Diversification and Portfolios

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-gtPortfolio

• 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

• Dont 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
Malkiels 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 b
• Part that is unique to the firm e
• Rm is the return of the market
• Experiment
• Choose two stocks and betas
• 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 bv
• Security 2 x 0.1 -bv
• v is random
• Portfolio (1/2) each
• port 0.5(0.1bv) 0.5(0.1-bv)
• 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 1950s
• 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
Portfolio Risk (Variance or standard deviation)
Diversifiable Risk (unsystematic)
Nondiversifiable Risk (systematic)
Number of Securities
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/(WL)
• As you play more and more this gets very close to
  0.5

22
Diversification HistogramsDistribution 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 betas

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?Malkiels
experiment

• Assume the risk measure that people care about is
  related to the total (nondiversifiablediversifiab
  le) 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 Malkiels 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 BorrowLike 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. Whats going on?
• Taking on greater risk
• Buying on margin

38
Risk Versus ReturnBuilding your own Betas
Expected Return
Market
Borrowing
Risk Free
Beta
0
1
39
Risk Versus ReturnSlope (E(Rm)-Rf)/1
Expected Return
Market
E(Rm)-Rf
Risk Free
Beta
0
1
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
Slope (E(Rm)-RF)/std(Rm)
Expected Return
a1
0.10
Market
Borrowing
Risk Free
a0
std(R)
0
0.20
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 AgainNot 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
Expected Return




Slope (Rm-Rf)
Market



Risk Free
Beta
0
1
48
CAPM Equation

• Required (expected) return and beta
• Stock j
• Rm market return
• Security market line
• Required (expected) return from CAPM

49
Risk Versus ReturnWhat if this didnt hold?
Expected Return

Stock X




Slope (Rm-Rf)

Market

Market


Risk Free
Beta
0
1
50
Beta Examples (2007)
Amazon.com 1.44
Ebay 1.48
Disney 1.08
General Motors 1.14
Nike 0.57
PepsiCo 0.61
51
CAPM CalculationsE(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
Malkiels Mutual FundsQuarterly 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?