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.
Merton[1] considers a continuous time market in equilibrium.
The change in wealth is: We can use dynamic programming to solve the problem.
For instance, if we consider a series of discrete time problems: Then, a Taylor expansion gives: where
Assuming that returns follow a Brownian motion: with: Then canceling out terms of second and higher order: Using Bellman equation, 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[2] , we have the following objective function: where