Portfolio Statistics Basics of expected returns, volatility, correlation and diversification

Transcription

1 Finance Portfolio Statistics Basics of expected returns, volatility, correlation and diversification Finance Fall 2016 Tepper School of Business Carnegie Mellon University c 2016 Chris Telmer. Some content from slides by Bryan Routledge. Used with permission :39

2 Questions :: Where does the 8.4% come from? Portfolio Stats 2

3 Questions :: You are an investment advisor. Your client is interested in two stocks: :: Which one is best? :: Perhaps she should hold some of both? :: She asks for a particular amount of return? :: She expands her horizons to include more stocks. What now? Portfolio Stats 3

4 Plan :: Data: what are the facts? :: Bonds, interest rates :: Stocks, Equity Premium :: Characterizing portfolios :: Characterizing portfolio risk and return :: Benefits of diversification :: Characterizing the distribution of returns with a statistical model :: Next 2 weeks: :: Optimal portfolio choice :: CAPM: Equilibrium asset pricing (where does 8.4% come from Portfolio Stats 4

5 Empirical Facts About Risk and Return (What is the Equity Premium Portfolio Stats 5

6 Question One-year interest rates :: Now: 0.25% :: Historical average: 3 or 4%. How much should you expect to get on stocks? Portfolio Stats 6

7 Spreadsheet See spreadsheet, eq prem data.xlsx :: Also, AEO data from problem set Portfolio Stats 7

8 U.S. Government Bonds :: U.S. Government bonds are called Treasury Securities :: Treasury Bills = mature 1, 3, 6, and 12 months No coupon payment (i.e., a zero coupon :: Treasury Notes = Mature 2 to 10 years Semi-annual coupons :: Treasury Bonds = Mature > 10 years Semi-annual coupons :: Others Portfolio Stats 8

9 Bond Prices [Source: Yahoo :: Typically quoted in terms of YTM: Portfolio Stats 9

10 Bond Yields (Prices [Source: St.Louis Fed ] Treasury Yields yield jan jan jan jan jan jan Year 10 Year 5 Year 20 Year GetTreasuryAndPlot.do Source: FRED 1962 to Constant Maturity US Treasury Portfolio Stats 10

11 Bond Yields (Prices [Source: St.Louis Fed ] Treasury Yields yield jan jan jan jan jan jan jan Month 1 Year 5 Year 10 Year 20 Year 30 Year GetTreasuryAndPlot.do Source: FRED 2000 to Constant Maturity US Treasury Portfolio Stats 11

12 Interest Rates, Nominal and Real The Fisher Equation: Nominal Interest Rate = Real Interest Rate + E ( Inflation :: Which interest rate should we use for FCF valuation? Portfolio Stats 12

13 Stocks - The Equity Risk Premium :: Excess Returns = The return on an asset (e.g., a stock above the risk-free rate = r i,t r f,t Portfolio Stats 13

14 Stocks - The Equity Risk Premium :: Excess Returns = The return on an asset (e.g., a stock above the risk-free rate = r i,t r f,t :: Expected Excess Returns = E[ r i,t r f,t ] = Compensation for the risk or a risk premium Portfolio Stats 13

15 Stocks - The Equity Risk Premium :: Excess Returns = The return on an asset (e.g., a stock above the risk-free rate = r i,t r f,t :: Expected Excess Returns = E[ r i,t r f,t ] = Compensation for the risk or a risk premium :: Equity Risk Premium = Expected excess return on a portfolio of all stocks = All stocks in US; weighted by value; similar to S&P 500 = E[ r m,t r f,t ] Portfolio Stats 13

16 Stocks - The Equity Risk Premium [Data: CRSP via Ken French ] T Data: CRSP Indicies :: Monthly Note: log scale (base e Invest $1 at Portfolio Value as of month end Market NYSE, AMEX, and NASDAQ [R] Risk Free Rate One month t bill [FF] continuously return Consumer Price Index for All Urban Consumers: All Items $ $20.25 $12.92 Portfolio Stats 14

