1. (6 pts total) A couple decides to set up a trust for their newborn child. They invest a lump sum into a mutual fund, which accumulates every month. They make no further investments in the fund. Each month, the distribution of the amount of interest the mutual fund bears is normal, with a mean of 0.5% and a standard deviation of 1%. The monthly returns are independent of one another.
(4 pts for setting up X & Y correctly) Let 𝑋, and 𝑌 represent the distribution of returns after 10 and 30 years, respectively. Create a Monte Carlo Simulation with 10,000 trials to model the behavior of the account, then answer the following questions:
a) (.5 pt) What are the mean and standard deviation of 𝑋?
b) (.5 pt) What are the mean and standard deviation of 𝑌?
c) (.5 pt) What is the correlation between 𝑋 and 𝑌?
d) (.5 pt) Of the distributions listed below, which best fits 𝑌?
i. Cauchy
ii. Exponential
iii. Gamma
iv. Logistic
v. Lognormal
vi. Normal
vii. Weibull
[Hints: You don’t need to create a dataframe and define random variables for each month. Instead, create a matrix, and use the apply function in conjunction with the prod function. Also, you can create a list of distribution-fit models using lapply, then call goodness-of-fit functions on that list of models directly.]
2. (6 pts total) Space Violence 5 is the latest hit video game, and Acme has won an exclusive contract to sell Space Violence themed umbrellas and water guns online. It costs Acme $12 to produce an umbrella and $20 to produce a water gun. The umbrellas retail for $18 and the water guns retail for $25. Once the month is over, Acme does not believe anyone will purchase Space Violence merchandise at full price, so they’ll sell any remaining umbrellas for $8 and water guns for $13. Acme’s market research indicates the following:
The Spearman correlation between umbrella demand and water gun demand is -0.7
Umbrella demand for the month follows a triangular distribution, with minimum 15,000; most
likely 45,000; and maximum 60,000.
Water gun demand for the month follows a gamma distribution, with shape parameter 100 and
scale parameter 1000
a) (1 pts) If Acme chooses to produce 50,000 umbrellas and 120,000 water guns, what would be their expected profit?
b) (2 pts) How many of each product should Acme produce if they wish to maximize their expected profit?
c) (2 pts) How many of each product should Acme produce if they wish to maximize their chance of earning at least 600,000?
d) (1 pts) How many of each product should Acme produce if they wish to maximize their expected profit, while ensuring they have at least a 5% chance of earning 760,000?
3. (Extra credit: 3 pts total) A hotel chain decides to offer an exciting new service to its Platinum Members: the “No Reservation No Problem” program. As part of this program, the hotel chain sets aside rooms it won’t book to other clients. If such a room is unoccupied, a Platinum Member can book it without making a reservation and stay as long as they like.
The hotel reasons that the number of nights a Platinum Member would stay in such a room follows a geometric distribution with parameter p=1/4, plus one night (since the minimum stay is one night). The amount of time the room would stay empty before the next Platinum Member would book the room also follows a geometric distribution with parameter p=1/5, though it is possible for the room to be empty for 0 days before a new Platinum Member books it. Demand does not depend on the day of the week. To defray the cost of the program, Platinum Members pay $30 more per night than the average revenue the hotel would make on the room. However, leaving a room empty loses the hotel $40 per night in opportunity costs and upkeep.
(1.5 pts for setting up the simulation) Assume a hotel sets aside one room for this program. Use a Monte Carlo simulation with 10,000 trials to determine the distribution of the hotel’s profit (or loss) on that room over the course of 365 days.
a) (.25 pts) On average, how much will the hotel make or lose by offering this program?
b) (.25 pts) What is the standard deviation of the hotel’s profit (or loss)?
c) (.25 pts) How many trials would they have to run to ensure their estimate of the mean were
within $15 with 90% confidence?
d) (.25 pts) What is the likelihood that the hotel will lose more than $1,500?
e) (.25 pts) What is the 𝑉𝑎𝑅.95 of the loss they’ll incur? (or 𝑉𝑎𝑅.05 of their profit?)
f) (.25 pts) What is the 𝐶𝑉𝑎𝑅.95 of the loss they’ll incur? (or 𝐶𝑉𝑎𝑅.05 of their profit?)
[Hint: The approach I would take to start solving this problem is setting up a matrix where odd columns represent the number of empty days and even columns represent the number of occupied days. Make sure there are enough columns to ensure that the sums of each row are over 365.]