Should I invest in project A or project B?

In mathematics the expected value should help one to take the decision on whether or not he/she

should invest in either project. The one with the higher expected value should be preferred.

We note the expected value as follows:

()= =1

Where:

- are the different possible outcomes

- are the probabilities associated to the outcomes

In other words, the expected value is the probability weighted-sum of the outcomes.

In 1738, Bernoulli and Cramer realized that people would not systematically choose the highest expected value. They called it the paradox of St Petersburg. To better understand the paradox, suppose someone ask you to choose between two things:

1/ Flip a coin and he will give you $100 if you get a head. He will give you nothing in case you get a tail;

2/ Or he immediately gives you $50 upfront

Question 1

1.1.What would be your preferred choice? Be honest and explain your point of view.

1.1) I would chose option 1, because the option of flipping a coin and getting $100 if the result is head since with this option there is a chance to earn twice of the second option.

1.2.Compute the expected value of each choice and compare them.

1.2) expected value:

Option 1: f(x) = n*p = $100x1/2=$50

Option 2: f(x) = $50

The expected value are equals.

1.3.From a mathematical point of view, what would be the best choice? Is it different from your ownchoice you answered in 1.1.? Why is that?

1.3) as the expected values are equals, the two options are expected to result in the same expected value in the long run. This is different from what I chose in 1.1. I chose option 1 because $100 is greater than $50 but the result show that the options have the same expected value.

Introducing utility functions Bernoulli concluded that something else was driving individuals in their decision making process. In fact, the optimal choice would ultimately depend on the willingness of the individual to take risks. A "risk adverse" investor would prefer a safer choice for instance. To mathematically model this phenomenon, Bernoulli, Cramer and John Von Neumann introduced the concept of utility illustrating the behavior of investors. Going back to the example above, we can analyze the situation as follows:

- The investor is willing to lose the opportunity to earn an extra $50 as long as $50 is locked with certainty in his account.

- In other words, the first $50 have a superior interest: they are "more valuable" than the next $50.

- However, no investors would refuse an extra $50 in exchange of another experiment.

Question 2

2.1. If you were to draw the utility function that would reflect the above statements, what would be the shape of the curve? Assuming that the x-axis represents the dollar amounts and the y-axis the utility of an investor. Draw an example of such function.

2.2. Explain the shape of the curve you just drew