17 Stocks - The Equity Risk Premium Portfolio Stats 15

18 Stocks - The Equity Risk Premium [Data: CRSP via Ken French ] Daily Returns Equity Index T Market all NYSE, AMEX, and NASDAQ [FF] Risk Free Rate One month t bill [FF] continuously return Data: CRSP Daily 1920 to 2012/9 Portfolio Stats 16

19 Stocks - The Equity Risk Premium [Data: CRSP via Ken French ] Daily Returns Equity Index T Market all NYSE, AMEX, and NASDAQ [FF] Risk Free Rate One month t bill [FF] continuously return Data: CRSP Daily 2000 to 2012/9 Portfolio Stats 17

20 Stocks - The Equity Risk Premium [Data: CRSP via Ken French ] Invest $1 at Portfolio Value as of month end $ T Market NYSE, AMEX, and NASDAQ [R] Risk Free Rate One month t bill [FF] continuously return Consumer Price Index for All Urban Consumers: All Items Data: CRSP Indicies :: Monthly Note: log scale (base e Portfolio Stats 18

21 Summary Estimating the discount rate (OCC: :: Get risk-free interest rate from UST securities: :: r f = 0.02 (10-yr UST spot rate :: Estimate equity risk premium: :: E ( r r f 0.06 :: Estimate firm/project β ( beta and use: :: E ( r i = rf + β ( E( r r f Portfolio Stats 19

22 Portfolio Construction Portfolio Stats 20

23 Portfolio Definitions (today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity :: p i0 = today s price-per-share, stock i ( ex-dividend Portfolio Stats 21

24 Portfolio Definitions (today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity :: p i0 = today s price-per-share, stock i ( ex-dividend :: p i = future price-per-share (random, also ex-dividend Portfolio Stats 21

25 Portfolio Definitions (today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity :: p i0 = today s price-per-share, stock i ( ex-dividend :: p i = future price-per-share (random, also ex-dividend :: n i = number of shares held, stock i Portfolio Stats 21

26 Portfolio Definitions (today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity :: p i0 = today s price-per-share, stock i ( ex-dividend :: p i = future price-per-share (random, also ex-dividend :: n i = number of shares held, stock i :: V 0 = N i=1 p i0n i, total funds invested Portfolio Stats 21

27 Portfolio Definitions (today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity :: p i0 = today s price-per-share, stock i ( ex-dividend :: p i = future price-per-share (random, also ex-dividend :: n i = number of shares held, stock i :: V 0 = N i=1 p i0n i, total funds invested :: V = N i=1 p in i realized portfolio payoff (random Portfolio Stats 21

28 Portfolio Definitions (today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity :: p i0 = today s price-per-share, stock i ( ex-dividend :: p i = future price-per-share (random, also ex-dividend :: n i = number of shares held, stock i :: V 0 = N i=1 p i0n i, total funds invested :: V = N i=1 p in i realized portfolio payoff (random :: γ i = ( p i0 n i /V0, portfolio share, stock i Portfolio Stats 21

29 Portfolio Definitions (today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity :: p i0 = today s price-per-share, stock i ( ex-dividend :: p i = future price-per-share (random, also ex-dividend :: n i = number of shares held, stock i :: V 0 = N i=1 p i0n i, total funds invested :: V = N i=1 p in i realized portfolio payoff (random :: γ i = ( p i0 n i /V0, portfolio share, stock i :: 1 + r i = p i /p i0 rate-of-return, stock i (random Portfolio Stats 21

30 Portfolio Definitions (today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity :: p i0 = today s price-per-share, stock i ( ex-dividend :: p i = future price-per-share (random, also ex-dividend :: n i = number of shares held, stock i :: V 0 = N i=1 p i0n i, total funds invested :: V = N i=1 p in i realized portfolio payoff (random :: γ i = ( p i0 n i /V0, portfolio share, stock i :: 1 + r i = p i /p i0 rate-of-return, stock i (random :: 1 + r p = V /V 0 rate-of-return, portfolio (random Portfolio Stats 21

