Questions tagged [utility-theory]
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55
questions
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1answer
91 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 ...
0
votes
1answer
87 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 ...
0
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0answers
52 views
CRRA Utility Function Problem
"Assume an investor with total wealth of $100 that has a constant relative risk aversion (CRRA) utility function. The functional formula for the CRRA utility function is given as
$\ U[W]=\frac{W^{1-θ}...
0
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0answers
48 views
Where does this inequality come from?
We consider a market in which you can buy at $t=0$ only a risk-free asset, its yield at $t=1$ is the real number $r_{f}$, and one risky asset, its yield at $t=1$ is described by the random variable $...
0
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0answers
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 ...
1
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0answers
76 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 ...
2
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0answers
83 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, ...
1
vote
1answer
80 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 ...
5
votes
0answers
189 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 ...
3
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0answers
33 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 ...
1
vote
0answers
21 views
Reference request: Terminal wealth distribution including deterministic contribution
I'm looking into some classical utility maximization problems. In particular, I'm interested in looking at the wealth evolution where you invest your money across $n$ assets and each time period you ...
19
votes
0answers
247 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 $Q-$ ...
4
votes
1answer
221 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 ...
1
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0answers
30 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?
5
votes
1answer
409 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$...
2
votes
1answer
233 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 ...
1
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0answers
431 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 ...
7
votes
1answer
810 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 ...
1
vote
0answers
78 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 ...
1
vote
0answers
134 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:$$dS_t=\mu S_tdt+\sigma ...
3
votes
1answer
197 views
Why is this utility function not picking up its penalty?
I was reading this seminal paper by Infanger. On page 40, Figure 11. was quite interesting. In particular I was interested in the top one, 19 Years and I wanted to reproduce this plot. To give some ...
2
votes
0answers
82 views
Why is utility concave?
I have read that the utility function is usually concave. I assume this requirement arises in order to meet the diversification effect:$$f(\lambda_1c_1+\lambda_2c_2)\ge \lambda_2 f(c_1)+\lambda_2f(c_2)...
3
votes
0answers
161 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 ...
3
votes
2answers
479 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-...
2
votes
1answer
87 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
votes
1answer
532 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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0answers
499 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 ...
0
votes
1answer
124 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 ...
1
vote
1answer
111 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
votes
1answer
180 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 ...
1
vote
1answer
941 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
2
votes
1answer
180 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 ...
1
vote
1answer
72 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 \...
7
votes
3answers
7k 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 ...
7
votes
2answers
517 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$. ...
2
votes
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 ...
2
votes
1answer
2k 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 ...
1
vote
2answers
164 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 ...
1
vote
1answer
203 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. ...
1
vote
2answers
188 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 ...
2
votes
0answers
71 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 ...
1
vote
2answers
238 views
Expected Utility and $\log$
I've just started reading about expected utility and utility functions and have the following question.
$\textbf{Question:}$ An investor has an initial wealth of 100 and a utility function of the ...
-2
votes
1answer
127 views
Implicit relation between risk and reward
I want to differentiate w.r.t. $\sigma^2$ the following equation
$u'(Y)\mu$ + $\frac{u''(Y)}{2}$$(\sigma^2 + \mu^2) = 0$
where we can consider $\mu$(reward) as an implicit function of $\sigma^2$(risk) ...
1
vote
0answers
51 views
Comparing cost of two alternative given their distribution
I have distribution for cost of two alternative through Monte Carlo simulation. The distributions are not normal. Given the benefit of the two alternatives is the same but ungiven, I want to choose ...
1
vote
1answer
34 views
convention in borrowing money in a multiperiod model
I have a question concerning the idea of consumption in multi period. The following is given
$$C_1=W_0-xS_1+B$$
$$C_2=xS_2-BR$$
where
$W_0$ is initial wealth
$x$ is the weight on an asset with ...
3
votes
3answers
465 views
Validity of CAPM
I came across some literature regarding "Framing Theory" or "Prospect Theory", and the validity of CAPM. I was wondering if you could shed some light on a few questions I have in this regard:
...
7
votes
1answer
1k views
Why maximize expected growth rate?
It seems to me that the optimality of the Kelly Criterion relies on the assumption that it is in an investor's best interest to maximize his portfolio's expected growth rate. Why would he care what ...
8
votes
1answer
496 views
Does risk-neutral measure have anything to deal with risk-neutrality in utility theory?
Or simply: why do we call equivalent martingale measures as risk-neutral measures?
In the utility or game theory, when we consider a person's preferences to certain outcomes, we often deal with the ...
5
votes
2answers
176 views
Critique against consumption-based asset pricing theory?
I find asset pricing theory very vague and full of assumptions, especially the consumption-based modern theory. In its essence, the theory states that asset prices depend on the covariance between ...
1
vote
1answer
458 views
Why are indifference equations in mean-variance portfolio theory convex shaped
As the title suggests why is the indifference equations in mean variance portfolio theory convex shaped?
Indifference Equation: https://en.wikipedia.org/wiki/Indifference_curve
A graph:
