Intertemporal CAPM

In mathematical finance, the intertemporal capital asset pricing model, or ICAPM, created by Robert C. Merton,[1] 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:

dX=μdt+sdZ

The investor maximizes his Von Neumann–Morgenstern utility:

Eo{∫oTU[C(t),t]dt+B[W(T),T]}

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 wi be the weight invested in the asset i. Then:

W(t+dt)=[W(t)−C(t)dt]∑i=0nwi[1+ri(t+dt)]

where ri is the return on asset i. The change in wealth is:

dW=−C(t)dt+[W(t)−C(t)dt]∑wi(t)ri(t+dt)

We can use dynamic programming to solve the problem. For instance, if we consider a series of discrete time problems:

max⁡E0{∑t=0T−dt∫tt+dtU[C(s),s]ds+B[W(T),T]}

Then, a Taylor expansion gives:

∫tt+dtU[C(s),s]ds=U[C(t),t]dt+12Ut[C(t*),t*]dt2≈U[C(t),t]dt

where t* is a value between t and t+dt.

Assuming that returns follow a Brownian motion:

ri(t+dt)=αidt+σidzi

with:

E(ri)=αidt;E(ri2)=var(ri)=σi2dt;cov(ri,rj)=σijdt

Then canceling out terms of second and higher order:

dW≈[W(t)∑wiαi−C(t)]dt+W(t)∑wiσidzi

Using Bellman equation, we can restate the problem:

J(W,X,t)=maxEt{∫tt+dtU[C(s),s]ds+J[W(t+dt),X(t+dt),t+dt]}

subject to the wealth constraint previously stated.

Using Ito's lemma we can rewrite:

dJ=J[W(t+dt),X(t+dt),t+dt]−J[W(t),X(t),t+dt]=Jtdt+JWdW+JXdX+12JXXdX2+12JWWdW2+JWXdXdW

and the expected value:

EtJ[W(t+dt),X(t+dt),t+dt]=J[W(t),X(t),t]+Jtdt+JWE[dW]+JXE(dX)+12JXXvar(dX)+12JWWvar[dW]+JWXcov(dX,dW)

After some algebra[2] , we have the following objective function:

max{U(C,t)+Jt+JWW[∑i=1nwi(αi−rf)+rf]−JWC+W22JWW∑i=1n∑j=1nwiwjσij+JXμ+12JXXs2+JWXW∑i=1nwiσiX}

where rf is the risk-free return. First order conditions are:

JW(αi−rf)+JWWW∑j=1nwj*σij+JWXσiX=0i=1,2,…,n

In matrix form, we have:

(α−rf𝟏)=−JWWJWΩw*W+−JWXJWcovrX

where α is the vector of expected returns, Ω the covariance matrix of returns, 𝟏 a unity vector covrX the covariance between returns and the state variable. The optimal weights are:

𝐰*=−JWJWWWΩ−1(α−rf𝟏)−JWXJWWWΩ−1covrX

Notice that the intertemporal model provides the same weights of the CAPM. Expected returns can be expressed as follows:

αi=rf+βim(αm−rf)+βih(αh−rf)

where m is the market portfolio and h a portfolio to hedge the state variable.

## See also

* Intertemporal portfolio choice

## References

1. ↑ Merton, Robert (1973). "An Intertemporal Capital Asset Pricing Model". Econometrica 41 (5): 867–887. doi:10.2307/1913811.
2. ↑ :E(dW)=−C(t)dt+W(t)∑wi(t)αidt

var(dW)=[W(t)−C(t)dt]2var[∑wi(t)ri(t+dt)]=W(t)2∑i=1∑i=1wiwjσijdt
∑i=onwi(t)αi=∑i=1nwi(t)[αi−rf]+rf

* 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

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