  Risk diversification is the basis of insurance and investment. It is thus
crucial to study the effects that could limit it. One of them is the existence
of systemic risk that affects all the policies at the same time. We introduce
here a probabilistic approach to examine the consequences of its presence on
the risk loading of the premium of a portfolio of insurance policies. This
approach could be easily generalized for investment risk. We see that, even
with a small probability of occurrence, systemic risk can reduce dramatically
the diversification benefits. It is clearly revealed via a non-diversifiable
term that appears in the analytical expression of the variance of our models.
We propose two ways of introducing it and discuss their advantages and
limitations. By using both VaR and TVaR to compute the loading, we see that
only the latter captures the full effect of systemic risk when its probability
to occur is low
