Reducing risk by merging counter-monotonic risks Ka Chun Cheung ∗ , † Jan Dhaene ‡ , Ambrose Lo § , Qihe Tang ¶ Abstract In this article, we show that some important implications concerning comonotonic couples and corresponding convex order relations for their sums cannot be translated to counter-monotonicity in general. In a financial context, it amounts to saying that merging counter-monotonic positions does not necessarily reduce the overall level of risk. We pro- pose a simple necessary and sufficient condition for such a merge to be effective. Natural interpretations and various characterizations of this condition are given. As applications, we develop cancellation laws for convex order and identify desirable structural properties of insurance indemnities that make an insurance contract universally marketable, in the sense that it is appealing to both the policyholder and the insurer. KEYWORDS: comonotonicity; counter-monotonicity; convex order; Tail Value-at- Risk 1 Introduction Consider a portfolio with random value X at the end of a given reference period. In order to have a better control of the risk involved, it is often desirable that an additional asset Z is added to the position X so that the overall risk is reduced in the sense that ρ(X + Z ) ≤ ρ(X + E(Z )), where ρ is an appropriate risk measure. Commonly used risk measures include Value-at-Risk, Tail Value-at-Risk (TVaR), and other distortion risk measures. We refer to Denuit et al. (2005), * Corresponding author. † Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. E-mail: kccg@hku.hk ‡ KU Leuven, Department of Accountancy, Finance and Insurance, Naamsestraat 69, B-3000 Leuven, Bel- gium. E-mail: Jan.Dhaene@kuleuven.be § Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. E-mail: amblo@hku.hk ¶ Department of Statistics and Actuarial Science, The University of Iowa, 241 Schaeffer Hall, Iowa City, IA 52242, USA. E-mail: qihe-tang@uiowa.edu 1