Statistics
Monte Carlo methods
A broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Their essential idea is using randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution. Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and, in math, evaluation of multidimensional definite integrals with complicated boundary conditions. In application to space and oil exploration problems, Monte Carlo-based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative "soft" methods. In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. By the law of large numbers, integrals described by the expected value of some random variable can be approximated by taking the empirical mean (a.k.a. the sample mean) of independent samples of the variable. When the probability distribution of the variable is parametrized, mathematicians often use a Markov Chain Monte Carlo (MCMC) sampler. The central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution. By the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler.
Real business-cycle theory
A class of New classical macroeconomics models in which business-cycle fluctuations to a large extent can be accounted for by real (in contrast to nominal) shocks. Unlike other leading theories of the business cycle, RBC theory sees business cycle fluctuations as the efficient response to exogenous changes in the real economic environment. That is, the level of national output necessarily maximizes expected utility, and governments should therefore concentrate on long-run structural policy changes and not intervene through discretionary fiscal or monetary policy designed to actively smooth out economic short-term fluctuations. According to RBC theory, business cycles are therefore "real" in that they do not represent a failure of markets to clear but rather reflect the most efficient possible operation of the economy, given the structure of the economy. Real business cycle theory categorically rejects Keynesian economics and the real effectiveness of monetary policy as promoted by monetarism and New Keynesian economics, which are the pillars of mainstream macroeconomic policy. RBC theory differs in this way from other theories of the business cycle such as Keynesian economics and monetarism. RBC theory is associated with freshwater economics (the Chicago School of Economics in the neoclassical tradition).
Statistical model
A class of mathematical model, which embodies a set of assumptions concerning the generation of some sample data, and similar data from a larger population. A statistical model represents, often in considerably idealized form, the data-generating process. The assumptions embodied by a statistical model describe a set of probability distributions, some of which are assumed to adequately approximate the distribution from which a particular data set is sampled. The probability distributions inherent in statistical models are what distinguishes statistical models from other, non-statistical, mathematical models. A statistical model is usually specified by mathematical equations that relate one or more random variables and possibly other non-random variables. As such, "a model is a formal representation of a theory" (Herman Adèr quoting Kenneth Bollen). All statistical hypothesis tests and all statistical estimators are derived from statistical models. More generally, statistical models are part of the foundation of statistical inference.
Differential equation
A mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
Finite-state machine (state machine)
A mathematical model of computation used to design both computer programs and sequential logic circuits. It is conceived as an abstract machine that can be in one of a finite number of states. The machine is in only one state at a time; the state it is in at any given time is called the current state. It can change from one state to another when initiated by a triggering event or condition; this is called a transition. A particular FSM is defined by a list of its states, its initial state, and the triggering condition for each transition.
Alpha (finance)
A measure of the active return on an investment, the performance of that investment compared to a suitable market index. An alpha of 1% means the investment's return on investment over a selected period of time was 1% better than the market during that same period, an alpha of -1 means the investment underperformed the market. Alpha is one of the five key measures in modern portfolio theory: alpha, beta, standard deviation, R-squared and the Sharpe ratio. In modern financial markets, where index funds are widely available for purchase, alpha is commonly used to judge the performance of mutual funds and similar investments. As these funds include various fees normally expressed in percent terms, the fund has to maintain an alpha greater than its fees in order to provide positive gains compared to an index fund. Historically, the vast majority of traditional funds have had negative alphas, which has led to a flight of capital to index funds and non-traditional hedge funds. It is also possible to analyze a portfolio of investments and calculate a theoretical performance, most commonly using the capital asset pricing model (CAPM). Returns on that portfolio can be compared to the theoretical returns, in which case the measure is known as Jensen's alpha. This is useful for non-traditional or highly focused funds, where a single stock index might not be representative of the investment's holdings.
Bayesian inference/method
A method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability".
Stochastic process
A random process evolving with time. More specifically, in probability theory, a time sequence representing the evolution of some system represented by a variable whose change is subject to a random variation. This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic, or random process, there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve. In many stochastic processes, the movement to the next state or position depends on only the current state, and is independent from prior states or values the process has taken. In the simple case of discrete time, as opposed to continuous time, a stochastic process is a sequence of random variables.
Bayesian hierarchical modeling
A statistical model written in multiple levels (hierarchical form) that estimates the parameters of the posterior distribution using the Bayesian method. The sub-models combine to form the hierarchical model, and the Bayes' theorem is used to integrate them with the observed data, and account for all the uncertainty that is present. The result of this integration is the posterior distribution, also known as the updated probability estimate, as additional evidence on the prior distribution is acquired. Frequentist statistics, the more popular foundation of statistics, has been known to contradict Bayesian statistics due to its treatment of the parameters as a random variable, and its use of subjective information in establishing assumptions on these parameters. However, Bayesians argue that relevant information regarding decision making and updating beliefs cannot be ignored and that hierarchical modeling has the potential to overrule classical methods in applications where respondents give multiple observational data. Moreover, the model has proven to be robust, with the posterior distribution less sensitive to the more flexible hierarchical priors. Hierarchical modeling is used when information is available on several different levels of observational units. The hierarchical form of analysis and organization helps in the understanding of multiparameter problems and also plays an important role in developing computational strategies.
