The Arrow-Pratt coefficient of absolute risk aversion
Definition (Arrow-Pratt coefficient of absolute risk aversion). Given a twice differentiable Bernoullio utility function \(u(\cdot)\),
$$ A_u(x):=-\frac{u”(x)}{u'(x)} $$
- Risk-aversion is related to concavity of \(u(\cdot)\); a “more concave” function has a smaller (more negative) second derivative hence a larger \(u”(x)\).
- Normalisation by \(u'(x)\) takes care of the fact that \(au(\cdot)+b\) represents the same preferences as \(u(\cdot)\).
- In probability premium
Consider a risk-averse consumer:
1. prefers \(x\) for certain to a 50-50 gamble between \(x+\epsilon\) and \(x-\epsilon\).
2. If we want to convince the agent to take the gamble, it could not be 50-50 – we need to make the \(x+\epsilon\) payout more likely.
3. Consider the gamble G such that the agent is indifferent between G and receiving x for certain, where
$$G= \begin{cases} x+\epsilon, & \text{with probability $\frac{1}{2}+\pi$}.\\ x-\epsilon, & \text{with probability $\frac{1}{2}-\pi$ } \end{cases}$$
4. It turns out that \(A_u(x)\) is proportional to \(\pi/\epsilon\) as \(\epsilon \rightarrow 0\); i.e., \(A_u(x)\) tells us the “premium” measured in probability that the decision-maker demands per unit of spread \(\epsilon\).
ARA.
Decreasing Absolute Risk Aversion. The Bernoulli function \(u\cdot)\) has decreasing absolute risk aversion iff \(A_u(\cdot)\) is a decreasing function of \(x\). Increasing Absolute Risk Aversion… Constant Absolute Risk Aversion – Bernoulli utility function has constant absolute risk aversion iff \(A_u(\cdot)\) is a constant function of \(x\).
Relative Risk Aversion
Definition (coefficient of relative risk aversion). Given a twice differentiable Bernoulli utility function \(u(\cdot)\),
$$ R_u(x):=-x\frac{u”(x)}{u'(x)}=xA_u(x) $$
There could be decreasing/increasing/constant relative risk aversion as above.
Implication: DARA means that if I take a 10 gamble when poor, I will take a10 gamble when risk. DRRA means that if I gamble 10% of my wealth when poor, I will gamble 10% when rich.