Editing Finance (section)

==Financial theory==
{| class="wikitable floatright" | width="250"
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|{{smalldiv|
:<math>
\sum_{t=1}^n \frac{FCFF_t}{(1+WACC_{t})^t} +
 \frac{\left[\frac{FCFF_{n+1}}{(WACC_{n+1}-g_{n+1})}\right]}{(1+WACC_{n})^n}
</math>
}}
{{small|[[Valuation using discounted cash flows#Basic formula for firm valuation using DCF model|DCF valuation formula]] widely applied in business and finance, since articulated [[The Theory of Investment Value|in 1938]]. Here, to get the [[business valuation|value of the firm]], its forecasted [[free cash flow]]s are discounted to the present using the [[weighted average cost of capital]] for the discount factor.
For [[stock valuation|share valuation]] investors use the related [[dividend discount model]].}}
|}
Financial theory is studied and developed within the disciplines of [[Management#Training and education|management]], [[financial economics|(financial)]] [[economics]], [[accountancy]] and [[applied mathematics]].
In the abstract,<ref name = "Drake_Fabozzi"/><ref name= "Fama and Miller"/> ''finance'' is concerned with the investment and deployment of [[asset]]s and [[Liability (financial accounting)|liabilities]] over "space and time"; i.e., it is about performing [[valuation (finance)|valuation]] and [[asset allocation]] today, based on the risk and uncertainty of future outcomes while appropriately incorporating the [[time value of money]].
Determining the [[present value]] of these future values, "discounting", must be at the [[required rate of return|risk-appropriate discount rate]], in turn, a major focus of finance-theory.<ref name="freedictionary">[https://financial-dictionary.thefreedictionary.com/finance "Finance"] {{Webarchive|url=https://web.archive.org/web/20191222201948/https://financial-dictionary.thefreedictionary.com/Finance |date=2019-12-22 }} Farlex Financial Dictionary. 2012</ref><!-- Participants in the [[market (economics)|market]] aim to price assets based on their risk level, fundamental value, and their expected [[rate of return]].
The core finance theories can be divided between the following categories: [[financial economics]], [[mathematical finance]] and [[Valuation (finance)|valuation theory]]. 
-->As financial theory has roots in many disciplines, including mathematics, statistics, economics, physics, and psychology, it can be considered a mix of an [[art]] and [[science]],<ref name=InvPediaHayes/> and there are ongoing related efforts to organize a [[list of unsolved problems in finance]].

===Managerial finance===
<!-- {{Unreferenced section|date=June 2023}} -->
[[Image:Manual decision tree.jpg|right|thumb|[[Decision tree]]s, a more sophisticated valuation-approach, sometimes applied to corporate finance [[Corporate finance#Investment and project valuation|"project" valuations]] (and a standard<ref>A. Pinkasovitch (2021). [https://www.investopedia.com/articles/financial-theory/11/decisions-trees-finance.asp Using Decision Trees in Finance] {{Webarchive|url=https://web.archive.org/web/20211210085522/https://www.investopedia.com/articles/financial-theory/11/decisions-trees-finance.asp |date=2021-12-10 }}</ref> in [[business school]] curricula); various scenarios are considered, and their discounted cash flows are probability weighted.]]
{{Main|Managerial finance}}
Managerial finance<ref name ="cfi">[https://corporatefinanceinstitute.com/resources/knowledge/finance/managerial-finance/ What is managerial finance?] {{Webarchive|url=https://web.archive.org/web/20220704085037/https://corporatefinanceinstitute.com/resources/knowledge/finance/managerial-finance/ |date=4 July 2022 }}, [[Corporate Finance Institute]]</ref>
is the branch of finance that deals with the financial aspects of the [[management]] of a company, and the financial dimension of managerial decision-making more broadly. It provides the [[Management#Theoretical scope|theoretical underpin]] for the practice [[#Corporate finance|described above]], concerning itself with the [[Management#Implementation of policies and strategies|managerial application]] of the various [[financial analysis|finance techniques]]. Academics working in this area are typically based in [[business school]] finance departments, in [[accounting]], or in [[management science]].

The tools addressed and developed relate in the main to [[managerial accounting]] and [[corporate finance]]:
the former allow management to better understand, and hence act on, financial information relating to [[profitability]] and performance; the latter, as above, are about optimizing the overall financial structure, including its impact on working capital. Key aspects of managerial finance thus include:
# Financial planning and forecasting
# Capital budgeting
# Capital structure
# Working capital management
# Risk management
# Financial analysis and reporting.
The discussion, however, extends to [[business strategy]] more broadly, emphasizing alignment with the company's overall strategic objectives; and similarly incorporates the [[Management#Theoretical scope|managerial perspectives]] of planning, directing, and controlling.

