Researcher course in fixed income
Copula Models for Credit Risk
How dependence between defaults is modelled: copulas and Sklar's theorem, the Gaussian and Student t copulas, the one-factor portfolio model behind Basel capital, CDO tranche pricing, and the failure of these models in 2008.
A copula describes how risks move together, separately from how each one behaves alone. In credit, that is the question that matters: one default is a loss, but many defaults at once is a crisis. This course teaches the copula models used for credit portfolios, CDO tranches and bank capital, and the limits that 2008 exposed.
Where are copula models used today?
In bank capital, where the one-factor Gaussian model underlies the Basel formula for credit risk; in credit portfolio management; in pricing index tranches; and in market risk wherever joint tail losses matter. Each unit works one of these on data.
How is the course taught?
Audio lessons give the general subject, explainer films show defaults clustering under different copulas, and the workbooks hold derivations, simulation notebooks and cheat sheets. Assignments and the project receive written feedback from the faculty.
Who is this course for?
Quants heading for credit, risk or model validation roles. It counts toward the Certificate in Quantitative Risk Management and so toward the Fellowship in Computational Finance.
What you will be able to do
- Explain Sklar's theorem and build Gaussian, Student t and Archimedean copulas.
- Simulate correlated defaults and compute the loss distribution of a credit portfolio.
- Price CDO tranches with the one-factor Gaussian copula and read base correlation.
- Judge the model risk of copula methods using the 2008 experience.
Syllabus, module by module
Dependence beyond correlation
Why linear correlation is not enough, rank correlation and tail dependence.
Copulas
Sklar's theorem, Gaussian, Student t, Clayton and Gumbel copulas, and fitting them.
Correlated defaults
Latent-variable models and simulation of defaults in a portfolio.
The one-factor model
The Vasicek large-portfolio limit and its use in Basel capital.
Pricing tranches
Synthetic CDOs, the Gaussian copula and base correlation.
What went wrong
2008, correlation smiles and how to use copulas with honest model risk.
How the certificate is earned
Graded assignments and a final project: loss distribution and tranche prices for a credit portfolio under two copulas, with a written model-risk discussion. On a pass you receive the Course certificate in Copula Models for Credit Risk, issued by the Institute with a verification code. Read how certification works.
Before you start
Credit default swaps and probability at the Institute's level.
This course counts toward
- Certificate in Quantitative Risk ManagementA financial risk management certification in one semester of eleven courses and a capstone on measuring and governing market and credit risk: VaR and expected shortfall, volatility models, stress tests, credit and counterparty risk, bank capital and the validation of the models themselves.
- Fellowship in Computational FinanceThe top credential on the researcher path. Four semesters, 38 courses, and a dissertation supervised in writing and examined by two faculty members.
Questions applicants ask
What is a copula in finance?
A copula is a function that joins the separate distributions of several variables into a joint distribution, so that the dependence between them can be modelled on its own. In credit it describes how defaults cluster.
Did the Gaussian copula cause the financial crisis?
It did not cause it, but it was used beyond its limits. It understated the chance of many defaults together, and markets relied on single correlation numbers. The course uses that history to teach model risk.