31 Portfolio Definitions (today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity :: p i0 = today s price-per-share, stock i ( ex-dividend :: p i = future price-per-share (random, also ex-dividend :: n i = number of shares held, stock i :: V 0 = N i=1 p i0n i, total funds invested :: V = N i=1 p in i realized portfolio payoff (random :: γ i = ( p i0 n i /V0, portfolio share, stock i :: 1 + r i = p i /p i0 rate-of-return, stock i (random :: 1 + r p = V /V 0 rate-of-return, portfolio (random :: 1 + r f = risk-free return (interest rate Portfolio Stats 21

32 Portfolio Return: Two-Stock Example Dividends = 0 for notational simplicty... incorporated soon Today s portfolio value: V 0 = p 1,0 n 1 + p 2,0 n 2 Next period s value V = p 1 n 1 + p 2 n 2 Portfolio Stats 22

33 Portfolio Return: Two-Stock Example Dividends = 0 for notational simplicty... incorporated soon Today s portfolio value: V 0 = p 1,0 n 1 + p 2,0 n 2 Next period s value V = p 1 n 1 + p 2 n 2 V V 0 = p 1,0n 1 V 0 p 1 p 1,0 + p 2,0n 2 V 0 p 2 p 2,0 Portfolio Stats 22

34 Portfolio Return: Two-Stock Example Dividends = 0 for notational simplicty... incorporated soon Today s portfolio value: V 0 = p 1,0 n 1 + p 2,0 n 2 Next period s value V = p 1 n 1 + p 2 n 2 V V 0 = p 1,0n 1 V 0 p 1 p 1,0 + p 2,0n 2 V 0 p 2 p 2,0 1 + r p = γ 1 ( 1 + r1 + γ2 ( 1 + r2 Portfolio Stats 22

35 Portfolio Return: Two-Stock Example Dividends = 0 for notational simplicty... incorporated soon Today s portfolio value: V 0 = p 1,0 n 1 + p 2,0 n 2 Next period s value V = p 1 n 1 + p 2 n 2 V V 0 = p 1,0n 1 V 0 p 1 p 1,0 + p 2,0n 2 V 0 p 2 p 2,0 1 + r p = γ 1 ( 1 + r1 + γ2 ( 1 + r2 r p = γ 1 r 1 + ( 1 γ 1 r2 :: Return on portfolio is value-weighted average of return on portfolio components Portfolio Stats 22

36 Many Stocks 1 + r p = γ 1 ( 1 + r1 + γ2 ( 1 + r γn ( 1 + rn Portfolio Stats 23

37 Many Stocks 1 + r p = γ 1 ( 1 + r1 + γ2 ( 1 + r γn ( 1 + rn r p = γ 1 r 1 + γ 2 r γ N r N Portfolio Stats 23

38 Many Stocks 1 + r p = γ 1 ( 1 + r1 + γ2 ( 1 + r γn ( 1 + rn r p = γ 1 r 1 + γ 2 r γ N r N = N γ i r i i=1 Note: :: Portfolio weights sum to one: N i=1 γ i = 1 :: If γ i < 0 we call this short selling stock i :: If γ i is restricted to be positive, we call this a no short sales restriction. Portfolio Stats 23

39 Dividends For stocks 1 and 2, dividends are d 1 and d 2. Nothing changes, except returns incorporate dividend income. Note that dividends are (typically random. Today s value: Next period s value V 0 = p 1,0 n 1 + p 2,0 n 2 V = (p 1 + d 1 n 1 + (p 2 + d 2 n 2 Portfolio Stats 24

40 Dividends For stocks 1 and 2, dividends are d 1 and d 2. Nothing changes, except returns incorporate dividend income. Note that dividends are (typically random. Today s value: Next period s value V 0 = p 1,0 n 1 + p 2,0 n 2 V = (p 1 + d 1 n 1 + (p 2 + d 2 n 2 V = p ( 1,0n 1 p1 + d 1 + p ( 2,0n 2 p2 + d 2 V 0 V 0 p 1,0 V 0 p 2,0 Portfolio Stats 24

