# Tag Info

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I believe you should look into the field of Utility Theory which aims to model how people actually understand and feel about gains and losses. Usually, the most interesting cases are when the outcomes of the experiment are actually random, or when the payment can occur at different times. A famous model for the utility function is Risk Aversion. You can ...

7

The von Neumann-Morgenstern utility axioms are normative criteria for rational choice. In contrast, he Artzner/Uryasev axioms are normative criteria that some argue must hold for any measure that aims to measure portfolio risk. What they have in common is simply that they are normative criteria. The substance of the axioms are different, however, since they ...

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This is the well known Euler's equation for optimality. The trick here is to setup the budget constraint correctly. Your initial wealth $W_0$ is irrelevant. The terminal (risky) wealth is, $$W = W_0( 1 + \pi_1 (R_1 - r_f) + \ldots + \pi_n (R_n - r_f) )$$ (Check that this can be written this way), where $\pi_i \in \mathbf{R}$ is the weight allocated to ...

4

The Kelly criterion is just one approach to portfolio construction (or bet sizing) that considers the risk-return tradeoff. There are many possible strategies (static or dynamic) that incorporate other criteria such as the maximum drawdown, probability of ruin, etc. As pointed out by @John, Kelly is maximizing the log of wealth, which is equivalent to ...

2

One approach is to use an exponential utility function: $U(x) = -e^{-\lambda x}$. Here, $\lambda$ records what is known as the absolute risk aversion. Exponential utility functions are nice because they have a wealth independence property (of course, this may be seen as a drawback). As we will see below, the initial capital $X$ plays no part in the ...

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An agent with utility function $U$ values a final position $X_T$ by $E\left[U(X_T)\right]$. You can think of this as a function mapping random variables to $\mathbb{R}$, $X_T \mapsto E \left[U(X_T)\right]$. A risk-neutral mapping should be a linear mapping of the kind above. In other words, $f$ should map some space of random variables to $\mathbb{R}$, ...

2

I can understand your concerns, but I think you are expecting too much from these theories. We cannot explain aggregate behavior from first principle based on a sound theory of individual decisions under uncertainty and I personally doubt that there will ever be such a Grand Unification in economics. Consumption-based asset pricing models are more related ...

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The best explanation I came across so far is the one in Gravelle and Rees (2003) chapter 17. I could exactly write here what they state, but that would be copying.

