# How to calculate the degree of risk aversion of a consumer in a lottery?

To give an example, say we start with 100 dollars and we enter a lottery. With probability $$\pi$$, this 100 dollars is reduced by 2 dollars. Otherwise our endowed 100 dollars does not change. Let's say our consumer's utility function is described by $$u(c_h)=ln(c_h)$$.

It's clear that the consumer is risk-averse, but how would I calculate the degree of risk aversion? How do you find the risk premium here?

Thanks.

• Are you aware that there are degrees of absolute risk aversion and of relative risk aversion? Which degree of risk aversion are you asking about? – Herr K. Dec 3 '19 at 4:54

The gamble:

$$g = \left(\pi \circ 98, (1-\pi)\circ100\right)$$

The expected value of the gamble:

$$\mathbb{E}[g]= \pi\cdot 98 + (1-\pi)\cdot 100$$

Expected utility:

$$u(g) = \pi \cdot \ln (98) + (1-\pi)\cdot \ln (100)$$

$$\ln \left(\mathbb{E}[g]-P\right) = \pi \cdot \ln (98) + (1-\pi)\cdot \ln (100)$$

Where $$P$$ is the risk premium. Solve the above equation for $$P$$. You can measure the degree of absolute risk aversion as follows:

$$ARA = -\frac{u''(c_h)}{u'(c_h)}$$

And the degree of relative risk aversion as:

$$RRA = - \frac{c_hu''(c_h)}{u'(c_h)}$$

• Thank you! I have one further question I was wondering if you could answer: say there was an offer to buy insurance that fully mitigates the risk of loss for 2 dollars. Would this be an actuarially fair offer? – Biff Dec 3 '19 at 3:43
• Being actuarially fair depends on the insurance premium. If the insurance premium is the expected loss (in your case), then yes. – InSung Cho Dec 3 '19 at 16:16
• It would be actuarially fair if the insurer has zero expected profit. If you set up the problem of the insurance company, you should find that an actuarially fair offer is one which the cost per dollar of coverage (the premium) is equal to the probability of loss, $\pi$. – lunar_props Dec 3 '19 at 19:50