Risk measure and management of insurance and financial portfolios

Abstract

This thesis discusses two crucial problems in modern actuarial science and finance, i.e., the evaluation problem of finite-time ruin probability in risk theory and the estimation problem of large-dimensional covariance matrix in portfolio management. The first part of this thesis focuses on the study of finite-time ruin prob- ability in the risk model driven by a Lévy subordinator by incorporating the popular Fourier-cosine method. To propose a general approximation for any specified precision provided that the characteristic function of the Lévy Pro- cess is known, an explicit integral expression, expressed in terms of the density function and the survival function, is derived for the finite-time ruin prob- ability. Moreover, the rearrangement inequality is utilized to improve the approximations further. In addition, with only mild and practically relevant iii assumptions, this thesis proves that the approximation error can be made ar- bitrarily small (actually an algebraic convergence rate up to 3, which is the fastest possible approximant known upon all in the literature), and the ap- proximation has a linear computation complexity in a number of terms of the Fourier-cosine expansion. The effectiveness of the results is demonstrated in various numerical studies; through these examples, the supreme power of the Fourier-cosine method is shown once. In the second part, a large-dimensional covariance matrix estimation prob- lem in multivariate analysis is studied under the case that the number of variables and sample size both go to infinity with a positive ratio. In the limiting case, the optimal shrinkage estimator for the Minimum Variance loss function can be expressed in terms of the Stieltjes transform of an unknown limiting empirical spectral distribution. To estimate this Stieltjes transform, an imaginary direction smoothed estimator is proposed, which turns out to coincide with a kernel-based estimation using Cauchy density. With the help of anisotropic local law in Random Matrix Theory, this thesis shows that the smoothed estimator converges to the original one uniformly in probability. Moreover, the maximum order of bandwidth choice is extended from 2/5 to one slightly less than the upper limit 1 in theory, and a corresponding local bandwidth selection approach is proposed in practice. Based on this smoothed estimator, this thesis further studies the spectral density estimation as a di- rect corollary, investigates the population moments estimation by deriving a iv

recursive formula for the high-order moments, discusses the covariance ma- trix estimation under various loss functions, and explores multivariate mean estimation under a slightly modified model setting. Finally, extensive Monte Carlo simulations are conducted to show the robustness and efficiency of the proposed estimators.