# 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 '19 at 17:32
• Fourth Edition - Chapter 1: A primer on portfolio theory - Exercise 2 page 43 – JeanGuillaume Jun 28 '19 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 '19 at 23:02

## 1 Answer

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).

Merton, Robert C. “An Analytic Derivation of the Efficient Portfolio Frontier.” The Journal of Financial and Quantitative Analysis, vol. 7, no. 4, 1972, pp. 1851–1872.

As for homogeneous expectations on the other hand, from sidebar link, investors are assumed rational and only use what data they are presented with.

Homogeneous expectations is an assumption in Harry Markowitz's Modern Portfolio Theory that all investors will have the same expectations and make the same choices given a particular set of circumstances. The assumption of homogeneous expectations states that all investors will have the same expectations regarding inputs used to develop efficient portfolios, including asset returns, variances, and covariances.

• 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 '19 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 '19 at 12:45