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+sdZ

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 return 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_{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:

\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}^{2}dt\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}dXdW

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_{W}C+{\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}{\mathbf {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, ${\mathbf {1}}$ a unity vector $cov_{rX}$ the covariance between returns and the state variable. The optimal weights are:

{{\mathbf {w}}^{*}}={\frac {-J_{W}}{J_{WW}W}}\Omega ^{-1}(\alpha -r_{f}{\mathbf {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.

References

↑ Merton, Robert (1973). "An Intertemporal Capital Asset Pricing Model". Econometrica. 41 (5): 867–887. JSTOR 1913811. doi:10.2307/1913811.
↑
$E(dW)=-C(t)dt+W(t)\sum w_{i}(t)\alpha _{i}dt$

$var(dW)=[W(t)-C(t)dt]^{2}var[\sum w_{i}(t)r_{i}(t+dt)]=W(t)^{2}\sum _{i=1}\sum _{i=1}w_{i}w_{j}\sigma _{ij}dt$

$\sum _{i=o}^{n}w_{i}(t)\alpha _{i}=\sum _{i=1}^{n}w_{i}(t)[\alpha _{i}-r_{f}]+r_{f}$

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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Intertemporal CAPM

Continuous time version

See also

References