Risk, Return, Portfolio Theory and CAPM

Risk, Return, Portfolio Theory and CAPM

TIP If you do not understand

anything,

ask me!

Where does the discount rate (for stock valuation) come from?

2

So far,

❚ We have taken the discount rate as given.❚ We have also learned the factors that

determine the discount rate in bond valuation.

❚ What is the appropriate discount rate in stock valuation?

3

Topics

❚ Risk and returns❙ 75 Years of Capital Market History❙ How to measuring risk

❘ Individual security risk❘ Portfolio risk

❙ Diversification ❘ Unique risk ❘ Systematic risk or market risk

❙ Measure market risk: beta❙ CAPM

4

The Value of an Investment of $1 in 1926

Source: Ibbotson Associates

0.1

10

1000

1925 1940 1955 1970 1985 2000

S&PSmall CapCorp BondsLong BondT Bill

Inde

x

Year End

1

6402

2587

64.1

48.9

16.6

5

0.1

10

1000

1925 1940 1955 1970 1985 2000

S&PSmall CapCorp BondsLong BondT Bill


Inde

x

Year End

1

660

267

6.6

5.0

1.7

Real returnsThe Value of an Investment of $1 in 1926

6

Rates of Return 1926-2000


-60

-40

-20

0

20

40

60

26 30 35 40 45 50 55 60 65 70 75 80 85 90 9520

00

Common Stocks

Long T-Bonds

T-Bills

Year

Per

cent

age

Ret

urn

7

Selected Realized Returns, 1926 – 2001

Average Standard Return Deviation

Small-company stocks 17.3% 33.2%Large-company stocks 12.7 20.2L-T corporate bonds 6.1 8.6L-T government bonds 5.7 9.4U.S. Treasury bills 3.9 3.2

Source: Ibbotson Associates.

8

Risk premium

❚ Risk aversion – in Finance we assume investors dislike risk, so when they invest in risky securities, they require a higher expected rate of return to encourage them to bear the risk.

❚ The risk premium is the difference between the expected rate of return on a risky security and the expected rate of return on a risk-free security, e.g., T-bills.

❚ Over the last century, the average risk premium is about 7% for stocks.

9

Measuring Risks

❚ How to measure the risk of a security? ❙ Stand-alone risk: when the return is analyzed

in isolation. This provides a starting point.

❙ Portfolio risk: when the return is analyzed in a portfolio. This is what matters in reality when people hold portfolios.

10

PART I: Standard alone risk

❚ The risk an investor would face if s/he held only one asset.

❚ Investment risk is related to the probability of earning a low or negative actual return.

❚ The greater the chance of lower than expected or negative returns, the riskier the investment.

❚ The greater the range of possible events that can occur, the greater the risk

11

Probability distributions Which firm is more likely to have a return closer to its expected

value?

❚ A listing of all possible outcomes, and the probability of each occurrence.

❚ Can be shown graphically.

Expected Rate of Return

Rate ofReturn (%)100150-70

Firm X

Firm Y

12

Measuring Risk

In financial markets, we use the volatility of a security return to measure its risk.

Variance – Weighted average value of squared deviations from mean.

Standard Deviation – Squared root of variance.

13

Some basic concepts

❚ Some basic formula for Expectation and Variance

❚ Let X be a return of a security in the next period. Then we have

∑=

==N

iiXipXEX

1)()(][ ∑

=−=

N

iXiXipXVar

1

2))()((][

14

Investment alternatives

43.0%30.0%-20.0%50.0%8.0%0.1Boom

29.0%45.0%-10.0%35.0%8.0%0.2Above avg

15.0%7.0%0.0%20.0%8.0%0.4Average

1.0%-10.0%14.7%-2.0%8.0%0.2Below avg

-13.0%10.0%28.0%-22.0%8.0%0.1Recession

MPUSRCollHTT-BillProb.Economy

15

Return: Calculating the expected return for each alternative

17.4% (0.1) (50%) (0.2) (35%) (0.4) (20%)

