We can group people’s attitudes towards risk into 3 distinct categories, based on the form of their respective Bernoulli utility functions. Let’s illustrate this with a simple gamble based on a coin toss, which pays $10 if the coin lands heads and $20 if the coin lands tails. The expected value of this gamble is, of course: (0.5 * 10) + (0.5 * 20) = $15.

1. **Risk-Averse:**

If a person’s utility of the expected value of a gamble is greater than their expected utility from the gamble itself, they are said to be risk-averse. This is a more precise definition of Bernoulli’s idea.

Risk-averse behaviour is captured by a concave Bernoulli utility function, like a logarithmic function. For the above gamble, a risk-averse person whose Bernoulli utility function took the form u(w) = log(w), where w was the outcome, would have an expected utility over the gamble of: 0.5 * log(10) + 0.5 * log(20) = 1.15, while their utility of the expected value of the gamble is log(15) = 1.176.

**2. Risk-loving:**

If a person’s utility of the expected value of a gamble is less than their expected utility from the gamble itself, they are said to be risk-loving. Note, however, that this does not capture normal gambling behaviour of the kind observed in casinos the world over. By this definition, a truly risk-loving person ought to be willing to stake all of their assets, everything they own, on a single roll of dice.

A convex Bernoulli utility function captures risk-loving behaviour; for example, an exponential function. For the above gamble, a risk-loving person whose Bernoulli utility function took the form u(w) = w2 would have an expected utility over the gamble of:

0.5 * 102 + 0.5 * 202 = 250,

while their utility of the expected value of the gamble is 152 = 225.

**3. Risk-neutral:**

If a person’s utility of the expected value of a gamble is exactly equal to their expected utility from the gamble itself, they are said to be risk-neutral. In practice, most financial institutions behave in a risk-neutral manner while investing.

Risk-neutral behaviour is captured by a linear Bernoulli function. For the above gamble, a risk-neutral person whose Bernoulli utility function took the form u(w) = 2w would have an expected utility over the gamble of: (0.5 * 2 * 10) + (0.5 * 2 * 20) = 30, while their utility of the expected value of the gamble is 2 * 15 = 30.

### A Note On The Certainty Equivalent

The certainty equivalent of any gamble g is that amount of money, call it CE, offered for certain, which gives the consumer exactly the same utility as the gamble.

On the same lines, the risk premium of any gamble is the difference between the expected value of the gamble and its certainty equivalent, i.e.: Risk Premium = E(g) – CE

It follows that a risk-averse person’s certainty equivalent will be less than the expected value of the gamble, and they will have a positive risk-premium. Simply put, risk-averse people need an additional incentive to make them want to take on the risk of the gamble.

A risk-neutral consumer will have a zero risk premium, and a certainty equivalent equal to the expected value of the gamble. Similarly, a risk-loving consumer will have a negative risk-premium, since she would need an extra incentive to accept the expected value, not the risky gamble, and her certainty equivalent would be greater than the expected value of the gamble.