# Finance Theory

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1 Finance Theory MIT Sloan MBA Program Andrew W. Lo Harris & Harris Group Professor, MIT Sloan School Lecture : : Risk Analytics and

2 Critical Concepts Motivation Measuring Risk and Reward Mean-Variance Analysis The Efficient Frontier The Tangency Portfolio Readings: Brealey, Myers, and Allen Chapters 7 and 8.1 Slide 2

3 Motivation What Is A Portfolio and Why Is It Useful? A portfolio is simply a specific combination of securities, usually defined by portfolio weights that sum to 1: Portfolio weights can sum to 0 (dollar-neutral portfolios), and weights can be positive (long positions) or negative (short positions). Assumption: Portfolio weights summarize all relevant information. Slide 3

4 Motivation Example: Your investment account of \$100,000 consists of three stocks: 200 shares of stock A, 1,000 shares of stock B, and 750 shares of stock C. Your portfolio is summarized by the following weights: Asset Shares Price/Share A B C Dollar Investment Portfolio Weight 200 \$50 \$10,000 10% 1,000 \$60 \$60,000 60% 750 \$40 \$30,000 30% Total \$100, % Slide 4

5 Motivation Example (cont): Your broker informs you that you only need to keep \$50,000 in your investment account to support the same portfolio of 200 shares of stock A, 1,000 shares of stock B, and 750 shares of stock C; in other words, you can buy these stocks on margin. You withdraw \$50,000 to use for other purposes, leaving \$50,000 in the account. Your portfolio is summarized by the following weights: Asset Shares Price/Share A B C Dollar Investment Portfolio Weight 200 \$50 \$10,000 20% 1,000 \$60 \$60, % 750 \$40 \$30,000 60% Riskless Bond \$50,000 \$1 \$50, % Total \$50, % Slide 5

6 Motivation Example: You decide to purchase a home that costs \$500,000 by paying 20% of the purchase price and getting a mortgage for the remaining 80% What are your portfolio weights for this investment? Asset Shares Price/Share Home Mortgage Dollar Investment Portfolio Weight 1 \$500,000 \$500, % 1 \$400,000 \$400, % Total \$100, % What happens to your total assets if your home price declines by 15%? Slide 6

7 Motivation Example: You own 100 shares of stock A, and you have shorted 200 shares of stock B. Your portfolio is summarized by the following weights: Stock Shares Price/Share A B Dollar Investment Portfolio Weight 100 \$50 \$5,000??? 200 \$25 \$5,000??? Zero net-investment portfolios do not have portfolio weights in percentages (because the denominator is 0) we simply use dollar amounts instead of portfolio weights to represent long and short positions Slide 7

8 Motivation Why Not Pick The Best Stock Instead of Forming a Portfolio? We don t know which stock is best! Portfolios provide diversification, reducing unnecessary risks. Portfolios can enhance performance by focusing bets. Portfolios can customize and manage risk/reward trade-offs. How Do We Construct a Good Portfolio? What does good mean? What characteristics do we care about for a given portfolio? Risk and reward Investors like higher expected returns Investors dislike risk Slide 8

9 Measuring Risk and Reward Reward is typically measured by return Higher returns are better than lower returns. But what if returns are unknown? Assume returns are random, and consider the distribution of returns. Several possible measures: Mean: central tendency. 75%: upper quartile. 5%: losses. 5% Mean 75% Slide 9

10 Measuring Risk and Reward How about risk? Likelihood of loss (negative return). But loss can come from positive return (e.g., short position). A symmetric measure of dispersion is variance or standard deviation. Variance Measures Spread: Blue distribution is riskier. Extreme outcomes more likely. This measure is symmetric. Slide 10

11 Measuring Risk and Reward Assumption Investors like high expected returns but dislike high volatility Investors care only about the expected return and volatility of their overall portfolio Not individual stocks in the portfolio Investors are generally assumed to be well-diversified Key questions: How much does a stock contribute to the risk and return of a portfolio, and how can we choose portfolio weights to optimize the risk/reward characteristics of the overall portfolio? Slide 11

12 Mean-Variance Analysis Objective Assume investors focus only on the expected return and variance (or standard deviation) of their portfolios: higher expected return is good, higher variance is bad Develop a method for constructing optimal portfolios Higher Utility Expected Return E[R p ] MRK GM MOT MCD T-Bills Standard Deviation of Return SD[R p ] Slide 12

13 Mean-Variance Analysis Basic Properties of Mean and Variance For Individual Returns: Basic Properties of Mean And Variance For Portfolio Returns: Slide 13

14 Mean-Variance Analysis Variance Is More Complicated: Slide 14

15 Mean-Variance Analysis Portfolio variance is the weighted sum of all the variances and covariances: There are n variances, and n 2 n covariances Covariances dominate portfolio variance Positive covariances increase portfolio variance; negative covariances decrease portfolio variance (diversification) Slide 15

