Bad Probability and Economic Disaster; or How Ignoring Bayes Theorem Caused the Mess

There is at least a little bit of interesting bath math
to learn from in the whole financial mess going on now. A couple
of commenters beat me to it, but I’ll go ahead and write about
it anyway.

One of the big questions that comes up again and again is: how did they get away with this? How could they find any way of
taking things that were worthless, and turn them into something that could be represented as safe?

The answer is that they cheated in the math.

The way that you assess risk for something like a mortgage bond is based on working out the probability of the underlying loans failing, and using that to compute the likelihood of the
entire bond package to end up losing.

The biggest problem is that the whole system of ratings and
insurance for mortgage (and other) bonds is based on probability
computations of how likely it is for the underlying loans to
default. The problem is in how they computed the probability of
default. They made the same mistake that we constantly see
creationists making in some of their stupid arguments: false
independence. They build up assessments of risk based on the the
assumption that for a given set of loans, the probabilities of
different loans failing are completely independent of one another.

Quick refresher on probability. Take two events – like
two loans defaulting. If the probability of the first
loan defaulting is p0, and the probability of the second
loan defaulting is p1. If the two events are independent, then the probability of both occuring – the probability of both loans defaulting – is p1×p2. But if they’re not
independent, that doesn’t work. Then the computation gets a
lot messier – because you need to work out the math describing
the relationship – which generally involves lots of analysis,
and lots and lots of use of Bayes’ theorem.

It’s easiest to describe how this works by using an example.
Suppose we’ve got a package of 100 mortgages, each of which
borrowed $100,000. So we’ve got 10 million dollars worth of
mortgages. Suppose, for simplicity, that the total interest earned
on the loans was going to be 150% – so at the end of the 30 year
term of the loan, the bonds were expected to pay $25 million.

Now, suppose that these were really lousy loans – they expected
that each loan had a 10% probability of defaulting, and that they’d lose the entire amount of the loan on a default.

By assuming that the probability of default for each of the loans is independent of the probability of default for any other loan, they can say that the probability of any particular loan defaulting is 10%. Assuming that defaults for loans are all independent, they can build an argument that probability predicts that only 10% of the loan will fail, and that it’s incredibly unlikely for the rate of failure to reach higher than 20%.
Based on that, they say that by using tranching to separate risk, they can claim that they’re being extra careful, and put 80% of
the mortgage bonds into a top tranch fund, which is supposed to be
super safe.

But the probability of defaults aren’t independent. Sure, there’s random failures, where someone gets sick and can’t work, and ends up defaulting on a loan. That kind of default generally
really is an independent event. But that’s not the story behind most defaults in low-quality loans. The garbage loans are almost always variable interest rate, and the most common cause of default is interest rate changes, which cause the loan payments
to become too large for the borrowers to pay. When that happens,
they’re not independent events. The same thing that causes one
loan to fail causes others to fail. Huge numbers fail at the same time, for the same cause. Depending on how the numbers work out,
you can get very different results for what’s likely to happen.
It’s not simple – there’s no one answer. And I don’t have the information that I would need to do the math. This isn’t back of
the envelope stuff.

But without doing anything complicated, I can say that they did it wrong. As I argued above, when you’ve got a huge collection of high-risk loans with variable interest rates, the pattern of default isn’t going to be independent. While the probability of any given loan defaulting, looked at individually, is only around 10%, the probability of 50% of the loans defaulting
isn’t 1 in 10something really big – because the same condition that causes one loan to fail is likely to cause many others to fail.

As a result of this, tranching didn’t work. They set up the tranches so that they partitioned things so that the top tranch was safe given the assumption of independence. But if independence
doesn’t hold – and it doesn’t in these – then even an extremely conservatively structured tranching package doesn’t guarantee
the safety of the top tier.

Plenty of people knew that this was wrong. But they were able to write up impressive looking risk assessments, full of pretty math showing how unlikely it was for anything bad to happen. The math was dreadful – but it looked good. And the fact that it looked good provided enough of an excuse for everyone to pretend that they believed it. (And for lots of people to actually really believe it; there’s no shortage of people who invested in this stuff without understanding it, who assumed that a “AAA” rating actually meant that someone had really checked to make sure that it was safe.)

0 thoughts on “Bad Probability and Economic Disaster; or How Ignoring Bayes Theorem Caused the Mess

  1. Aaron Bergman

    You really think that all the really smart people (I know some of them) working on Wall Street have never seen a fat tail or knew that correlations exist? LTCM was 10 years ago, and even that was more complicated than the simple fact of fat tails.

  2. llewelly

    You really think that all the really smart people (I know some of them) working on Wall Street have never seen a fat tail or knew that correlations exist?

    The alternative is that they deliberately set out to cause the current credit disaster. And that would explain the $2.5 billion bonus given to Lehman brothers execs right as Lehman brothers failed.

    But sometimes, even very smart people believe weird things.

  3. Anonymous

    Aaron Bergman remarked

    You really think that all the really smart people (I know some of them) working on Wall Street have never seen a fat tail or knew that correlations exist?

    I know some of them too, having traded financial derivatives for nearly 20 years (currently for a very large hedge fund). What the models still do not include is the fact that under stress, all the correlations go to 1.0. What seemed like a nice diversified portfolio, say under VAR assumptions, can in a matter of single digit days turn out to be a massively correlated aggregation of toxic waste. By the time a trailing estimator responds it’s too late. Everyone read The Black Swan, but my bet is that only Taleb and a few others acted on it.