Bayesian probability
An interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, assigned probabilities represent states of knowledge or belief. The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses, i.e., the propositions whose truth or falsity is uncertain. In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist inference, a hypothesis is typically tested without being assigned a probability. Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies some prior probability, which is then updated to a posterior probability in the light of new, relevant data (evidence). The Bayesian interpretation provides a standard set of procedures and formulae to perform this calculation. Broadly speaking, there are two views on Bayesian probability that interpret the probability concept in different ways. According to the objectivist view, the rules of Bayesian statistics can be justified by requirements of rationality and consistency and interpreted as an extension of logic. According to the subjectivist view, probability quantifies a "personal belief".
Fama-French three-factor model
In asset pricing and portfolio management, a model designed by Eugene Fama and Kenneth French to describe stock returns. The three factors are 1. company Size, 2. company Price-to-Book Ratio, and 3. Market Risk. The traditional asset pricing model, known formally as the capital asset pricing model (CAPM) uses only one variable to describe the returns of a portfolio or stock with the returns of the market as a whole. In contrast, the Fama-French model uses three variables. Fama and French started with the observation that two classes of stocks have tended to do better than the market as a whole: (i) small caps and (ii) stocks with a low Price-to-Book ratio (P/B, customarily called value stocks, contrasted with growth stocks). They then added two factors to CAPM to reflect a portfolio's exposure to these two classes. r is the portfolio's expected rate of return, Rf is the risk-free return rate, and Km is the return of the market portfolio. The "three factor" β is analogous to the classical β but not equal to it, since there are now two additional factors to do some of the work. SMB stands for "Small [market capitalization] Minus Big" and HML for "High [book-to-market ratio] Minus Low"; they measure the historic excess returns of small caps over big caps and of value stocks over growth stocks. These factors are calculated with combinations of portfolios composed by ranked stocks (BtM ranking, Cap ranking) and available historical market data.
Deterministic algorithm
In computer science, an algorithm which, given a particular input, will always produce the same output, with the underlying machine always passing through the same sequence of states. By far the most studied and familiar kind of algorithm, as well as one of the most practical, since they can be run on real machines efficiently. Formally, it computes a mathematical function; a function has a unique value for any input in its domain, and the algorithm is a process that produces this particular value as output.
Monte Carlo algorithm
In computing, a randomized algorithm whose running time is deterministic, but whose output may be incorrect with a certain (typically small) probability.
Systematic risk
In finance and economics, vulnerability to events which affect aggregate outcomes such as broad market returns, total economy-wide resource holdings, or aggregate income. Often called aggregate risk or undiversifiable risk in economics. In many contexts, events like earthquakes and major weather catastrophes pose aggregate risks—they affect not only the distribution but also the total amount of resources. If every possible outcome of a stochastic economic process is characterized by the same aggregate result (but potentially different distributional outcomes), then the process has no aggregate risk. Systematic risk plays an important role in portfolio allocation. Risk which cannot be eliminated through diversification commands returns in excess of the risk-free rate (while idiosyncratic risk does not command such returns since it can be diversified). Over the long run, a well-diversified portfolio provides returns which correspond with its exposure to systematic risk; investors face a trade-off between returns and systematic risk. Therefore, an investor's desired returns correspond with their desired exposure to systematic risk and corresponding asset selection. Investors can only reduce a portfolio's exposure to systematic risk by sacrificing returns.
Arbitrage pricing theory
In finance, a general theory of asset pricing that holds that the expected return of a financial asset can be modeled as a linear function of various macro-economic factors or theoretical market indices, where sensitivity to changes in each factor is represented by a factor-specific beta coefficient. The model-derived rate of return will then be used to price the asset correctly - the asset price should equal the expected end of period price discounted at the rate implied by the model. If the price diverges, arbitrage should bring it back into line. Under the APT, an asset is mispriced if its current price diverges from the price predicted by the model.
Heston model
In finance, a mathematical model describing the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process.
Capital asset pricing model
In finance, a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. The model takes into account the asset's sensitivity to non-diversifiable risk (also known as systematic risk or market risk), often represented by the quantity beta (β) in the financial industry, as well as the expected return of the market and the expected return of a theoretical risk-free asset. CAPM assumes a particular form of utility functions (in which only first and second moments matter, that is risk is measured by variance, for example a quadratic utility) or alternatively asset returns whose probability distributions are completely described by the first two moments (for example, the normal distribution) and zero transaction costs (necessary for diversification to get rid of all idiosyncratic risk). Under these conditions, CAPM shows that the cost of equity capital is determined only by beta. Despite it failing numerous empirical tests, and the existence of more modern approaches to asset pricing and portfolio selection (such as arbitrage pricing theory and Merton's portfolio problem), the CAPM still remains popular due to its simplicity and utility in a variety of situations.
