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How the Gaussian Copula Function humbled a Giant Financial System


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.

Quantumrun Foresight
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