  4. President George W. Bush

    What do “fat tails” have to do with independence?
    I know that “long tail” is a trendy term used in analysis of web sites. But if by “fat tails” you mean heavy-tailed probability distributions like the Laplace distribution, you can still have, say, 100 Laplace random variables all be independent of each other. This article has nothing to do with fat tails.

  5. T

    The really smart people (I know some of them) working on Wall Street

    Are evidently not quite so smart. If they were, they wouldn’t have caused this mess (assuming it was unintentional), or at least wouldn’t have been caught (in the, in my opinion, much more likely case that it was deliberate). The only solution to this mess is to let the companies crash and burn and spend some of the $700,000,000,000.00 on the investigation and incarceration of those responsible. Any other course of action (especially the current plan which been reported as potentially giving some of these companies a profit) will ensure that this happens again; say what you want about the markets, but they’re usually pretty good at taking free money.

  6. Dunc

    Are evidently not quite so smart. If they were, they wouldn’t have caused this mess (assuming it was unintentional), or at least wouldn’t have been caught (in the, in my opinion, much more likely case that it was deliberate).

    Hey, they’ve still got their multi-million dollar bonuses and their villas in the Hamptons. My pension plan’s screwed. Who’s smarter – them or me?
    The mistake was in assuming that a bunch of people motivated solely by personal greed were going to act in the common good, rather than shaft everybody else and run off with all the money. They’ve done just fine out of all this – none of them are going to jail, and none of them will be paying any of the money back. Seems pretty smart to me…

  7. Aaron Bergman

    Really smart people are smart enough to find new ways to screw up massively as opposed to simply rehashing the old ones.

  8. Ahcuah

    There is also the question as to whether the decision makers even listened to their risk evaluators who got it right. Many a time I’ve seen management totally ignore the mathematical analyses of their technical employees and just pick the answer they want.
    Seriously, if you follow the technical advice, and get a $10,000 bonus, or ignore that advice and get a $10,000,000 bonus, which will the less technically astute manager choose?

  9. Dunc

    It doesn’t matter if the manager is technically astute or not if he will not personally suffer any negative consequences for making the wrong decision, or if those personal consequences do not outweigh the personal benefits. I’d take the 10 mil even if it means losing my job six months down the line (with generous golden parachute, obviously) – who needs a job when you’re rich?

  10. Jarrett

    I just needed to +1 Ahcuah’s comment. I too have seen managers ignore the math and picked the option they wanted without concern for risk or future events unrelated to them. What happens when managers ignore these things? Do they get reported up the chain? No. If my bosses don’t tell their bosses, then my analysis is dead.

  11. Mark C. Chu-Carroll

    Re #9, #10:
    That’s exactly my point about regulation.
    Free markets are great at some things. But they’re not a panacea. There are numerous situations where a pure free market approach leads to things like the current situation.
    Looked at from a viewpoint of pure self-interest – which is the basis of free-market approaches – cheating can be highly beneficial.
    Telling the truth about the risk of a set of loans would result in the business and the individual broker making a lot less money. Covering up information about the risk, or just looking the other way when someone else covered it up, results in getting immensely rich. There’s no downside to going along with the scam. The people who got rich on this aren’t going to lose any of those ill-gotten gains. They’re not going to jail. They’re not going to be fined. There’s absolutely no penalty for them. Looked at from a free-market perspective, they acted perfectly rationally.
    That’s why regulation is so important. You need rules to create the downside for these kinds of situations. If you take the free market assumption that people are rational actors who make decisions in their own self interest, then you need to make sure that there’s at least some counterbalance to the rewards of cheating.

  12. Dunc

    Yep, absolutely – but it’s not enough if regulation only operates at the institutional level, as it currently tends to. If we were to start putting CEOs in prison for, say, 25 years at a time (or even just fining them into penury), you’d soon see a change in behaviour. Aren’t the GOP supposed to be the party of personal responsibility? 😉
    Then, of course, there is also the problem that many of the fines currently levied for violating regulations are smaller than the profits to be gained – that makes it just another cost of doing business.

  13. MPL

    The other thing to remember, is that when there’s a bubble going on, the “right thing” usually loses money compared to a few, trendy “wrong things”. The problem is, as the bubble goes on, and the people doing the wrong thing keep winning, they get bigger…and bigger…and bigger. Pretty soon the people who are actually doing the right thing look like losers, and have a smaller portion of the market.
    So even if everyone is sincere, the bubble will keep amplifying the strength of the people doing something that is fundamentally very foolish, until the bubble finally bursts, and all that money has to vanish.

  14. KeithB

    What is “bath math”? Is that one of those problems from algebra with faucets and drains?
    Would it be fair to say that the purpose of regulation is to protect us from the “Problem of the Commons”?

  15. Matt Springer

    “They’re not going to jail. They’re not going to be fined. There’s absolutely no penalty for them. Looked at from a free-market perspective, they acted perfectly rationally.”
    In the absense of bailouts, they’d lose their banks. That might not be something an exec with a golden parachute would care about, but the board of directors and stockholders certainly have an incentive to not let this mess happen.
    I think one thing they should try that might help would be strictly performance-based CEO pay, and the abolition of golden parachutes. The bank collapses under your leadership, you forfeit 90% (or something) of your salary back to the bank. I don’t know if it would be practical, but I’d like to see it tried.

