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.
105-667: [REDACTED] Look up rra in Wiktionary, the free dictionary. RRA can refer to: Arrow-Pratt measure of relative risk-aversion Radioreceptor assay , a test to determine the binding affinity of a radioactive-labelled substance to its receptor Rahanweyn Resistance Army , also known as the Reewin Resistance Army, an armed faction in Somalia Russell Reynolds Associates ,
210-521: A b x f ( x ) d x = ∫ a b x x 2 + π 2 d x = 1 2 ln b 2 + π 2 a 2 + π 2 . {\displaystyle \int _{a}^{b}xf(x)\,dx=\int _{a}^{b}{\frac {x}{x^{2}+\pi ^{2}}}\,dx={\frac {1}{2}}\ln {\frac {b^{2}+\pi ^{2}}{a^{2}+\pi ^{2}}}.} The limit of this expression as
315-401: A weighted average of the x i values, with weights given by their probabilities p i . In the special case that all possible outcomes are equiprobable (that is, p 1 = ⋅⋅⋅ = p k ), the weighted average is given by the standard average . In the general case, the expected value takes into account the fact that some outcomes are more likely than others. Informally,
420-421: A → −∞ and b → ∞ does not exist: if the limits are taken so that a = − b , then the limit is zero, while if the constraint 2 a = − b is taken, then the limit is ln(2) . To avoid such ambiguities, in mathematical textbooks it is common to require that the given integral converges absolutely , with E[ X ] left undefined otherwise. However, measure-theoretic notions as given below can be used to give
525-401: A Microsoft API and server software Richardson Racing Automobiles , a defunct British racing car constructor Religious Research Association , an association of researchers and religious professionals Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title RRA . If an internal link led you here, you may wish to change
630-577: A database term used by the RRD Editor Red Ribbon Army , a fictional military organization that appears in the Dragon Ball metaseries Road Records Association , a British cycle racing organisation Rock River Arms , an American firearms manufacturing company Internal Revenue Service Restructuring and Reform Act of 1998 Rural Reconstruction Association , a British agricultural group Routing and Remote Access ,
735-678: A global executive-search, and leadership and succession consulting firm. Rainbow Routes Association , a non-profit organization in Canada dedicated to sustainable and active transportation Rail America Inc. , NYSE ticker code RRA Rapid Rural Appraisal, a term for Participatory rural appraisal Reiulf Ramstad Arkitekter , a Norwegian architecture and design studio in Oslo Revenue Reconciliation Act of 1993 , US tax reform legislation Round Robin Archive,
840-439: A measure of risk aversion. An individual that is risk averse has a certainty equivalent that is smaller than 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
945-488: A multidimensional random variable, i.e. a random vector X . It is defined component by component, as E[ X ] i = E[ X i ] . Similarly, one may define the expected value of a random matrix X with components X ij by E[ X ] ij = E[ X ij ] . Consider a random variable X with a finite list x 1 , ..., x k of possible outcomes, each of which (respectively) has probability p 1 , ..., p k of occurring. The expectation of X
1050-463: 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
1155-570: A real number μ {\displaystyle \mu } if and only if the two surfaces in the x {\displaystyle x} - y {\displaystyle y} -plane, described by x ≤ μ , 0 ≤ y ≤ F ( x ) or x ≥ μ , F ( x ) ≤ y ≤ 1 {\displaystyle x\leq \mu ,\;\,0\leq y\leq F(x)\quad {\text{or}}\quad x\geq \mu ,\;\,F(x)\leq y\leq 1} respectively, have
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#17327719331971260-537: 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
1365-459: A small circle of mutual scientific friends in Paris about it. In Dutch mathematician Christiaan Huygens' book, he considered the problem of points, and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens published his treatise in 1657, (see Huygens (1657) ) " De ratiociniis in ludo aleæ " on probability theory just after visiting Paris. The book extended
1470-411: 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
1575-417: 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
1680-616: A systematic definition of E[ X ] for more general random variables X . All definitions of the expected value may be expressed in the language of measure theory . In general, if X is a real-valued random variable defined on a probability space (Ω, Σ, P) , then the expected value of X , denoted by E[ X ] , is defined as the Lebesgue integral E [ X ] = ∫ Ω X d P . {\displaystyle \operatorname {E} [X]=\int _{\Omega }X\,d\operatorname {P} .} Despite
1785-409: A utility function u ( c ) where c represents the value that he might receive in money or goods (in 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
