# Tag Info

9

Recursive Utility The traditional approach to consumption-based asset pricing includes time separable (additive) expected utility functions, $$U(C_t,C_{t+1})=u(C_t)+\beta \mathbb{E}_t[u(C_{t+1})],$$ where $\beta<1$ measures impatience (subjective discount factor). That's the first equation in Chapter 1.1. in Cochrane's stellar asset pricing book. This ...

8

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 ...

6

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 ...

6

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

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 ...

5

The risky and riskless assets follow processes, $$\frac{dS_t}{S_t}= \mu \, dt + \sigma \, dB_t, \,\,\, \frac{dM_t}{M_t}= r \, dt$$ If the proportion invested in the risky asset at time $t$ is $p_t$, then the wealth process is $$\frac{dX_t}{X_t}= p_t \frac{dS_t}{S_t}+ (1-p_t)\frac{dM_t}{M_t}= (r + p_t(\mu -r)) dt + p_t \sigma dB_t$$ Finding the process for a ...

4

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 \...

4

In most economic models the risk aversion coefficient is definitely related to the equity premium. Assuming utility is CRRA (as you mention): $$U(C_t) = \frac{C_t^{1-\gamma}}{1-\gamma}$$ Also assume the agent has access to an equity claim and risk free. So that his portfolio follows: $W_{t+1} = [\alpha_t R_{t+1} + (1-\alpha_t)... 3 In most settings, utility functions are defined up to an affine transformation: if$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$around 0. So the next step is just to drop off higher order terms. The investor is also usually ... 3 The assumption of quadratic utility function is very convenient in ptf Theory because it is possible to demonstrate that also if the ptf return are not normally distributed the mean-variance approach is still the best. The best in the sense that any other distributional properties is amenable into mean and variance. For converse, if the return are normally ... 3 All the parameters of the solution need to be estimated for your specific stock. Stochastic process-specific parameters, i.e.$\mu$,$\sigma$, have to be estimated by some classical method (e.g. MLE, minimum contrast, etc.). No parameter of tick size is incorporated in the model, you will have to decide at the end whether the in-between quote shall be ... 3 I think the usual argument is that if an investor is maximizing expected log wealth, then this implies preference for higher odd order moments (mean return, skew, etc.) and for lower even order moments (volatility, kurtosis, etc.). This comes from the Taylor expansion of the log. However, if one wishes to maximize the probability that returns over a given ... 3 The efficient frontier is defined as the set of portfolios which have the highest return for a given measure of volatility, i.e.$\{S: s \in P \; s.t. \nexists \; t \in P \; \text{where} \;R(s) < R(t) \; \text{and} \; \sigma(s)=\sigma(t) \}$, where$Pis the set of all validly constructed portfolios. Therefore this also holds for the efficient frontier ... 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 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 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 $$\sup_{ \pi_1, \ldots, \pi_n } E[ U(W) ] = \sup_{ \pi_1, \ldots, \... 2 One approach is to use an exponential utility function: U(x) = -e^{-\lambda x}. Here, \lambda records what is known as the absolute risk aversion. Exponential utility functions are nice because they have a wealth independence property (of course, this may be seen as a drawback). As we will see below, the initial capital X plays no part in the ... 2 Try to formulate the problem as a constrained optimization problem, and examine the KKT (Karush-Kuhn-Tucker) complementary slackness conditions. 2$$-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$$2 An Insurance premium typically focuses solely on the downside of your Risk. An Insurance pays if you suffered some damage, but you do not give them some share of your profit if things are good. That means you have to get rid of the positive part of X, which has than of course a non-zero mean. Apart from that, I think you are correct, in that you can see \... 2 The functions are set up so that maximizing the expectation of the utility function will obey the von Neumann-Morgenstern axioms (completeness, transitivity, independence, continuity). As you've pointed out, convexity is also a desired behavior and certainly you can come up with other functions. 2 There are a few papers out there. A good starting point os the (always relevant) Grossman and Stiglitz (1980) "On the Impossibility of Informationally Efficient Markets" which shows that markets cannot be perfectly efficient without analysts being rewarded for their efforts (and is mentioned by many of the following articles as explaining some of ... 2 I want to add two comments to this. 1. Empirical utility-based-optimization and moments I would argue that comparing different degrees of a Taylor approximated utility optimization (a.k.a. a moment based model with two, three, four, ..., infinite moments) adds additional assumptions to your model when working with an asset universe whose statistics are not ... 1 I hope my computations are correct. Let u(t,x)=\max_{(f_s)_{s\geq t}}\mathbb{E}[(b+X^{f_.}_T)^\gamma]. Using HJB (you have to prove that it is ok to use it).$$0=\max_{f}\partial_t u(t,x)+(\mu f x - C)\partial_x u(t,x)+\frac{\sigma^2}{2}f^2x^2\partial_{xx}u(t,x)Since \partial_{xx}u(t,x)<0 (prove it), maximum is hit at f=\frac{\mu x \partial_xu(... 1 Assuming that W_0 is the initial wealth and that \alpha and \beta are yields of return, the final wealth is a discrete random variable \begin{align} W_T = \begin{cases} W_0(1+\alpha) &\text{ with probability }\quad p=0.6\\ W_0(1-\beta) &\text{ with probability }\quad 1-p=0.4 \end{cases} \end{align} The investment should be avoided iff the ... 1 What do you mean by "rigorous approach for finding them"? You have the four conditions and every function which fulfills those conditions is a risk-averse utility function. This is all there is; what else do you need? If you are looking for a description of this set in terms of elementary functions (+,.,polynomials, exp and such) you will be disappointed. ... 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 ...

Only top voted, non community-wiki answers of a minimum length are eligible