Sharpe Ratio Distribution from Uniform Random Portfolios
Summary
The document summarizes a geometric approach to the Sharpe ratio distribution of uniformly sampled portfolios. Because multiplying portfolio weights by a positive constant leaves the Sharpe ratio unchanged, portfolios can be represented by directions on a sphere. After transforming weights using the Cholesky factor of the return covariance matrix, the optimal portfolio direction becomes an axis, and portfolios with Sharpe ratios above a threshold correspond to a spherical cap around that axis.
The proposed distribution is obtained from the ratio of the cap’s surface area to the full sphere’s surface area. The question challenges whether this area calculation remains valid when the original sphere is distorted by the covariance transformation. Scale invariance explains why normalization in transformed space does not change individual Sharpe ratios, but the excerpt does not resolve whether the resulting uniform sampling measure or cap-area ratio matches the original portfolio distribution. That distinction is a key assumption to examine before using the derived distribution.
Key ideas
- Sharpe ratio is unchanged when portfolio weights are multiplied by a positive scalar.
- Covariance-based transformation maps portfolio directions into a space where Sharpe ranking is expressed by angle to the optimal direction.
- Portfolios above a Sharpe threshold form a spherical cap in the transformed representation.
- A cap-to-sphere area ratio gives a distribution only under the assumed uniform sampling measure.
- Scale invariance preserves Sharpe values under normalization but does not by itself preserve uniform area measure under a transformation.
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# Sharpe ratio and uniformly distributed random portfolio
# 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!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.