1890-418: 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
1995-507: A value in any given open interval is given by the integral of f over that interval. The expectation of X is then given by the integral E [ X ] = ∫ − ∞ ∞ x f ( x ) d x . {\displaystyle \operatorname {E} [X]=\int _{-\infty }^{\infty }xf(x)\,dx.} A general and mathematically precise formulation of this definition uses measure theory and Lebesgue integration , and
2100-589: 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
2205-469: A variety of stylizations: the expectation operator can be stylized as E (upright), E (italic), or E {\displaystyle \mathbb {E} } (in blackboard bold ), while a variety of bracket notations (such as E( X ) , E[ X ] , and E X ) are all used. Another popular notation is μ X . ⟨ X ⟩ , ⟨ X ⟩ av , and X ¯ {\displaystyle {\overline {X}}} are commonly used in physics. M( X )
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#17327719331972310-516: 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
2415-640: Is a Borel function ), we can use this inversion formula to obtain E [ g ( X ) ] = 1 2 π ∫ R g ( x ) [ ∫ R e − i t x φ X ( t ) d t ] d x . {\displaystyle \operatorname {E} [g(X)]={\frac {1}{2\pi }}\int _{\mathbb {R} }g(x)\left[\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt\right]dx.} If E [ g ( X ) ] {\displaystyle \operatorname {E} [g(X)]}
2520-484: Is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration . In the axiomatic foundation for probability provided by measure theory , the expectation is given by Lebesgue integration . The expected value of a random variable X is often denoted by E( X ) , E[ X ] , or E X , with E also often stylized as E {\displaystyle \mathbb {E} } or E . The idea of
2625-461: 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
2730-546: Is any random variable with finite expectation, then Markov's inequality may be applied to the random variable | X −E[ X ]| to obtain Chebyshev's inequality P ( | X − E [ X ] | ≥ a ) ≤ Var [ X ] a 2 , {\displaystyle \operatorname {P} (|X-{\text{E}}[X]|\geq a)\leq {\frac {\operatorname {Var} [X]}{a^{2}}},} where Var
2835-462: Is as in the previous example. A number of convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below. The probability density function f X {\displaystyle f_{X}} of a scalar random variable X {\displaystyle X} is related to its characteristic function φ X {\displaystyle \varphi _{X}} by
2940-635: Is called the probability density function of X (relative to Lebesgue measure). According to the change-of-variables formula for Lebesgue integration, combined with the law of the unconscious statistician , it follows that E [ X ] ≡ ∫ Ω X d P = ∫ R x f ( x ) d x {\displaystyle \operatorname {E} [X]\equiv \int _{\Omega }X\,d\operatorname {P} =\int _{\mathbb {R} }xf(x)\,dx} for any absolutely continuous random variable X . The above discussion of continuous random variables
3045-408: Is defined as E [ X ] = x 1 p 1 + x 2 p 2 + ⋯ + x k p k . {\displaystyle \operatorname {E} [X]=x_{1}p_{1}+x_{2}p_{2}+\cdots +x_{k}p_{k}.} Since the probabilities must satisfy p 1 + ⋅⋅⋅ + p k = 1 , it is natural to interpret E[ X ] as
3150-407: Is easily obtained by setting Y 0 = X 1 {\displaystyle Y_{0}=X_{1}} and Y n = X n + 1 − X n {\displaystyle Y_{n}=X_{n+1}-X_{n}} for n ≥ 1 , {\displaystyle n\geq 1,} where X n {\displaystyle X_{n}}
3255-564: Is equivalent to the representation E [ X ] = ∫ 0 ∞ ( 1 − F ( x ) ) d x − ∫ − ∞ 0 F ( x ) d x , {\displaystyle \operatorname {E} [X]=\int _{0}^{\infty }{\bigl (}1-F(x){\bigr )}\,dx-\int _{-\infty }^{0}F(x)\,dx,} also with convergent integrals. Expected values as defined above are automatically finite numbers. However, in many cases it
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3360-436: Is finite if and only if E[ X ] and E[ X ] are both finite. Due to the formula | X | = X + X , this is the case if and only if E| X | is finite, and this is equivalent to the absolute convergence conditions in the definitions above. As such, the present considerations do not define finite expected values in any cases not previously considered; they are only useful for infinite expectations. The following table gives
3465-626: Is finite, changing the order of integration, we get, in accordance with Fubini–Tonelli theorem , E [ g ( X ) ] = 1 2 π ∫ R G ( t ) φ X ( t ) d t , {\displaystyle \operatorname {E} [g(X)]={\frac {1}{2\pi }}\int _{\mathbb {R} }G(t)\varphi _{X}(t)\,dt,} where G ( t ) = ∫ R g ( x ) e − i t x d x {\displaystyle G(t)=\int _{\mathbb {R} }g(x)e^{-itx}\,dx}