41 Dividends For stocks 1 and 2, dividends are d 1 and d 2. Nothing changes, except returns incorporate dividend income. Note that dividends are (typically random. Today s value: Next period s value V 0 = p 1,0 n 1 + p 2,0 n 2 V = (p 1 + d 1 n 1 + (p 2 + d 2 n 2 V = p ( 1,0n 1 p1 + d 1 + p ( 2,0n 2 p2 + d 2 V 0 V 0 p 1,0 V 0 p 2,0 1 + r p = γ 1 ( 1 + r1 + γ2 ( 1 + r2 Portfolio Stats 24

42 Dividends For stocks 1 and 2, dividends are d 1 and d 2. Nothing changes, except returns incorporate dividend income. Note that dividends are (typically random. Today s value: Next period s value V 0 = p 1,0 n 1 + p 2,0 n 2 V = (p 1 + d 1 n 1 + (p 2 + d 2 n 2 V = p ( 1,0n 1 p1 + d 1 + p ( 2,0n 2 p2 + d 2 V 0 V 0 p 1,0 V 0 p 2,0 1 + r p = γ 1 ( 1 + r1 + γ2 ( 1 + r2 r p = γ 1 r 1 + ( 1 γ 1 r2 :: Same as before: Return on portfolio is value-weighted average of return on portfolio components Portfolio Stats 24

43 Numerical Examples :: You have 120 shares of Stock 1 and 50 shares of Stock 2. Today s prices are p 1,0 = 25 and p 2,0 = 40. Returns are 15% for Stock 1 and 2% for Stock 2. Dividends are zero. :: What are the portfolio shares. What is the portfolio return. What are the realized (next period stock prices. What is today s wealth and tomorrow s wealth? (Answers: 60%, 40%, 9.8%, $28.75, $40.80, $5,000, $5,490 :: Dividends are now 0.5 and 1.6. :: Same questions. (Answers: 60%, 40%, 9.8%, $28.25, $39.2 $5,000, $5,490. Also, dividend yields are d 1 /p 1,0 = 0.02 and d 2 /p 2,0 = 0.04 (this is a common convention for expressing dividends... dividend per dollar of share price :: Same data as previous question, but future prices are 32 and 35 and you must solve for returns. :: (Answers: 60%, 40%, 14.6%, 30%, 8.50%, $5,000, $5,730, for portfolio shares, portfolio return, stock returns, initial and terminal wealth, respectively Portfolio Stats 25

44 Statistical Reminder Portfolio Stats 26

45 Notation and Statistical Reminder :: x and y are random variables. :: Notation (µ, σ, σ xy, ϕ xy :: Expected value: µ x = E ( x, µ y = E ( y :: Variance: σ 2 x = Var ( x = E ( x µ x 2 = E ( x 2 µ 2 x :: Standard Deviation: σ x = σ 2 x :: Covariance: σ xy = E ( x µ x ( y µy = E ( x y E ( x E ( y :: Correlation: ϕ xy = σ xy / ( σ x σ y Portfolio Stats 27

46 Reminder :: Linear combinations of random variables: if x and y are random, and a, b, c are constants, and z = a + bx + cy, then E ( z = a + be ( x + ce ( y = a + bµ x + cµ y Var ( z = Var ( a + bx + cy = b 2 Var ( x + c 2 Var ( y + 2bcCov ( x, y = b 2 σ 2 x + c 2 σ 2 y + 2bcσ xy = b 2 σ 2 x + c 2 σ 2 y + 2bcσ xσ y ϕ xy :: Linear combinations of normals are normal. If x and y are normal random variables, x N ( µ x, σx 2 y N ( µ y, σy 2 then, for z = a + bx + cy, z N ( µ z, σz 2 N ( a + bµ x + cµ y, b 2 σx 2 + c 2 σy 2 + 2bcσ xy Portfolio Stats 28

