Tangency Portfolios, Sharpe Ratio Scaling, and Homogeneity
Summary
The document asks how the maximum-Sharpe portfolio relates to linear homogeneity and its economic interpretation. It presents a portfolio formula based on the inverse covariance matrix and expected returns, along with a corresponding Sharpe ratio, then questions why the ratio is described as linear homogeneous. A response points out that a risk-free rate should be included when defining excess returns and sketches a tangency-portfolio solution with weights constrained to sum to one.
The discussion connects the constraint to the Lagrange multiplier derivation and distinguishes this mathematical issue from homogeneous expectations, an assumption that investors share the same beliefs about returns and risk. It cites portfolio theory as context but does not fully establish the derivation or economic interpretation. The formulas and terminology in the exchange should therefore be checked against the chosen Sharpe-ratio convention and constraint setup.
Key ideas
- A tangency portfolio is defined using expected excess returns when a risk-free rate is included.
- The covariance matrix and expected returns determine the unconstrained direction of the maximum-Sharpe allocation.
- A sum-to-one constraint affects the derivation of portfolio weights.
- Linear homogeneity of a portfolio expression is distinct from the assumption of homogeneous investor expectations.
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# Sharpe Ratio - Linear Homogeneous
# 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 !
## Answer by develarist (score 1)
https://quant.stackexchange.com/a/46305
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.Shown 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.