3570-1055: Is fundamental to be able to consider expected values of ±∞ . This is intuitive, for example, in the case of the St. Petersburg paradox , in which one considers a random variable with possible outcomes x i = 2 , with associated probabilities p i = 2 , for i ranging over all positive integers. According to the summation formula in the case of random variables with countably many outcomes, one has E [ X ] = ∑ i = 1 ∞ x i p i = 2 ⋅ 1 2 + 4 ⋅ 1 4 + 8 ⋅ 1 8 + 16 ⋅ 1 16 + ⋯ = 1 + 1 + 1 + 1 + ⋯ . {\displaystyle \operatorname {E} [X]=\sum _{i=1}^{\infty }x_{i}\,p_{i}=2\cdot {\frac {1}{2}}+4\cdot {\frac {1}{4}}+8\cdot {\frac {1}{8}}+16\cdot {\frac {1}{16}}+\cdots =1+1+1+1+\cdots .} It
3675-409: Is given the choice between two scenarios: one with 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
3780-472: Is natural to say that the expected value equals +∞ . There is a rigorous mathematical theory underlying such ideas, which is often taken as part of the definition of the Lebesgue integral. The first fundamental observation is that, whichever of the above definitions are followed, any nonnegative random variable whatsoever can be given an unambiguous expected value; whenever absolute convergence fails, then
3885-562: 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
3990-642: Is otherwise available. For example, in the case of an unweighted dice, Chebyshev's inequality says that odds of rolling between 1 and 6 is at least 53%; in reality, the odds are of course 100%. The Kolmogorov inequality extends the Chebyshev inequality to the context of sums of random variables. The following three inequalities are of fundamental importance in the field of mathematical analysis and its applications to probability theory. The Hölder and Minkowski inequalities can be extended to general measure spaces , and are often given in that context. By contrast,
4095-411: 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
4200-604: Is the Fourier transform of g ( x ) . {\displaystyle g(x).} The expression for E [ g ( X ) ] {\displaystyle \operatorname {E} [g(X)]} also follows directly from the Plancherel theorem . The expectation of a random variable plays an important role in a variety of contexts. In statistics , where one seeks estimates for unknown parameters based on available data gained from samples ,
4305-482: Is the variance . These inequalities are significant for their nearly complete lack of conditional assumptions. For example, for any random variable with finite expectation, the Chebyshev inequality implies that there is at least a 75% probability of an outcome being within two standard deviations of the expected value. However, in special cases the Markov and Chebyshev inequalities often give much weaker information than
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4410-1673: Is then natural to define: E [ X ] = { E [ X + ] − E [ X − ] if E [ X + ] < ∞ and E [ X − ] < ∞ ; + ∞ if E [ X + ] = ∞ and E [ X − ] < ∞ ; − ∞ if E [ X + ] < ∞ and E [ X − ] = ∞ ; undefined if E [ X + ] = ∞ and E [ X − ] = ∞ . {\displaystyle \operatorname {E} [X]={\begin{cases}\operatorname {E} [X^{+}]-\operatorname {E} [X^{-}]&{\text{if }}\operatorname {E} [X^{+}]<\infty {\text{ and }}\operatorname {E} [X^{-}]<\infty ;\\+\infty &{\text{if }}\operatorname {E} [X^{+}]=\infty {\text{ and }}\operatorname {E} [X^{-}]<\infty ;\\-\infty &{\text{if }}\operatorname {E} [X^{+}]<\infty {\text{ and }}\operatorname {E} [X^{-}]=\infty ;\\{\text{undefined}}&{\text{if }}\operatorname {E} [X^{+}]=\infty {\text{ and }}\operatorname {E} [X^{-}]=\infty .\end{cases}}} According to this definition, E[ X ] exists and
4515-514: Is thus a special case of the general Lebesgue theory, due to the fact that every piecewise-continuous function is measurable. The expected value of any real-valued random variable X {\displaystyle X} can also be defined on the graph of its cumulative distribution function F {\displaystyle F} by a nearby equality of areas. In fact, E [ X ] = μ {\displaystyle \operatorname {E} [X]=\mu } with
4620-466: Is used in Russian-language literature. As discussed above, there are several context-dependent ways of defining the expected value. The simplest and original definition deals with the case of finitely many possible outcomes, such as in the flip of a coin. With the theory of infinite series, this can be extended to the case of countably many possible outcomes. It is also very common to consider
4725-403: Is worth just such a Sum, as wou'd procure in the same Chance and Expectation at a fair Lay. ... If I expect a or b, and have an equal chance of gaining them, my Expectation is worth (a+b)/2. More than a hundred years later, in 1814, Pierre-Simon Laplace published his tract " Théorie analytique des probabilités ", where the concept of expected value was defined explicitly: ... this advantage in
4830-674: 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
4935-619: 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
5040-454: 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
