# What is the mathematical difference between Mean-Variance Optimization and CAPM?

I have spent some time going through the maths of both Mean-Variance Optimization and CAPM, and I'm trying to pin down the mathematical differences between them. For both, let $$p$$ be a portfolio consisting of risky assets with returns $$\mathbf{r}=( r_1,r_2,\dots,r_m)$$ and weights $$\mathbf{w} = (w_1,w_2,\dots,w_m)$$, along with a risk-free asset with return $$r_f$$, so that

$$r_p = \mathbf{w}^T\mathbf{r}+(1 - \mathbf{w}^T\mathbf{1}_m)r_f$$

• Mean Variance Optimization:

\begin{align} \text{Minimize:}\ \ & \frac{1}{2}\mathbf{w}^T\boldsymbol{\Sigma}\mathbf{w} \\ \text{Subject to: }\ \ & \mathbb{E}[r_p] = \mathbf{w}^T\mathbb{E}[\mathbf{r}]+(1 - \mathbf{w}^T\mathbf{1}_m)r_f \end{align}

Source: Mathematics for Finance, MIT (I actually couldn't find them mentioning CAPM, just MVO)

• CAPM:

\begin{align} \text{Minimize:}\ \ & \sqrt{\mathbf{w}^T\boldsymbol{\Sigma}\mathbf{w}} \\ \text{Subject to: }\ \ & \mathbb{E}[r_p] = \mathbf{w}^T\mathbb{E}[\mathbf{r}]+(1 - \mathbf{w}^T\mathbf{1}_m)r_f \\ \text{Then:}\ \ & \text{Use the specific case where }\mathbf{w}^T\mathbf{1}_m=1 \end{align}

Source: QuantPy video which I believe is based on these lecture notes.

So it seems like MVO is minimizing variance, whereas CAPM minimizes standard deviation (as well as assuming full investment in the market). I have never heard the difference explained this way, so is my understanding correct? What are the consequences of choosing to minimize variance vs minimize standard deviation?

The CAPM is an asset pricing model, while mean-variance optimization is a type of optimization. These are objects from two different categories. When you characterize the CAPM as an optimization problem, you are not spelling out what the CAPM is. Instead of your characterization of the CAPM, I would write $$\mathrm{E}(R_i)-r_f=\beta_i[\mathrm{E}(R_m)-r_f]$$ where $$R_i$$ is return on asset $$i$$ and $$R_m$$ is return on the market.