Questions tagged [utility-theory]

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103 views

How to evaluate the following integral?

I stumbled across an expression and I wonder how to evaluate this: $-\int_ {0} ^ {+\infty} {v(x)} dw^{+} (1-p(x))$ where $v(x)$ is some utility function and $w(p(x)) $ is a decision weighting function,...
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37 views

Expected value of stepwise function of Prospect Theory as improper integral

I am looking for a sufficiently precise solution algorithm to evaluate the following integral: $$\int_ {-\infty} ^ {\infty} {v(x)} {f(x)} dx$$ where $v(x)$ is the Prospect Theory value function and $f(...
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12 views

How does this imply that a Pareto optimum maximizes a weighted average of utility functions?

I'm reading a passage from Asset Pricing and Portfolio Choice Theory by Kerry Back, and I don't understand some of it. I would appreciate any help anyone could provide me. In the passage, Back is ...
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2answers
160 views

Deriving the risk-aversion coefficient

By considering the parametrised formulation of the mean-variance criterion by Markowitz, the risk aversion coefficient $\lambda$ can be derived as follow. As suggested by Arrow and Pratt, given the ...
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31 views

Iso-expected-value locus

I was considering the following indifference curve/Bernoulli utility function u: For $ p_1 x_1 + p_2 x_2 = w $ it says, that this is an iso-expected-value locus. Can somebody explain what an iso-...
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48 views

finding optimal weight using Kelly criterion

Question: Suppose you have two strategies. Strategy 1 gains 8% with probability p, and loses 5% with probability 1-p, where p = 0.53. Strategy 2 gains 8% with probability q, and loses 5% with ...
3
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2answers
237 views

How to compare mean-variance-skewness-kurtosis portfolios obtained by expected utility maximization?

Suppose I have some portfolios which are the result of maximizing the expected utility of different approximations of a utility function, how do you test these portfolio's out-of-sample and how do you ...
3
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1answer
502 views

Anyone has detailed explanation on how to use epstein-zin preferences in asset pricing models

I'd be interested to know how Epstein-Zin preferences are used in, say, consumption-based asset pricing models. I'm looking for specific derivations (how you get the SDF) and possible numerical ...
3
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1answer
207 views

ETF pricing papers

May I request for research paper recommendations, if any, on existing models that study how the presence of ETFs affect equilibrium prices of the underlying assets? I am exploring a project on a ...
3
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1answer
201 views

Application of Ito's Lemma in expected utility theory

An investor with utility curve $U(.)$ has wealth $X_t$ at time t. He invests A proportion $p$ of his wealth in a risky asset that follows a geometric Brownian motion, with parameters $\mu$ and $\...
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66 views

How do I maximize my expected utility of wealth?

Suppose I have a utility function say $U(p)=p^{1/2}$ and I bet on a basketball game. I have my initial investment, payouts and probabilities of winning, how can I determine the maximum I need to bet ...
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1answer
41 views

Closed Form Solution for Implied Risk Aversion with Two Assets under Quadratic Utility

So I believe there should be a closed form solution for implied risk aversion for two assets but I'm not sure how to get there. Say you have Quadratic Utility $U$ on a fully invested portfolio of two ...
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1answer
252 views

Fixes of quadratic utility when probability of decreasing utility is large

In finance and specifically portfolio theory, a popular utility function is quadratic utility $$ u(x)=x-\frac{\lambda}{2}(x-\mu_X)^2 $$ where $x$ is wealth and $\lambda$ is the parameter of risk ...
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1answer
129 views

Prove that the portfolio that maximizes utility lies on the efficient frontier

When maximizing mean-variance utility in a portfolio optimization framework $max \{R - \lambda \sigma ^2\}$ where R is portfolio return, $\lambda$ is a risk aversion parameter, and $\sigma^2$ is ...
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30 views

Compute Utility From Portfolio Holdings Over Time

I have a dataset comprising daily stock holdings for individual investors over a one year-period. I only know about the individuals' investment in stocks. I have no information on any other wealth of ...
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0answers
126 views

Risk-aversion parameters estimation in utility functions

Are there any "typical" risk-aversion parameters for power utility function and exponential utility function? Once I've seen an articel, in which author stated that for extremely risky person gamma in ...
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0answers
123 views

