We are offered a proposal to invest in a very innovative software start-up. The founder has a very atypical professional and academic background. As the start-up is still at the project stage, we have no data on historical cash flows. Moreover, the project is so innovative that using start-ups from the same sector as benchmarks would be irrelevant.

We trust our feeling, we buy it.

What has happened in our brain to make such a decision ?

To explore the cognitive mechanisms at the origin of this financial choice, we will use the Cumulative Prospect Theory framework ( Kahneman and Tversky, 1992) which is one of the most famous model in behavioral economics for financial decisions.

## First step: Risk evaluation

We make a subjective evaluation on the probability of success for the start-up

In some cases, statistical analysis or probability laws objectively provides us the repartition of probabilities for particular events, or at least an estimation. We speak in this case of objective probabilities. Here, we have absolutely no data to value this start-up. Therefore, we have to subjectively evaluate the repartition of potential gains and losses based on our intuition. There are a lot of points to compute, but we know that our system 1 is able to treat a tremendous quantity of data outside of our awareness. Let’s oversimplify by supposing that our subjective evaluation of the profitability of this start-up follows a normal distribution around 2 (we hope, on average, to multiply your initial investment by 2).

## Second step: Biased perception of the evaluated risk

0 ,1% or 1%…same thing?

Why do we gamble and have a ton of insurance while we know that the mathematical probability of winning or seeing our house destroyed by an earthquake is extremely low? Because we are not as rational as classic economic theories would like us to be. We tend to overweight low probabilities of extreme events. As a result, we will overestimate the low probability of the exceptional success of the start-up that will make us earn millions and underestimate the high probability of bankruptcy more common to start-ups. The way we over- or underestimate risk depends on many parameters, such as our optimism or familiarity with the source of investment. We will be more optimistic if we personally know the founder or the sector of the start-up, for instance.

## Third step: Gain evaluation

How do we evaluate the potential gains drawn from this investment?

After evaluating the risk, let’s try to understand how we evaluate gains. We already know about the direct financial benefits that may be drawn from dividends and from the sale of shares for the start-up. Nevertheless, there may be additional ”hidden” gains such as potential synergy with your own company or simply the pride of having helped a young start-up. For our final investment decision, our brain will thus evaluate and take into account all these gains.

## Fourth step: Biased perception of gains

We hate losing twice as much as we enjoy winning.

As for the probabilities of success, we unconsciously transform the estimated gains. If we are asked to gamble and we have a 50% chance of winning 100€ (or nothing) or we have the chance to go back home with 50€, most of us will prefer to take the 50€. Now, if we are asked to gamble where we could lose 100€ with a 50% chance of losing nothing or to pay a fee of 50€, most of us will try their luck to avoid losing. Generally speaking, we hate losing twice as much as we like winning. As a consequence, we will underweight the potential gains of the start-up and overweight its potential losses.

## Conclusion

In the end, we have started from a subjective evaluation of risk and gains that we have unconsciously transformed according to our degree of risk aversion. This cognitive process has been modelled by many behavioral economic theories, the most famous being the Cumulative Prospect Theory (CPT) whose model is given below.

U is the utility we attribute to the start-up.

v is the utility we attribute to each potential gains and losses, including non-monetary one

w represents the way we transform our subjective evaluation of risk whose mass function is F.

In our example, time has not been voluntarily taken into account to simplify the problem despite its major impact on financial decisions. You can have a look at the article dedicated to inter-temporal choices to learn more about this subject:

**Reference:**

Prospect theory for time and ambiguity, Wakker