1

Today

Risk and Return

• Risk and Return • Portfolio Theory • Capital Asset Pricing Model

Reading

• Brealey, Myers, and Allen, Chapters 7 and 8

2 Measuring Risk •

Variance

- Average value of squared deviations from mean. A measure of volatility.

•

Standard Deviation

- Average value of squared deviations from mean. A measure of volatility.

• Variance measures ‘ Total Risk ’

3 Measuring Risk • •

Unique Risk

- Risk factors affecting only that firm. Also called “ diversifiable risk.

”

Market Risk

- Economy-wide sources of risk that affect the overall stock market. Also called “ systematic risk.

” •

Diversification

reduce risk - Strategy designed to by spreading the portfolio across many investments.

4 Measuring Risk

0 Unique risk Market risk 5 10 Number of Securities 15

5 Portfolio Risk The variance of a two stock portfolio is the sum of these four boxes Stock 1 Stock 2 Stock 1 x 2 1 σ 1 2 x 1 x 2 σ 12  x 1 x 2 ρ 12 σ 1 σ 2 x 1 x 2 σ Stock 2  12 x 1 x 2 ρ 12 σ 1 σ 2 x 2 2 σ 2 2

6 Portfolio Risk

Example

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The expected return on your portfolio is:

7 Portfolio Risk

Example

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%.

The standard deviation of their annualized daily returns are 18.2% and 27.3%, respectively. Assume a correlation coefficient of 1.0 and calculate the portfolio variance.

Exxon Coca Mobil Cola Exxon Mobil x 1 2 σ 1 2  (.

60 ) 2  ( 18 .

2 ) 2 x 1 x 2 ρ 12 σ 1 σ 2  1  18 .

2  .

40  27 .

3  .

60 x 1 x Coca 2 ρ 12 σ 1 σ 2 Cola  .

40  .

60  1  18 .

2  27 .

3 x 2 2 σ 2 2  (.

40 ) 2  ( 27 .

3 ) 2

8 Portfolio Risk

Example

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The standard deviation of their annualized daily returns are 18.2% and 27.3%, respectively. Assume a correlation coefficient of 1.0 and calculate the portfolio variance.

9 Portfolio Return and Risk Expected Portfolio Return  (x 1 r 1 )  ( x 2 r 2 ) Portfolio Variance  x 1 2 σ 1 2  x 2 2 σ 2 2  2 ( x 1 x 2 ρ 12 σ 1 σ 2 )

10 Portfolio Risk

The shaded boxes contain variance terms; the remainder contain covariance terms.

STOCK 1 2 3 4 5 6 To calculate portfolio variance add up the boxes N 1 2 3 4 5 6 STOCK N

11 Beta and Unique Risk

1. Total risk = diversifiable risk + market risk 2. Market risk is measured by beta, the sensitivity to market changes Expected stock return 10% -10% + 10% Copyright 1996 by The McGraw-Hill Companies, Inc beta Expected market return

12 Beta and Unique Risk Market Portfolio - Portfolio of all assets in the economy. In practice a broad stock market index, such as the S&P Composite, is used to represent the market.

Beta - Sensitivity of a stock ’ s return to the return on the market portfolio.

13 Beta and Unique Risk 

i

  

im

2

m

Covariance with the market Variance of the market

14 Markowitz Portfolio Theory • Combining stocks into portfolios can reduce standard deviation, below the level obtained from a simple weighted average calculation.

• Correlation coefficients make this possible.

• The various weighted combinations of stocks that create this standard deviations constitute the set of

efficient portfolios

.

