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Homework 5 Hints and Monte Carlo Simulation

Homework 5 Hints and Monte Carlo Simulation

Problem 1: Hawaii Trip Savings

  • Scenario: Jane saves for a Hawaii trip by investing in a stock fund.
  • Objective: Calculate the probability she'll have enough money by year four.
  • Parameters:
    • Annual savings: 5,000
    • Average stock return: 6%
    • Stock return standard deviation: 7%
    • Trip cost: 16,000
  • Setup: Model the total fund at the end of each year.

Year 1

  • Added fund: 5,000
  • Cumulative fund (beginning of year 1): 5,000
  • Stock fund return rate (random variable):
    • Use inverse CDF of normal distribution: NORM.INV(RAND(), mean, standard_dev)
    • NORM.INV(RAND(), 0.06, 0.07)
  • Total fund (end of year 1): (1 + \text{return rate}) \times \text{cumulative fund}

Year 2

  • Cumulative fund: Total fund from year 1 + 5,000
  • Stock fund return rate: Another random variable (different from year 1).
    • NORM.INV(RAND(), 0.06, 0.07)
  • Total fund (end of year 2): (1 + \text{return rate}) \times \text{cumulative fund}

Year 3

  • Cumulative fund: Total fund from year 2 + 5,000
  • Stock fund return rate: Another random variable.
    • NORM.INV(RAND(), 0.06, 0.07)
  • Total fund (end of year 3): (1 + \text{return rate}) \times \text{cumulative fund}
  • Compare total fund at the end of year 3 with 16,000 to determine if Jane can afford the trip.

Modifying the Mean

  • Adding 1 to the mean (e.g., using 106% instead of 6% in the NORM.INV function):
    • If 1 is added to the mean within the NORM.INV function, it should not be added again when calculating the total fund.
    • The addition should occur either within the NORM.INV function or when calculating the total fund, but not both.

Homework Problem: Jennifer's College Fund

  • Scenario: Jennifer's parents save for her college education.
  • Objective: Accumulate 100,000 by the time she starts college in five years.
  • Initial Investment: 25,000
  • Annual Savings: 10,000 for the next four years, plus 10,000 in the fifth year.
  • Investment Split: Investments are split evenly between a stock fund and a bond fund.
  • Stock Fund:
    • Average annual return: 8%
    • Standard deviation: 6%
  • Bond Fund:
    • Average annual return: 4%
    • Standard deviation: 3%
  • Key Differences from Mini Example:
    • Initial fund.
    • Separate investment at the end of the investment period.
    • Two investment tools (stock and bond funds).
  • Setup:
    • Initial fund is 12,500 for each of the stock fund and the bond fund.
    • For each year, generate random variables for both stock and bond fund return rates.
    • Five random variables representing the return rate for each year are required for each fund.
    • The final 10,000 (i.e., 5,000 for each fund) added at the end of the fifth year does not undergo any further investment return.

Problem 2: Project Bidding

Hint Problem: RPI Road Construction

  • Scenario: RPI bids on a county road construction project.
  • Objective: Calculate the probability that RPI will win the bid.
  • Parameters:
    • RPI's estimated cost: 5,000,000
    • RPI's bidding price: 5,700,000
    • Competitor's bidding price:
      • Normally distributed
      • Mean: 30% over the cost
      • Standard deviation: 10% of the cost
  • Calculations:
    • Competitor's mean bidding price: 5,000,000 * 1.3 = 6,500,000
    • Competitor's standard deviation: 5,000,000 * 0.1 = 500,000
    • Generate random variable for competitor's bidding price:
      • NORM.INV(RAND(), mean, standard_dev)
      • NORM.INV(RAND(), 6500000, 500000)
    • Determine if RPI wins: IF(RPI's bidding price < Competitor's bidding price, 1, 0).
    • Calculate profit: IF(Win, RPI's bidding price - Cost, 0).
  • Additional Information:
    • To avoid negative bidding prices, use MAX(NORM.INV(…), 0). This is especially useful when the mean is small compared to standard deviation.

Actual Homework Problem

  • Differences from Hint Problem:
    • Four competitors instead of one.
    • Cost of putting together a bid is estimated to be 50,000.
  • Considerations:
    • Model bids submitted by each of the four competitors.
    • Determine if RPI wins based on all four competitors' bids.
  • Cost:
    • The 50,000 cost of bid should not be included when estimating the bidding price.
    • The bidding price is a normal distribution centered at 20% over the cost (i.e., 5,000,000).
    • The 50,000 cost is reflected in the profit/payoff calculation.
    • If RPI loses the bid, the payoff is negative ($-50,000).

Problem 3: Bonus - New Jersey Electricity Power

  • Scenario: New Jersey electric power (NJEP) manages power supply during a heatwave.
  • Objective: Decide how many peaker plants to activate to minimize brownouts or rolling blackouts.
  • Parameters:
    • Baseline plants: 18,000 MW
    • Peaker plants: Three plants, each adding 500 MW
    • Brownout reduction: Up to 4% reduction in overall power consumption.
    • Peak load demand (normally distributed):
      • Varies each day (mean and standard deviation given in table).
  • Decision Variable: Number of peaker plants to fire up.
  • Objective: Reducing probability of brownouts and rolling blackouts.
  • Key Information:
    • Number of peaker plants must be decided before the five-day heatwave period due to ramp-up time.
  • Modeling:
    • Calculate the probability of brownouts and rolling blackouts for each day considering:
      • Electricity demand (random variable).
      • Thresholds for brownout and blackout.
  • LP (Linear Programming) is not Needed:
    • No linear constraints.
    • The objective function modeling the relationship between number of peaker plants and brownout probability cannot be expressed using linear functions.