5 in the united states with the average stock having

stock having zero exposure to this factor
IQ,
r ² R ³
i F
β
mkt
RMRF ³ β
size
SMB ³ β
value
HML ³
β liq L
i i i i

Example: Fama–French Model
mkt size value
i
r ² R ³ β RMRF ³ β SMB ³ β HML
² 3% ³ 1.20(5%) ³ 0.12(2.2%) ³ 0.34(3.8%)
² 10.56%
i F i i
Positive size beta
• Small-cap stock because stock
has positive exposure to the size
premium.
• Negative beta would indicate a
larger than average stock
Positive value beta
• Value stock because stock has
positive exposure to the value
premium
• A negative beta would indicate a
growth stock
Examining the sign of the betas, helps
ascertain the stock’s characteristics
1b
|
Risk-free rate 3.0%
Equity risk premium 5.0%
Beta 1.20
Size premium 2.2%
Size beta 0.12
Value premium 3.8%
Value beta 0.34
Positive value beta
• Value stock because stock
has positive exposure
• to the value
premium
• A negative beta would indicate
a growth stock

ANOTHER Example:
Fama–French Model
Risk-free rate 3.0%
Equity risk premium 5.0%
Beta 1.20
Size premium 2.2%
Size beta 0.12
Value premium 3.8%
Value beta 0.34

4-FACTOR MODELS
Selecting the Portfolios
Fama-French-Carhart (FFC) Factor Specifications
1
1
[ ] ( [ ] ) [ ]
[ ] [ ]
² ³ ´ ³
³ ³
Mkt SMB
s f s Mkt f s SMB
HML PR YR
s HML s PR YR
E R r E R r E R
E R E R
b b
b b

EXAMPLE: 4-FACTOR MODELS
FFC Portfolio Average Monthly Returns, 1926–2008

Example (cont'd)

Multi-factor models (3/3) – Macro economic model (APT)
1b
Factor sensitivity is asset ’ s sensitivity to a particular factor (holding all other factors constant)
Factor risk premium is the factor ’ s expected return in excess of the risk-free rate
Arbitrage Pricing Theory (APT)
Given N factor portfolios with returns R
F1
, . . . , R
FN
, the expected
return of asset s is defined as:
|
ε F β F β F β Rf R
S S GNP GNP I I
³ ³ ³ ³ ²
β
1
…. β
N
are the factor betas.

Arbitrage Pricing Theory (APT)
For example, suppose we have identified three systematic
risks: Inflation, GNP growth, and the dollar-euro spot
exchange rate, S ($,€).
The model is:
beta rate exchange spot the is
beta GNP the is
beta inflation the is
S
GNP
I
S S GNP GNP I I
β
β
β
F β F β F β Rf R
ε m Rf R
³ ³ ³ ²
³ ³ ²

Arbitrage Pricing Theory (APT)
Suppose we have made the following estimates:
1. b
I
= -2.30
2. b
GNP
= 1.50
3. b
S
= 0.50
ε F β F β F β Rf R
S S GNP GNP I I
³ ³ ³ ³ ²
% 1 ² ε
S GNP I
F F F Rf R ¶ ³ ¶ ³ ¶ ´ ² 50 . 0 50 . 1 30 . 2

Arbitrage Pricing Theory (APT)
We must decide what surprises took place in the systematic
factors.
If it were the case that the inflation rate was expected to be 3%, but
in fact was 8% during the time period, then:
F
I
= Surprise in the inflation rate = actual – expected
= 8% – 3% = 5%
S GNP I
F F F Rf R ¶ ³ ¶ ³ ¶ ´ ² 50 . 0 50 . 1 30 . 2
S GNP
F F Rf R ¶ ³ ¶ ³ ¶ ´ ² 50 . 0 50 . 1 % 5 30 . 2

Arbitrage Pricing Theory (APT)
If it were the case that the rate of GNP growth was expected to be
4%, but in fact was 1%, then:
F
GNP
= Surprise in the rate of GNP growth
= actual – expected = 1% – 4%
= – 3%
S GNP
F F Rf R ¶ ³ ¶ ³ ¶ ´ ² 50 . 0 50 . 1 % 5 30 . 2
S
F R R ¶ ³ ´ ¶ ³ ¶ ´ ² 50 . 0 %) 3 ( 50 . 1 % 5 30 . 2