47 Reminders :: Important special cases of z = a + bx + cy: :: b = c = 0: Var ( z = Var ( a = 0 :: c = 0: Var ( z = Var ( a + bx = b 2 σ 2 x :: We typically work with data on correlations not covariances. But, we sometimes need to use the covariance to do calculations. Just remember: Correlation : ϕ xy = σ xy ( σx σ y :: What is the standard deviation? :: Sometimes called volatility :: Given an intuitive answer, in terms of the normal distribution. i.e., if x N ( µ x, σ 2 x, and µx = 10 and σ x = 2, what does σ x = 2 mean? (answer: the realized value of x is very unlikely to be outside of the interval (6, 14. Portfolio Stats 29

48 Discrete Random Variables There are S states of nature, each indexed with s. The probability of state s being realized is p s. x and y are discrete-valued random variables than take on values x s and p s, respectively. We refer to the means, standard deviations and the correlation of these random variables as (some of their moments. Formulae for the moments are below. This is review from your stats class. In order to check that you remember what you need to remember, here is a test. There are two states of nature, State 1 and State 2. Probs are 0.4 and 0.6, respectively. x and y take on values (18,9 and (25,10 in States 1 and 2, respectively. What are the means and the standard deviations of x and y and what is their correlation? Here is spreadsheet and a coding-up of the formulae that appear below. Mean: E(x = s psxs Standard Deviation: Var(x = E(x 2 E(x 2 = s psx 2 s ( s psxs2 Covariance: Cov(x, y = E(xy E(xE(y = s psxsys s psxs s psys Correlation: Corr(x, y = Cov(x, y/(σ xσ y Notation: µ x E(x, σ x Stdev(x = Var(x, σ 2 x Var(x, σ xy Cov(x, y, ϕ xy Corr(x, y = σ xy /(σ xσ y. Portfolio Stats 30

49 Statistical Properties of Portfolio Returns Portfolio Stats 31

50 Portfolio Behavior :: We now know what portfolio returns are. :: What next? Consider an investor. What does she care about? :: Higher returns are better than lower ones? :: But returns are random, uncertain, stochastic... :: We ll focus on moments of the return distribution: :: Mean ( expected return :: Variance, standard deviation ( volatility :: Why? Graph: tradeoff Portfolio Stats 32

51 Graph: Expected Return vs Volatility Same as Risk versus Return. µ σ space. Consider portfolios, first dominant then not Portfolio Stats 33

52 Data Deviations: Inflation Adjusted Returns: Arithmetic Returns Average Emerging Market Equity Small Cap Equity Mid Cap Equity REITs High Yield Bonds Large Cap Equity International Equity International Govt Bonds Long Term Bond Commodities Intermediate Term Bond Fixed Income Std_Dev Source: Burton Hollifield Portfolio Stats 34

53 Optimal Portfolio Choice :: Choose portfolio with maximal expected return, conditional on given amount of risk (volatility. :: Sometimes called: :: Mean-Variance Analysis :: Markowitz Diversification :: The property of investors desiring mean-variance-efficient portfolios :: Maximize the Sharpe Ratio Portfolio Stats 35

54 Portfolio Moments r p = γ 1 r 1 + ( 1 γ 1 r2 :: Expected return: E r p = γ 1 E r 1 + ( 1 γ 1 E r2 Portfolio Stats 36

55 Portfolio Moments r p = γ 1 r 1 + ( 1 γ 1 r2 :: Expected return: E r p = γ 1 E r 1 + ( 1 γ 1 E r2 µ p = γ 1 µ 1 + ( 1 γ 1 µ2 Portfolio Stats 36

56 Portfolio Moments r p = γ 1 r 1 + ( 1 γ 1 r2 :: Expected return: E r p = γ 1 E r 1 + ( 1 γ 1 E r2 µ p = γ 1 µ 1 + ( 1 γ 1 µ2 :: Variance and volatility (using γ 2 = (1 γ 1 to avoid clutter Var ( r p = γ 2 1 Var ( r 1 + γ 2 2 Var ( r 2 + 2γ1 γ 2 Cov ( r 1, r 2 Portfolio Stats 36

