<p>If you have quadratic preferences then your utility function is:
<span class="math-container">$$ U(W) =  W - \lambda W^2$$</span>
this implies your expected utility function looks like:
<span class="math-container">$$ E[U(W)] =  E[W - \lambda W^2] =  E[W] - \lambda E[W^2]$$</span>
<span class="math-container">$$ =  E[W] - \lambda E[W^2 - E[W]^2 + E[W]^2]$$</span>
<span class="math-container">$$ =  E[W] - \lambda E[W^2 - E[W]^2] - \lambda E[E[W]^2]]$$</span>
<span class="math-container">$$ =  \mu_w - \lambda \sigma_w^2 - \lambda \mu_w^2$$</span></p>
<p>Therefore, we have established that expected utility depends only only the mean <span class="math-container">$(\mu_W)$</span> and variance <span class="math-container">$(\sigma^2_W)$</span> of wealth and the risk parameter <span class="math-container">$(\lambda)$</span>.</p>

CC BY-SA 4.0
2021-11-05T14:00:21.427Z
Line 3 doesn't distribute the negative from $-\lambda$ to the final term. The solution should be $=\mu_w-\lambda\sigma_w^2-\lambda\mu_w^2$. [See for reference.](https://som.yale.edu/sites/default/files/files/05%20Mean-Variance%20Theory%20and%20the%20CAPM.pdf)
J.R.
CC BY-SA 4.0
2021-11-05T18:43:41.760Z
Good catch and now fixed.
BKay
CC BY-SA 4.0
2019-10-19T17:33:49.960Z
2021-11-05T18:43:29.817Z
2021-11-05T18:43:29.817Z
BKay
<p>Why does investors having quadratic utility function mean that their optimal portfolios can be chosen by only considering mean and variance of returns i.e. imply <span class="math-container">$\mu-\sigma$</span> preference?</p>

CC BY-SA 4.0
2019-10-18T19:19:28.970Z
Hint: the quadratic utility function exhibits increasing absolute risk aversion
Ali
<utility><preferences>
Why does quadratic utility function imply $\mu-\sigma$ preference?
Aqqqq
# Why does quadratic utility function imply $\mu-\sigma$ preference?

(Asked by: username_0 on 2019-10-16)

Why does investors having quadratic utility function mean that their optimal portfolios can be chosen by only considering mean and variance of returns i.e. imply $\mu-\sigma$ preference?

**Tags:** utility, preferences
> **Comments:**
> **username_1**: Hint: the quadratic utility function exhibits increasing absolute risk aversion  

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## 1 answers

### Answer by username_2 on 2019-10-19. Score: 8 | Accepted answer

If you have quadratic preferences then your utility function is:
$$ U(W) = W - \lambda W^2$$
this implies your expected utility function looks like:
$$ E[U(W)] = E[W - \lambda W^2] = E[W] - \lambda E[W^2]$$
$$ = E[W] - \lambda E[W^2 - E[W]^2 + E[W]^2]$$
$$ = E[W] - \lambda E[W^2 - E[W]^2] - \lambda E[E[W]^2]]$$
$$ = \mu\_w - \lambda \sigma\_w^2 - \lambda \mu\_w^2$$

Therefore, we have established that expected utility depends only only the mean $(\mu\_W)$ and variance $(\sigma^2\_W)$ of wealth and the risk parameter $(\lambda)$.

> **Comments:**
> **username_3**: Line 3 doesn't distribute the negative from $-\lambda$ to the final term. The solution should be $=\mu_w-\lambda\sigma_w^2-\lambda\mu_w^2$. [See for reference.](https://som.yale.edu/sites/default/files/files/05%20Mean-Variance%20Theory%20and%20the%20CAPM.pdf)  
> **username_2**: Good catch and now fixed.  

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