Within mathematical finance , the intertemporal capital asset pricing model , or ICAPM , is an alternative to the CAPM provided by Robert Merton . It is a linear factor model with wealth as state variable that forecasts changes in the distribution of future returns or income .
In the ICAPM investors are solving lifetime consumption decisions when faced with more than one uncertainty. The main difference between ICAPM and standard CAPM is the additional state variables that acknowledge the fact that investors hedge against shortfalls in consumption or against changes in the future investment opportunity set.
Continuous time version
edit
Merton [ 1] Brownian motion :
d
X
=
μ
d
t
+
s
d
Z
{\displaystyle dX=\mu dt+sdZ}
The investor maximizes his Von Neumann–Morgenstern utility :
E
o
{
∫
o
T
U
[
C
(
t
)
,
t
]
d
t
+
B
[
W
(
T
)
,
T
]
}
{\displaystyle E_{o}\left\{\int _{o}^{T}U[C(t),t]dt+B[W(T),T]\right\}}
where T is the time horizon and B[W(T),T] the utility from wealth (W).
The investor has the following constraint on wealth (W).
Let
w
i
{\displaystyle w_{i}}
W
(
t
+
d
t
)
=
[
W
(
t
)
−
C
(
t
)
d
t
]
∑
i
=
0
n
w
i
[
1
+
r
i
(
t
+
d
t
)
]
{\displaystyle W(t+dt)=[W(t)-C(t)dt]\sum _{i=0}^{n}w_{i}[1+r_{i}(t+dt)]}
where
r
i
{\displaystyle r_{i}}
d
W
=
−
C
(
t
)
d
t
+
[
W
(
t
)
−
C
(
t
)
d
t
]
∑
w
i
(
t
)
r
i
(
t
+
d
t
)
{\displaystyle dW=-C(t)dt+[W(t)-C(t)dt]\sum w_{i}(t)r_{i}(t+dt)}
We can use dynamic programming to solve the problem. For instance, if we consider a series of discrete time problems:
max
E
0
{
∑
t
=
0
T
−
d
t
∫
t
t
+
d
t
U
[
C
(
s
)
,
s
]
d
s
+
B
[
W
(
T
)
,
T
]
}
{\displaystyle \max E_{0}\left\{\sum _{t=0}^{T-dt}\int _{t}^{t+dt}U[C(s),s]ds+B[W(T),T]\right\}}
Then, a Taylor expansion gives:
∫
t
t
+
d
t
U
[
C
(
s
)
,
s
]
d
s
=
U
[
C
(
t
)
,
t
]
d
t
+
1
2
U
t
[
C
(
t
∗
)
,
t
∗
]
d
t
2
≈
U
[
C
(
t
)
,
t
]
d
t
{\displaystyle \int _{t}^{t+dt}U[C(s),s]ds=U[C(t),t]dt+{\frac {1}{2}}U_{t}[C(t^{*}),t^{*}]dt^{2}\approx U[C(t),t]dt}
where
t
∗
{\displaystyle t^{*}}
Assuming that returns follow a Brownian motion :
r
i
(
t
+
d
t
)
=
α
i
d
t
+
σ
i
d
z
i
{\displaystyle r_{i}(t+dt)=\alpha _{i}dt+\sigma _{i}dz_{i}}
with:
E
(
r
i
)
=
α
i
d
t
;
E
(
r
i
2
)
=
v
a
r
(
r
i
)
=
σ
i
2
d
t
;
c
o
v
(
r
i
,
r
j
)
=
σ
i
j
d
t
{\displaystyle E(r_{i})=\alpha _{i}dt\quad ;\quad E(r_{i}^{2})=var(r_{i})=\sigma _{i}^{2}dt\quad ;\quad cov(r_{i},r_{j})=\sigma _{ij}dt}
Then canceling out terms of second and higher order:
d
W
≈
[
W
(
t
)
∑
w
i
α
i
−
C
(
t
)
]
d
t
+
W
(
t
)
∑
w
i
σ
i
d
z
i
{\displaystyle dW\approx [W(t)\sum w_{i}\alpha _{i}-C(t)]dt+W(t)\sum w_{i}\sigma _{i}dz_{i}}
Using Bellman equation , we can restate the problem:
J
(
W
,
X
,
t
)
=
m
a
x
E
t
{
∫
t
t
+
d
t
U
[
C
(
s
)
,
s
]
d
s
+
J
[
W
(
t
+
d
t
)
,
X
(
t
+
d
t
)
,
t
+
d
t
]
}
{\displaystyle J(W,X,t)=max\;E_{t}\left\{\int _{t}^{t+dt}U[C(s),s]ds+J[W(t+dt),X(t+dt),t+dt]\right\}}
subject to the wealth constraint previously stated.
