Chapter 8 Risk, Return, and Portfolio Theory

1. Chapter 8Risk, Return, and Portfolio Theory

2. 8.4 The Efficient Frontier Chapter 8 Outline • Modern Portfolio Theory • Harry Markowitz • Efficient portfolio

3. 8.1 Measuring Returns on Investment Definitions: • Ex post return-past or historical returns • Ex ante returns-expected returns • Income yield-return earned by investors as a periodic cash flow • Capital gain-measures the appreciation (or depreciation) in the price of the asset from some starting price

4. Where CF1 is the expected cash flows to be received, P0 is the purchase price today, and P1 = selling price 1 year from today

5. Total return Total return-the sum of the income yield and the capital gain (or loss) yield Example: At the end of 2009, IBM had a stock price of $130.90 and at the end of 2010 IBM had a stock price $146.76. During 2010, IBM paid four dividends, totaling $2.50. The return on IBM stock for 2010 is:

6. Total Return Answer: =1.191% + 12.12% = 14.03%

7. Measuring Average Returns Arithmetic mean or average mean-sum of all observations divided by the total number of observations Geometric mean-average or compound growth rate over multiple time periods

8. Why do the arithmetic mean return and the geometric mean return differ? The arithmeticmean simply averages the annual rates of return without taking into account that the amount invested varies across time. The geometric mean is a better average return estimate when we are interested in the rate of return performance of an investment over time.

9. Estimating Expected Returns on Investment The expected return is often estimated based on historical averages, but the problem is that there is no guarantee that the past will repeat itself.

10. Estimating Expected Returns on Investment Example: Suppose you have two possible returns on investment: The expected return is the weighted average of the possible returns, where the weights are the probabilities: Expected return, E(r) = 0.07

11. Estimating expected returns What is the expected value return, based on these estimates? Suppose we have the following estimates for the return on an investment:

12. Estimating expected returns Expected return

13. 8.2 Measuring Risk • Riskis the probability of incurring harm, and for financial managers, harm generally means losing money or earning an inadequate rate of return. • Riskis the probability that the actual return from an investment is less than the expected return. This means that the more variable the possible return, the greater the risk.

14. Methods of Measuring Risk Standard Deviation: Ex Post Uses information that has occurred (ex post) Ex post standard deviation (σ) = Where σ is the standard deviation, is the average return, xiis the observation in year i, and N is the number of observations

15. Methods of Measuring Risk The Standard Deviation: Ex Ante Formulated based on expectations about the future cash flows or returns of an asset. Ex ante standard deviation (σ) = Where r is one of the possible outcomes, E(r) is the calculated expected value of the possible outcomes, and piis the probability of the occurrence

16. Ex-ante risk

17. Methods of Measuring Risk

18. 8.3 Expected Return and Risk for Portfolios A portfolio is a collection of assets, such as stocks and bonds, that are combined and considered a single asset. Investors should diversify their investments so that they are not unnecessarily exposed to a single negative event “Don’t put all your eggs in one basket”

19. Calculating a Portfolio’s Return The expected return on a portfolio is the weighted average of the expected returns on the individual assets in the portfolio: where E(rp)represents the expected return on the portfolio, E(ri) represents the expected return on asset i, and wi represents the portfolio weight of asset i.

20. Calculating a Portfolio’s Return Expected return on A Expected return on B

21. Calculating a Portfolio’s Return Estimating standard deviation: where σpisthe portfolio standard deviation and COVxyis the covariance of the returns on X and Y Covariance-a statistical measure of the degree to which two or more series move together

22. Estimating a standard deviations of returns Standard deviation of A = 3.1225% Standard deviation of B = 9.3226%

23. Estimating a Covariance Covariance 15%-10.5% 20%-10.7%

24. Correlation Coefficient Correlation coefficient- a statistical measure that identifies how asset returns move in relation to one another; denoted by p • Ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation) • Related to covariance and individual standard deviations: where ρxy is the correlation coefficient, COV is covariance, subxy are the variables, and σ is the standard deviation

25. Correlation Examples No correlation:

26. Correlation Examples Perfect positive and perfect negative correlation:

27. Correlation Examples Positive and negative correlation:

28. Correlation From our example,

29. Word of Caution The essential problem is that our models are still too simple to capture the full array of governing variables that drive global economic reality. A model, of necessity, is an abstraction from the full detail of the real world.

30. 8.4 The Efficient Frontier There is a mix of assets that minimizes a portfolio’s standard deviation

31. Modern Portfolio Theory Modern portfolio theory is a set of theories that explain how rational investors, who are risk averse, can select a set of investments that maximize the expected return for a given level of risk Harry Markowitzis the “father” of modern portfolio theory, was awarded the 1990 Nobel Prize in Economics

32. Modern Portfolio Theory • Harry Markowitz showed investors how to diversify their portfolios based on several assumptions: • Investors are rational decision makers • Investors are risk averse • Investor preferences are based on a portfolio’s expected return and risk (as measured by variance or standard deviation) • Introduced the notion of an efficient portfolio

33. The Efficient Portfolio The efficient portfolio is a collection of investments that offers the highest expected return for a given level of risk, or, equivalently, offers the lowest risk for a given expected return

34. The Efficient Frontier

35. The Efficient Frontier Portfolios are either: • Attainable (lie on the minimum variance frontier), or • Dominated (lower level of expected return for a given level of risk than another portfolio)

36. The Efficient Frontier Rational, risk-averse investors are interested in holding only those portfolios, such as Portfolio II, that offer the highest expected return for their given level of risk. A more aggressive (i.e., less risk averse) investor might choose Portfolio II, whereas a more conservative (i.e., more risk averse) investor might prefer Portfolio V (i.e., the MVP).

37. 8.5 Diversification The reduction of risk by investing funds across several assets

38. Types of Risk Randomdiversification or naïve diversification is the act of randomly buying assets without regard to relevant investment characteristics (e.g., “dartboard”)

39. Types of Risk Unique risk, nonsystematic risk, or diversifiable risk-a company-specific part of total risk that is eliminated by diversification Market risk, systematic risk,beta risk, or nondiversifiable risk-a systematic part of total risk that cannot be eliminated by diversification

40. Types of Risk Total risk = Market risk + Unique risk

41. Summarizing Risks

42. Summary In financial decision making and analysis, we use both ex post and ex ante returns on investments: We use ex post returns when we look back at what has happened, and we use ex ante returns when we look forward, into the future. One measure of risk is the standard deviation, which is a measure of the dispersion of possible outcomes.

43. Summary When we invest in more than one investment, there may be some form of diversification, which is the reduction in risk from combining investments whose returns are not perfectly correlated. If we consider all possible investments and their respective expected return and risk, there are sets of investments that are better than others in terms of return and risk. These sets make up the efficient frontier.

44. Summary If we consider a company as a portfolio of investments, diversification plays a role in financial decision making. Financial managers need to consider not only what an investment looks like in terms of its return and risk as a stand-alone investment, but more important, how it fits into the company’s portfolio of investments.

45. Problem #1 Scenario Probability Outcome Good 30% $40 Normal 50% $20 Bad 20% $10 What is the expected value and standard deviation for this probability distribution?

46. Problem #2 The Key Company is evaluating two projects: • Project 1 has a 40% chance of generating a return of 20% and a 60% change of generating a return of -10%. • Project 2 has a 20% chance of generating a return of 30% and an 80% chance of return of -5%. Which project is riskier? Why?

47. Problem #3 Suppose the covariance between the returns on project A and B is -0.0045. And suppose the standard deviations of A and B are 0.1 and 0.3, respectively. What is the correlation between A and B’s returns?