5145-400: 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
5250-408: The sample mean serves as an estimate for the expectation, and is itself a random variable. In such settings, the sample mean is considered to meet the desirable criterion for a "good" estimator in being unbiased ; that is, the expected value of the estimate is equal to the true value of the underlying parameter. For a different example, in decision theory , an agent making an optimal choice in
5355-420: The weighted average . Informally, the expected value is the mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would "expect" to get in reality. The expected value of a random variable with a finite number of outcomes
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#17327719331975460-531: The Jensen inequality is special to the case of probability spaces. In general, it is not the case that E [ X n ] → E [ X ] {\displaystyle \operatorname {E} [X_{n}]\to \operatorname {E} [X]} even if X n → X {\displaystyle X_{n}\to X} pointwise. Thus, one cannot interchange limits and expectation, without additional conditions on
5565-398: The Lebesgue theory of expectation is identical to the summation formulas given above. However, the Lebesgue theory clarifies the scope of the theory of probability density functions. A random variable X is said to be absolutely continuous if any of the following conditions are satisfied: These conditions are all equivalent, although this is nontrivial to establish. In this definition, f
5670-531: 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,
5775-534: 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
5880-437: 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
5985-508: 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
6090-507: The concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem (e.g., for three or more players), and can be seen as the first successful attempt at laying down the foundations of the theory of probability . In the foreword to his treatise, Huygens wrote: It should be said, also, that for some time some of the best mathematicians of France have occupied themselves with this kind of calculus so that no one should attribute to me
6195-409: 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 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
6300-462: The corresponding theory of absolutely continuous random variables is described in the next section. The density functions of many common distributions are piecewise continuous , and as such the theory is often developed in this restricted setting. For such functions, it is sufficient to only consider the standard Riemann integration . Sometimes continuous random variables are defined as those corresponding to this special class of densities, although
6405-450: 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
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#17327719331976510-490: The distinct case of random variables dictated by (piecewise-)continuous probability density functions , as these arise in many natural contexts. All of these specific definitions may be viewed as special cases of the general definition based upon the mathematical tools of measure theory and Lebesgue integration , which provide these different contexts with an axiomatic foundation and common language. Any definition of expected value may be extended to define an expected value of
6615-413: 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
6720-527: The expectation of a random variable with a countably infinite set of possible outcomes is defined analogously as the weighted average of all possible outcomes, where the weights are given by the probabilities of realizing each given value. This is to say that E [ X ] = ∑ i = 1 ∞ x i p i , {\displaystyle \operatorname {E} [X]=\sum _{i=1}^{\infty }x_{i}\,p_{i},} where x 1 , x 2 , ... are
6825-432: 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 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
6930-492: The expected value can be defined as +∞ . The second fundamental observation is that any random variable can be written as the difference of two nonnegative random variables. Given a random variable X , one defines the positive and negative parts by X = max( X , 0) and X = −min( X , 0) . These are nonnegative random variables, and it can be directly checked that X = X − X . Since E[ X ] and E[ X ] are both then defined as either nonnegative numbers or +∞ , it
7035-494: The expected value operator is not σ {\displaystyle \sigma } -additive, i.e. E [ ∑ n = 0 ∞ Y n ] ≠ ∑ n = 0 ∞ E [ Y n ] . {\displaystyle \operatorname {E} \left[\sum _{n=0}^{\infty }Y_{n}\right]\neq \sum _{n=0}^{\infty }\operatorname {E} [Y_{n}].} An example
