Modern Portfolio Theory -Markowitz Model

Transcription

1 Modern Portfolio Theory -Markowitz Model Rahul Kumar Project Trainee, IDRBT 3 rd year student Integrated M.Sc. Mathematics & Computing IIT Kharagpur rahulkumar641@gmail.com Project guide: Dr Mahil Carr Associate Professor IDRBT, Hyderabad

2 CONTENTS Topic Page Certificate 3 Declaration by the candidate 4 Acknowledgement 5 Abstract 6 Introduction 7 Returns 8 Variance of a portfolio 9 Diversification 10 Efficient Frontier 11 Quadratic Programming 14 Example solved using MATLAB 15 Conclusion and future scope 22 Bibliography 23 2

3 Certificate This is to certify that the project report titled Modern Portfolio Theory- Markowitz Model submitted by Mr Rahul Kumar, 3 rd year student enrolled for the Integrated M.Sc. course in Mathematics & Computing in the Department of Mathematics, Indian Institute of Technology Kharagpur, is a record of the bona fide work carried out by him under my guidance during the period 6 th May 2011 to 6 th July 2011 at the Institute for Development & Research in Banking Technology (IDRBT), Hyderabad. The project work is a research study, which has been successfully completed as per the set objectives. I observed Mr RAHUL KUMAR as sincere, hardworking and having capability and aptitude for independent research work. I wish him every success in life. Dr MAHIL CARR Associate Professor IDRBT Hyderabad 6 th July

4 Declaration by the candidate I declare that the summer internship project report titled Modern Portfolio Theory- Markowitz Model is my own work conducted under the supervision of Dr Mahil Carr at the Institute for Development & Research in Banking Technology, Hyderabad. I have put in 62 days of attendance with my supervisor at IDRBT and have been awarded a project fellowship. I further declare that, to the best of my knowledge, the report does not contain any part of any work which has been submitted for the award of any degree either in this institute or in any other university without proper citation. Rahul Kumar 3 rd year student Integrated M.Sc. Mathematics & Computing IIT Kharagpur 6 th July

5 Acknowledgement I would like to thank Mr B. Sambamurthy, Director of IDRBT, for giving me this opportunity. I gratefully acknowledge the guidance from Dr Mahil Carr, who helped me sort out all the problems in concept clarifications & without whose support the project would not have reached its present state. Rahul Kumar 3 rd year student Integrated M.Sc. Mathematics & Computing IIT Kharagpur 6 th July

6 Abstract This report goes into the details of Modern Portfolio Theory and gives an approach for portfolio selection using Markowitz s mean variance model. A software has been made using MATLAB to solve any portfolio selection problem using this approach. An example has been solved to understand the execution of the software. 6

7 Introduction Modern portfolio theory (MPT) was introduced by Harry Markowitz with his paper "Portfolio Selection," which appeared in the 1952 Journal of Finance. Thirty-eight years later, he shared a Nobel Prize with Merton Miller and William Sharpe for what has become a broad theory for portfolio selection. Prior to Markowitz's work, investors focused on assessing the risks and rewards of individual securities in constructing their portfolios. Standard investment advice was to identify those securities that offered the best opportunities for gain with the least risk and then construct a portfolio from these. Following this advice, an investor might conclude that railroad stocks all offered good risk-reward characteristics and compile a portfolio entirely from these. Intuitively, this would be foolish. Markowitz formalized this intuition. Detailing mathematics of diversification, he proposed that investors focus on selecting portfolios based on their overall risk-reward characteristics instead of merely compiling portfolios from securities that each individually has attractive risk-reward characteristics. In a nutshell, inventors should select portfolios, not individual securities. If we treat single-period returns for various securities as random variables, we can assign them expected values, standard deviations and correlations. Based on these, we can calculate the expected return and standard deviation of any portfolio constructed with those securities. We may treat standard deviation and expected return as proxies for risk and reward. Out of the entire universe of possible portfolios, certain ones will optimally balance risk and reward. These comprise what Markowitz called an efficient frontier of portfolios. An investor should select a portfolio that lies on the efficient frontier. James Tobin (1958) expanded on Markowitz's work by adding a risk-free asset to the analysis. This made it possible to leverage or deleverage portfolios on the efficient frontier. This leads to the notion of a super-efficient portfolio and the capital market line. Through leverage, portfolios on the capital market line are able to outperform portfolio on the efficient frontier. Sharpe (1964) formalized the capital asset pricing model (CAPM). This makes strong assumptions that lead to interesting conclusions. Not only does the market portfolio sit on the efficient frontier, but it is actually Tobin's super-efficient portfolio. According to CAPM, all investors should hold the market portfolio, leveraged or de-leveraged with positions in the risk-free asset. CAPM also introduced beta and relates an asset's expected return to its beta. Portfolio theory provides a broad context for understanding the interactions of systematic risk and reward. It has profoundly shaped how institutional portfolios are managed, and motivated the use of passive investment management techniques. The mathematics of portfolio theory is used extensively in financial risk management and was a theoretical precursor for today's value-at-risk measures. 7

