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146-469: Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in insurance , pension , finance , investment and other industries and professions. Actuaries are professionals trained in this discipline. In many countries, actuaries must demonstrate their competence by passing a series of rigorous professional examinations focused in fields such as probability and predictive analysis. Actuarial science includes

292-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects

438-440: A benchmark for evaluating the effects of factors outside the model that do affect value. The mechanism for determining (corporate) value is provided by John Burr Williams ' The Theory of Investment Value , which proposes that the value of an asset should be calculated using "evaluation by the rule of present worth". Thus, for a common stock, the "intrinsic" , long-term worth is the present value of its future net cashflows, in

584-514: A broad range of goals. Copulas have lately been applied here ; recently this is the case also for genetic algorithms and Machine learning, more generally . (Tail) risk parity focuses on allocation of risk, rather than allocation of capital. See Portfolio optimization § Improving portfolio optimization for other techniques and objectives, and Financial risk management § Investment management for discussion. Interpretation: Analogous to Black-Scholes, arbitrage arguments describe

730-409: A correlation between security and market returns, without (numerous) other economic assumptions. It is useful in that it simplifies the estimation of correlation between securities, significantly reducing the inputs for building the correlation matrix required for portfolio optimization. The arbitrage pricing theory (APT) similarly differs as regards its assumptions. APT "gives up the notion that there

876-400: A formal study of the financial markets themselves, especially market microstructure and market regulation . It is built on the foundations of microeconomics and decision theory . Financial econometrics is the branch of financial economics that uses econometric techniques to parameterise the relationships identified. Mathematical finance is related in that it will derive and extend

1022-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of

1168-637: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)

1314-454: A large number of researchers have used options to extract state-prices for a variety of other applications in financial economics. For path dependent derivatives , Monte Carlo methods for option pricing are employed; here the modelling is in continuous time, but similarly uses risk neutral expected value. Various other numeric techniques have also been developed. The theoretical framework too has been extended such that martingale pricing

1460-429: A market for every time period and forward prices for every commodity at all time periods. A direct extension, then, is the concept of a state price security, also called an Arrow–Debreu security, a contract that agrees to pay one unit of a numeraire (a currency or a commodity) if a particular state occurs ("up" and "down" in the simplified example above) at a particular time in the future and pays zero numeraire in all

1606-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were

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1752-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of

1898-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it

2044-554: A number of interrelated subjects, including mathematics, probability theory , statistics, finance, economics , financial accounting and computer science . Historically, actuarial science used deterministic models in the construction of tables and premiums. The science has gone through revolutionary changes since the 1980s due to the proliferation of high speed computers and the union of stochastic actuarial models with modern financial theory. Many universities have undergraduate and graduate degree programs in actuarial science. In 2010,

2190-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes

2336-541: A range of applications outside the traditional fields of insurance, pensions, etc. One notable example is the use in some US states of actuarial models to set criminal sentencing guidelines. These models attempt to predict the chance of re-offending according to rating factors which include the type of crime, age, educational background and ethnicity of the offender. However, these models have been open to criticism as providing justification for discrimination against specific ethnic groups by law enforcement personnel. Whether this

2482-487: A result, actuarial science developed along a different path, becoming more reliant on assumptions, as opposed to the arbitrage-free risk-neutral valuation concepts used in modern finance. The divergence is not related to the use of historical data and statistical projections of liability cash flows, but is instead caused by the manner in which traditional actuarial methods apply market data with those numbers. For example, one traditional actuarial method suggests that changing

2628-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as

2774-565: A set of prices exists that will result in an overall equilibrium. (This is in contrast to partial equilibrium, which only analyzes single markets.) The two concepts are linked as follows: where market prices do not allow profitable arbitrage, i.e. they comprise an arbitrage-free market, then these prices are also said to constitute an "arbitrage equilibrium". Intuitively, this may be seen by considering that where an arbitrage opportunity does exist, then prices can be expected to change, and they are therefore not in equilibrium. An arbitrage equilibrium

