Chapter 7 Portfolio Theory

Prepared By : Wael Shams EL-Din

Background In the early 1960s, the investment community talked about risk, but there was no specific measure for the term, however investors had to quantify their risk variable. The basic portfolio model was developed by Harry Markowitz, who derived the expected rate of return for a portfolio of assets and an expected risk measure. While William Sharp originated in his article (capital asset prices): New Theory of market equilibrium under conditions of risk which appeared in September 1964.

What is the Portfolio? A group of individual assets held in combination. An asset that would be relatively risky if held in isolation may have little or even no risk if held in a well diversified portfolio.

So stocks risk can be eliminated by diversification, so rational investors should hold a portfolio of stocks rather than just one stock also there are different models that link risk and required rate of return.

Efficient portfolio provides the highest expected rate of return for the lowest degree of risk.

The science of risk-efficient portfolios is associated with a couple of scientists (both are Nobel Prize holders) named Harry Markowitz and William Sharpe.What is the Efficient Portfolio?

Markowitz & Efficient Frontier The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of return.

Frontier will be portfolios of investments rather than individual securities.

5-Assumptions of Markowitz

1. Investors consider each investment alternative as being represented by a probability distribution of expected returns over Some Holding Period. 2. Investors Maximize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth.

3. Investors estimate the Risk of the portfolio on the basis of the variability of expected returns

4. Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the expected variance (or standard deviation) of returns only.

5. For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected return, investors prefer less risk to more risk

How is the Risk of a Portfolio Measured?

The risk of portfolio is measured by the standard deviation of its returns.Example

Correlation =0.35 What is the expected Rate of return for this Portfolio? What is the risk of this portfolio ?

Stock A Stock B Expected ( R )10%16%Risk () 20%40%Weight 30% 70%

AnswerExpected rate of Return = 14.20% ______________________________ P = w2A2A + wB22B + 2wAwBAB r:a,b

Stock Return Weight Weighted Avg.A10%30%3%B16%70%11.20%Total 14.20%

__________________________________P = (0.30)2X (0.20)2 + (0.70)2X (0.40)2+2(0.30) (0.70) (0.20) (0.40) X0.35

________________________P = (0.09) (0.04) + (0.49)(0.16)+ 0.01176 _______________________________P = 0.0036+0.0784+0.01176 __________P= 0.09376 = .306 = 30.60%

A Three-Asset PortfolioExample

Correlations r: A,B = 0.25r: A,C = 0.08r: B,C = 0.15What is the Expected Rate of Return for this Portfolio? What is the Risk of this Portfolio ?

Stock RWeight A12%20%60%B8%10%30%C4%3 %10%Total 100%

Answer Expected Rate of Return = Weight X Return R= 0.60 X12%+0.30 X 8% +0.10X4% R = 7.20% + 2.40% + 0.40% = 10%Risk of The Portfolio 2= [WA2 A2 + WB2 B2 + WC2 C2] + [2WA WB A B rA, B + 2WA WC AC rA, C + 2WB WC B C rB,C] _____p = 2

Risk of The Portfolio

2 = [(0.6)2(0.20)2 + (0.3)2(0.10)2 + (0.1)2(0.03)2]+ {[2(0.6) (0.3) (0.20) (0.10) (0.25)] + [2(0.6) (0.1) (0.20) (0.03) (0.08)]+ [2(0.3) (0.1) (0.10) (0.03) (0.15)]}

= [0.015309] + {[0.0018] + [0.0000576] + [0.000027]}= 0.0170784 ___________p = 0.0170784 = 0.1307 = 13.07%

Efficient FrontierA BC203040 R510

Portfolio A Dominated Portfolio C Return =5% while Risk = 20% @ point A Return =5% while Risk = 40% @ Point C Portfolio B Dominated Portfolio C Return =10% while Risk = 40% @ point B Return = 5% while Risk = 40% @ Point C

Return Risk CV Portfolio A 5%20% 4Portfolio B 10%40%4

Efficient FrontierIf we limit our self to low-risk securities, we will be limiting our self to investments that tend to have low rates of return. So what we really want to do is include some higher growth, higher risk securities in our portfolio, but they should be combined in a smart way, so that some of their fluctuations cancel each other out. In statistical terms, we are looking for a combined standard deviation that's low, relative to the standard deviations of the individual securities). The result should give us a high average rate of return, with less of the harmful fluctuations

The feasible set of portfolio represent all portfolios that can be constructed from a given set of stocks.An efficient portfolio is one that offer most return for a given amount of risk or the least risk for a given amount of return.The collection of efficient portfolios is called the efficient frontier. Optimal Portfolio: is defined by the Tangency Point between the efficient set and the investors indifference curve.

Optimal PortfolioAn individual investors utility curves specify the trade-off between expected return and risk. These utility curves determine which particular portfolio on the efficient frontier best suits an individual investor. Two investors chosen the same portfolio from the efficient set only if their utility curves are identical.

CAPM The Idea of CAPM is Building on that only one factor which Risk Free Rate ( RFR). RRR= RFR+( Rm RFR)Beta RRR Required Rate of ReturnRFR Risk Free Rate Rm Average Return of the Market Beta : The Relationship between the moves in the asset return and the moves in the market return.

If Beta = 1.0, stock is average risk.If Beta > 1.0, stock is riskier than average.If Beta < 1.0, stock is less risky than average.Most stocks have betas in the range of 0.5 to 1.5

Thank You

*************