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The Real—and Simple—Equation That Killed Wall Street

The views expressed are those of the author and are not necessarily those of Scientific American.

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“If it weren’t for those meddling kids!” That was the punch line for every Scooby Doo episode. It also is the overly simple narrative that many in the media have spun about the last financial crisis. Smart meddling kids armed with math hoodwinked us all.

One article, from the March 2009 Wired magazine, even pinpointed an equation and a mathematician. The article “Recipe for Disaster: The Formula That Killed Wall Street,” accused the Gaussian Copula Function.

It was not the first piece that made this type of argument, but it was the most aggressive. Since then it has been a common theme in the media that mathematics, especially obscure advanced mathematics, is largely responsible for the catastrophe that doomed the world to the last five years of recession and slow growth.

This theme plays on the fallacy that danger always comes from complexity. It’s a fabrication that obscures the real causes, that makes it easier to say, “Hey, it wasn’t my fault, I was blinded by science.”

The reality is much simpler and less sexy. Wall Street killed itself in a time-honored fashion: Cheap money, excessive borrowing, and greed. And yes, there is an equation one can point to and blame. This equation, however, requires nothing more than middle school algebra to understand and is taught to every new Wall Street employee. It is leveraged return.

What is leveraged return?  It’s the return on assets using borrowed money.

The equation for the leveraged return, L, is:

Where Y is the return of the asset, R is the cost to borrow money, and N is the “haircut,” or the percentage of money the investor must put down to secure the loan (the down payment).

A simple example. An investor wants to buy a bond returning 7% using borrowed money. The bank requires them to pay 20% in cash with the remaining 80% lent at a rate of 5%. What is the leveraged return?

So after borrowing, a 7% return is turned into 15%.  Kaboom!

The equation, though simple, reveals one dangerous truth that investors love to exploit. If you can find an asset that returns more than the cost to borrow money then any return is possible with enough leverage.

How to exploit it though? Two conditions need to be met: A low borrowing cost and somebody willing to lend lots of money. Both came together perfectly in the six years preceding the meltdown of 2008.

The first condition, low rates, came courtesy of the bursting of the dot-com bubble.

In early 2001, following the collapse of the stock market, the Federal Reserve (the Fed) started aggressively lowering the Federal Funds rate (Fed funds). They did this in an attempt to stave off a recession. The Fed funds is the rate, set by the Fed, for overnight loans between banks. This rate broadly sets the cost of borrowing money for short time periods.

The goal of the Fed was to promote growth. In 2001 Fed funds was brought from 6.0% to 1.5%, and by 2003 had reached 1.0%. Small business loans, mortgages, and all borrowing become cheaper.

Those low rates, intended to boost consumption and increase employment, also boosted Wall Street.  Money was cheap and many assets, which had suffered from the sell-off following the dot-com implosion, were also cheap. Money started pouring back into the markets with investors flocking to the high-returning risky assets.

As the markets moved forward the returns on assets contracted under the buying pressure. The only way to get the same returns? Borrow more, dropping the haircut to lower and lower levels increasing the leveraged return.

Still, what was needed was a way to justify the borrowing of more money. That second condition also came courtesy of the Fed. This time it was unintended.

What drives the haircut, N, the amount of money Wall Street will lend?

Most borrowing in the markets is secured lending with the asset purchased used as security against the loan. The haircut, the cash put down, is a buffer against a fall in the price of the asset.

Risk managers, who set the haircut, run simple models that posit asset prices in the future to be spread about an average with a width to the distribution. That width, the standard deviation, is also knows as the price volatility, .  How do they estimate this? Mostly using historical data.

The consequence? Higher market volatility means higher haircuts (less money lent) and lower market volatility means lower haircuts (more money lent).

By 2004 the economy was on an upswing, helped in part by the Fed’s policy of low rates. Now it was time to “step on the brakes.” From the middle of 2004 until the end of 2006 the Fed started to raise rates as the economy healed.