2920-399: A simplified example see Rational pricing § Risk neutral valuation , where the economy has only two possible states – up and down – and where q u p {\displaystyle q_{up}} and q d o w n {\displaystyle q_{down}} ( = 1 − q u p {\displaystyle 1-q_{up}} ) are

3066-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of

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3212-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating

3358-521: A study published by job search website CareerCast ranked actuary as the #1 job in the United States. The study used five key criteria to rank jobs: environment, income, employment outlook, physical demands, and stress. In 2024, U.S. News & World Report ranked actuary as the third-best job in the business sector and the eighth-best job in STEM . Actuarial science became a formal mathematical discipline in

3504-448: A variety of theoretical and practical reasons. Elementary mutual aid agreements and pensions arose in antiquity. Early in the Roman empire , associations were formed to meet the expenses of burial, cremation, and monuments—precursors to burial insurance and friendly societies . A small sum was paid into a communal fund on a weekly basis, and upon the death of a member, the fund would cover

3650-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to

3796-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry

3942-507: Is flat " and "a field is always a ring ". Financial economics The subject is concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment". It therefore centers on decision making under uncertainty in the context of the financial markets, and the resultant economic and financial models and principles, and is concerned with deriving testable or policy implications from acceptable assumptions. It thus also includes

4088-471: Is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as

4234-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example

4380-683: Is consistent with the above , but with the expectation based on "the market" (i.e. arbitrage-free, and, per the theorem, therefore in equilibrium) as opposed to individual preferences. Continuing the example, in pricing a derivative instrument , its forecasted cashflows in the above-mentioned up- and down-states X u p {\displaystyle X_{up}} and X d o w n {\displaystyle X_{down}} , are multiplied through by q u p {\displaystyle q_{up}} and q d o w n {\displaystyle q_{down}} , and are then discounted at

4526-509: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of

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4672-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module

4818-492: Is expressed as the Black–Scholes equation, a partial differential equation describing the changing price of the option over time; it is derived assuming log-normal, geometric Brownian motion (see Brownian model of financial markets ). The key financial insight behind the model is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk", absenting

4964-467: Is generally provided on a renewable period, (such as a yearly). Coverage can be cancelled at the end of the period by either party. Property and casualty insurance companies tend to specialize because of the complexity and diversity of risks. One division is to organize around personal and commercial lines of insurance. Personal lines of insurance are for individuals and include fire, auto, homeowners, theft and umbrella coverages. Commercial lines address

5110-487: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as

5256-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example

5402-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,

5548-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of

5694-547: Is often held to be Archimedes ( c.  287  – c.  212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and

5840-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it

5986-452: Is one right portfolio for everyone in the world, and ...replaces it with an explanatory model of what drives asset returns." It returns the required (expected) return of a financial asset as a linear function of various macro-economic factors, and assumes that arbitrage should bring incorrectly priced assets back into line. The linear factor model structure of the APT is used as the basis for many of

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6132-554: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after

6278-544: Is statistically correct or a self-fulfilling correlation remains under debate. Another example is the use of actuarial models to assess the risk of sex offense recidivism. Actuarial models and associated tables, such as the MnSOST-R, Static-99, and SORAG, have been used since the late 1990s to determine the likelihood that a sex offender will re-offend and thus whether he or she should be institutionalized or set free. Traditional actuarial science and modern financial economics in

6424-476: Is the focus of "asset pricing models" as follows. Fisher's formulation of the theory here - developing an intertemporal equilibrium model - underpins also the below applications to uncertainty; see for the development. For "choice under uncertainty" the twin assumptions of rationality and market efficiency , as more closely defined, lead to modern portfolio theory (MPT) with its capital asset pricing model (CAPM) – an equilibrium-based result – and to

6570-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,

6716-491: Is the risk-free rate plus the market premium multiplied by beta ( ρ i , m σ i σ m {\displaystyle \rho _{i,m}{\frac {\sigma _{i}}{\sigma _{m}}}} ), the asset's correlated volatility relative to the overall market m {\displaystyle m} . Applying the above economic concepts, we may then derive various economic- and financial models and principles. As above,

