Unformatted text preview: Estimating Loss Distributions Estimating
With reliable historical data available: In some cases risk managers have data on past experiences of losses, or on others’ (e.g. industry) experiences of losses Construct summary measures of losses Loss frequency Loss severity Loss per exposure unit Analyze historical loss patterns from firm and/or industry
Analyze causes of loss Analyze timing of claims Analyze locational or interpersonal patterns of claims Estimating Loss Distributions Estimating Using data on loss frequency and severity, construct estimates of loss frequency and severity distributions Apply statistical analysis to estimate probability distribution of losses Two main approaches:
Approximate losses using a known probability distribution Computer simulation of loss distributions Loss Frequency and Severity Loss
Loss Frequency (F) The number of losses per exposure unit per time period Loss Severity (S) The dollar amount of loss per accident or occurrence Loss per Exposure (X) The dollar amount of loss per exposure unit per time period (F and S combine to determine X) Frequency and Severity Frequency
10,000 employees 300 injuries $600,000 in total injury costs Loss frequency (F): Loss severity (S): Loss per employee (X): Monte Carlo Simulation Monte Make assumptions about distributions for Make frequency and severity of individual losses frequency Based on data or constructed estimates Randomly draw from each distribution and Randomly calculate the firm’s total losses under alternative risk management strategies risk Redo step two many times to obtain a distribution Redo for total losses under each of the alternative strategies strategies Compare strategies (distributions) Simulation Example – HN Text Simulation
1. Assume loss frequency follows a Poisson 1. distribution distribution Important property: Poisson distribution gives the Important probability of 0 claims, 1 claim, 2 claims, etc. 2. Assume that the expected value of the probability distribution of loss frequency depends on other uncertain events Expected value equals Expected 20 with probability 20 30 with probability 40 with probability 1/3 1/3 1/3 Example, Cont. Example,
3. Assume loss severity follows a Lognormal 3. distribution with distribution
expected value = $100,000 standard deviation = $300,000 note skewness Simulation Example Simulation
Frequency Distribution w ith Ex pected Va lue Equal to 30
PROBABILITY 0.25 0.2 0.15 0.1 0.05 0 24 12 18 30 36 42 48 Num be r of Claim s S ample Frequency Distribution w ith Uncertain Expe cte d Value (1000 trials)
PROBABILITY 0.25 0.2 0.15 0.1 0.05 0 24 12 18 30 36 42 48 Num be r of Claim s Sample Loss Severity Distribution (1000 tria ls)
PROBABILITY 0.12 0.1 0.08 0.06 0.04 0.02 0 0.0075 0.6 1.2 1.8 2.4 Los s in M illions 54 3 0 6 54 0 6 Simulation Example Results Simulation Example Results
No Insurance
0.2 0.18 PROBABILITY 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 10 12 13 0 1. 4. 7. 0 3 6 9 5 5 Values in Millions 5 .5 .5 Simulation Example  Results Simulation
Statistic Mean value of losses Standard deviation of losses Maximum probable loss at Maximum 95% confidence level 95% Maximum value of losses Losses (without insurance) Losses $3,042 $1,839 $6,462 $18,898 Probability that losses ≤ $6 million 92.7% Probability Example Example A firm has 120 automobiles It observed the following number of accidents per automobile last year:
Accidents/auto 0 1 2 # Autos Experiencing 60 36 24 What is the expected value of accident frequency and the variance of accident frequency? Sample Statistics Estimates of the expected value and variance of a probability distribution when the true probability distribution is not observed. Sample mean: [x1 + x2 + … + xN]/N Sample variance: Σ(xiE(X))2/N
where in this case E(X) is the sample mean we sometimes divide by N1 to account for that Note: these statistics can and usually will differ from population expected value and standard deviation. Coin flip example: H = +1, T = 1 Example, Cont. Example, A firm has 120 automobiles It observed the following number of accidents per automobile last year:
Accidents/auto 0 1 2 # Autos Experiencing 60 36 24 What is the expected value of accident frequency and the variance of accident frequency? Example, continued Example, The firm observed a total of 84 accidents last year (36*1 + 24*2) In those accidents, it observed the following costs per accident:
Cost/Accidents $500 $3,000 $12,500 $35,000 # Accidents Experiencing 34 25 17 8 What is the expected value of accident severity and the variance of accident severity? Example, continued Example, What is the expected value and variance of losses per car (X)? Formulas for relationship between F, S and X:
E(X) = E(F)*E(S) Var(X) = E(F)*Var(S) + [E(S)]2*Var(F) Why F and S Separately? Why This is what the risk manager observes based on accidents during some period of time (i.e. available) Loss forecasting: disaggregate data allow for better identification of trends Trends in F and S are not likely to be the same S must be adjusted for inflation Loss control: disaggregate data allow for better identification of patterns Number and timing of loss events Relative number of large versus small accidents Forecasting Expected Losses Forecasting
Year Number of Exposures Total Number of Accidents Total Accident Costs 2003 2004 2005 2006 2007 2008 1,000 1,000 1,000 1,200 1,200 1,200 25 35 31 29 27 34 $52,500 $105,000 $86,800 $89,900 $86,400 $108,800 Forecasting Expected Losses Forecasting
Year Disaggregate the data Disaggregate into average loss frequency and severity. frequency Then apply trend or Then regression or other statistical analysis to estimate expected value of future losses. of Number of Number Accidents Accidents Average ExampleAverage Cost per Accident Accident 2003 2004 2005 2006 2007 2008 25 35 31 29 27 34 $2,100 $3,000 $2,800 $3,100 $3,200 $3,200 Forecasting Issues Forecasting Constant number of exposure units per year? If yes, data on total number of accidents may be used. If no, calculate historical accidents per exposure unit. Probability distributions stable? If yes, average values of accidents and costs may be used If no, calculate trend or predictors of the variables (e.g. using regression If analysis) analysis) Adjust for factors affecting loss patterns Inflation rate zero? If yes, historical average severity may be used If no, construct constant dollar severity averages and apply projected If inflation rate in severity forecast inflation ...
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 Spring '07
 TENNYSON
 Normal Distribution, Variance, Probability theory

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