8 Returns a) For an individual security Return = Money gained/money invested For example: Return on stock (for a year) = closing price of the year closing price of the previous year + dividends for the year (closing price of the previous year) Now calculate mean return for each security from the return series of that security. b) For a portfolio Portfolio return is the proportion-weighted combination of the constituent securities. r p = n X i r i i=1 where, r p = Expected return on the portfolio X i = fraction invested in the i th of the n securities r i = mean return of the i th security 8

9 Variance of a portfolio σ p 2 = n i=1 n X i j=1 X j σ ij where, σ p 2 = Variance of the portfolio having n securities σ ij = Covariance between securities i & j Covariance matrix is σ 11 σ 1j σ 1n.. σ i1 σ ij σ in.. σ n1 σ nj σ nn 9

10 Diversification An investor can reduce portfolio risk simply by holding combinations of instruments which are not perfectly positively correlated (correlation coefficient, -1 ρ ij <1). In other words, investors can reduce their exposure to individual security risk by holding a diversified portfolio of securities. Diversification may allow for the same portfolio expected return with reduced risk. If all the security pairs have correlations of 0 they are perfectly uncorrelated the portfolio's return variance is the sum over all securities of the square of the fraction held in the asset times the asset's return variance (and the portfolio standard deviation is the square root of this sum). 10

11 Plot r p vs σ p using critical line algorithm Given: Efficient Frontier r p = n X i r i i=1 σ p 2 = n i=1 n X i j=1 X j σ ij n i=1 X i = 1 X i 0 (no short selling allowed) We get the Efficient Frontier as given below: Expected Return(r p ) All portfolios on the line are efficient Standard deviation (σ p ) Combinations along this curve represent portfolios for which there is lowest risk for a given level of expected return. Equivalently, a portfolio lying on the efficient frontier represents the combination offering the best possible expected return for given risk level. Modern Portfolio Theory assumes that investors are risk-averse, meaning that given two portfolios that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher expected returns must accept more risk. The exact 11

12 trade-off will be the same for all investors, but different investors will evaluate the trade-off differently based on individual risk aversion characteristics. The implication is that a rational investor will not invest in a portfolio if a second portfolio exists with a more favourable riskexpected return profile i.e., if for that level of risk an alternative portfolio exists which has better expected returns. It deals with a single period case only. 12

13 For a 3 security portfolio, we get X 3 = 1 - X 1 X 2 σ p 2 = X 1 2 σ 11 2σ 13 + σ 33 + X 2 2 σ 22 2σ 23 + σ X 1 X 2 σ 12 σ 13 σ 23 + σ 33 + [2X 1 σ 13 σ X 2 σ 23 σ 33 + σ 33 r p = X 1 r 1 r 3 + X 2 r 2 r 3 + r 3 r p r p r 1,r 2,r 3 13

