How maths killed Lehman Brothers

Horatio Boedihardjo
June 2009

This article is the winner in the university category of the Plus new writers award 2009.

On a sunny morning in 2001, a piece of investment plan landed on the desk of Dick Fuld, the then Chief Executive of Lehman Brothers. The document, compiled by a team of maths and physics PhDs, included a calculation to show how the bank will always end up with a profit if they invest on the real estate markets. Fuld was impressed. The next five years saw the bank borrowing billions of dollars to invest in the housing market. It worked. The housing market boom had turned Lehman Brothers from a modest firm into the world's fourth largest investment bank.

Lehman Brothers headquarters

The Lehman Brothers headquarters, Rockefeller Center, New York, before the collapse. Image: David Shankbone.

But as the housing market started to shrink, the assumptions that the PhDs made began to break down one by one. The investment now became a mistake, resulting in a stunning loss of $613 billion. On September 15 2008 Lehman Brothers collapsed — "The largest bankruptcy in the US history," as described by Wikipedia.

The money making model

Imagine that you are working for Lehman Brothers and one morning you receive a phone call from HSBC.

"Hi! A hundred customers have each borrowed $1 million from us for a year. We would like to buy an insurance from you which will cover us in the case of any of them defaulting. From their application forms we reckon they each have a 3% chance of default. How much will the insurance cost?"

You can in fact calculate it, easily. The 100 customers each have a 3% chance of defaulting, so you expect three customers to default next year. That is, you will need to pay $3 million next year. Assuming the interest rate is about 3% each year, next year's $3 million would be worth 3/(3/100+1)=3/1.03=2.91 million now.

Therefore HSBC will have to pay you at least $2.91 million for the insurance. Obviously Lehman Brothers wasn't a charity and so, to make money, they would double the price to $5.82 million and expect to make $2.91 million out of each of these deals on average. This kind of insurance is called a credit default swap (CDS).

The legendary CDO

After putting down the phone, you might be quite worried about what would happen if ten of the borrowers defaulted, because then you would have to pay $10 million back! In this case, consider this deal: how about paying me a certain premium, and if more than ten defaults occur, you will only need to pay for ten of them and I will pay for the rest. If less than ten defaults occur, you will have to pay for all the defaults and I won't pay anything. The type of deal that I am offering is called the senior tranche of a collateralised debt obligation (CDO) contract, while the one you are getting is called the junior tranche of the CDO.

Looks like a safe investment? Better think twice!

Looks like a safe investment? Better think twice!

The attractiveness of the senior tranche is that almost all of the time I don't have to pay anything, just pocketing my premium. Imagine how unlikely it is to have more than ten borrowers defaulting together! Senior tranches were generally considered to be almost as safe as borrowing money from the government. Since the senior tranche offers a better return than, but seems to be just as safe as, putting the money in the bank, the investors just loved it. In 2004 there were only $157.4 billion of CDO being issued, but by 2007 the amount grew to $481.6 billion.

But don't you find it a bit unfair that you can have something as safe as bank deposits, that offers a higher return?

The pitfalls

Yes, it is unfair! In fact, CDO is a lot riskier than bank deposits, but Lehman Brothers, like many investors, didn't seem to know that. Let's go back to our original model. The first source of error is that we have assumed that each investor has a 3% chance of defaulting. How do we know that? It must be from historical data. The problem is, there hasn't been a national drop of housing price since the great depression in the 1920s, so the chance that a borrower defaults was calculated on the basis of a good period when the housing prices surged. However, the housing market crashed in 2007. Many borrowers' properties are now worth even less than the loan they have to pay in the future, so many of them refuse to pay. To worsen the situation, 22% of these borrowers are the so-called subprime borrowers — those who had little income and had little hope of returning money. Banks were not afraid of lending money to them because even if they defaulted, the insurance would pay them back. The participation of the subprime borrowers makes lending much riskier than before.

In fact, the default probability in the US has quadrupled from the 3% as assumed in the model to 12% since 2007, making it four times riskier. This means that investors like Lehman Brothers will be hit four times harder than they have anticipated.

Actually it is worse than that. The profitability, or lack of it, of financial products more complicated than CDS and CDO may depend on the square of the default probability, rather than the probability itself. Now as the default probability rises from 0.03 to 0.12, the square of the probability increased from 0.0009 to 0.0144 — that's an increase by a factor of 16!

Lehman Brothers headquarters

The Lehman Brothers headquarters on the night of September 15, 2008. Image: Robert Scoble.

