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128-427: 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

256-479: A 3% change in GDP after one year, and one gave almost no change, with the rest spread between. Partly as a result of such experiments, modern central bankers no longer have as much confidence that it is possible to 'fine-tune' the economy as they had in the 1960s and early 1970s. Modern policy makers tend to use a less activist approach, explicitly because they lack confidence that their models will actually predict where

384-502: A Leontiev model, see the Phillips reference below. All through the 18th century (that is, well before the founding of modern political economy, conventionally marked by Adam Smith's 1776 Wealth of Nations ), simple probabilistic models were used to understand the economics of insurance . This was a natural extrapolation of the theory of gambling , and played an important role both in the development of probability theory itself and in

512-439: 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

640-514: 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

768-597: A clear basis for soundness, namely the validity of the supporting model. Economic models in current use do not pretend to be theories of everything economic ; any such pretensions would immediately be thwarted by computational infeasibility and the incompleteness or lack of theories for various types of economic behavior. Therefore, conclusions drawn from models will be approximate representations of economic facts. However, properly constructed models can remove extraneous information and isolate useful approximations of key relationships. In this way more can be understood about

896-409: 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

1024-400: 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

1152-633: A fundamental limit to their predictive powers: chaos . Although the modern mathematical work on chaotic systems began in the 1970s the danger of chaos had been identified and defined in Econometrica as early as 1958: It is straightforward to design economic models susceptible to butterfly effects of initial-condition sensitivity. However, the econometric research program to identify which variables are chaotic (if any) has largely concluded that aggregate macroeconomic variables probably do not behave chaotically. This would mean that refinements to

1280-467: A guaranteed payoff, and one with a risky payoff with same average value. In the former scenario, the person receives $ 50. In the uncertain scenario, a coin is flipped to decide whether the person receives $ 100 or nothing. The expected payoff for both scenarios is $ 50, meaning that an individual who was insensitive to risk would not care whether they took the guaranteed payment or the gamble. However, individuals may have different risk attitudes . A person

1408-451: 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

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1536-429: 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

1664-476: A model of behavior, so that an economist can differentiate between changes in relative prices and changes in price that are to be attributed to inflation. In addition to their professional academic interest, uses of models include: A model establishes an argumentative framework for applying logic and mathematics that can be independently discussed and tested and that can be applied in various instances. Policies and arguments that rely on economic models have

1792-561: A more practical level, quantitative modelling is applied to many areas of economics and several methodologies have evolved more or less independently of each other. As a result, no overall model taxonomy is naturally available. We can nonetheless provide a few examples that illustrate some particularly relevant points of model construction. Most economic models rest on a number of assumptions that are not entirely realistic. For example, agents are often assumed to have perfect information, and markets are often assumed to clear without friction. Or,

1920-463: A quadratic utility function exhibiting IARA. The Arrow–Pratt measure of relative risk aversion (RRA) or coefficient of relative risk aversion is defined as Unlike ARA whose units are in $ , RRA is a dimensionless quantity, which allows it to be applied universally. Like for absolute risk aversion, the corresponding terms constant relative risk aversion (CRRA) and decreasing/increasing relative risk aversion (DRRA/IRRA) are used. This measure has

2048-407: A reasoned choice of which variables and which relationships between these variables are relevant and which ways of analyzing and presenting this information are useful. Selection is important because the nature of an economic model will often determine what facts will be looked at and how they will be compiled. For example, inflation is a general economic concept, but to measure inflation requires

2176-537: A risk-averse, expected-utility-maximizing individual who, from any initial wealth level [...] turns down gambles where she loses $ 100 or gains $ 110, each with 50% probability [...] will turn down 50–50 bets of losing $ 1,000 or gaining any sum of money. Rabin criticizes this implication of expected utility theory on grounds of implausibility—individuals who are risk averse for small gambles due to diminishing marginal utility would exhibit extreme forms of risk aversion in risky decisions under larger stakes. One solution to

2304-564: 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

2432-399: 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

2560-411: A smaller commodity share of the bargain. This is because their utility function concaves hence their utility increases at a decreasing rate while their non-risk averse opponents may increase at a constant or increasing rate. Intuitively, a risk-averse person will hence settle for a smaller share of the bargain as opposed to a risk-neutral or risk-seeking individual. Attitudes towards risk have attracted