7140-573: The expected value originated in the middle of the 17th century from the study of the so-called problem of points , which seeks to divide the stakes in a fair way between two players, who have to end their game before it is properly finished. This problem had been debated for centuries. Many conflicting proposals and solutions had been suggested over the years when it was posed to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré in 1654. Méré claimed that this problem could not be solved and that it showed just how flawed mathematics
7245-488: The expected values of some commonly occurring probability distributions . The third column gives the expected values both in the form immediately given by the definition, as well as in the simplified form obtained by computation therefrom. The details of these computations, which are not always straightforward, can be found in the indicated references. The basic properties below (and their names in bold) replicate or follow immediately from those of Lebesgue integral . Note that
7350-1221: 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
7455-646: 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,
7560-408: The guaranteed payment or the gamble. However, individuals may have different risk attitudes . A person 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
7665-401: The honour of the first invention. This does not belong to me. But these savants, although they put each other to the test by proposing to each other many questions difficult to solve, have hidden their methods. I have had therefore to examine and go deeply for myself into this matter by beginning with the elements, and it is impossible for me for this reason to affirm that I have even started from
7770-430: The inclination to agree to a situation with a lower average payoff that 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
7875-1193: The indicator function of the event A . {\displaystyle A.} Then, it follows that X n → 0 {\displaystyle X_{n}\to 0} pointwise. But, E [ X n ] = n ⋅ Pr ( U ∈ [ 0 , 1 n ] ) = n ⋅ 1 n = 1 {\displaystyle \operatorname {E} [X_{n}]=n\cdot \Pr \left(U\in \left[0,{\tfrac {1}{n}}\right]\right)=n\cdot {\tfrac {1}{n}}=1} for each n . {\displaystyle n.} Hence, lim n → ∞ E [ X n ] = 1 ≠ 0 = E [ lim n → ∞ X n ] . {\displaystyle \lim _{n\to \infty }\operatorname {E} [X_{n}]=1\neq 0=\operatorname {E} \left[\lim _{n\to \infty }X_{n}\right].} Analogously, for general sequence of random variables { Y n : n ≥ 0 } , {\displaystyle \{Y_{n}:n\geq 0\},}
7980-399: The infinite sum is a finite number independent of the ordering of summands. In the alternative case that the infinite sum does not converge absolutely, one says the random variable does not have finite expectation. Now consider a random variable X which has a probability density function given by a function f on the real number line . This means that the probability of X taking on
8085-427: 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
8190-552: The inversion formula: f X ( x ) = 1 2 π ∫ R e − i t x φ X ( t ) d t . {\displaystyle f_{X}(x)={\frac {1}{2\pi }}\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt.} For the expected value of g ( X ) {\displaystyle g(X)} (where g : R → R {\displaystyle g:{\mathbb {R} }\to {\mathbb {R} }}
8295-447: The letters "a.s." stand for " almost surely "—a central property of the Lebesgue integral. Basically, one says that an inequality like X ≥ 0 {\displaystyle X\geq 0} is true almost surely, when the probability measure attributes zero-mass to the complementary event { X < 0 } . {\displaystyle \left\{X<0\right\}.} Concentration inequalities control
8400-447: The likelihood of a random variable taking on large values. Markov's inequality is among the best-known and simplest to prove: for a nonnegative random variable X and any positive number a , it states that P ( X ≥ a ) ≤ E [ X ] a . {\displaystyle \operatorname {P} (X\geq a)\leq {\frac {\operatorname {E} [X]}{a}}.} If X
8505-403: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=RRA&oldid=986660936 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Risk aversion#Relative risk aversion Risk aversion explains
8610-406: 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
8715-403: 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
8820-432: The newly abstract situation, this definition is extremely similar in nature to the very simplest definition of expected values, given above, as certain weighted averages. This is because, in measure theory, the value of the Lebesgue integral of X is defined via weighted averages of approximations of X which take on finitely many values. Moreover, if given a random variable with finitely or countably many possible values,
