1. Copyright © 2000 by Harcourt, Inc. All rights reserved. Introduction In the next three chapters, we will examine different aspects of capital market theory, including: Bringing risk and return into the picture of investment management – Markowitz optimization Modeling risk and return – CAPM, APT, and variations Estimating risk and return – the Single-Index Model (SIM) and risk and expected return factor models These provide the framework for both modern finance, which we have briefly discussed already, as well as for quantitative investment management, which will be the subject of the next section of the course

2. Lecture Presentation Software to accompany Investment Analysis and Portfolio Management Seventh Edition by Frank K. Reilly & Keith C. Brown Chapter 7

3. Copyright © 2000 by Harcourt, Inc. All rights reserved. Background Assumptions As an investor you want to maximize the returns for a given level of risk. Your portfolio includes all of your assets and liabilities The relationship between the returns for assets in the portfolio is important. A good portfolio is not simply a collection of individually good investments.

4. Copyright © 2000 by Harcourt, Inc. All rights reserved. Risk Aversion Given a choice between two assets with equal rates of return, risk averse investors will select the asset with the lower level of risk. –Investors prefer a sure thing to a “fair game” –$1 preferred to 50:50 chance of $2 or nothing Risk aversion is a consequence of decreasing marginal utility (e.g., 2 nd BMW is less valuable to you than the 1 st one; next two cheeseburgers give you less value than the prior two did). All investors are assumed to be risk-averse.

5. Copyright © 2000 by Harcourt, Inc. All rights reserved. Evidence That Investors are Risk-Averse Many investors purchase insurance for: Life, Automobile, Health, and Disability Income. The purchaser trades known costs for unknown risk of loss Yield on bonds increases with risk classifications from AAA to AA to A….

6. Copyright © 2000 by Harcourt, Inc. All rights reserved. Are all investors risk-averse? Risk preference may have to do with amount of money involved - risking small amounts with expected, though small, losses (e.g., lottery tickets), but insuring against large losses But, outcomes aren’t comparable – winning lottery tickets can put purchaser into a whole new class of consumption (e.g., not just a 2 nd car, but 1 st Lamborghini vs. 1 st Ford Focus) Also, risk aversion doesn’t mean you don’t take on risks, but that you are willing to take on risks only with the expectation of a potential higher return So, again, all investors are assumed to be risk-averse

7. Copyright © 2000 by Harcourt, Inc. All rights reserved. Definition of Risk Everyone has an intuitive idea about risk –“Don’t put all your eggs in one basket” But, how can you define risk precisely in order to make it operational? Two possibilities include: 1.Uncertainty of future outcomes 2.Probability of an adverse outcome We will consider several measures of risk that are used in developing portfolio theory

8. Copyright © 2000 by Harcourt, Inc. All rights reserved. Alternative Measures of Risk Range of returns Returns below expectations –Semivariance - measure expected returns below some target –Intended to minimize the damage Variance or standard deviation of returns –Generally works as well as the other measures, but mathematically more convenient to use –Gives us information about the uncertainty of returns, and can be used to determine the probability of adverse outcomes

9. Copyright © 2000 by Harcourt, Inc. All rights reserved. Markowitz Portfolio Theory Quantifies risk Derives the expected rate of return for a portfolio of assets and an expected risk measure Shows the variance of the rate of return is a meaningful measure of portfolio risk Derives the formula for computing the variance of a portfolio, showing how to effectively diversify a portfolio Provides both: –the foundation for Modern Finance –a key tool for Haugen’s New Finance

10. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Markowitz Portfolio Theory 1.Investors view potential investments in terms of the distributions of their returns over some given holding period

11. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Markowitz Portfolio Theory 2. Investors maximize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth. –I.e., investors like higher returns, but they are risk-averse in seeking those returns

12. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Markowitz Portfolio Theory 3. Investors measure the risk of their portfolios in terms of the variance of the portfolio’s returns. –I.e., variance is the key measure of risk

13. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Markowitz Portfolio Theory 4. Investors base decisions solely on expected return and risk, due to either: –Investors’ utility curves are functions of only expected return and the variance (or standard deviation) of returns. –Stocks’ returns are normally distributed or follow some other distribution that is fully described by mean and variance.

14. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Markowitz Portfolio Theory 5. For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected returns, investors prefer less risk to more risk.

15. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Markowitz Portfolio Theory Using these five assumptions, a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.

