# 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:

$$w^* = (\iota \Sigma^{-1} \mu)^{-1} \Sigma^{-1} \mu$$

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
Feb 12, 2019 at 10:01
• @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 $r_f$ Oct 30, 2020 at 15:04

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! Aug 3, 2013 at 4:45
• 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? Jan 29, 2017 at 17:25
• @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.
– John
Jan 30, 2017 at 17:32
• 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 Nov 9, 2020 at 19:36
• @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.
– John
Nov 9, 2020 at 20:01

Check out following link. In page 23 you'll find the derivation. http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.pdf

• It is advisable that you also quote the relevant part instead of simply referring to an external link. External references are not permanent and have a tendency to become unreachable as time passes. Feb 10, 2014 at 22:24
• this paper often comes up but relies on the $Ax=b$ relationship. is there a different derivation out there that clearly shows the substitutions happening between the Lagrangean formulas (on p. 8 of the pdf), also in matrix algebra, but without resorting to and stopping short at the $Ax=b$ crutch? Jun 28, 2019 at 15:04

Merton, Robert, 1972, An Analytic Derivation of the Efficient Portfolio Frontier, Journal of Financial and Quantitative Analysis