Utility-based portfolio optimization

I think I can't get the idea of optimization based on utility. For some reasons, I should choose one of several common utility functions (exponential, isoelastic function and some others). Obviously, ...
2
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1answer
88 views

Risk neutrality coherence with risk aversion

I haven't been able to find an understandable explanation why the risk neutrality is coherent with the risk aversion implication of the expected utility hypothesis. I can see that when using the risk ...
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272 views

Merton's portfolio problem with constraints

Suppose the investor can invest in a Black-Scholes market with one risky asset $S$ with drift $\alpha$ and volatility $\sigma$ and a riskless asset $B$ with a riskless rate of return $r$, and the ...
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39 views

Utility Maximization on a finite Probability Space. Possible mistakes in a paper?

I am currently reading this paper on utility maximization in a financial market model. On page 5 the author starts with the case of a finite probability space and on page 19 he considers the ...
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441 views

Is there a relationship between Risk Neutral Pricing framework and Nash Equilibria?

Based on the Fundamental Theorem of Asset Pricing, the risk neutral price of a contingent claim on an asset in a liquid, arbitrage free market can be determined by switching to an equivalent $Q-$ ...
5
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1answer
297 views

Is positive skewness preferences rational or irrational?

Is positive skewness preference rational or irrational? I have a great trouble understanding why investors should prefer positive skewness over negative one. Sometimes it is argued that preference ...
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0answers
36 views

Hedger's utility function associated with minimizing value at risk

What must be the general shape of a hedger's utility function if the hedger is minimizing value at risk? What is a simple example of such a utility function?
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1answer
672 views

What is the relation between Relative Risk Aversion and Market Price of Risk

If we assume that the preferences of investors in a market aggregate to display the following utility function $$u(W)=\dfrac{1}{1-\gamma}W^{1-\gamma},\quad \gamma>0,\quad \gamma\neq1$$ then from$...
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1answer
275 views

Does CRRA-utility imply higher risk-aversion for lower wealth?

Consider the utility function $u(W)=\dfrac{1}{1-\gamma}W^{1-\gamma}$, where $\gamma=0.5$ Since this function will exhibit decreasing marginal utility of wealth, is it correct to say that for any ...
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623 views

Debreu's Representation Theorem proof

In microeconomy this theorem states that : given a consumption set $X\subseteq\mathbb{R}^n$, if the preference relation $\succcurlyeq$ is complete, transitive and continuous there exist a utility ...
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1answer
1k views

Finding parameters of an utility function in a market making strategy to apply it in practice

I am reading this paper below about optimal bid-ask spread in a market making strategy. It finds an approximation for optimal solution, but I cannot understand how it's practice to set the parameters ...
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91 views

How to derive Expected utility approximation with power function

My question concern how to derive the expected utility of a power function in the following model: I have two normally distributed risky assets X and Y and a risk-free asset B, for which : rA = 0.5y ...
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138 views

optimizing the expected utility

The market consist of one single stock and call options with different strike price based on the given stock.Suppose the market believes the stock follows the following GBM:$$dS_t=\mu S_tdt+\sigma ...
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1answer
213 views

Why is this utility function not picking up its penalty?

I was reading this seminal paper by Infanger. On page 40, Figure 11. was quite interesting. In particular I was interested in the top one, 19 Years and I wanted to reproduce this plot. To give some ...
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0answers
91 views

Why is utility concave?

I have read that the utility function is usually concave. I assume this requirement arises in order to meet the diversification effect:$$f(\lambda_1c_1+\lambda_2c_2)\ge \lambda_2 f(c_1)+\lambda_2f(c_2)...
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173 views

Martingale method for utility maximization - is the optimal strategy also a martingale?

The Martingale Method for utility maximization (seen in e.g. Björk's book) is based on separating the optimization problem $E^\mathbb{P}[U(X_T)]$ over a class of admissible strategies into the static ...
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2answers
621 views

Why are some utility functions widely used?

There are some von-Neumann utility functions that I come across quite often in different articles / books like: $ U(x)=\ln(x)$, $U(x)= \frac {1}{\gamma}x^\gamma$ with $\gamma <1$ and $U(x)=\frac {1-...
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1answer
106 views

Utility-optimal leverage with costs

Say I have a portfolio, $X_t$, using a leverage of $f$, such that the dynamics are given by \begin{equation} dX_t = \mu f X_t dt + \sigma f X_t dW_t \end{equation} I want to optimize the expected ...
2
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1answer
848 views

Are Insurance and Risk premium totally different?