16. Copyright © 2000 by Harcourt, Inc. All rights reserved. Expected Rates of Return Individual risky asset –Sum of probability times possible rate of return Portfolio –Weighted average of expected rates of return for the individual investments in the portfolio

17. Copyright © 2000 by Harcourt, Inc. All rights reserved. Computation of Expected Return for an Individual Risky Investment Table 8.1

18. Copyright © 2000 by Harcourt, Inc. All rights reserved. Computation of the Expected Return for a Portfolio of Risky Assets Table 8.2

19. Copyright © 2000 by Harcourt, Inc. All rights reserved. Variance (Standard Deviation) of Returns for an Individual Investment Standard deviation is the square root of the variance Variance is a measure of the variation of possible rates of return R i, from the expected rate of return [E(R i )]

20. Copyright © 2000 by Harcourt, Inc. All rights reserved. Variance (Standard Deviation) of Returns for an Individual Investment where P i is the probability of the possible rate of return, R i

21. Copyright © 2000 by Harcourt, Inc. All rights reserved. Variance (Standard Deviation) of Returns for an Individual Investment Standard Deviation

22. Copyright © 2000 by Harcourt, Inc. All rights reserved. Variance (Standard Deviation) of Returns for an Individual Investment Table 8.3 Variance ( 2 ) =.0050 Standard Deviation ( ) =.02236

23. Copyright © 2000 by Harcourt, Inc. All rights reserved. But what if you don’t know the true probability distribution (including the true probabilities for each of the possible outcomes) for the stocks you are studying? One alternative – use the ex post, time series distribution –Assumes past is representative of the future, with each past period’s outcome an equally likely event (probability = 1 / T) –Naïve, limited information approach –Nonetheless, can be complicated – many choices and tradeoffs must be made to implement this approach (will be discussed more later)

24. Copyright © 2000 by Harcourt, Inc. All rights reserved. Ex-Post Estimation of Mean, Variance, and Covariance Computation of Monthly Rates of Return Table 8.4

25. Copyright © 2000 by Harcourt, Inc. All rights reserved. Time Series Returns for Coca-Cola: 1998 Figure 8.1

26. Copyright © 2000 by Harcourt, Inc. All rights reserved. Times Series Returns for Exxon: 1998 Figure 8.2

27. Copyright © 2000 by Harcourt, Inc. All rights reserved. Times Series Returns for Coca-Cola and Exxon: 1998

28. Copyright © 2000 by Harcourt, Inc. All rights reserved. Juxtaposing these two plots on one graph helps to illustrate why portfolio management isn’t simply about selecting a collection of individually good stocks Co-movements between the stocks in the portfolio can have important implications for the portfolio as a whole One measure of such co-movements is covariance, or its standardized analogue, correlation …

29. Copyright © 2000 by Harcourt, Inc. All rights reserved. Variance (Standard Deviation) of Returns for a Portfolio For two assets, i and j, the covariance of rates of return is defined as: Cov ij = E{[R i - E(R i )][R j - E(R j )]} =  s [(R is - E(R i ))(R js - E(R j ))] P s, where s refers to the possible state of the economy

30. Copyright © 2000 by Harcourt, Inc. All rights reserved. Estimation of Covariance of Returns for Coca-Cola and Exxon: 1998 Table 6.5

31. Copyright © 2000 by Harcourt, Inc. All rights reserved. Scatter Plot of Monthly Returns for Coca-Cola and Exxon: 1998 Figure 8.3

32. Copyright © 2000 by Harcourt, Inc. All rights reserved. Covariance and Correlation Covariance is an absolute measure of co- movement, correlation is a relative or standardized measure of co-movement Correlation coefficient varies from -1 to +1

33. Copyright © 2000 by Harcourt, Inc. All rights reserved. Estimation of Standard Deviation of Returns for Coca-Cola and Exxon: 1998 Table 8.6

34. Copyright © 2000 by Harcourt, Inc. All rights reserved. Parameters vs. Estimates No one knows the true values for the expected return and variance and covariance of returns These must be estimated from the available data The most basic way to estimate these is the naïve or unconditional estimate using the sample mean, sample variance, and sample covariance from a time series sample of stock returns Two typical time series used: –Last 60 months’ worth of monthly returns –Last 3 years’ worth of weekly returns

35. Copyright © 2000 by Harcourt, Inc. All rights reserved. Sample Estimates Sample Mean: Sample Variance: Sample Covariance:  =

36. Copyright © 2000 by Harcourt, Inc. All rights reserved. Now, the key equation for Markowitz portfolio optimization Takes us up from the level of the individual stock to the level of the portfolio as a whole

37. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Standard Deviation Formula

38. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Standard Deviation Calculation Any asset of a portfolio may be described by two characteristics: –The expected rate of return –The expected standard deviations of returns The correlation, measured by covariance, affects the portfolio standard deviation Low correlation reduces portfolio risk while not affecting the expected return For a well-diversified portfolio, the main source of portfolio risk is covariance risk; the lower the covariance risk, the lower the portfolio risk!

39. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining Stocks with Different Returns and Risk Case Correlation Coefficient Covariance a +1.00.0070 b +0.50.0035 c 0.00.0000 d -0.50 -.0035 e -1.00 -.0070 1.10.50.0049.07 2.20.50.0100.10

40. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining Stocks with Different Returns and Risk Assets may differ in expected rates of return and individual standard deviations Negative correlation reduces portfolio risk If correlation = -1.0 between two stocks, portfolio can be constructed with zero risk (s.d. = 0) and certain return

41. Copyright © 2000 by Harcourt, Inc. All rights reserved. Constant Correlation with Changing Weights 1.10 r ij = 0.00 2.20

42. Copyright © 2000 by Harcourt, Inc. All rights reserved. Constant Correlation with Changing Weights

43. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = +1.00 1 2 With two perfectly correlated assets, it is only possible to create a two asset portfolio with risk- return along a line between either single asset

44. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = +1.00 f g h i j k 1 2 With uncorrelated assets it is possible to create a two asset portfolio with lower risk than either single asset

45. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = +1.00 R ij = +0.50 f g h i j k 1 2 With correlated assets it is possible to create a two asset portfolio between the first two curves

46. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = +1.00 R ij = -0.50 R ij = +0.50 f g h i j k 1 2 With negatively correlated assets it is possible to create a two asset portfolio with much lower risk than either single asset

47. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = +1.00 R ij = -1.00 R ij = +0.50 f g h i j k 1 2 With perfectly negatively correlated assets it is possible to create a two asset portfolio with almost no risk R ij = -0.50 Figure 8.7

48. Copyright © 2000 by Harcourt, Inc. All rights reserved. Markowitz Optimization and Efficient Diversification Use info. on E(R i )’s and  ij ’s to obtain maximum benefits from diversification Key question to answer: –What proportion of portfolio to invest in each asset/stock? Look at issue similarly to Ben Graham: –“First control your risks, then worry about the returns” –Markowitz follows this process quantitatively …

49. Copyright © 2000 by Harcourt, Inc. All rights reserved. Markowitz Model Modern Portfolio Theory (MPT) Minimize portfolio risk: subject to (a) minimum required return and (b) full investment: yields the “Efficient Frontier”

50. Copyright © 2000 by Harcourt, Inc. All rights reserved. The 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 –Only one exception – the highest expected return point on the efficient frontier will be the single stock that has the highest expected return among all of the stocks

51. Copyright © 2000 by Harcourt, Inc. All rights reserved. Efficient Frontier for Alternative Portfolios Efficient Frontier A B C Figure 8.9 E(R) Standard Deviation of Return

52. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Efficient Frontier and Investor Utility Once we have our efficient frontier, we’re faced with the next question: –Out of all the portfolios that lie on the efficient frontier, which one should we choose? –Will come back to this question in the next chapter, but for now the question will be resolved by looking at investor utility curves. –The single best portfolio will vary from investor to investor, depending upon how risk averse or risk tolerant the individual investor is!

53. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Efficient Frontier and Investor Utility An individual investor’s utility curve specifies the trade-offs he is willing to make between expected return and risk The slope of the efficient frontier curve decreases steadily as you move upward These two interactions will determine the particular portfolio selected by an individual investor

54. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Efficient Frontier and Investor Utility The optimal portfolio has the highest utility for a given investor It lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility

55. Copyright © 2000 by Harcourt, Inc. All rights reserved. Selecting an Optimal Risky Portfolio X Y U3U3 U2U2 U1U1 U 3’ U 2’ U 1’ Figure 8.10

56. Copyright © 2000 by Harcourt, Inc. All rights reserved. Use of Markowitz Optimization The key areas of application of Markowitz portfolio optimization include: –Enhanced Index Funds –Determination of Asset Allocations But, in general, applications of Markowitz portfolio optimization have been relatively limited. Why? Answer leads to the subject of implementation …

57. Copyright © 2000 by Harcourt, Inc. All rights reserved. Markowitz Portfolio Optimization: Implementation  If Markowitz Diversification leads to the best, dominant portfolios, why doesn’t everyone use it?  Requires many inputs:  expected return and standard deviation of returns for each stock in the portfolio [2n]  correlations or covariances among all of the stocks in the portfolio [n(n-1)/2]  expands exponentially as the number of stocks in the portfolio increases

58. Copyright © 2000 by Harcourt, Inc. All rights reserved. Markowitz Portfolio Optimization: Implementation  This presents a couple of problems.  Very data-intensive  [2n+n(n-1)/2] inputs required  100 stock portfolio requires 5,150 items  500 stock portfolio requires 125,750 items! known  Moreover, none of these inputs is known!  Not only do we not know what the actual return for a stock will be, we don’t even know for sure what the distribution of its possible returns is, including its expected return, standard deviation of returns, and the correlation or covariance of its returns with other stocks’ returns estimated  All of these items must be estimated!