  16. natural cynic

    The rational free market would have the shareholders getting together and kidnapping the ex-CEO to hold him for a ransom equal to the golden parachute + the last two years bonuses + half of the salary from the last two years.[/snark]

  17. Tulse

    In the absense of bailouts, they’d lose their banks. That might not be something an exec with a golden parachute would care about, but the board of directors and stockholders certainly have an incentive to not let this mess happen.

    Which is why all the whining about how exec compensation limits would prevent companies from participating in the bailout is pure BS. There is no way shareholders wouldn’t force a CEO to take a paycut if it meant their shares didn’t lose all their value.

  18. Uncle Al

    Uncle Al embraces Bush the Lesser and his Gang of Thieves: “If stupidity got us into this mess, then why can’t it get us out?”
    Grant Secretary of the Treasury Henry Paulson unlimited and unregulated power to donate $6 trillion to his friends. The man has character.
    “Decisions by the Secretary [of the Treasury] pursuant to the authority of this Act are non-reviewable and committed to agency discretion, and may not be reviewed by any court of law or any administrative agency.”

  19. Jon L

    If raising interest rates is what causes most loans to fail, wouldn’t the government imposing strict interest caps (and perhaps minimum payments) immediately and retroactively on the effected loans solve the problem (or at least make it much more manageable).As long as the bond owners got the full amount its better than the debtors declaring bankruptcy and the debtors get to keep there collateral. That way, the only people are are actually going to loose money are the brokers who bet on a corrupt system.
    Well, that’s if I understand this right which I might not.

  20. travc

    Great posts on the financial mess… Most economists understand economics less than you do, so don’t worry about not being a supposed expert.
    As a complex adaptive systems fan (and researcher), there are a few heuristic principles the econ/financial peoples could learn from my discipline (alas, a fuzzy discipline because it is by definition complex.)
    1) Emergent behaviour happens, and it normally isn’t what you expect
    2) You get exactly what you select for… so make sure you are selecting for the right high level goal
    3) Exploits and pathological cases will occur and have to be detected and dealt with on a case-by-case basis (you simply cannot predict everything.)
    4) Allowed interactions need to be kept as simple as possible if you want any sort of predictability and stability.
    It is pretty simple how this applies to finance. For example, “innovative devices” should be allowed only in a very conservative manner (small scale for a long time until proven), or just banned if they don’t serve a clear purpose. Regulations can of course have specific rules, but they must also include general rules/principles (like don’t lie) which can be applied in odd cases. If they aren’t those odd cases become the norm really really fast.
    BTW: I think Wall Street should adopt a new symbol, one more apt than the bulls and bears: the invisible pink unicorn. A huge chunk of the current financial sector is based on imaginary money. I’m actually leaning towards the “let it crash”. The real economy will suffer some collateral damage, but will be better off without all the fictional financial crap sucking resources out of it in the mid/long term.

  21. Jeff Guy

    Thank you for your extraordinarily clear explanation of what the polititians in charge are failing to explain. Of course the media too is failing to report this since they assume all of us are too stupid to understand it. I’ve heard a few talking heads (and Democrats) talking about how deregulation is responsible, but without your clear explanation it isn’t obvious just how true that really is, and how culpable the pro-deregulation advocates (like Bush and McCain and Greenspan) have been.
    In addition to the non-independent risk associated with a rise in interest rates you discuss, I think there is another non-independent risk that was not considered: namely that house prices were being inflated by an unrealistic bubble of purchases. House prices were being bid up unrealistically by the easy bad-loan money that was readily available. We’ve all seen the stories of homeowners in Miami and elsewhere buying multiple properties as “investments” – that is clearly a bad sign that a bubble is occurring.
    Bubbles always deflate, and the deflation of a bubble is clearly not an independent risk. When this bubble deflated it left a lot of home values underwater. And a borrower with an underwater property who has no equity in the property (thanks to “no money down” terms of those same loans) is much more likely to walk away from a home than someone who has invested 10% or 20% of their own money and is living in the house. The availability of bad loans – those without proof of ability to repay and minimal money down – is another case of lack of regulation leading to the overvaluation of the equities built from these mortgages, and thus leading to the current mess.

  22. Marc

    I am surprised that you think for one second that the people involved in CDO pricing ignored correlation. This is plain wrong. In fact the prices of CDO tranches is viewed as an opinion on correlation (especial the more senior tranches). Your contempt of the people involved in this huge mess seems to cloud your judgment.
    One of the mistake people made is assuming a “nice” correlation dynamic, if they didn’t simply assume it constant. In reality, as noted in one of the comments, correlation as a strong tendency to jump very high in time of crisis.
    An other mistake some (but not all) people did is assuming that recovery and default where uncorrelated. When someone defaults on a loan, the bank sells the house to recoup it’s loses. After expenses, and depending on the housing market condition, it expects to be able to recover at most e.g. %80 of the house value. This %80 is the recovery factor. It is negatively correlated with the number of defaults: when many people default, may houses are on sale so their price goes down. But this was not taken into account by some/many people pricing CDO tranches.

  23. jayh

    “That’s why regulation is so important. You need rules to create the downside for these kinds of situations. ”
    The problem is that regulations have no intelligence. Regulations consist of relatively capricious rules by people who have no particular insight into the future. Regulations tight enough to prevent most ‘bad things’ are also restrictive enough to prevent constructive flexibility.
    It is ironic that the prez and Congress, the people who brought us trillion dollar deficits and the Iraq war, collapsed bridges and sparkling Katrina response are somehow going to have the wisdom to regulate this problem out of existence.

