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 forecast changes in the distribution of future
returns or
income
Income is the consumption and saving opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms. Income is difficult to define conceptually and the definition may be different across fields. Fo ...
.
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
An investor is a person who allocates financial capital with the expectation of a future return (profit) or to gain an advantage (interest). Through this allocated capital most of the time the investor purchases some species of property. Type ...
hedge against shortfalls in consumption or against changes in the future
investment opportunity set.
Continuous time version
Merton considers a continuous time market in equilibrium.
The state variable (X) follows a
brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
:
:
The investor maximizes his
Von Neumann–Morgenstern utility:
:
where T is the time horizon and B
(T),Tthe utility from wealth (W).
The investor has the following constraint on wealth (W).
Let
be the weight invested in the asset i. Then:
:
where
is the return on asset i.
The change in wealth is:
:
We can use
dynamic programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. ...
to solve the problem. For instance, if we consider a series of discrete time problems:
:
Then, a
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
gives:
:
where
is a value between t and t+dt.
Assuming that returns follow a
brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
:
:
with:
:
Then canceling out terms of second and higher order:
:
Using
Bellman equation
A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. It writes the "value" of a decision problem at a certain point in time ...
, we can restate the problem:
:
subject to the wealth constraint previously stated.
Using
Ito's lemma we can rewrite:
:
and the expected value:
:
After some algebra
[:
:
:]
, we have the following objective function:
:
where
is the risk-free return.
First order conditions are:
:
In matrix form, we have:
:
where
is the vector of expected returns,
the
covariance matrix of returns,
a unity vector
the covariance between returns and the state variable. The optimal weights are:
:
Notice that the intertemporal model provides the same weights of the
CAPM. Expected returns can be expressed as follows:
:
where m is the market portfolio and h a portfolio to hedge the state variable.
See also
*
Intertemporal portfolio choice Intertemporal portfolio choice is the process of allocating one's investable wealth to various assets, especially financial assets, repeatedly over time, in such a way as to optimize some criterion. The set of asset proportions at any time defines ...
References
{{Reflist
* 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
Mathematical finance
Finance theories
Financial economics
Financial models