There is also a second and more subtle source of error. Whether you can make money from selling the CDO insurance for the bank depends on whether the borrowers return the money, which in turns depends on the economy. So if the economy goes down, you are a lot more likely to lose money. If you are an active investor, then you probably have invested in the stock market as well. Now if the market crashes you lose both the money invested in the stock market and in the CDO. Suppose, on the other hand, that instead of spending the money on CDO, you bet on whether Manchester United will win the European Champion League. This time in order to lose all your money you need both the market to go down and Manchester United to lose their match — this is less likely than just having the market go down. Therefore, investing on CDO is a riskier choice than betting for Manchester United. The error in our model is that we have not taken into account this extra risk due to its dependence of CDO on the market.

These two errors were sufficient to mask the risk in CDO. In fact, the errors are so serious that 27 out of 30 of the CDOs issued by Merril Lynch were downgraded from AAA (the safest investment) to "junk" when the errors were spotted.

The fall of Lehman Brothers

Lehman Brothers, unaware of the hidden risks, decided to invest big on CDO. It even had a 35 to 1 debt to equity ratio, that is, it only owned $1 out of every $36 in its bank account — the other $35 were borrowed from somewhere. This meant that a loss of just 3% of the money on its balance sheet, would have meant the loss of all the money it owned. After suffering heavy losses (more than 3% of the money in its balance) from CDO, borrowers began to lose confidence and called back the loans. As Lehman had always relied on short-term loans, its lenders were able to pull their cash back quickly. Now the bank was in trouble. It borrowed much more than it was able to return and soon found itself unable to pay back. On 15th September 2008, the world's fourth biggest investment bank was gone, forever.

About the author

Horatio is studying for a doctorate in probability at Oxford University. He enrolled at Oxford University as a maths undergraduate when he was 15 and had to reside with a guardian. After spending four months unsuccessfully searching for one, his dad decided to quit his management job in Hong Kong to accompany his study. Horatio would therefore like to dedicate this article to his loving father for all his sacrifice. He would also like to thank Iris for her encouragement when he almost gave up writing this article.


I enjoyed reading your article, thanks.

great article. congratulations and best of luck with your studies. you should be very proud.

very interesting article! good job

very nice analysis. good luck for you.

I have read the article and liked the analysis. this is wonderful. we actually need more from our young minds and we shall change the world. keep it up man. elijah kalii - nairobi, kenya

Many thanks for your great article ^^

Very Good explanation and truly independent view of the report. Please follow the good path and make a niche for yourself in chosen field, that is the best way forward

I like how you described this - I have an understanding of how this happened but it would be great if you can write an article about where this money has now shifted too. The billions that came from "somewhere" went "somewhere" via Lehman etc. Who lost and who gained in the grand scheme of things? Please explain the mechanism that the government and the fed have shifted private debt to the public with a nice simple explanation as you have illustrated for the collapse of Lehman Bros (that's if it was a collapse at all??? - I think it was just a way to consolidate financial power). Thanks again.

Doing a case study of Lehman Brothers, was totally lost till I read this - you break it down to laymans terms, most enjoyable


My confusion regarding CDS has resolved today

very good example..


Thanks for your lucid report.

It's beauty lies in its simplicity.

Simple so that any common fellow has the opportunity to get a grasp on the major part of the economic downfall.

This article is superb. Explanation with example about CDS was a great ease for reader to understand the concept. Thank You!!

Thank you for this clear, comprehensible article. It improved my understanding of the Lehman situation a great deal. It worries me that you almost gave up on writing this. Please do not be discouraged -- instead, write many more! We laypeople wish to understand the financial implosion, but to do so we need intelligent, dedicated people like you. Please thank your father for making your studies -- and therefore this article -- possible.

I wonder how someone in his clear mind (HSBC in this case (but do bankers have a clear mind?)) would pay 5.82 M$ to avoid the risk of losing 3 M$? Can someone explain this to me?

HSBC is willing to pay 5 Mill to insure 100 Mill. It's just like an auto insurance policy, the probability of a person getting into an accident is about 0.189%, the probability of that person getting into an accident that would exceed 1 million in 3rd party liability is 0.035%. That's why if you want to increase your 3rd party liability from 1 million to 5 million would only be a difference of roughly 20$ a month. But most people would get that because it means they could walk away from any accident without resulting in any financial reprecussions.