6862-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of

7008-559: Is therefore consistent with the equilibrium based capital asset pricing. Both models, in turn, are ultimately consistent with the Arrow–Debreu theory, and can be derived via state-pricing – essentially, by expanding the fundamental result above – further explaining, and if required demonstrating, this consistency. Here, the CAPM is derived by linking Y {\displaystyle Y} , risk aversion, to overall market return, and setting

7154-515: Is thus a precondition for a general economic equilibrium. "Complete" here means that there is a price for every asset in every possible state of the world, s {\displaystyle s} , and that the complete set of possible bets on future states-of-the-world can therefore be constructed with existing assets (assuming no friction ): essentially solving simultaneously for n (risk-neutral) probabilities, q s {\displaystyle q_{s}} , given n prices. For

7300-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with

7446-574: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It

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7592-545: The Black–Scholes–Merton theory (BSM; often, simply Black–Scholes) for option pricing – an arbitrage-free result. As above, the (intuitive) link between these, is that the latter derivative prices are calculated such that they are arbitrage-free with respect to the more fundamental, equilibrium determined, securities prices; see Asset pricing § Interrelationship . Briefly, and intuitively – and consistent with § Arbitrage-free pricing and equilibrium above –

7738-545: The CAPM (or extensions) as will be seen under § Uncertainty . The difference is explained as follows: By construction, the value of the derivative will (must) grow at the risk free rate, and, by arbitrage arguments, its value must then be discounted correspondingly; in the case of an option, this is achieved by "manufacturing" the instrument as a combination of the underlying and a risk free "bond"; see Rational pricing § Delta hedging (and § Uncertainty below). Where

7884-620: The Ellsberg paradox . The New Palgrave Dictionary of Economics (2008, 2nd ed.) also uses the JEL codes to classify its entries in v. 8, Subject Index, including Financial Economics at pp. 863–64. The below have links to entry abstracts of The New Palgrave Online for each primary or secondary JEL category (10 or fewer per page, similar to Google searches): Tertiary category entries can also be searched. The concepts of arbitrage -free, "rational", pricing and equilibrium are then coupled with

8030-551: The Financial Accounting Standards Board , (FASB) in the US and Canada, which regulates pensions valuations and funding. Historically, much of the foundation of actuarial theory predated modern financial theory. In the early twentieth century, actuaries were developing many techniques that can be found in modern financial theory, but for various historical reasons, these developments did not achieve much recognition. As

8176-753: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during

8322-505: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity

8468-524: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of

8614-694: The asset allocation mix of investments can change the value of liabilities and assets (by changing the discount rate assumption). This concept is inconsistent with financial economics . The potential of modern financial economics theory to complement existing actuarial science was recognized by actuaries in the mid-twentieth century. In the late 1980s and early 1990s, there was a distinct effort for actuaries to combine financial theory and stochastic methods into their established models. Ideas from financial economics became increasingly influential in actuarial thinking, and actuarial science has started to embrace more sophisticated mathematical modelling of finance. Today,

8760-510: The best guess of the future : the assumption of rational expectations . The EMH does allow that when faced with new information, some investors may overreact and some may underreact, but what is required, however, is that investors' reactions follow a normal distribution – so that the net effect on market prices cannot be reliably exploited to make an abnormal profit. In the competitive limit, then, market prices will reflect all available information and prices can only move in response to news:

8906-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object

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9052-459: The deterministic methods they had used in the past. The introduction and development of the computer further revolutionized the actuarial profession. From pencil-and-paper to punchcards to current high-speed devices, the modeling and forecasting ability of the actuary has rapidly improved, while still being heavily dependent on the assumptions input into the models, and actuaries needed to adjust to this new world . Mathematics Mathematics

9198-532: The random walk hypothesis . This news, of course, could be "good" or "bad", minor or, less common, major; and these moves are then, correspondingly, normally distributed; with the price therefore following a log-normal distribution. Under these conditions, investors can then be assumed to act rationally: their investment decision must be calculated or a loss is sure to follow; correspondingly, where an arbitrage opportunity presents itself, then arbitrageurs will exploit it, reinforcing this equilibrium. Here, as under

