intertemporal CAPM

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mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that requi ...
, 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 forecast changes in the distribution of future
returns Return may refer to: In business, economics, and finance * Return on investment (ROI), the financial gain after an expense. * Rate of return, the financial term for the profit or loss derived from an investment * Tax return, a blank document or ...
or
income Income is the consumption and saving opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms. Income is difficult to define conceptually and the definition may be different across fields. For ...
. 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 An investor is a person who allocates financial capital with the expectation of a future return (profit) or to gain an advantage (interest). Through this allocated capital most of the time the investor purchases some species of property. Typ ...
hedge against shortfalls in consumption or against changes in the future
investment Investment is the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance, the purpose of investing is ...
opportunity set.

# Continuous time version

Merton considers a continuous time market in equilibrium. The state variable (X) follows a
brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
: :$dX = \mu dt + s dZ$ The investor maximizes his Von Neumann–Morgenstern utility: :$E_o \left\$ where T is the time horizon and B (T),Tthe 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: : where $r_i$ is the return on asset i. The change in wealth is: : We can use
dynamic programming Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. ...
to solve the problem. For instance, if we consider a series of discrete time problems: :$\max E_0 \left\$ Then, a
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
gives: : where $t^*$ is a value between t and t+dt. Assuming that returns follow a
brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
: :$r_i\left(t+dt\right) = \alpha_i dt + \sigma_i dz_i$ with: :$E\left(r_i\right) = \alpha_i dt \quad ;\quad E\left(r_i^2\right)=var\left(r_i\right)=\sigma_i^2dt \quad ;\quad cov\left(r_i,r_j\right) = \sigma_dt$ Then canceling out terms of second and higher order: : Using
Bellman equation A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. It writes the "value" of a decision problem at a certain point in tim ...
, we can restate the problem: :$J\left(W,X,t\right) = max \; E_t\left\$ subject to the wealth constraint previously stated. Using Ito's lemma we can rewrite: : and the expected value: : After some algebra:$E\left(dW\right)=-C\left(t\right)dt + W\left(t\right) \sum w_i\left(t\right) \alpha_i dt$ : : , we have the following objective function: :$max \left\$ where $r_f$ is the risk-free return. First order conditions are: :$J_W\left(\alpha_i-r_f\right)+J_W \sum_^n w^*_j \sigma_ + J_ \sigma_=0 \quad i=1,2,\ldots,n$ In matrix form, we have: :$\left(\alpha - r_f \right) = \frac \Omega w^* W + \frac cov_$ where $\alpha$ is the vector of expected returns, $\Omega$ the
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements o ...
of returns,  a unity vector $cov_$ the covariance between returns and the state variable. The optimal weights are: :$= \frac\Omega^\left(\alpha - r_f \right) - \frac\Omega^ cov_$ Notice that the intertemporal model provides the same weights of the CAPM. Expected returns can be expressed as follows: :$\alpha_i = r_f + \beta_ \left(\alpha_m - r_f\right) + \beta_\left(\alpha_h - r_f\right)$ where m is the market portfolio and h a portfolio to hedge the state variable.