Mean-Variance Optimization and Signal Risk Scaling

Summary

The document compares two ways to set portfolio weights to a target ex-ante volatility. Mean-variance optimization uses the inverse covariance matrix to adjust the signal and chooses a risk-aversion parameter to reach the target. Direct risk scaling instead multiplies the signal by a scalar based on its covariance risk. The question notes that both constructions are intended to produce the same target risk, while potentially assigning different weights across assets.

The document provides formulas but no empirical comparison or conclusion about which method is superior. Their relative allocations depend on how the signal relates to the covariance matrix: mean-variance optimization accounts for covariance when shaping weights, while direct scaling preserves the signal's relative weights. The comparison also depends on consistent definitions of expected returns, risk, and the signal; as written, the risk expression shown for direct scaling may require clarification because covariance-based portfolio variance is normally evaluated as weights transposed, times the covariance matrix, times weights. No performance evidence or implementation details are supplied.

Key ideas

• Mean-variance optimization shapes a signal using the inverse covariance matrix.
• A risk-aversion parameter can be selected to target a specified ex-ante risk level.
• Direct risk scaling adjusts the signal's overall size while preserving its relative weights.
• Equal target risk does not imply equal asset allocations or expected returns.
• The document poses the comparison but offers no evidence that one approach is superior.

Tags

# Mean Variance Optimization vs Risk Scaling


# 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.

- 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 $\sigma$.

- 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: $h^TVh$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.