An Intertemporal Capital Asset Pricing Model

Source: Merton, R. C. (1973). Econometrica 41(5), 867–887.

TL;DR

Generalizes the CAPM to a dynamic, continuous-time world. When the investment opportunity set shifts

over time, investors care not only about market risk but also about assets that hedge adverse changes

in those opportunities. The result — the Intertemporal CAPM (ICAPM) — is a multi-factor model in

which state variables that drive time-varying opportunities are additional priced risks.

The static CAPM assumes a single period and a constant investment opportunity set. But interest rates,

expected returns, and volatilities change over time. If a lifetime-utility-maximizing investor knows

opportunities will shift, how does that reshape optimal portfolios and equilibrium expected returns?

An investor maximizes E₀∫u(C,t)dt + B(W(T),T) (eqn 1) by choosing consumption and weights over one

instantaneously risk-free asset and n risky assets whose drifts/vols depend on a state variable x that

follows its own Itô process (eqns 2–5); x makes the opportunity set stochastic. Solving the HJB

optimality conditions, optimal weights contain a standard mean-variance term plus a hedging term

(proportional to J_Wx) absent in the constant-opportunity-set case. With one state variable a

Three-Fund Theorem holds: investors hold (1) the riskless asset, (2) the mean-variance ("market")

portfolio, and (3) a portfolio with maximum absolute correlation with x — the best hedge against

changes in the state variable. Equilibrium then gives expected excess return linear in the market beta

and a beta on each state variable (eqn 24, generalizing to multiple state variables, one extra beta

per variable).

Expected excess return E[Rᵢ] − R_f is linear in the market beta and in betas with respect to the

state variables describing time-varying opportunities — a multi-factor pricing equation.

A hedging demand arises whenever an asset covaries with the state variable (σ_ix ≠ 0); it vanishes

only if x is uncorrelated with returns or utility is log (J_Wx = 0), recovering the static CAPM.

Assets that pay off when opportunities deteriorate command lower premia (they are valuable hedges).

Provides the theoretical justification for multi-factor models distinct from APT: factors should

be (or proxy) state variables forecasting the future investment opportunity set.

Underlies interpretations of term, default, and volatility factors — and some readings of value — as

hedges against bad states (e.g., Campbell 1996; Fama–French factors).

The same pricing relations later emerge in the Cox–Ingersoll–Ross (1985) general-equilibrium model,

extending their reach beyond Merton's partial-equilibrium setup.

The theory does not pin down which state variables matter or how many — leaving wide empirical

discretion (the door to data-mined "ICAPM" factors).

Partial equilibrium: asset supplies and the price-process forms are taken as given.

Idealizations: continuous, costless trading and known dynamics.

Merton, R. (1973) — An Intertemporal Capital Asset Pricing Model — Econometrica

Cox, J., Ingersoll, J. & Ross, S. (1985) — An Intertemporal General Equilibrium Model of Asset Prices — Econometrica

Campbell, J. (1996) — Understanding Risk and Return — Journal of Political Economy

Fama, E. & French, K. (1993) — Common Risk Factors in the Returns on Stocks and Bonds — Journal of Financial Economics

Provenance: summarized from secondary material (lecture notes), not the original paper's full text — figures indicative.