# Application Of Portfolio Optimization: A Statistical Approach

## Application Of Portfolio Optimization: A Statistical Approach

ABSTRACT

The main objective of this thesis is to look at how the Markowitz Mean-Variance assets selection model performs with distribution free model, Gini-Mean Difference model and highlight statistical approach to portfolio optimization in terms of risk reduction; interrelationships of diversified assets, assets rebalancing and stochastic dominance etc. In the study, the Mean-Variance model tends to slightly outperform the Gini-Mean Difference model in return/risk characteristics. More sophisticated investors can use the Gini-Mean Difference model if they posses the skills as it involves more complex computations in the procedure, and further captures the data than Mean-Variance approach since its distribution free model.

CHAPTER ONE

INTRODUCTION

1.1 Background of the Study

Nigeria stock market is a place where investors buy shares of assets or securities mostly for them to earn reward (returns) at a given level of “tolerable” risk. The word “portfolio” is used to mean a mix or combinatory pool of assets, securities or investment of financial or physical nature, in which an investor holds to meet his/her set objectives. On the other hand, portfolio selection is choosing the most suitable combination of assets by risk averse or rational investor to maximize return while at the same time minimizes risk.

Due to abrupt unpleasant surprises of global financial crisis, and inferior market downturn experienced by financial markets in different times, investors are becoming highly concerned about the risk of their investments coupled seeking ways to achieve more attractive risk/return characteristics and better capital protection in difficult environments. The main plan of portfolio management is to form diverse securities in a portfolio that meets the needs of investors and thereafter manage the portfolio in other to obtain required goals (Vigdiset al., 2011). Many players in the financial market have been frequently looking for strategic ways of investing which is capable of meeting this aforesaid protest of the market.

Modern Portfolio Theory(MPT) pioneered by Markowitz (1952, 1970, 1987) seminal work specifically solves the tradeoff between risk and return using formed curve in graph called efficient frontier. This frontier solves the problem by considering the risk, return of invested assets and the correlation that exist between the asset return. The curve identifies those that are at maximum return for a given level of risk, or at their minimum risk for a given level of return.

Markowitz’s work to a great extent changes the behavior of investors and financial managers by providing more insight on the issue of assets selection. Nowadays, players in the market are embracing the framework viewed as the most standard model for today‟s investment management even with the model‟s unrealistic questionable assumptions.

Markowitz (1991), as well asElton and Gruber (1997) talk more on the main issue an investor faces when investing, of them is how to distribute resources among different assets alternatives. Almost all investors are being posed with this same problem, with the added sufferings and complications needed to clearly include the properties of the liabilities in the analysis (Bodieet al. 2004). The problems are different in terms of structure, but that can be categorized to the portfolio theory.

MPT can be confidently termed as the opposite to traditional asset picking brought into light by economist persevering to look and understand the dynamics of the market in it totality, rather than basing attention at things that make an investment opportunity unique. The problem of spreading one‟s fund into different asset alternatives is one of the forefront basic concerns of financial theory (Cohen and Natoli 2003). Risk as well as asset allocation are important parts of the MPT, with investment being explained statistically with respect to their long-term return rate and their short-term expected volatility.

Risk (volatility) is the likelihood average bad years of an investment. The objective and aim of an investor is to find acceptable risk tolerance of his investment, while at the same time identify portfolio with maximum expected return with that level of risk (Elton and Gruber 1997).

As pointed out by Elton and Gruber (1997), market players or investors intend to form „perfect investment‟ and attribute it to have high return with no risk coming with it. This type of investment in reality is almost unachievable. With no surprise, these investors spend much of their time and energy coming up with methods and theories that come close to the „„perfect investment‟‟. Among all the theories and methods none is with high popularity and power of the MPT. That been said, it would be highly important that market players and professionals get themselves familiar with how to make use of that theory to develop portfolio which best suit client wishes and risk tolerance. It would also be highly beneficial if financial managers understand the forces that drive risk and return of portfolio and know how they can be amended to tune with clients aspirations for maximum benefit.

Kristein et al. (2006) stated that modern portfolio theory holds that spreading funds among different assets which is termed as diversification increases return at given level of risk or at minimum level of risk. The theory uses the volatility of returns implied by market price fluctuations as the composite of risks. It clearly solved the dilemma on how risk-averse financial investors can form master assets(portfolio) in such a way that optimize market risk for a given expected returns, with emphasis that volatility or risk is an inherent aspect of higher reward (return).

Diversification is one of the strategies financial investors use when constructing low risky portfolio (Bodie et al.2004). It is a wisely offensive style to the market swings or a defensive technique to investment risk. As the popular adage “don’t put all your eggs into one basket” is quite simple to say but more hard to practically perform in real life. Diversification is highly regarded these days due to the heat coming with global financial crisis and as proof of this; Harry Markowitz was awarded the famous Nobel Prize in Economics in recognition of his outstanding research (Markowitz, 1991).

1.2 Statement of the Problem

During global financial crisis at different point in time, investors face the risk of loosing their capital not to talk of the excess return they garners. Due to this reoccurring problem at different point in time, investors are always in search of ways to secure their capital or at least the ways their investments would come with tolerable risk. The question that arises is how do these investors spread their/diversify their wealth in presence of multiple investments to choose from that have attributes of their expectations? And when they have successfully achieved it, how do they maintain it so that the market does not take them by surprise? We will compare two approaches, Markowitz mean-variance and Gini-Mean approaches to portfolio optimization, using variance and gini index as the measures of risk to see the superiority of one over the other, and thereafter derive some useful statistics risk-averse investors would be interested to know.

1.3 Purpose of the Study

The set target of this study is to examine whether there is any possible improvement when applying MPT (Markowitz mean-variance) investing strategy than using the naïve index investing. On doing that, we would be exploring the statistical analysis of portfolio diversification, correlation among market assets and Gini-Mean Difference statistic as a substitute for variance (risk) statistic in Markowitz mean-variance model.

1.4 Significance of the study

The significance of the study is its highly advocacy of the MPT as a way of yielding good investment returns at an acceptable risk as against the traditional (naïve) index investing strategy that high return stocks always come with high risk.

1.5 Aim and Objectives of the Study

The aim of this research is to compare the two portfolio optimization approaches, Markowits Mean-variance and Gini-Mean difference models while the objectives include:

i. Use of Variance and Gini index as two risk measures in the objective functions of the two approaches.

ii. Assets selection rebalancing in the optimization engine using five different sectors of the Nigeria Stock Market.

1.6 Limitation of the Study

As a result of major concern of resources constraints, all the parameters cannot be taken into consideration when examining the assets portfolio performance. The main point of view is solely performance and power results using the Nigerian Capital Market in asset selection. Also, other constraints such as tax-efficiency, positive and negative leverage and are overlooked in the study

As mentioned above, no amount of money going to be invested rather the real performance is to be examined from selected assets market index as its going to help the investor in achieving a nicer and more all-encompassing outcome of the findings. For us to be able to get more true observational result in the study, the benchmark index is narrowed down. The index is the general market performance index for the thirty most capitalized stocks in the Nigerian Stock Exchange; this index mimics the market movement of the stocks in total. The difficulty comes from the fact that as the change in market condition happens, the risk and expected return of the various stocks change; and as a result of this constantly shifting market conditions, we only used short period of market history that encompass good and bad market cycles to look at the effectiveness of the approaches.