As a CFO, we can choose to invest in between three options:

- We get €500.000
- We invest in a stock with a 50% of chance of earning €1M or nothing
- We invest in an industrial project on which we have little information except that we can earn €1M, if it is successful or we earn nothing.

** What do we choose?**

If you have read the articles on the Ellsberg and Allais paradox, you already know that our biases which will make us prefer the first option over the others and the second one over the third one: **certainty bias, risk aversion and ambiguity aversion**. Why? Because we try as much as possible to avoid uncertainty when making a financial investment.

But let’s further explore how the brain works to understand the mechanisms of these biases.

**Certain case:**

Easy one, we know that we get €500.000 for sure so P(option 1 wins)=1

**Risky case:**

The investment is risky but we know the probability of success for the investment so there is a lower degree of uncertainty. Nevertheless, we will not mathematically make our choice based on the expected value and risk of the investment. Our brain, which is uncertainty averse, will transform the probability of success as following.

Probabilities of positive events under 1/3 will be overweighted while probabilities of positive outcomes over 1/3 will be underweighted. In other words, we will be risk-seeking when the probability of success is low (Have you ever gambled at a casino?) and risk-averse otherwise. The more pessimistic we are, the more underweighted large probabilities and the least overweighted small probabilities will be. Similarly, our likelihood insensitivity will be on average 1/3 but varies depending on numerous psychological factors. In the end, the utility that we will attribute to the risky option will be (based on a Cumulative Prospect Theory framework) :

U(Stock)=w(1/2)*u(€1M)+(1-w(1/2))*u(€0)

With:

u = the utility function which measures our own evaluation of gains and which is less influenced by the type of investment

w = the weighting function which measures our own evaluation of risk and which highly depends on the type of investment and on our individual biases.

Since 1/2>1/3, we will get w(1/2)<1-w(1/2) so that the risk of losing will be overestimated while the probability of success will be underestimated.

The difference between the minimal preferred certain option and the expected value of the rejected risky option is called risk aversion.

**Ambiguous case:**

Let’s move to the third option where the probability of success is unknown. As we have little information, we will automatically compute subjective probabilities of success based on our own experience. This process, of course, will be highly biased. Nevertheless, the most influential bias does not come from this hypothesis on gain distribution but on how we will transform them. Once we have evaluated the potential probabilities of the success of the project, we will, as for the risky option, weight this probability following the same weighting function (overweighting small probabilities and underweighting large probabilities for positive issues). However, because our brain is uncertainty averse (activation of the amygdala fear system), we will be more pessimistic than in the risky case. This is ambiguity aversion. The more familiar and attractive the source of the investment is to our brain, the less pessimistic, and thus risk-averse, we will be. For instance, we will be less pessimistic, as French, to bet on the decrease or on the increase of temperature in Paris than in other countries or than in CAC40 (home bias).

At the end, the utility that we will attribute to the risky option will be (based on a Cumulative Prospect Theory framework) :

U(Stock)=ws(P(S))*u(€1M)+(1-ws(P(F))*u(€0)

With:

u = the utility function which will measure our own evaluation of gains and which is less influenced by the degree of uncertainty

ws = the weighting function which measures how we transform the probabilities we have subjectively computed and which is highly dependent on the type S of investment (how familiar or attractive is the source of uncertainty?) and on personal biases.

**References:**

Prospect theory, for risk and ambiguity (2011), Wakker

Abdellaoui, M., Baillon, A., Placido, L., & Wakker, P. (2009). The rich domain of uncertainty. American Economic Review.