7

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 ...


7

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 ...


5

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 $\mathbb{R}$, ...


5

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 ...


5

These are a natural and easiest (most tractable mathematically) choice. A utility function is defined up to a positive affine transformation: economically there is no difference between the utility functions $U(x)$ and $\tilde{U}(x)=Au(x)+B$. Hence, a measure of risk aversion that remains constant w.r.t. affine transformations would be useful. How does one ...


3

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 \...


3

CARA Utility function $u(c)=\frac{-e^{-ac}}{a}$ for $a>0.$ Expected utility $E(u(c))=\int_{-\infty}^{\infty} u(c) f(c) dc,$ where f is a density. Example f(10)=0.3, f(20)=0.7, else f=0 and a=2. Then $E(u(c))=0.3\times u(10)+0.7\times u(20)=0.3\times \frac{-e^{-2*10}}{2}+0.7\times \frac{-e^{-2*20}}{2}$ Now, for normal density $f(c)=\frac{1}{\sqrt{2\pi}}\...


3

In most economic models the risk aversion coefficient is definitely related to the equity premium. Assuming utility is CRRA (as you mention): \begin{equation} U(C_t) = \frac{C_t^{1-\gamma}}{1-\gamma} \end{equation} Also assume the agent has access to an equity claim and risk free. So that his portfolio follows: $W_{t+1} = [\alpha_t R_{t+1} + (1-\alpha_t)...


3

I think the usual argument is that if an investor is maximizing expected log wealth, then this implies preference for higher odd order moments (mean return, skew, etc.) and for lower even order moments (volatility, kurtosis, etc.). This comes from the Taylor expansion of the log. However, if one wishes to maximize the probability that returns over a given ...


2

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. ...


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 ...


2

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.


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 ...


2

Try to formulate the problem as a constrained optimization problem, and examine the KKT (Karush-Kuhn-Tucker) complementary slackness conditions.


2

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.


2

In most settings, utility functions are defined up to an affine transformation: if $u(x)$ defines the preference of an investor, then so does $a*u(x)+b.$ This implies, you can normalize the Taylor expantion of any smooth utility function to $u(x)=x+a*x^2+\ldots$ around 0. So the next step is just to drop off higher order terms. The investor is also usually ...


2

The assumption of quadratic utility function is very convenient in ptf Theory because it is possible to demonstrate that also if the ptf return are not normally distributed the mean-variance approach is still the best. The best in the sense that any other distributional properties is amenable into mean and variance. For converse, if the return are normally ...


2

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

$$-x \frac{d^2U}{dx^2}=(ax+b)\frac{dU}{dx}$$ Letting $\frac{dU}{dx}=v$ $$-x \frac{dv}{dx}=(ax+b)v$$ Rearranging $$-\frac{1}{v} dv=\left(a+\frac{b}{x}\right)dx$$ $$ -\int(\ln{v}) dv= \int \left(a+\frac{b}{x}\right) dx $$ $$ v=C e^{-ax}x^b$$ Where $C$ is a constant. By taking $t$ as a dummy variable. $$U(x)= C\int ^x_0e^{-at}t^{-b}\, dt$$


2

An Insurance premium typically focuses solely on the downside of your Risk. An Insurance pays if you suffered some damage, but you do not give them some share of your profit if things are good. That means you have to get rid of the positive part of X, which has than of course a non-zero mean. Apart from that, I think you are correct, in that you can see $\...


2

All the parameters of the solution need to be estimated for your specific stock. Stochastic process-specific parameters, i.e. $\mu$, $\sigma$, have to be estimated by some classical method (e.g. MLE, minimum contrast, etc.). No parameter of tick size is incorporated in the model, you will have to decide at the end whether the in-between quote shall be ...


1

Here's a try/start: Let $A,B$, and $C$ be three possible events, and let $U(event)$ be the utility derived from each event. For example, if event $A$ corresponds to the event of winning the lottery, then $U(A)$ will presumably be a very large value. By contrast, if event $C$ corresponds to the event of falling off a ladder and breaking an arm, $U(C)$ will ...


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. \end{...


1

What you learn in school are models, meant to illustrate the concepts and methods of the field. Later you will learn about other forms of utility functions (power utility most prominently). With such families of utility functions the computations aren't as clean as with quadratic utility, but by then you will have understood the concepts and methods, and you ...


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 given $$dS_u=\sigma dW_u$$ Integrate both side: $$\int_t^T dS_u=\int_t^T dW_u$$ $$S_T-S_t=\sigma\big(W_T-W_t\big)$$ 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 $q^...


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.


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