2688-501: A solution of the paradoxical Saint Petersburg problem . All of these developments were summarized by Laplace in his Analytical Theory of Probabilities (1812). Thus, by the time David Ricardo came along he had a well-established mathematical basis to draw from. In the late 1980s, the Brookings Institution compared 12 leading macroeconomic models available at the time. They compared the models' predictions for how

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2816-417: A specific utility u {\displaystyle u} can be assigned to both outcomes. Now the function becomes; E U ( A ) = 0.3 u ( $ 100 , 000 ) + 0.7 u ( $ 0 ) {\displaystyle EU(A)=0.3u(\$ 100,000)+0.7u(\$ 0)} For a risk averse person, u {\displaystyle u}  would equal

2944-418: A utility function based on how they weigh different outcomes can be deduced. In applying this model to risk aversion, the function can be used to show how an individual’s preferences of wins and losses will influence their expected utility function. For example, if a risk-averse individual with $ 20,000 in savings is given the option to gamble it for $ 100,000 with a 30% chance of winning, they may still not take

3072-589: A value that means that the individual would rather keep their $ 20,000 in savings than gamble it all to potentially increase their wealth to $ 100,000. Hence a risk averse individuals’ function would show that; E U ( A ) ≺ $ 20 , 000 ( k e e p i n g s a v i n g s ) {\displaystyle EU(A)\prec \$ 20,000(keepingsavings)} Using expected utility theory's approach to risk aversion to analyze small stakes decisions has come under criticism. Matthew Rabin has showed that

3200-516: Is CRRA (see below), as c A ( c ) = 1 / a = c o n s t {\displaystyle cA(c)=1/a=const} . See and this can hold only if u ‴ ( c ) > 0 {\displaystyle u'''(c)>0} . Therefore, DARA implies that the utility function is positively skewed; that is, u ‴ ( c ) > 0 {\displaystyle u'''(c)>0} . Analogously, IARA can be derived with

3328-461: Is an indication of risk-seeking behavior in negative prospects and eliminates other explanations for the certainty effect such as aversion for uncertainty or variability. The initial findings regarding the reflection effect faced criticism regarding its validity, as it was claimed that there are insufficient evidence to support the effect on the individual level. Subsequently, an extensive investigation revealed its possible limitations, suggesting that

3456-683: 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

3584-492: 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

3712-409: Is more predictable rather than another situation with a less predictable payoff that is higher on average. For example, a risk-averse investor might choose to put their money into a bank account with a low but guaranteed interest rate, rather than into a stock that may have high expected returns, but also involves a chance of losing value. A person is given the choice between two scenarios: one with

3840-562: Is needed rather than just the second derivative of u ( c ) {\displaystyle u(c)} . One such measure is the Arrow–Pratt measure of absolute risk aversion ( ARA ), after the economists Kenneth Arrow and John W. Pratt , also known as the coefficient of absolute risk aversion , defined as where u ′ ( c ) {\displaystyle u'(c)} and u ″ ( c ) {\displaystyle u''(c)} denote

3968-627: Is now the standard approach. Economic model An economic model is a theoretical construct representing economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified, often mathematical , framework designed to illustrate complex processes. Frequently, economic models posit structural parameters . A model may have various exogenous variables , and those variables may change to create various responses by economic variables. Methodological uses of models include investigation, theorizing, and fitting theories to

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4096-453: 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

4224-411: Is risk averse: a sure amount would always be preferred over a risky bet having the same expected value; moreover, for risky bets the person would prefer a bet which is a mean-preserving contraction of an alternative bet (that is, if some of the probability mass of the first bet is spread out without altering the mean to form the second bet, then the first bet is preferred). There are various measures of

4352-408: Is said to be: The average payoff of the gamble, known as its expected value , is $ 50. The smallest guaranteed dollar amount that an individual would be indifferent to compared to an uncertain gain of a specific average predicted value is called the certainty equivalent , which is also used as a measure of risk aversion. An individual that is risk averse has a certainty equivalent that is smaller than

4480-491: Is that the market's invisible hand guides an economy to prosperity more efficiently than central planning using an economic model. One reason, emphasized by Friedrich Hayek , is the claim that many of the true forces shaping the economy can never be captured in a single plan. This is an argument that cannot be made through a conventional (mathematical) economic model because it says that there are critical systemic-elements that will always be omitted from any top-down analysis of

