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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.

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 [

There is a broad literature which addresses the question of how to reduce estimation risk in portfolio optimization. De Miguel et al. compare portfolio strategies which differ in the treatment of estimation risk in [

The main aim of this paper is to compare the efficient frontier for real data based on corrected estimators of [

We carry out our analysis for three scenarios namely the Indian stock market, London Stock market and U.S stock market to facilitate a comparative study and to conclude about the uniformity of our results. We use constituent stocks of NSE CNX 100, FTSE 100 and S&P 100 respectively for the three scenarios as our data base taking daily data from 1st Jan 2013 to 1st Jan 2014 time span. The daily returns data are publicly available from NSE India and yahoo finance. Thus we have at our disposal, 100 stocks for each country with 250 observations per stock. In other words, considering p to be the number of assets and n to be the number of observations per asset, we arrive at a large p, large n setting which in modern statistical parlance can be considered to be a high dimensional setting.

The rest of the paper is organized as follows. Section 2 is committed to explaining the modern portfolio theory. Section 3 deals with identifying the underestimation factors and the bias inherent in the plug in estimators and subsequently eliminating them from the empirical optimized portfolio, to arrive at the final error-free optimized weights. Section 4 deals with norm constrained models. In section 5, we present the empirical results of comparing the efficient frontiers obtained from Markowitz portfolio to error-free efficient frontier and norm constrained portfolio efficient frontiers. We present our conclusions in section 6.

Markowitz portfolio theory [

There is an opportunity to invest in p assets

The mean returns are represented by a p-dimensional vector

The covariance matrix of the returns is denoted by

The aim is to create a portfolio with guaranteed mean return

The problem is to find the weights or amount allocated to various assets of the portfolio.

Note that

Here 1_{p} is a p-dimensional vector with one in every entry.

In practice, Σ and µ are unknown. The most common procedure known as plug-in implementation replaces them with their sample estimators as follows to obtain the optimal weights.

With

If

where

The curve

In the Markowitz setting, let us assume that the returns have normal distribution. We shall assume n and p both go to infinity and each X_{i} ~ N_{p} (µ, S) independently and identically. The parameters of the distribution are estimated using sample estimators defined in (2).

We have from Corollary 3.3 of [

where

at hand while

in the factor

Also when

The estimator

It is also shown in Theorem 5.1 of [

where

We use the 95% confidence intervals for the variance of a single Normal variable with unknown mean µ and standard deviation σ given by:

where

The short sale constrained minimum-variance portfolio,

The 1-norm-constrained portfolio, _{1}-norm of the portfolio-weight vector be smaller than or equal to a certain threshold c; that is,

1-norm constrained portfolio problem can be summarized as

Markowitz risk minimization problem can be recast as a regression problem.

By using the fact that the sum of total weights is one, we have

where R = Return vector,

Finding the optimal weight w is the same as finding the regression coefficient

where

does not. Efron et al. developed an efficient algorithm in [

Here our objective is to minimize the out-of-sample portfolio variance. To choose c we use leave-one-out- cross validation (see [

The 2-norm-constrained portfolio, _{2}-norm of the portfolio-weight vector is smaller than or equal to a certain threshold c; that is,

2-norm constrained portfolio problem can be summarized as

Similar to the 1-norm constrained portfolio finding the optimal weight w in this case is the same as finding the regression coefficient

The gross-exposure constraint

where

The whole solution pat

To choose c we use cross validation, as in the case of 1-norm constrained portfolio.

Below 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.

In Figures 1-3, we present the efficient frontiers using the different methods. The dashed lines represent the empirical 95% confidence intervals computed for a fixed expected return. The x-axis is variance and y-axis is

expected returns. We have considered the same set of µ’s and Σ’s, for each individual country to keep the results comparable. It can be concluded from the relative positions of the corrected and uncorrected efficient frontiers that the risk is indeed underestimated in case of high dimensional data. But comparing to 2-norm and 1-norm constrained portfolios as they outperform the corrected frontiers. The constrained portfolios are, in general, less efficient than the corrected portfolio, in the sense that they have higher variance for each fixed level of return. Of course constrained portfolios have their own advantages due to sparsity that might out-weigh the loss in efficiency. For the 1-norm and 2-norm portfolios, the choice of the asset Y is important. We have chosen Y to be the no short sale portfolio in all our computations. For each country, the 2-norm portfolio is most efficient among the constrained portfolios and the 1-norm is not monotone.

The amount of shrinkage or regularization is directly related to the number of stocks included in the optimal portfolio. In _{1} norm. For almost all values of c, the number of stocks in the portfolio is highest for the Indian market and lowest for the US market. Results for the L_{2}norm are similar.

For out of sample performance we first created portfolios for all the three datasets using the return data for the first 230 trading days. These portfolios are then held for one month and rebalanced at the end next month. The summary statistics of these portfolios are presented for the three datasets as box-plots in Figures 5-7. 1-norm constrained portfolios were created for c = 2 and c = 3 for all the three nations. 2-norm constrained portfolios were created for the optimal c chosen by cross validation, as mentioned in Section 4. This value equals 1.2544, 1.14 and 1.0739 respectively for US, UK and Indian data.

The out-of-sample performance is very different for the three markets. For the US data, the 2-normcon-

strained, corrected and no-short-sale portfolios have close to zero average returns while the other methods yield negative average returns. The variances are almost same for all methods except the Markowitz, which has a lower variance. For UK data, the 1-norm with c = 3 and corrected portfolios have significantly negative average return while others have small positive or zero average returns. The variances are almost all the same. For the Indian data, all portfolios except the Markowitz have high positive average returns. In particular, the corrected portfolio has very high average returns, but the variance is also quite high. Overall, from the out-of-sample results, the 2-norm constrained portfolio has higher average and comparable variance to the Markowitz portfolio in all the markets.

In this paper we study the effect of high dimension on the efficient frontier with real data on three markets. In particular we study how the recently suggested methods of corrected frontier based on normality assumptions and norm-constrained methods perform relative to Markowitz portfolio optimization. We observe that the Markowitz solution indeed leads to biased estimates of risk that can be improved with the corrected estimates. The norm-constrained methods are comparable and need less model assumptions. Alternative methods of improving the covariance matrix estimation are Bayesian shrinkage approach [

RituparnaSen,PulkitGupta,DebanjanaDey, (2016) High Dimensionality Effects on the Efficient Frontier: A Tri-Nation Study. Journal of Data Analysis and Information Processing,04,13-20. doi: 10.4236/jdaip.2016.41002