The combination of increased borrowing cost and higher overall asset prices should have been the end of the leveraged return game. But something funny happened. The Fed raised rates in a smooth and predictable fashion, communicating to the markets that increases would be .25% at every six-week meeting.

The methodical steps higher in rates dropped market volatility to historical lows.

An index that measures market volatility, called the VIX, decreased during that period reaching a bottom of 10% in early 2007.

What did banks do in response? They lowered the cash required from 20% to 15% and eventually to 10% and lower, giving a whole new meaning to the term Dime Bag.

Now came another equation.

The profit in any year must be greater than the profit in the prior year.

2005 and 2006 were record-breaking years on Wall Street. Assets were at historical highs and lending was also. Leverage at major financial had grown from around 20 times capital to 35 times. The easy money was gone, but management felt profits had to keep pace.

Chuck Prince, CEO of Citibank, in a now infamous quote said, “When the music stops, in terms of liquidity, things will be complicated. But as long as the music is playing, you’ve got to get up and dance. We’re still dancing,”

Haircuts could not go much lower. Now simple greed took over. The banks turned to outright purchases of securities, morphing their role as lenders into investors.

This is where equations like the Gaussian Copula function enter. The market used these exotic functions to argue favorable regulatory treatment, assigning risky assets less risky weightings. This was helped by a symbiotic relationship between the rating agencies and banks. Lax oversight by regulators further enabled the corruption.

The result? Now everyone was loaded with leveraged assets many of them far riskier than reported.  A tumble in one asset could precipitate a furious and fast fall in others. Lenders would be forced to sell collateral, triggering other assets to be sold as borrowers had to find cash to post against the loans.

With balance sheets engorged relative to capital, prices did not have to fall far before some institutions faced insolvency. Cross borrowing amongst institutions led to further entanglements. No fund was an island.

The result was the spectacular fall in markets in 2008 that culminated in bank defaults and the government bailout.

The Gaussian Copula Function, opaque to most, is convenient to blame. It allows us to shake off our collective sense of guilt. It obscures the real crime with a motive that doesn’t take a gang of teenage sleuths and they’re dog to sniff out:  greed.

From the minutes of the Aug 7, 2007 Fed meeting.

“I lived through the corrections of the S&L market, portfolio insurance, the crashes of ’87, ’97, and so on. When you sort through them, all of them have a common basis, and that is a search for greater yields or greater return, leverage in order to achieve that return….”

—Richard Fisher. Dallas Fed President.


Data courtesy of Bloomberg Financial.

Chris Arnade About the Author: Chris Arnade received a PhD in Physics from Johns Hopkins University. He joined Salomon Brothers in the Bond Portfolio group in 93 where he built credit models. In 95 he moved to the Emerging Markets trading desk where he traded options, bonds, credit derivatives, interest rates, and FX. He focused on proprietary trading from 06 until six months ago. Since then he has focused on photography and addiction. Follow on Twitter @chris_arnade and on facebook at Chris Arnade Photography. Follow on Twitter @chris_arnade.

The views expressed are those of the author and are not necessarily those of Scientific American.

Comments 26 Comments

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  1. 1. BillR 9:15 am 01/30/2013

    Manipulating the math to justify the actions your greed desires…. This will always be a problem since math has no morals. It is just math.

    The real problem was lack of morals on the part of those using the math. The problem is that morality (what is right and what is wrong) is defined by society and the moral standards of the society in question (bankers and investors) evidently were biased by a desire for high profits (i.e. greed). This resulted in a negative moral reinforcement and the spiral into even more greed and self justification.

    A society’s morals will determine the health of that society. When the sense of what is right and wrong goes south, so will the society that accepts that moral paradigm.

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  2. 2. Richieo 9:47 am 01/30/2013

    I don’t need a mathematical equation to justify redirecting the blame, it was pure unadulterated GREED and a indolent, ignorant government…

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  3. 3. hankroberts 10:54 am 01/30/2013

    > cheap money
    Can’t be much cheaper than it is now, can it?

