# Markowitz Mean-Variance Implied Returns

What is the closed form solution for the following inverse Markowitz problem?

Given a mean-variance optimized fully invested portfolio $X$, a risk aversion parameter $\lambda$ and a var-covar matrix $C$. What is the formula for the (implied) returns $\mu_{impl}$ that must have been used to build the portfolio $X$?

I have a working paper (Kritzman et. al. 2008) claiming a closed form solution $\mu_{impl}$ that depends on the "expected returns" $\mu$ which seems a bit circular and is perhaps a typo.

$$\mu_{impl} = \lambda C X^T + \frac{-\lambda + 1 C^{-1} \mu^{T}}{1 C^{-1} 1^T} 1^T$$

• References are always appreciated. Feb 24 '15 at 20:33
• Indeed, references are always appreciated ;-) ..."I have a working paper"... Feb 24 '15 at 20:34
• fair complaint :) Feb 24 '15 at 20:42
• They say in the paper that table 2 has their return assumptions, but it doesn't have any returns... Feb 24 '15 at 23:40
• The paper seems not finished, SSRN hints at an updated version but its not available for download,... papers.ssrn.com/sol3/papers.cfm?abstract_id=1340013 Feb 25 '15 at 5:33

The formula is $$\mu = \lambda CX$$ in your notation. You find it in many places, e.g. here.
The assumption is that you know $\lambda$ which is a strong assumption. Furthermore it only holds if investors are unconstrained (long/short not long only).