Using Ito's lemma we can rewrite:
d
J
=
J
[
W
(
t
+
d
t
)
,
X
(
t
+
d
t
)
,
t
+
d
t
]
−
J
[
W
(
t
)
,
X
(
t
)
,
t
+
d
t
]
=
J
t
d
t
+
J
W
d
W
+
J
X
d
X
+
1
2
J
X
X
d
X
2
+
1
2
J
W
W
d
W
2
+
J
W
X
d
X
d
W
{\displaystyle dJ=J[W(t+dt),X(t+dt),t+dt]-J[W(t),X(t),t+dt]=J_{t}dt+J_{W}dW+J_{X}dX+{\frac {1}{2}}J_{XX}dX^{2}+{\frac {1}{2}}J_{WW}dW^{2}+J_{WX}dXdW}
and the expected value:
E
t
J
[
W
(
t
+
d
t
)
,
X
(
t
+
d
t
)
,
t
+
d
t
]
=
J
[
W
(
t
)
,
X
(
t
)
,
t
]
+
J
t
d
t
+
J
W
E
[
d
W
]
+
J
X
E
(
d
X
)
+
1
2
J
X
X
v
a
r
(
d
X
)
+
1
2
J
W
W
v
a
r
[
d
W
]
+
J
W
X
c
o
v
(
d
X
,
d
W
)
{\displaystyle E_{t}J[W(t+dt),X(t+dt),t+dt]=J[W(t),X(t),t]+J_{t}dt+J_{W}E[dW]+J_{X}E(dX)+{\frac {1}{2}}J_{XX}var(dX)+{\frac {1}{2}}J_{WW}var[dW]+J_{WX}cov(dX,dW)}
After some algebra[ 2]
m
a
x
{
U
(
C
,
t
)
+
J
t
+
J
W
W
[
∑
i
=
1
n
w
i
(
α
i
−
r
f
)
+
r
f
]
−
J
W
C
+
W
2
2
J
W
W
∑
i
=
1
n
∑
j
=
1
n
w
i
w
j
σ
i
j
+
J
X
μ
+
1
2
J
X
X
s
2
+
J
W
X
W
∑
i
=
1
n
w
i
σ
i
X
}
{\displaystyle max\left\{U(C,t)+J_{t}+J_{W}W[\sum _{i=1}^{n}w_{i}(\alpha _{i}-r_{f})+r_{f}]-J_{W}C+{\frac {W^{2}}{2}}J_{WW}\sum _{i=1}^{n}\sum _{j=1}^{n}w_{i}w_{j}\sigma _{ij}+J_{X}\mu +{\frac {1}{2}}J_{XX}s^{2}+J_{WX}W\sum _{i=1}^{n}w_{i}\sigma _{iX}\right\}}
where
r
f
{\displaystyle r_{f}}
J
W
(
α
i
−
r
f
)
+
J
W
W
W
∑
j
=
1
n
w
j
∗
σ
i
j
+
J
W
X
σ
i
X
=
0
i
=
1
,
2
,
…
,
n
{\displaystyle J_{W}(\alpha _{i}-r_{f})+J_{WW}W\sum _{j=1}^{n}w_{j}^{*}\sigma _{ij}+J_{WX}\sigma _{iX}=0\quad i=1,2,\ldots ,n}
In matrix form, we have:
(
α
−
r
f
1
)
=
−
J
W
W
J
W
Ω
w
∗
W
+
−
J
W
X
J
W
c
o
v
r
X
{\displaystyle (\alpha -r_{f}{\mathbf {1} })={\frac {-J_{WW}}{J_{W}}}\Omega w^{*}W+{\frac {-J_{WX}}{J_{W}}}cov_{rX}}
where
α
{\displaystyle \alpha }
Ω
{\displaystyle \Omega }
covariance matrix of returns,
1
{\displaystyle {\mathbf {1} }}
c
o
v
r
X
{\displaystyle cov_{rX}}
w
∗
=
−
J
W
J
W
W
W
Ω
−
1
(
α
−
r
f
1
)
−
J
W
X
J
W
W
W
Ω
−
1
c
o
v
r
X
{\displaystyle {\mathbf {w} ^{*}}={\frac {-J_{W}}{J_{WW}W}}\Omega ^{-1}(\alpha -r_{f}{\mathbf {1} })-{\frac {J_{WX}}{J_{WW}W}}\Omega ^{-1}cov_{rX}}
Notice that the intertemporal model provides the same weights of the CAPM . Expected returns can be expressed as follows:
α
i
=
r
f
+
β
i
m
(
α
m
−
r
f
)
+
β
i
h
(
α
h
−
r
f
)
{\displaystyle \alpha _{i}=r_{f}+\beta _{im}(\alpha _{m}-r_{f})+\beta _{ih}(\alpha _{h}-r_{f})}
where m is the market portfolio and h a portfolio to hedge the state variable.
