Editing Financial economics (section)

===Present value, expectation and utility===
Underlying all of financial economics are the concepts of [[present value]] and [[Expected value|expectation]].<ref name="Rubinstein"/>

Calculating their present value, <math>X_{sj}/r</math> in the first formula, allows the decision maker to aggregate the [[cashflow]]s (or other returns) to be produced by the asset in the future to a single value at the date in question, and to thus more readily compare two opportunities; this concept is then the starting point for financial decision making.{{NoteTag|Its history is correspondingly early: [[Fibonacci]] developed the concept of present value already in 1202 in his ''[[Liber Abaci]]''. [[Compound interest]] was discussed in depth by [[Richard Witt]] in 1613, in his ''Arithmeticall Questions'',<ref>C. Lewin (1970). [https://www.actuaries.org.uk/system/files/documents/pdf/0121-0132.pdf An early book on compound interest] {{Webarchive|url=https://web.archive.org/web/20161221163926/https://www.actuaries.org.uk/system/files/documents/pdf/0121-0132.pdf |date=2016-12-21 }}, Institute and Faculty of Actuaries</ref> and was further developed by [[Johan de Witt]] in 1671 <ref>James E. Ciecka. 2008. [https://fac.comtech.depaul.edu/jciecka/deWitt.pdf "The First Mathematically Correct Life Annuity"]. Journal of Legal Economics 15(1): pp. 59-63</ref> and by [[Edmond Halley]] in 1705.<ref>James E. Ciecka (2008). [https://fac.comtech.depaul.edu/jciecka/Halley.pdf "Edmond Halley’s Life Table and Its Uses"]. ''Journal of Legal Economics'' 15(1): pp. 65-74.</ref>}}
(Note that here, "<math>r</math>" represents a generic (or arbitrary) [[Discounted cash flow#Discount rate|discount rate]] applied to the cash flows, whereas in the valuation formulae, the [[risk-free rate]] is applied once these have been "adjusted" for their riskiness; see below.)

An immediate extension is to combine probabilities with present value, leading to the [[Expected value|expected value criterion]] which sets asset value as a function of the sizes of the expected payouts and the probabilities of their occurrence, <math>X_{s}</math> and <math>p_{s}</math> respectively.{{NoteTag|These ideas originate with [[Blaise Pascal]] and [[Pierre de Fermat]] in 1654; see {{slink|Problem of points#Pascal and Fermat}}.}}

This decision method, however, fails to consider [[risk aversion]]. In other words, since individuals receive greater [[Utility#Applications|utility]] from an extra dollar when they are poor and less utility when comparatively rich, the approach is therefore to "adjust" the weight assigned to the various outcomes, i.e. "states", correspondingly: <math>Y_{s}</math>. See [[indifference price]]. (Some investors may in fact be [[risk seeking]] as opposed to [[Risk aversion|risk averse]], but the same logic would apply.)

Choice under uncertainty here may then be defined as the maximization of [[expected utility]]. More formally, the resulting [[expected utility hypothesis]] states that, if certain axioms are satisfied, the [[Subjective theory of value|subjective]] value associated with a gamble by an individual is ''that individual''{{'}}s [[Expected value|statistical expectation]] of the valuations of the outcomes of that gamble.

The impetus for these ideas arises from various inconsistencies observed under the expected value framework, such as the [[St. Petersburg paradox]] and the [[Ellsberg paradox]].{{NoteTag|The development here is originally due to [[Daniel Bernoulli]] in 1738; it was later formalized by [[John von Neumann]] and [[Oskar Morgenstern]] in 1947.}}