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AD 616: Enterprise Risk Analytics

Assignment 2

What to submit?

Please submit (i) a word file explaining in detail your answers to each question (you can use screenshots of the R Code to explain your answers) AND (ii) an R file for each question. For each question, make sure you develop the model and present the simulation results – the R code should be self-explanatory. The assessment of your work will include both the accuracy and the clarity of your word file and the R Code.

1) Answer the following two questions.

a) According to the central limit theorem, the sum of n independent identically distributed random variables will start to resemble a normal distribution as n grows large. The mean of the resulting distribution will be n times the mean of the summands, and the variance n times the variance of the summands. Demonstrate this property using Monte Carlo simulation.  Over 10,000 trials, take the sum of 100 uniform random variables (with min=0 and max=1). Note: the variance of the uniform distribution with min 0 and max 1 is 1/12. Include:

i) A histogram of the results of the MC simulation

ii) A density plot of a normal distribution with the appropriate mean and standard deviation

iii) The mean and standard deviation of the MC simulation. 

b) According to probability theory, if  are independent and uniform distributed random variables with min=0 and max=1, then the k^th lowest will follow a beta distribution with parameters shape1=k, shape2=n+1-k. Demonstrate this property using Monte Carlo simulation. Simulate 10,000 trials. For each trial, generate 10 uniform random variables and select the 3^rd lowest. Include:

i) A histogram of the results of the MC simulation

ii) A density plot of the beta distribution with the appropriate parameters

iii) The mean and standard deviation of the MC simulation

1. Camer Pharmaceuticals is testing a new product in the market¹. The demand for the new product is estimated to be normally distributed with a mean 1,800,000 and standard deviation 220,000. The demand is estimated to grow at a rate of 3% per year. The R&D costs are estimated to be between $550 millions of dollars and $850 millions of dollars with a most likely value of $700 millions of dollars. Clinical trial costs are estimated to be between $130 millions of dollars and $170 millions of dollars with a most likely value of $150 millions of dollars. There are competitors in the market, and Camer Pharmaceuticals estimates that their market share in the first year will be any number between 4% and 11%, with each number being equally likely. The company estimates that their market share will grow by 20% each year. A monthly prescription is estimated to generate a revenue of $240. The variable costs are estimated to be $40. Develop a simulation model that calculates the net present value (NPV) of this project over 3 years assuming an interest rate of 10%. Run the simulation for 10000 iterations and answer the following questions.

What is the distribution of the NPV (mean and standard deviation)?

What are the first quartile, median, and third quartile of NPV?

What is the probability that NPV will not be positive?

We can be 90% certain the NPV will be at least x. Solve for x. 

We can be 90% certain the cumulative net profit in the third year will be at least y. Solve for y.

What is the 95% confidence interval for the mean NPV? Interpret the resulting confidence interval.

What is the number of iterations needed if we want to estimate the NPV within $1,500,000 with 95% confidence?

What are the correlations between the net present value of the project and the four modelled sources of uncertainty (demand, initial market share, R&D costs, and clinical trial costs?)