How to Calculate Absolute Risk Aversion

Absolute Risk Aversion Formula:

\[ ARA = -\frac{U''(W)}{U'(W)} \]

Absolute Risk Aversion (ARA):

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1. What is Absolute Risk Aversion?

Absolute Risk Aversion (ARA) is an economic measure that quantifies an individual's aversion to risk. It represents how much an individual dislikes uncertainty about their wealth level, with higher values indicating greater risk aversion.

2. How Does the Calculator Work?

The calculator uses the Arrow-Pratt Absolute Risk Aversion formula:

\[ ARA = -\frac{U''(W)}{U'(W)} \]

Where:

$ U''(W) $ — Second derivative of utility function with respect to wealth
$ U'(W) $ — First derivative of utility function with respect to wealth
$ W $ — Wealth in dollars ($)
$ ARA $ — Absolute Risk Aversion (1/$)

Explanation: The formula measures the curvature of the utility function, indicating how much utility decreases as risk increases.

3. Importance of ARA Calculation

Details: Absolute Risk Aversion is crucial in portfolio theory, insurance decisions, and investment analysis. It helps determine optimal asset allocation and risk management strategies.

4. Using the Calculator

Tips: Enter the second derivative (U''(W)) and first derivative (U'(W)) of your utility function, along with current wealth. Ensure U'(W) is not zero to avoid division by zero errors.

5. Frequently Asked Questions (FAQ)

Q1: What does a positive ARA value mean?
A: A positive ARA indicates risk aversion - the individual prefers certain outcomes over uncertain ones with the same expected value.

Q2: How does ARA differ from Relative Risk Aversion?
A: ARA measures absolute dollar risk aversion, while Relative Risk Aversion (RRA) measures risk aversion relative to wealth level (RRA = ARA × W).

Q3: What are typical ARA values?
A: ARA typically decreases with wealth. Values range from 0.0001 to 0.1 for most individuals, with higher values indicating greater risk aversion.

Q4: Can ARA be negative?
A: Negative ARA indicates risk-seeking behavior, where individuals prefer uncertain outcomes over certain ones.

Q5: How is ARA used in financial modeling?
A: ARA is used in portfolio optimization, derivative pricing, and determining insurance premiums to model investor behavior under uncertainty.