  24. eddie

    Nobody’s going to jail. They’re not going to pay any or the money back. The ‘bailout’ is not subject to review.
    Isn’t this what you guys have your 2nd ammendment for?

  25. jim

    It might help if the government would restructure those variable loans. Perhaps put the interest rate back down (have it fluctuate with a slower surprise max of 75% of the cpi per year in the upward direction) and demand a percentage of the gross sale price when the owner sells the home.(whenever that might be)
    At least the home owner could continue to live in their home – thus having less of a disruptive locally economic affect. Forclosure means that the home would sell at a signifigant discount and thus force the price of homes down in that area. This could have a cascading effect.

  26. Seamus

    Great site. Intelligent analysis; gotta love it.
    If CEOs know they can act w/impunity, and walk w/a parachute that exceeds the amounts of any fines, they will. Any arguments to the contrary are simply pimping the free-market in the abscence of data. See Angelo Mozilo and his fraudulent stock-sales at Countrywide.
    Clearly, boards of directors reward CEOs, and act in their own self-interest just as much as anyone else; not the good of the company, the employees, or the economy at large.
    Jayh: regulations can always be updated and amended. How is a lack of regulation any more “intelligent,” than regulation? We’ve let the CDS market balloon to $43 trillion, much of which is speculative, and was never intended to make any positive contribution to our economy.
    Maybe America should go back to actually making products, instead of just pushing paper around and paying folks to provide services to the paper-pushers. Just a thought.

  27. Lord

    The worst of this would have been avoided by verifying incomes, qualifying borrowers on fully adjusted market rates, and limiting leverage. This ties prices to incomes and prevents a substantial bubble from forming. The problem with this is most would not be able to borrow as much under these innovations, making them harder to sell and less profitable. The truth is the finance industry is not very profitable without these in the short run and not very profitable with them in the long run.

  28. mike

    Doesn’t anyone care that Mark is completely wrong on this?
    The probabilities of these defaults are indeed independent of one another. Having the same cause does not make the events dependent on one another. It would be as if saying all defaults due to illness are not independent because they have the same cause. One person defaulting on a variable-interest-rate mortgage does not affect the probability of a different person defaulting on their variable-interest-rate mortgage (both might be high for the same reason, but independent nonetheless).
    All of the focus on middlemen and ‘top executives’ only distracts from the real causes, namely that credit was given to undeserving people and housing prices fell. Keep in mind that none of these problems would exist if borrows pay their mortgages and if the collateral behind the loans was still worth the value of the mortgages.
    Also, the government is partially to blame for the widespread purchase of risky mortgages. The United States Congress mandated that starting in 1992 Freddie Mac and Fannie Mae increase their purchases of mortgages for low-income and medium-income borrowers. If Fannie and Freddie were not pressured to buy the risky mortgages, then lenders would not have been so inclined to issue loans (that is according to the ‘oh-so-imperfect freemarket’ rules).

  29. Mark C. Chu-Carroll

    That’s rubbish.
    In terms of probability, independence doesn’t imply causality – just relation.
    If you’ve got two events, A and B; and you’ve got an event C, where the occurrence of C
    increases the probability of both A and B, then A and B are not independent.
    The probability of defaults on variable-rate loans is related to the interest rate. If the interest rate increases, the probability of default on all of the loans increases. They aren’t independent.
    Further, there’s a lot more to the loan situation than “people took loans they couldn’t afford”. Mortgage brokers were pressuring people into variable rate loans. In fact, in many cases, mortgage brokers refused to approve people for anything but variable-rate loans, even if they qualified for them.
    And how did all of those undeserving people get loans? Who was out there offering them loans? Who was deliberately approaching people and offering them loans?
    Wasn’t it Alan Greenspan, just a couple of years ago, testifying that more people should take variable-rate loans rather than conventional fixed-rate mortgages?
    There’s a mind-boggling amount of stupidity here. I don’t mean to minimize the role of individual responsibility in this. There are a lot of people out there who knowingly took loans that they couldn’t afford if anything went wrong. There are also a lot of people who are innumerate, and aren’t capable of working out what they can afford for themselves, and so trusted whatever the mortgage agent told them. There are also a lot of people who just got screwed by mortgage agents.
    But there’s clearly two sides to it – the people who took loans, and the people who gave loans. You’re arguing that the entire blame belongs on the people who took loans – but they couldn’t have taken loans that no one wanted to give them. The banks and investment firms wouldn’t be in trouble now unless they made a decision to deliberately give loans that couldn’t realistically be paid back, and then
    lied about the nature of those loans to make it possible to sell them as securities.

  30. mike

    Two events are dependent if and only if the occurrence of one event affects the probability of the occurrence of the other.
    If one person defaults on their variable-interest-rate mortgage then the probability of another person defaulting is unchanged, even though the circumstances are related. Hence the two events are independent.
    If the probability of a person defaulting on their variable-interest-rate mortgage is wrong because the changing interest rate was not accounted for (which is not likely), then that is not because the mortgages defaults are dependent on one another.
    Also, I did not mean to write that borrowers are solely to blame as I did mention that Freddie and Fannie were pressured to buy high-risk mortgages in turn creating incentive for lenders to push loans on people knowing they could sell them. The government, which you would like to see more involved, is partially to blame.

