# 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. – rhaskett Feb 24 '15 at 20:33
• Indeed, references are always appreciated ;-) ..."I have a working paper"... – vonjd Feb 24 '15 at 20:34
• fair complaint :) – rhaskett 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... – rhaskett 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 – pyCthon Feb 25 '15 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. – rhaskett Feb 26 '15 at 19:44
• This Herold paper is excellent. Thanks again. – rhaskett Feb 26 '15 at 21:48