深入理解Risk aversion||风险偏好||Risk utility function

• Risk aversion

  From wikipedia, in economics and finance, risk aversion is the behavior of humans (especially consumers and investors), who, when exposed to uncertainty, attempt to lower that uncertainty.

  It is the hesitation(犹豫) of a person to agree to asituation with an unknown payoff tather than another situation with a more predictable payoff but possibly lower expected payoff.

  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.

• Example

  A person is given the choice between two scenarios(情景), one with a guranteed payoff and one without. In the guaranteed 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 guranteed payment or the gamble.

  However, individuals may have different risk attitudes:

  1. risk averse (or risk avoiding) - if they would accept a certain payment (certainty equivalent) of less than $50 (for example, $40), rather than taking the gamble and possibly receiving nothing.
  2. risk neutral - if they are indifferent between the bet and a certain $50 payment.
  3. risk loving (or risk seeking) - if they would accept the bet even when the guranteed payment is more than $50 (like $60).

  The average payoff of the gamble, known as its expected value, is $50.

  certainty equivalent : the smallest dollar amount that the individual would accept instead of the bet.

  risk premium(风险溢价): the difference between the expected value and the certainty equivalent :

  • risk-averse individuals, it is positive (50-40)

  • risk-neutral individuals, it is zero (50-50)

  • risk-loving individuals, it is negative (50-60)

    Type	Expected Value	Certainty Equivalent	Risk Premium
    Risk aversion	50	40	10
    Risk neutral	50	50	0
    Risk loving	50	60	-10

• Utility of money

  In expected utility theory, an agent has a utility function $u(x)$ 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 the conclusion.

  An agent possesses risk aversion if and only if the utiltiy function is concave(凹). For instance $u(0)$ could be $0$, $u(100)$ might be $10$, $u(40)$ might be $5$, 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
$$E(u)=\frac{u(0)+u(100)}{2}$$
and if the person has the utility function with $u(0)=0,u(40)=5,u(100)=10$ 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) = $10, or in proportional terms :
KaTeX parse error: Can't use function '$' in math mode at position 10: \frac{$̲50-$40}{$40}
or 25% (where $50 is the expected value of the risky bet). 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 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 $4.0001$ and $u(50)$ might be $5.0001$.

The utility function for perceived gains has two key properties:

• an upward slope

  The upward slope implies that 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)

• concavity

  The concavity of the utility function implies that the person 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).

• Measures of risk aversion under expected utility theory

  There are multiple measures of the risk aversion expressed by a given utility function. Several functional forms often used for utility functions are expressed in terms of these measures.

  • Absolute risk aversion

    The higher the curvature of $u(c)$, the higher the risk aversion.

    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 :
    $$A(c)=-\frac{u^{{}^{\prime \prime }}(c)}{u^{{}^{\prime }}(c)}$$
    $u^{{}^{\prime \prime }}$ : the second derivatives with respect to $c$ of $u(c)$;

    $u^{{}^{\prime }}$ : the first derivatives with respect to $c$ of $u(c)$;

  • Relative risk aversion

    Arrow-Pratt measure of relative risk aversion (RRA) or coefficient of relative risk aversion is defined as :
    $$R(c)=cA(c)=-\frac{c{u}^{{}^{\prime \prime }}(c)}{u^{{}^{\prime }}(c)}$$
    RRA is a dimension-less quantity.

  • Portfolio Theory

    In MPT, risk aversion is measured as the additional expected reward an investor requires to accept additional risk.

    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 the n-th root of the n-th central moment.

    The symbol used for risk aversion is $A$ or $A_n$ :
    $$A=\frac{dE(c)}{d\sigma }$$

    $$A_n=\frac{dE(c)}{d{\sqrt{\mu }}_n}$$

$$\begin{gathered} lambda=\frac{dE(c)}{d\sigma } \\ or \\ \frac{dE(c)}{d\sigma }里面包含着lambda \end{gathered}$$

• 《理解Utility效用in economy》

  对于风险规避（Risk Aversion）的主体，$u(40)$就可以得到$5$的满意度。

• 《理解期望效用假设Expected Utility Hypothesis》