Portfolio Theory and the Capital Asset Pricing Model

1. Portfolio Theory and the Capital Asset Pricing Model 723g28 LinköpingsUniversitet, IEI

2. We have learned from last chapter risk and return: (that for an individual investor) Combining stocks into portfolios can reduce standard deviation, below the level obtained from a simple weighted average calculation. Rational investors maximize the expected return given risks. Or minimize risks given expected return.

3. Markowitz Portfolio Theory • Efficient portfolio provides the highest return for a given level of risk, or least risk for a given level of return. The market portfolio is the one that has the highest Sharpe ratio with the return and risk. • The Sharpe ratiois a measure of risk premium per unit of risk in an investment asset or a trading strategy

4. Effect of diversification on variance Assuming the following: • N independent assets, i.i.d. with covariance=0, • σ=std of the return • r= expected return • Equally weighted portfolio, Then, we have: the more the assets are in, the lower the standard deviation σ. σ portfolio =

5. Markowitz Portfolio Theory Price changes vs. Normal distribution IBM - Daily % change 1988-2008 Proportion of Days Daily % Change

6. Markowitz Portfolio Theory Standard Deviation VS. Expected Return Investment A % probability % return

7. Markowitz Portfolio Theory Standard Deviation VS. Expected Return Investment B % probability % return

8. Markowitz Portfolio Theory Standard Deviation VS. Expected Return Investment C % probability % return

9. Markowitz Portfolio Theory Expected Returns and Standard Deviations vary given different weighted combinations of the stocks Boeing 40% in Boeing Expected Return (%) Campbell Soup Standard Deviation

10. A two asset portfolio constructed with % of both assets, allow short selling of one assets

11. Efficient Frontier Try graph the efficient frontier and find the market portfolio with the highest Sharpe Ratio!

12. Efficient Frontier 4 Efficient Portfolios all from the same 10 stocks

13. Efficient Frontier Lending or Borrowing at the risk free rate (rf) allows us to exist outside the efficient frontier. Expected Return (%) S Lending Borrowing rf Minimum variance portfolio T Standard Deviation The red line is the Capital Market Line, where you can hold a combination of the risk free assets and the market portfolio and get any returns you like.

14. Efficient Frontier Another Example Correlation Coefficient = .4 Stocks s % of Portfolio Avg Return ABC Corp 28 60% 15% Big Corp 42 40% 21% Standard Deviation = weighted avg = 33.6 Standard Deviation = Portfolio = 28.1 Return = weighted avg = Portfolio = 17.4% Let’s Add stock New Corp to the portfolio

15. Efficient Frontier Return B A Risk (measured as s)

16. Efficient Frontier Return B AB A Risk

17. Efficient Frontier Return B N AB A Risk

18. Efficient Frontier Return B N ABN AB A Risk

19. Efficient Frontier Goal is to move up and left. WHY? Return B N ABN AB A Risk

20. Efficient Frontier The ratio of the risk premium to the standard deviation is the Sharpe ratio. In a competitive market, the expected risk premium varies in proportion to portfolio standard deviation. P denotes portfolio. Along the CapitalMarket Line one holds the risky assets and a risk free loan.

21. Capital Asset Pricing Model CAPM

22. Market Return = rm Security Market Line Stock Return ri . Market Portfolio rf Risk Free Return = (Treasury bills) 1.0 BETA risk 2,0

23. Efficient Frontier Return Low Risk High Return High Risk High Return Low Risk Low Return High Risk Low Return Risk

24. Market Return = rm CapitalMarket Line Return Tangent portfolio . Market Portfolio rf Risk Free Return = (Treasury bills) Risk

25. Market Return = rm Security Market Line Return . Market Portfolio rf Risk Free Return = (Treasury bills) 1.0 BETA

26. Market Risk Premium: Example Example: Market Portfolio (market return = 12%)

27. Security Market Line: depicts the CAPM Return Security Market Line SML rf BETA 1.0 SML Equation = rf + β( rm - rf )

28. Expected Returns These estimates of the returns expected by investors in February 2009 were based on the capital asset pricing model. We assumed 0.2% for the interest rate r f and 7 % for the expected risk premium r m − r f .

29. SML Equilibrium • In equilibrium no stock can lie below the security market line. For example, instead of buying stock A, investors would prefer to lend part of their money and put the balance in the market portfolio. And instead of buying stock B, they would prefer to borrow and invest in the market portfolio. (lend=save, borrow is leveraging.) risk free assets and the market portfolio can span the whole Security market line) Higher risk lowerreturn

30. Testing the CAPM Beta vs. Average Risk Premium: low beta portfolio fared better than high beta portfolio 1931-2008 Average Risk Premium 1931-2008 20 12 0 SML Investors Market Portfolio Portfolio Beta 1.0

31. Testing the CAPM Beta vs. Average Risk Premium Average Risk Premium 1966-2008 12 8 4 0 Investors SML Market Portfolio Portfolio Beta 1.0

32. Testing the CAPM: Return vs. Book-to-Market Cumulated difference of Small minus big firm stocks Cumulated difference of High minus low book-to-market firm stocks Dollars (log scale) High-minus low book-to-market 2008 Small minus big http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html