Question 6. Consider a firm that has a project. There are two possible outcomes of the project in each period: good or bad. The payment of the project is for good and bad outcome, respectively. The quality of the project is unknown. If the quality of a project is , then the probability that observing a good outcome is , i.e. . Suppose there are two qualities: high ( ) and low ( ). Let . Suppose that investors are risk neutral and update their beliefs using Bayes' rule. In the beginning, they believe that the probability that the project has high quality is . Let the risk-free interest rate be and assume there are only two periods (the project ends and does not generate any outcome after period 3). Observe that a bad outcome fully reveals to quality of the project since (i.e. a high-quality project never generates a bad outcome). This implies that . Let be the history of our past observations. For example, if the outcome is good in the first period, then . If the outcome is good in the first period and bad in the second period, then . Let be the probability of a good outcome after some history . a) [10 points] Calculate and . Specifically, you have to calculate numerically: . . x =g10,x =b5 Pr(g) = H L =H1, =L0.1 Pr( ) =H0.5 r =f0.1 =H1 Pr( b) =H0 h h= g h= gb h h g b =gPr(gg) = HPr( g)+H LPr( g)L =bPr(gb) = HPr( b)+H LPr( b)L g = b =

b) [10 points] What is the price of the project after a good outcome ( ) and a bad outcome ( )? Assume that we calculate the price after the payment of the first outcome and the next payment will be in the next period. In other words, the price does not incorporate the first payment but incorporates the discounted value of the second payment.