Sharpe ratio
In finance, the Sharpe ratio (also known as the reward-to-variability ratio) is a way to examine the performance of an investment by adjusting for its risk. The ratio measures the excess return (or risk premium) per unit of deviation in an investment asset or a trading strategy, typically referred to as risk (and is a deviation risk measure). The Sharpe ratio characterizes how well the return of an asset compensates the investor for the risk taken. When comparing two assets versus a common benchmark, the one with a higher Sharpe ratio provides better return for the same risk (or, equivalently, the same return for lower risk). However, like any other mathematical model, it relies on the data being correct. Pyramid schemes with a long duration of operation would typically provide a high Sharpe ratio when derived from reported returns, but the inputs are false. When examining the investment performance of assets with smoothing of returns (such as with-profits funds) the Sharpe ratio should be derived from the performance of the underlying assets rather than the fund returns. Sharpe ratios, along with Treynor ratios and Jensen's alphas, are often used to rank the performance of portfolio or mutual fund managers.
Beta (finance)
In finance, the beta (β) of an investment indicates whether the investment is more or less volatile than the market. In general, a beta less than 1 indicates that the investment is less volatile than the market, while a beta more than 1 indicates that the investment is more volatile than the market. Volatility is measured as the fluctuation of the price around the mean: the standard deviation. Beta is a measure of the risk arising from exposure to general market movements as opposed to idiosyncratic factors. The market portfolio of all investable assets has a beta of exactly 1. A beta below 1 can indicate either an investment with lower volatility than the market, or a volatile investment whose price movements are not highly correlated with the market. An example of the first is a treasury bill: the price does not go up or down a lot, so it has a low beta. An example of the second is gold. The price of gold does go up and down a lot, but not in the same direction or at the same time as the market. A beta greater than one generally means that the asset both is volatile and tends to move up and down with the market. An example is a stock in a big technology company. Negative betas are possible for investments that tend to go down when the market goes up, and vice versa. There are few fundamental investments with consistent and significant negative betas, but some derivatives like put options can have large negative betas. Beta is important because it measures the risk of an investment that cannot be reduced by diversification. It does not measure the risk of an investment held on a stand-alone basis, but the amount of risk the investment adds to an already-diversified portfolio. In the capital asset pricing model, beta risk is the only kind of risk for which investors should receive an expected return higher than the risk-free rate of interest.
Tracking error
In finance, tracking error or active risk is a measure of the risk in an investment portfolio that is due to active management decisions made by the portfolio manager; it indicates how closely a portfolio follows the index to which it is benchmarked. The best measure is the standard deviation of the difference between the portfolio and index returns.
Partial differential equation
In mathematics, a differential equation that contains unknown multivariable functions and their partial derivatives. (A special case are ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model.
Ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.
Bayes' rule
In probability theory and applications, Bayes's rule relates the odds of event A1 to the odds of event A2, before (prior to) and after (posterior to) conditioning on another event B. The odds on A1 to event A2 is simply the ratio of the probabilities of the two events. The prior odds is the ratio of the unconditional or prior probabilities, the posterior odds is the ratio of conditional or posterior probabilities given the event B. The relationship is expressed in terms of the likelihood ratio or Bayes factor, Λ (Lambda). By definition, this is the ratio of the conditional probabilities of the event B given that A1 is the case or that A2 is the case, respectively. The rule simply states: posterior odds equals prior odds times Bayes factor.
Bayes' theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule) describes the probability of an event, based on conditions that might be related to the event. For example, if cancer is related to age, then, using Bayes' theorem, a person's age can be used to more accurately assess the probability that they have cancer. One of the many applications of Bayes' theorem is Bayesian inference, a particular approach to statistical inference.
Markov chain
In probability theory and statistics, a stochastic process that satisfies the Markov property (usually characterized as "memorylessness"). Loosely speaking, this is when predictions for the future of the process can be made based solely on its present state just as well as one could knowing the process's full history. i.e., conditional on the present state of the system, its future and past are independent. The term "Markov chain" refers to the sequence of random variables such a process moves through, with the Markov property defining serial dependence only between adjacent periods (as in a "chain"). It can thus be used for describing systems that follow a chain of linked events, where what happens next depends only on the current state of the system. Markov chains and other random walks are not deterministic systems, because their development depends on random choices.
Modern portfolio theory
Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk, defined as variance. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return. Economist Harry Markowitz introduced MPT in a 1952 essay, for which he was later awarded a Nobel Prize in economics.