16 Mean-Variance Analysis Consider The Special Case of Two Assets: As correlation increases, overall portfolio variance increases Slide 16

17 Mean-Variance Analysis Example: From , Motorola had an average monthly return of 1.75% and a std dev of 9.73%. GM had an average return of 1.08% and a std dev of 6.23%. Their correlation is How would a portfolio of the two stocks perform? w Mot w GM E[R P ] var(r P ) stdev(r P ) Slide 17

18 Mean-Variance Analysis Mean/SD Trade-Off for Portfolios of GM and Motorola 2.1% 1.7% Motorola Expected Return 1.3% 0.9% GM 0.5% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% Standard Deviation of Return Slide 18

19 Mean-Variance Analysis Example (cont): Suppose the correlation between GM and Motorola changes. What if it equals 1.0? 0.0? 1.0? Std dev of portfolio w Mot w GM E[R P ] corr = -1 corr = 0 corr = % 6.23% 6.23% 6.23% Slide 19

20 Mean-Variance Analysis Mean/SD Trade-Off for Portfolios of GM and Motorola 2.2% 1.8% Expected Return 1.4% 1.0% corr=-1 corr=-.5 corr=0 corr=1 0.6% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Standard Deviation of Return Slide 20

21 Mean-Variance Analysis Example: In 1980, you were thinking about investing in GD. Over the subsequent 10 years, GD had an average monthly return of 0.00% and a std dev of 9.96%. Motorola had an average return of 1.28% and a std dev of 9.33%. Their correlation is How would a portfolio of the two stocks perform? w Mot w GD E[R P ] var(r P ) stdev(r P ) Slide 21

22 Mean-Variance Analysis Mean/SD Trade-Off for Portfolios of GD and Motorola 2.0% 1.5% Expected Return Motorola GD 1.0% 0.5% 0.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% -0.5% Standard Deviation of Return -1.0% Slide 22

23 Mean-Variance Analysis Example: You are trying to decide how to allocate your retirement savings between Treasury bills and the stock market. The T-Bill rate is 0.12% monthly. You expect the stock market to have a monthly return of 0.75% with a standard deviation of 4.25%. w Stk w Tbill E[R P ] var(r P ) stdev(r P ) Slide 23

24 Mean-Variance Analysis Mean/SD Trade-Off for Portfolios of T-Bills and The Stock Market 1.0% 0.8% Stock Market Expected Return 0.6% 0.4% 0.2% 50/50 portfolio T-Bills 0.0% 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% Standard Deviation of Return Slide 24

25 Mean-Variance Analysis Summary Observations E[R P ] is a weighted average of stocks expected returns SD(R P ) is smaller if stocks correlation is lower. It is less than a weighted average of the stocks standard deviations (unless perfect correlation) The graph of portfolio mean/sd is nonlinear If we combine T-Bills with any risky stock, portfolios plot along a straight line Slide 25

26 Mean-Variance Analysis The General Case: Portfolio variance is the sum of weights times entries in the covariance matrix Slide 26

27 Mean-Variance Analysis The General Case: Covariance matrix contains n 2 terms n terms are variances n 2 n terms are covariances In a well-diversified portfolio, covariances are more important than variances A stock s covariance with other stocks determines its contribution to the portfolio s overall variance Investors should care more about the risk that is common to many stocks; risks that are unique to each stock can be diversified away Slide 27

28 Mean-Variance Analysis Special Case: Consider an equally weighted portfolio: For portfolios with many stocks, the variance is determined by the average covariance among the stocks Slide 28

29 Mean-Variance Analysis Example: The average stock has a monthly standard deviation of 10% and the average correlation between stocks is If you invest the same amount in each stock, what is variance of the portfolio? What if the correlation is 0.0? 1.0? Slide 29

30 Mean-Variance Analysis Example (cont): 12% 10% ρ = 1.0 SD of Portfolio 8% 6% 4% ρ = 0.4 2% ρ = 0.0 0% Number of Stocks Slide 30

31 Mean-Variance Analysis Eventually, Diversification Benefits Reach A Limit: Remaining risk known as systematic or market risk Due to common factors that cannot be diversified Example: S&P 500 Other sources of systematic risk may exist: Credit Liquidity Volatility Business Cycle Value/Growth Portfolio Variance Total risk of typical stock Risk eliminated by diversification Undiversifiable or systematic risk Number of Stocks in Portfolio Provides motivation for linear factor models Slide 31

32 The Efficient Frontier Given Portfolio Expected Returns and Variances: How Should We Choose The Best Weights? All feasible portfolios lie inside a bullet-shaped region, called the minimum-variance boundary or frontier The efficient frontier is the top half of the minimum-variance boundary (why?) Rational investors should select portfolios from the efficient frontier Slide 32