  31. Jonathan Birge

    I really don’t think the problem was assumed independence. I think it was simply bad assumptions on the risks, period, based on lack of understanding of how poorly the loan origination process was managing risk. The system got so disconnected and distributed that nobody really knew what was going on. Furthermore, they assumed that government would back Fannie and Freddie, and they did. In that sense, regulation was part of what got us in this, as is our fractional reserve system “managed” by the Fed, which has repeatedly inflated asset bubbles. All things caused by regulation and distortions of the free market.
    To say that this is dishonesty that will be fixed by more regulation is naive. Did this work out well for anybody involved? Hell no! Clearly, this wasn’t so much people trying to get away with anything as it was stupidity and hubris. Now, if the people whose money was actually at risk (e.g. Lehman, for one) didn’t know what the hell was going on, how well do you think some government beaurocrat with no skin in the game are going to keep track of things?
    My guess is congress will fall over themselves creating new regulations, and will only make the problem worse. They will only succeed and destroying the dollar’s status as the world’s reserve currency.

  32. Mark C. Chu-Carroll

    I think you’re being naive.
    Did this work out well for anyone involved? Hell yes! Just look at the bonus statements for the brokers at Lehman for last year. Lots of people made absolutely mind-blowing amounts of money through these scams.
    The point of regulation, as I see it, is to get rid of the incentive for participating in this. The brokers made huge amounts of money doing stuff that they knew was wrong. But they also knew that there was very little downside in it for them. Sell a half-billion dollars worth of bad loans, and they make a couple of million in commissions. And when it goes bad, they get to keep those millions. They don’t go to jail, they don’t lose any money. At worst, they end up out of a job for a while. Even if that
    happens, they’re still filthy stinking rich – far more rich than they would be had they been honest.
    What regulation can do is produce rules that penalize people for doing that. Sell someone an investment based on faked information, and you’ll lose the money you made on it, or you’ll go to jail. Something, *anything* that creates a real downside, a
    penalty to be paid.

  33. ramona

    “Two events are dependent if and only if the occurrence of one event affects the probability of the occurrence of the other.
    If one person defaults on their variable-interest-rate mortgage then the probability of another person defaulting is unchanged, even though the circumstances are related. Hence the two events are independent.”
    Mike, there you go confusing ‘dependence’ and ‘causality’ again…

  34. MartinB

    “Two events are dependent if and only if the occurrence of one event affects the probability of the occurrence of the other.”
    Even under this definition, if both events have a common cause observing one will change the probability of observing the other. Oberserving me with a raincoat increases the probability of observing a wet street.

  35. BaldApe

    I vote for “they knew exactly what they were doing.” That’s why I oppose any bailout at all.
    It’s a classic example of moral hazard. They took big risks expecting to be bailed out.

    My guess is congress will fall over themselves creating new regulations, and will only make the problem worse.

    Yeah, that’s not too hard to figure out, is it? (please don’t take my sarcasm personally, it’s not meant that way)

  36. Anonymous

    I write that independence means that occurrences do not affect the probability of each other. Events causing each other is different. In fact, I mean that independence is completely different from causality. In order to show that two events are dependent one must show that the occurrence of one event affects the probability of the other event occurring (without necessarily showing they cause one another).
    Given it is raining, the probabilities of two people wearing rain coats is independent and depends only on how likely one is to prepare for the weather.
    If we do not know whether it will rain, then the probability of two people wearing rain coats depends on the probability of rain and the probability of a person to prepare for the weather (Bayes Theorem).

  37. MartinB

    Your examples are correct, but I think they do not pertain to the situation. Probably my fault because I was rushed and could not finish my post properly.
    I understand the situation as follows: You say that the probability of some credit busting is 10%. So the probability of two credits busting is 1%. But this is only if different credits busting never have a common cause.
    Take again the rain example: The probability of one person wearing a raincoat (not knowing anything about the weather) is, say 20%.
    So the probability of 2 people doing so is 4%, for 3 its 0.8% and so on.
    But of course that’s nonsense – if it rains, we’ll all wear raincoats. So the probability of everyone wearing a raincoat is at least equal to the probability of rain. In this sense different people wearing raincoats are not independent.
    The same with credits: If interest rates raise incredibly high, every credit goes bust. So the probability of every credit going bust cannot be smaller than the probability of interest rates going incredibly high (whatever number you want to put to it, it’s just for illustration). If you assume that credits busting are independent of each other, you could make the probability of all of them failing infinitely small by having infinitely many credits. But since they will all fail simultaneously if the rates get incredibly high, that’s a wrong assumption.
    Hope this was clearer now.

  38. Daryl McCullough

    Mike, to say that two events A and B are independent is simply to say that P(A&B) = P(A)*P(B). That is not the case with A = “Alice wears a raincoat tomorrow” and B = “Bob wears a raincoat tomorrow”.
    I think what you are saying is that if you know C = “It is going to rain tomorrow”, then the conditional probabilities of A and B may be independent. That is,
    P(A&B|C) = P(A|C)*P(B|C). But that’s not what it means for two events to be independent. They are independent if and only if P(A&B) = P(A)*P(B).