===Financial economics===
[[File:markowitz frontier.jpg|right|thumb|The "[[efficient frontier]]", a prototypical concept in portfolio optimization. Introduced [[Markowitz model|in 1952]], it remains "a mainstay of investing and finance".<ref>W. Kenton (2021). [https://www.investopedia.com/terms/h/harrymarkowitz.asp "Harry Markowitz"] {{Webarchive|url=https://web.archive.org/web/20211126113108/https://www.investopedia.com/terms/h/harrymarkowitz.asp |date=2021-11-26 }}, investopedia.com</ref>
An "efficient" portfolio, i.e. combination of assets, has the best possible expected return for its level of risk (represented by the standard deviation of return).]]
[[Image:MM2.png|thumb|right|[[Modigliani–Miller theorem]], a foundational element of finance theory, introduced in 1958; it forms the basis for modern thinking on [[capital structure]]. Even if [[leverage (finance)|leverage]] ([[Debt to equity ratio|D/E]]) increases, the [[weighted average cost of capital]] (k0) stays constant.]]
{{Main|Financial economics}}
Financial economics<ref name="Sharpe">For an overview, see [http://www.stanford.edu/~wfsharpe/mia/int/mia_int2.htm "Financial Economics"] {{Webarchive|url=https://web.archive.org/web/20040604105441/http://www.stanford.edu/~wfsharpe/mia/int/mia_int2.htm |date=2004-06-04 }}, [[William F. Sharpe]] (Stanford University manuscript)</ref> is the branch of [[economics]] that studies the interrelation of financial [[Variable (mathematics)|variables]], such as [[price]]s, [[interest rate]]s and shares, as opposed to [[Real vs. nominal in economics|real]] economic variables, i.e. [[goods and services]]. It thus centers on pricing, decision making, and risk management in the [[financial market]]s,<ref name="Sharpe"/><ref name= "Fama and Miller"/> and produces many of the commonly employed [[financial model]]s. ([[Financial econometrics]] is the branch of financial economics that uses econometric techniques to parameterize the relationships suggested.)

The discipline has two main areas of focus:<ref name= "Fama and Miller">See the discussion re finance theory by Fama and Miller under {{slink||Notes}}.</ref> [[asset pricing]] and corporate finance; the first being the perspective of providers of capital, i.e. investors, and the second of users of capital; respectively:
# Asset pricing theory develops the models used in determining the risk-appropriate discount rate, and in pricing derivatives; and includes the [[outline of finance#Portfolio theory|portfolio-]] and [[investment theory]] applied in asset management. The analysis essentially explores how [[homo economicus|rational investors]] would apply risk and return to the problem of [[investment]] under uncertainty, producing the key "[[Fundamental theorem of asset pricing]]". Here, the twin assumptions of [[rational pricing|rationality]] and [[efficient-market hypothesis|market efficiency]] lead to [[modern portfolio theory]] (the [[Capital asset pricing model|CAPM]]), and to the [[Black–Scholes model|Black–Scholes]] theory for [[Valuation of options|option valuation]]. At more advanced levels—and often in response to [[financial crisis|financial crises]]—the study [[Financial economics#Extensions|then extends]] these [[Neoclassical economics#Rational Behavior Assumptions|"neoclassical" models]] to incorporate phenomena where their assumptions do not hold, or to more general settings.
# Much of [[Outline of finance#Corporate finance theory|corporate finance theory]], by contrast, considers investment under "[[certainty]]" ([[Fisher separation theorem]], [[The Theory of Investment Value|"theory of investment value"]], and [[Modigliani–Miller theorem]]). Here, theory and methods are developed for the decisioning about funding, dividends, and capital structure discussed above. A recent development is [[Financial economics#Corporate finance theory|to incorporate uncertainty]] and [[contingent claim valuation|contingency]]—and thus various elements of asset pricing—into these decisions, employing for example [[real options analysis]].