(0.2) (-2%) (0.1) (-22.%) k

P k k

return of rate expected k

HT

^

n

1iii

^

^

=++++=

=

=

∑=

16

Risk: Calculating the standard deviation for each alternative

deviation Standard=σ

2Variance σ==σ

i2

n

1ii P)k̂k(∑

=−=σ

17

Standard deviation calculation

15.3% 18.8% 20.0% 13.4% 0.0%

(0.1)8.0) - (8.0 (0.2)8.0) - (8.0 (0.4)8.0) - (8.0

(0.2)8.0) - (8.0 (0.1)8.0) - (8.0

P )k (k

M

USRHT

CollbillsT

2

22

22

billsT

n

1ii

2^

i

=====

+++

+=

−=

−

−

=∑

σσσσσ

σ

σ

21

18

Comparing risk and return

15.3%15.0%Market

18.8%13.8%USR

13.4%1.7%Coll

20.0%17.4%HT

0.0%8.0%T-bills

Risk, σExpected returnSecurity

19

Comments on standard deviation as a measure of risk

❚ Standard deviation (σi) measures “total”, or stand-alone, risk.

❚ The larger the σi , the lower the probability that actual returns will be closer to expected returns, that is : the larger the stand-alone risk.

20

PART II: Risk in a portfolio context

❚ Portfolio risk is more important because in reality no one holds just one single asset.

❚ The risk & return of an individual security should be analyzed in terms of how this asset contributes the risk and return of the whole portfolio being held.

21

Portfolio

❚ A portfolio is a set of securities and can be regarded as a security.

❚ If you invest W dollars in a portfolio of n securities, let Wi be the money invested in security i, then the portfolio weight on stock i is

, with property W

Wx ii = 1

1=∑

=

n

iix

22

Example 1

❚ Suppose that you want to invest $1,000 in a portfolio of IBM and GE. You spend $200 on IBM and the other $800 on GE. What is the portfolio weight on each stock?

8.01000/800

2.01000/200

====

GE

IBM

x

x

23

Portfolio return and risk of two stocks

)rx()r(x Return Portfolio Expected 2211 +=

)σσρxx(2σxσxVariance Portfolio 21122122

22

21

21 ++=

24

In a portfolio…

❚ In English, we say:❚ The expected return = weighted

average of each stock’s expected return.

❚ But the portfolio standard deviation is <= the weighted average of each stock’s standard deviation.

❚ I Mathematics, we say:

25

(Not required) Suppose a portfolio is made up of x1 shares of stock 1 and x2 shares of stock 2.

2 2 21 1 2 2 1 1 2 2 1 1 2 2

1 2 1 2

2 2 2 21 1 2 2 1 2 1 2

21 1 2 2

1 1 2 2 1 1 2 2

( ) ( ) ( ) ( )

2 ( ) ( )

2

( ) 1 1

( )

'

x r x r Var x r x r x Var r x Var r

x x Var r Var r

x x x x

x x because

Thus x r x r x x

We say that a portfolio s standard deviation is a convex funct

σ

ρ

σ σ ρ σ σσ σ ρσ σ σ

+ ≡ + = +

+

= + +

≤ + − ≤ ≤+ ≤ +

' .

ion

of its components standard deviations

1 1 2 2 1 1 2 2( ) ( ) ( )E x r x r x E r x E r+ = +

26

Suppose you invest 50% in HT and 50% in Coll. What are the expected returns and standard deviation for the 2-stock portfolio?

13.4%1.7%Coll*

20.0%17.4%HT

Risk, σExpected returnSecurity

-20.0%50.0%0.1Boom

-10.0%35.0%0.2Above avg

0.0%20.0%0.4Average

14.7%-2.0%0.2Below avg

28.0%-22.0%0.1Recession

CollHTProb.Economy

27

Calculating portfolio expected return

9.6% (1.7%) 0.5 (17.4%) 0.5 k

kw k

:average weighted a is k

p

^

n

1i

i

^

ip

^

p

^

=+=

= ∑=

28

An alternative method for determining portfolio expected return

15.0%15.0%-20.0%50.0%0.1Boom12.5%12.5%-10.0%35.0%0.2Above avg10.0%10.0%0.0%20.0%0.4Average6.4%6.4%14.7%-2.0%0.2Below avg3.0%3.0%28.0%-22.0%0.1RecessionPort.Port.CollHTProb.Economy

9.6% (15.0%) 0.10 (12.5%) 0.20 (10.0%) 0.40 (6.4%) 0.20 (3.0%) 0.10 kp

^

=++++=

29

Calculating portfolio standard deviation

3.3%

9.6) - (15.0 0.10

9.6) - (12.5 0.20

9.6) - (10.0 0.40

9.6) - (6.4 0.20

9.6) - (3.0 0.10

21

2

2

2

2

2

p =

++++

=σ

30

Comments on portfolio risk measures

❚ σp = 3.3% is lower than the weighted average of HT and Coll.’s σ (16.7%). This is true so long as the two stocks’ returns are not perfectly positively correlated.