15 Markowitz Portfolio Theory Price changes vs. Normal distribution 0.14

0.12

0.1

0.08

0.06

0.04

0.02

0 -9

Coca Cola - Daily % change 1987-2004

-7 -5 -3 -1 0 2 4 6 7 Daily % Change

16 Markowitz Portfolio Theory Standard Deviation VS. Expected Return

20 18 16 14 12 10 8 6 4 2 0 -50

Investment A

0 50

% return

17 Markowitz Portfolio Theory Standard Deviation VS. Expected Return

20 18 16 14 12 10 8 6 4 2 0 -50

Investment B

0 50

% return

18 Markowitz Portfolio Theory  Expected Returns and Standard Deviations vary given different weighted combinations of the stocks Expected Return (%) Coca Cola 40% in Coca Cola Exxon Mobil Standard Deviation

19 Efficient Frontier •Each half egg shell represents the possible weighted combinations for two stocks.

•The composite of all stock sets constitutes the efficient frontier Expected Return (%) Standard Deviation

20 Efficient Frontier •Lending or Borrowing at the risk free rate ( r f ) allows us to exist outside the efficient frontier.

Expected Return (%) T r f S Standard Deviation

21 Efficient Frontier Example Stocks ABC Corp Big Corp  28 42 Correlation Coefficient = .4

% of Portfolio 60% 40% Avg Return 15% 21% Standard Deviation = weighted avg =

Standard Deviation = Portfolio = Return = weighted avg = Portfolio =

22 Efficient Frontier Example Stocks ABC Corp Big Corp  28 42 Correlation Coefficient = .4

% of Portfolio 60% 40% Avg Return 15% 21% Standard Deviation = weighted avg = Standard Deviation = Portfolio = Return = weighted avg = Portfolio = Let ’ s Add stock New Corp to the portfolio

23 Efficient Frontier Example Stocks Portfolio New Corp  28.1

30 Correlation Coefficient = .3

% of Portfolio 50% 50% Avg Return 17.4% 19% NEW Standard Deviation = weighted avg = NEW Standard Deviation = Portfolio = NEW Return = weighted avg = Portfolio = NOTE: Higher return & Lower risk How did we do that?

DIVERSIFICATION

24 Efficient Frontier

Return A B Risk (measured as



)

25 Efficient Frontier

Return AB A B Risk

26 Efficient Frontier

Return AB A N B Risk

27 Efficient Frontier

Return ABN AB A N B Risk

28 Efficient Frontier

Return Goal is to move up and left.

WHY?

ABN AB A N B Risk

29 Efficient Frontier

Return Low Risk High Return High Risk High Return Low Risk Low Return High Risk Low Return Risk

30 Efficient Frontier

Return Low Risk High Return High Risk High Return Low Risk Low Return High Risk Low Return Risk

31 Efficient Frontier

Return ABN AB A N B Risk

32 Security Market Line

Return Market Return = r m Risk Free Return = r f .

Efficient Portfolio Risk

33 Security Market Line

Return Market Return = r m Risk Free Return = r f .

1.0

Efficient Portfolio BETA

34 Security Market Line

Return Risk Free Return = r f .

Security Market Line (SML) BETA

35 Security Market Line

Return SML r f BETA 1.0

SML Equation = r f + β ( r m - r f )

36 Capital Asset Pricing Model

R = r

f

+

β

( r

m

- r

f

)

CAPM

37 Testing the CAPM Beta vs. Average Risk Premium

Avg Risk Premium 1931-2002

30

SML

Investors 20 10 0 Market Portfolio

Portfolio Beta 1.0

38 Testing the CAPM Beta vs. Average Risk Premium

Avg Risk Premium 1931-65 SML

30 20 Investors 10 0 Market Portfolio

Portfolio Beta 1.0

39 Testing the CAPM Beta vs. Average Risk Premium

Avg Risk Premium 1966-2002

30 20 Investors 10 0

SML 1.0

Market Portfolio

Portfolio Beta

40 Arbitrage Pricing Theory

Alternative to CAPM

Expected Risk Premium = r - r f = B factor1 (r factor1 - r f ) + B f2 (r f2 - r f ) + … Return = a + b factor1 (r factor1 ) + b f2 (r f2 ) + …

41 Arbitrage Pricing Theory

Estimated risk premiums for taking on risk factors (1978-1990)

Factor Yield spread Interest rate Exchange rate Real GNP Inflation Market Estimated Risk Premium (r factor  5.10%

r f

) .61

.59

.49

.83

6.36