Since the financial crisis of 2007-2009, there has been a renewed interest toward quantifying more appropriately the risks involved in financial positions. Popular risk measures such as variance and value-at-risk have been found inadequate as we now give more importance to properties such as monotonicity, convexity, translation invariance, scale invariance, and law invariance. Unfortunately, the challenge remains that it is unclear how to choose a risk measure that faithfully represents the decision maker's true risk attitude. In this work, we show that one can account precisely for (neither more nor less than) what we know of the risk preferences of an investor/policy maker when comparing and optimizing financial positions. We assume that the decision maker can commit to a subset of the above properties (the use of a law invariant convex risk measure for example) and that he can provide a series of assessments comparing pairs of potential risky payoffs. Given this information, we propose to seek financial positions that perform best with respect to the most pessimistic estimation of the level of risk potentially perceived by the decision maker. We present how this preference robust risk minimization problem can be solved numerically by formulating convex optimization problems of reasonable size. Numerical experiments on a portfolio selection problem, where the problem reduces to a linear program, will illustrate the advantage of accounting for the fact that the information about risk perception is limited.
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