Actuarial Risk Matrices: The Nearest Positive Semidefinite Matrix Problem

The manner in which a group of insurance risks are interrelated is commonly presented via a correlation matrix. Actuarial risk correlation matrices are often constructed using output from disparate modeling sources and can be subjectively adjusted, for example, increasing the estimated correlation b...

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Vydáno v:North American actuarial journal Ročník 21; číslo 4; s. 552 - 564
Hlavní autoři: Cutajar, Stefan, Smigoc, Helena, O'Hagan, Adrian
Médium: Journal Article
Jazyk:angličtina
Vydáno: Routledge 02.10.2017
Taylor & Francis
ISSN:1092-0277, 2325-0453
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Abstract The manner in which a group of insurance risks are interrelated is commonly presented via a correlation matrix. Actuarial risk correlation matrices are often constructed using output from disparate modeling sources and can be subjectively adjusted, for example, increasing the estimated correlation between two risk sources to confer reserving prudence. Hence, while individual elements still obey the assumptions of correlation values, the overall matrix is often not mathematically valid (not positive semidefinite). This can prove problematic in using the matrix in statistical models. The first objective of this article is to review existing techniques that address the nearest positive semidefinite matrix problem in a very general setting. The chief approaches studied are Semidefinite Programming (SDP) and the Alternating Projections Method (APM). The second objective is to finesse the original problem specification to consider imposition of a block structure on the initial risk correlation matrix. This commonly employed technique identifies off-diagonal subsets of the matrix where values can or should be set equal to some constant. This may be due to similarity of the underlying risks and/or with the goal of increasing computational efficiency for processes involving large matrices. Implementation of further linear constraints of this nature requires adaptation of the standard SDP and APM algorithms. In addition, a new Shrinking Method is proposed to provide an alternative solution in the context of this increased complexity. "Nearness" is primarily considered in terms of two summary measures for differences between matrices: the Chebychev Norm (maximum element distance) and the Frobenius Norm (sum of squared element distances). Among the existing methods, adapted to function appropriately for actuarial risk matrices, APM is extremely efficient in producing solutions that are optimal in the Frobenius norm. An efficient algorithm that would return a positive semidefinite matrix that is optimal in Chebychev norm is currently unknown. However, APM is used to highlight the existence of matrices close to such an optimum and exploited, via the Shrinking Method, to find high-quality solutions. All methods are shown to work well both on artificial and real actuarial risk matrices provided under collaboration with Tokio Marine Kiln (TMK). Convergence speeds are calculated and compared and sample data and MATLAB code is provided. Ultimately the APM is identified as being superior in Frobenius distance and convergence speed. The Shrinking Method, building on the output of the APM algorithm, is demonstrated to provide excellent results at low computational cost for minimizing Chebychev distance.
AbstractList The manner in which a group of insurance risks are interrelated is commonly presented via a correlation matrix. Actuarial risk correlation matrices are often constructed using output from disparate modeling sources and can be subjectively adjusted, for example, increasing the estimated correlation between two risk sources to confer reserving prudence. Hence, while individual elements still obey the assumptions of correlation values, the overall matrix is often not mathematically valid (not positive semidefinite). This can prove problematic in using the matrix in statistical models. The first objective of this article is to review existing techniques that address the nearest positive semidefinite matrix problem in a very general setting. The chief approaches studied are Semidefinite Programming (SDP) and the Alternating Projections Method (APM). The second objective is to finesse the original problem specification to consider imposition of a block structure on the initial risk correlation matrix. This commonly employed technique identifies off-diagonal subsets of the matrix where values can or should be set equal to some constant. This may be due to similarity of the underlying risks and/or with the goal of increasing computational efficiency for processes involving large matrices. Implementation of further linear constraints of this nature requires adaptation of the standard SDP and APM algorithms. In addition, a new Shrinking Method is proposed to provide an alternative solution in the context of this increased complexity. "Nearness" is primarily considered in terms of two summary measures for differences between matrices: the Chebychev Norm (maximum element distance) and the Frobenius Norm (sum of squared element distances). Among the existing methods, adapted to function appropriately for actuarial risk matrices, APM is extremely efficient in producing solutions that are optimal in the Frobenius norm. An efficient algorithm that would return a positive semidefinite matrix that is optimal in Chebychev norm is currently unknown. However, APM is used to highlight the existence of matrices close to such an optimum and exploited, via the Shrinking Method, to find high-quality solutions. All methods are shown to work well both on artificial and real actuarial risk matrices provided under collaboration with Tokio Marine Kiln (TMK). Convergence speeds are calculated and compared and sample data and MATLAB code is provided. Ultimately the APM is identified as being superior in Frobenius distance and convergence speed. The Shrinking Method, building on the output of the APM algorithm, is demonstrated to provide excellent results at low computational cost for minimizing Chebychev distance.
