Markowitz Model ¾ Introduction ¾ Assumptions of Markowitz Theory ¾ Markowitz Diversification ¾ Criteria of Dominance ¾ Markowitz Model ¾ Measurement of Risk ¾ Portfolio Risk ¾ Markowitz Vs Sharp Model ¾ Optimal Portfolio of Sharp ¾ Arbitrage Pricing Theory ¾ Asset Selection in APT ¾ Components of Expected Return ¾ Empirical Testing of APT Introduction

Harry M. Markowitz, introduced new concepts of risk management and their application in selection of portfolios. His model is theoretical framework for analysis of risk and return and their inter relationship. He generated number of portfolios within a given amount of money and given preferences of investors for risk and return. Markowitz emphasised that quality of a portfolio will be different from the quality of individual assets within it. Assumptions of Markowitz Theory

Markowitz’s is based on following assumptions; 1. Investors are rational and wants to maximise their return 2. They have free access to fair and correct information 3. The markets are efficient and absorb the information 4. Investors are risk averse and try to minimise the risk 5. They prefer higher returns to lower returns for a given level of risk. Markowitz Diversification

The diversification theory of portfolio by Markowitz attaches importance to standard deviation, to reduce it to zero, if possible, covariance to have as much as possible negative interactive effect among the securities within the portfolio. The theory also postulated that diversification should not only aim at reducing the risk of security by reducing its variability or standard deviation, but by reducing the covariance of two or more securities in portfolio. Parameters of Markowitz Diversification For building up the efficient set of portfolio, we need to look into following important parameters; 1. Expected return 2. Covariance of one asset return to other asset returns. 3. Variability of returns as measured by standard deviation from the mean. Whatever is the risk of the individual securities in isolation, the total risk of the portfolio of all securities may be lower, if the covariance of their return is negative or negligible. Criteria of Dominance The superiority of one portfolio over the other, if with the same return, risk is lower or higher, is known as dominance. This principle includes trade off between risk and return. For two security portfolio, minimise the portfolio risk by the equation

2 2 σp = Wa σa + Wb σb + 2(Wa Wb σa σb σab) E (Rp) = WaE (Ra) + Wb E (Rb) R refers to return, E (Rp) is the expected return, W refers to the proportion invested in each security σa and σb are the std. dev. of a and b securities and σab is the covariance of the security returns. Markowitz Model In Markowitz model risk is measured by standard deviation of the return over the Mean for a number of observations. There are two types of risks

Types of Risk

Systematic ( Risk) Unsystematic (Company Risk) e.g. Market Risk, Inflation Risk, e.g. Labour Problems, Liquidity Risk, Rate Risk etc. Financial Risk, Management Problems Measurement of Risk Let’s take following example to calculate total risk. Observations : 10%, -5%, 20%, 35%, -10% 10% will be the mean Standard Deviation from the Square of Deviation = 2 ∑d ÷5 Mean Deviation 10 - 10 = 00 0 1350 ÷ 5 -5 - 10 = -15 225 20 - 10 = 10 100 270 35 - 10 = 25 625 -10 - 10 = -20 400 ∑d2 = 1350 Portfolio Risk When two or more assets are combined in a portfolio, their covariance or interactive risk is to be considered. Mathematically covariance is defined as, Cov x Y = (1 ÷ N) ∑ (Rx – Rx) (Ry – Ry) Where, Rx = return on security x, Ry = return on security Y Rx and Ry are expected returns on them respectively and N is the number of observations. Markowitz Vs Sharp Model Markowitz Model Sharp Model of a portfolio is risk adjusted Optimal portfolio is set up by using the return. It is equal to portfolio return minus single index model of sharp. The risk penalty. desirability of any stock is directly related Where, to its excess return to Beta ratio, namely Risk penalty = (Risk Squared ÷ Risk Tolerance) Sharp Index = (RJ – RF) ÷ BJ It is portfolio risk, relative to the Where, RJ is expected return on the investor’s risk tolerance. The optimal stock, RF is the risk free return, and BJ is portfolio is one on the the Beta relating the J stock to the market that maximises utility. return. To generate efficient portfolios the Then rank all the stocks in their order Markowitz Model requires – (a) expected of the index value. return on each assets (b) Standard In this model, the return on any stock deviation of returns as a measure risk of depends on some constant (α) called alpha each asset, and (c) the covariance or plus coefficient (β) called Beta, times the correlation coefficient as a measure of value of a stock Index (I), plus random inter-relationship between the returns on component. The equation in Sharp is assets considered. RJ = αJ + βJI +eJ Sharp Model Sharp model, having simplified process of compilation of expected return, standard deviations, variance of each security to every other security in the portfolio through relating the return in a security to single market Index. As against this Markowitz model had serious practical limitations in compiling, due to rigours calculations. The sharp model is useful because of following reasons, i] This will theoretically reflect all well traded security ii] It will reduce the work involve in compiling. Optimal Portfolio of Sharp This portfolio of sharp also called Single Index Model. It is directly related to Beta. If Ri is expected return on stock I and Rf is risk free rate, then the excess return is Ri – Rf. This has to be adjusted to βi. so the equation for ranking Stocks in the order of their return adjusted for risk is as below,

(Ri – Rf) ÷ (β1) Arbitrage Pricing Theory APT CAPM a) Investors do not look at expected returns and a) Investors look at the expected return and standard deviations. Based on the law of one price, if accompanying risks measured by standard deviation. the price of an asset is different in different markets, b) Investors are risk averse and risk-return analysis is arbitrage brings them to the same price. necessary. b) Investors prefer higher wealth/returns to lower c) Investors maximise wealth for a given level of risk. wealth. c) APT is based on the return generated by factor models. The APT is an equilibrium model of asset pricing like CAPM. But this theory assumes that the returns are generated by a factor model.

Factor Models in APT

Single Factor Model Multiple Factor Model Asset Selection in APT

In APT model, many types of strategies can be selected. If there are many securities to be selected and the amount is fixed for investment, the investor can choose in a manner that he can aim at zero non-factor risk (ei = 0). Although, theoretically it would be possible to create “pure factor” portfolios that are sensitive to only one factor and have insignificant non factor risk. But in practice only impure factor portfolios can be created. Components of Expected Return

There are two parts of expected return. i] risk free rate of return ii] pure factor portfolio. Although theory claims that the non-factor risk can be reduced to zero, it is not possible in real life. Therefore, in practical investment, it is better to combine CAPM and APT model. The trade off between risk and return is not considered by APT model. So synthesis between CAPM and APT is more realistic. Empirical Testing of APT The practical experience tell us that other things being equal, securities with large ex-ante betas will have relatively large expected returns. Investment is made on expectation and hence investors use betas, despite the fact that exact value of the beta is may not really give an indication of actual returns in future.

In the graph, OM is risk free return. Actual CAPM line is shown in the Graph to vary from the Zero Beta Line to a Substantial extent. Problem on Arbitrage Pricing Theory

Question: From the below given figures, calculate the expected rate of return of the stock.

λ1 = 1.8 Risk free Rate λ0 = 7%

λ2 = 1.25 b1 = 1.1

λ3 = 0.50 b2 = 1.5

b3 = -0.75 Answer: The equation for APT Model

Eri = λ0 + λ1 b1 + λ2 b2 + λ3 b3 ……. Eri = 7+1.8(1.2) + 1.25(1.5) + .50(-0.75) = 7 + 2.16 + 1.875-0.375 = 10.66

The Expected Rate of Return is 10.66%.