Project Report on STUDY OF PORTFOLIO OPTIMZATION MODELS Project Supervisor: Dr V.N.Sastry Associate Professor, IDRBT Submitted by: Chander Prakash Jumrani 2007MT50436 3rd Year Undergraduate Student IIT Delhi JULY 2010 Institute for Development and Research in Banking Technology Castle Hills, Masab Tank Hyderabad – 500 057 Phone: 90-40-23534981 (8 Lines); Fax: 90-40-23535157 Web: http://www.idrbt.ac.in CERTIFICATE This is to certify that the Summer Internship project work entitled “Study of Portfolio Optimization Models”, submitted for partial fulfilment of the requirements for the award of the Integrated Degree of Master of Technology in Mathematics and Computing to the Indian Institute of Technology Delhi is a record of bonafide project work carried out by Mr. Chander Prakash Jumrani (Entry No. 2007MT50436) of third year at IDRBT (Institute for Development and Research in Banking Technology), Hyderabad, for a period of 10 weeks under my guidance during May-July 2010. The subject matter embodied in the project report has not been submitted for the award of any other degree or diploma. V.N.Sastry Associate Professor, IDRBT, Castle Hills, Masab Tank, Hyderabad – 500 057. e-mail: [email protected] Abstract: Portfolio Optimization is the technique of adjusting the ratio of securities in the portfolio to meet ones needs in the best possible way. The objective of the project is to study the various single objective and multi-objective based portfolio optimization models which includes: • Markowitz Mean Variance Model • Capital Asset Pricing Model • Goal Programming based Model These models were tested on a basic two stock and three stock portfolios and solved using quadratic and goal programming techniques. The results were analysed and inference was made. The study also included the use of tools such as Solver in MS EXCEL and the Optimization Toolbox in MATLAB to solve these optimization problems. CONTENTS OF THE REPORT: 1. Introduction 2. Chapter 1 : Single objective based portfolio optimization models. 3. Chapter 2: Multi-objective based portfolio optimization models. 4. References INTRODUCTION Optimization: It refers to choosing the best element from some set of available alternatives. The decisions are based on certain conditions that must be met. Portfolio : It is a set of investments held by an individual or an organization. It includes risky (stocks, options and other derivative securities) as well as non -risky (bonds) assets. Optimization of Portfolio : It is technique of changing the ratio and quantity of assets present in a portfolio in order to meet certain objective in the most optimal way. Objectives may include • Risk Minimization • Return Maximization • Other objective includes liquidity, market dynamics etc. These objectives and constraints may be linear or non-linear; accordingly we may get LPP or Non-LPP. We have considered Quadratic Programming and Non-linear Multi-objective portfolio optimization problems. Chapter 1: Single objective portfolio optimization models 1.1 The Markowitz Mean Variance Model Markowitz Mean Variance Model also known as the Modern portfolio theory (MPT) is a theory of investment which tries to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets. Markowitz portfolio shows that as we add assets to an investment portfolio the total risk of that portfolio - as measured by the variance (or standard deviation) of total return - declines continuously, but the expected return of the portfolio is a weighted average of the expected returns of the individual assets. In other words, by investing in portfolios rather than in individual assets, investors could lower the total risk of investing without sacrificing return. The fundamental concept behind MPT is that the assets in an investment portfolio cannot be selected individually, each on their own merits. Rather, it is important to consider how each asset changes in price relative to how every other asset in the portfolio changes in price. Investing is a tradeoff between risk and expected return. In general, assets with higher expected returns are riskier. For a given amount of risk, MPT describes how to select a portfolio with the highest possible expected return. Or, for a given expected return, MPT explains how to select a portfolio with the lowest possible risk MPT is therefore a form of diversification. Under certain assumptions and for specific quantitative definitions of risk and return, MPT explains how to find the best possible diversification strategy. The Markowitz model is based on several assumptions concerning the behaviour of investors and financial markets: 1. A probability distribution of possible returns over some holding period can be estimated by investors (here it is assumed to be normally distributed.). 