8925-401: The ones who are able to solve problems for themselves. Children need to have a certain amount of risk taking ... so they'll know how to get out of situations." Expected value In probability theory , the expected value (also called expectation , expectancy , expectation operator , mathematical expectation , mean , expectation value , or first moment ) is a generalization of
9030-883: 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
9135-408: 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,
9240-437: 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
9345-405: 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
9450-688: 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 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
9555-502: The possible outcomes of the random variable X and p 1 , p 2 , ... are their corresponding probabilities. In many non-mathematical textbooks, this is presented as the full definition of expected values in this context. However, there are some subtleties with infinite summation, so the above formula is not suitable as a mathematical definition. In particular, the Riemann series theorem of mathematical analysis illustrates that
9660-419: 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
9765-405: 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
9870-672: The random variables. To see this, let U {\displaystyle U} be a random variable distributed uniformly on [ 0 , 1 ] . {\displaystyle [0,1].} For n ≥ 1 , {\displaystyle n\geq 1,} define a sequence of random variables X n = n ⋅ 1 { U ∈ ( 0 , 1 n ) } , {\displaystyle X_{n}=n\cdot \mathbf {1} \left\{U\in \left(0,{\tfrac {1}{n}}\right)\right\},} with 1 { A } {\displaystyle \mathbf {1} \{A\}} being
9975-464: 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
10080-406: 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
10185-438: The same finite area, i.e. if ∫ − ∞ μ F ( x ) d x = ∫ μ ∞ ( 1 − F ( x ) ) d x {\displaystyle \int _{-\infty }^{\mu }F(x)\,dx=\int _{\mu }^{\infty }{\big (}1-F(x){\big )}\,dx} and both improper Riemann integrals converge. Finally, this
10290-449: The same fundamental principle. The principle is that the value of a future gain should be directly proportional to the chance of getting it. This principle seemed to have come naturally to both of them. They were very pleased by the fact that they had found essentially the same solution, and this in turn made them absolutely convinced that they had solved the problem conclusively; however, they did not publish their findings. They only informed
10395-418: The same principle. But finally I have found that my answers in many cases do not differ from theirs. In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the expectations of random variables . Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes: That any one Chance or Expectation to win any thing
10500-452: 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
10605-514: The sum hoped for. We will call this advantage mathematical hope. The use of the letter E to denote "expected value" goes back to W. A. Whitworth in 1901. The symbol has since become popular for English writers. In German, E stands for Erwartungswert , in Spanish for esperanza matemática , and in French for espérance mathématique. When "E" is used to denote "expected value", authors use
10710-464: The term is used differently by various authors. Analogously to the countably-infinite case above, there are subtleties with this expression due to the infinite region of integration. Such subtleties can be seen concretely if the distribution of X is given by the Cauchy distribution Cauchy(0, π) , so that f ( x ) = ( x + π ) . It is straightforward to compute in this case that ∫
10815-411: The theory of chance is the product of the sum hoped for by the probability of obtaining it; it is the partial sum which ought to result when we do not wish to run the risks of the event in supposing that the division is made proportional to the probabilities. This division is the only equitable one when all strange circumstances are eliminated; because an equal degree of probability gives an equal right for
10920-411: The value of certain infinite sums involving positive and negative summands depends on the order in which the summands are given. Since the outcomes of a random variable have no naturally given order, this creates a difficulty in defining expected value precisely. For this reason, many mathematical textbooks only consider the case that the infinite sum given above converges absolutely , which implies that
11025-434: Was when it came to its application to the real world. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all. He began to discuss the problem in the famous series of letters to Pierre de Fermat . Soon enough, they both independently came up with a solution. They solved the problem in different computational ways, but their results were identical because their computations were based on
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