Consider the following situation

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Consider the following situation. Engineers working for the government of Mexico have recently discovered an offshore area potentially rich in oil and gas. To raise revenue, the Mexican government decides to auction off drilling rights to private companies. The auction is structured as a first price sealed bid auction. First price means the highest bidder will win the auction. Sealed bid means the bidders submit bids simultaneously and secretly – no bidder knows what its rivals have bid until the auction is over and the winner is declared. Three companies formally enroll in the auction and prepare to submit their bids: Exxon, Shell, and Pemex.

 

1. You are hired as a consultant for Exxon to help them prepare their bid. Due to the sealed bid structure, the bids of Shell and Pemex can be thought of as continuous random variables. They are denoted BShell

   

    and BPemex

   

    respectively. Using the notation of a joint cdf, write down a simple formula for the probability Exxon will win the auction if it submits a bid of 80 million dollars.

 

2. Suppose your strategic analysis of Shell and Pemex leads you to believe that their bids will be independent in a probabilistic sense. Furthermore you believe that FBShell70m=0.4

   

    and FBPemex70m=0.6

   

   . Given this information, what is the probability Exxon will win the auction if it submits a bid of 70 million dollars?

 

Problem Two. Suppose X

 and Y

 are two independent random variables. Prove that,

 

FX|Yxy=FX(x)

 

 

Problem Three. You are a consultant hired to advise a bank. The bank has made a loan and wishes for you to perform an analysis on the creditworthiness of the borrower. In particular the bank wishes to know if the borrower will default on their loan in the next hundred days.

 

1. Common to credit modeling, you decide to use an exponential probability distribution to describe the “time to default” random variable T

   

   . In this type of credit analysis it is assumed that every borrower will eventually default if given enough time, and this default occurs at a time T

   

    from the present. A bigger T

   

    means the borrower will default later on, while a smaller T

   

    means the borrower will default sooner. The exponential probability distribution gives a cdf for T

   

    of,

 

FTt=1-e-λt

 

 

Where the term λ

 is a parameter that describes how creditworthy the borrower is. A higher λ

 corresponds to a riskier borrower and a lower λ

 corresponds to a safer borrower. According to your analysis, you estimate that λ

 is equal to 0.002. What then is the probability that the borrower will default in the next hundred days, i.e. that FT100

?

 

 

2. What is the probability the borrower will default in the next two hundred days?

 

3. Suppose the economy goes into a sudden and unexpectedly harsh recession due to the Covid-19 pandemic. This causes you to revise your estimate of λ

   

    to 0.003. How does this change your answer to part a) and b) above?