AD 616: Enterprise Risk Analytics
Assignment 5
What to submit?
Please submit (i) a Word file explaining in detail your answers to each question (you can use screenshots of the Python to explain your answers) AND (ii) an ipynb file with a separation for each question. For each question, make sure you develop the model and present the simulation results – the ipynb file should be self-explanatory. The assessment of your work will include both the accuracy and the clarity of your Word file and the Python Code.
Question 1: Bayesian Investment Decision Problem: Real estate decision with a consultant
Hint: You do not need Python code to answer this question
A real estate speculator is considering buying property at a resort island for $400 thousand. The local government is considering a proposal to rezone the property for commercial use, which has the potential to increase its value drastically. Once the government makes its decision, the speculator would lose the chance to purchase the property. As it stands, there’s a 30% chance the property will be rezoned. If it were rezoned, there’d be a 20% chance the speculator can incite a bidding war over the property and sell it for $3 million; there’d be a 40% chance he could interest a developer in the property and sell it for $1.8 million, and even if neither possibility played out, he could still sell it for $700 thousand. If the property isn’t rezoned, there’s a 25% chance he could resell it and recoup $300 thousand, but failing that, he would be stuck with a useless property, which would increase his liability by an additional $100 thousand.
The real estate developer discusses his options with a trusted consultant, who offers to look into the political situation on the island for a flat fee of 50 thousand dollars. The consultant has a good reputation; she has a 90% probability of correctly identifying that a property is going to be rezoned, and a 70% probability of correctly identifying that the motion to rezone a property will fail.
While he’s considering her offer, she mentions another possibility: if he’s willing to pay her an additional $75 thousand (for a total of $125 thousand), she’ll use that money to make some generous campaign contributions to some of the island’s key government officials in exchange for future considerations. She estimates, with this strategy, the chance the property would be rezoned after the speculator’s purchase would increase to 60%, but there would be a 5% chance one of the island’s less enterprising functionaries would catch wind of her efforts. In this instance, the speculator would end up losing the amount he paid her, eating the cost of the property, and paying an additional $1.5 million in fines.
Question 1 tasks:
1. (1pt) What is the expected value (EV) of buying the property outright without hiring the consultant?
2. (2pts) What is the EV if the speculator only hires the consultant for 50K (report only)? Assume the speculator will only buy when the report is positive.
3. (2 pts) What is the EV if the speculator hires the consultant for both reporting and campaigns (125K)? Which option should the speculator choose?
Question 2, Markov Chain Monte Carlo:
Imagine you are working for a financial analytics firm and are tasked with assessing the expected return of a volatile stock. The stock's return is believed to follow a normal distribution with unknown mean μ and known standard deviation σ = 2% (i.e., 0.02). You are given the following historical monthly return data (in decimal format):
0.021, 0.017, -0.005, 0.023, 0.019, 0.025, -0.003, 0.020, 0.018, 0.021
You are to construct a Bayesian model to infer the posterior distribution of the unknown mean return μ. Assume the following prior: μ ~ Normal(0, 0.01)
1. (1 pt) Write the unnormalized log-posterior function for μ, given the data and assumptions. Hint: The formula for the unnormalized log-posterior is
2. (2 pts) Implement the Metropolis-Hastings algorithm to sample from the posterior distribution of μ, using a normal proposal Run the sampler for 10,000 iterations with τ=0.00, starting at μ=0.
3. (1 pt) Plot the histogram of the samples and report the posterior mean and 95% credible interval for μ.
4. (1 pt) Comment on how prior beliefs influence the posterior. What happens if you use a wider prior