4608-414: Is the expected value of the risky bet: ( 1 2 0 + 1 2 100 {\displaystyle {\tfrac {1}{2}}0+{\tfrac {1}{2}}100} ). This risk premium means that the person would be willing to sacrifice as much as $ 10 in expected value in order to achieve perfect certainty about how much money will be received. In other words, the person would be indifferent between

4736-474: 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

4864-491: 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,

4992-562: 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

5120-515: 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

5248-545: 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 –

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5376-549: 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

5504-626: 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

5632-674: The Winton Professorship of the Public Understanding of Risk , a role described as outreach rather than traditional academic research by the holder, David Spiegelhalter . Children's services such as schools and playgrounds have become the focus of much risk-averse planning, meaning that children are often prevented from benefiting from activities that they would otherwise have had. Many playgrounds have been fitted with impact-absorbing matting surfaces. However, these are only designed to save children from death in

5760-509: 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:

5888-619: The elasticity of intertemporal substitution often cannot be disentangled from the coefficient of relative risk aversion. The isoelastic utility function exhibits constant relative risk aversion with R ( c ) = ρ {\displaystyle R(c)=\rho } and the elasticity of intertemporal substitution ε u ( c ) = 1 / ρ {\displaystyle \varepsilon _{u(c)}=1/\rho } . When ρ = 1 , {\displaystyle \rho =1,} using l'Hôpital's rule shows that this simplifies to

6016-454: The n-th root of the n-th central moment . The symbol used for risk aversion is A or A n . The von Neumann-Morgenstern utility theorem is another model used to denote how risk aversion influences an actor’s utility function. An extension of the expected utility function, the von Neumann-Morgenstern model includes risk aversion axiomatically rather than as an additional variable. John von Neumann and Oskar Morgenstern first developed

6144-400: The prospect theory , in the behavioral economics domain. The reflection effect is an identified pattern of opposite preferences between negative as opposed to positive prospects: people tend to avoid risk when the gamble is between gains, and to seek risks when the gamble is between losses. For example, most people prefer a certain gain of 3,000 to an 80% chance of a gain of 4,000. When posed

6272-531: 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

6400-630: 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

6528-402: The above bet (with a 50% chance of receiving 100 and a 50% chance of receiving 0) is and if the person has the utility function with u (0)=0, u (40)=5, and u (100)=10 then the expected utility of the bet equals 5, which is the same as the known utility of the amount 40. Hence the certainty equivalent is 40. The risk premium is ($ 50 minus $ 40)=$ 10, or in proportional terms or 25% (where $ 50

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6656-538: The above example c could be $ 0 or $ 40 or $ 100). The utility function u ( c ) is defined only up to positive affine transformation – in other words, a constant could be added to the value of u ( c ) for all c , and/or u ( c ) could be multiplied by a positive constant factor, without affecting the conclusions. An agent is risk-averse if and only if the utility function is concave . For instance u (0) could be 0, u (100) might be 10, u (40) might be 5, and for comparison u (50) might be 6. The expected utility of

6784-485: 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} ,

6912-406: 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

7040-531: The advantage that it is still a valid measure of risk aversion, even if the utility function changes from risk averse to risk loving as c varies, i.e. utility is not strictly convex/concave over all c . A constant RRA implies a decreasing ARA, but the reverse is not always true. As a specific example of constant relative risk aversion, the utility function u ( c ) = log ⁡ ( c ) {\displaystyle u(c)=\log(c)} implies RRA = 1 . In intertemporal choice problems,

7168-506: 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

7296-422: 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,

7424-534: The behavior implied by the utility function) is: where R = 1 / a {\displaystyle R=1/a} and c s = − b / a {\displaystyle c_{s}=-b/a} . Note that when a = 0 {\displaystyle a=0} , this is CARA, as A ( c ) = 1 / b = c o n s t {\displaystyle A(c)=1/b=const} , and when b = 0 {\displaystyle b=0} , this

