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# Factor Models: Announcements, Surprises, and Expected Returns

## 1.

11.1 Factor Models: Announcements, Surprises, andExpected Returns

11.2 Risk: Systematic and Unsystematic

11.3 Systematic Risk and Betas

11.4 Portfolios and Factor Models

11.5 Betas and Expected Returns

11.6 The Capital Asset Pricing Model and the Arbitrage

Pricing Theory

11.7 Parametric Approaches to Asset Pricing

11.8 Summary and Conclusions

## 2. 11.1 Factor Models: Announcements, Surprises, and Expected Returns

• The return on any security consists of two parts.1) the expected or normal return: the return that

shareholders in the market predict or expect

2) the unexpected or risky return: the portion that comes

from information that will be revealed .

Examples of relevant information:

– Statistics Canada figures (e.g., GNP)

– A sudden drop in interest rates

– News that the company’s sales figures are higher than

expected

## 3. 11.1 Factor Models: Announcements, Surprises, and Expected Returns

• A way to write the return on a stock in the comingmonth is:

R R U

where

R is the expected part of the return

U is the unexpected part of the return

## 4. 11.1 Factor Models: Announcements, Surprises, and Expected Returns

• Any announcement can be broken down into twoparts, the anticipated or expected part and the

surprise or innovation:

• Announcement = Expected part + Surprise.

• The expected part of any announcement is part of

the information the market uses to form the

expectation, R of the return on the stock.

• The surprise is the news that influences the

unanticipated return on the stock, U.

## 5. 11.2 Risk: Systematic and Unsystematic

• A systematic risk is any risk that affects a large numberof assets, each to a greater or lesser degree.

• An unsystematic risk is a risk that specifically affects a

single asset or small group of assets.

• Unsystematic risk can be diversified away.

• Examples of systematic risk include uncertainty about

general economic conditions, such as GNP, interest

rates, or inflation.

• On the other hand, announcements specific to a

company, such as a gold mining company striking gold,

are examples of unsystematic risk.

## 6. 11.2 Risk: Systematic and Unsystematic

We can break down the risk, U, of holding a stock into twocomponents: systematic risk and unsystematic risk:

Total risk; U

R R U

becomes

R R m ε

Nonsystematic Risk;

Systematic Risk; m

where

m is the systematic risk

ε is the unsystematic risk

n

## 7. 11.2 Risk: Systematic and Unsystematic

Systematic risk is referred to as market risk.m influences all assets in the market to some extent.

ε

Is specific to the company and unrelated to the

specific risk of most other companies.

Corr ( ε ε ) 0

i,

j

## 8. 11.3 Systematic Risk and Betas

• The beta coefficient, b, tells us the response of thestock’s return to a systematic risk.

• In the CAPM, b measured the responsiveness of a

security’s return to a specific risk factor, the return

on the market portfolio.

bi

Cov( Ri , RM )

( RM )

2

• We shall now consider many types of systematic risk.

## 9. 11.3 Systematic Risk and Betas

• For example, suppose we have identified threesystematic risks on which we want to focus:

1. Inflation

2. GDP growth

3. The dollar-pound spot exchange rate, S($,£)

• Our model is:

R R m ε

R R β I FI βGDP FGDP βS FS ε

β I is the inflation beta

βGDP is the GDP beta

βS is the spot exchange rate beta

ε is the unsystematic risk

## 10. Systematic Risk and Betas: Example

R R βI FI βGDP FGDP βS FS ε• Suppose we have made the following

estimates:

1. bI = -2.30

2. bGDP = 1.50

3. bS = 0.50.

• Finally, the firm

ε was

1% able to attract a

“superstar” CEO and this unanticipated

development contributes 1% to the return.

R R 2.30 FI 1.50 FGDP 0.50 FS 1%

## 11. Systematic Risk and Betas: Example

R R 2.30 FI 1.50 FGDP 0.50 FS 1%We must decide what surprises took place in the

systematic factors.

If it was the case that the inflation rate was expected

to be 3%, but in fact was 8% during the time

period, then

FI = Surprise in the inflation rate

= actual – expected

= 8% - 3%

= 5%

R R 2.30 5% 1.50 FGDP 0.50 FS 1%

## 12. Systematic Risk and Betas: Example

R R 2.30 5% 1.50 FGDP 0.50 FS 1%If it was the case that the rate of GDP growth

was expected to be 4%, but in fact was

1%, then

FGDP = Surprise in the rate of GDP growth

= actual – expected

= 1% - 4%

R R 2.30 5% 1.50 ( 3%) 0.50 FS 1%

= -3%

## 13. Systematic Risk and Betas: Example

R R 2.30 5% 1.50 ( 3%) 0.50 FS 1%If it was the case that dollar-pound spot exchange

rate, S($,£), was expected to increase by 10%,

but in fact remained stable during the time

period, then

FS = Surprise in the exchange rate

= actual – expected

= 0% - 10%

R= -10%

R 2.30 5% 1.50 ( 3%) 0.50 ( 10%) 1%

## 14. Systematic Risk and Betas: Example

R R 2.30 5% 1.50 ( 3%) 0.50 FS 1%Finally, if it was the case that the expected

return on the stock was 8%, then

R 8%

R 8% 2.30 5% 1.50 ( 3%) 0.50 ( 10%) 1%

R 12%

## 15. 11.4 Portfolios and Factor Models

• Now let us consider what happens to portfolios of stockswhen each of the stocks follows a one-factor model.

