.

And why is this so bad?

A few of the modeling decisions I’ve described above are unquestionably poor ones, though it may not be obvious how to improve them. Here is my brief rundown:

The Gaussian dependence structure – what’s wrong with it? What alternatives are there? Why are they better?
The single factor – is it really sufficient to describe the behavior?
The single correlation number – is it sufficient to describe the behavior? Can we reliably estimate more relationships? Is correlation the right metric in the first place?

I’ll attempt to answer all these and more in part III…

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More reverse convertibles: cutting the nose to spite the face?

June 18, 2009 in Finance

Felix is back at the forefront of the “ban reverse convertibles” charge. He makes some salient points, but continues to encourage a slippery slope form of regulation that would ultimately handicap an industry to protect the naive daytrader.

Referring to embedded short options in general, he notes:

But retail-facing financial instruments should never embed such bets, because retail investors, as a rule, lack the sophistication necessary to be making such bets in the first place.

I suppose then that we should ban nearly every corporate bond (which contain embedded short calls at par), as well as the practice of “covered calls” (broker-jargon for a collateralized short put)? In fact, why not ban levered ETFs as well, since they are mathematically destined to wind up worthless? You can’t ban products just because someone isn’t capable of understanding it. Imagine if all industries bent to the will of the most naive user!

He continues:

Fernando looks at those investors and says it’s “their responsibility to be skeptical buyers”. No. It’s the stockbroker’s responsibility to act in the investor’s best interest, and it’s the government’s responsibility to prevent the sale of products which can end up being extremely damaging to those who buy them. As Elizabeth Warren famously says, no one has a problem with the government banning the sale of dangerous toasters, and dangerous financial products cause much more damage than dangerous toasters ever do.

Who is still foolish enough to think that their broker’s interest is in “looking out for them” as opposed to “selling stuff to make a commission”? And as for toaster ovens, well, it’s very easy to describe a bad toaster oven – it spontaneously combusts, or in some way fails to do something other than its stated purpose of charring bread. How can we tell if a financial instrument is deviating from its stated purpose? What is its stated purpose in the first place?

Imagine getting a call from your broker, saying that, “The stock you bought has dropped too much, so we’re going to refund you everything you invested because this isn’t how stocks are supposed to work.” That would be great – unless you were on the other end: “You sold stock to us, but it’s gone down a lot and we’re going to need you to reimburse us, since we didn’t buy it with the intention of losing money.”

Felix concludes:

The fact is that reverse converts, in particular, are the kind of product which can cause a great deal of harm; what’s more, they exist entirely so that banks can use their stockbroking arms to rip off their clientele. Banning them would do much less harm than selling them. So let’s ban them, and their ilk.

I can’t think of any product that doesn’t have the potential to cause “a great deal of harm.” Stocks, bonds – just about anything except a Treasury note (and even there…) can hurt investors. And as for the “exist solely” part, what exactly does Felix think stockbrokers do? Transaction costs are famously egregious! Oh, that’s right, he thinks they look out for the interests of their clients and try to protect them.

Basically, Felix’s plan is to ban anything that could hurt a naive investor. I have a better idea: ban naive investors from products they don’t understand. This has worked rather well with CDS: only qualified institutional investors may trade the product. In fact, there are many examples of products which require institutional status to trade. There is no need to cut off the nose to spite the face by banning certain products rather than simply restricting access.

Besides, let’s be honest, if you ban one bad product another will just pop up to take its place. The world is full of double-knock-in snowball ratchet options, and there’s always another salesman waiting to pitch them.

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Reverse convertibles

June 17, 2009

Ever since the WSJ published this article on the front page of section C, a lot of people are talking about “reverse convertible notes.”
James Kwak and Felix Salmon led a charge to ban the instruments but Felix, at least, seems to have backed off a little bit after these responses.
I’ve seen many varieties of these notes [...]

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CNBC thinks Taleb is a bad trader

June 17, 2009

More on Taleb: CNBC is running a piece called “Swan Song: Why Nassim Taleb is Still Wrong.”
The crux of the argument seems to be this paragraph:
Arguing against Taleb is a little embarrassing; who among us wants to side with the plodders when for the price of a paperback you can join the elect? But the [...]

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Deconstructing the Gaussian copula, part I

June 5, 2009

A number of misconceptions about the Gaussian copula are addressed.

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The Inviolate Principle…?

May 29, 2009

Those naive financial journalists at The Atlantic are back! Andrew Gelman pointed me toward this misguided look at the latest auto bankruptcy (you know the one I mean). Key quote:
Purists — and virtually every academic economist one happens to encounter — wonder what happened to the once inviolate principle of rewarding risk-takers.

You’ll have to excuse [...]

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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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Something doesn’t add up at the PEC

May 1, 2009

AK points out a story from our favorite financial journalist, Jeff Goldberg, regarding the Palestinian Electric Company’s surprising 2008 profit of $6mm. What Goldberg fails to note (citing instead a quote on nebulous “corruption”) is that the PEC is paid an unconditional annual fee of $29mm by the Palestinian Authority, meaning they actually hemorrhaged $23mm. [...]

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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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FRN’s & negative duration

April 16, 2009

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

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It’s been tried

March 17, 2009

Today’s Dilbert is excellent:

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Diversification

March 11, 2009

On a comics roll:

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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 [...]