7552-475: 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

7680-481: The bet and a guarantee of $ 40, and would prefer anything over $ 40 to the bet. In the case of a wealthier individual, the risk of losing $ 100 would be less significant, and for such small amounts his utility function would be likely to be almost linear. For instance, if u(0) = 0 and u(100) = 10, then u(40) might be 4.02 and u(50) might be 5.01. The utility function for perceived gains has two key properties: an upward slope, and concavity. (i) The upward slope implies that

7808-437: The case of log utility , u ( c ) = log c , and the income effect and substitution effect on saving exactly offset. A time-varying relative risk aversion can be considered. The most straightforward implications of increasing or decreasing absolute or relative risk aversion, and the ones that motivate a focus on these concepts, occur in the context of forming a portfolio with one risky asset and one risk-free asset. If

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7936-508: The case of direct falls on their heads and do not achieve their main goals. They are expensive, meaning that less resources are available to benefit users in other ways (such as building a playground closer to the child's home, reducing the risk of a road traffic accident on the way to it), and—some argue—children may attempt more dangerous acts, with confidence in the artificial surface. Shiela Sage, an early years school advisor, observes "Children who are only ever kept in very safe places, are not

8064-734: 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,

8192-553: 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

8320-457: 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

8448-499: 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

8576-450: The cost of losing the utility of the risky activity. It is important to consider the opportunity cost when mitigating a risk; the cost of not taking the risky action. Writing laws focused on the risk without the balance of the utility may misrepresent society's goals. The public understanding of risk, which influences political decisions, is an area which has recently been recognised as deserving focus. In 2007 Cambridge University initiated

8704-425: 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

8832-629: The development of actuarial science . Many of the giants of 18th century mathematics contributed to this field. Around 1730, De Moivre addressed some of these problems in the 3rd edition of The Doctrine of Chances . Even earlier (1709), Nicolas Bernoulli studies problems related to savings and interest in the Ars Conjectandi . In 1730, Daniel Bernoulli studied "moral probability" in his book Mensura Sortis , where he introduced what would today be called "logarithmic utility of money" and applied it to gambling and insurance problems, including

8960-486: The economy is going, or the effect of any shock upon it. The new, more humble, approach sees danger in dramatic policy changes based on model predictions, because of several practical and theoretical limitations in current macroeconomic models; in addition to the theoretical pitfalls, ( listed above ) some problems specific to aggregate modelling are: Complex systems specialist and mathematician David Orrell wrote on this issue in his book Apollo's Arrow and explained that

9088-451: The economy would respond to specific economic shocks (allowing the models to control for all the variability in the real world; this was a test of model vs. model, not a test against the actual outcome). Although the models simplified the world and started from a stable, known common parameters the various models gave significantly different answers. For instance, in calculating the impact of a monetary loosening on output some models estimated

9216-406: The economy. Risk aversion In economics and finance , risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more certain outcome. Risk aversion explains the inclination to agree to a situation with a lower average payoff that

9344-413: The effect is most prevalent when either small or large amounts and extreme probabilities are involved. Numerous studies have shown that in riskless bargaining scenarios, being risk-averse is disadvantageous. Moreover, opponents will always prefer to play against the most risk-averse person. Based on both the von Neumann-Morgenstern and Nash Game Theory model, a risk-averse person will happily receive

9472-458: 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

9600-489: The exact form of these equations. This is because complex systems like the economy or the climate consist of a delicate balance of opposing forces, so a slight imbalance in their representation has big effects. Thus, predictions of things like economic recessions are still highly inaccurate, despite the use of enormous models running on fast computers. See Unreasonable ineffectiveness of mathematics § Economics and finance . Economic and meteorological simulations may share

9728-533: The extent that it accurately mirrors the relationships that it purports to describe. Creating and diagnosing a model is frequently an iterative process in which the model is modified (and hopefully improved) with each iteration of diagnosis and respecification. Once a satisfactory model is found, it should be double checked by applying it to a different data set. According to whether all the model variables are deterministic, economic models can be classified as stochastic or non-stochastic models; according to whether all

9856-1221: The first and second derivatives with respect to c {\displaystyle c} of u ( c ) {\displaystyle u(c)} . For example, if u ( c ) = α + β l n ( c ) , {\displaystyle u(c)=\alpha +\beta ln(c),} so u ′ ( c ) = β / c {\displaystyle u'(c)=\beta /c} and u ″ ( c ) = − β / c 2 , {\displaystyle u''(c)=-\beta /c^{2},} then A ( c ) = 1 / c . {\displaystyle A(c)=1/c.} Note how A ( c ) {\displaystyle A(c)} does not depend on α {\displaystyle \alpha } and β , {\displaystyle \beta ,} so affine transformations of u ( c ) {\displaystyle u(c)} do not change it. The following expressions relate to this term: The solution to this differential equation (omitting additive and multiplicative constant terms, which do not affect