• We will create portfolios from a list of N stocks and will

capture the systematic risk with a 1-factor model.

• The ith stock in the list have returns:

Ri R i βi F εi

## 16. Relationship Between the Return on the Common Factor & Excess Return

Relationship Between the Return on theCommon Factor & Excess Return

Excess

return

i

Ri R i βi F εi

If we assume

that there is no

unsystematic

risk, then i = 0

The return on the factor F

## 17. Relationship Between the Return on the Common Factor & Excess Return

Relationship Between the Return on theCommon Factor & Excess Return

Excess

return

Ri R i βi F

If we assume

that there is no

unsystematic

risk, then i = 0

The return on the factor F

## 18. Relationship Between the Return on the Common Factor & Excess Return

Relationship Between the Return on theCommon Factor & Excess Return

Excess

return

β A 1.5 β B 1.0

βC 0.50

Different

securities will

have different

betas

The return on the factor F

## 19. Portfolios and Diversification

• We know that the portfolio return is the weightedaverage of the returns on the individual assets in the

portfolio:

RP X 1 R1 X 2 R2 X i Ri X N RN

Ri R i βi F εi

RP X 1 ( R1 β1 F ε1 ) X 2 ( R 2 β2 F ε2 )

X N ( R N βN F εN )

RP X 1 R1 X 1 β1 F X 1ε1 X 2 R 2 X 2 β2 F X 2 ε2

X N R N X N βN F X N εN

## 20. Portfolios and Diversification

The return on any portfolio is determined by threesets of parameters:

1. The weighed average of expected returns.

2. The weighted average of the betas times the factor.

3. The weighted average of the unsystematic risks.

R

X

R

X

R

X

R

1

2

N

P

1

2

N

( X 1 β1 X 2 β2 X N β N ) F

X 1ε1 X 2 ε2 X N ε N

In a large portfolio, the third row of this equation

disappears as the unsystematic risk is diversified away.

## 21. Portfolios and Diversification

So the return on a diversified portfolio isdetermined by two sets of parameters:

1. The weighed average of expected returns.

2. The weighted average of the betas times the

factor

RP F. X 1 R1 X 2 R 2 X N R N

( X 1 β1 X 2 β2 X N β N ) F

In a large portfolio, the only source of uncertainty is the

portfolio’s sensitivity to the factor.

## 22. 11.5 Betas and Expected Returns

RP X 1 R1 X N R N ( X 1 β1 X N β N ) FβP

RP

Recall that

and

R P X 1 R1 X N R N

β P X 1 β1 X N β N

The return on a diversified portfolio is the sum of the

expected return plus the sensitivity of the portfolio to the

factor.

RP R P βP F

## 23. Relationship Between b & Expected Return

Relationship Between b & Expected Return• The relevant risk in large and well-diversified

portfolios is all systematic, because unsystematic risk

is diversified away.

• If shareholders are ignoring unsystematic risk, only

the systematic risk of a stock can be related to its

expected return.

R R β F

P

P

P

## 24. Relationship Between b & Expected Return

Expected returnRelationship Between b & Expected Return

RF

SML

A

D

B

C

b

R RF β ( R P RF )

## 25. 11.6 The Capital Asset Pricing Model and the Arbitrage Pricing Theory

• APT applies to well diversified portfolios and notnecessarily to individual stocks.

• With APT it is possible for some individual stocks to

be mispriced---not lie on the SML.

• APT is more general in that it gets to an expected

return and beta relationship without the assumption

of the market portfolio.

• APT can be extended to multifactor models.

## 26. Multi-factor APT

R R ( R R ) β ( R R ) β ... ( R R ) βF

1

F

1

2

F

2

k

F

Example: A Canadian study (Otuteye, CIR 1991)

with five factors:

1. the rate of growth in industrial production

2. the changes in the slope of the term structure of

interest rates

3. the default risk premium for bonds

4. inflation

5. The value-weighted return on the market portfolio

(TSE 300)

k

## 27. 11.7 Empirical Approaches to Asset Pricing

• Both the CAPM and APT are risk-based models. Thereare alternatives.

• Empirical methods are based less on theory and more

on looking for some regularities in the historical record.

• Be aware that correlation does not imply causality.

• Related to empirical methods is the practice of

classifying portfolios by style e.g.,

– Value portfolio

– Growth portfolio

## 28. 11.8 Summary and Conclusions

• The APT assumes that stock returns aregenerated according to factor models such as:

R R βI FI βGDP FGDP βS FS ε

· As securities are added to the portfolio, the unsystematic

risks of the individual securities offset each other. A fully

diversified portfolio has no unsystematic risk.

· The CAPM can be viewed as a special case of the APT.

· Empirical models try to capture the relations between

returns and stock attributes that can be measured directly

from the data without appeal to theory.