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How the Gaussian Copula Function humbled a Giant Financial System
Formulas are the alchemy the finance industry uses to turn numbers into profits. But what happens when a formula—so trusted by all—becomes a nightmare that crashes the market? This is the story of an elegant formula that may have done just that.
The Formula that Crippled Wall Street
Pawan J. Shamdasani, Staff Writer
Design by: Jeff Fritz
(First published in the Arbitrage Magazine Issue 2)
In 1998, David X. Li invented the Gaussian Copula Function, a complex formula used to measure and model risk with greater ease and accuracy. A Canadian financial economist and mathematician by trade, Li previously worked in companies like CIBC, Barclays and JPMorgan Chase, but with all this experience, he did not see the impact his formula would have on the financial industry.
For over the past few years, it has been heavily used by some of America’s biggest financial institutions, making it possible for them to sell a substantial range of new securities. Additionally, it was widely used by bond investors, Wall Street banks, rating agencies and financial regulators, which allowed them to make a lot of money—that is, as long as the warnings about the formula’s limitations were ignored.
Unfortunately, those limitations couldn’t be ignored forever, especially when the financial markets began to deteriorate rapidly in 2008, swallowing trillions of dollars and causing one of the biggest recessions since the Great Depression. How did one formula help to humble the giant financial sector? The answer remains with the bond market.
There are various types of bonds, including government bonds, municipal bonds and corporate bonds. Bond investors deem these as relatively safe because historical data shows a low default risk upon maturation. However, a more complicated element of the bond market is mortgage pools, consisting of hundreds or even thousands of mortgages, all carrying different interest rates, maturity dates and default risks. Therefore, investing in mortgage pools is different from investing in the aforesaid bonds for several reasons: (1) there is no guaranteed interest rate, (2) there is no fixed maturity date and (3) there’s no way to assign a single probability to the chance of default.
Quantitative analysts at Wall Street (or better known as quants) invented mortgage tranches as a solution to the risk factors of investing in mortgage pools by diversifying the risk. Tranches are composed of hundreds of mortgages and classified as tranche A, B, C and so on.
Tranche A are composed of first lien assets and are given a rating of AAA, AA or A. This implies an extremely safe investment. Tranche B are composed of second lien assets and given a rating of BBB, BB or B. This signals a safe investment but with a slightly higher risk of default than tranche A, and so on. Investors who possess investments of tranche B can thus charge a higher interest rate than those who have tranche A, as they bear the slightly higher default risk. The issue with mortgage tranches and their ratings is that they don’t take into account that millions of homeowners could default on their loans at the same time.
For example, if housing prices fall in your area, you lose some of your equity. You choose to default on your mortgage and so there is a high chance that your neighbours will default too. This concept is known as correlation, the measure of the degree to which one variable moves in relation to the other variable.
In particular, bond investors and mortgage lenders desire to know how to measure, model and price correlation so that they can determine how risky mortgage bonds and their corresponding tranches are. How can this be done when the correlations vary so much?
The solution was provided by Li in his paper “On Default Correlation: A Copula Function Approach,” published in the Journal of Fixed Income, where he explained a way to model default correlation without using historical default data. He chose to rely on the prices of credit default swaps (CDS) instead; these swaps being a credit derivative contract where “the buyer of a credit swap receives credit protection, whereas the seller of the swap guarantees the credit worthiness of the product (investopedia.com).”
Li’s idea was that he would use historical prices from the CDS market instead of waiting to gather enough historical data about actual defaults, which rarely happen, insinuating that prices of credit default swaps and the probability of defaults were positively correlated. In other words, when the price of a credit default swap increases, default risk rises as well.
“Li wrote a model that used price rather than real-world default data as a shortcut (making an implicit assumption that financial markets in general, and CDS markets in particular, can price default risk correctly),” said Felix Salmon, writer of Recipe for Disaster: The Formula that Killed Wall Street in Wired Magazine.
The formula opened a new door of opportunities for many large financial firms helping them to market the complex financial instruments known as credit derivatives, including what risk they faced, what strategies they needed to minimize the risk and what return they should demand. Using Li’s formula, rating agencies (such as Moody’s) could determine how safe or risky a tranche was based on a single correlation number. Bankers were able to bundle anything and create a triple A-rated bond from car loans, bank loans and corporate bonds, to mortgage-backed securities. These pools were better known as collateralized debt obligations (CDOs).
“The CDS and CDO markets grew together, feeding on each other. At the end of 2001, there was $920 billion in credit default swaps outstanding. By the end of 2007, that number had skyrocketed to more than $62 trillion. The CDO market, which stood at $275 billion in 2000, grew to $4.7 trillion by 2006,” said Salmon.
However, cautionary advice against using the Gaussian copula function and against using correlation as a deciding factor for investment or as a tool for risk management were clearly voiced by many in the finance industry. The main argument against the formula was that it made no allowance for unpredictability and assumed correlation was a constant rather than a variable, resulting in a lot of uncertainty in the end-result. This also suggests that the implications of using the formula were well understood before the financial crisis began, yet managers continued to misuse the formula because it mainly allowed them to capitalize on huge returns.
Another flaw with the copula formula was that it used CDS prices to calculate correlation in a time, when house prices were rising and thus where default correlations were at their lowest. But, when the housing bubble popped, house prices began to drop across the country and default correlations started to escalate. This meant that the formula was highly sensitive to changes in house prices. Bankers took advantage of the formula’s sensitivity to house-price appreciations by continuing to create more CDOs.
Many bankers were unaware that large changes in the correlation number were a result of small changes in their underlying assumptions. The results they were seeing were not as volatile as they should have been, suggesting the risk shifted elsewhere. The main problem was that the managers were responsible for all asset-allocation decisions and lacked the quantitative skills to understand the models in terms of what they did and how they worked. They also made their decisions based on outputs from “black-box” computer models and not on common sense.
“Very few people understand the essence of the model. The most dangerous part is when people believe everything coming out of it,” Li once said of his own model.
In all, most in the finance industry fail to realise the reality, nor understand the impact, their figures can have on the markets. Worse, many bankers, especially managers, simply don’t grasp the mathematical workings of such financial models like the Gaussian Copula function. No model can perfectly represent reality. Therefore, common sense must come into play when making sound investment and asset-allocation decisions.
The copula function was an elegant way to model risk, provided the underlying assumptions were met. A debate continues as to whether Li or those who misinterpreted his formula should be personally blamed for the Housing Meltdown. But in reality, this may be just the tip of the iceberg.
Arbitrage Magazine
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