### Subsistence-(Threshold) payoff and truncated risk preferences

#### Abstract

**Abstract.** 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.

**Keywords.** Subsistence-payoff; Non-expected utility theories; Constrained expected utility; Truncated preferences towards risk; Maxmin, maxmax; Trimmed preferences towards risk; Focal points; Mean – variance(-utility) preferences; “Trimmed” normal (with tail focal points) distribution; Triangular distribution; Triangular preferences.

**JEL.**D81; C10; C16; C24; D11.

#### Keywords

#### References

Arrow, K.J. (1965). *Some Aspects of the Theory of Risk-Bearing.* Yrjo Jahnssonin Saatio, Helsinki.

Black, F. (1972). Capital market equilibrium with restricted borrowing. *Journal of Business.* 45, 444-455.

Drèze, J.H. (1987). *Essays on Economic Decisions under Uncertainty. *Cambridge: Cambridge University Press.

Eichner, T., and Wagener, A. (2003). Variance vulnerability, background risks, and mean-variance preferences. *The Geneva Papers on Risk and Insurance Theory,* 28, 173-184. doi. 10.1023/A:1026396922206

Gollier, C. (2001). *The Economics of Risk and Time. *Cambridge: Massachusetts Institute of Technology.

Hirshleifer, J., and Riley, J.G. (1992). *The Analytics of Uncertainty and Information. *Cambridge: Cambridge University Press.

Johnson, N.L. and Kotz, S. (1970). *Continuous Univariate Distributions - 1.* New York: Wiley.

Johnson, N.L. and Kotz, S. (1970a). *Continuous Univariate Distributions - 2.* New York: Wiley.

Karni, E., and Schmeidler, D. (1991). Utility Theory with Uncertainty. In *Handbook of Mathematical Economics. *Vol. 4. Edited by Werner Hildenbrand and Hugo Sonnenschein. Amsterdam: North-Holland.

Kelsey, D., and Quiggin, J. (1992). Theories of choice under ignorance and uncertainty. *Journal of Economic Surveys.* 6(2), 133-153. doi. 10.1111/j.1467-6419.1992.tb00148.x

Laffont, J.-J. (1989). *The Economics of Uncertainty and Information. *Cambridge: The Massachusetts Institute of Technology.

Lajeri-Chaherli, F. (2002). More on properness: The case of mean-variance preferences. *The Geneva Papers on Risk and Insurance Theory.* 27, 49-60.

Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. *Review of Economics and Statistics.* 47, 13-37. doi. 10.2307/1924119

Markowitz, H. (1959). *Portfolio Selection. *Cowles Foundation Monograph N. 16. New York: John Wiley & Sons, Inc.

Martins, A.P. (2004). Multivariate risk-exposure: Risk-premium, optimal decisions and mean-variance implications. Unpublished manuscript.

Ormiston, M.B. and Schlee, E.E. (2001). Mean-variance preferences and investor behaviour. *The Economic Journal.* 111, 849-861. doi. 10.1111/1468-0297.00662

Pratt, J.W. (1964). Risk aversion in the small and in the large. *Econometrica.* 32, 122-136. doi. 10.2307/1913738

Rieskamp, J., Busemeyer, J.R., and Mellers, B.A. (2006). Extending the bounds of rationality: Evidence and theories of preferential choice. *Journal of Economic Literature.* 44(3), 631-661. doi. 10.1257/jel.44.3.631

Sharpe, W. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. *Journal of Finance.* 19, 425-442. doi. 10.1111/j.1540-6261.1964.tb02865.x

Starmer, C. (2000). Developments in non-expected utility theory. *Journal of Economic Literature.* 38(2), 332-382. doi. 10.1257/jel.38.2.332

Tobin, J. (1958). Liquidity preference as behavior towards risk. *Review of Economic Studies.* 25, 68-85. doi. 10.2307/2296205

DOI: http://dx.doi.org/10.1453/ter.v10i1-2.2447

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