33 The Efficient Frontier 2.4% Efficient Frontier Expected Return 1.8% 1.2% 0.6% Minimum-Variance Boundary 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Standard Deviation of Return Slide 33

34 The Efficient Frontier Example: You can invest in any combination of GM, IBM, and MOT. What portfolio would you choose? Variance / covariance Stock Mean Std dev GM IBM Motorola GM IBM Motorola Slide 34

35 The Efficient Frontier Example: You can invest in any combination of GM, IBM, and MOT. What portfolio would you choose? Variance / covariance Stock Mean Std dev GM IBM Motorola GM IBM Motorola Slide 35

36 The Efficient Frontier Example: You can invest in any combination of GM, IBM, and MOT. What portfolio would you choose? Variance / covariance Stock Mean Std dev GM IBM Motorola GM IBM Motorola E[R P ] = (w GM 1.08) + (w IBM 1.32) + (w Mot 1.75) Slide 36

37 The Efficient Frontier Example: You can invest in any combination of GM, IBM, and MOT. What portfolio would you choose? Variance / covariance Stock Mean Std dev GM IBM Motorola GM IBM Motorola E[R P ] = (w GM 1.08) + (w IBM 1.32) + (w Mot 1.75) var(r P ) = (w GM ) + (w IBM 40.21) + (w Mot ) + (2 w GM w IBM 16.13) + (2 w GM w Mot 22.43) + (2 w IBM w Mot 23.99) Slide 37

38 The Efficient Frontier Example: You can invest in any combination of GM, IBM, and MOT. What portfolio would you choose? Variance / covariance Stock Mean Std dev GM IBM Motorola GM IBM Motorola E[R P ] = (w GM 1.08) + (w IBM 1.32) + (w Mot 1.75) var(r P ) = (w GM ) + (w IBM 40.21) + (w Mot ) + (2 w GM w IBM 16.13) + (2 w GM w Mot 22.43) + (2 w IBM w Mot 23.99) Slide 38

39 The Efficient Frontier Example (cont): Feasible Portfolios 2.1% 1.8% Motorola Expected Return 1.5% 1.2% 0.9% IBM GM 0.6% 3.0% 5.0% 7.0% 9.0% 11.0% Standard Deviation of Return Slide 39

40 The Tangency Portfolio The Tangency Portfolio If there is also a riskless asset (T-Bills), all investors should hold exactly the same stock portfolio! All efficient portfolios are combinations of the riskless asset and a unique portfolio of stocks, called the tangency portfolio.* In this case, efficient frontier becomes straight line * Harry Markowitz, Nobel Laureate Slide 40

41 The Tangency Portfolio 2.4% 1.8% Motorola Expected Return 1.2% 0.6% IBM GM P Tbill 0.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Standard Deviation of Return Slide 41

42 The Tangency Portfolio 2.4% Expected Return 1.8% 1.2% 0.6% Tangency portfolio IBM GM Motorola Tbill 0.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Standard Deviation of Return Slide 42

43 The Tangency Portfolio Sharpe ratio A measure of a portfolio s risk-return trade-off, equal to the portfolio s risk premium divided by its volatility: Sharpe Ratio E[R p] r f σ p (higher is better!) The tangency portfolio has the highest possible Sharpe ratio of any portfolio Aside: Alpha is a measure of a mutual fund s risk-adjusted performance. The tangency portfolio also maximizes the fund s alpha. Slide 43

44 The Tangency Portfolio Slope = Sharpe Ratio (why?) 2.4% Expected Return 1.8% 1.2% 0.6% Tangency portfolio IBM GM Motorola Tbill 0.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Standard Deviation of Returns Slide 44

45 Key Points Diversification reduces risk. The standard deviation of a portfolio is always less than the average standard deviation of the individual stocks in the portfolio. In diversified portfolios, covariances among stocks are more important than individual variances. Only systematic risk matters. Investors should try to hold portfolios on the efficient frontier. These portfolios maximize expected return for a given level of risk. With a riskless asset, all investors should hold the tangency portfolio. This portfolio maximizes the trade-off between risk and expected return. Slide 45

46 Additional References Bernstein, 1992, Capital Ideas. New York: Free Press. Bodie, Z., Kane, A. and A. Marcus, 2005, Investments, 6th edition. New York: McGraw-Hill. Brennan, T., Lo, A. and T. Nguyen, 2007, : A Review, to appear in Foundations of Finance. Campbell, J., Lo, A. and C. MacKinlay, 1997, The Econometrics of Financial Markets. Princeton, NJ: Princeton University Press. Grinold, R. and R. Kahn, 2000, Active Portfolio Management. New York: McGraw-Hill. Slide 46

47 MIT OpenCourseWare Finance Theory I Fall 2008 For information about citing these materials or our Terms of Use, visit:

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