8-1 CHAPTER 8 Risk and Rates of Return This chapter is relatively important.

8-1

CHAPTER 8Risk and Rates of Return

This chapter is relatively important.

8-2

Investment returns

The rate of return on an investment can be calculated as follows:

(Amount received – Amount invested)

Return = ________________________

Amount invested

For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is:

($1,100 - $1,000) / $1,000 = 10%.

8-3

What is investment risk? 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 The Chinese definition Two types of investment risk

Stand-alone risk (when the return is analyzed in isolation.)

Portfolio risk (when the return is analyzed in a portfolio.)

8-4

PART I: Standard alone risk

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

8-5

Probability distributions 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

8-6

Which firm is more likely to have a return

closer to its expected value?

Firm X? Firm Y?

8-7

Investor attitude towards risk Risk aversion – assumes investors dislike

risk and require higher rates of return to encourage them to hold riskier securities.

Who wants to be a millionaire?

Risk premium – the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities.

8-8

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: Based on Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2002 Yearbook (Chicago: Ibbotson Associates, 2002), 28.

8-9

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

8-10

Rates of Return 1926-2000

Source: Ibbotson Associates

-60

-40

-20

0

20

40

60 Common Stocks

Long T-Bonds

T-Bills

Year

Per

cent

age

Ret

urn

8-11

Suppose there are 5 possible outcomes over the investment horizon for the following securities:

Economy Prob. T-Bill HT Coll USR MP

Recession

0.1 5.5% -27.0%

27.0% 6.0% -17.0%

Below avg

0.2 5.5% -7.0% 13.0% -14.0%

-3.0%

Average 0.4 5.5% 15.0% 0.0% 3.0% 10.0%

Above avg

0.2 5.5% 30.0% -11.0%

41.0% 25.0%

Boom 0.1 5.5% 45.0% -21.0%

26.0% 38.0%

8-12

Why is the T-bill return independent of the economy?

T-bills will return the promised 5.5%, regardless of the economy.

T-bills are risk-free in the default sense of the word.

8-13

How do the returns of HT and Coll. behave in relation to the market?

HT – Moves with the economy, and has a positive correlation. This is typical.

Coll. – Is countercyclical with the economy, and has a negative correlation. This is unusual.

8-14

Calculating the expected return

12.4% (0.1) (45%)

(0.2) (30%) (0.4) (15%)

(0.2) (-7%) (0.1) (-27%) r

P r r

return of rate expected r

HT

^

N

1iii

^

^

8-15

Summary of expected returns

Expected returnHT 12.4%Market 10.5%USR 9.8%T-bill 5.5%Coll. 1.0%

HT has the highest expected return, and appears to be the best investment alternative, but is it really? Have we failed to account for risk?

8-16

Risk: Calculating standard deviation

deviation Standard

2Variance

i2

N

1ii P)r(rσ

ˆ

8-17

Standard deviation for each investment

15.2%

18.8% 20.0%

13.2% 0.0%

(0.1)5.5) - (5.5

(0.2)5.5) - (5.5 (0.4)5.5) - (5.5

(0.2)5.5) - (5.5 (0.1)5.5) - (5.5

P )r (r

M

USRHT

CollbillsT

2

22

22

billsT

N

1ii

2^

i

21

8-18

Comparing standard deviations

USR

Prob.T - bill

HT

0 5.5 9.8 12.4 Rate of Return (%)

8-19

Comments on standard deviation as a measure of risk

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

The larger σi is, the lower the probability that actual returns will be closer to expected returns.

Larger σi is associated with a wider probability distribution of returns.

8-20

Comparing risk and return

Security Expected return, r

Risk, σ

T-bills 5.5% 0.0%

HT 12.4% 20.0%

Coll* 1.0% 13.2%

USR* 9.8% 18.8%

Market 10.5% 15.2%* Seem out of place.

^

8-21

(Not required) Coefficient of Variation (CV)

A standardized measure of dispersion about the expected value, that shows the risk per unit of return.

^

k

Meandev Std

CV

8-22

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.