===Financial mathematics===
{| class="wikitable floatright" | width="250"
|- style="text-align:left;"
|{{smalldiv|
:<math>\begin{align}
  C(S, t) &= N(d_1)S - N(d_2) Ke^{-r(T - t)} \\
     d_1 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\
     d_2 &= d_1 - \sigma\sqrt{T - t} \\
\end{align}</math>
[[Black–Scholes model#Black–Scholes formula|The Black–Scholes formula]] for the value of a [[call option]]. Although lately its use is [[Financial economics#Departures from normality|considered naive]], it has underpinned the development of derivatives-theory, and financial mathematics more generally, since its introduction in 1973.<ref>[https://priceonomics.com/the-history-of-the-black-scholes-formula/ "The History of the Black-Scholes Formula"] {{Webarchive|url=https://web.archive.org/web/20211126125626/https://priceonomics.com/the-history-of-the-black-scholes-formula/ |date=2021-11-26 }}, priceonomics.com</ref>}}
|}
[[File:OAS valuation tree (es).png|thumb|[[Lattice model (finance)|"Trees"]] are widely applied in mathematical finance; here used in calculating an [[option adjusted spread|OAS]]. Other common pricing-methods are [[Monte Carlo methods in finance#Overview|simulation]] and [[Finite difference methods for option pricing|PDEs]]. These are used for settings beyond [[Black–Scholes model#Fundamental hypotheses|those envisaged]] by Black-Scholes.
[[Valuation of options#Post crisis|Post crisis]], even in those settings, banks use [[local volatility|local]] and [[stochastic volatility]] models to incorporate the [[volatility surface]], while the [[xVA]] adjustments accommodate [[counterparty credit risk|counterparty]] and capital considerations.]]
{{Main|Mathematical finance}}
{{see also|Quantitative analysis (finance)|Financial modeling#Quantitative finance}}
Financial mathematics<ref name="SIAM2">[https://www.siam.org/research-areas/detail/financial-mathematics-and-engineering# Research Area: Financial Mathematics and Engineering] {{Webarchive|url=https://web.archive.org/web/20220516114958/https://www.siam.org/research-areas/detail/financial-mathematics-and-engineering |date=2022-05-16 }}, Society for Industrial and Applied Mathematics</ref> is the field of [[applied mathematics]] concerned with [[financial market]]s;
[[Louis Bachelier#The doctoral thesis|Louis Bachelier's doctoral thesis]], defended in 1900, is considered to be the first scholarly work in this area. The field is largely focused on the [[Outline of finance#Derivatives pricing|modeling of derivatives]]—with much emphasis on [[Interest rate derivative|interest rate-]] and [[Credit derivative|credit risk modeling]]—while other important areas include [[actuarial science|insurance mathematics]] and [[Outline of finance#Mathematical techniques|quantitative portfolio management]]. Relatedly, the techniques developed [[contingent claim analysis|are applied]] to pricing and hedging a wide range of [[Asset-backed security|asset-backed]], [[Government bond|government]], and [[Capital structure|corporate]]-securities.

As [[#Quantitative finance|above]], in terms of practice, the field is referred to as quantitative finance and / or mathematical finance, and comprises primarily the three areas discussed.
The [[Outline of finance#Mathematical tools|main mathematical tools]] and techniques are, correspondingly:
* for derivatives,<ref name="Mastro">For a survey, see [https://catalogimages.wiley.com/images/db/pdf/9781118487716.excerpt.pdf "Financial Models"] {{Webarchive|url=https://web.archive.org/web/20211113134700/https://catalogimages.wiley.com/images/db/pdf/9781118487716.excerpt.pdf |date=2021-11-13 }}, from Michael Mastro (2013). ''Financial Derivative and Energy Market Valuation'', John Wiley & Sons. {{ISBN| 978-1-118-48771-6}}.</ref> [[Itô calculus|Itô's stochastic calculus]], [[Monte Carlo methods in finance|simulation]], and [[partial differential equation]]s; see aside boxed discussion re the prototypical [[Black-Scholes model]] and [[Valuation of options#Pricing models|the various numeric techniques]] now applied
* for risk management,<ref name="DeMeo">See generally, Roy E. DeMeo (N.D.) [https://mathfinance.charlotte.edu/sites/mathfinance.charlotte.edu/files/media/An%20Introduction%20to%20Value%20At%20Risk%20New.pdf Quantitative Risk Management: VaR and Others] {{Webarchive|url=https://web.archive.org/web/20211112082350/https://mathfinance.charlotte.edu/sites/mathfinance.charlotte.edu/files/media/An%20Introduction%20to%20Value%20At%20Risk%20New.pdf |date=2021-11-12 }}</ref> [[value at risk]], [[stress test (financial)|stress testing]] and [[PnL Explained#Sensitivities method|"sensitivities" analysis]] (applying the "greeks"); the underlying mathematics comprises [[Mixture model#A financial model|mixture models]], [[Principal component analysis#Quantitative finance|PCA]], [[volatility clustering]] and [[Copula (probability theory)#Quantitative finance|copulas]].<ref>See for example III.A.3, in Carol Alexander, ed. (2005). ''The Professional Risk Managers' Handbook''. PRMIA Publications. {{ISBN|978-0-9766097-0-4}}</ref>
* in both of these areas, and particularly for portfolio problems, quants employ [[Outline of finance#Mathematical techniques|sophisticated optimization techniques]]

