The Dark Magic of Structured Finance

In Too Big To Save Robert Pozen gives a clever example, based on an excellent paper by Coval, Jurek and Stafford, which explains both the lure of structured finance and why the model exploded so quickly.

Suppose we have 100 mortgages that pay $1 or $0.  The probability of default is 0.05 (assume independence).  We pool the mortgages and then prioritize them into tranches such that tranche 1 pays out $1 if no mortgage defaults and $0 otherwise, tranche 2 pays out $1 if 1 or fewer mortgages defaults, $0 otherwise.  Tranche 10 then pays out $1 if 9 or fewer mortgages default and $0 otherwise.  Tranche 10 has a probability of defaulting of 2.82 percent.  A fortiori tranches 11 and higher all have lower probabilities of defaulting.  Thus, we have transformed 100 securities each with a default of 5% into 9 with probabilities of default greater than 5% and 91 with probabilities of default less than 5%.

Now let's try this trick again.  Suppose we take 100 of these type-10 tranches and suppose we now pool and prioritize these into tranches creating 100 new securities.  Now tranche 10 of what is in effect a CDO will have a probability of default of just 0.05 percent, i.e. p=.000543895 to be exact.  We have now created some "super safe," securities which can be very profitable if there are a lot of investors demanding triple AAA.

To review we have assumed that the underlying mortgages each have a probability
of default of p=.05 and by pooling and prioritizing we have created a tranche with a probability of
default of just p=.0282 and a CDO with a probability of default of
p=.0005.  In this way, structured finance was able to create many triple AAA securities from a pool of securities none of which were triple AAA.  This point is widely understood.  Now here is a much less well understood consequence.

Suppose that we misspecified the underlying probability of mortgage default and we later discover the true probability is not .05 but .06.  In terms of our original mortgages the true default rate is 20 percent higher than we thought–not good but not deadly either.  However, with this small error, the probability of default in the 10 tranche jumps from p=.0282 to p=.0775, a 175% increase.  Moreover, the probability of default of the CDO jumps from p=.0005 to p=.247, a 45,000% increase! 

The dark magic of structured finance conjured many low-risk securities
out of many risky securities.  Like all dark magic, however, the
conjuring came at a price because if you didn't get the spell exactly
correct it was easy to create something much more risky and dangerous
than you were likely to have ever imagined.

Here is an excel file, StructuredFinanceMath, with the calculations.

Addendum: Adding in correlation among mortgage defaults makes the math more difficult but doesn't change the bottom line that I wanted to illustrate which is that small changes in the underling default risk (or correlation) are highly amplified in the tranches and CDOs.

Comments

Excellent explanataion!

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"the probability of default of the CDO jumps from p=.0005 to p=24.7"

There is a typo at that second probability.

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Good catch, Marcos!

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Good catch, Marcos!

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It is the essence of risk management to recognize the probabilities used for calculations are nearly unknowable. This is why Fischer Black is a crank first and a genius second. He knew, but chose to ignore that small changes in probability and correlation can mean vastly different outcomes. He used the idea that everything goes to equilibrium to drive real world dynamics and human error out of finance - a crime of the highest order.

A huge majority of our financial world and economics world doesn't recognize this idea and refuse to understand it when they do recognize it. We've optimized an economic system for speed, but it is not robust.

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I think garbage in, garbage out is a better description than 'black magic'.

Assume all securities in a pool have the same rate of default? That (as pointed out above) that defaults are independent? Take the leftover junk from one pool and use them to create another pool with supposedly AAA tranches? Don't conduct due diligence into the securities going into these pools? Ignore news reports of fraud in mortgage lending and discussions of real estate bubbles? Use a corporate debt risk model for residential mortgages?

All idiotic moves on the part of those putting money down on the long side.

It isn't so much that structured finance is the underlying problem or cause of the crash, it is that the dummies hid behind structured finance.

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But none of this securitization adds or subtracts any total risk from the economy, it just reapportions it. Total default losses are the same as before securitization.

So you still haven't explained why securitization is supposedly so harmful to the economy overall.

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AAA targets a 1 in 10k chance of default, the classic 'act of God'. People who model these things should anticipate temporal volatility in the mean default rate of the underlying collateral that is lognormally distributed. They didn't.

A funny thing about structured finance is that when structuring deals where objective data is not available, people make multiple conservative adjustments in safety margins because they know there assumptions have standard errors, which is why AAA securities for corporates actually have very low default rates. When the data on the underlying are available, however, people don't add the safety margins, or not as many (eg, make the AAA robust to 3 times the default assumption).

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So you still haven't explained why securitization is supposedly so harmful to the economy overall.

Because it's fraud. Securitization is a way of convincing people to accept risk by telling them they are not accepting risk. If the bullshit is convincing enough, they will buy.

Worse, if the buyers are fiduciaries, they can buy for the return, and even if they actually *are* aware of the risk, pretend they're not. Then they look like financial geniuses if the bomb doesn't go off and their ass is covered if it does -- nobody could have predicted, etc. Unless they're as dumb as Abramoff and leave smoking guns lying around their office, this is practically impossible to prove even after the damage has been done.

