In mathematical finance, the intertemporal capital asset pricing model, or ICAPM, created by Robert C. Merton, is an alternative to the capital asset pricing model (CAPM). 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 Merton considers a continuous time market in equilibrium. The state variable (X) follows a 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}} be the weight invested in the asset i. Then:
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}} is the return on asset i. The change in wealth is:
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^{*}} is a value between t and t+dt. 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:
… excerpt ends here. Continue reading the full article.
