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Evolution of Modern Portfolio TheoryBy MUSHTAQ AHMAD SHAHResearch scholar

Department of Management studies Guru Ghasidas vishwavidalaya

Evolution of Modern Portfolio Theory

Efficient FrontierSingle Index ModelCapital Asset Pricing Model (CAPM)Arbitrage Pricing Theory (APT)

Evolution of Modern Portfolio TheoryEfficient FrontierMarkowitz, H. M., Portfolio Selection, Journal of Finance (December 1952).Rather than choose each security individually, choose portfolios that maximize return for given levels of risk (i.e., those that lie on the efficient frontier). Problem: When managing large numbers of securities, the number of statistical inputs required to use the model is tremendous. The correlation or covariance between every pair of securities must be evaluated in order to estimate portfolio risk.

Evolution of Modern Portfolio Theory(Continued)Single Index ModelSharpe, W. F., A Simplified Model of Portfolio Analysis, Management Science (January 1963).Substantially reduced the number of required inputs when estimating portfolio risk. Instead of estimating the correlation between every pair of securities, simply correlate each security with an index of all of the securities included in the analysis.

Evolution of Modern Portfolio Theory (Continued)Capital Asset Pricing Model (CAPM)Sharpe, W. F., Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk, Journal of Finance (September 1964).

Instead of correlating each security with an index of all securities included in the analysis, correlate each security with the efficient market value weighted portfolio of all risky securities in the universe (i.e., the market portfolio). Also, allow investors the option of investing in a risk-free asset.

Evolution of Modern Portfolio Theory (Continued)Arbitrage Pricing Theory (APT)Ross, S. A., The Arbitrage Theory of Capital Asset Pricing, Journal of Economic Theory (December 1976).

Instead of correlating each security with only the market portfolio (one factor), correlate each security with an appropriate series of factors (e.g., inflation, industrial production, interest rates, etc.).

Markowitzs ContributionHarry Markowitzs Portfolio Selection Journal of Finance article (1952) set the stage for modern portfolio theoryThe first major publication indicating the important of security return correlation in the construction of stock portfolios

Markowitz showed that for a given level of expected return and for a given security universe, knowledge of the covariance and correlation matrices are required8

Harry Markowitz Model

Harry Max Markowitz (born August 24, 1927) is an American economist.He is best known for his pioneering work in Modern Portfolio Theory.Harry Markowitz put forward this model in 1952. Studied the effects of asset risk, return, correlation and diversification on probable investment portfolio returnsHarry Markowitz Essence of Markowitz ModelAn investor has a certain amount of capital he wants to invest over a single time horizon. He can choose between different investment instruments, like stocks, bonds, options, currency, or portfolio. The investment decision depends on the future risk and return. The decision also depends on if he or she wants to either maximize the yield or minimize the risk

Essence of Markowitz ModelMarkowitz model assists in the selection of the most efficient by analysing various possible portfolios of the given securities.By choosing securities that do not 'move' exactly together, the HM model shows investors how to reduce their risk. The HM model is also called Mean-Variance Model due to the fact that it is based on expected returns (mean) and the standard deviation (variance) of the various portfolios.

Diversification and Portfolio RiskPortfolio RiskNumber of Shares5101520

Total RiskSRUSR

p

SR: Systematic RiskUSR: Unsystematic Risk

AssumptionsAn investor has a certain amount of capital he wants to invest over a single time horizon.He can choose between different investment instruments, like stocks, bonds, options, currency, or portfolio.The investment decision depends on the future risk and return.The decision also depends on if he or she wants to either maximize the yield or minimize the risk.The investor is only willing to accept a higher risk if he or she gets a higher expected return.

Efficient Frontier Construct a risk/return plot of all possible portfolios

Those portfolios that are not dominated constitute the efficient frontier12

Efficient Frontier (contd)13

Standard DeviationExpected Return

100% investment in security with highest E(R)100% investment in minimum variance portfolioPoints below the efficient frontier are dominatedNo points plot above the lineAll portfolios on the line are efficient

Efficient Frontier (contd)When a risk-free investment is available, the shape of the efficient frontier changes

The expected return and variance of a risk-free rate/stock return combination are simply a weighted average of the two expected returns and variance

The risk-free rate has a variance of zero

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Efficient Frontier (contd)15

Standard DeviationExpected Return

RfABC

Efficient Frontier (contd)The efficient frontier with a risk-free rate: Extends from the risk-free rate to point B

The line is tangent to the risky securities efficient frontier Follows the curve from point B to point C16

The Sharpe Index Model

Need for Sharpe ModelIn Markowitz model a number of co-variances have to be estimated.If a financial institution buys 150 stocks, it has to estimate 11,175 i.e., (N2 N)/2 correlationco-efficients.Sharpe assumed that the return of a security is linearly related to a single index like the market index.