9344-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry

9490-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not

9636-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and

9782-631: The CAPM. In general, then, while portfolio theory studies how investors should balance risk and return when investing in many assets or securities, the CAPM is more focused, describing how, in equilibrium, markets set the prices of assets in relation to how risky they are. This result will be independent of the investor's level of risk aversion and assumed utility function , thus providing a readily determined discount rate for corporate finance decision makers as above , and for other investors. The argument proceeds as follows : If one can construct an efficient frontier – i.e. each combination of assets offering

9928-620: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,

10074-651: The US have different practices, which is caused by different ways of calculating funding and investment strategies, and by different regulations. Regulations are from the Armstrong investigation of 1905 , the Glass–Steagall Act of 1932 , the adoption of the Mandatory Security Valuation Reserve by the National Association of Insurance Commissioners , which cushioned market fluctuations, and

10220-485: The above pricing formulae. DCF valuation formula , where the value of the firm , is its forecasted free cash flows discounted to the present using the weighted average cost of capital , i.e. cost of equity and cost of debt , with the former (often) derived using the below CAPM. For share valuation investors use the related dividend discount model . The expected return used when discounting cashflows on an asset i {\displaystyle i} ,

10366-408: The above to derive various of the "classical" (or "neo-classical" ) financial economics models. Rational pricing is the assumption that asset prices (and hence asset pricing models) will reflect the arbitrage-free price of the asset, as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to

10512-509: The application of decision making under uncertainty to the financial markets . The underlying economic logic yields the fundamental theorem of asset pricing , which gives the conditions for arbitrage -free asset pricing. The various "fundamental" valuation formulae result directly. Underlying all of financial economics are the concepts of present value and expectation . Calculating their present value, X s j / r {\displaystyle X_{sj}/r} in

10658-422: The approach is therefore to "adjust" the weight assigned to the various outcomes, i.e. "states", correspondingly: Y s {\displaystyle Y_{s}} . See indifference price . (Some investors may in fact be risk seeking as opposed to risk averse , but the same logic would apply.) Choice under uncertainty here may then be defined as the maximization of expected utility . More formally,

10804-574: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During

10950-471: The best possible expected level of return for its level of risk, see diagram – then mean-variance efficient portfolios can be formed simply as a combination of holdings of the risk-free asset and the " market portfolio " (the Mutual fund separation theorem ), with the combinations here plotting as the capital market line , or CML. Then, given this CML, the required return on a risky security will be independent of

11096-487: The calculations as easy as possible, for example "commutation functions" (essentially precalculated columns of summations over time of discounted values of survival and death probabilities). Actuarial organizations were founded to support and further both actuaries and actuarial science, and to protect the public interest by promoting competency and ethical standards. However, calculations remained cumbersome, and actuarial shortcuts were commonplace. Non-life actuaries followed in

11242-735: 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 criterion which sets asset value as a function of the sizes of the expected payouts and the probabilities of their occurrence, X s {\displaystyle X_{s}} and p s {\displaystyle p_{s}} respectively. This decision method, however, fails to consider risk aversion . In other words, since individuals receive greater utility from an extra dollar when they are poor and less utility when comparatively rich,

11388-550: The certainty-case above, the specific assumption as to pricing is that prices are calculated as the present value of expected future dividends, as based on currently available information. What is required though, is a theory for determining the appropriate discount rate, i.e. "required return", given this uncertainty: this is provided by the MPT and its CAPM. Relatedly, rationality – in the sense of arbitrage-exploitation – gives rise to Black–Scholes; option values here ultimately consistent with

11534-458: The commercial risk systems employed by asset managers. As regards portfolio optimization , the Black–Litterman model departs from the original Markowitz model – i.e. of constructing portfolios via an efficient frontier . Black–Litterman instead starts with an equilibrium assumption, and is then modified to take into account the 'views' (i.e., the specific opinions about asset returns) of

11680-503: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,

11826-499: The corporation will be the maximization of its present value, regardless of the preferences of its shareholders. Related is the Modigliani–Miller theorem , which shows that, under certain conditions, the value of a firm is unaffected by how that firm is financed, and depends neither on its dividend policy nor its decision to raise capital by issuing stock or selling debt. The proof here proceeds using arbitrage arguments, and acts as