  39. mike

    Your writing is logical, persuasive and well thought-out, but I simply must explain this.
    First, I reject the idea that my definition of independence is anything but canonical.
    Let me try to be clear now.
    Raincoats: Assume that any given day has a certain probability of rainy weather and that everyone wears a rain coat if and only if it rains. The probability of two people both wearing a raincoat *on any given day* is equal to the product of the probability of rain in the first person’s day and the probability of rain in the second person’s day.
    Example: Person A walks outside on a day with a 50% chance of rain, and Person B walks outside on a day with a 70% chance of rain. Knowing that they will wear raincoats if and only if it rains, the probability of Person A and Person B both wearing raincoats is 50% multiplied by 70%.
    It is important that we allow the people to possibly exist in different days in order for them to qualify as separate events.
    Why? Well, consider the example of tossing a coin (analogous to our raincoats).
    A coin has a 50% probability of landing heads (probability of rain). There is also the probability of whether any time you look at the coin a face will star back at you (probability of raincoat). Considering one specific coin toss, either every viewing of the coin will have a face starring at you or no viewings will result in a face starring at you. But that does not mean separate coin toss events will all be heads or all be tails. Actually, if we consider the the toss landing on X, where X = 1 if heads and 0 if tails, then the probability of a face starring at you will be 100% likely to be X. If we say that a viewing can equally likely be heads or tails, then we must flip the coin with every viewing because we used the probability associated with the tossing.
    Now consider this counter example:
    Assume everyone is more likely to wear a rain coat when it is raining and more likely not to wear a rain coat when it is not raining.
    Person A is not wearing a raincoat. We are concerned with how likely this is. Not knowing what anyone else is wearing or the weather we may safely assume that the likelihood of this event is based on the probability of rain and the preparedness of Person A.
    But what I forgot to mention was that 10,000 people are walking right next to Person A and all wearing raincoats with nobody else not wearing a raincoat. Assuming that wearing raincoats is not independent in the sense that one raincoat increases the probability of another, we may conclude that Person A not wearing a raincoat is very unlikely.
    But I also forgot to mention that it is not raining. Thus because every person is more likely not to wear a raincoat when it is not raining, it is a likely event that Person A is not wearing a raincoat.
    Now one may accuse me of living in nuance and overlooking the probability of the group of 10,000 all wearing a raincoat on a dry day, but keep in mind that we are only concerned with the likelihood of Person A not wearing a raincoat and how the raincoats of the 10,000 affect that likelihood.
    Notice that each person wears a raincoat for more or less the exact same reason and cause. Thus I fiercely disagree with your statement, “But this is only if different credits busting never have a common cause.” Two events may indeed have the same cause but still be independent. With your example of credits bursting you assumed that event probabilities and occurrences affect each other if they have the same cause, namely high interest rates. MarkCC made the same mistake. Nobody has explained quite how one default on a variable-interest-rate mortgage can affect the probability of another through that connection.
    I feel like I am the only one not wearing a raincoat on a sunny day.

  40. Stephen Wells

    Mike, your definition of “independence” re. probabilities is wrong, and Mark’s is right. You’re talking about causality, which is different. Two mortgages can fail for the same reason (interest rates); they’re not statistically independent.

  41. MartinB

    I’ll try once again – please don’t be annoyed if I don’t come back to this discussion, I’m leaving on a journey.
    “Nobody has explained quite how one default on a variable-interest-rate mortgage can affect the probability of another through that connection.”
    Let’s make this as simple as possible. Let’s assume that we have 10000 people with exactly the same income etc. Each of them is financially sound unless the interest rates raise beyond a certain value, say 10%.
    If I observe one of them defaulting in their credit, these conditions make *sure* that everyone else will also default.
    The total probability of everyone defaulting the credit is thus equal to the probability of one of them defaulting and also equal to the probability of interest rates going beyond 10%.
    If the probability of interest rates going beyond 10% is, say 0.05, then you cannot say: The probability of all credits defaulting simultaneously is 0.05^10000. The probability of all credits defaulting is the same as that of one credits defaulting, 0.05, and observing one defaulting makes the probability of all of them defaulting equal to 1.
    If this is still not clear, then I hope someone else can make it clearer.

  42. mike

    “Mike, your definition of ‘independence’ re. probabilities is wrong, and Mark’s is right.”
    Ok, how so?
    By the way, “independence” in that quote referred to statistical independence. Stephen, if you and MarkCC mean another type of independence then I am simply ignorant.
    A,B Independent iff P(A^B)=P(A)*P(B) iff P(A|B)=P(A)
    This is the definition I use in my examples and counterexamples, and I did mention how independence is not about causality.

  43. mike

    I can be patient when it comes to math.
    I think what you are saying in your example is that all interest rates must rise or fall together and that there is a certain probability of that occurring, and if that occurs then everyone has their own probability of defaulting (in this case equal for simplicity, say x). Thus the overall probability of every loan defaulting is the probability of any rate increasing multiplied by x.
    Another way to write it:
    A – one default B – another default
    C – interest rates rise
    A,B not independent.
    But this is a strawman built on the assumption that all interest rates rise and fall together, an assumption with inherent dependency.
    If events A and B are caused by C and D respectively, where C and D are the same but independent events, then
    , which does not imply A and B are not independent.
    I think that what I mean is given two variable-interest-rate loans which are both bound to have rising interest rates let
    A – one default B – another default
    Therefore, of course, P(A&B)=P(A)*P(B)

  44. Stephen Wells

    Mike, your “strawman” is in fact what happened. Lots of rates depended on the US base rate. That spiked. Lots of defaults. Not a strawman.

