Weighted Quasi-Arithmetic Means and a Risk Index for Stochastic Environments

In this paper, the weighted quasi-arithmetic means are discussed from the viewpoint of utility functions and downward risks in economics. Representing the weighting functions by probability density functions and the conditional expectations, an index for downward risks in stochastic environments is derived. This paper discusses the relation among the index, the first-order stochastic dominance and the risk premium in economics, and further it investigates the relation between the index and value-at-risks which are known as another estimation for downward risks in finance. Finally, this paper shows a lot of examples of the weighted quasi-arithmetic mean and the aggregated mean ratio for various typical utility functions with various typical utility functions and probability density functions.