Intertemporal capital asset pricing model

What Is Intertemporal Capital Asset Pricing Model?

The Intertemporal Capital Asset Pricing Model (ICAPM) is an asset pricing model that extends the traditional Capital Asset Pricing Model (CAPM) by acknowledging that investors are concerned not only with their end-of-period wealth but also with future investment opportunities. This model, falling under the broader category of portfolio theory, recognizes that changes in the investment environment, such as shifts in interest rates or volatility, affect an investor's well-being over time. Unlike single-period models, the ICAPM accounts for an investor's desire to hedge against adverse changes in the future state variables that characterize these opportunities.

History and Origin

The Intertemporal Capital Asset Pricing Model was developed by Nobel laureate Robert C. Merton in his seminal 1973 paper, "An Intertemporal Capital Asset Pricing Model."⁸, ⁹, ¹⁰ Merton's work built upon the foundational portfolio choice theories, notably those of Harry Markowitz, and extended the single-period CAPM into a multi-period framework. The conventional CAPM assumes investors make decisions for a single period, focusing solely on the mean and variance of their returns.⁷ However, Merton's innovation was to incorporate a dynamic view, where investors continuously rebalance their portfolios and consider how current decisions impact their future utility function and consumption possibilities. This recognized that investors might exhibit specific hedging demands for assets that protect them from unfavorable shifts in their future investment landscape.

Key Takeaways

• The Intertemporal Capital Asset Pricing Model (ICAPM) is a multi-period asset pricing model that extends the traditional CAPM.
• It accounts for investors' desire to hedge against changes in future investment opportunities, such as shifts in interest rates or market volatility.
• The ICAPM includes additional risk factors beyond just market risk, reflecting the covariance of an asset's return with these state variables.
• It implies that assets providing a hedge against unfavorable changes in future investment opportunities may offer lower expected returns.
• Empirical testing of the ICAPM can be challenging due to the difficulty in identifying and accurately measuring all relevant state variables.

Formula and Calculation

The Intertemporal Capital Asset Pricing Model extends the core idea of the CAPM by including additional risk premia related to changes in the investment opportunity set. While the full derivation involves stochastic calculus and continuous-time finance, a generalized form of the ICAPM can be represented as:

$E[R_i] - R_f = \beta_{i,M}(E[R_M] - R_f) + \sum_{k=1}^K \beta_{i,k} \text{RiskPremium}_k$

Where:

• (E[R_i]) = The expected return on asset (i).
• (R_f) = The risk-free rate.
• (\beta_{i,M}) = The market risk beta of asset (i), measuring its sensitivity to the market portfolio.
• (E[R_M]) = The expected return on the market portfolio.
• (\beta_{i,k}) = The sensitivity of asset (i)'s return to changes in the (k^{th}) state variable.
• (\text{RiskPremium}_k) = The risk premium associated with the (k^{th}) state variable, representing compensation for bearing risk related to changes in future investment opportunities.
• (K) = The number of relevant state variables.

This formula indicates that an asset's expected return in excess of the risk-free rate is determined not only by its exposure to market risk but also by its exposure to various other sources of systematic risk related to shifts in the investment environment.

Interpreting the Intertemporal Capital Asset Pricing Model

The Intertemporal Capital Asset Pricing Model suggests that investors are not merely concerned with the static trade-off between risk and return over a single period but also how their portfolio performs across changing economic conditions. Therefore, when interpreting the ICAPM, the key lies in understanding the additional beta factors and their associated risk premia. If an asset's returns negatively covary with a state variable that signals a worsening of future investment opportunities (e.g., rising interest rates or increased volatility), that asset provides a hedge. Investors would be willing to accept a lower expected return for such an asset because it offers valuable insurance against these adverse future changes. Conversely, assets that perform poorly when investment opportunities improve would require a higher expected return to compensate investors for that exposure. This framework provides a richer perspective on the drivers of expected return compared to single-factor models.

Hypothetical Example

Consider an investor, Sarah, who is planning for retirement over several decades. Sarah understands that the future economic landscape will change, affecting her ability to save and invest. She is particularly concerned about unexpected increases in inflation, which could erode the purchasing power of her future retirement income.

