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Does Risk Diversification Always Work? The Answer Through Simple Modelling

With a simple example of throwing a die, we show how to price an insurance policy. We further study how this price decreases when many similar policies are sold. The diversification benefits increase with the number of policies and similarly the risk loading of the premium required for the risk decreases tending to zero. This is true as long as the risks are completely independent. However, when introducing in addition a biased die played by a crooked croupier, a non-diversifiable risk does appear. Indeed, we can show analytically that, with the biased die, there exists an additional term in the variance, which does not decrease with the number of policies in the portfolio and leads to a limit to diversification. We propose and study analytically three cases of introducing the non-diversifiable risk. For each of them, the behavior of the risk loading based on the underlying risk process is examined and a numerical illustration is provided. Then the results are discussed in view of the risk loading. Such a modelling could be used to study particular investment choices under uncertainty.