# explanation of factor tilts that uses mathematical notation

Can anyone provide a definition of "factor tilt" that uses mathematical notation?

Perhaps factor tilts are based on factor regression models. Let's say that our returns vector $$\mathbf{y}_t$$ can be expressed in terms of a market return $$x_t$$: $$\mathbf{y}_t = \mathbf{B}x_t + \mathbf{v}_t$$ where $$\mathbf{B}$$ is a loadings vector containing "betas" and $$\mathbf{v}_t$$ is iid noise.

A model like this is often advertised on its ability to reduce the number of parameters that need to be estimated in order to specify a forecast's covariance matrix. It also suggests, depending on one's strategy, a vector of portfolio weights $$\mathbf{w}_t$$.

Does "factor tilting" somehow manually override the weights suggested by an optimization routine? Is this idea even necessarily tied to factor-based modeling?

• Factor tilting does not override the weights suggested by an optimization routine, it overrides the weights suggested by Market Cap allocation, i.e. the weights typically used by an Index Fund. It does this in order to achieve higher loading than the index fund on certain factors the proponents of factor tilting consider particularly desirable for long term performance. The index fund has loadings $B_0$ and the tilt fund has loadings $B_1$ with higher values for all but the first entry, the CAPM loading, which is the same. – Alex C Jun 7 '19 at 21:16
• There is a website that will caculate factor exposures of any ETF or fund portfoliovisualizer.com/factor-analysis and thus allow you to see if a fund has a factor tilt vs an index fund or not. – Alex C Jun 8 '19 at 14:09