14 minimize σ p 2 = subject to: n n i=1 n j=1 X i X j σ ij a) i=1 X i = 1 n b) i=1 X i r i = r p c) X i 0 (no short selling allowed) where, r p is the desired rate of return for the portfolio Quadratic Programming which is same as minimize σ 2 p = X T CX subject to: 1 a) X T = 1 1 b) X T r = r c) X 0 where, C is the covariance matrix X is a column vector r is a row vector of the mean return of individual securities 14

15 Example solved using MATLAB We have taken the adjusted monthly closing prices of the stocks of four companies listed on the NSE Infosys Ltd TATA Steel Ltd Reliance Industries Ltd (RIL) State Bank of India (SBI) Note that the adjusted closing prices take into consideration dividends paid and stock splits, and hence return could be calculated directly. The data is in a file named example.xlsx and the data is listed column wise, i.e. 1 st column has data of Infosys, 2 nd column is TATA Steel and so on. The data of each security in a column is listed with January 2006 on top and June 2011 in the bottom. Keep the data file and the MATLAB file in the same folder. 15

16 The monthly data is from January 1, 2006 to June 1,

17 MATLAB Code clear all clc %read excel file containing price data of securities filename=input('excel filename enclosed in single quotes = '); N=xlsread(filename); m=size(n); %form return series for j=1:m(2) for i=1:m(1)-1 N(i,j)=(N(i+1,j)-N(i,j))/N(i,j); end end N(i+1,:)=[]; %calculate correlation coefficients, mean returns and standard deviation ExpCovariance=cov(N); Correlation_coefficient = corrcoef(n) ExpReturn=mean(N); Expected_Return = ExpReturn Standard_deviation = sqrt(diag(expcovariance)') %plot return graph for each security n=m(2); for i=1:n figure('name','return Plot Window','NumberTitle','off') plot(n(:,i)); xlabel('time'); ylabel('return(in fraction)'); title(sprintf('security %d',i)); end %plot graph for efficient frontier NumPorts=50; frontcon(expreturn,expcovariance,numports); %solve quadratic programming problem H=2*ExpCovariance; Aones=ones(1,n); Aeq=[Aones;ExpReturn]; DesiredRet=input('Return desired for the entire portfolio = '); fprintf('\n'); beq=[1;desiredret]; lb=zeros(n,1); ub=ones(n,1); options = optimset('largescale','off'); [x,fval,exitflag]=quadprog(h,[],[],[],aeq,beq,lb,ub,[],options); fprintf('\n'); %output allocation to each security if(exitflag==1) for i=1:n fprintf('security %d allocation= %2.2f%%\n',i,x(i)*100); end else fprintf('no feasible solution exits'); end 17

18 Infosys monthly returns TATA Steel monthly returns 18

19 Reliance Industries monthly returns State Bank of India monthly returns 19

20 Plot of Efficient Frontier If we input the desired rate of return as 0.02 (i.e. a monthly rate of return of 2%), we get Table: Allocation of amount to particular company s shares Company Percentage allocation Infosys 30.96% TATA Steel 0.00% Reliance Industries 0.02% State Bank of India 69.01% 20

21 Output 21

22 Conclusion and future scope Markowitz model is used today extensively in financial management. Markowitz was even awarded a Nobel Prize in Economic Sciences in 1990 for his work and its impact. James Tobin (1958) expanded on Markowitz's work by adding a risk-free asset to the analysis. William Sharpe (1964) formalized the capital asset pricing model (CAPM). One can work on these developments to delve deeper into the modern portfolio theory. 22

23 Bibliography Markowitz, H.M., Portfolio selection. The Journal of Finance 7 (1), 77 91, March. Markowitz, H.M., Portfolio Selection: Efficient Diversification of Investments. Wiley, Yale University Press, 1970, second ed., Basil Blackwell, Tobin, James [1958b], Liquidity Preference as Behaviour Towards Risk, Review of Economic Studies, 25, No. 67, pp Sharpe, William F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (3),