^ Merton, Robert (1973). "An Intertemporal Capital Asset Pricing Model". Econometrica . 41 (5): 867–887. doi :10.2307/1913811 . JSTOR 1913811 . ^ :
E
(
d
W
)
=
−
C
(
t
)
d
t
+
W
(
t
)
∑
w
i
(
t
)
α
i
d
t
{\displaystyle E(dW)=-C(t)dt+W(t)\sum w_{i}(t)\alpha _{i}dt}
v
a
r
(
d
W
)
=
[
W
(
t
)
−
C
(
t
)
d
t
]
2
v
a
r
[
∑
w
i
(
t
)
r
i
(
t
+
d
t
)
]
=
W
(
t
)
2
∑
i
=
1
∑
i
=
1
w
i
w
j
σ
i
j
d
t
{\displaystyle var(dW)=[W(t)-C(t)dt]^{2}var[\sum w_{i}(t)r_{i}(t+dt)]=W(t)^{2}\sum _{i=1}\sum _{i=1}w_{i}w_{j}\sigma _{ij}dt}
∑
i
=
o
n
w
i
(
t
)
α
i
=
∑
i
=
1
n
w
i
(
t
)
[
α
i
−
r
f
]
+
r
f
{\displaystyle \sum _{i=o}^{n}w_{i}(t)\alpha _{i}=\sum _{i=1}^{n}w_{i}(t)[\alpha _{i}-r_{f}]+r_{f}}
Merton, R.C., (1973), An Intertemporal Capital Asset Pricing Model. Econometrica 41, Vol. 41, No. 5. (Sep., 1973), pp. 867–887
"Multifactor Portfolio Efficiency and Multifactor Asset Pricing" by Eugene F. Fama, (The Journal of Financial and Quantitative Analysis ), Vol. 31, No. 4, Dec., 1996 
![{\displaystyle E_{o}\left\{\int _{o}^{T}U[C(t),t]dt+B[W(T),T]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e613cbd824424c5d2306d6e668c4a3d1faeae1e)
![{\displaystyle W(t+dt)=[W(t)-C(t)dt]\sum _{i=0}^{n}w_{i}[1+r_{i}(t+dt)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b3b677781f1c972004dcd693cb5521ff476ee2)
![{\displaystyle dW=-C(t)dt+[W(t)-C(t)dt]\sum w_{i}(t)r_{i}(t+dt)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf62015ff2eb5196b6b00be413913348c5502b45)
![{\displaystyle \max E_{0}\left\{\sum _{t=0}^{T-dt}\int _{t}^{t+dt}U[C(s),s]ds+B[W(T),T]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a6900036ac4577e1595b01b4b11d92ef769c74a)
![{\displaystyle \int _{t}^{t+dt}U[C(s),s]ds=U[C(t),t]dt+{\frac {1}{2}}U_{t}[C(t^{*}),t^{*}]dt^{2}\approx U[C(t),t]dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb7e91e10b07dbc3e602f321f3157f287641939)
![{\displaystyle dW\approx [W(t)\sum w_{i}\alpha _{i}-C(t)]dt+W(t)\sum w_{i}\sigma _{i}dz_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6659388859678ed334ade93acbd9cddd97d9cf1f)
![{\displaystyle J(W,X,t)=max\;E_{t}\left\{\int _{t}^{t+dt}U[C(s),s]ds+J[W(t+dt),X(t+dt),t+dt]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/727aa63687a7522b063223cd684c70d0959c41b2)
![{\displaystyle dJ=J[W(t+dt),X(t+dt),t+dt]-J[W(t),X(t),t+dt]=J_{t}dt+J_{W}dW+J_{X}dX+{\frac {1}{2}}J_{XX}dX^{2}+{\frac {1}{2}}J_{WW}dW^{2}+J_{WX}dXdW}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71ba6ca6d619cbddcf66b8685301da655be0c3cb)
![{\displaystyle E_{t}J[W(t+dt),X(t+dt),t+dt]=J[W(t),X(t),t]+J_{t}dt+J_{W}E[dW]+J_{X}E(dX)+{\frac {1}{2}}J_{XX}var(dX)+{\frac {1}{2}}J_{WW}var[dW]+J_{WX}cov(dX,dW)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9effb1c061fd7713e0b61acd75bada19eb0e044)
![{\displaystyle max\left\{U(C,t)+J_{t}+J_{W}W[\sum _{i=1}^{n}w_{i}(\alpha _{i}-r_{f})+r_{f}]-J_{W}C+{\frac {W^{2}}{2}}J_{WW}\sum _{i=1}^{n}\sum _{j=1}^{n}w_{i}w_{j}\sigma _{ij}+J_{X}\mu +{\frac {1}{2}}J_{XX}s^{2}+J_{WX}W\sum _{i=1}^{n}w_{i}\sigma _{iX}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b0e5f5be78e2d9c0cca2925f708cd452b50f5a6)
![{\displaystyle var(dW)=[W(t)-C(t)dt]^{2}var[\sum w_{i}(t)r_{i}(t+dt)]=W(t)^{2}\sum _{i=1}\sum _{i=1}w_{i}w_{j}\sigma _{ij}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb00cbf418506504433388b8bde33c1bfede4590)
![{\displaystyle \sum _{i=o}^{n}w_{i}(t)\alpha _{i}=\sum _{i=1}^{n}w_{i}(t)[\alpha _{i}-r_{f}]+r_{f}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74ec60224f700a9b1da99b36bfaecb8611663781)