Author Cutajar, Stefan
Smigoc, Helena
O'Hagan, Adrian
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  organization: School of Mathematics and Statistics, University College Dublin
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10.1080/01621459.1983.10477029
10.1023/A:1018363021404
10.1017/CBO9780511810817
10.1137/090776718
10.1137/1022058
10.1093/imanum/22.3.329
10.1098/rspl.1895.0041
10.1017/9780511811487
10.1137/0909059
10.1007/BF01582221
10.1080/01630569408816580
10.1080/10556789908805766
10.2307/1412159
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References Deutsch F. (cit0008) 1995; 454
cit0011
cit0022
Mishra S. K. (cit0021) 2007
cit0010
(cit0001) 2015
Von Neumann J. (cit0027) 1950
Luenberger D. G. (cit0019) 1969
Golub G. (cit0012) 1996
Boyle J. P. (cit0003) 1986; 37
Cantrell C. D. (cit0006) 2000
cit0009
cit0017
cit0007
cit0018
Shih-Ping H. (cit0023) 1988; 40
Burtschell X. (cit0004) 2005
cit0026
cit0005
cit0002
cit0013
cit0024
cit0025
References_xml – year: 2005
  ident: cit0004
  publication-title: ISFA Actuarial School and BNP Parisbas
– ident: cit0026
  doi: 10.1017/S0962492901000071
– volume: 454
  start-page: 107
  year: 1995
  ident: cit0008
  publication-title: NATO ASI Series C Mathematical and Physical Sciences-Advanced Study Institute
– volume-title: The MOSEK Optimization Toolbox for MATLAB Manual. Version 7.1 (Revision 28).
  year: 2015
  ident: cit0001
– volume: 37
  start-page: 4
  issue: 28
  year: 1986
  ident: cit0003
  publication-title: Lecture Notes in Statistics
– ident: cit0010
  doi: 10.1080/01621459.1983.10477029
– ident: cit0018
  doi: 10.1023/A:1018363021404
– ident: cit0017
  doi: 10.1017/CBO9780511810817
– ident: cit0002
  doi: 10.1137/090776718
– year: 1950
  ident: cit0027
  publication-title: Functional Operators: Measures and Integrals (Vol. 1)
– ident: cit0011
  doi: 10.1137/1022058
– ident: cit0013
  doi: 10.1093/imanum/22.3.329
– ident: cit0022
  doi: 10.1098/rspl.1895.0041
– volume: 40
  start-page: 1
  year: 1988
  ident: cit0023
  publication-title: Mathematical Programming
– volume-title: Modern Mathematical Methods for Physicists and Engineers
  year: 2000
  ident: cit0006
  doi: 10.1017/9780511811487
– ident: cit0005
  doi: 10.1137/0909059
– year: 2007
  ident: cit0021
  publication-title: Available at SSRN 980403
– ident: cit0007
  doi: 10.1007/BF01582221
– ident: cit0009
  doi: 10.1080/01630569408816580
– ident: cit0025
  doi: 10.1080/10556789908805766
– volume-title: Matrix Computations
  year: 1996
  ident: cit0012
– ident: cit0024
  doi: 10.2307/1412159
– volume-title: Optimization by Vector Space Methods
  year: 1969
  ident: cit0019
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Snippet The manner in which a group of insurance risks are interrelated is commonly presented via a correlation matrix. Actuarial risk correlation matrices are often...
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Title Actuarial Risk Matrices: The Nearest Positive Semidefinite Matrix Problem
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