8-23

In a portfolio…

The expected return is the weighted average of each individual stock’s expected return.

However, the portfolio standard deviation is generally lower than the weighted average of each individual stock’s standard deviation.

8-24

Portfolio construction:Risk and return

Assume a two-stock portfolio is created with $50,000 invested equally in both HT and Collections. That is, you invest 50% in each. What are the expected returns and standard deviation for the portfolio?

A portfolio’s expected return is a weighted average of the returns of the portfolio’s component assets.

Standard deviation is a little more tricky and requires that a new probability distribution for the portfolio returns be devised.

8-25

Calculating portfolio expected return

6.7% (1.0%) 0.5 (12.4%) 0.5 r

rw r

:average weighted a is r

p

^

N

1i

i

^

ip

^

p

^

8-26

An alternative method for determining portfolio expected return

Economy Prob.

HT Coll Port.Port.

Recession

0.1 -27.0%

27.0% 0.0%0.0%

Below avg

0.2 -7.0% 13.0% 3.0%3.0%

Average 0.4 15.0% 0.0% 7.5%7.5%

Above avg

0.2 30.0% -11.0%

9.5%9.5%

Boom 0.1 45.0% -21.0%

12.0%12.0%6.7% (12.0%) 0.10 (9.5%) 0.20

(7.5%) 0.40 (3.0%) 0.20 (0.0%) 0.10 rp

^

8-27

Calculating portfolio standard deviation

12 2

2

2p

2

2

0.10 (0.0 - 6.7)

0.20 (3.0 - 6.7)

0.40 (7.5 - 6.7) 3.4%

0.20 (9.5 - 6.7)

0.10 (12.0 - 6.7)

8-28

Earlier info for the two stocks:

Security Expected return

Risk, σ

HT 12.4% 20.0%

Coll* 1.0% 13.2%

8-29

Comments on portfolio risk measures σp = 3.4% is much lower than the σi of either

stock (σHT = 20.0%; σColl. = 13.2%). This is not generally true.

σp = 3.4% is lower than the weighted average of HT and Coll.’s σ (16.6%). This is usually 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!

8-30

General comments about risk

Most stocks are positively (though not perfectly) correlated with the market.

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

generally lowers risk.

8-31

Returns distribution for two perfectly negatively correlated stocks

-10

15 15

25 2525

15

0

-10

Stock W

0

Stock M

-10

0

Portfolio WM

8-32

Returns distribution for two perfectly positively correlated stocks

Stock M

0

15

25

-10

Stock M’

0

15

25

-10

Portfolio MM’

0

15

25

-10

8-33

Returns A stock’s realized return is often different

from its expected return. Total return= expected return +

unexpected return

Unexpected return=systematic portion + unsystematic portion

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

8-34

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.

8-35

Unsystematic Risk

This unsystematic 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.

8-36

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.

8-37

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

σp decreases as stocks added, because stocks usually would not be perfectly correlated with the existing portfolio.

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%.

8-38

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, p

p (%)35

8-39

Breaking down sources of total risk (stand-alone risk)

Stand-alone risk = Market risk + Firm-specific risk

Market risk (systematic risk, non-diversifiable risk) – portion of a security’s stand-alone risk that cannot be eliminated through diversification.

Firm-specific risk (unsystematic risk, diversifiable 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.

8-40

Failure to diversify If an investor chooses to hold just one stock in

her/his portfolio (thus exposed to more risk than a diversified investor), should the investor be compensated for the firm-specific risk (earn higher returns)?

No.

An analogy, food diversification

Firm-specific risk is not important to a well-diversified investor, who only cares about the systematic risk.

8-41

So, Rational, risk-averse investors are

concerned with σp, which is based upon market risk.

No compensation should be earned for holding unnecessary, diversifiable risk.

Only systematic risk will be compensated.

8-42

How do we measure systematic risk? Beta

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

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

Measure of a firm’s market risk or the risk that remains after diversification

Beta will decide a stock’s required rate of return.

8-43

Calculating betas

Run a regression of past returns of a security against past market returns. (Market return is the weighted average of all stocks’ returns at a certain time.)

