Constant Absolute Risk Aversion (CARA) is a key concept in modern portfolio theory and utility-based investment decisions. It describes a specific investor behavior where the degree of risk aversion remains constant regardless of changes in wealth. Understanding CARA is essential for constructing optimal asset allocation strategies that align with an investor’s risk preferences and maximize expected utility.
Understanding CARA
Absolute Risk Aversion (ARA) measures how much an investor dislikes risk in absolute terms. For CARA, the investor’s risk aversion does not change with wealth:
ARA(W) = -\frac{U''(W)}{U'(W)} = \text{constant}Where:
- U(W) = utility function of wealth W
- U'(W) = marginal utility of wealth
- U''(W) = second derivative of utility with respect to wealth
A commonly used CARA utility function is the negative exponential utility function:
U(W) = -e^{-\alpha W}- \alpha = constant coefficient of absolute risk aversion
- This utility function implies that the investor’s willingness to bear risk does not change with wealth.
Implications for Asset Allocation
1. Fixed Risk Tolerance
- CARA investors maintain the same absolute level of risk exposure regardless of wealth fluctuations.
- The proportion of wealth invested in risky assets decreases as wealth increases, but the dollar amount remains constant.
2. Optimal Portfolio Choice
For an investor with CARA utility facing risky assets with normally distributed returns, the optimal investment in a risky asset is:
x^* = \frac{E[R] - R_f}{\alpha \sigma^2}Where:
- x^* = dollar amount invested in the risky asset
- E[R] = expected return of the risky asset
- R_f = risk-free rate
- \sigma^2 = variance of the risky asset return
- \alpha = constant absolute risk aversion coefficient
Key Insight: The dollar amount invested in risky assets is independent of the investor’s wealth, unlike constant relative risk aversion (CRRA), where the proportion invested depends on wealth.
3. Diversification Across Multiple Assets
For multiple risky assets with expected returns vector \mathbf{E[R]} and covariance matrix \Sigma, the optimal allocation is:
\mathbf{x^*} = \frac{1}{\alpha} \Sigma^{-1} (\mathbf{E[R]} - R_f \mathbf{1})- \mathbf{x^*} = vector of dollar amounts to invest in each risky asset
- Ensures that more capital is allocated to assets with higher risk-adjusted expected returns
- Dollar allocation is constant with wealth, consistent with CARA behavior
Example
Assume an investor has CARA utility with \alpha = 2, a risk-free rate of 2%, and a risky asset with:
- Expected return: 8%
- Standard deviation: 10% (\sigma = 0.1)
Optimal investment in the risky asset:
x^* = \frac{0.08 - 0.02}{2 \times 0.1^2} = \frac{0.06}{0.02} = 3- The investor should invest $3 (per unit scale) in the risky asset, regardless of their total wealth.
4. Comparison with CRRA
| Feature | CARA | CRRA |
|---|---|---|
| Risk Aversion | Constant absolute | Constant relative (proportional) |
| Wealth Effect | Dollar amount invested fixed | Fraction of wealth invested fixed |
| Utility Function | Negative exponential | Power utility U(W) = \frac{W^{1-\gamma}}{1-\gamma} |
| Portfolio Scaling | Independent of wealth | Scales with wealth |
Practical Considerations
- Wealth Independence: CARA is most relevant for investors with stable financial obligations or moderate wealth, where absolute risk tolerance is more appropriate than relative risk tolerance.
- Portfolio Rebalancing: Since the optimal dollar allocation does not depend on wealth, CARA investors frequently rebalance to maintain the desired absolute exposure to risky assets.
- Limitations: Real-world investors often exhibit decreasing absolute risk aversion, meaning they become more willing to take risks as wealth increases, which is not captured by CARA.
Applications in Investment Strategy
- Hedging: CARA assumptions are commonly used in derivative pricing and hedging strategies, where the focus is on dollar exposure rather than proportional wealth allocation.
- Fixed Dollar Investment Plans: Suitable for strategies like dollar-cost averaging in a fixed amount rather than a fixed percentage of wealth.
- Institutional Asset Allocation: Pension funds or endowments may adopt CARA-like strategies when their liabilities are fixed in absolute terms.
Conclusion
Constant Absolute Risk Aversion (CARA) provides a framework for understanding investors who maintain a fixed tolerance for risk regardless of wealth changes. The CARA utility function leads to asset allocations where the dollar amount invested in risky assets is constant, simplifying investment decisions and rebalancing strategies. While less common than CRRA in personal finance, CARA is particularly useful in institutional settings, hedging, and scenarios where absolute exposure to risk is the key consideration.
