Subsistence-(Threshold) payoff and truncated risk preferences

A measure of aversion to a risk akin to the risk premium is the required payoff truncation – a probability level, or a point of the distribution range – of the – null mean – risk distribution that allows an expected utility equal to the deterministic level. For a small risk – a noise of null expected value – added to the argument of an utility function, it is straight-forward to show that – for a risk-averse individual – such subsistence probability equals the conventional risk-premium over the symmetric of the worst possible outcome. However, both measures do not take into account aversion (or proneness) to risk in the utility distribution itself – they apply to expected utility maximizers. Maxmin behaviour and quantile preferences, applicable in the presence of uncertainty (or non-cooperative opponents) rather than risk, can be suggested to circumvent the problem. An alternative theory – constrained expected utility – relies on the use the expected utility over the upper truncated distribution (lower – or doubly truncated – in case of risk-loving behavior) at a given (individual specific) truncation point, or probability level. Then, a conventionally defined risk-premium weighs both the truncation bias and risk dispersion. Such distinction also applies if preference truncation – or rather, “trimming” – is (instead) accompanied by a switch of probability mass to tail “focal” points. Then, if the latter are sufficiently extreme, the effect on attitude towards risk may be reversed relative to standard preference truncation: lower trimming enhancing risk-aversion, upper one reducing it. Applications of truncated principles to mean-variance “utility” preferences – and risk-loving attitudes – were also briefly outlined. Illustrations for normal and uniform risks were often appended.