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

8-44

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

.

.

.ri

_

rM

_-5 0 5 10 15 20

20

15

10

5

-5

-10

Regression line:

ri = -2.59 + 1.44 rM^ ^

Year rM ri

1 15% 18%

2 -5 -10

3 12 16

8-45

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.

8-46

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 index has a beta=1. Most stocks have betas in the range of 0.5 to

1.5.

8-47

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 highly unlikely.

A stock that delivers higher return in recession is generally more valuable to investors, thus required rate of return is lower.

8-48

Beta coefficients for HT, Coll, and T-Bills

ri

_

kM

_

-20 0 20 40

40

20

-20

HT: b = 1.30

T-bills: b = 0

Coll: b = -0.87

8-49

Comparing expected returns and beta coefficients

Security Expected Return Beta HT 12.4% 1.32Market 10.5 1.00USR 9.8 0.88T-Bills 5.5 0.00Coll. 1.0 -0.87

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

8-50

Until now… We have argued that well-diversified

investors only cares about a stock’s systematic risk (measured by beta).

The higher the systematic risk (non-diversifiable 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.

8-51

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.

In expectation, (ri –rRF)/ βi = (rM – rRF)/ βm

8-52

The higher the beta, the higher the risk premium.

Market beta=1 (ri –rRF) / (rM – rRF)= βi /1 Thus, we have ri = rRF + (rM – rRF) βi

You’ve got CAPM!

8-53

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.

8-54

The Security Market Line (SML):Calculating required rates of return

SML: ri = rRF + (rM – rRF) βi SML is a graphical representation of

CAPM Assume rRF = 5.5% and rM = 10.5%. The market risk premium is rM – rRF =

10.5% – 5.5% = 5%. If a stock has a beta=1.5, how much is

its required rate of returns?

8-55

Risk-Free Rate

Required rate of return for risk-less investments

Typically measured by U.S. Treasury Bill Rate

8-56

What is the market risk premium?

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.

Varies from year to year, but most estimates suggest that it ranges between 4% and 8% per year.

8-57

Calculating required rates of return

rHT = 5.5% + (5.0%)(1.32)

= 5.5% + 6.6% = 12.10% rM = 5.5% + (5.0%)(1.00) = 10.50% rUSR = 5.5% + (5.0%)(0.88) = 9.90% rT-bill = 5.5% + (5.0%)(0.00) = 5.50% rColl = 5.5% + (5.0%)(-0.87)= 1.15%

8-58

Expected vs. Required returns

r) r( Overvalued 1.2 1.0 Coll.

r) r( uedFairly val 5.5 5.5 bills-T

r) r( Overvalued 9.9 9.8 USR

r) r( uedFairly val 10.5 10.5 Market

r) r( dUndervalue 12.1% 12.4% HT

r r

^

^

^

^

^

^

8-59

If market is fully efficient Then there are no under-valued or over- valued

stocks. And the expected returns should be equal to required returns.

If many people believe that the a stock’s expected return is higher than required return, they would bid for that stock, pushing up the stock price, hence lowering the expected return.

Market competition will lead to: expected returns = required returns.

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

8-60

Illustrating the Security Market Line

..Coll.

.HT

T-bills

.USR

SML

rM = 10.5

rRF = 5.5

-1 0 1 2

.

SML: ri = 5.5% + (5.0%) βi

ri (%)

Risk, βi

8-61

Security Market Line

What does the slope of SML mean? Market risk premium= rM – rRF

What variable is in the horizontal line?

Beta

8-62

An example: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 (1.32) + 0.5 (-0.87)

βP = 0.225

8-63

Calculating portfolio required returns

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

rP = wHT rHT + wColl rColl

rP = 0.5 (12.10%) + 0.5 (1.15%)

rP = 6.63% Or, using the portfolio’s beta, CAPM can be used

to solve for expected return.

rP = rRF + (rM – rRF) βP

rP = 5.5% + (5.0%) (0.225)

rP = 6.63%

8-64

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.

ki = kRF + (kM – kRF) βi + ???