Abstract
Markowitz Portfolio theory under-estimates the risk associated with the return of a portfolio in case of high dimensional data. El Karoui mathematically proved this in [1] and suggested improved estimators for unbiased estimation of this risk under specific model assumptions. Norm constrained portfolios have recently been studied to keep the effective dimension low. In this paper we consider three sets of high dimensional data, the stock market prices for three countries, namely US, UK and India. We compare the Markowitz efficient frontier to those obtained by unbiasedness corrections and imposing norm-constraints in these real data scenarios. We also study the out-of-sample performance of the different procedures. We find that the 2-norm constrained portfolio has best overall performance.
Highlights
The need for solutions to optimization problems in a high dimensional setting is increasing in the finance industry with huge amount of data being generated every day
Many empirical studies indicate that minimum variance portfolios in general lead to a better out-of-sample performance than stock index portfolios [2] [3]
We provide an overview of our results of Markowitz efficient frontier, corrected frontier using Gaussian assumption, 1-norm and 2-norm constrained efficient frontiers for the 3 countries
Summary
The need for solutions to optimization problems in a high dimensional setting is increasing in the finance industry with huge amount of data being generated every day. Many empirical studies indicate that minimum variance portfolios in general lead to a better out-of-sample performance than stock index portfolios [2] [3]. When implementing portfolio optimization according to [4], one needs to estimate the expected asset returns as well as the corresponding variances and covariances. One computed using population data and another estimated from sample data. This relationship is important and relevant for high dimensional data where one suspects that the difference between the two may be considerable
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