❚ Perfect correlation means the returns of two stocks will move exactly in same rhythm.

❚ Portfolio provides average return of component stocks, but lower than average risk.

❚ The more negatively correlated the two stocks, the more dramatic reduction in portfolio standard deviation.

31

Returns distribution for two perfectly negatively correlated stocks (ρ = -1.0)

-10

15 15

25 2525

15

0

-10

Stock W

0

Stock M

-10

0

Portfolio WM

32

Returns distribution for two perfectly positively correlated stocks (ρ = 1.0)

Stock M

0

15

25

-10

Stock M’

0

15

25

-10

Portfolio MM’

0

15

25

-10

33

General comments about risk

❚ Most stocks are positively correlated with the market (ρk,m ≈ 0.65).

❚ σ ≈ 35% for an average stock.❚ Combining stocks in a portfolio

generally lowers risk.

34

Total Risk

❚ A stock’s realized return is often different from its expected return.

❚ Total return= expected return + unexpected return

❚ Unexpected return=systematic portion + unsystematic portion

❚ Thus: Total return= expected return +systematic portion + unsystematic portion

❚ Total risk (stand-alone risk)= systematic portion + unsystematic portion

35

Systematic Risk


❚ The systematic portion will be affected by factors such as changes in GDP, inflation, interest rates, etc.

❚ This portion is not diversifiable because the factor will affect all stocks in the market.

❚ Such risk factors affect a large number of stocks. Also called Market risk, non-diversifiable risk, beta risk.

36

Unsystematic Risk


❚ This portion is affected by factors such as labor strikes, part shortages, etc, that will only affect a specific firm, or a small number of firms.

❚ Also called diversifiable risk, firm specific risk.

37

Diversification

❚ Portfolio diversification is the investment in several different classes or sectors of stocks.

❚ Diversification is not just holding a lot of stocks.

❚ For example, if you hold 50 internet stocks, you are not well diversified.

38

Creating a portfolio:Beginning with one stock and adding randomly selected stocks to portfolio

❚ σp decreases in general as stocks added. ❚ Expected return of the portfolio would

remain relatively constant.❚ Diversification can substantially reduce the

variability of returns with out an equivalent reduction in expected returns.

❚ Eventually the diversification benefits of adding more stocks dissipates (after about 10 stocks), and for large stock portfolios, σp tends to converge to ≈ 20%.

39

Illustrating diversification effects of a stock portfolio

# Stocks in Portfolio10 20 30 40 2,000+

Company-Specific Risk

Market Risk

20

0

Stand-Alone Risk, σp

σp (%)35

40

Breaking down sources of total riskTotal risk= systematic portion (market risk) + unsystematic

portion (firm-specific risk)❚ Market risk (systematic risk, non-diversifiable risk, beta risk) –

portion of a security’s stand-alone risk that cannot be eliminated through diversification. It is affected by economy-wide sources of risk that affect the overall stock market.

❚ Firm-specific risk (unsystematic risk, diversifiable risk, idiosyncratic risk) – portion of a security’s stand-alone risk that can be eliminated through proper diversification.

❚ If a portfolio is well diversified, unsystematic is very small. Rational, risk-averse investors are just concerned with portfolio standard deviation σp, which is based upon market risk. That is: investor care little about a stock’s firm–specific risk.

41

How do we measure a tock’s systematic (market) risk? Beta

❚ Measures a stock’s market risk, and shows a stock’s volatility relative to the market (i.e., degree of co-movement with the market return.)

❚ Indicates how risky a stock is if the stock is held in a well-diversified portfolio.

42

Calculating betas

❚ Run a regression of past returns of a security against past market returns.

❚ Market return is the return of market portfolio.

❚ 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.

❚ The slope of the regression line (called the security’s characteristic line) is defined as the beta for the security.

43

Illustrating the calculation of beta (security’s characteristic line)

.

.