9984-429: 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

10112-538: 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

10240-504: 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

10368-423: 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

10496-646: The gamble in fear of losing their savings. This does not make sense using the traditional expected utility model however; E U ( A ) = 0.3 ( $ 100 , 000 ) + 0.7 ( $ 0 ) {\displaystyle EU(A)=0.3(\$ 100,000)+0.7(\$ 0)} E U ( A ) = $ 30 , 000 {\displaystyle EU(A)=\$ 30,000} E U ( A ) > $ 20 , 000 {\displaystyle EU(A)>\$ 20,000} The von Neumann-Morgenstern model can explain this scenario. Based on preference relations,

10624-522: 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,

10752-427: The interest of the field of neuroeconomics and behavioral economics . A 2009 study by Christopoulos et al. suggested that the activity of a specific brain area (right inferior frontal gyrus) correlates with risk aversion, with more risk averse participants (i.e. those having higher risk premia) also having higher responses to safer options. This result coincides with other studies, that show that neuromodulation of

10880-489: 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

11008-400: 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

11136-877: 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

11264-444: 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

11392-480: The major problems addressed by economic models has been understanding economic growth. An early attempt to provide a technique to approach this came from the French physiocratic school in the eighteenth century. Among these economists, François Quesnay was known particularly for his development and use of tables he called Tableaux économiques . These tables have in fact been interpreted in more modern terminology as

11520-451: 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

11648-406: The model in their book Theory of Games and Economic Behaviour . Essentially, von Neumann and Morgenstern hypothesised that individuals seek to maximise their expected utility rather than the expected monetary value of assets. In defining expected utility in this sense, the pair developed a function based on preference relations. As such, if an individual’s preferences satisfy four key axioms, then

11776-413: The model may omit issues that are important to the question being considered, such as externalities . Any analysis of the results of an economic model must therefore consider the extent to which these results may be compromised by inaccuracies in these assumptions, and a large literature has grown up discussing problems with economic models , or at least asserting that their results are unreliable. One of

11904-399: 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,

12032-535: The models could ultimately produce reliable long-term forecasts. However, the validity of this conclusion has generated two challenges: More recently, chaos (or the butterfly effect) has been identified as less significant than previously thought to explain prediction errors. Rather, the predictive power of economics and meteorology would mostly be limited by the models themselves and the nature of their underlying systems (see Comparison with models in other sciences above). A key strand of free market economic thinking

12160-403: The more the relative risk aversion increases, the more money demand shocks will impact the economy. In modern portfolio theory , risk aversion is measured as the additional expected reward an investor requires to accept additional risk. If an investor is risk-averse, they will invest in multiple uncertain assets, but only when the predicted return on a portfolio that is uncertain is greater than

12288-883: The opposite directions of inequalities, which permits but does not require a negatively skewed utility function ( u ‴ ( c ) < 0 {\displaystyle u'''(c)<0} ). An example of a DARA utility function is u ( c ) = log ⁡ ( c ) {\displaystyle u(c)=\log(c)} , with A ( c ) = 1 / c {\displaystyle A(c)=1/c} , while u ( c ) = c − α c 2 , {\displaystyle u(c)=c-\alpha c^{2},} α > 0 {\displaystyle \alpha >0} , with A ( c ) = 2 α / ( 1 − 2 α c ) {\displaystyle A(c)=2\alpha /(1-2\alpha c)} would represent

12416-455: 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

12544-408: The person experiences an increase in wealth, he/she will choose to increase (or keep unchanged, or decrease) the fraction of the portfolio held in the risky asset if relative risk aversion is decreasing (or constant, or increasing). In one model in monetary economics , an increase in relative risk aversion increases the impact of households' money holdings on the overall economy. In other words,

