Derivation of the tangency (maximum Sharpe Ratio) portfolio in Markowitz Portfolio Theory?

  • I have seen the following formula for the tangency portfolio in Markowitz portfolio theory but couldn't find a reference for derivation, and failed to derive myself. If expected excess returns of $N$ securities is the vector $\mu$ and the covariance of returns is $\Sigma$, then the tangent portfolio (maximum Sharpe Ratio portfolio) is:

    \begin{equation} w^* = (\iota \Sigma^{-1} \mu)^{-1} \Sigma^{-1} \mu \end{equation}

    Where $\iota$ is a vector of ones. Anyone know a source of the derivation?

    Hi, would you also elaborate a bit on why such a portfolio is called max Sharpe portfolio? Does it maxmise $w^T r / \sqrt{w^T\Sigma w}$?

    @Vim it maximizes $\frac{w^T \mu - r_f}{\sqrt{w^T\Sigma w}}$ which is yes the Sharpe ratio. the op re-stated $\mu$ to be excess returns inclusive of $r_f$

  • John

    John Correct answer

    8 years ago

    The unconstrained mean-variance problem $$w_{mv,unc}\equiv argmax\left\{ w'\mu-\frac{1}{2}\lambda w'\Sigma w\right\} $$ can easily be found by taking the derivative $$\frac{\partial}{\partial w}\left(w'\mu-\frac{1}{2}\lambda w'\Sigma w\right)=\mu-\lambda\Sigma w $$ setting it to zero, and solving for $w$. This gives $$w_{mv,unc}\equiv\frac{1}{\lambda}\Sigma^{-1}\mu $$ To find the portfolio constraining all the weights to sum to $1$, it is as simple as dividing by the sum of the portfolio weights $$w_{mv,c}\equiv\frac{w_{mv,unc}}{1'w_{mv,unc}}=\frac{\Sigma^{-1}\mu}{1'\Sigma^{-1}\mu} $$which after canceling out the risk aversion variables gives what you have above.

    For more general constraints, such that $Aw=b$, the formula is more complex. I often refer to the derivation in this paper for the formula.

    Thank you so very much. I never thought it would be so simple. I think everyone is familiar with the unconstrained optimal portfolio, but for some reason I never understood how to put the constraint in. Thanks again!

    I know it's late, but why is the tangency optimization problem $argmax\{w'\mu - \frac{ \lambda w' \Sigma w}{2} \}$ instead of $argmax \frac{w'\mu}{\sqrt{w'\Sigma w}}$? We are trying to find the portfolio on the efficient frontier that maximizes the sharpe ratio, the ratio of return to standard deviation, are we not?

    @Marie.P. If you want to maximize the Sharpe ratio, then that's generally the formula you would use. It's more difficult than standard mean variance. Under some assumptions, the optimal mean variance portfolio fully invested will equal the maximum Sharpe ratio portfolio. I just wanted to give a simple derivation of the formula the OP was asking about. I'm sure it would be useful to post other derivations here, if you want to add another.

    i think she also means that there is a $\lambda$ in the first formula, whereas there is no $\lambda$ in the second, making them look even more unequivalent

    @develarist The simple reason why I did it the way I did it is that it makes the math easier and produces the same result. The easiest way to know that is because if you take the formula for the unconstrained mean-variance weights and plug that into the formula for the Sharpe ratio you get a constant and the risk aversion parameter cancels out. In other words, it should always have the same Sharpe ratio, the maximum. This doesn't work when you add in inequality constraints though.

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Content dated before 7/24/2021 11:53 AM