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Edit: The original follow fill in the details answer to my answer by James is still wrong (despite many hints). I'll just go in to fix it to avoid detracting future readers of this post. Just to fill in the details from the answer that has been accepted already: It is required to maximize \sup_{ \pi_1, \ldots, \pi_n } E[ U(W) ] = \sup_{ \pi_1, \ldots, ... 2 Try to formulate the problem as a constrained optimization problem, and examine the KKT (Karush-Kuhn-Tucker) complementary slackness conditions. 1 We assume that \begin{align*} \frac{dX_t}{X_t} &= (r+\pi Y_t)dt + \pi\sigma dW_t,\tag{1}\\ dY_t &= -\lambda Y_t + dB_t.\tag{2} \end{align*} From (2), \begin{align*} Y_t = Y_0 e^{-\lambda t}+ e^{-\lambda t}\int_0^t e^{\lambda u} dB_u. \end{align*} Moreover, from (1), \begin{align*} \ln X_T &= \ln X_0 + (r-\frac{1}{2}\pi^2\sigma^2)T + \pi ... 1 I assume that the problem is\max_{\pi} E\left(\ln Z_T^{\Pi} \right).Note that \ln Z_t^{\Pi} = \ln X_t^{\Pi} -\ln X_t^{\rho}. Moreover, \begin{align*} d\ln Z_t^{\Pi} &= d\ln X_t^{\Pi} -d\ln X_t^{\rho}\\ &=\Big[\big(\mu \pi - \frac{1}{2}\sigma^2 \pi^2\big) - \big(\mu \rho- \frac{1}{2}\sigma^2 \rho^2\big) \Big]dt + \sigma(\pi-\rho)dW_t. ... 1 Under uncertainty we have to deal with "lotteries" where for example with 75% chance you get A and with 25% chance you get B and you have to compute expected utility 0.75*U(A)+0.25*U(B). It is clear that transformations of the utility function are going to create problems (i.e. different outcomes) unless they are linear. 1 Your calculation seems to be correct. I found this document here:http://home.uchicago.edu/rmyerson/teaching/util206.pdf. You can see that in P10, the certainty equivalence formula has that 1/2 factor there. 1 The answer is relatively straightforward if you assume that x is normally distributed - x \sim N(\mu_x,\sigma^2_x). If x is normally distributed then maximizing U(x)=−e^{ax} is the same as maximizing a mean variance utility: U = E(W) - 0.5a Var(W) . Now given that: E(W) = s\mu_x + (W-s)  where s is the amount of money on the risky stock and ... 1 First, you need to derive the distribution of S_T. You have givendS_u=\sigma dW_u$$Integrate both side:$$\int_t^T dS_u=\int_t^T dW_uS_T-S_t=\sigma\big(W_T-W_t\big)$$We know that W_t-W_t \sim N(0, T-t), so$$S_T-S_t \sim N(0, \sigma^2 (T-t))S_T|S_t\sim N(S_t, \sigma^2(T-t))$$It is given that initial value of S_t is s, so we have ... 1 To answer your question you need to look at what the distribution is of S_T. Since dS = \sigma dW we have that S_T is normal distribution with variance of \sigma^2 (T). Now you get E(-exp(-\gamma(x + qS_T))) = -exp(-\gamma x) E (exp(-\gamma q S_T)). Given that S_T is normal -\gamma q S_T is normal as well. You just scale the variance with ... 1 This is a strange question. Usually questions about expected utility involve some uncertainty about the future wealth of the investor. If there is no uncertainty in the outcome and the investor is not doing anything which might change his or her future wealth then the expectation of utility is a constant, that is E[U(w)] = U(w). 1 Your question depends on the discount factor you wish to use for pricing. If u use the risk-free rate (from the bond), it wouldn't be in line with the no-abitrage condition to assume an risk neutral agent can't/wouldn't invest in bonds to carry money into next period. To understand this: just assume a 1 period model with two outcomes for S, where both ... 1 That is true. Utility would not be concave anymore under prospect theory (only for gains), but convex for losses, which is evidence against CAPM. CAPM is valid either : -if the utility function is quadratic (which is nonsense in terms of economic interpretation, and in general, Von Neumann- Morgenstern utility describes poorly reality and should be ... 1 The general problem of the investor is:$$ \max_{w\in[0,1]^n} U(\mu_p(w),\sigma_p(w))\quad s.t. \sum_{i=1}^n w_i=1 where $w$ being the portfolio weights, and $U$ utility function of portfolio risk $\sigma_p$ and return $\mu_p$. CAPM assumes investors with concave utility function $U=\mu_p-\frac{1}{2}\sigma_p^2$, from which then follows that all ...

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Consumption-based asset pricing theories are about representative agents, not necessarily about traders and investors in financial institutions. The agents are assumed to follow behaviors based on how people generally would decide whether to invest and how much to invest. The idea is that the average person in the economy does not invest for the joy of ...

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I agree with @MattWolf The graph you show is confusing and evil, it makes me feel dumb every time I look at it. So I inverted the axis. Now we see the familiar shape of an utility curve, discussed in your previous question. It is upward sloping at a declining rate. In this case $u$ takes the place of $R_p$ and the general form of mean variance utility is ...

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The problem is to find the best functional form of the utility function plus estimate its parameters. A good starting point is the following draft chapter from an upcoming book which gives a good intuition and many examples: Preferences by Andrew Ang

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See the paper "On the conditional value at risk Probability dependent utility function" by Alexandre Street, on Theory and Decision, 2010. It shows that the well know CVAR fails in the independence axiom but it also provides good insights for that. The CVAR (redefined for revenues and not for losses - see the above paper) is convex in the probability set. ...

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