  45. mike

    That sounds nice, but it’s an oversimplification.
    Mortgages are different with some having adjustable interest rates and some not.
    Adjustable rate mortgages are different with different limits on interest rate changes each occurring at different times.
    Interest rate changes are different in how much they are subject to change.
    Interest rate changes are different in how much they depend on their respective indices.
    Indices differ in what rates they incorporate.
    Different banks use different indices. The Cost of Funds Index affecting an interest rate on a loan might even depend only on the costs of a single bank.
    (Not to mention different people’s ability to refinance!)
    If you suggest that a Bayesian calculation should be used because some loans commonly have adjustable rates subject to change and some of their banks base a part of loan interest rate changes on indices which include an index that includes the Federal Reserve prime rate, thus commonly affecting to some extent multiple borrowers’ ability to repay the loans, then I suggest we also incorporate a Bayesian calculation to determine the effect of changing prime rates on borrowers’ ability to repay loans other than through mortgage interest rates, the price of bread too while we are at it (horoscopes?).
    Also, that was not the argument I originally commented on or that MarkCC used. The original blog post only noted that variable-interest-rate mortgages have changing rates and posited that this relationship meant that such loans are not independent.
    “But the probability of defaults aren’t independent,” wrote Mark. “…The garbage loans are almost always variable interest rate, and the most common cause of default is interest rate changes, which cause the loan payments to become too large for the borrowers to pay. When that happens, they’re not independent events…”

  46. Alex

    There are actually a few minor errors in your post.
    First and foremost, the people pricing mortgage-backed securities never assumed independence of defaults. They assumed certain correlation structures (typically based on the Gaussian copula) which, nevertheless, turned out the be quite incorrect.
    Second, there’s a much simpler way that a lot of the rating agencies and banks cheated. For any security based on a pool of mortgages, you can essentially decompose its value as being driven by two factors: default rates and house price appreciation (HPA). The default rates were very wrong, but the worst issues were on the HPA side.
    Agencies like S&P and Moody’s only used data since 2000-2001 to calculate their HPA projections when rating mortgage-backed securities. Hence, those AAA ratings were being given out based on the assumption that home prices would keep growing from the next 20-30 years as they had during the largest housing boom in US history.
    I learned all about these methods while working in the fixed income division of a large investment bank last summer; my jaw pretty much dropped to the floor at the time. I’m out of there now, but I’m still amazed how shortsighted the industry was.

  47. Daryl McCullough

    I think what you seem to be saying is that if we hold all causal influence fixed, then the probability of two people wearing a raincoat or defaulting on their loans is independent. That’s true. That’s saying that the conditional probabilities are independent.
    Let’s go through the example once again. Suppose that we know that
    (1) If it is raining, then both Alice and Bob will wear a raincoat. (2) If it is not raining, then neither will wear a raincoat. (3) On any given day, there is a 50% chance that it will rain.
    If I tell you that I saw Alice a few weeks ago, then what is the probability that she was wearing a raincoat? The answer is 50% If I tell you that I saw Bob on another occasion, then what is the probability that he was wearing a raincoat? The answer is 50%. Now, if I tell you that there was yet another occasion that I saw Bob and Alice together, then what was the probability that they were both wearing raincoats? The answer is 50%.
    So the events A and B are not independent! P(A&B) is not equal to P(A)*P(B).
    Your counterargument is that if you knew whether it was raining, then you would either have P(A) = 0, P(B) = 0, P(A&B) = 0, or you have P(A) = 1, P(B) = 1, P(A&B) = 1. That’s true. But we’re talking about the case in which you don’t know the truth of whether it is raining or not. If you don’t know the complete situation about the causal factors, then the subjective probabilities are dependent.

  48. mike

    Right. Absolutely. You are clear and I really agree.
    Except this is not an accurate paraphrasing:
    “Your counterargument is that if you knew whether it was raining, then you would either have P(A) = 0, P(B) = 0, P(A&B) = 0, or you have P(A) = 1, P(B) = 1, P(A&B) = 1.”
    What I meant was that even if we don’t know the weather, the probability on a specific day of two people both wearing coats equals X, where X is the probability of rain on that day. The difference may be trivial here, but I don’t want to be misunderstood.
    The problem is that dependency is inherent in the assumptions and comes only from the assumptions. If one cannot show that P(A|B)=/=P(A) without assuming it then the argument boils down to the jejune fact, “nothing is independent.”
    That is not good enough to use to cast blame on people for the economic downturn, especially when other factors would be more at fault, like housing prices, government and deadbeats.

  49. Stephen Wells

    Mike, if the situation is such that A and B are not independent, that’s not an assumption, it’s a description. You can hardly deny that any given mortgage provider handed out a lot of mortgages which depended on the same rate- that provider’s variable rate; the performance of thsoe mortgages are of course not statistically independent. What is your complaint?

  50. Nancy Lebovitz

    House prices aren’t entirely independent. Foreclosures in a neighborhood can drive down the prices of nearby houses, even if their mortgage payments were current.
    If a substantial proportion of people have mortgages which they can only sustain when house prices are going up, then I don’t see how the values of the mortgages can be independent of each other.

  51. anontrol

    Not to put too fine a point on this, but subprime and the resulting spread to everything else has little to do with math.
    The closest that you can come to a generalization is that the people assembling the CDOs were driving with the rear view mirror. Getting the full picture is difficult, but this piece:
    does a good job of taking a very representative sample of what was actually happening. The folks analyzing the risk were using old data, bad assumptions and poor due diligence. Throw in a dash of bonus incentives, zero exposure to “moral hazard” and no particular reason to worry about what might happen.
    I don’t know what they teach in high finance courses that business students headed to Wall St. take, but there had to be a little cult of personality built up around traunched CDOs as a mechanism to reduce risk. That mathematical mechanism failed… not because of math, but because they were plugging fictional numbers into it. The numbers map the math to reality.

