Correlation stress testing for value-at-risk: an unconstrained convex optimization approach
Correlation stress testing is employed in several financial models for determining the value-at-risk (VaR) of a financial institution’s portfolio. The possible lack of mathematical consistence in the target correlation matrix, which must be positive semidefinite, often causes breakdown of these models. The target matrix is obtained by fixing some of the correlations (often contained in blocks of submatrices) in the current correlation matrix while stressing the remaining to a certain level to reflect various stressing scenarios. The combination of fixing and stressing effects often leads to mathematical inconsistence of the target matrix. It is then naturally to find the nearest correlation matrix to the target matrix with the fixed correlations unaltered. However, the number of fixed correlations could be potentially very large, posing a computational challenge to existing methods. In this paper, we propose an unconstrained convex optimization approach by solving one or a sequence of continuously differentiable (but not twice continuously differentiable) convex optimization problems, depending on different stress patterns. This research fully takes advantage of the recently developed theory of strongly semismooth matrix valued functions, which makes fast convergent numerical methods applicable to the underlying unconstrained optimization problem. Promising numerical results on practical data (RiskMetrics database) and randomly generated problems of larger sizes are reported.
KeywordsStress testing Convex optimization Newton’s method Augmented Lagrangian functions
Unable to display preview. Download preview PDF.
- 1.Alexander, C.: Market Models: A Guide to Financial Data Analysis. Wiley, New York (2001) Google Scholar
- 5.Bhansali, V., Wise, B.: Forecasting portfolio risk in normal and stressed market. J. Risk 4(1), 91–106 (2001) Google Scholar
- 12.Fender, I., Gibson, M.S., Mosser, P.C.: An international survey of stress tests. Federal Reserve Bank of New York, Current Issues in Economics and Finance, vol. 7, No. 10 (2001) Google Scholar
- 13.Finger, C.: A methodology for stress correlation. In: Risk Metrics Monitor, Fourth Quarter (1997) Google Scholar
- 20.J.P. Morgan/Reuters: RiskMetrics™—Technical Document, 4th edn. New York (1996) Google Scholar
- 25.Rebonato, R., Jäckel, P.: The most general methodology for creating a valid correlation matrix for risk management and option pricing purpose. J. Risk 2(2), 17–27 (2000) Google Scholar
- 34.Turkay, S., Epperlein, E., Christofides, N.: Correlation stress testing for value-at-risk. J. Risk 5(4), 75–89 (2003) Google Scholar
- 35.Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory I and II. In: Zarantonello, E.H. (ed.) Contributions to Nonlinear Functional Analysis, pp. 237–424. Academic, New York (1971) Google Scholar