2. Variability about the possible values of return is used by investors to measure risk. 3. Investors care only about the means and variance of the returns of their portfolios over a particular period. 4. Expected return and risk as used by investors are measured by the first two moments of the probability distribution of returns-expected value and variance. 5. The correlation factors are assumed to be fixed/ constant. MEASURMENT OF RETURN AND RISK Throughout this chapter, investors are assumed to measure the level of return by computing the expected value of the distribution, using the probability distribution of expected returns for a portfolio. Risk is assumed to be measurable by the variability around the expected value of the probability distribution of returns. The most accepted measures of this variability are the variance and standard deviation. Return Given any set of risky assets and a set of weights that describe how the portfolio investment is split, the general formula of expected return for n assets is: n Er(P ) = ∑ wEr i() i (1) i=1 Where: n ∑ wi = 1.0; i=1 n = the number of securities; wi = the proportion of the funds invested in security i; th ri, r P = the return on i security and portfolio p; and E ( ) = the expectation of the variable in the parentheses. The return computation is nothing more than finding the weighted average return of the securities included in the portfolio. Risk The variance of a single security is the expected value of the sum of the squared deviations from the mean, and the standard deviation is the square root of the variance. The variance of a portfolio combination of securities is equal to the weighted average covariance of the returns on its individual securities: n n 2 Var()()rp=σ p = ∑∑ ww ij Cov, rr ij (2) i=1 j = 1 Covariance can also be expressed in terms of the correlation coefficient as follows: Cov(rij , r ) =ρσσ ijij = σ ij (3) where ρij = correlation coefficient between the rates of return on security i, ri , and the rates of return on security j, rj , and σ i , and σ j represent standard deviations of ri and rj respectively. Therefore: n n Var ()rp= ∑∑ w ijijij w ρ σ σ (4) i=1 j = 1 Overall, the estimate of the mean return for each security is its average value in the sample period; the estimate of variance is the average value of the squared deviations around the sample average; the estimate of the covariance is the average value of the cross-product of deviations. 1.1.2 TWO RISKY ASSET PORTFOLIOS THEORY: Two risky asset portfolio means that we are limiting our portfolio to just two stocks say A(µ 1, σ1) and B(µ 2, σ2).Let the mean and variance of the portfolio be (µ, σ) respectively. Let us take w 1 units of A and w 2 units of B such that w 1+ w 2 = 1. The expected rate of return on the portfolio is Er()P= wEr AA () + wEr BB () (5) The variance of the rate of return on the two-asset portfolio is 2 22222 σP=+(ww AA σσ BB ) = w AA σ + w BB σ + 2 ww ABABAB ρσσ (6) (A) The Correlation Factor ( ƥ) We have the Correlation factor ( ƥ) between A and B defined as ƥ = (Covariance A,B )/( σA. σB) It is calculated using the past data and determines the relationship between the two stocks and its expected behaviors w.r.t the fluctuations in the market. We have • -1< ƥ < 1 • ƥ=0 => uncorrelated • ƥ=1 => perfect positively correlated • ƥ=-1 => perfectly negatively correlated • ƥ< 0 => the stocks are negatively correlated i.e. the increase in the value of one causes a decrease in the value of the other. • ƥ>0 => the stocks are positively correlated i.e. the increase in the value of one causes a increase in the value of the other. Taking w 2=s and w 1=1- s and substituting in the equations of µ and σ, we get µ = µ 1(1-s) + µ2s …….(A) 2 2 2 2 σ = { σ1 (1-s) + σ2 s + 2 ƥ σ1σ2s(1-s)} ^ 0.5 …….(B) Using the above relation and taking s as the parameter a graph is plotted showing the relation between return and risk for two stocks and for different values of ƥ. ρ = − 1 asset B ρ = 0 ρ =1 0<ρ < 1 Expected Return asset A ρ = − 1 0 Standard Deviation Figure.1 Investment opportunity sets for asset A and asset B with various correlation coefficients INFERENCE FROM THE GRAPH 1. In Figure.1 the opportunity set with perfect positive correlation – is a straight line through the component assets. No portfolio can be discarded as inefficient in this case, and the choice among portfolios depends only on risk preference. Diversification in the case of perfect positive correlation is not effective. 2. With any correlation coefficient less than 1.0( ρ <1), there will be a diversification effect, the portfolio standard deviation is less than the weighted average of the standard deviations of the component securities.