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 aversion. For $x>\mu_X+\frac{1}{\lambda}$ the utility is decreasing in $x$. This is undesirable as we do not think investors derive disutility from a high-enough return on investment. Is this a common problem? Let us consider an example.
An investor holds shares of a company worth around \$$100$ with $\mu_x\approx\$100$. Daily fluctuations of share prices of around 0.25% (corresponding to $\pm$\$0.25) and larger are not uncommon. Given a reasonable value of $\lambda=4$ (see "Typical risk aversion parameter value for mean-variance optimization"), this means $x>\mu_X+\frac{1}{\lambda}$ will not be uncommon, i.e. a sufficiently large gain in wealth will lead to a reduction in utility quite frequently. If the investor holds shares worth \$$10,000$ instead, close to half of the days will show $x>10,000+\frac{1}{4}$. Hence, the problem seems to be very common.
Are there any common approaches in the literature to fixing this flaw while sticking to quadratic utility? What are they?
(I could come up with some simple modifications of the utility function myself, but I would like to follow the relevant literature instead, if there is any.)
Edit: I am not sure whether $\mu_X$ belongs in the function. It could (should?) be $u(x)=x−\frac{\lambda'}{2}x^2$ or $u(x)=x−\frac{\lambda''}{2}(x−c)^2$ for some $c$ that reflects an agent's preferences. Ideally, $c$ would be greater than $\max(x)$, but if the support of $x$ extends to $+\infty$, such a $c$ does not exist, which is likely the root of the problem.
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