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In finance, a common problem is selection of an optimal portfolio given some constraints (e.g. budget constraint and perhaps nonnegative allocation constraint). One can define the optimization problem based on maximization of expected utility. For example, if the utility function is $u(x)=x-\frac{A}{2}x^2$, then (in absence of constraints aside from the budget constraint) this yields the well-known mean-variance portfolio optimization.

Questions:

  1. What is $x$? Is it (portfolio) return (in \$), % or logarithmic return, total wealth (initial wealth + portfolio return) or yet something else?
  2. Which choice is typically used in portfolio optimization?
  3. Which choice makes most sense from the perspective of utility theory?

I am interested in this in the context of trying to pick a sensible value of $A$ to represent preferences of a "typical" agent. In my experience, $A=2$ as commonly found in the literature does not seem to align well with $x$ being logarithmic return (yields funny portfolios). Thoughts on this will also be appreciated.

Given the popularity of mean-variance portfolio theory and how basic my questions are, I suppose they have been considered before (perhaps even treated in textbooks). References would be appreciated.

Edit: In addition to the answer I got for Question 2, I am still highly interested in obtaining an answer for Question 3 (as well as alternative answers to Question 2).

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In mean-variance optimisation I have typically seen the below quadratic utility function where $๐ธ[๐‘…]$ is the expected return (or mean return) of a possible portfolio, $\sigma^{2}$ is the return variance of that portfolio and $A$ is a parameter that the determines the sensitivity to variance.

$$๐‘ˆ=๐ธ[๐‘…]โˆ’\frac{1}{2}A\sigma^{2}$$

As an aside, maximising % return, also maximises $ return, log return and total wealth. They might not be equivalent when optimising for mean and variance though.

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  • $\begingroup$ Thank you for your answer! I do not think \$ return, % return and other are interchangeable when optimizing mean and variance, because mean is linear in the quantity of $x$ while variance is quadratic. Also, from the perspective of utility theory, would it not be common to consider $x$ as total wealth rather than increments in wealth (returns) (Question 3.)? $\endgroup$ – Richard Hardy Jan 21 at 14:07
  • $\begingroup$ I see the question has been extended. I think when mean variance optimisation is applied to real life assets like global equities, bonds, commodities it suggests very strange portfolio weights that are not observed in practice. This ties in a bit with your other questions. This is a very abstract model (I guess because of its assumption that humans can be described with a two variable utility function), which means it can be formulated in a number of different ways. I don't know more, sorry. $\endgroup$ – M3RS Jan 21 at 14:21
  • $\begingroup$ Sorry, I meant to notify you about updating my question but forgot to. Also, +1 for the input. I think the questions I pose above must have been considered in the body of research on mean-variance portfolio optimization, but I am unaware of the right references. I would expect there have been explanations to very strange portfolio weights, and the model has likely been formulated so as to yield sensible weights. I would assume that mean-variance optimization is too popular an approach for these issues to have been neglected. So I guess I am looking for some references, too. $\endgroup$ – Richard Hardy Jan 21 at 14:46
  • $\begingroup$ Very good point. $\endgroup$ – M3RS Jan 21 at 14:57
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    $\begingroup$ I think it's a very famous theory, but I don't think practitioners use it very much. If we take the biggest pension fund in the US, for example, this is not how they would decide on their portfolio weights. One reason is that it is very hard to estimate future returns and variances, but portfolio weights suggested by this theory are highly sensitive to these inputs. So you results are highly sensitive to something that's highly tenuous. $\endgroup$ – M3RS Jan 21 at 14:58

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