Cara ##VERIFIED## Download Eos Utility
Richelle Raridon
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 the above example c could be $0 or $40 or $100).
An agent is risk-seeking 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.
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.
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 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 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).
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 the case of log utility, u(c) = log c, and the income effect and substitution effect on saving exactly offset.
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.[5][6] If 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.
Using expected utility theory's approach to risk aversion to analyze small stakes decisions has come under criticism. Matthew Rabin has showed that a risk-averse, expected-utility-maximizing individual who,
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 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.[18]
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.[21] Based on both the von Neumann-Morgenstern and Nash Game Theory model, a risk-averse person will happily receive a smaller commodity share of the bargain.[22] 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.[23] 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.
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 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 the Winton Professorship of the Public Understanding of Risk, a role described as outreach rather than traditional academic research by the holder, David Spiegelhalter.[26]
The design of experiments, to determine at what increase of wealth or income would an individual change their employment status from a position of security to more risky ventures, must include flexible utility specifications with salient incentives integrated with risk preferences.[34] The application of relevant experiments can avoid the generalisation of varying individual preferences through the use of this model and its specified utility functions.
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For a function $u$, we define the risk aversion function by $r_u(x):=-\fracu''(x)u'(x)$. In your utility function, $r_u(x) = \lambda$; hence, it is a constant absolute risk aversion utility and $\lambda$ is the "coefficient of risk aversion," not the "risk coefficient aversion".
Your problem is the basic, canonical portfolio choice model with utility over final wealth. The guy in your problem just consumes what he has in the end of the time period. This is also important to bear in mind.
if $a^*\rightarrow \infty$, i.e. if the optimal solution is unbounded, then the derivative of the expected utility evaluated at the optimal solution is zero - and since this doesn't make any sense, you have to rephrase it as
Edit: Can you point out a reference where you got this model from? In your framework investor is trying to maximize his wealth. Naturally he invests everything (using leverage) to get maximum return. If you want to run unconstrained optimization, I think that you target function should look a bit different, as you are solving multi-period optimization problem and investor want to maximize his total utility. Usually in economic theory they consider 2 period optimization problem like: $\ \mathop \arg \max \limits_a E[U(C_0) + \frac11 + DF U(C_1))]\ $, where$\ C$ is his wealth/consumption,$\ DF$ is discount factor (investor prefers wealth today, rather than tomorrow).
First, your statement that your utility function goes to infinity is wrong. It's minus exponenta. You can think of it as a minimum of $e^f(x)$ which is bounded below by zero whatever $f(x)$ is. In other words, your utility function is bounded above by 0.
The Constant Absolute Risk Aversion (CARA) utility function is a measure of risk aversion. It is characterized by a constant absolute risk aversion coefficient, meaning risk aversion is the same for all levels of wealth. This is unrealistic, as wealthy investors are less affected by financial losses.
A risk aversion coefficient is a measure of how much utility a person gains or loses as their wealth increases or decreases. It is a measure of risk aversion under the expected utility theory in finance and economics.
The utility function is a mathematical representation of your preference over different uncertain outcomes. The derivative of the utility function with respect to wealth (or other parameters) measures how much the utility changes as the outcome changes.
Your risk aversion determines how much utility (benefit) you get from wealth. Instead of wealth, it can also measure the satisfaction you get from other forms of consumption such as different bundles of goods, services, or investments.
Cox and Huang (1989) provided an answer to this question by applying the Harrison-Kreps-Pliska martingale methodology and obtained an explicit solution (cf. Example 3.4 in their paper). However, their solution is too complicated to analyze and it is also hard to extend their solution to the exponential case. Also Zeng and Taksar (2013) commented that the optimal strategy for the exponential utility function may not be obtained by letting \(\gamma \rightarrow \infty \) in the optimal strategy for the power utility function.
As a result, the value function (3.3) is indeed a solution to the HJB equation (2.6) when both the utility function U and the bequest function B are of the HARA form. This completes the proof.
Therefore, the value function (3.7) is indeed a solution to the original HJB equation (2.6) when both the utility function U and the bequest function B are of exponential form as (2.11). This completes the proof.
Factoring out a negative alpha, and equating the remaining part of the exponential as the certainty equivalent of a random wealth (I might not be explaining that well, but I am almost certain this is the correct path), I can maximize utility by maximizing the utility of the certainty equivalent, which is done by maximizing the certainty equivalent itself.
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