According to the Intertemporal Capital Asset Pricing Model, Sarah would not only consider the market beta of an asset but also its sensitivity to changes in inflation. If a particular asset, say an inflation-protected bond fund, tends to perform well when inflation unexpectedly rises (i.e., it has a positive beta with respect to inflation surprises), it provides a valuable hedge against this specific risk. Even if this fund has a lower expected return than a standard equity fund with similar market risk, Sarah might still invest in it because the inflation hedge provides a significant benefit for her long-term portfolio choice. The ICAPM helps explain why investors might hold assets that seem suboptimal in a single-period, mean-variance framework but are rational given their intertemporal concerns.

Practical Applications

The Intertemporal Capital Asset Pricing Model, while complex, has several practical applications in finance, particularly in portfolio management and asset pricing models research. It provides a theoretical basis for multi-factor models, which are widely used in practice to explain the cross-section of asset returns. For instance, institutional investors and pension funds, with long investment horizons, often use frameworks that implicitly or explicitly consider how changing economic conditions might affect their long-term liabilities and desired future consumption.

The model helps explain why certain assets might be held for their hedging properties against specific macroeconomic risks, rather than purely for their market risk-adjusted return. For example, some assets might offer a hedge against changes in interest rates or inflation, which are critical state variables for long-term investors. Research by institutions like the National Bureau of Economic Research (NBER) and the Federal Reserve often explores how macroeconomic risks influence bond and equity returns, aligning with the ICAPM's multi-factor approach.⁴, ⁵, ⁶

Limitations and Criticisms

Despite its theoretical advancements, the Intertemporal Capital Asset Pricing Model faces several practical limitations and criticisms. A primary challenge is the identification and measurement of the relevant state variables. While Merton's model implies that investors hedge against adverse changes in their investment opportunities, precisely defining and quantifying all such variables in the real world is difficult.³ Researchers often rely on proxies like interest rates, inflation, or the market price of risk, but these may not fully capture the complete set of factors influencing an investor's intertemporal utility.

Empirical tests of the ICAPM have yielded mixed results, partly due to the difficulty in precisely specifying the model's factors. Some studies find that various factors beyond market beta are significant in explaining asset returns, supporting the ICAPM's multi-factor nature. However, other research highlights that selecting the right factors remains a challenge, and simpler models can sometimes provide comparable explanatory power for the cross-section of returns.¹, ² The model's complexity, involving stochastic processes and continuous-time analysis, also makes it less intuitive for many practitioners compared to the more straightforward Capital Asset Pricing Model. Furthermore, the assumption of risk aversion and rational investor behavior, fundamental to the ICAPM, may not always hold in real markets.

Intertemporal Capital Asset Pricing Model vs. Capital Asset Pricing Model

The key distinction between the Intertemporal Capital Asset Pricing Model (ICAPM) and the Capital Asset Pricing Model (CAPM) lies in their underlying assumptions about investor behavior and the investment horizon.

While the CAPM focuses solely on the relationship between an asset's expected return and its systematic risk relative to the market, the ICAPM expands this by recognizing that investors also value assets that protect them from unfavorable changes in their future investment opportunities. This hedging motive introduces additional risk factors beyond just market movements, making the ICAPM a more comprehensive, albeit more complex, framework for asset pricing models.

FAQs

What problem does the ICAPM address that the CAPM does not?

The ICAPM addresses the limitation of the traditional CAPM, which assumes investors only care about their wealth at the end of a single period. The ICAPM recognizes that investors make decisions over multiple periods and are concerned about changes in their investment opportunities over time, such as shifts in interest rates or market volatility.

What are "state variables" in the context of the ICAPM?

State variables in the ICAPM are economic factors or market conditions that can change unexpectedly and influence an investor's future investment opportunities or wealth. Examples might include inflation rates, interest rate levels, the market price of risk, or other macroeconomic indicators that affect future returns or consumption possibilities.

Is the ICAPM used in practice?

While the full mathematical rigor of the ICAPM can be complex, its core insight – that investors hedge against changes in future investment opportunities – is widely applied in practice. It provides the theoretical foundation for many multi-factor asset pricing models and explains why investors or fund managers might hold assets that offer protection against specific economic risks, even if those assets don't solely optimize for mean-variance efficiency in a single period.

How does the ICAPM relate to arbitrage pricing theory?

Both the ICAPM and the Arbitrage Pricing Theory (APT) are multi-factor models that extend beyond the single-factor CAPM. The APT is an empirical model that identifies multiple systematic risk factors that explain asset returns, without necessarily specifying what those factors are. The ICAPM, in contrast, provides a theoretical basis for these factors, positing that they arise from investors' desire to hedge against changes in their investment opportunities. Thus, the ICAPM can be seen as providing theoretical underpinnings for the factors observed in APT.