I've been studying various aspects of utility function and I came across the definition of risk premium and insurance, which are mathematically very different from each other. In the book "Theory of ...
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708 views

Examples of risk-seeking utility functions?

In the past, most literature assumed a risk-averse investor to model utility preferences. This includes the CRRA and CARA utility functions. In recent papers, researchers state that investors may be ...
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1answer
134 views

Utility function for avoiding investment

An investor has initial wealth $30000$ and utility function $\ln{x}$. He is planning to invest in a project where he has $60%$ chance of gaining $\alpha%$ and $40%$ chance of losing $\beta%$. Express ...
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1answer
120 views

Differentiating risk aversion based on utility theory

A utility function $U$ whose corresponding relative risk aversion function is a linear, increasing function satisfies the differential equation $$-x\frac{U''(x)}{U'(x)}=ax+b$$ for some constants $a&...
3
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1answer
203 views

List of risk-averse utility functions

In the context of optimal portfolio allocation, I am looking for a (possibly exhaustive) list of risk-averse utility functions verifying part of the so-called Inada conditions. Essentially, I am ...
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1answer
1k views

Explain Four Basic Axioms of Maximising Expected Utility

I begin learn PRM , Someone help me understand Four Basic Axioms of Maximising Expected Utility most intuitive way .Thank you very much
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1answer
245 views

example Hamilton-Jacobi-Bellman Equation - clarification of $dX_t$ derivation using $\pi_t$, $\Pi_t$

I have a market with safe rate r and risky asset S $$ \frac{dS_t}{S_t}=(r+Y_t)dt+\sigma dW_t \quad \quad (1)$$ $$ dY_t = - \lambda Y_t +dB_t \quad \quad (2)$$ where W, B are Brownian Motions with ...
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1answer
87 views

optimal strategy problem (using Jensen's inequality)

I have a strategy in Samuelson model with zero safe rate defined as $$Z_t^{\Pi}=\frac{X_t^{\Pi}}{X_t^{\rho}} \quad \quad (1)$$ where $$\frac{dX_t^{\Pi}}{X_t^{\Pi}} = \mu \pi dt + \sigma \pi \ dW_t \...
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3answers
11k views

Why do we assume quadratic utility in portfolio theory?

In my text (Investments by BKM), the investor's mean-variance utility (given as $U = E[R] - \frac12A\sigma^2$) is stated to be the objective function we wish to maximize. Upon further digging, it ...
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2answers
626 views

Utility Theory and portfolio optimization - Proof of a lemma

I have a question on the following problem from chapter 9 of D. Luenberger, Investment Science, International Edition: (Portfolio Optimization) Suppose an investor has utility function $U$. There are ...
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1answer
2k views

Utility Theory - Certainty equivalent approximation formula derivation

I have a question on an exercise from chapter 9 of D. Luenberger, Investment Science, International Edition, where I suspect there may be a typo. Exercise 8 (Certainty approximation) There is a ...
2
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1answer
3k views

Utility Theory - How to show that this exponential utility function is wealth-independent?

I have a question on the following exercise from chapter 9 of D. Luenberger, Investment Science, International Edition. Exercise 2 (Wealth Independence) Suppose an investor has exponential utility ...
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2answers
178 views

How to arrive at expectation of negative utility function via Taylor series expansion

I'm attempting to follow an author's steps in an argument and having trouble seeing how Taylor series expansion can be applied to give the stated result. The scenario is as follows. The mid price ...
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1answer
261 views

Investor choice problem

Guys I'm stuck with a problem... Consider the portfolio choice problem of a risk-averse individual with a strictly increasing utility function. There is a single risky asset, and a risk free asset. ...
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2answers
236 views

Expected Utility

We know that under certainty, any increasing monotone transformation of a utility function is also a utility function representing the same preferences. Under uncertainty, we must restrict this ...
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0answers
84 views

Financial theory

Ok guys, I'm studying from Danthine and Donaldson - Intermediate Financial Theory. The book itself doesn't have a lot of worked examples, and I'm lacking the basics for understanding some concepts ...