1. 1 Estimating Return and Risk Chapter 7 Jones, Investments: Analysis and Management

2. 2 Investment Decisions Involve uncertainty Focus on expected returns – Estimates of future returns needed to consider and manage risk Goal is to reduce risk without affecting returns – Accomplished by building a portfolio – Diversification is key

3. 3 Dealing With Uncertainty Risk that an expected return will not be realized Investors must think about return distributions, not just a single return Use probability distributions – A probability should be assigned to each possible outcome to create a distribution – Can be discrete or continuous

4. 4 Calculating Expected Return Expected value – The single most likely outcome from a particular probability distribution – The weighted average of all possible return outcomes – Referred to as an ex ante or expected return

5. 5 Calculating Risk Variance and standard deviation used to quantify and measure risk – Measures the spread in the probability distribution – Variance of returns:  2 =  (R i - E(R)) 2 pr i – Standard deviation of returns:  =(  2 ) 1/2 – Ex ante rather than ex post  relevant

6. 6 Portfolio Expected Return Weighted average of the individual security expected returns – Each portfolio asset has a weight, w, which represents the percent of the total portfolio value – The expected return on any portfolio can be calculated as:

7. 7 Portfolio Risk Portfolio risk not simply the sum of individual security risks Emphasis on the risk of the entire portfolio and not on risk of individual securities in the portfolio Individual stocks are risky only if they add risk to the total portfolio

8. 8 Portfolio Risk Measured by the variance or standard deviation of the portfolio’s return – Portfolio risk is not a weighted average of the risk of the individual securities in the portfolio

9. 9 Risk Reduction in Portfolios Assume all risk sources for a portfolio of securities are independent The larger the number of securities the smaller the exposure to any particular risk – “Insurance principle” Only issue is how many securities to hold

10. 10 Risk Reduction in Portfolios Random diversification – Diversifying without looking at relevant investment characteristics – Marginal risk reduction gets smaller and smaller as more securities are added A large number of securities is not required for significant risk reduction International diversification is beneficial

11. Portfolio Risk and Diversification  p % 35 20 0 Number of securities in portfolio 10203040......100+ Portfolio risk Market Risk

12. 12 Markowitz Diversification Non-random diversification – Active measurement and management of portfolio risk – Investigate relationships between portfolio securities before making a decision to invest – Takes advantage of expected return and risk for individual securities and how security returns move together

13. 13 Measuring Comovements in Security Returns Needed to calculate risk of a portfolio: – Weighted individual security risks » Calculated by a weighted variance using the proportion of funds in each security » For security i: (w i   i ) 2 – Weighted comovements between returns » Return covariances are weighted using the proportion of funds in each security » For securities i, j: 2w i w j   ij

14. 14 Correlation Coefficient Statistical measure of relative co- movements between security returns  mn = correlation coefficient between securities m and n –  mn =+1.0 = perfect positive correlation –  mn =-1.0 = perfect negative (inverse) correlation –  mn =0.0 = zero correlation

15. 15 When does diversification pay? –Combining securities with perfect positive correlation provides no reduction in risk » Risk is simply a weighted average of the individual risks of securities – Combining securities with zero correlation reduces the risk of the portfolio – Combining securities with negative correlation can eliminate risk altogether Correlation Coefficient

16. 16 Covariance Absolute measure of association – Not limited to values between -1 and +1 – Sign interpreted the same as correlation – The formulas for calculating covariance and the relationship between the covariance and the correlation coefficient are:

17. 17 Calculating Portfolio Risk Encompasses three factors – Variance (risk) of each security – Covariance between each pair of securities – Portfolio weights for each security Goal: select weights to determine the minimum variance combination for a given level of expected return

18. 18 Generalizations – The smaller the positive correlation between securities, the better – As the number of securities increases: » The importance of covariance relationships increases » The importance of each individual security’s risk decreases Calculating Portfolio Risk

19. 19 Simplifying Markowitz Calculations Markowitz full-covariance model – Requires a covariance between the returns of all securities in order to calculate portfolio variance – Full-covariance model becomes burdensome as the number of securites in a portfolio grows » n(n-1)/2 unique covariances for n securities Therefore, Markowitz suggests using an index to simplify calculations

20. Efficient Portfolios An efficient portfolio has the smallest portfolio risk for a given level of expected return Alternatively, an efficient portfolio maximizes the expected return for a given level of portfolio risk Porfolios located on efficient frontier dominate all other portfolios 20

21. Diversifiable vs. Nondiversifiable Risk Diversifiable, or nonsystematic, risks are those risks which are unique to individual companies –Adding nonperfectly correlated stocks to a portfolio reduces its nonsystematic risk Nondiversifiable risks are called systematic risks –systematic risks include interest rate risk, market risk and inflation risk 21

22. Capital Asset Pricing Model (CAPM) Beta –Beta is a measure of the systematic risk of a security that cannot be avoided through diversification` –The overall market has a beta of 1 »Riskier stocks, those which are more volatile than the market have Betas greater than 1 »Less risky stocks have Betas less than 1

23. CAPM Required rate of return –k i = Risk free rate + Risk premium »=RF + ß i [E(R m ) - RF] Security Market Line (SML) is the linear relationship between an asset’s risk and its required rate of return