Introduction
Understanding how individuals make decisions over time is crucial in finance and economics. The Kreps-Porteus Preference theory, developed by David M. Kreps and Evan L. Porteus, provides a framework that separates intertemporal substitution from risk aversion, addressing the limitations of traditional expected utility models. In this article, I explore the mathematical underpinnings, applications, and implications of the Kreps-Porteus preference model.
Table of Contents
Traditional Expected Utility and Its Limitations
The standard expected utility model, developed by von Neumann and Morgenstern, assumes that individuals evaluate uncertain outcomes based on a single utility function, incorporating both risk and intertemporal considerations. The traditional model is expressed as:
U=∑t=0Tβtu(ct)U = \sum_{t=0}^{T} \beta^t u(c_t)
where:
- UU is the expected utility,
- β\beta is the discount factor,
- u(ct)u(c_t) is the instantaneous utility of consumption at time tt.
A key limitation of this model is that it imposes a rigid structure on intertemporal preferences, tying risk aversion directly to the elasticity of intertemporal substitution (EIS). Kreps and Porteus introduced a new approach to relax this restriction.
Kreps-Porteus Recursive Utility Model
The Kreps-Porteus model allows for a distinct separation between risk aversion and intertemporal substitution. This is achieved through a recursive formulation of utility:
Vt=[(1−β)u(ct)+β(Et[Vt+11−ρ])11−ρ]V_t = \left[ (1-\beta) u(c_t) + \beta \left( \mathbb{E}_t [V_{t+1}^{1-\rho}] \right)^{\frac{1}{1-\rho}} \right]
where:
- ρ\rho governs risk aversion,
- β\beta is the discount factor,
- VtV_t is the continuation value,
- u(ct)u(c_t) is the utility of consumption.
Unlike the expected utility model, the Kreps-Porteus formulation allows ρ\rho, which governs risk aversion, to be independent of the elasticity of intertemporal substitution (EIS), which is given by 1/ψ1/\psi, where ψ\psi controls substitution over time.
Key Features of Kreps-Porteus Preferences
- Separation of Risk Aversion and EIS
- Traditional models tie risk aversion directly to intertemporal substitution, which can be unrealistic.
- Kreps-Porteus preferences allow individuals to exhibit high risk aversion while still being willing to substitute consumption across time.
- Recursive Structure
- The model uses a dynamic recursive framework rather than a simple summation of discounted utilities.
- Future utility is evaluated in expectation but with an adjustable curvature that reflects risk attitudes.
- Empirical Implications
- This model better aligns with observed behavior in financial markets, where individuals often exhibit a strong preference for smooth consumption paths but still fear uncertain outcomes.
Comparison with Traditional Expected Utility
The table below highlights key differences between the traditional expected utility model and the Kreps-Porteus model.
| Feature | Expected Utility Model | Kreps-Porteus Model |
|---|---|---|
| Risk Aversion | Tied to EIS | Independent of EIS |
| Utility Structure | Additive | Recursive |
| Empirical Fit | Often inconsistent with data | More flexible |
| Application in Finance | Limited in explaining asset prices | Better fits risk premia |
Applications in Finance
The Kreps-Porteus model is particularly useful in explaining financial phenomena that the traditional model struggles with, such as:
1. Asset Pricing
- The model provides a framework to explain the equity premium puzzle. By allowing a high degree of risk aversion (ρ\rho) while maintaining reasonable EIS values (ψ\psi), it aligns better with observed risk premia in markets.
- The Euler equation for asset pricing under Kreps-Porteus preferences is: Pt=Et[β(Vt+1Vt)ρ−1Rt+1]P_t = \mathbb{E}_t \left[ \beta \left( \frac{V_{t+1}}{V_t} \right)^{\rho-1} R_{t+1} \right] where PtP_t is the price of an asset and Rt+1R_{t+1} is the return.
2. Consumption-Savings Decisions
- The separation of EIS from risk aversion allows for better modeling of consumption choices, particularly under uncertain income streams.
- Empirical studies suggest that individuals exhibit moderate intertemporal substitution but strong risk aversion, a behavior well captured by Kreps-Porteus preferences.
3. Life-Cycle Models
- This model improves predictions in life-cycle consumption choices, where individuals plan for retirement while facing uncertain investment returns.
Example Calculation
To illustrate how the Kreps-Porteus preference model works in practice, consider an investor choosing between two assets: a risk-free bond with a 2% return and a risky stock with an expected return of 8% but a standard deviation of 15%.
Let’s assume:
- β=0.95\beta = 0.95,
- ρ=3\rho = 3 (high risk aversion),
- ψ=0.5\psi = 0.5 (low intertemporal substitution).
Using the recursive utility function, we calculate the certainty equivalent return as:
CE=[(1−β)u(ct)+β(Et[Vt+11−ρ])11−ρ]CE = \left[ (1-\beta) u(c_t) + \beta \left( \mathbb{E}_t [V_{t+1}^{1-\rho}] \right)^{\frac{1}{1-\rho}} \right]
Given high risk aversion (ρ=3\rho = 3), the investor will require a substantial risk premium over the bond to hold the risky asset. This preference structure leads to a higher equity premium than that predicted by the traditional model, helping to explain observed market behavior.
Conclusion
The Kreps-Porteus preference theory represents a significant advancement in our understanding of intertemporal decision-making under uncertainty. By decoupling risk aversion from intertemporal substitution, it provides a more realistic representation of economic behavior. This framework has profound implications in finance, particularly in asset pricing, consumption-savings decisions, and risk assessment. Future research continues to refine its applications, integrating it with behavioral insights and empirical data to enhance its predictive power.