Single Index ModelCasual observation of the stock prices over a period of time reveals that most of the stock prices move with the market index.When the Sensex increases, stock prices also tend to increase and vice versa.This indicates that some underlying factors affect the market index as well as the stock prices.

Stock prices are related to the market index and this relationship could be used to estimate the return of stock.Ri = ai + bi Rm + eiwhere Ri expected return on security i ai intercept of the straight line or alpha co-efficient bi slope of straight line or beta co-efficient Rm the rate of return on market index ei error term

BetaWhat Is Beta and How Is It Calculated?

BetaA coefficient measuring a stocks relative volatility

Beta measures a stocks sensitivity to overall market movements

Source:UBS Warburg Dictionary of Finance and Investment Terms

In practice, Beta is measured by comparing changes in a stock price to changes in the value of the S&P 500 index over a given time period

The S&P 500 index has a beta of 1

A Generic ExampleStock XYZ has a beta of 2

The S&P 500 index increases in value by 10%

The price of XYZ is expected to increase 20% over the same time period

Beta can be NegativeStock XYZ has a beta of 2

The S&P 500 index INCREASES in value by 10%

The price of XYZ is expected to DECREASE 20% over the same time period

If the beta of XYZ is 1.5

And the S&P increases in value by 10%

The price of XYZ is expected to increase 15%

A beta of 0 indicates that changes in the market index cannot be used to predict changes in the price of the stock

The companys stock price has no correlation to movments in the market index

CompanyBetaAMGN0.82BRK.B0.73C1.37XOM0.10MSFT1.80MWD2.19NOK2.05PXLW1.93TXN1.70VIA.B1.39

Source: taken from yahoo.finance.com, except PXLW from bloomberg.com

If beta is a measure of risk, then investors who hold stocks with higher betas should expect a higher return for taking on that risk

What does this remind you of?

Beta and RiskBeta is a measure of volatility

Volatility is associated with risk

How to Calculate Beta

Beta = Covariance(stock price, market index)Variance(market index)

**When calculating, you must compare the percent change in the stock price to the percent change in the market index**

How to Calculate BetaEasily calculated using Excel and Yahoo! Finance Use COVAR and VARP worksheet functionsAn example: Calculate the beta of Citigroup stock over the 5-yr time period from Jan. 1, 1997 Dec. 31, 2001

RiskSystematic risk= bi2 variance of market index = bi2 sm2Unsystematic risk= Total variance Systematic riskei2= si2 Systematic riskThus the total risk= Systematic risk + Unsystematic risk= bi2 sm2 + ei2

Portfolio Variance

where 2p = variance of portfolio

2m = expected variance of market index e2i= Unsystematic risk

xi = the portion of stock i in the portfolio

ExampleThe following details are given for x and y companies stocks and the Sensex for a period of one year. Calculate the systematic and unsystematic risk for the companies stock. If equal amount of money is allocated for the stocks , then what would be the portfolio risk ? X stock Y stock SensexAverage return 0.15 0.25 0.06Variance of return 6.30 5.86 2.25eta 0.71 0.27

Company XSystematic risk= bi2 variance of market index = bi2 sm2 = ( 0.71)2 x 2.25 = 1.134Unsystematic risk= Total variance Systematic riskei2= si2 Systematic risk = 6.3 1.134 =5.166Total risk= Systematic risk + Unsystematic risk= bi2 sm2 + ei2 = 1.134 + 5.166 = 6.3

Company YSystematic risk= bi2 variance of market index = bi2 sm2 = ( 0.27)2 x 2.25 = 0.1640Unsystematic risk= Total variance Systematic riskei2= si2 Systematic risk = 5.86 1.134 =5.166

2p = [ ( .5 x .71 + .5 x .27)2 2.25 ] + [ ( .5)2 (5.166) + (.5 )2 ( 5.696) ]

= [ ( .355 + .135 )2 2.25 ] + [ ( 1.292 + 1.424 ) ]