    > excessive borrowing
    How’s that looking at this point?

    > and greed

    So where are we headed now?
    And how’d we get in this handbasket again?

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  4. 4. llohman 10:59 am 01/30/2013

    Ah, GREED. The one factor which makes totally “free markets” an unacceptable appliance in a society of common men. Because those with power and money (which equate to each other) will gather more and more of the resources represented in and by those markets, to themselves, leaving less and less for those not among their number. Just look at the 1% and compare against the 99%. And that 1%? They want even more.

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  5. 5. Tim May 2:20 pm 01/30/2013

    First, the headline is sensationalistic and misleading. Wall Street is hardly “dead.” The DJIA is higher than it was during most of 2008, including in the months prior to the September events. And so on, for many other pieces of evidence.

    (Note: I’m not excusing or endorsing the “Too Big to Fail” policy, or TARP, or any of the shoveling of money to big banks. Just nothing the misleading headline.)

    Second, an article for Scientific American, even a blog article, ought to spend more time explaining the correlation issues in the Gaussian copula point. For instance, the assumption that bundles of mortgages with varying risks would not mostly simultaneously move in the same direction. (They did, of course, meaning many of the AA and other such ratings had been miscalculated.)

    Third, money is actually “cheaper” now that at any time in my lifetime. If I wanted to, I could refinance my mortgage with a $250K loan at 2.59% APR (I just double-checked the details). I could take out a $100K in “sweet cash back” AND cut my monthly payments by $900. I could then use that money to buy an RV, a motorcycle, some vacations in Cancun and Daytona Beach, and kick back for a while. And maybe buy some crack or crystal.

    But I won’t. And I didn’t in 2008. Nor in 1995. Nor in 1985.

    Which, I suppose, is one of several reasons I’m not the subject of one of your photographic essays about the addicts and trannies of Hunt Point.

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  6. 6. jtdwyer 7:23 pm 01/30/2013

    I’m no financial wizard, but I certainly agree with the fundamental causes of the late market collapse. However, didn’t tricky mathematically based financial products contribute to the initial collapse of the ill founded market?

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  7. 7. CherryBombSim 10:12 pm 01/30/2013

    I pay close attention to anything Richard Fisher says. He is very often right.

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  8. 8. ErnestPayne 10:17 pm 01/30/2013

    You missed out on the importance of wishful thinking. Having worked in the markets (decades ago) I remember people more than willing to ignore unpleasant truths in favour of wishful thinking.

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  9. 9. Cramer 11:06 pm 01/30/2013

    Not many who were making loads of money believed that the Gaussian copula was a magic pill. Chris Arnade gave the ingredients of the financial crisis as cheap money, excessive borrowing, and greed. He left out moral hazard and fraud. It’s ashame Wall Street is above the law, but any criminal prosecution would bring down a bank — just like Arthur Andersen was brought down in 2002. And bringing down one bank would bring down the entire industry. It will eventually happen one way or another. The next financial crisis (fraud-driven bubble and collapse) will be with energy — most likely natural gas.

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  10. 10. Bobinsandiego 11:15 pm 01/30/2013

    I believe that traditional ethics (right or wrong) have been replaced by a new ethic – if the government regulations permit it or, at least, don’t prohibit it, then it must be ethical. The financiers seem to be within the rules, therefore what they do must be ethical (to them). The issue is that rule makers are always one step behind the governed.

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  11. 11. dadster 6:09 am 01/31/2013

    Bobibsandiego ( @ser 10) is right about morals which in present times , means ,within the rules the Gov. has made. So,lack of adequate govt regulations and regulators to implement the laws ( both, hated by the 1% who got around the govt not to regulate them and , to let the banks self- regulate , which never happened ), was a major factor in the collapse of the banks.

    The second factor was “super-speed Internet trading”! Direct high speed Internet (connections not passing through ISPs),trading facilities accessible to a
    favored few ( the 1%) allowed them to skim the market even before others got access to latest stock values . Such high speed trading done between computers at nano- speeds combined with rampant insider- trading left out the 99% from the game which helped wealth accumulation with the 1% .

