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</li><li>Averagestockreturn:6<li>Stockreturnstandarddeviation:7<li>Tripcost:5,000</li> <li>Average stock return: 6%</li> <li>Stock return standard deviation: 7%</li> <li>Trip cost:16,000
  • Setup: Model the total fund at the end of each year.
Year 1
  • Added fund: 5,000</li><li>Cumulativefund(beginningofyear1):5,000</li> <li>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)NORM.INV(RAND(), 0.06, 0.07)
  • Total fund (end of year 1): (1+return rate)×cumulative fund(1 + \text{return rate}) \times \text{cumulative fund}
Year 2
  • Cumulative fund: Total fund from year 1 + 5,000</li><li>Stockfundreturnrate:Anotherrandomvariable(differentfromyear1).<ul><li>5,000</li> <li>Stock fund return rate: Another random variable (different from year 1).<ul> <li>NORM.INV(RAND(), 0.06, 0.07)</li></ul></li><li>Totalfund(endofyear2):</li></ul></li> <li>Total fund (end of year 2):(1 + \text{return rate}) \times \text{cumulative fund}</li></ul><h5id="year3">Year3</h5><ul><li>Cumulativefund:Totalfundfromyear2+</li> </ul> <h5 id="year3">Year 3</h5> <ul> <li>Cumulative fund: Total fund from year 2 +5,000
  • Stock fund return rate: Another random variable.
    • NORM.INV(RAND(),0.06,0.07)NORM.INV(RAND(), 0.06, 0.07)
  • Total fund (end of year 3): (1+return rate)×cumulative fund(1 + \text{return rate}) \times \text{cumulative fund}
  • Compare total fund at the end of year 3 with 16,000todetermineifJanecanaffordthetrip.</li></ul><h4id="modifyingthemean">ModifyingtheMean</h4><ul><li>Adding1tothemean(e.g.,using106<li>If1isaddedtothemean<em>within</em>the<code>NORM.INV</code>function,itshould<em>not</em>beaddedagainwhencalculatingthetotalfund.</li><li>Theadditionshouldoccureitherwithinthe<code>NORM.INV</code>functionorwhencalculatingthetotalfund,butnotboth.</li></ul></li></ul><h4id="homeworkproblemjenniferscollegefund">HomeworkProblem:Jennifer′sCollegeFund</h4><ul><li><strong>Scenario:</strong>Jennifer′sparentssaveforhercollegeeducation.</li><li><strong>Objective:</strong>Accumulate16,000 to determine if Jane can afford the trip.</li> </ul> <h4 id="modifyingthemean">Modifying the Mean</h4> <ul> <li>Adding 1 to the mean (e.g., using 106% instead of 6% in the <code>NORM.INV</code> function):<ul> <li>If 1 is added to the mean <em>within</em> the <code>NORM.INV</code> function, it should <em>not</em> be added again when calculating the total fund.</li> <li>The addition should occur either within the <code>NORM.INV</code> function or when calculating the total fund, but not both.</li></ul></li> </ul> <h4 id="homeworkproblemjenniferscollegefund">Homework Problem: Jennifer's College Fund</h4> <ul> <li><strong>Scenario:</strong> Jennifer's parents save for her college education.</li> <li><strong>Objective:</strong> Accumulate100,000 by the time she starts college in five years.
  • Initial Investment: 25,000</li><li><strong>AnnualSavings:</strong>25,000</li> <li><strong>Annual Savings:</strong>10,000 for the next four years, plus 10,000inthefifthyear.</li><li><strong>InvestmentSplit:</strong>Investmentsaresplitevenlybetweenastockfundandabondfund.</li><li><strong>StockFund:</strong><ul><li>Averageannualreturn:8<li>Standarddeviation:6<li><strong>BondFund:</strong><ul><li>Averageannualreturn:4<li>Standarddeviation:3<li><strong>KeyDifferencesfromMiniExample:</strong><ul><li>Initialfund.</li><li>Separateinvestmentattheendoftheinvestmentperiod.</li><li>Twoinvestmenttools(stockandbondfunds).</li></ul></li><li><strong>Setup:</strong><ul><li>Initialfundis10,000 in the fifth year.</li> <li><strong>Investment Split:</strong> Investments are split evenly between a stock fund and a bond fund.</li> <li><strong>Stock Fund:</strong><ul> <li>Average annual return: 8%</li> <li>Standard deviation: 6%</li></ul></li> <li><strong>Bond Fund:</strong><ul> <li>Average annual return: 4%</li> <li>Standard deviation: 3%</li></ul></li> <li><strong>Key Differences from Mini Example:</strong><ul> <li>Initial fund.</li> <li>Separate investment at the end of the investment period.</li> <li>Two investment tools (stock and bond funds).</li></ul></li> <li><strong>Setup:</strong><ul> <li>Initial fund is12,500 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.,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</li><li>RPI′sbiddingprice:5,000,000</li> <li>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=5,000,000 * 1.3 =6,500,000
    • Competitor's standard deviation: 5,000,000∗0.1=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)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.</li></ul></li><li><strong>Considerations:</strong><ul><li>Modelbidssubmittedbyeachofthefourcompetitors.</li><li>DetermineifRPIwinsbasedonallfourcompetitors′bids.</li></ul></li><li><strong>Cost:</strong><ul><li>The50,000.</li></ul></li> <li><strong>Considerations:</strong><ul> <li>Model bids submitted by each of the four competitors.</li> <li>Determine if RPI wins based on all four competitors' bids.</li></ul></li> <li><strong>Cost:</strong><ul> <li>The50,000 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).</li><li>The5,000,000).</li> <li>The50,000 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.