# Mean Variance Optimization vs Risk Scaling

What would be the difference between the following. Both techniques will result is an ex-ante risk of $$\sigma$$. However, that would be achieved via two different values of h. I want to understand which might be superior or better.

1. Doing a mean variance optimization :

$$h = \frac{V^{-1} \alpha}{2 \lambda}$$ choosing

$$\lambda = \sqrt{\frac{\alpha^T V ^{-1} \alpha}{4 \sigma^2}}$$

which is just risk targeting to $$\sigma$$.

1. Scaling your weights (signal) to risk target.

$$h = \frac{\sigma}{\sqrt{\alpha^T V \alpha}} \alpha$$

Notations:

$$h$$ : Final weights

$$V$$ : Covariance matrix

$$\alpha$$: Signal ( assume it to be normal in the cross section)

If we calculate the ex-ante risk we will get $$\sigma$$ from both ex-Ante risk: $$h^TVh$$

• Both techniques target an ex-ante volatility of $\sigma$. $\lambda$ is the Lagrangian multiplier and for a specific solution, one would have to specify the value of $\lambda$ – whisperer Aug 29 '19 at 20:54
• Ok, I see. Sort of. I'll think about it. – Alex C Aug 29 '19 at 21:16