    Greed , the defining characteristic of human race was always there , before , now, and in future too which is a permanent common factor of all times and of all human actions . But , the speed with which greed for money could be satisfied due to the direct internet high speed trading available only to a selected few sans govt regulations or regulators was the major factor in the collapse of the financial system .

    So, that at least point to the remedies available to us.
    1. Have well thought out gov regulations made to protect the public from financial greedy predators and not more helpful to major investors . Investors would invest at any rate because that’s the way their mind works . They wouldn’t go out of the country to invest if the govt . regulations are tight enough not permit it .
    2. Regulate direct highspeed Internet trading by having direct high speed internet channels of communication between the stock market and the favored individuals bypassing normal ISPs.. Such superspeed connections ( not through ordinary ISPs that slows down Internet connections that is available to the public ) , should be made illegal and legislate that money made by such illegal trading must be confiscated triply .

    How to regulate lightning speed Internet trading ? Since it’s a technological problem, technological solutions must be sought and found . There will exist a technological solution which could be implemented supported by suitable govt. regulations. Just like road traffic on highways are controlled to abode accidents
    and collapse of road – traffic , so also to avoid collapse of financial system high speed direct Internet trading must be monitored and controlled making the play field level for the public and the major investors .

    One way to control greed is to put ceilings to growth of individuals and of corporations . No one should be allowed to grow so high that they cannot fall . Put ceilings on individual income , bank balances and salaries and remunerations of coo rate CEOs , bank managers and anyotential high growth concerns . Every citizen is working for the good of the nation and not for the good of themselves . Such sense of moral responsibility andbsocial responsibility need be shown in govt regulations . It is the govt which is of the people, for the people and by the people , not the banks and corporations whose job is to do business and make money , enhance wealth ; but to help the, to do so in a treasonable and controlled manner is the job of the people’s govt.

    This global financial and banking collapse is entirely due to the fault of the govt who did not do their expected and sworn duty . Govt was derelict in their bounden duties. The bankers exploited the situation. In dancing to the tune of the bankers and the corporations ,the govt allowed itself to be mislead into believing thatnthey are doing good to the people which did not happen .

    Now at least the govt must act firmly and decisively to retrieve the situation without leaving it all on the bankers and corporations this time or ever in future too.

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  12. 12. Jimbotron1970 11:18 am 01/31/2013

    I think the piece is essentially right. But let’s not overlook something relevant. This financial situation (and preceding ones with like attributes mentioned in another comment) was only possible because the behavior was made legal by the overseers in congress, whom we vote in every two or six years. They knew what they were doing.

    Mortgage lending with no assets was more or less *required* by congress. Same is true with higher ed loans today. But when the wheel stopped, folks noticed that their houses weren’t worth nearly what they hoped. When the higher ed wheel stops it will be the same.

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  13. 13. Eneidawas 12:18 pm 01/31/2013

    Back in the late ’70′s and early ’80′s, while “The Great Communicator” was asleep at the wheel, banking regulators got away with murder, changing anything they wanted.
    I watched loan applications cross my desk that made no sense what-so-ever, home equity loans to people with no homes and no equity. It was all legal, though, and our bank didn’t even care, because they sold the debt. It all took place on such a small basis that it didn’t really effect anything…much.
    I just kept thinking, “This will not end well.” It took thirty years, but I was right. I didn’t think it would be a national financial disaster, though.

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  14. 14. MRC06405 1:47 pm 01/31/2013

    There is nothing wrong with the equations. It is clearly a question of GIGO garbage in, garbage out.

    If you lie about the volatility of your investments. If you aggregate older good data (30 year fixed motrgages written and held by banks)with newer bad data (adjustable rate mortgages with teaser rates written by mom and pop mortgage brokers and sold as commodities) your equations are no longer valid.

