# 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. Commented Feb 24, 2015 at 20:33
• Indeed, references are always appreciated ;-) ..."I have a working paper"... Commented Feb 24, 2015 at 20:34
• fair complaint :) Commented Feb 24, 2015 at 20:42
• They say in the paper that table 2 has their return assumptions, but it doesn't have any returns... Commented Feb 24, 2015 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 Commented Feb 25, 2015 at 5:33

## 1 Answer

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).

It is intuitive as it says that given the weighting the return expectation increases with risk aversion and risk.

The case with the full allocation constraint (sum of weights is one) is covered by Herold but I also think that this is a bit circular ...

• From the Herold paper "The budget constraint implicitly changes the original set of expected returns." This appears to be what they are trying to do in the paper. Many thanks. Commented Feb 26, 2015 at 19:44
• This Herold paper is excellent. Thanks again. Commented Feb 26, 2015 at 21:48