11972-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is

12118-512: The debate. Financial economists argue that pension benefits are bond-like and should not be funded with equity investments without reflecting the risks of not achieving expected returns. But some pension products do reflect the risks of unexpected returns. In some cases, the pension beneficiary assumes the risk, or the employer assumes the risk. The current debate now seems to be focusing on four principles: Essentially, financial economics state that pension assets should not be invested in equities for

12264-499: The decision of one individual) with the effect that the problems are then mathematically tractable. With this measure in place, the expected, i.e. required , return of any security (or portfolio) will then equal the risk-free return, plus an "adjustment for risk", i.e. a security-specific risk premium , compensating for the extent to which its cashflows are unpredictable. All pricing models are then essentially variants of this, given specific assumptions or conditions. This approach

12410-428: The derivative value – true in fact for "every security" – as a linear combination of its state-prices; i.e. back-solve for the state-prices corresponding to observed derivative prices. These recovered state-prices can then be used for valuation of other instruments with exposure to the underlyer, or for other decision making relating to the underlyer itself. Using the related stochastic discount factor - also called

12556-553: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely

12702-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of

12848-567: The expenses of rites and burial. These societies sometimes sold shares in the building of columbāria , or burial vaults, owned by the fund—the precursor to mutual insurance companies . Other early examples of mutual surety and assurance pacts can be traced back to various forms of fellowship within the Saxon clans of England and their Germanic forebears, and to Celtic society. However, many of these earlier forms of surety and aid would often fail due to lack of understanding and knowledge. The 17th century

12994-464: The father of demography , John Graunt , who showed that there were predictable patterns of longevity and death in a group, or cohort , of people of the same age, despite the uncertainty of the date of death of any one individual. This study became the basis for the original life table . One could now set up an insurance scheme to provide life insurance or pensions for a group of people, and to calculate with some degree of accuracy how much each person in

13140-429: The first formula, allows the decision maker to aggregate the cashflows (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. (Note that here, " r {\displaystyle r} " represents a generic (or arbitrary) discount rate applied to

13286-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",

13432-540: The focus on cash flows from securities over time," valuation based on a term structure of interest rates is in fact consistent with arbitrage-free pricing. Indeed, a corollary of the above is that " the law of one price implies the existence of a discount factor"; correspondingly, as formulated, ∑ s π s = 1 / r {\textstyle \sum _{s}\pi _{s}=1/r} . Whereas these "certainty" results are all commonly employed under corporate finance, uncertainty

13578-592: The following 200 years. Equitable Life was the first to use the word "actuary" for its chief executive officer in 1762. Previously, "actuary" meant an official who recorded the decisions, or "acts", of ecclesiastical courts. Other companies that did not use such mathematical and scientific methods most often failed or were forced to adopt the methods pioneered by Equitable. In the 18th and 19th centuries, calculations were performed without computers. The computations of life insurance premiums and reserving requirements are rather complex, and actuaries developed techniques to make

13724-490: The footsteps of their life insurance colleagues during the 20th century. The 1920 revision for the New-York based National Council on Workmen's Compensation Insurance rates took over two months of around-the-clock work by day and night teams of actuaries. In the 1930s and 1940s, the mathematical foundations for stochastic processes were developed. Actuaries could now begin to estimate losses using models of random events, instead of

13870-504: The form of dividends . What remains to be determined is the appropriate discount rate. Later developments show that, "rationally", i.e. in the formal sense, the appropriate discount rate here will (should) depend on the asset's riskiness relative to the overall market, as opposed to its owners' preferences; see below. Net present value (NPV) is the direct extension of these ideas typically applied to Corporate Finance decisioning. For other results, as well as specific models developed here, see

14016-425: The fourth formula (see above regarding the absence of a risk premium here). For a continuous random variable indicating a continuum of possible states, the value is found by integrating over the state price "density". State prices find immediate application as a conceptual tool (" contingent claim analysis "); but can also be applied to valuation problems. Given the pricing mechanism described, one can decompose