12672-437: The person experiences an increase in wealth, he/she will choose to increase (or keep unchanged, or decrease) the number of dollars of the risky asset held in the portfolio if absolute risk aversion is decreasing (or constant, or increasing). Thus economists avoid using utility functions such as the quadratic, which exhibit increasing absolute risk aversion, because they have an unrealistic behavioral implication. Similarly, if

12800-405: The person feels that more is better: a larger amount received yields greater utility, and for risky bets the person would prefer a bet which is first-order stochastically dominant over an alternative bet (that is, if the probability mass of the second bet is pushed to the right to form the first bet, then the first bet is preferred). (ii) The concavity of the utility function implies that the person

12928-409: 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

13056-419: The predicted return on one that is not uncertain will the investor will prefer the former. Here, the risk-return spectrum is relevant, as it results largely from this type of risk aversion. Here risk is measured as the standard deviation of the return on investment, i.e. the square root of its variance . In advanced portfolio theory, different kinds of risk are taken into consideration. They are measured as

13184-432: The prediction of uncertain gains. The risk premium is the difference between the expected value and the certainty equivalent. For risk-averse individuals, risk premium is positive, for risk-neutral persons it is zero, and for risk-loving individuals their risk premium is negative. In expected utility theory, an agent has a utility function u ( c ) where c represents the value that he might receive in money or goods (in

13312-755: 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}} ] :

13440-480: 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

13568-407: 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

13696-407: 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

13824-405: The problem observed by Rabin is that proposed by prospect theory and cumulative prospect theory , where outcomes are considered relative to a reference point (usually the status quo), rather than considering only the final wealth. Another limitation is the reflection effect, which demonstrates the reversing of risk aversion. This effect was first presented by Kahneman and Tversky as a part of

13952-663: 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

14080-413: The relationships in question than by trying to understand the entire economic process. The details of model construction vary with type of model and its application, but a generic process can be identified. Generally, any modelling process has two steps: generating a model, then checking the model for accuracy (sometimes called diagnostics). The diagnostic step is important because a model is only useful to

14208-404: 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

14336-458: 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

14464-414: 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

14592-404: 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

14720-464: The risk aversion expressed by those given utility function. Several functional forms often used for utility functions are represented by these measures. The higher the curvature of u ( c ) {\displaystyle u(c)} , the higher the risk aversion. However, since expected utility functions are not uniquely defined (are defined only up to affine transformations ), a measure that stays constant with respect to these transformations

14848-405: 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

14976-406: The same area results in participants making more or less risk averse choices, depending on whether the modulation increases or decreases the activity of the target area. In the real world, many government agencies, e.g. Health and Safety Executive , are fundamentally risk-averse in their mandate. This often means that they demand (with the power of legal enforcement) that risks be minimized, even at

15104-452: The same problem, but for losses, most people prefer an 80% chance of a loss of 4,000 to a certain loss of 3,000. The reflection effect (as well as the certainty effect ) is inconsistent with the expected utility hypothesis. It is assumed that the psychological principle which stands behind this kind of behavior is the overweighting of certainty. Options which are perceived as certain are over-weighted relative to uncertain options. This pattern

15232-573: 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

15360-435: 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 the decision of one individual) with

15488-407: 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

15616-520: 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

15744-581: 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,

15872-547: 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

16000-399: 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

16128-494: The variables are quantitative, economic models are classified as discrete or continuous choice model; according to the model's intended purpose/function, it can be classified as quantitative or qualitative; according to the model's ambit, it can be classified as a general equilibrium model, a partial equilibrium model, or even a non-equilibrium model; according to the economic agent's characteristics, models can be classified as rational agent models, representative agent models etc. At

16256-415: The weather, human health and economics use similar methods of prediction (mathematical models). Their systems—the atmosphere, the human body and the economy—also have similar levels of complexity. He found that forecasts fail because the models suffer from two problems: (i) they cannot capture the full detail of the underlying system, so rely on approximate equations; (ii) they are sensitive to small changes in

16384-682: The world. In general terms, economic models have two functions: first as a simplification of and abstraction from observed data, and second as a means of selection of data based on a paradigm of econometric study. Simplification is particularly important for economics given the enormous complexity of economic processes. This complexity can be attributed to the diversity of factors that determine economic activity; these factors include: individual and cooperative decision processes, resource limitations, environmental and geographical constraints, institutional and legal requirements and purely random fluctuations. Economists therefore must make

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