1. Diversification and Portfolios Economics 71a: Spring 2007 Mayo chapter 8 Malkiel, Chap 9-10 Lecture notes 3.2b

2. Goals  Portfolios and correlations  Diversifiable versus nondiversifiable risk  CAPM and Beta Capital asset pricing model  Is the CAPM really useful?  Asset allocation

3. Risk: Individual->Portfolio  Early models Risk is based on each individual stock  Modern approaches Consider how it effects “portfolio” of holdings Markowitz Modern portfolio theory Diversification

4. Diversification and Portfolios  “Don’t put all your eggs in one basket”  Buying a large set of securities can reduce risk

5. What is the return of a portfolio?  $ values in assets 1 and 2 = h1 and h1  R1 and R2 are returns of assets 1 and 2  Rp is the return of the portfolio  Ending portfolio = End  Starting value = Start

6. In words  The return of a portfolio is equal to a weighted average of the returns of each investment in the portfolio  The weight is equal to the fraction of wealth in each investment

7. Malkiel’s Example of Risk Reduction Umbrella Company Resort Company Rainy Season+50%-25% Sunny Season-25%+50%

8. Portfolio 50/50 in Each  Return = Rain : (0.5) (0.50) + (0.5)(-0.25) = 12% Shine: (0.5) (-0.25) + (0.5)(0.50) = 12% = 12% rain or shine  No risk  This is the beauty of diversification  Simple risk management  Quirk: Need “negative” relation

9. What is going on?  Asset returns have perfect “negative correlation”  They move exactly opposite to each other  Is this always necessary? No

10. Diversification Experiment  Assume the following framework for stock returns  Two parts Part that moves with market:  Part that is unique to the firm: e  Rm is the return of the market  Experiment: Choose two stocks and beta’s Beta determines how closely the stock move with each other Combine two stocks as x and (1-x) fractions Return = x R1 + (1-x) R2 Example portfolio variance

11. Web Examples  See multi-Beta scatter plots Portfolio 2

12. Quick Application: A perfect hedge  Security 1: y = 0.1 + b*v  Security 2: x = 0.1 + -b*v  v is random  Portfolio: (1/2) each port = 0.5(0.1+b*v) + 0.5(0.1-b*v) port = 0.1 + 0.5(b-b)*v = 0.1 Risk free  Perfect negative correlation

13. Summary: Portfolio Theory  A radically new approach to risk In the 1950’s  Two key points Diversification matters Worry about how an investment moves with the rest of your portfolio Worry more about correlations than standard deviations and variances

14. Goals  Portfolios and correlations  Diversifiable versus nondiversifiable risk  CAPM and Beta Capital asset pricing model  Is the CAPM really useful?

15. Nondiversifiable Risk  Many equity returns are positively correlated  What does that mean to our new thoughts on risk?

16. Individual Equity Return Structure  Assume the following framework for stock returns  R(j) is the return on some stock  a(j) is a constant  R(m) is the return on the market  e(j) is random noise, special for stock j: Mean or expectation of e(j) = 0, E(e(j)) = 0

17. What does a portfolio of 2 stocks look like?  Call these two stock 1 and stock 2  Hold 50/50 of each

18. Portfolio of N stocks  Sum with 1/N weight on each

19. What about diversifiable risk?  The part of the portfolio related to diversifiable risk is  The critical aspect of diversification is that as N gets big this random number gets close to zero  “Law of large numbers”

20. Risk Reduction Number of Securities Portfolio Risk (Variance or standard deviation) Nondiversifiable Risk (systematic) Diversifiable Risk (unsystematic)

21. Why?  This is a little like going to a casino, and playing roulette  You bet on red many, many times  Keep track of W/(W+L)  As you play more and more this gets very close to 0.5

22. Diversification Histograms Distribution of mean(e) for portfolios of sizes 1, 5, 20

23. What About Beta?  Beta (nondiversifiable risk) is the mean over all the individual stock beta’s

24. Key issue  For equities the diversifiable part of risk can be eliminated  All that remains is the part that moves with the market, or the nondiversifiable risk  This depends on beta ONLY

25. Java Example  See web example

26. Estimating Beta  Statistics: Use linear regression to estimate beta  Problems Not stable over time Nonlinear relationships

27. Goals  Portfolios and correlations  Diversifiable versus nondiversifiable risk  CAPM and Beta Capital asset pricing model  Is the CAPM really useful?

