Q1. AUTOMOBILE INSURANCE: MORAL HAZARD AND ADVERSE SELECTION
Data from an automobile insurance company show that claims frequency and size are affected by what type of insurance is chosen by an individual. The data are from an Israeli insurer for the years 1994-1999. Table 3.1 provides relevant data. Model the claims frequency using a Poisson distribution to allow for more than one claim per policy holder) and use a Lognormal distribution for the size of claims (with the mean given in Table 3.1 and standard deviation equal to the mean). Assume that 1,000 drivers are insured by the policies, in the proportions indicated in Table 3.1. Also assume that there is no correlation between policy types. Build a simulation model to
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Table 3.1 Insurance Policy Types
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Policy Type
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Average
Premium
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Deductible
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Claim
Frequency
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Average
Damage/Claim
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Percentage
Choosing
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Regular
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2,800
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1,400
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21.33%
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11,433
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76.72
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Low deductible
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3,640
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840
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27.76%
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10,233
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21.96
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High deductible
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1,960
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2,520
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15.05%
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13,600
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0.74
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Very high deductible
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1,920
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3,640
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11.37%
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10,750
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0.6
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a. Estimate the contribution margin (revenue minus direct costs from claims-that is, what is left to cover overhead costs and profits) per insured driver-for each policy type and for the total of all drivers insured by this company.
b. See whether the differences in premiums appear to match the differences in expected costs across policies.
c. Determine the overall probability that this company will have a positive contribution. What level of contribution do you estimate there is a 95% likelihood of achieving or exceeding?
Q2. AIRLINE DELAYS
The Bureau of Transportation Statistics provides on-time data for airlines in the United States. Exercise4-2.xlsx contains all flight data for January 2010, and Exercise4-2short.xlsx contains the data for flights departing from Denver International Airport during that month. (Use the larger file if you want to test your database skills for extracting the Denver data; the smaller file already contains the extracted data for those that do not want to download the large file.)
a. Count the number of flights with departure delays (including the cancelled flights) and count the number of flights with arrival delays. Based on the count, determine a probability that a flight will have a departure delay and the probability that a flight will have an arrival delay. Assuming that, during a typical January, there are 20,000 departure flights, build a binomial distribution object to simulate the number of delays.
b. Fit distributions (without parameter uncertainty) for the length of the departure delays and the length of the arrival delays (contingent on there being a delay). Choose distributions that visually appear to fit the data best.
c. Estimate the total monthly departure and arrival delays in minutes of flights leaving DIA (for a "typical" January). Provide a 90% confidence interval for this estimate.
Q3. AIRFARES
Exercise5-2.xlsx contains data for all continental U.S. domestic air routes that average at least 10 passengers per day.' Data are provided for the originating and terminating city (and airport code) for each route, along with the average one-way distance, average number of passengers per day, market share of the largest carrier, code for the largest carrier, and the average fare charged on the route.
a. Build a multiple regression model explaining airfares as a function of distance, passengers, and market share of the largest carrier. Interpret the meaning and statistical significance of each coefficient.
b. Suppose that, due to consolidation in the airline industry, we expect that the market shares of the largest carriers will increase on average by 10% and that the average number of passengers per day will increase by two (due to better scheduling and flight connections). Provide a point estimate of the impact on the average of the airfares across all routes that would result, using your regression model. Simulate the average fare across all routes without accounting for parameter uncertainty (Hint: following Section 5.7 use the standard error of the whole regression for this. Use the fact that the regression model goes through the means of all the variables; that is, the average fare as a function of average distance, average passengers, and average market share are a point on the regression line.)
c. Using the uncertainty inherent in the data, perform a parametric Bootstrap to simulate the resulting average airfare. Compare the probability that airfares will rise as a result of consolidation) for your models with and without parameter uncertainty.
Q4. AIRLINE LOAD FACTORS
Airline load factors are a primary driver of airline profitability (Of course, prices and costs also matter: A full plane will lose money if price is less than average cost and a half-empty flight can be profitable if price is greater than average cost.) Some low-cost airlines may be profitable at load factors as low as 64%, while others may require load factors of 100% or more.' Exercise6-2.xlsx contains data for the U.S. airline industry for passenger load factors for domestic and international operations over the period of 2000 through September 2010.
a. Inspect the data and comment on whether there appear to be any trends, seasonality, or randomness in the two series.
b. Fit a time series model to each series separately and predict the next 10 months of load factors (October 2010-July 2011).
c. Fit a multivariate time series to the two sets of data and compare and discuss your forecast 10 months out with your results for part b.
Assignment Files - https://www.dropbox.com/s/6z9vhonc2j6biqr/Assignment%20Files.rar?dl=0