Intertemporal CAPM

The Intertemporal Capital Asset Pricing Model, or ICAPM, was an alternative to the CAPM provided by Robert Merton. It is a linear factor model with wealth and state variable that forecast 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[1] considers a continuous time market in equilibrium. The state variable (X) follows a brownian motion:

dX = \mu dt + s dZ

The investor maximizes his Von Neumann–Morgenstern utility:

E_o \left\{\int_o^T U[C(t),t]dt + B[W(T),T] \right\}

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

W(t+dt) = [W(t) -C(t) dt]\sum_{i=0}^n w_i[1+ r_i(t+ dt)]

where r_i is the retutn on asset i. The change in wealth is:

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_o \left\{\sum_{t=o}^{T-dt}\int_t^{t+dt} U[C(s),s]ds + B[W(T),T] \right\}

Then, a Taylor expansion gives:

\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^* is a value between t and t+dt.

Assuming that returns follow a brownian motion:

r_i(t+dt) = \alpha_i dt + \sigma_i dz_i

with:

E(r_i) = \alpha_i dt \quad ;\quad E(r_i^2)=var(r_i)=\sigma_i^2dt \quad ;\quad cov(r_i,r_j) = \sigma_{ij}dt

Then canceling out terms of second and higher order:

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) = 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:

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} dX dW

and the expected value:

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] , we have the following objective function:

max \left\{ U(C,t) + J_t + J_W W [\sum_{i=1}^n w_i(\alpha_i-r_f)+r_f] - J_WC + \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 is the risk-free return. First order conditions are:

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:

(\alpha - r_f {\bold 1}) = \frac{-J_{WW}}{J_W} \Omega w^* W + \frac{-J_{WX}}{J_W} cov_{rX}

where \alpha is the vector of expected returns, \Omega the covariance matrix of returns, {\bold 1} a unity vector cov_{rX} the covariance between returns and the state variable. The optimal weights are:

{\bold w^*} = \frac{-J_W}{J_{WW} W}\Omega^{-1}(\alpha - r_f {\bold 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:

\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.

See also

References

  1. ↑ Merton, Robert (1973). "An Intertemporal Capital Asset Pricing Model". Econometrica 41 (5): 867–887. JSTOR 1913811.
  2. ↑
    E(dW)=-C(t)dt + W(t) \sum w_i(t) \alpha_i dt
    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
    \sum_{i=o}^n w_i(t) \alpha_i = \sum_{i=1}^n w_i(t)[\alpha_i - r_f] + r_f