59. Copyright © 2000 by Harcourt, Inc. All rights reserved. Markowitz Portfolio Optimization: Implementation  Inputs to Markowitz diversification must be estimated, but there are a number of problems standing in the way of good estimates! 1. Too little data  Typically use last 60 months worth of returns for each stock for estimation  For 500 stocks, this would be 30,000 observations  But, for a 500 stock portfolio, would have 125,750 items to estimate! estimation error  More items to estimate than observations with which to do the estimation leads to estimation error!

60. Copyright © 2000 by Harcourt, Inc. All rights reserved. Markowitz Portfolio Optimization: Implementation 2. Distributional parameters are continually (though typically gradually) changing as economic conditions evolve and the corporations’ markets and finances evolve. moving target  Thus, estimation procedures are trying to hit a moving target! 3. In addition to and at least partially because of the above, there are different estimation procedures that could be used to estimate the inputs, each of which would give somewhat different results, hence would lead to different estimates of the optimal composition of efficient portfolios.  Thus, two different analysts, working with the same data for the same stocks, could end up with two different estimates for the composition and location of the efficient frontier!

61. Copyright © 2000 by Harcourt, Inc. All rights reserved. Markowitz Portfolio Optimization: Implementation  A potential solution to the first problem, too many individual correlations to estimate, is to ask an economic question:  What is/are the main factor(s) driving stocks to move in the same direction?  Main factor = what the market as a whole is doing!  Thus, a market factor is the main factor driving correlations between stocks.

62. Copyright © 2000 by Harcourt, Inc. All rights reserved. Markowitz Portfolio Optimization: Implementation  This leads to the use of the “Single-Index Model” in portfolio optimization  Esimated covariance =  Beta of first stock times  Beta of second stock times  Variance of market index’s returns  I.e., co-variance between two stocks’ returns  variance of market factor’s returns times the product of the stocks’ sensitivities to the market’s returns  So, only need one beta estimate for each stock in the portfolio plus an estimate of the variance of the market’s returns to be able to obtain values for all the covariances within the portfolio.

63. Copyright © 2000 by Harcourt, Inc. All rights reserved. With the assumption that stock returns can be described by a single index model, such as the Market Model, the number of correlations required reduces to one plus the number of assets, or (n+1) Single index market model: b i = the slope coefficient that relates the returns for security i to the returns for the aggregate stock market R m = the returns for the aggregate stock market Markowitz Portfolio Optimization: Implementation

64. Copyright © 2000 by Harcourt, Inc. All rights reserved. Under the assumptions of the Market Model: Covariance of returns of different stocks is a function of their betas Total variance of the returns for a single risky security is equal to market + idiosyncratic variance Markowitz Portfolio Optimization: Implementation

65. Copyright © 2000 by Harcourt, Inc. All rights reserved.  The SIM is the most basic and simplistic risk-factor model (see Haugen, Chapters 1 & 2).  But, the market as a whole is not the only source of co-movements between individual stocks’ returns:  Changes in oil prices, for example, would tend to push oil company and airline stocks in opposite directions.  The Arbitrage Pricing Theory (APT – see Chapters 9 & 10) is a multiple-factor (or multiple-index) model that takes other macroeconomic factors into account (beyond just what the market or the economy as a whole are doing) in estimating the expected returns and covariances for stocks. Markowitz Portfolio Optimization: Implementation

66. Copyright © 2000 by Harcourt, Inc. All rights reserved.  The most widely used APT model is Fama and French’s 3-factor model, which incorporates the following three factors: 1. Company size 2. Price-to-book ratio 3. Beta  In general, quantitative portfolio managers use risk-factor models that include a variety of risk factors to model the correlations between the stocks in the portfolio. Markowitz Portfolio Optimization: Implementation

67. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Internet Investments Online www.pionlie.com www.investmentnews.com www.micropal.com www.riskview.com www.altivest.com

68. Copyright © 2000 by Harcourt, Inc. All rights reserved. Future topics Modeling risk and return CAPM & APT – models, testing, and extensions Estimating risk and return References: –Reilly, Chs. 8 – 9 –Haugen, Chs. 2 – 5