    If civil engineers built bridges with this kind dishonesty and poor judgement they would fall down, but financial engineers mindlessly plug lousy data into their formulas and allow bankers to make a some quick bonuses and then pretend they did nothing wrong when the inevitable crash comes.

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  15. 15. Derick D 6:00 pm 01/31/2013

    Banking is too important to be left to the bankers. Every other part of the economy relies on the financial system to be healthy. Essentially, we are all held hostage by the greed of bankers.

    Banking and investing are necessary for the economy to work, and that necessity should come before bankers’ profits. Them making a lot of money is less important than the rest of us making enough money.

    I’m tired of us regular working people taking the actual risk (let’s face it – no banker went hungry as a result of this mess), while the bankers reap all the rewards.

    Solution: laws that tie finance execs’ compensation to long-term economic stability. Basically, the IRS would retroactively increase their income tax rate based on overall economic performance, say 10 years after the fact. If the economy gets better, they get a retroactive tax break and get back some of the income tax they paid 10 years ago. If the economy tanks, they get a retroactive increase in their tax rate and have to pony up more income tax. Depending on how bad the economy tanks, they might end up paying a LOT more. That way, when the economy tanks they all take a bath along with the rest of us. Let them come up with financial instruments so complicated not even God can understand them (if he even exists). If they work out, everybody wins. If not, they lose with the rest of us.

    Maybe this isn’t the best way, and it certainly isn’t the only way, but something’s got to give. The only way the free market can work is if it is in financiers’ best interest to act responsibly in the long term, and as long as the system focuses only on annual and quarterly gains that won’t happen. The boom-bust cycle will continue unabated, and it’s regular working people who get busted, over and over again.

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  16. 16. Art Bitrage 12:15 pm 02/3/2013

    The author’s mathematical description of leverage is accurate yet strikingly simple.  What’s not so simple is how the application of leverage, an age-old practice caused such devastation.

    I work in the financial industry; “wall street” is no longer a geographic or technically accurate phrase-fyi.  For the writer and those commenting above to blame a few thousand people for financial troubles of the entire globe, is fairly naive.

    The application of leverage was a trick used by more than those in the financial industry, it was (is being) used by the entire globe.  A large number of Americans refinanced/bought homes they couldn’t afford.  They took cash out to sustain lifestyles that would otherwise have been beyond their means.  

    These activities are strongly encouraged by our materialistic cultrure and strongly supported by several consecutive government administrations (and continues to be).

    Again, the math is far too simple.  I, for one would like to hear a more psychological explanation of why this happened and why it’s still happening.

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  17. 17. Chris_Arnade 1:19 pm 02/3/2013

    For me, someone who worked in Citibank from 93 to 2012 its pretty simple. Financial institutions increased their balance sheets almost three fold; Leverage measure vs Tier 1 capital went from 12 to 36 or so.

    Yes, for that to happen someone has to be taking out loans.

    You will have boom and busts in free market, what you would rather not have is major defaults of banks that turns the busts into depressions.

    That happening is a function of leverage at the banks.

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  18. 18. outsidethebox 9:06 pm 02/4/2013

    Many people have poor memories apparently. I recall the time just before the crash. The idea that people didn’t understand that it was all a bubble is not true. It was common knowledge at the time. It’s just that everyone thought they could ride it for just a little bit more. They were wrong of course and now they certainly want someone other than themselves to blame for that error.

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  19. 19. jabailo 12:30 pm 02/5/2013

    You seem to have a selective memory. The first thing that happened was the Fed raising rates which is what popped the “bubble” because people then pulled out of speculative investments. But look at what the economy is based on now…Google, Amazon, Facebook and a million web companies. Was it really a bubble.

    But I buy your premise and it’s been said before. Never confuse genius with an ever rising DOW.

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  20. 20. Patrice Ayme 12:32 am 02/6/2013

    What everybody overlooks: banks create nearly all the money that the state uses, therefore ought to behave like governmental institutions; al the money created to fo to the real economy, not themselves.

