<p>In finance and specifically portfolio theory, a popular utility function is quadratic utility 
<span class="math-container">$$
u(x)=x-\frac{\lambda}{2}(x-\mu_x)^2
$$</span> 
where <span class="math-container">$x$</span> is wealth and <span class="math-container">$\lambda$</span> is the parameter of risk aversion. For <span class="math-container">$x&gt;\mu_x+\frac{1}{\lambda}$</span> the utility is decreasing in <span class="math-container">$x$</span>. 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. </p>

<p>An investor holds shares of a company worth around \$<span class="math-container">$100$</span> with <span class="math-container">$\mu_x\approx\$100$</span>. Daily fluctuations of share prices of around 0.25% (corresponding to <span class="math-container">$\pm$</span>\$0.25) and larger are not uncommon. Given a reasonable value of <span class="math-container">$\lambda=4$</span> (see <a href="https://quant.stackexchange.com/questions/8405/typical-risk-aversion-parameter-value-for-mean-variance-optimization">"Typical risk aversion parameter value for mean-variance optimization"</a>), this means <span class="math-container">$x&gt;\mu_x+\frac{1}{\lambda}$</span> 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 \$<span class="math-container">$10,000$</span> instead, close to half of the days will show <span class="math-container">$x&gt;10,000+\frac{1}{4}$</span>. Hence, the problem seems to be very common. </p>

<p>Are there any common approaches in the literature to fixing this flaw while sticking to quadratic utility? What are they?</p>

<p>(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.)</p>

CC BY-SA 4.0
2019-08-15T17:40:45.463Z
The mean-variance criterion is [known to be incompatible](https://www.jstor.org/stable/2330243) with expected utility maximization, unless the utility function and the set of lotteries are [restricted to be a certain type](https://m.tau.ac.il/~spiegel/teaching/corpfin/mean-variance.pdf). [Nakamura (2015)](https://www.sciencedirect.com/science/article/pii/S0022053115001799?via%3Dihub) derives axioms that delineate the boundaries of applicability of this type of utility function.
Herr K.
CC BY-SA 4.0
2019-08-15T19:18:04.363Z
@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
CC BY-SA 4.0
2019-08-15T22:18:52.230Z
Isnt your $\lambda = 4$ simply too big?
Grada Gukovic
CC BY-SA 4.0
2019-08-16T06:28:30.453Z
@GradaGukovic, [this thread](https://quant.stackexchange.com/questions/8405/typical-risk-aversion-parameter-value-for-mean-variance-optimization/9431#9431) 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
CC BY-SA 4.0
2019-08-16T06:30:40.670Z
References given in the thread are Fabozzi et al. ["Robust Portfolio Optimization and Management"](https://www.amazon.com/Robust-Portfolio-Optimization-Management-Fabozzi/dp/047192122X) p. 35 and Ang ["Asset Management: A Systematic Approach to Factor Investing"](https://www.amazon.com/Asset-Management-Systematic-Investing-Association/dp/0199959323).
Richard Hardy
CC BY-SA 4.0
2019-10-31T23:13:32.100Z
Richard, is lambda referring to absolute or relative risk aversion coefficient? I am under the impression that absolute risk aversion is equal to lambda only when evaluated at the specific value x=mu?
Matifou
CC BY-SA 4.0
2019-11-01T12:50:30.627Z
@Matifou, could you elaborate a bit more? What would a "yes" or a "no" answer to your question imply? (I am glad you joined the thread. As you can see, there has not been much progress here...)
Richard Hardy
CC BY-SA 4.0
2019-11-01T17:16:28.943Z
@RichardHardy this is just a side-point actually, but I am not sure whether the mentionned post is talking about absolute or relative RA coefficients, and I guess these numbers make only sense for a function that is constant (i.e. for any x) in rel/absolute RA? And it seems the MV does not have these properties? Now to the main point, I remember seeing people set the parameters of the MV such that its max occurs at the max of x, so that never decreasing. But indeed sounds like a dirty fix (what is max is infinity?)
Matifou
CC BY-SA 4.0
2020-01-21T19:01:23.860Z
A somewhat related question: [Utility theory and portfolio optimization: utility of what exactly?](https://economics.stackexchange.com/questions/33688).
Richard Hardy
<utility><finance>
Fixes of quadratic utility when probability of decreasing utility is large
Richard Hardy
# Fixes of quadratic utility when probability of decreasing utility is large

(Asked by: username_0 on 2019-08-15)

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"](https://quant.stackexchange.com/questions/8405/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.)

**Tags:** utility, finance
> **Comments:**
> **username_1**: The mean-variance criterion is [known to be incompatible](https://www.jstor.org/stable/2330243) with expected utility maximization, unless the utility function and the set of lotteries are [restricted to be a certain type](https://m.tau.ac.il/~spiegel/teaching/corpfin/mean-variance.pdf). [Nakamura (2015)](https://www.sciencedirect.com/science/article/pii/S0022053115001799?via%3Dihub) derives axioms that delineate the boundaries of applicability of this type of utility function.  
> **username_0**: @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.  
> **username_2**: Isnt your $\lambda = 4$ simply too big?  
> **username_0**: @GradaGukovic, [this thread](https://quant.stackexchange.com/questions/8405/typical-risk-aversion-parameter-value-for-mean-variance-optimization/9431#9431) 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?  
> **username_0**: References given in the thread are Fabozzi et al. ["Robust Portfolio Optimization and Management"](https://www.amazon.com/Robust-Portfolio-Optimization-Management-Fabozzi/dp/047192122X) p. 35 and Ang ["Asset Management: A Systematic Approach to Factor Investing"](https://www.amazon.com/Asset-Management-Systematic-Investing-Association/dp/0199959323).  
> **username_3**: Richard, is lambda referring to absolute or relative risk aversion coefficient? I am under the impression that absolute risk aversion is equal to lambda only when evaluated at the specific value x=mu?  
> **username_0**: @Matifou, could you elaborate a bit more? What would a "yes" or a "no" answer to your question imply? (I am glad you joined the thread. As you can see, there has not been much progress here...)  
> **username_3**: @RichardHardy this is just a side-point actually, but I am not sure whether the mentionned post is talking about absolute or relative RA coefficients, and I guess these numbers make only sense for a function that is constant (i.e. for any x) in rel/absolute RA? And it seems the MV does not have these properties? Now to the main point, I remember seeing people set the parameters of the MV such that its max occurs at the max of x, so that never decreasing. But indeed sounds like a dirty fix (what is max is infinity?)  
> **username_0**: A somewhat related question: [Utility theory and portfolio optimization: utility of what exactly?](https://economics.stackexchange.com/questions/33688).  

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## 0 answers

