Deriving the Fully Invested Tangency Portfolio
Summary
The document derives the fully invested tangency portfolio from the unconstrained mean-variance solution. Maximizing expected excess return minus a risk-aversion-weighted variance penalty yields weights proportional to the inverse covariance matrix multiplied by the expected excess-return vector. Dividing those unconstrained weights by their sum imposes the constraint that portfolio weights add to one.
The resulting normalized portfolio is the familiar maximum-Sharpe allocation under the stated setup, with the risk-aversion parameter canceling during normalization. The explanation provides the optimization objective, its first-order condition, and the normalization step as its evidence. It assumes an invertible covariance matrix and does not include additional constraints such as bounds, transaction costs, or other linear restrictions; the document notes that more general constraints require a more complex formula.
Key ideas
- The unconstrained mean-variance optimum is proportional to the inverse covariance matrix times expected excess returns.
- Normalizing the unconstrained weights by their sum makes the portfolio fully invested.
- The risk-aversion parameter cancels when the unconstrained weights are normalized.
- Additional constraints generally change the closed-form portfolio solution.
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# Derivation of the tangency (maximum Sharpe Ratio) portfolio in Markowitz Portfolio Theory?
# 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?
## Answer by John (score 14, accepted)
https://quant.stackexchange.com/a/8620
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.
## Answer by Hamed (score 7)
https://quant.stackexchange.com/a/10162
Check out following link. In page 23 you'll find the derivation. http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.pdf
## Answer by papdog (score -1)
https://quant.stackexchange.com/a/8617
Merton, Robert, 1972, An Analytic Derivation of the Efficient Portfolio Frontier, Journal of Financial and Quantitative AnalysisShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.