Saturday, February 6, 2016

optimization - 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.


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



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 ...


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