  52. Doug Spoonwood

    Mike you said:
    “Two events are dependent if and only if the occurrence of one event affects the probability of the occurrence of the other.”
    No. Look, independence of two events a and b means that p(a)*p(b)=p(a and b) as stated I don’t know how many times already. Dependence means NOT independent. In other words p(a)*p(b) does not equal p(a and b). This says nothing about probabilities changing. Additionally, say one event ‘a’ occurs and this ensures that the other event b occurs. In such a case we have p(a)=1 and p(b)=1, since one event actually occuring means it has probability 1, and since the other event b has gotten ensured to occur, it also has probability 1. p(a)*p(b)=1*1=1, so p(a) and p(b) come out as independent. Consequently, they don’t qualify as dependent even though one event has affected the probability of the occurance of another.

  53. D Becker

    I don’t agree that this is driven entirely by statistics, but even there the causality/correlation arguments above miss the mark.
    Mortgage defaults are correlated, and not just by the common cause of interest rate increases. Imagine the case of a new far-suburban starter-house community. Everyone in a neighborhood is just barely making their payments, and the perceived value of their houses is about the outstanding mortgages. Their best course of action is to not default. Imagine if there is one default, and the house is sold at foreclosure at 75% of it’s “perfect” value. (This is typical — people skip upkeep when things get tight, have no motivation to create curb appeal and strip a house when evicted.) Now all of the houses are worth a little less, and some of the owners will be underwater far enough to be motivated to default. At some point this cascades, putting the whole community/region/country into a different operational condition, even when little has changed but the perception of the situation.
    Now move away from the strictly financial decision, to the human element. Before, no one had been foreclosed upon. Defaulting and losing your home seemed socially unthinkable. But now, “everyone” is doing it. The stigma is gone. We can flip back to the financial view: ‘the barrier to rational economic decisions has been removed’. But it really is the human “everyone is doing it, it must be OK (or maybe even the right thing).”
    The bottom line is unrelated to both: the money has already been extracted from the market by people selling marginal (zero down, no doc) mortgages, and firms packaging those up into “securities”. That money has been pocketed, moved away, can’t be recovered. The people responsible would sell another hundred zero-down mortgages and bundle them into a security today if they could find a buyer. And they would pocket the 12% and still couldn’t be touched.

  54. Thomas Bayes

    I’m coming late to this discussion (a good one, by the way, not much name calling!), but I wanted to clarify a mistake in Doug Spoonwood’s post.
    Doug said (a bit impaciently): “No. Look, independence of two events a and b means that p(a)*p(b)=p(a and b) as stated I don’t know how many times already.”
    This is correct. What is not correct is the implied statement that Mike’s original sentence was wrong.
    Mike said:
    “Two events are dependent if and only if the occurrence of one event affects the probability of the occurrence of the other.”
    Ok, take the negative of Mike’s sentence:
    “Two events are INdependent if and only if the occurrence of one event DOES NOT affect the probability of the occurrence of the other.” This sentence has the same validity as the original – either both are false or both are true.
    Translate the negative of Mike’s sentence into something closer to symbolic math:
    “A and B are independent” iff “A occurred” does not change the probability of “B will occur”.
    Go for another round of translation:
    “A and B are independent” if Prob[“B will occur” | “A occurred”] = Prob[“B will occur, regardless of whether A occurred or not”]
    And finally:
    “A and B are independent” iff P(B|A) = P(B)
    Multiplying both sides of the previous equality by P(A), and applying Bayes formula, gives:
    “A and B are independent” iff P(A and B) = P(A)P(B)
    which is what Doug correctly said. So, in conclusion, both Mike and Doug were correct, and Doug was wrong in implying that Mike didn’t get it. No flames intended here, just wanted to clarify the point.
    The best recent comment was Nancy Lebovitz’s comment, which, in my opinion, nailed it. The reason why P(“John defaults” | “Bill defaults”) is not equal to P(“John defaults, regardless of Bill”) has nothing to do with the underlying cause of either defaulting first, but rather with the fact that “Bill defaults” implies “many things, including house price decline and reduced market confidence” implies “John defaults”.
    The dominos were stacked on the table. A small shake of the table caused a few dominoes to fall, which cascaded into many of the other dominoes being knocked out by their fellow falling dominoes. The mathematical mistake referred to in the main text is the fact that financial analysts forgot (or ignored on purpose) that a tipping domino piece, in a crowded table of dominoes, will bring everybody down. Again, P[“domino i falls” | “domino j fell” and “table shaken”] is not equal to the simpler case P[“domino i falls, regardless of domino j” | “table shaken”]. Notice that in the last equation I kept the underlying common cause, just to emphasize that the whole discussion is about “default A” impacting “default B” beyond the effect of the common factors.

  55. Stephen Lund

    Again a late comment here, I think there should be some accounting for the probability of a default caused by the market value of a home becoming less than the ammount owed. If the interest rises enough that enough individuals independently conclude that they can no longer make the payments and are forced to default, and if the concurance of these possibly independent events has the effect of changing the supply and demand situation, lowering the prices and setting up another round. It seems reasonable to think there is a feed-back loop operating and the probabilities are not completly independent but rather partly dependent.


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