Chance-constrained portfolio selection

Chance-constrained portfolio selection is an approach to portfolio selection under loss aversion.

The formulation assumes that (i) investor's preferences are representable by the expected utility of final wealth, and that (ii) they require that the probability of their final wealth falling below a survival or safety level must to be acceptably low.

The chance-constrained portfolio problem is then to find:

:Max $\backslash sum\_\{j\}$w_jE(X_j), subject to Pr($\backslash sum\_\{j\}$ w_jX_j < s) ≤ {{math|α}}, $\backslash sum\_\{j\}$w_j = 1, w_j ≥ 0 for all j,

::where s is the survival level and {{math|α}} is the admissible probability of ruin; w is the weight and x is the value of the jth asset to be included in the portfolio.

The original implementation is based on the seminal work of Abraham Charnes and William W. Cooper on chance constrained programming in 1959,A. Chance and W. W. Cooper (1959), "Chance-Constrained Programming," Management Science, 6, No. 1, 73-79. [https://www.jstor.org/stable/2627476?Search=yes&resultItemClick=true&searchText=((Management%20Science)%20AND%20(Charnes%20and%20cooper))%20AND%20(Chance%20Constrained%20programming)&searchUri=%2Faction%2FdoBasicSearch%3FQuery%3DChance%2BConstrained%2Bprogramming%26cty_journal_facet%3Dam91cm5hbA%253D%253D%26prq%3D(Management%2BScience)%2BAND%2B(Charnes%2Band%2Bcooper)%26swp%3Don&ab_segments=0%2Fbasic_search_SYC-5455%2Fcontrol&refreqid=fastly-default%3A78e869b443ea6c0e104737db22b5a854]. Retrieved September 24, 2020

and was first applied to finance by Bertil Naslund and Andrew B. Whinston in 1962Naslund, B. and A. Whinston (1962), "A Model of Multi-Period Investment under Uncertainty," Management Science, 8, No. 2, 184-200. [https://www.jstor.org/stable/2627501?Search=yes&resultItemClick=true&searchText=(management%20science)%20AND%20(Naslund%20Bertil)&searchUri=%2Faction%2FdoBasicSearch%3FQuery%3DNaslund%2BBertil%26prq%3Dmanagement%2Bscience%26swp%3Don&ab_segments=0%2Fbasic_search_SYC-5455%2Fcontrol&refreqid=fastly-default%3A99711b4afe59bd073fa1d58b5788eec2] Retrieved September 24, 2020.

and in 1969 by N. H. Agnew, et al.Agnew, N.H, R.A. Agnes, J. Rasmussen and K. R. Smith (1969), "An Application of Chance-Constrained Programming to Portfolio Selection in a Casualty Insurance Firm," Management Science, 15, No. 10, 512-520. [https://www.jstor.org/stable/2628459?Search=yes&resultItemClick=true&searchText=an%20application%20of%20chance%20constrained%20programming&searchUri=%2Faction%2FdoBasicSearch%3FQuery%3Dan%2Bapplication%2Bof%2Bchance%2Bconstrained%2Bprogramming&ab_segments=0%2Fbasic_search_SYC-5455%2Fcontrol&refreqid=fastly-default%3Ac66a912b771798f92fc98226d695291d]. Retrieved September 24, 2020.

For fixed {{math|α}} the chance-constrained portfolio problem represents lexicographic preferences and is an implementation of capital asset pricing under loss aversion.

In general though, it is observedBorch, K. H. (1968), The Economics of Uncertainty, Princeton University Press, Princeton. [https://press.princeton.edu/books/hardcover/9780691649313/the-economics-of-uncertainty-psme-2-volume-2]. Retrieved September 24, 2020. that no utility function can represent the preference ordering of chance-constrained programming because a fixed {{math|α}} does not admit compensation for a small increase in {{math|α}} by any increase in expected wealth.

For a comparison to mean-variance and safety-first portfolio problems, see;Seppälä, J. (1994), “The diversification of currency loans: A comparison between safety-first and mean-variance criteria,” European Journal of Operations Research, 74, 325-343. [https://www.sciencedirect.com/science/article/pii/0377221794901007]. Retrieved September 25, 2020. for a survey of solution methods here, see;Bay, X., X. Zheng and X. Sun (2012), "A survey on probabilistic constrained optimization problems," Numerical Algebra, Control and Optimization, 2, No. 4, 767-778. [https://www.aimsciences.org/article/doi/10.3934/naco.2012.2.767]. Retrieved September 25, 2020. for a discussion of the risk aversion properties of chance-constrained portfolio selection, see.Pyle, D. H. and Stephen J. Turnovsky (1971), “Risk Aversion in Chance Constrained Portfolio Selection, Management Science,18, No. 3, 218-225.[https://www.jstor.org/action/doBasicSearch?Query=Pyle+and+Turnovsky&prq=management+science&swp=on]. Retrieved September 24, 2020.