.ki

_

kM

_-5 0 5 10 15 20

20

15

10

5

-5

-10

Regression line:

ki = -2.59 + 1.44 kM^ ^

Year kM ki

1 15% 18% 2 -5 -10

3 12 16

44

Security Character Line

❚ What does the slope of SCL mean?❚ Beta

❚ What variable is in the horizontal line?❚ Market return.

❚ The steeper the line, the more sensitive the stock’s return relative to the market return, that is, the greater the beta.

45

Comments on beta

❚ A stock with a Beta of 0 has no systematic risk❚ A stock with a Beta of 1 has systematic risk equal

to the “typical” stock in the marketplace❚ A stock with a Beta greater than 1 has systematic

risk greater than the “typical” stock in the marketplace

❚ A stock with a Beta less than 1 has systematic risk less than the “typical” stock in the marketplace

❚ The market return has a beta=1 (why?).❚ Most stocks have betas in the range of 0.5 to 1.5.

46

Can the beta of a security be negative?❚ Yes, if the correlation between Stock i and

the market is negative.❚ If the correlation is negative, the regression

line would slope downward, and the beta would be negative.

❚ However, a negative beta is very rare.❚ A stock with negative beta will give your

higher return in recession and hence is more valuable to investors, thus required rate of return is lower.

47

Beta coefficients for HT, Coll, and T-Bills

ki

_

kM

_

-20 0 20 40

40

20

-20

HT: β = 1.30

T-bills: β = 0

Coll: β = -0.87

48

Comparing expected return and beta coefficients

Security Exp. Ret. Beta HT 17.4% 1.30Market 15.0 1.00USR 13.8 0.89T-Bills 8.0 0.00Coll. 1.7 -0.87

Riskier securities have higher returns, so the rank order is OK.

49

Until now...

❚ We argued that well-diversified investors only cares about a stock’s systematic risk (measured by beta).

❚ The higher the systematic risk, the higher the rate of return investors will require to compensate them for bearing the risk.

❚ This extra return above risk free rate that investors require for bearing the non-diversifiable risk of a stock is called risk premium.

50

Beta and risk premium

❚ That is: the higher the systematic risk (measured by beta), the greater the reward (measured by risk premium).

❚ risk premium =expected return - risk free rate.❚ In equilibrium, all stocks must have the same

reward to systematic risk ratio.❚ For any stock i and stock m: ❚ (ri-rf)/beta(i)=(rm-rf)/beta(m)

51

The higher the beta, the higher the risk premium.

❚ The above equation should hold for any two securities. It should also hold if m stands for market portfolio.

❚ (ri –rf ) / (rM – rf)= βi /1 ❚ Thus, we have❚ ri = rf + (rM – rf) βi

❚ You’ve got CAPM!❚ Yes, this ri decides the discount rate in stock

valuation.

52

Capital Asset Pricing Model (CAPM)

❚ Model based upon concept that a stock’s required rate of return is equal to the risk-free rate of return plus a risk premium that reflects the riskiness of the stock after diversification.

53

Calculating required rates of return according to CAPM

ri = rf + (rM – rf) βi

❚ Assume rf = 8% and rM = 15%.❚ The market (or equity) risk premium is rM – rf

= 15% – 8% = 7%.❚ If a stock has a beta=1.5, how much is its

required rate of returns?

54

Risk-Free Rate

❚ Required rate of return for risk-less investments

❚ Typically measured by yield on 90 days U.S. Treasury Bills.

55

What is the market risk premium (rm-rf)?

❚ Additional return over the risk-free rate needed to compensate investors for assuming an average amount of systematic risk.

❚ Its size depends on the perceived risk of the stock market and investors’ degree of risk aversion.

❚ Historically between 4% and 8%.

56

Comparing expected return and beta coefficientsSecurity Exp. Ret. Beta HT 17.4% 1.30Market 15.0 1.00USR 13.8 0.89T-Bills 8.0 0.00Coll. 1.7 -0.87Assume kRF = 8% and kM = 15%.❚The market risk premium is kM – kRF = 15% – 8%❚Please find the require rates of return for each security.