14162-403: The group should contribute to a common fund assumed to earn a fixed rate of interest. The first person to demonstrate publicly how this could be done was Edmond Halley (of Halley's comet fame). Halley constructed his own life table, and showed how it could be used to calculate the premium amount someone of a given age should pay to purchase a life annuity. James Dodson 's pioneering work on

14308-522: The instantaneous change in the bond price P {\displaystyle P} for changes in the (risk-free) short rate r {\displaystyle r} ; the analyst selects the specific short-rate model to be employed. In pricing derivatives, the binomial options pricing model provides a discretized version of Black–Scholes, useful for the valuation of American styled options . Discretized models of this type are built – at least implicitly – using state-prices ( as above ); relatedly,

14454-623: The insurance needs of businesses and include property, business continuation, product liability, fleet/commercial vehicle, workers compensation, fidelity and surety, and D&O insurance. The insurance industry also provides coverage for exposures such as catastrophe, weather-related risks, earthquakes, patent infringement and other forms of corporate espionage, terrorism, and "one-of-a-kind" (e.g., satellite launch). Actuarial science provides data collection, measurement, estimating, forecasting, and valuation tools to provide financial and underwriting data for management to assess marketing opportunities and

14600-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before

14746-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and

14892-490: The investor in question to arrive at a bespoke asset allocation. Where factors additional to volatility are considered (kurtosis, skew...) then multiple-criteria decision analysis can be applied; here deriving a Pareto efficient portfolio. The universal portfolio algorithm applies machine learning to asset selection, learning adaptively from historical data. Behavioral portfolio theory recognizes that investors have varied aims and create an investment portfolio that meets

15038-400: The investor's utility function , and solely determined by its covariance ("beta") with aggregate, i.e. market, risk. This is because investors here can then maximize utility through leverage as opposed to pricing; see Separation property (finance) , Markowitz model § Choosing the best portfolio and CML diagram aside. As can be seen in the formula aside, this result is consistent with

15184-435: The late 17th century with the increased demand for long-term insurance coverage such as burial, life insurance , and annuities. These long term coverages required that money be set aside to pay future benefits, such as annuity and death benefits many years into the future. This requires estimating future contingent events, such as the rates of mortality by age, as well as the development of mathematical techniques for discounting

15330-878: The list of "Equity valuation" topics under Outline of finance § Discounted cash flow valuation . Bond valuation , in that cashflows ( coupons and return of principal, or " Face value ") are deterministic, may proceed in the same fashion. An immediate extension, Arbitrage-free bond pricing , discounts each cashflow at the market derived rate – i.e. at each coupon's corresponding zero rate , and of equivalent credit worthiness – as opposed to an overall rate. In many treatments bond valuation precedes equity valuation , under which cashflows (dividends) are not "known" per se . Williams and onward allow for forecasting as to these – based on historic ratios or published dividend policy – and cashflows are then treated as essentially deterministic; see below under § Corporate finance theory . For both stocks and bonds, "under certainty, with

15476-525: The long term insurance contracts under which the same premium is charged each year led to the formation of the Society for Equitable Assurances on Lives and Survivorship (now commonly known as Equitable Life ) in London in 1762. William Morgan is often considered the father of modern actuarial science for his work in the field in the 1780s and 90s. Many other life insurance companies and pension funds were created over

15622-444: The lower case is used, as in the above equations.) With the above relationship established, the further specialized Arrow–Debreu model may be derived. This result suggests that, under certain economic conditions, there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy. The Arrow–Debreu model applies to economies with maximally complete markets , in which there exists

15768-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term

15914-452: The mathematical or numerical models suggested by financial economics. Whereas financial economics has a primarily microeconomic focus, monetary economics is primarily macroeconomic in nature. Four equivalent formulations, where: Financial economics studies how rational investors would apply decision theory to investment management . The subject is thus built on the foundations of microeconomics and derives several key results for

16060-399: The model. With intertemporal portfolio choice , the investor now repeatedly optimizes her portfolio; while the inclusion of consumption (in the economic sense) then incorporates all sources of wealth, and not just market-based investments, into the investor's calculation of required return. Whereas the above extend the CAPM, the single-index model is a more simple model. It assumes, only,

