The Gaussian Copula Function, Part Two: Broken Heart Syndrome

“This thesis is dedicated to my deceased parents who would be very happy to see their youngest child completing his PhD Studies”

-David X. Li, then Xiang Lin Li, Dedication of PhD Thesis “An Estimating Function Approach to Credibility Theory.”

So we left off with the description of the climate that made an end, all be all function to determine risk so tantalizing. Rather than comb through a company’s books and look for problems, rather than relying on the instincts and the gut feelings of traders, a bond rating agency could apply use the copula function approach to analyze risk. In this post I’ll discuss where David Li was coming from when he developed the function and what precisely he sought to do with it.

David X. Li was born Xiang Lin Li in rural China. He, despite his family’s poverty, was able to attend the highly prestigious Nankai University because of his work ethic and intelligence. In 1987, the Chinese government under Deng Xiaoping sought to increase the state’s understanding of Western capitalism and made it a policy to send especially promising graduates to Western Universities in order to make China more marketable on the international stage. So, having received a Masters in Economics from Nankai, Li was sent to Laval University in Quebec to earn his MBA. Li quickly learned French and sought to immerse himself in the business culture. He then sought to go to the University of Waterloo where he achieved his Ph. D in actuarial science.

It was in fast paced Ontario that Li decided to remain in the West to make his fortune. After some digging, I was able to find Li’s thesis, called “An estimating function approach to credibility theory.” It is, needless to say, slightly over my head. The main topic of the thesis is all too familiar, though; Insurance companies must assess an individual a premium, and the only way to do so is to use some combination of the prior claims of the individual in addition to the mean claims associated with a hypothetical individual in that person’s class. For instance, an individual person’s medical insurance premiums are a reflection both of that person’s individual past medical history  and a factor of who that person is on paper, involving weights such as age, weight, family history, race, smoking habits, and the like. The same goes for auto insurance, where the person’s past history of claims is combined with that person’s demographic probability of making a future claim. Li’s background in Actuarial Science led him to a career in the financial industry and this background offered him insight into cracking one of the toughest eggs in financial science.

By the late eighties, Wall Street had fully accepted the concept of having analysts in the back room develop and analyze algorithms which could be used by traders on the floor. The industry, as a whole, had accepted that the analysis of data outweighed the instincts of a trader. Ever since Robert Rubin, the future Clinton Treasury Secretary, pioneered the concept of data driven investment while at Goldman Sachs in 1987, Wall Street investment banks had been all to eager to hire PhDs looking to make a fortune analyzing markets. Li stumbled on to the scene just when it was getting popular. He began at the Canadian Imperial Bank of Commerce in 1997. He worked for the JP Morgan spin off CreditMetrics group, and through them in 2000 he published his famous and influential paper “On Default Correlation: A Copula Function Approach.”

In Actuarial science, there was an interesting observation that, statistically speaking, a individual was more likely to die after the loss of a beloved spouse. “Broken Heart Syndrome” was a familiar concept across the medical industry, studied in the Medical and diagnostic circles because they sought to understand it and cure it, and in the life insurance circles because the actuaries sought to maximize profits and set premiums to a ideal level. So, after analysis, it was decided that life insurance packages in which premiums increase after one partner dies, called Last Survivor Annuities, were overpriced, while Joint Life Annuities, packages where premiums remain the same after one partner dies, were underpriced. Now, the evidence was entirely statistical and, looking at a single couple, was not a factor, over entire, broad, sweeping markets, this was the case. By understanding the Broken Heart syndrome, the house always won.

Li’s background in actuarial sciences led him to the conclusion that a dying person was no different than a defaulted loan, in some ways. Granted, any two given loans were not necessarily bound together, but there was likelihood that there was at least some, if only tangential, connection, or ‘statistical correlation’, that allowed two loans to be compared by some factor. The function that was able to determine the connectedness of these loans, or bonds, or Collaterized Debt Obligation, was a ‘copula function.’ Perhaps the mortgages were issued by the same bank, and if the bank started calling in the loans then both mortgages would default. Maybe both mortgages were in the same city which was struck by the same hurricane. Maybe both loans were for small businesses in the same industries, even states apart. Either way, there is a statistical correlation between each of these that went unnoticed before the copula function approach was introduced. The thirty page paper caused a complete paradigm shift in how analysts viewed companies. I’ll get into the nitty gritty of what precisely the function is in my next post.

I will however make some notes on the research process itself. During the course of the research, I’ve noticed a  major theme in the coverage of both the large scale of the crisis and the actual function itself. There are only two types of coverage; the supremely detailed papers for professionals and PhDs in that specific niche of the financial industry, and then the watered down, oversimplified and often misrepresented viewpoints intended for the general public, coming even from highly respected newspapers. A reliance on broad platitudes and irresponsible metonymy undermine many of the articles about the crisis. It’s also very interesting to see how little many of the financial publications understand this sophisticated market. As I continue to research into the topic and into the financial industry, I’m beginning to see how grossly incorrect many perceptions about this crisis, the data, and the steps being taken to respond to it are. Reading the in house papers spread around the industry, I’m seeing different perspectives on the crisis and some very different viewpoints about how it was handled, especially by the US government. It’s also interesting to see how close we were at many points to a much larger problem and how quick responses on the part of the Banks, the New York Fed, and the Federal government really helped mitigate the crisis. I’ll get into that next time if the comments ask


  1. Adam Lerner says:

    Walter, will you please tell us how close we were at many points to a much larger problem and how quick responses on the part of the Banks, the New York Fed, and the Federal government really helped mitigate the crisis?

  2. That’s really cool that you’ve researched enough to be able to tell what’s incorrect in the watered-down accounts of financial dealings. Maybe you could make a career of providing concise and simple ways of understanding what’s truly going on in finance to ordinary people like me who struggle to understand what’s going on. Well at least it sounds like you’re learning a lot and enjoying yourself too. Good luck with the research Walter!