Mathematically, these separate into [[Mathematical finance#History: Q versus P|two analytic branches]]: derivatives pricing uses [[Risk-neutral measure|risk-neutral probability]] (or [[rational pricing|arbitrage-pricing]] probability), denoted by "Q"; while risk and portfolio management generally use physical (or actual or actuarial) probability, denoted by "P". These are interrelated through the above "[[Fundamental theorem of asset pricing]]".

The subject has a close relationship with financial economics, which, as outlined, is concerned with much of the underlying theory that is involved in financial mathematics:  generally, financial mathematics will derive and extend the [[mathematical model]]s suggested. [[Computational finance]] is the branch of (applied) [[computer science]] that deals with problems of practical interest in finance, and especially<ref name="SIAM2"/> emphasizes the [[numerical methods]] applied here.

===Experimental finance===
{{Main|Experimental finance}}

[[Experimental finance]]<ref name=ssrn>Bloomfield, Robert and Anderson, Alyssa. [http://fasri.net/wp-content/uploads/2009/10/experimental-finance-chapter-_wiley_.pdf "Experimental finance"] {{Webarchive|url=https://web.archive.org/web/20160304033849/http://fasri.net/wp-content/uploads/2009/10/experimental-finance-chapter-_wiley_.pdf |date=2016-03-04 }}. In Baker, H. Kent, and Nofsinger, John R., eds. Behavioral finance: investors, corporations, and markets. Vol. 6. John Wiley & Sons, 2010. pp. 113–131. {{ISBN|978-0-470-49911-5}}</ref> aims to establish different market settings and environments to experimentally observe and provide a lens through which science can analyze agents' behavior and the resulting characteristics of trading flows, information diffusion, and aggregation, price setting mechanisms, and returns processes. Researchers in experimental finance can study to what extent existing financial economics theory makes valid predictions and therefore prove them, as well as attempt to discover new principles on which such theory can be extended and be applied to future financial decisions. Research may proceed by conducting trading simulations or by establishing and studying the behavior of people in artificial, competitive, market-like settings.

===Behavioral finance===
{{Main|Behavioral economics}}

[[Behavioral finance]] studies how the ''[[psychology]]'' of investors or managers affects financial decisions and markets<ref>Glaser, Markus and Weber, Martin and Noeth, Markus. (2004). [https://madoc.bib.uni-mannheim.de/2770/1/dp03_14.pdf "Behavioral Finance"] {{Webarchive|url=https://web.archive.org/web/20230209093116/https://madoc.bib.uni-mannheim.de/2770/1/dp03_14.pdf |date=2023-02-09 }}, pp. 527–546 in ''Handbook of Judgment and Decision Making'', Blackwell Publishers {{ISBN|978-1-405-10746-4}}</ref>
and is relevant when making a decision that can impact either negatively or positively on one of their areas. With more in-depth research into behavioral finance, it is possible to bridge what actually happens in financial markets with analysis based on financial theory.<ref>{{Cite journal |last1=Zahera |first1=Syed Aliya |last2=Bansal |first2=Rohit |date=2018-05-08 |title=Do investors exhibit behavioral biases in investment decision making? A systematic review |url=https://www.emerald.com/insight/content/doi/10.1108/QRFM-04-2017-0028/full/html |journal=Qualitative Research in Financial Markets |language=en |volume=10 |issue=2 |pages=210–251 |doi=10.1108/QRFM-04-2017-0028 |issn=1755-4179 |access-date=2022-04-08 |archive-date=2022-04-08 |archive-url=https://web.archive.org/web/20220408144635/https://www.emerald.com/insight/content/doi/10.1108/QRFM-04-2017-0028/full/html |url-status=live }}</ref> Behavioral finance has grown over the last few decades to become an integral aspect of finance. Nowadays there is a need for more theory and testing of the effects of feelings on financial decisions. Especially, because now the time has come to move beyond behavioral finance to social finance, which studies the structure of social interactions, how financial ideas spread, and how social processes affect financial decisions and outcomes.<ref>{{cite book|last1=Shefrin|first1=Hersh|title=Beyond greed and fear: Understanding behavioral finance and the psychology of investing|date=2002|publisher=Oxford University Press|location=New York|isbn=978-0-19-530421-3|page=ix|url=https://archive.org/details/beyondgreedfearu00shef|url-access=registration|quote=growth of behavioral finance.|access-date=8 May 2017}}</ref><ref>{{Cite journal |last=Hirshleifer |first=David |date=2015 |title=Behavioral Finance |journal=Annual Review of Financial Economics |language=en |volume=7 |pages=133–159 |doi=10.1146/annurev-financial-092214-043752 |doi-access=free |issn=1941-1367}}</ref>