16206-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to

16352-451: The nature of the risks. Actuarial science often helps to assess the overall risk from catastrophic events in relation to its underwriting capacity or surplus. In the reinsurance fields, actuarial science can be used to design and price reinsurance and retrocession arrangements, and to establish reserve funds for known claims and future claims and catastrophes. There is an increasing trend to recognize that actuarial skills can be applied to

16498-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains

16644-456: The other states. The price of this security is the state price π s {\displaystyle \pi _{s}} of this particular state of the world; also referred to as a "Risk Neutral Density". In the above example, the state prices, π u p {\displaystyle \pi _{up}} , π d o w n {\displaystyle \pi _{down}} would equate to

16790-514: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000  BC , when

16936-410: The preceding , equaling the riskless return plus an adjustment for risk. A more modern, direct, derivation is as described at the bottom of this section; which can be generalized to derive other equilibrium-pricing models. Black–Scholes provides a mathematical model of a financial market containing derivative instruments, and the resultant formula for the price of European-styled options . The model

17082-755: The present values of $ q u p {\displaystyle \$ q_{up}} and $ q d o w n {\displaystyle \$ q_{down}} : i.e. what one would pay today, respectively, for the up- and down-state securities; the state price vector is the vector of state prices for all states. Applied to derivative valuation, the price today would simply be [ π u p {\displaystyle \pi _{up}} × X u p {\displaystyle X_{up}} + π d o w n {\displaystyle \pi _{down}} × X d o w n {\displaystyle X_{down}} ] :

17228-480: The price, is consistent with the equation, as the formula is the solution to the equation.) Since the formula is without reference to the share's expected return, Black–Scholes inheres risk neutrality; intuitively consistent with the "elimination of risk" here, and mathematically consistent with § Arbitrage-free pricing and equilibrium above. Relatedly, therefore, the pricing formula may also be derived directly via risk neutral expectation. Itô's lemma provides

17374-408: The pricing kernel - the asset price is computed by "discounting" the future cash flow by the stochastic factor m ~ {\displaystyle {\tilde {m}}} , and then taking the expectation; the third equation above. Essentially, this factor divides expected utility at the relevant future period - a function of the possible asset values realized under each state - by

17520-407: The pricing of derivative instruments. Economic equilibrium is a state in which economic forces such as supply and demand are balanced, and in the absence of external influences these equilibrium values of economic variables will not change. General equilibrium deals with the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that

17666-431: The profession, both in practice and in the educational syllabi of many actuarial organizations, is cognizant of the need to reflect the combined approach of tables, loss models, stochastic methods, and financial theory. However, assumption-dependent concepts are still widely used (such as the setting of the discount rate assumption as mentioned earlier), particularly in North America. Product design adds another dimension to

17812-654: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been

17958-664: The relationship between rationality and efficiency is as follows. Given the ability to profit from private information , self-interested traders are motivated to acquire and act on their private information. In doing so, traders contribute to more and more "correct", i.e. efficient , prices: the efficient-market hypothesis , or EMH. Thus, if prices of financial assets are (broadly) efficient, then deviations from these (equilibrium) values could not last for long. (See earnings response coefficient .) The EMH (implicitly) assumes that average expectations constitute an "optimal forecast", i.e. prices using all available information are identical to

18104-406: The required models. This derivation begins with the assumption of "no uncertainty" and is then expanded to incorporate the other considerations. (This division sometimes denoted " deterministic " and "random", or " stochastic ".) The starting point here is "Investment under certainty", and usually framed in the context of a corporation. The Fisher separation theorem , asserts that the objective of

18250-455: The resulting expected utility hypothesis states that, if certain axioms are satisfied, the subjective value associated with a gamble by an individual is that individual ' s 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

18396-415: The return on security j {\displaystyle j} as X j / P r i c e j {\displaystyle X_{j}/Price_{j}} ; see Stochastic discount factor § Properties . The Black-Scholes formula is found, in the limit, by attaching a binomial probability to each of numerous possible spot-prices (i.e. states) and then rearranging for

