# Sharpe ratio and uniformly distributed random portfolio

I am currently working on this paper which derives the Sharpe ratio distribution of uniformly random porfolios: https://www.researchgate.net/publication/271833345_A_Uniformly_Distributed_Random_Portfolio \

I'll give a rough summary of the second chapter, but you probably have to read this yourself to answer the question. This is fun though, as the topic is very interesting.
In section 2.1 Kim & Lee guide the distribution of all feasible portfolios from the perspective of the Sharpe ratio. With the invariance property of the Sharpe ratio: $$k>0$$: $$SR(kw \mid \mu ,\Sigma )= SR(w \mid \mu ,\Sigma )$$
It is possible to distribute a random variable w uniformly on a unit hypersphere. In section 2.2 this relationship between two portfolios is derived: $$w^*=\Sigma ^{-1} \mu \in \mathbb{R}^n$$ optimal portfolio of the market. Then:
$$SR(w_1 \mid \mu ,\Sigma ) \geq SR(w_2 \mid \mu ,\Sigma ) \Leftrightarrow \theta_1 \leq \theta_2$$

$$\theta_i=arccos \biggl( \frac{ (L_\Sigma ^T w_i)^T L_\Sigma ^T w^* }{\lVert L_\Sigma ^T w_i \rVert_2 \lVert L_\Sigma ^T w^* \rVert_2} \biggl)$$ angle between $$L_\Sigma ^T w_i$$ und $$L_\Sigma ^T w^*$$. $$\Sigma = L_\Sigma L_\Sigma ^T$$ Cholesky decomposition.

This means that all portfolios that have a Sharpe ratio higher than s can be displayed on the hyperspherical cap in an $$L_\Sigma ^T$$-transformed space with axis $$L_\Sigma ^T w^*$$ and colatitude angle $$\theta_s$$ In addition, the Sharpe ratio distribution of equally distributed portfolios can be determined by dividing the surface of the cap by the surface of the entire sphere. On Page 299 they quote:\ "†A unit hypersphere in the original space does not become a unit hypersphere in a $$L_\Sigma ^T$$-transformed space. However, due to the scale- invariance property of the Sharpe ratio, considering a unit hyper- sphere in a $$L_\Sigma ^T$$-transformed space does not affect our analysis."
I don't understand how scale invariance contributes to this. The further analysis is based on the area ratio of the unit hypersphere and the hyperspherical cap (axis $$L_\Sigma ^T w^*$$ and colatitude angle $$\theta_s$$). This area ratio is different if you place it with the same axis and the same colatitude angle on a unit hypersphere in an $$L_\Sigma ^T$$-transformed space and a unit hypersphere (in the original space) which was mapped in the $$L_\Sigma ^T$$-transformed space. Or do I get something wrong?\

Thank you very much!

• I don't think I buy it either, but this is an interesting find. Based on a historical sample of data, the sample portfolio should be 'near' the optimal one, and closer for larger $n$. (I used similar mathematics to find a Cramer Rao bound on portfolio Sharpe.) Oct 5 at 17:06
• Do you think it could be a "good" approximation? Oct 5 at 19:26
• I am not sure what you are asking, but the approximate distribution of that angle $\theta_i$ is, if I understand correctly, implied by equation (25) in my paper. There is no reason to think the sample portfolio would be uniformly distributed on the sphere unless the direction was picked without observing any historical data, which seems like a bad investing strategy. Oct 5 at 19:35
• In my Paper the excess return and the covariance matrix are fixed. Only the portfolios are uniformly distributed. This is equal to the enumeration of all feasible portfolios in a given market. With the Sharpe ratio distribution of uniformly random portfolios you are able to create a performance ranking of all portfolios in this market. my question is why can we consider an unit hypersphere in an $L_\Sigma ^T$-transformed space? An this because of the invariance property of the sharpe ratio. Oct 5 at 20:12
• By "invariance property" I think they mean that the constrain that sum of weights equal 1 is not important, portfolio with weights (1.2,0.8) will have same Sharpe ratio as one with weights (0.6,0.4). Oct 6 at 12:35