57 Portfolio Moments r p = γ 1 r 1 + ( 1 γ 1 r2 :: Expected return: E r p = γ 1 E r 1 + ( 1 γ 1 E r2 µ p = γ 1 µ 1 + ( 1 γ 1 µ2 :: Variance and volatility (using γ 2 = (1 γ 1 to avoid clutter Var ( r p = γ 2 1 Var ( r 1 + γ 2 2 Var ( r 2 + 2γ1 γ 2 Cov ( r 1, r 2 σ 2 p = γ 2 1σ γ 2 2σ γ 1 γ 2 σ 12 Portfolio Stats 36

58 Portfolio Moments r p = γ 1 r 1 + ( 1 γ 1 r2 :: Expected return: E r p = γ 1 E r 1 + ( 1 γ 1 E r2 µ p = γ 1 µ 1 + ( 1 γ 1 µ2 :: Variance and volatility (using γ 2 = (1 γ 1 to avoid clutter Var ( r p = γ 2 1 Var ( r 1 + γ 2 2 Var ( r 2 + 2γ1 γ 2 Cov ( r 1, r 2 σ 2 p = γ 2 1σ γ 2 2σ γ 1 γ 2 σ 12 = γ 2 1σ γ 2 2σ γ 1 γ 2 σ 1 σ 2 ϕ 12 Portfolio Stats 36

59 Portfolio Moments r p = γ 1 r 1 + ( 1 γ 1 r2 :: Expected return: E r p = γ 1 E r 1 + ( 1 γ 1 E r2 µ p = γ 1 µ 1 + ( 1 γ 1 µ2 :: Variance and volatility (using γ 2 = (1 γ 1 to avoid clutter Var ( r p = γ 2 1 Var ( r 1 + γ 2 2 Var ( r 2 + 2γ1 γ 2 Cov ( r 1, r 2 σ 2 p = γ 2 1σ γ 2 2σ γ 1 γ 2 σ 12 = γ1σ γ2σ γ 1 γ 2 σ 1 σ 2 ϕ 12 σ p = ( γ1σ γ2σ /2 + 2γ 1 γ 2 σ 1 σ 2 ϕ 12 Portfolio Stats 36

60

61 Graphs: Mean and Variance vs. γ 1 Mean is linear, variance (stdev are quadratic... unless ϕ 12 = 1. Diversification reduces risk Portfolio Stats 37

62 Graph: Distribution of Portfolio Returns How does diversification reduce portfolio risk? Suppose two stocks have same marginal distribution. Portfolio Stats 38

63 Business Day Dow Hits Record Heights WORLD U.S. N.Y. / REGION BUSINESS TECHNOLOGY SCIENCE HEA LTH SPORTS OPINION A RTS STYLE TRA V EL JOBS REAL ESTATE AUTOS Global DealBook Markets Economy Energy Media Technology Personal Tech Small Business Your Money March 2013, Dow hits new record. Note that many Dow stocks went down! Advertise on NYTimes.com Published: March 5, 2013 FACEBOOK TWITTER GOOGLE+ SHARE The Dow s Movers Change in each company s stock price since the Dow s previous high on Oct. 9, Related Article» HOME DEPOT IBM MCDONALD S WAL-MART WALT DISNEY TRAVELERS +108% +75% +67% +63% +59% +53% COCA-COLA CHEVRON JOHNSON & JOHNSON VERIZON UNITED TECHNOLOGIES UNITEDHEALTH +34% +27% +17% +12% +12% +10% PFIZER CATERPILLAR 3M PROCTER & GAMBLE JPMORGAN CHASE AMERICAN EXPRESS +10% +9% +9% +8% +4% +3% DUPONT 2% EXXON MOBIL 3% MICROSOFT 6% AT&T 13% INTEL 17% MERCK 19% BOEING 22% CISCO 36% GENERAL 44% HEWLETT- 61% BANK OF 78% ALCOA SYSTEMS ELECTRIC PACKARD AMERICA 79% Source: Bloomberg 2013 The New York Times Company Site Map Privacy Your Ad Choices Advertise Terms of Sale Terms of Service Work With Us RSS Help Contact Us Site Feedback Portfolio Stats 39

