# Sharpe Ratio - Linear Homogeneous

In the book "Portfolio Construction and Risk Budgeting" of Sherer, there is an excercise with the following prompt:

"Use matrix algebra to find the maximum Sharpe ratio portfolio. Show that the Sharpe ratio is linear homogeneous. What does this mean economically ? "

I have found this portfolio ( $$w = \frac{\Sigma^{-1}\mu}{\mu '\Sigma^{-1}\mu}$$ ) and computed its sharpe ratio ( $$\sqrt{\mu '\Sigma^{-1}\mu}$$). So, its is homogeneous but for me it is not linear. Has someone an explanation about this linear homogeneous feature and its economic meaning ? Thank you !

• Could you provide the reference to the exercise please? Edition and page number please – Magic is in the chain Jun 27 at 17:32
• Fourth Edition - Chapter 1: A primer on portfolio theory - Exercise 2 page 43 – JeanGuillaume Jun 28 at 7:52
• The homogeneous property is meant in the leverage/deleverage sense as in the definition of coherent risk measure. I have seen this homogeneous property in the allocation sense but can’t find the definition used in the book as it seems to come out of nowhere! – Magic is in the chain Jun 28 at 23:02

The formula you came up with doesn't appear to account for the riskless asset. isn't the maximum Sharpe ratio portfolio $$\boldsymbol{\omega}= \frac{\mathbf{\Sigma}^{-1}(\boldsymbol{\mu}-r_f\cdot \boldsymbol{\iota}_N)}{{\boldsymbol{\iota}_N\mathbf{\Sigma}}^{-1}(\boldsymbol{\mu}-r_f\cdot \boldsymbol{\iota}_N)}$$ because the Sharpe ratio is $$\frac{\boldsymbol{\omega^{\top}\mu}-r_f}{\boldsymbol{\omega}^{\top} \mathbf{\Sigma} \boldsymbol{\omega}}$$?
I think linear homogeneity has to do with the constraint that individual portfolio weights must sum to 1: $$\boldsymbol\iota_N^{\top}\boldsymbol\omega=1$$, or $$\omega_1+\omega_2+\dots+\omega_N=1$$. The constraint is made homogenous in the Lagrangean derivation of the tangency portfolio by using $$\omega_1+\omega_2+\dots+\omega_N -1=0$$ instead, which is linear with the rest of the Lagrangean equation in the above analytical solution's derivation due to multiplier $$\lambda$$ placed at the front of it (see here and search 'homog' then tangency).
• In the book, the maximum sharpe ratio portfolio is the portfolio minimising the variance given the condition that $w' \mu = 1$. The formula I gave is correct if I consider $\mu$ as the expected excess return, isn't it ? There is no such constraint of the sum of the weights equal to one. As long as the sum is positive, I can divide by a constant to get this condition without changing the Sharpe Ratio. That is the meaning of homogeneous for me. – JeanGuillaume Jun 26 at 11:43
• if weights summing to 1 ex-ante for me is pretty much the equivalent of you scaling weights ex-post, then how we are defining homogeneous correspond to one another (somehow). $\boldsymbol\iota_N^{\top}\boldsymbol\omega=1$, and for you its $\boldsymbol\omega=\boldsymbol\omega/\boldsymbol\iota_N^{\top}\boldsymbol\omega$. so your solution probably is just derived from a different angle, but mine and your constraints on the weights are imposed at different times during the process – develarist Jun 26 at 12:45