28. Capital Asset Pricing Model (CAPM)  Risk depends on Beta alone  If there is a payoff of higher return for higher risk, then alpha, the expected return, depends on Beta only  In the CAPM world Beta is the key component of risk

29. What would happen in a non CAPM world? Malkiel’s experiment  Assume the risk measure that people care about is related to the total (nondiversifiable+diversifiable) risk  Stocks with higher e(j) variance pay higher returns  Build two stock portfolios High e(j) variance, Beta = 1 Low e(j) variance, Beta = 1

30. More on Malkiel’s Experiment  Since this is a nonCAPM world The first portfolio earns a higher return  However, the risk of the two portfolios is the same They have the same beta e(j) risk is diversified away  Investors will load up on high e(j) risk stocks This drives the price up, and expected returns will fall on these stocks until they are equal to the others

31. Beta is Key  In the CAPM world: No reward for holding stocks with lots of diversifiable risk Only beta matters as a measure of risk

32. Adjusting Beta Using a Risk Free Asset  Market Portfolio Expected return = 10% Beta = 1  Risk free (bank account) Expected return = 4% Beta = 0  Combine these two

33. Combinations  All risk free Beta = 0, expected return = 4%

34. Combinations  50/50 Market/Risk free  Beta = 0.5  Expected return =  0.5 (4%) + 0.5 (10%) = 7%  More Beta, more risk, more expected return

35. Fully Invested in Market  Easy Beta = 1 Expected return = 10%

36. More risk: Borrow Like buying on margin  Borrow $0.50 at 4% risk free  Invest $1.50 in the market  What does this portfolio look like at end?  Beta = 1.5, riskier than 1

37. What is the expected return?  -0.5(4%)+1.5(10%) = 13%  Wow! Greater than the expected return on the market. What’s going on?  Taking on greater risk Buying on margin

38. Risk Versus Return Building your own Beta’s Beta Expected Return 0 Risk Free 1 Market Borrowing

39. Risk Versus Return Slope = (E(Rm)-Rf)/1 Beta Expected Return 0 Risk Free 1 Market E(Rm)-Rf

40. Constructing Variance Risk  Market Portfolio Expected return = 10% Variance = 20%  Risk free (bank account) Expected return = 4% Variance = 0  Combine these two

41. Portfolio for $1 a = fraction in stock market

42. Risk Versus Return std(R) Expected Return 0 Risk Free 0.20 Market Borrowing a=1 a=0 0.10 Slope = (E(Rm)-RF)/std(Rm)

43. Returns and Borrowing  By borrowing more (leverage) can increase returns  Also, increase risk  It is easy to be on the line (if you have enough credit)  Simply reporting returns alone is never enough  Would a return of 20% per year be amazing?

44. Sharpe Ratio Again Not affected by leverage

45. CAPM: Two views  Simple risk measure, Beta  Perfect CAPM world Market equilibrium linking beta and expected returns

46. Perfect CAPM World  Beta (and Beta alone) is risk measure  Everyone holds market portfolio and some amount of risk free  Individual stock returns and Beta are linearly related

47. Risk Versus Return Beta Expected Return 0 Risk Free 1 Market * * * * * * * Slope = (Rm-Rf)

48. CAPM Equation  Required (expected) return and beta Stock j Rm = market return Security market line  Required (expected) return from CAPM

49. Risk Versus Return What if this didn’t hold? Beta Expected Return 0 Risk Free 1 Market * * * * * * * Slope = (Rm-Rf) * * Market Stock X

50. Beta Examples (2007) Amazon.com1.44 Ebay1.48 Disney1.08 General Motors1.14 Nike0.57 PepsiCo0.61

51. CAPM Calculations E(Rm) = 8%, Rf = 2%  Amazon (beta = 1.4) Required return = 0.02 + 1.4(0.08-0.02) Required return = 0.104 = 10.4%

52. Common Notation  e(j) = noise (mean zero)  beta(j)(Rm-Rf) (CAPM required return)  alpha(j) extra beyond CAPM  “Chasing alpha”

53. Goals  Portfolios and correlations  Diversifiable versus nondiversifiable risk  CAPM and Beta Capital asset pricing model  Is the CAPM really useful?

54. How well does the CAPM work?  Results: Fama and French Malkiel  Construct portfolios of stocks  Estimate betas  Plot beta versus expected return  No relationship

55. Malkiel’s Mutual Funds Quarterly Returns 1981-91(page 234) Beta

56. Is Beta Dead?  Older research showed a weak relationship between beta and expected return  Recent evidence shows that there is probably no relationship  Premier model of asset pricing  Should or do we still care?

57. Reasons to Still think about Beta  Diversification and portfolio theory is still important Beta is informative about how a security moves with the market  If the CAPM is not working, should try to “beat it” Load up on low beta stocks Should be lower risk, and higher return

58. Problems with CAPM  Beta is very unstable over time Hard to estimate  Market inefficiency  Diversification  Attitudes toward risk  Important side message Look at other stuff

59. Goals  Portfolios and correlations  Diversifiable versus nondiversifiable risk  CAPM and Beta Capital asset pricing model  Is the CAPM really useful?