In HSBC's case, even if all 100 of them defaulted (a very low probability but it's still probable), the bank wouldn't lose a single penny! This is also known as credit default swap or CDS which are just contracts between 2 parties.

Now CDO's are a little different, they are backed by collateral, in this case, real assests which includes subprime mortgages.

Lehman brother's failed because they were over leveraged almost over 40:1. They did not have a diversified portfolio like other investment banks, and only focused on selling CDOs. More importantly, they were too greedy. They were aware of the underlying inherent risks yet they ignored it. Unlike Goldman Sachs, Morgan Stanley and other investment banks, they did not properly hedge the risks involved with CDOs so when the market crashed, they crashed with it while the other investment banks made millions and millions.

One thing I would like to add is that all the rating agencies like S&P, Moody and Fitch are paid to by the big banks to give AAA ratings so they can sell them with a higher premium. There is a conflict of interest at stake.

The banks along with it's conglomerates and rating agencies conspired to rip off investors to line their own pockets. Greed is bad.

Nice job, keep it up Horatio. You should definetely start your own blog. Wish you good luck.

This is a brilliant article - very clearly and concisely explained, thank you! and keep writing :)

Nice article. Please can you simplify a little on the following - what lead to the fall in the real estate market? Coz from what I've read the real estate market has grown exponentially globally. Especially in India, property prices have only risen. & then yet again, dont banks have pre-checks for the borrowers which accordingly determine the category the borrowers fall into (for example - prime, sub prime). We here in India, the monthly payments for a housing loan is an ACH debit from our bank accounts, mostly at the beginning of the month for salaried borrowers. Maybe you could reference a follow up done for Lehman Brothers.

Thanks for this article which helps put things very clearly in perspective without too much jargon.

Very well written and easy to understand article Horatio! You should certainly continue educating people since most documentaries and other articles do not explain it in such an easy to understand language. Now, would you please explain derivatives?

Thank you and good luck with your PhD!


At last I feel I now understand what a CDS is, this is one of the best written articles I have read in a very long time. Please write more, you should have your own blog !!!!

Now, would you please explain derivatives? Please, they give me a headache !!!!


Most people thought CDO's is rocket science. You nailed it budd. Eloquently written. Well we now know that we need to assume the new math that 1+1 doesn't add up to 2 anymore in Stock Markets. Haratio, I am sure we can expect more down to earth articles from you in future and your dad is simply the Greatest.

I finally think i understand the collapse and subsequently the reasons for the credit crunch.
Congratulations Horatio, I enjoyed reading this article, usually when reading accounting articles (although actually studying it) I switch off halfway through and start thinking of what Im going to make for my tea!
Please could you re write all my AAT Level 4 text books so they are less boring and clearer to understand, whilst making me want to read more?
And as for your dad, What a star!!
Nicola x Liverpool

Great clear article - thank u

Than you very much .Well done and simply written.Good Luck!

This is really an informative article which open eyes for the common people.

Mr. Horatio...You did a very good job. I have been trying to understand many of these financial dramas. Your article cleared most of my doubts. After reading your article, I am able to co-relate the transactions they made and understood how they ended up into bankruptcy. I hope, by this time you have received your P hd. Wishing you and your father good luck..

This article describes the failure to focus on the "what if" there is a plunge.

The plunge was not the only reason for an increased default rate. It was a secondary reason.

The primary reason happened prior to the plunge.

There was a housing "bubble" based on loose lending criteria and temporarily low interest rates.

The Adjustable rate mortgage was underwritten to the fixed rate portion only and not to potential market rates after this "teaser" rate expired. These mortgages "reset" to much higher rates because Greenspan started increasing interest rates in 2005-06.

Increased defaults and foreclosures crashed the CDO market months before the credit freeze and housing price crash.

For more details check out:

This is complex to understand but I do understand that the top people in the investment banks behavior lead to a world wide crisis that is still harming people and all of these maggots got to walk away with millions and not a single inditement of any of them has been made. They knew damn well they were buying bad mortgages and didn't care because they repackaged them as AAA bonds and sold them to suckers.

Congratulations on your article. I enjoyed reading it as it helped organize my ideas regarding the housing crisis, which is so often talked about.

I very much enjoyed this very clear and engaging article which explains simple probability in a clear manner; statistics professors should take note. Your explanations of the boundary breakdowns without resorting to technical jargon is brilliant. You have done your father proud and, given your talents, should have a very bright future indeed.

So you wrote Business Knowledge for It in Global Investment Banking Chapter 6?