Pythagorean theorem of Sharpe ratio

2017/03/08 by Shinzato, Takashi
#Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Economics and business #FOS: Mathematics #FOS: Physical sciences #Optimization and Control (math.OC) #Portfolio Management (q-fin.PM) #Risk Management (q-fin.RM)

paper · doi:10.48550/arxiv.1703.02777

Abstract

In the present paper, using a replica analysis, we examine the portfolio optimization problem handled in previous work and discuss the minimization of investment risk under constraints of budget and expected return for the case that the distribution of the hyperparameters of the mean and variance of the return rate of each asset are not limited to a specific probability family. Findings derived using our proposed method are compared with those in previous work to verify the effectiveness of our proposed method. Further, we derive a Pythagorean theorem of the Sharpe ratio and macroscopic relations of opportunity loss. Using numerical experiments, the effectiveness of our proposed method is demonstrated for a specific situation.

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