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Analytic solution to variance optimization with no short selling
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We have performed numerical simulations to support the analytic results and found perfect agreement for N/T<2. Numerical experiments on finite size samples of symmetrically distributed returns show that above this critical point the probability of finding solutions with zero in-sample variance increases rapidly with increasing N, becoming one in the large N limit. However, these are not legitimate solutions of the optimization problem, as they are infinitely sensitive to any change in the input parameters, in particular they will wildly fluctuate from sample to sample. With some narrative license we may say that the regularizer takes care of the longitudinal fluctuations of the optimal weight vector, but does not eliminate the divergent transverse fluctuations.
We also calculate the distribution of the optimal weights over the random samples and show that the regularizer preferentially removes the assets with large variances, in accord with one's natural expectation.
Author(s):
Imre Kondor
London Mathematical Laboratory
United Kingdom
Gabor Papp
Eötvös University, Budapest, Institute for Physics
Hungary
Fabio Caccioli
University College London, Department of Computer Science
United Kingdom