Correlation matrices with average constraints☆
Introduction
Correlation matrices are a crucial model component in many disciplines including statistics, risk management, finance and insurance. For instance, they appear in portfolio optimization models (e.g., Markowitz’ framework), models for setting capital requirements (e.g., Solvency II directive), and credit risk portfolio management (e.g., KMV’s credit risk portfolio framework). Clearly, it is often difficult to accurately estimate all pairwise correlations, i.e., full information is rarely available. For instance, in a credit risk context the pairwise default correlation between the default events of two obligors is essentially equivalent to the probability they default together. As defaults are rare events such joint probability is very hard to estimate with a reasonable degree of precision. Simulating correlation matrices that are consistent with all available, yet incomplete, information is then useful to provide insight into the sensitivity of the model used. Indeed, while pairwise correlations might not be (all) known, information might be available at a more aggregate level. Specifically, the average correlation might be known with sufficient degree of accuracy. In this paper, we aim to describe the set of all correlation matrices that are consistent with one or more average constraints. Specifically, we provide an algorithm to randomly simulate from this set.
We are not the first to describe sets of correlation matrices that satisfy some constraints. Marsaglia and Olkin (1984) discuss how to generate correlation matrices with given eigenvalues. Joe (2006) proposes a method to generate correlation matrices based on partial correlations, which was extended in Lewandowski et al. (2009) using a vine method. To account for the uncertainty on the input correlations, Hardin et al. (2013) propose to add in a controlled manner a noisy matrix to the input correlation matrix. Our paper is closest to the one of Hüttner and Mai (2019) who propose an algorithm to generate correlation matrices with Perron–Frobenius property. As this property is known to often hold in financial data series such algorithm is highly relevant for applications in finance. In this letter, we propose an algorithm to generate random correlation matrices with given average correlation. We extend the results to the case of multiple average constraints. To the best of our knowledge, we are the first to study this problem.
The rest of the letter is organized as follows. In Section 2, we develop an algorithm to generate correlation matrices with given average. Section 3 provides the extension to further account for multiple average constraints. We provide pseudo-code and numerical examples showing that our algorithms work well. Section 4 concludes the paper.
Section snippets
Problem formulation
Let be an integer. We denote by the set consisting of matrices in . A correlation matrix is a positive semidefinite matrix that is symmetric and which has ones on the diagonal. We denote by the standard inner product on , i.e., for all , . Furthermore, denotes the corresponding (Euclidean) norm, i.e., . In the sequel, we will also refer to the norm of a vector as its length. For , the so-called triangle inequalities are
Multiple average constraints
We now want to generate random correlation matrices with some blocks (sub-matrices) each satisfying an average constraint. To make this more precise, suppose that are all positive integers and that we are given an correlation matrix for each , where each satisfies an average constraint, as discussed in the previous section.
In this section, we specifically denote by the lower triangular matrix such that . The rows of each are denoted as vectors
Conclusion
In this letter, we propose an algorithm to generate correlation matrices subject to a given average correlation. We extend the algorithm to account for correlation matrices with block structures. Our algorithms are useful for stress-testing models. Specific applications to model risk assessment are considered in further research. Finally, we wish to point out that the problem of finding correlation matrices with given average that we consider in this paper is strongly related to the so-called
Acknowledgement
Jing Yao acknowledges the National Natural Science Foundation of China (No. 11971506).
References (15)
- et al.
Model uncertainty and VaR aggregation
J. Bank. Finance
(2013) Generating random correlation matrices based on partial correlations
J. Multivariate Anal.
(2006)- et al.
Generating random correlation matrices based on vines and extended onion method
J. Multivariate Anal.
(2009) - et al.
The complete mixability and convex minimization problems with monotone marginal densities
J. Multivariate Anal.
(2011) - et al.
Value-at-risk bounds with variance constraints
J. Risk Insurance
(2017) - et al.
Risk bounds for factor models
Finance Stoch.
(2017) - et al.
How robust is the value-at-risk of credit risk portfolios?
Eur. J. Finance
(2017)
Cited by (3)
Generic features in the spectral decomposition of correlation matrices
2021, Journal of Mathematical Physics
- ☆
All authors contribute to the paper equally.
- 1
Please contact me when you wish to obtain the R-code for the algorithms we provide in this letter.