# Fixes of quadratic utility when probability of decreasing utility is large

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.) • The mean-variance criterion is known to be incompatible with expected utility maximization, unless the utility function and the set of lotteries are restricted to be a certain type. Nakamura (2015) derives axioms that delineate the boundaries of applicability of this type of utility function. – Herr K. Aug 15 '19 at 17:40 • @HerrK., thanks, I will take a look! Nevertheless, it is a standard approach in finance, so I would like to see if there are any good fixes within the framework of quadratic utility. – Richard Hardy Aug 15 '19 at 19:18 • Isnt your$\lambda = 4$simply too big? – Grada Gukovic Aug 15 '19 at 22:18 • @GradaGukovic, this thread on Quantitative Finance SE indicates a reasonable range of$\lambda$values as either$2\leq\frac{\lambda}{2}\leq 4$or$1\leq\frac{\lambda}{2}\leq 10$, so my choice of$\lambda=4$(implying$\frac{\lambda}{2}=2\$) is clearly on the lower end of the range. Or am I misunderstanding something? – Richard Hardy Aug 16 '19 at 6:28
• References given in the thread are Fabozzi et al. "Robust Portfolio Optimization and Management" p. 35 and Ang "Asset Management: A Systematic Approach to Factor Investing". – Richard Hardy Aug 16 '19 at 6:30