Behavioral finance includes such topics as:
# Empirical studies that demonstrate significant deviations from classical theories;
# Models of how psychology affects and impacts trading and prices;
# Forecasting based on these methods;
# Studies of experimental asset markets and the use of models to forecast experiments.

A strand of behavioral finance has been dubbed [[quantitative behavioral finance]], which uses mathematical and statistical methodology to understand behavioral biases in conjunction with valuation.
<!-- Some of these endeavors has been led by [[Gunduz Caginalp]] (Professor of Mathematics and Editor of [[Journal of Behavioral Finance]] during 2001–2004) and collaborators including [[Vernon L. Smith|Vernon Smith]] (2002 Nobel Laureate in Economics), David Porter, Don Balenovich, Vladimira Ilieva, Ahmet Duran). Studies by Jeff Madura, Ray Sturm, and others have demonstrated significant behavioral effects in stocks and exchange-traded funds. Among other topics, quantitative behavioral finance studies behavioral effects together with the non-classical assumption of the finiteness of assets. --><!--
Doesn't really belong under the theory section...
===Intangible asset finance===
{{Main|Intangible asset finance}}

Intangible asset finance is the area of finance that deals with intangible assets such as patents, trademarks, goodwill, reputation, etc.

Intangible assets are divided into indefinite or definite. The brand name of a company is an indefinite asset; it stays with the company throughout its existence. A patent, however, granted to that company for a limited amount of time would be a definite intangible asset.<ref>{{Cite web|url=http://www.investopedia.com/terms/i/intangibleasset.asp|title=Intangible Asset Definition {{!}} Investopedia|last=root|date=2003-11-20|language=en-US|access-date=2016-06-27}}</ref>
--><!-- === Environmental finance ===
{{excerpt|Environmental finance|paragraphs=1|file=no}} -->
=== Quantum finance ===
{{ref improve section|date=April 2025}}
{{Main|Quantum finance}}
Quantum finance involves applying quantum mechanical approaches to financial theory, providing novel methods and perspectives in the field.<ref>{{Cite journal |last1=Focardi |first1=Sergio |last2=Fabozzi |first2=Frank J. |last3=Mazza |first3=Davide |date=2020-08-31 |title=Quantum Option Pricing and Quantum Finance |url=http://pm-research.com/lookup/doi/10.3905/jod.2020.1.111 |journal=The Journal of Derivatives |language=en |volume=28 |issue=1 |pages=79–98 |doi=10.3905/jod.2020.1.111 |issn=1074-1240}}</ref>  ''Quantum'' ''finance'' is an interdisciplinary field, in which theories and methods developed by ''quantum'' physicists and economists are applied to solve financial problems. It represents a branch known as econophysics. Although ''quantum'' computational methods have been around for quite some time and use the basic principles of physics to better understand the ways to implement and manage cash flows, it is mathematics that is actually important in this new scenario<ref>{{Cite journal |last=Ristic |first=Kristijan |date=2–3 December 2021 |title=New Financial Future: Digital Finance As a key Aspect of Financial Innovation |url=https://www.proquest.com/docview/2616890742 |journal=75th International Scientific Conference on Economic and Social Development |pages=283–288 |id={{ProQuest|2616890742}} }}</ref> Finance theory is heavily based on financial instrument pricing such as [[stock option]] pricing. Many of the problems facing the finance community have no known analytical solution. As a result, numerical methods and computer simulations for solving these problems have proliferated. This research area is known as [[computational finance]]. Many computational finance problems have a high degree of computational complexity and are slow to converge to a solution on classical computers. In particular, when it comes to option pricing, there is additional complexity resulting from the need to respond to quickly changing markets. For example, in order to take advantage of inaccurately priced stock options, the computation must complete before the next change in the almost continuously changing stock market. As a result, the finance community is always looking for ways to overcome the resulting performance issues that arise when pricing options. This has led to research that applies alternative computing techniques to finance. Most commonly used quantum financial models are quantum continuous model, quantum binomial model, multi-step quantum binomial model etc.