= 0.540 + 2.716

= 3.256

Sharpes optimal portfolio

The selection of any stock is directly related to its excess return to beta ratio.

where Ri = the expected return on stock i Rf = the return on a risk less asset bi = Systematic risk

Capital Asset Pricing Model (CAPM)Introduction

Systematic and unsystematic risk

Fundamental risk/return relationship revisited40

Introduction

The Capital Asset Pricing Model (CAPM) is a theoretical description of the way in which the market prices investment assetsThe CAPM is a positive theory41

Systematic and Unsystematic RiskUnsystematic risk can be diversified and is irrelevant

Systematic risk cannot be diversified and is relevantMeasured by betaBeta determines the level of expected return on a security or portfolio (SML)42

CAPMThe more risk you carry, the greater the expected return:

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CAPM (contd)The CAPM deals with expectations about the future

Excess returns on a particular stock are directly related to:The beta of the stockThe expected excess return on the market

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CAPM (contd)CAPM assumptions:Variance of return and mean return are all investors care aboutInvestors are price takers, they cannot influence the market individuallyAll investors have equal and costless access to informationThere are no taxes or commission costs

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CAPM (contd)CAPM assumptions (contd):Investors look only one period ahead

Everyone is equally adept at analyzing securities and interpreting the news

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SML and CAPMIf you show the security market line with excess returns on the vertical axis, the equation of the SML is the CAPM

The intercept is zero

The slope of the line is beta47

Note on the CAPM AssumptionsSeveral assumptions are unrealistic:People pay taxes and commissionsMany people look ahead more than one periodNot all investors forecast the same distribution

Theory is useful to the extent that it helps us learn more about the way the world actsEmpirical testing shows that the CAPM works reasonably well48

Stationarity of BetaBeta is not stationaryEvidence that weekly betas are less than monthly betas, especially for high-beta stocksEvidence that the stationary of beta increases as the estimation period increases

The informed investment manager knows that betas change49

Equity Risk PremiumEquity risk premium refers to the difference in the average return between stocks and some measure of the risk-free rateThe equity risk premium in the CAPM is the excess expected return on the market

Some researchers are proposing that the size of the equity risk premium is shrinking50

Using A Scatter Diagram to measure BetaCorrelation of returns

Linear regression and beta

Importance of logarithms

Statistical significance51

Correlation of ReturnsMuch of the daily news is of a general economic nature and affects all securitiesStock prices often move as a group

Some stock routinely move more than the others regardless of whether the market advances or declines

Some stocks are more sensitive to changes in economic conditions 52

Linear Regression and BetaTo obtain beta with a linear regression:

Plot a stocks return against the market return

Use Excel to run a linear regression and obtain the coefficientsThe coefficient for the market return is the beta statisticThe intercept is the trend in the security price returns that is inexplicable by finance theory

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Statistical SignificancePublished betas are not always useful numbersIndividual securities have substantial unsystematic risk and will behave differently than beta predicts

Portfolio betas are more useful since some unsystematic risk is diversified away54

Arbitrage Pricing TheoryAPT backgroundThe APT modelComparison of the CAPM and the APT55

APT BackgroundArbitrage pricing theory (APT) states that a number of distinct factors determine the market returnRoll and Ross state that a securitys long-run return is a function of changes in:InflationIndustrial productionRisk premiumsThe slope of the term structure of interest rates56

APT Background (contd)Not all analysts are concerned with the same set of economic information

A single market measure such as beta does not capture all the information relevant to the price of a stock57

The APT Model General representation of the APT model:

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APT59

Replicating the RandomnessLets try to replicate the random component of security A by forming a portfolio with the following weights:60

Key Point in ReasoningSince we were able to match the random components exactly, the only terms that differ at this point are the fixed components.

But if one fixed component is larger than the other, arbitrage profits are possible by investing in the highest yielding security (either A or the portfolio of factors) while short-selling the other (being long in one and short in the other will assure an exact cancellation of the random terms).61

Comparison of the CAPM and the APTThe CAPMs market portfolio is difficult to construct:Theoretically all assets should be included (real estate, gold, etc.)Practically, a proxy like the S&P 500 index is used

APT requires specification of the relevant macroeconomic factors62

Comparison of the CAPM and the APT (contd) The CAPM and APT complement each other rather than competeBoth models predict that positive returns will result from factor sensitivities that move with the market and vice versa63