❚(Sorry I use K and r interchangeably)

57

Calculating required rates of return

❚ kHT = 8.0% + (15.0% - 8.0%)(1.30)

= 8.0% + (7.0%)(1.30)= 8.0% + 9.1% = 17.10%

❚ kM = 8.0% + (7.0%)(1.00)= 15.00%

❚ kUSR = 8.0% + (7.0%)(0.89)= 14.23%

❚ kT-bill = 8.0% + (7.0%)(0.00)= 8.00%

❚ kColl = 8.0% + (7.0%)(-0.87)= 1.91%

58

Expected vs. Required returns

k) k( Overvalued 1.9 1.7 Coll.

k) k( uedFairly val 8.0 8.0 bills-T

k) k( Overvalued 14.2 13.8 USR

k) k( uedFairly val 15.0 15.0 Market

k) k( dUndervalue 17.1% 17.4% HT

k k

^

^

^

^

^

^

<

=

<

=

>

59

Expected v. required rate of return

❚ In short run, there might be mis-valued stocks and expected return may be different from the required return. In the long run and in an efficient market , expected returns = required returns.

❚ If many people believe that the a stock’s expected return is higher than required return (stock is undervalued), they would bid for that stock, pushing up the stock price, hence lowering the expected return, until market competition will lead to: expected returns = required returns.

60

Security Market Line (SML): ri = rf + (rM – rf) βi SML (redline) is a graphical representation of CAPM

..Coll.

.HT

T-bills

.USR

SML

kM = 15

kRF = 8

-1 0 1 2

.

SML: ki = 8% + (15% – 8%) βi

ki (%)

Risk, βi

61

Security Market Line

❚ What does the slope of SML mean?❚ Market risk premium= kM- kRF

❚ What variable is in the horizontal line?❚ Beta

❚ Which stock is over valued from previous graph?❚ Coll.

62

CAPM—Efficient frontier (won’t test)

❚ In addition to saying that ri = rf + (rM – rf)βi , CAPM also says each investor should hold a combination of the market portfolio and risk free security.

63


❚ If an investor has a preference that is decided by the expected return and the standard deviation of a portfolio, then he or she will choose a portfolio that has the highest expected return given the standard deviation of the portfolio.

❚ The set of these portfolios are called the efficient portfolio frontier.

64

CAPM—Efficient frontier (won’t test) Example with two stocks

Coca Cola

Reebok

Standard Deviation

Expected Return (%)

35% in Reebok

Expected Returns and Standard Deviations vary given different weighted combinations of the stocks

S

65

Efficient Frontier with n stocks (Won’t test)

Standard Deviation

Expected Return (%)A

S

66

Efficient Frontier with a risk-free security (won’t test)

Standard Deviation

Expected Return (%)

•Lending or Borrowing at the risk free rate (rf) will change the efficient frontier when there are only risky securities

rf

Lending

BorrowingT

S

A

67


❚ The red line from previous graph gives the efficient frontier if a risk free security is available.

❚ Each investor should hold a portfolio on the red line from previous graph.

❚ That is: each investor should invest in a S&P composite index and some T-bills (bonds). This perhaps is too strong a prediction of CAPM.

68

How to get portfolio beta: Equally-weighted two-stock portfolio

❚ Create a portfolio with 50% invested in HT and 50% invested in Collections.

❚ The beta of a portfolio is the weighted average of each of the stock’s betas.

βP = wHT βHT + wColl βColl βP = 0.5 (1.30) + 0.5 (-0.87)βP = 0.215

69

Calculating portfolio required returns

❚ The required return of a portfolio is the weighted average of each of the stock’s required returns.

kP = wHT kHT + wColl kColl kP = 0.5 (17.1%) + 0.5 (1.9%)kP = 9.5%

❚ Or, using the portfolio’s beta, CAPM can be used to solve for expected return.

kP = kRF + (kM – kRF) βP kP = 8.0% + (15.0% – 8.0%) (0.215)kP = 9.5%

70

Verifying the CAPM empirically

❚ The CAPM has not been verified completely.

❚ Statistical tests have problems that make verification almost impossible.

❚ Some argue that there are additional risk factors, other than the market risk premium, that must be considered.

71

More thoughts on the CAPM

❚ Investors seem to be concerned with both market risk and other risk factors. Therefore, the SML may not produce a correct estimate of ki.

ki = kRF + (kM – kRF) βi + ???

❚ CAPM/SML concepts are based upon expectations, but betas are calculated using historical data. A company’s historical data may not reflect investors’ expectations about future riskiness.

72

On the midterm

❚ Please prepare for the scantron, pencil, eraser, and calculator.

❚ There are 20 multiple choice problems to be finished in 1.25 hours.

Documents

Risk, Return, Portfolio Theory and CAPM