18542-406: The risk adjustment from the pricing ( V {\displaystyle V} , the value, or price, of the option, grows at r {\displaystyle r} , the risk-free rate). This hedge, in turn, implies that there is only one right price – in an arbitrage-free sense – for the option. And this price is returned by the Black–Scholes option pricing formula. (The formula, and hence

18688-405: The risk-free interest rate; per the second equation above. In pricing a "fundamental", underlying, instrument (in equilibrium), on the other hand, a risk-appropriate premium over risk-free is required in the discounting, essentially employing the first equation with Y {\displaystyle Y} and r {\displaystyle r} combined. This premium may be derived by

18834-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become

18980-561: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and

19126-573: The terms corresponding to N ( d 1 ) {\displaystyle N(d_{1})} and N ( d 2 ) {\displaystyle N(d_{2})} , per the boxed description; see Binomial options pricing model § Relationship with Black–Scholes . More recent work further generalizes and extends these models. As regards asset pricing , developments in equilibrium-based pricing are discussed under "Portfolio theory" below, while "Derivative pricing" relates to risk-neutral, i.e. arbitrage-free, pricing. As regards

19272-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,

19418-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in

19564-402: The two corresponding probabilities, and in turn, the derived distribution, or "measure" . The formal derivation will proceed by arbitrage arguments. The analysis here is often undertaken assuming a representative agent , essentially treating all market participants, " agents ", as identical (or, at least, assuming that they act in such a way that the sum of their choices is equivalent to

19710-504: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in

19856-409: The two usual areas of focus are Asset Pricing and Corporate Finance, the first being the perspective of providers of capital, the second of users of capital. Here, and for (almost) all other financial economics models, the questions addressed are typically framed in terms of "time, uncertainty, options, and information", as will be seen below. Applying this framework, with the above concepts, leads to

20002-520: The underlying is itself being priced, such "manufacturing" is of course not possible – the instrument being "fundamental", i.e. as opposed to "derivative" – and a premium is then required for risk. (Correspondingly, mathematical finance separates into two analytic regimes : risk and portfolio management (generally) use physical (or actual or actuarial) probability, denoted by "P"; while derivatives pricing uses risk-neutral probability (or arbitrage-pricing probability), denoted by "Q". In specific applications

20148-567: The underlying mathematics , and, with Itô calculus more generally, remains fundamental in quantitative finance. As implied by the Fundamental Theorem, the two major results are consistent . Here, the Black Scholes equation can alternatively be derived from the CAPM, and the price obtained from the Black–Scholes model is thus consistent with the assumptions of the CAPM. The Black–Scholes theory, although built on Arbitrage-free pricing,

20294-542: The use of capital, "Corporate finance theory" relates, mainly, to the application of these models. The majority of developments here relate to required return, i.e. pricing, extending the basic CAPM. Multi-factor models such as the Fama–French three-factor model and the Carhart four-factor model , propose factors other than market return as relevant in pricing. The intertemporal CAPM and consumption-based CAPM similarly extend

20440-399: The utility due to today's wealth, and is then also referred to as "the intertemporal marginal rate of substitution ". Bond valuation formula where Coupons and Face value are discounted at the appropriate rate, "i": typically a spread over the (per period) risk free rate as a function of credit risk ; often quoted as a " yield to maturity ". See body for discussion re the relationship with

20586-430: The value of funds set aside and invested. This led to the development of an important actuarial concept, referred to as the present value of a future sum. Certain aspects of the actuarial methods for discounting pension funds have come under criticism from modern financial economics . Actuarial science is also applied to property , casualty , liability , and general insurance . In these forms of insurance, coverage

20732-453: Was a period of advances in mathematics in Germany, France and England. At the same time there was a rapidly growing desire and need to place the valuation of personal risk on a more scientific basis. Independently of each other, compound interest was studied and probability theory emerged as a well-understood mathematical discipline. Another important advance came in 1662 from a London draper ,

20878-457: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until

21024-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"

21170-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to

21316-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In

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