The riskier the assets, the riskier the portfolios: pitfalls and misinterpretations.

As the dependence structure (i.e. the copula) among the assets is fixed, one might think that the riskier the assets, the riskier the portfolio. Surprisingly enough, this conjecture turns out to be false even for coherent risk measures and normal returns. We show that two conditions are able to preserve risk ordering under the portfolio: convexity for the risk measure and conditional increasingness for the copula. Eventually, conditional increasingness is checked for the most popular families of copulas used in financial modelling and actuarial sciences.