View source for Chance-constrained portfolio selection

'''Chance-constrained portfolio selection''' is an approach to [[Finance:Portfolio optimization|portfolio selection]] under [[Philosophy:Loss aversion|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 analysis|survival or safety level]] must to be acceptably low.
The chance-constrained portfolio problem is then to find:
:Max <math>\sum_{j}</math>w<sub>j</sub>E(X<sub>j</sub>), subject to Pr(<math>\sum_{j}</math> w<sub>j</sub>X<sub>j</sub> < s) ≤ {{math|''&alpha;''}}, <math>\sum_{j}</math>w<sub>j</sub> = 1, w<sub>j</sub> ≥ 0 for all j,
::where '''s''' is the survival level and {{math|''&alpha;''}} is the admissible [[Ruin theory|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 [[Stochastic programming|stochastic programming]] in 1959,<ref>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</ref>
and was first applied to finance by Bertil Naslund and [[Biography:Andrew B. Whinston|Andrew B. Whinston]] in 1962<ref>Naslund, 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.</ref> 
and in 1969 by N. H. Agnew, et al.<ref>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.</ref>

For fixed {{math|''&alpha;''}} the chance-constrained portfolio problem represents [[Finance:Lexicographic preferences|lexicographic preferences]] and is an implementation of [[Finance:Capital asset pricing model|capital asset pricing]] under loss aversion.
In general though, it is observed<ref>Borch, 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.</ref> that no utility function can represent the preference ordering of chance-constrained programming because a fixed {{math|''&alpha;''}} does not admit compensation for a small increase in {{math|''&alpha;''}} by any increase in expected wealth.

For a comparison to [[Finance:Markowitz model|mean-variance]] and [[Finance:Roy's safety-first criterion|safety-first portfolio]] problems, see;<ref>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.</ref> for a survey of solution methods here, see;<ref>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.</ref> for a discussion of the [[Risk aversion|risk aversion]] properties of chance-constrained portfolio selection, see.<ref>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.</ref>

==See also==
*[[Finance:Capital asset pricing model|Capital asset pricing model]]
*Expected utility theory
*[[Finance:Lexicographic preferences|Lexicographic preferences]]
*[[Philosophy:Loss aversion|Loss aversion]]
*[[Finance:Portfolio optimization|Portfolio optimization]]
*Post modern portfolio theory
*[[Stochastic programming]]

==References==
<references />

[[Category:Portfolio theories]]
[[Category:Stochastic optimization]]
[[Category:Financial economics]]
[[Category:Actuarial science]]

{{Sourceattribution|Chance-constrained portfolio selection}}