3. The theoretical motivation
We know that the first-order condition of any consumer-investor’s optimization problem can be written as:
E[M
t
R
i,t
V
t − 1
] = 1, Ö
i = 1 ······N 1
where M
t
is known as a stochastic discount factor or an intertemporal marginal rate of substitution; R
i,t
is the gross return of asset i at time t and V
t − 1
is market information known at time t − 1.
Without specifying the form of M
t
, Eq. 1 has little empirical content since it is easy to find some random variable M
t
for which the equation holds. Thus, it is the specific form of M
t
implied by an asset pricing model that gives Eq. 1 further empirical content Ferson, 1995. Since this paper focuses on the pricing of market,
interest rate and exchange rate risks on the commercial bank stock returns, it assumes that M
t
and R
i,t
have the following factor representations: M
t
= a + b
W
F
W,t
+ b
INT
F
INT,t
+ b
FX
F
FX,t
+ u
t
2 r
i,t
= a
i
+ b
iW
F
W,t
+ b
iINT
F
INT,t
+ b
iFX
F
FX,t
+ o
i,t
Ö i = 1 ······ N
3 where r
i,t
= R
i,t
− R
0,t
is the raw returns of asset i in excess of the risk-free rate, R
0,t
, at time t, E[u
t
F
k,t
V
t − 1
] = E[u
t
V
t − 1
] = E[o
i,t
F
k,t
V
t − 1
] = E[o
i,t
V
t − 1
] = 0Öi.k,F
k,t
k = W, INT, FX are three common risk factors world market, interest rate, and exchange rate which capture systematic risk affecting all assets r
i,t
including M
t
, b
ik
k = W, INT, FX are the associated time-invariant factor loadings which measure the sensitivities of the asset to the three common factors, while u
t
is an innovation and o
i,t
’s are idiosyncratic terms which reflect unsystematic risk.
2
The risk-free rate, R
0,t − 1
, must also satisfy Eq. 1 E[M
t
R
0,t − 1
V
t − 1
] = 1 4
Subtract Eq. 4 from Eq. 1, we obtain E[M
t
r
i,t
V
t − 1
] = 0 Ö
i = 1 ······ N 5
Apply the definition of covariance to Eq. 5, obtaining: E[r
i,t
V
t − 1
] = Co6r
i,t
, − M
t
V
t − 1
E[M
t
V
t − 1
] Ö
i = 1 ······ N 6
Substitute Eq. 2 into Eq. 6: E[r
i,t
V
t − 1
] =
k
− b
k
E[M
t
V
t − 1
] Co6r
i,t
,F
k,t
V
t − 1
=
k
d
k,t − 1
Co6r
i,t
,F
k,t
V
t − 1
Ö k = W, INT, FX
7
2
The empirical studies of Ferson and Harvey 1993 and Ferson and Korajczyk 1995 consistently show that movements in factor exposuresbetas account for only a small fraction of the predictable
change in expected returns, in both the domestic and the international context. Thus, to simplify the model, a time-invariant factor beta seems to be reasonable.
where d
k,t − 1
is the time-varying price of factor risk. Eq. 7 is the conditional three-factor asset pricing model derived from the
intertemporal consumption-investment optimization problem which will be esti- mated and tested via pricing kernel approach in Section 6.
Alternatively the conditional three-factor asset pricing model can also be derived based on arbitrage arguments, such as Arbitrage Pricing Theory APT by Ross
1976. Substituting the factor model Eq. 3 into the right hand side of Eq. 6 and assuming that Co6o
i,t
; M
t + 1
= 0 implies E[r
i,t
V
t − 1
] =
k
b
ik
Co6F
k,t
− M
t
V
t − 1
E[M
t
V
t − 1
]
n
=
k
b
ik
l
k,t − 1
Ö k
= W, INT, FX
8 where l
k,t − 1
is the time-varying risk premium per unit of beta risk. Assuming F
k,t
= E[F
k,t
V
t − 1
] + o
k,t
, where o
k,t
is the factor innovation with E[o
k,t
V
t − 1
] = 0, and E[o
k,t
o
j,t
V
t − 1
] Ö
k j, then Eq. 3 can be rewritten as: r
i,t
= a
i
+
k
b
ik
E[F
k,t
V
t − 1
] +
k
b
ik
o
k,t
+ o
i,t
Ö i = 1 ······ N; Ök = W, INT, FX
9 Taking conditional expectation on both sides of Eq. 9 and compare it with Eq.
8, then under the null hypothesis of a
i
= 0 obtaining:
E[F
k,t
V
t − 1
] = l
k,t − 1
Ö k = W, INT, FX
10 Substituting Eq. 10 into Eq. 9, and assuming a
i
= 0, Eq. 9 becomes
r
i,t
=
k
b
ik
l
k,t − 1
+ o
k,t
+ o
i,t
Ö i = 1 ······N; Ök = W, INT, FX
11 Eq. 11 is the conditional three-factor asset pricing model based on the arbitrage
arguments which will be estimated and tested using MGRACH-M methodology in Section 6.
To compare with previous studies in testing an unconditional multi-factor model, the unconditional version of Eq. 11 where expected factor risk premia, l
k,t − 1
’s are restricted to be time invariant will also be estimated and tested in Section 6.
4. Econometric methodologies
Kategori