Explicit solutions to correlation matrix completion problems, with an application to risk management and insurance

We derive explicit solutions to the problem of completing a partially specified correlation matrix. Our results apply to several block structures for the unspecified entries that arise in insurance and risk management, where an insurance company with many lines of business is required to satisfy certain capital requirements but may have incomplete knowledge of the underlying correlation matrix. Among the many possible completions, we focus on the one with maximal determinant. This has attractive properties and we argue that it is suitable for use in the insurance application. Our explicit formulae enable easy solution of practical problems and are useful for testing more general algorithms for the maximal determinant correlation matrix completion problem.