64 Graph of Variance vs. Number of Securities Diversification reduces risk... but typically only up to a point Portfolio Stats 40

65 portfolio vs. the number of stocks Data std=40%, correlation=0.28 Standard deviation of an equally-weighted portfolio against number of stocks, σ = 0.40, ϕ = 0.28 Source: Burton Hollifield Portfolio Stats 41

66 Systematic vs Idiosyncratic Risk Why are there limits to risk-reducing diversification? Portfolio Stats 42

67 Statistical Model of Prices/Returns (Moments versus Models Portfolio Stats 43

68 Why Have a Model? :: Moments: :: Statistical moments are things like mean, variance, covariance, skewness, kurtosis, etc. :: They are properties of statistical distributions. :: Up to now, we ve just studied moments of returns :: We can use data to estimate them :: For some questions it is necessary to develop a full model of the statistical distribution. :: Example: Value-at-Risk. What is the most I might lose on my portfolio, with 95% probability? Portfolio Stats 44

69 Continuous Distributions Normal, Lognormal, Fat-Tailed, Skewed, etc. Portfolio Stats 45

70 Discrete Distributions There are three states of nature and three assets. The asset s prices and the distribution of the asset s payoffs are as follows State of Nature (s Price Probability (p s Bond Asset 1 (c 1s Asset 2 (c 2s :: Moments: mean, standard deviation, covariance (i = 1, 2 E(c i = p sc is s Stdev(c i = ( E(ci 2 E(c i 2 1/2 = ( p scis 2 E(c i 2 1/2 Cov(c 1 c 2 = E(c 1c 2 E(c 1E(c 2 = p sc 1sc 2s E(c 1E(c 2 s s Portfolio Stats 46

71 Moments of Discrete Distributions E(c Stdev(c Cov(c 1 c 2 Corr(c 1 c 2 Asset Asset Portfolio Stats 47

72 Returns Distribution: State of Nature (s Probability (p s Bond Asset 1 (r 1s Asset 2 (r 2s Moments: 1 E(r E(r r f Stdev(r Cov(r 1 r 2 Corr(r 1 r 2 Asset Asset r f = is the risk-free interest rate, so that E(r r f is the equity premium: the excess expected return on equity (excess means in excess of the risk-free rate Portfolio Stats 48

73 Summary Don t confuse moments with models :: Moments: Expected Return Standard Deviation Riskless Bond Stock Stock Correlation 0.50 :: What is the expected return and volatility of a portfolio? :: Models (here are two examples: r 1 N ( µ 1, σ1 2 r 2 N ( µ 2, σ2 2 State of Nature 1 2 Probability Asset Asset :: What is Prob ( r p < 0.10? Portfolio Stats 49

74 Summary: Risk and Return, Diversification Portfolio Stats 50

75 Summary :: Historical equity premium is about 6%. :: Compute OCC by adding risk-adjusted equity premium to today s riskless rate :: Construct portfolio returns as value-weighted averages :: Portfolio statistics (moments :: Mean and variance of linear combinations... correlations show up. :: Diversification reduces portfolio risk... but only up to a point: systematic risk :: Modeling the distribution, versus estimating the moments. :: Discrete distributions. :: Moments are probability-weighted averages of realizations. Portfolio Stats 51

76 Importance of Principle of Diversification Phillips from FT, Aug Portfolio Stats 52

77 Expected Returns Don t confuse expected returns with realized returns :: Example: Sony s Blu-Ray wins technological contest against Toshiba s HD-DVD. Portfolio Stats 53

78 Spaces It is important to understand the various spaces that the various graphs appear in. :: Time-series space (return against time :: Frequency space (return frequency distribution :: Mean-variance (or mean-standard deviation space :: Captures the risk-return tradeoff :: See spreadsheets Portfolio Stats 54