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It's a nice mathematical demonstration. However, it serves to bolster the pretense that what happened here was that something quite complicated went horribly wrong, when it's more accurate to describe it as a very simple control fraud. 'Mistakes were made' is in this case a de facto apologia for what was actually a massive crime.

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"The probability of default is 0.05."
Isn't that 0.05% or 0.0005?
This is a math example; please proof-read.

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It seems like everyone focuses on the senior tranches. What about the lower tranches? By separating the risk into layers, you are putting most of the risk into the bottom tranches. What happens to those?

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I'm an economist, not a Wall Street type, but surely Wall Street knows about Samuelson's famous "Fallacy of Large Numbers"? Insurance does NOT work by pooling risk, but rather by spreading risk. The fact that pooled securities decreases the mean chance of default but increases the variance so much that risk averse individuals are worse off is well known. The situation here isn't entirely analogous, but it's not that different....

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The Angry Bear takes umbrage at this post.

http://www.angrybearblog.com/2010/05/find-mistake.html

Dueling economists. I don't even know whose side to take, let alone determine the efficiency of firing at 10 paces...

I want emphasize that I do not understand the mathematical models. So I would not invest.

What is important is that the investors understand the models. That is, they must understand the risk and be able to do the probabilities. Otherwise, it is a sham and the entire thing falls apart when there is a default (but the issuer makes lots of money).

IMHO the whole problem with CDOs was that the investors and ratings agencies did not properly assess the risks (had they done so, they would have paid less for the products).

I am not soothed by this posting.

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What I'd like to know is whether there is any merit in structured products from a finance theory perspective. I'm not an expert, but it would seem that the effects of "tranching" could be replicated with far more flexibility by purchasing call options on a diversified market fund. An option which is far in the money will be "safe", one which is far out of the money will be "unsafe", and both will be exposed to market risk.

Am I missing something? Does "tranching" have any real advantages compared to derivative trading? Are the transaction costs lower?

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It's not "securitization," it's "secretization." Nobody knows what anything's worth because it's a secret.

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Is it possible they really didn't do ANY sensitivity analysis?!!? I don't buy it. They MUST have had a very solid estimate of p, probably analyzing millions of mortgages over time. Their confidence intervals were surely small enough to make it sound.

The REAL killer is that those probabilities are surely correlated--and the law of large numbers FAILS for correlated random variables!!

Of course, they don't seem correlated during normal market activity.

If I were to model it, I would have some underlying markov chain driving the states of correlation between the foreclosures.

In that event, you wouldn't even need a mis-estimate of the probabilities--the correlation is enough to mess it all up. And it looks uncorrelated if you look at historical data. There have been few real estate crashes historically.

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Tell us how the credit snobs figure in to the dark magic, Alex.

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Eric, One of the primary drivers of defaults in the models was the owners' equity in the underlying properties. The natural result is that if home prices fall on loans with high LTV, defaults spike across the board. If you use these default models to test the CDO using a stochastic process for home prices, you would get high concentrated defaults in the tail of the distribution where home prices are falling. You could see the same type of result with simple stress tests where home prices stagnate or fall in the first few years after the loans were made.

See the quoted exchange between a Fitch representative and a customer on page 23 of the Coval, Jurek and Stafford paper that Alex linked. Sensitivity testing was done; it just didn't extend beyond "normal" annual price appreciation below the mid-single digits. The ratings agencies knew that their ratings were not robust to downturns in home prices, as did any purchasers who performed a reasonable degree of due diligence.

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Alex, response to Waldman?

Also, even if the risk amplification is as high as your numbers suggest, doesn't it really come down to what bell curve they use to estimate the underlying risk?

Good reuters/NYT piece on this,

http://www.nytimes.com/2010/05/11/business/11views.html

but it didn't make clear what distribution curves were and are used by different parties.

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For this who care, Waldman flipped out that I didn't state that I was assuming independence. Frankly, I thought that was obvious. I have added an addendum, however.

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>doesn't change the bottom line

But is Waldman correct that adding correlation greatly reduces the risk amplification?

If yes, it strikes me that it does change the bottom line -- changing the amplitude off the effect (significantly?) but obviously not alleviating it.

This is far above my pay grade. Which is why I'm asking.

(He really does off the handle, though; seems out of line to me.)

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Here's a wacky idea: instead of trying to solve the "tricky math" in closed form, why not run a Monte Carlo instead to investigate the outcomes as we vary the conditional probability of multiple concomitant defaults given any single mortgage defaulting, from zero (equivalent to full independence as assumed here) on upward to something more likely to be representative of reality?

Where is the knee in the curve? It would seem that at the other extreme (full dependence) we will have turned 100 mortgages with 0.05 failure probability into 100 tranches with ~1.0000 failure probability, and the "safest" tranches will have paid a huge premium for the illusion of safeness.

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