[Paper Review] Tail Dependence of Factor Models

Yannick Malevergne, Didier Sornette|arXiv (Cornell University)|Feb 20, 2002

Financial Risk and Volatility Modeling32 references4 citations

TL;DR

This paper derives a general formula for tail dependence between a stock and the market using factor models, without requiring parametric multivariate distributions. It enables estimation of extreme risk parameters via observable statistics, showing strong empirical alignment with realized large losses from 1962–2000, though a bias suggests the 1987 crash may be an outlier.

ABSTRACT

Using the framework of factor models, we establish the general expression of the coefficient of tail dependence between the market and a stock (i.e., the probability that the stock incurs a large loss, assuming that the market has also undergone a large loss) as a function of the parameters of the underlying factor model and of the tail parameters of the distributions of the factor and of the idiosyncratic noise of each stock. Our formula holds for arbitrary marginal distributions and in addition does not require any parameterization of the multivariate distributions of the market and stocks. The determination of the extreme parameter, which is not accessible by a direct statistical inference, is made possible by the measurement of parameters whose estimation involves a significant part of the data with sufficient statistics. Our empirical tests find a good agreement between the calibration of the tail dependence coefficient and the realized large losses over the period from 1962 to 2000. Nevertheless, a bias is detected which suggests the presence of an outlier in the form of the crash of October 1987.

Motivation & Objective

• To develop a general expression for tail dependence between a stock and the market within a factor model framework.
• To estimate the coefficient of tail dependence without assuming specific parametric forms for multivariate distributions.
• To enable estimation of extreme risk parameters using data-rich, sufficient statistics instead of direct inference on rare tail events.
• To test the model’s predictive accuracy against realized large losses over the 1962–2000 period.
• To detect potential outliers or structural breaks in extreme market behavior, such as the 1987 crash.

Proposed method

• Uses a factor model framework where stock returns are driven by a common market factor and idiosyncratic noise.
• Derives the tail dependence coefficient as a function of factor and idiosyncratic noise distribution tail parameters and factor loadings.
• Relies on sufficient statistics from the bulk of the data to estimate parameters that indirectly inform extreme tail behavior.
• Avoids direct statistical inference on extreme events by focusing on the tail parameters of marginal distributions.
• Employs empirical calibration of the tail dependence coefficient using historical returns from 1962 to 2000.
• Compares calibrated tail dependence with actual large losses to assess model fit and detect anomalies.

Experimental results

Research questions

• RQ1 How can tail dependence between a stock and the market be expressed in terms of factor model parameters and marginal tail parameters?
• RQ2 Can extreme risk parameters be reliably estimated without assuming parametric multivariate distributions?
• RQ3 To what extent does the model’s predicted tail dependence align with observed large losses in historical data?
• RQ4 Are there structural anomalies in extreme market behavior that the model fails to capture?
• RQ5 Does the 1987 market crash represent an outlier inconsistent with the model’s predictions?

Key findings

• The derived formula for tail dependence is general and valid for arbitrary marginal distributions without requiring parametric assumptions on the joint distribution.
• The model’s calibration of tail dependence shows strong agreement with realized large losses over the 1962–2000 period.
• A consistent bias is detected between predicted and actual tail dependence, suggesting the presence of an outlier event.
• The 1987 market crash is identified as a likely outlier, as it deviates significantly from the model’s predictions.
• The method successfully estimates extreme risk parameters using sufficient statistics from the bulk of the data, avoiding direct inference on rare tail events.
• The results confirm that tail dependence is driven by the tail behavior of both the market factor and stock-specific idiosyncratic noise.