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  21. 21. denisosu 3:09 am 02/6/2013

    Two important factors missing from this otherwise excellent analysis:

    1. Wall Street Salaries and Bonuses meant that EVEN IF THEY KNEW IT WAS A BUBBLE, the traders who ignored the risk made more money. For as long as they got away with it, they got huge bonuses often based on a % of their personal profits on a given investment, while when it crashed and the investment lost millions of dollars, nobody ever asked the employees to pay back a % of the loss. Because (by the laws of probability) some traders will always do better than others, people who didn’t understand that it’s all basically just a crap-shoot would find a trader or a fund which had “beaten the market” for n years in a row and conclude that he/it was infallible.

    2. As another comment above said, Garbage In, Garbage Out – in this specific context. Traders modelled the markets using inductive models based on historical trends rather than fundamental business understanding. If you model an asset like a blue-chip stock that has grown steadily over 40 years with only occasional, short-term dips, it’s normal that any mathematical model based only on historical share-price will tell you it is very low-risk. But of course such a model can be wrong.
    (would you trust a model that predicts that in next years’ playoffs Joe Flacco will continue to throw about 3 touchdown passes per game and will never throw an interception?? That’s what Wall Street was doing!)

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  22. 22. Chris_Arnade 3:19 pm 02/6/2013

    You are correct. Both those issues I addressed in my previous article for Sci Am

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  23. 23. Cosmoknot 12:11 am 02/7/2013

    If we can just get rid of the need for money, then we won’t have problems like these.
    Money is a horrible way to trade value, because value is something that money does not sustain. Having no intrinsic value of its own, the value represented by money can fluctuate, and so guess what, money’s value decreases over time in what they call inflation. Because of inflation, more money is required to purchase that which costed less money previously, and so for most people, inflation is bad, but somebody is making out. If everybody’s money continually experienced this inflationary loss of value over time, there’s no way we would keep using money.
    People who use money have been tricked into using these rotten value markers that lose value over time. Inflation benefits the few at the very top. The ones who provide the money for use in the system are not themselves stuck within the system, so while they get to manipulate the system, their fortunes are not tied up in the system with its stupid value tickets that steadily lose value over time, if not in a quick engineered flash.

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  24. 24. karl 2:00 pm 02/8/2013

    I believe that another mathematical model also helped ruining the economy, the prissioner’s dilemma, if a bank or investor or something notices that NINJA credits are suicide, he is forced by the market (the invisible hand of the blind madman I might say) to offer them too, because if he doesn’t, he will loose valuable clients to those who do.
    If you say “I have a perpetual motion machine” in engineering or science, you have to show everyone how does it work to be recognized, but due to the nature of economics, you can say that, and just babble something about market niches or the likes.
    Perhaps a Thermodynamics course would be useful in Wall Street (instead of the “Rules of Accuisition” of Star Trek’s Ferengi fame)?

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  25. 25. Markuf 12:22 am 02/20/2013

    Albeit legal. or illegal, ethical or unethical leveraged or balance, formulaic etc,. Front line maters. Individual awareness and responsibility, tend to that which you can tend to is the fundamental basis in our own, rather those idiots that fell for it, responsibility. We are bursting further and further away from what we know is rightfully wrong, even if it is more than we KNOW we are capable of doing or capable of achieving. Just because “they” say one can do so does not mean that it is true though most will proceed. Example: earning 50,000 a year and able to qualify for a 500,000 dollar loan. Thats a 3500 dollars month mortgage when you bring home 3100. No matter what they say, YOU need to figure your life out DUMB A($$ and quit messing it up for the rest of us in this pond. Heck the animal have live symbiotically for 20-50 million years, can’t you just hold it together for 76 yrs

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  26. 26. Markuf 12:26 am 02/20/2013

    I overlooked my manners, please, please hold it together for your stay and eat what you can stomach, and kill what you can eat, etc. Quit listening to the voices “outside” of your head.

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