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USE OF BASIC STATISTICS IN INSURANCE PRICING AND RISK ASSESSMENT March 2017

USE OF BASIC STATISTICS IN INSURANCE PRICING AND RISK ASSESSMENT March 2017. Agenda. Statistical concepts relating to insurance: (1) Risk data (2) Presentation of risk data (3) Statistical measurement (4) Deriving probability (5) Probability distributions (6) Regression and correlation

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USE OF BASIC STATISTICS IN INSURANCE PRICING AND RISK ASSESSMENT March 2017

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  1. USE OF BASIC STATISTICS IN INSURANCE PRICING AND RISK ASSESSMENT March 2017

  2. Agenda Statistical concepts relating to insurance: (1) Risk data (2) Presentation of risk data (3) Statistical measurement (4) Deriving probability (5) Probability distributions (6) Regression and correlation Practical examples to illustrate the above concepts

  3. (1) Risk data

  4. Risk data Database • Use existing database, or • Create a database by: (i) using published data (ii) gathering data - direct observation - interviews - experiments - questionnaires

  5. Risk data Compare: • Population • Stratified sampling • Random sampling

  6. (2) Presentation of risk data

  7. Presentation of risk data TABLES • Simple tables • Compound tables • Relative figures • Frequency and severity • Effect of mix change

  8. Presentation of risk data Claims Analysis: a comparison of frequency and severity Cost (£) Frequency % Severity % 0<1,000 9,000 60% £3.00m 22% <2,000 3,750 25% £4.25m 31% <3,000 1,500 10% £3.10m 23% <4,000 450 3% £1.75m 13% <5,000 300 2% £1.45m 11% Total 15,000 100% £13.55m 100%

  9. Presentation of risk data Effect of mix change It is important to look at the elements which make up a total, not just the total itself

  10. Presentation of risk data GRAPHS • Single line graphs • Multiple line graphs • X-Y graphs

  11. Presentation of risk data BAR CHARTS • Simple bar charts • Multiple bar charts • Component bar charts

  12. Presentation of risk data PIE CHARTS • Simple pie charts • Enhanced pie charts PICTOGRAMS

  13. Presentation of risk data FREQUENCY DISTRIBUTIONS Unordered array Ordered array Frequency distribution Group frequency distribution

  14. Presentation of risk data To prepare a grouped frequency distribution from raw data: • the highest and lowest values (the range) • decide upon the number of classes (usually 4, 5 or 6) • divide the entire range by the number of classes • decide how to display the data (continuous or discrete; size of classes) • allocate the individual values to the classes

  15. Presentation of risk data Discrete Variables can only be whole numbers, fractions are impossible. Therefore they can be represented by class limits of 0-9; 10-19; 20-29 and so on. Continuous Variables can be any fraction of a number, represented by class limits 0<10, 10<20 and so on.

  16. Presentation of risk data Relative Frequency the percentage of recorded figures in each class as a measure of the total number of raw data items. Cumulative frequency the addition of each frequency class to the total of its predecessors. Compare “less than” and “more than” Histogram the area represents the actual values for each frequency distribution. Horizontal axis (continuous scale) records class limits, vertical axis records frequency values

  17. Presentation of risk data Relative and Cumulative Frequencies Cost Frequency Relative Cumulative Frequency Frequency 0<150 20 50% 20 40 150<300 10 25% 30 20 300<450 5 12.5% 35 10 450<600 3 7.5% 38 5 600<750 2 5% 40 2 Total 40 100%

  18. Presentation of risk data Ogives are graphs of cumulative frequency distributions horizontal axis represents class limits vertical axis represents cumulative frequency Frequency polygon • add a class interval at each end of the histogram • mark the mid points at the tops of each of the distribution rectangles • join the mid points with a straight line • the area of a frequency polygon = area of the histogram Frequency curve is a free-hand drawing of shape of the curve

  19. (3) Statistical measurement

  20. Location - where the data is placed in the whole span of possible data variables available Dispersion - the extent to which individual items in a set of values differ from each other in magnitude Skew - the clustering of the data around the mean, median and mode Statistical measurement

  21. LOCATION Arithmetic Mean Geometric Mean Median Mode Statistical measurement

  22. Arithmetic mean for list of values Add up all the data items and dividing by the number of items Arithmetic mean = x n Statistical measurement

  23. Arithmetic mean for frequency distribution e.g. time it takes to settle a claim in ABC Insurance plcTime in days Number of claims x f fx 20 50 1,000 21 65 1,365 26 40 1,040 29 20 580 31 5 155 f = 180 fx = 4,140 Arithmetic mean = fx = 4,140 = 23 days f 180 Statistical measurement

  24. Arithmetic mean for grouped data Travel claims costs £No 150<160 10 160<170 20 170<180 15 180<190 12 mid-pts f x fx 150<160 10 155 1550 160<170 20 165 3300 170<180 15 175 2625 180<190 12 185 2220 _ 57 9,695 So x = f x = 9,695 = £170.09 f 57 Statistical measurement

  25. Important Features of the Arithmetic mean • Most commonly used form of average • Involves all values in a distribution • Advantage: completeness • Disadvantage: easily distorted by extreme values. • To avoid this disadvantage, use the median or the mode • Used to work out the standard deviation • Can produce ‘impossible values’ in the case of discrete data (example: average of 2.1 children in a family) • Not applicable where relative changes are being averaged. In such cases the geometric mean must be used. Statistical measurement

  26. Geometric mean Used when relative changes in one variable are being averaged Statistical measurement

  27. Median for list of values (1) Arrange the data values in numerical order (2) For odd-numbered array of values, take the middle number (3) For even-numbered array of values, take the two middle values and calculate the arithmetic mean Statistical measurement

  28. Statistical measurement Median for a frequency distribution No. of clerksFrequencyCumulative Frequency 25 1 1 26 2 3 27 1 4 28 5 9 29 9 18 30 11 29 29 Median number of clerks is 29

  29. Statistical measurement Median for grouped data Claims costs (£) FrequencyCumulativeFrequency 140 - 159 7 7 160 - 179 20 27 180 - 199 33 60 200 - 219 25 85 220 - 239 11 96 240 - 259 4 100 100

  30. Statistical measurement Finding the median for grouped data • Find the class having the median value in it • Decide on the class boundaries • Find out how far in to the class you must go (consult the cumulative frequency distribution) • Determine the class interval and multiply this by the distance you wish to move into it • Add this to the lower boundary of the class See formula

  31. Mode for list of values The most frequently occurring value Mode for grouped data Mode = mean – 3(mean – median) Statistical measurement

  32. Statistical measurement DISPERSION Important to know the distribution or variance of the data values around the location Range the difference between the highest and lowest values. Quartile deviation The difference between the top and bottom quarters of the values indicates the inter-quartile range. Divide by 2 to get quartile deviation.

  33. Statistical measurement Standard Deviation the most satisfactory measure of distribution as it uses all the values Coefficient of variation used to compare the relative variability, or dispersion, of two or more sets of figures

  34. Statistical measurement Standard deviation For a simple list: _ s= (x- x)2 n For grouped data (adapted for midpoints) _ s= f(x- x)2 f s= fx2 - f x 2 f f

  35. Statistical measurement Distribution with many small values (example: claims under household policies) SKEW Symmetrical Distribution: Distribution with many large values (example: claims aviation insurance) The difference between mean and median can be used to measure Skew: Skew = 3 (mean - median) standard deviation

  36. Statistical measurement Normal distribution a frequency distribution which shows a symmetrical curve peaking at the centre. Mean median and mode coincide Positive Distribution the peak lies to the left of centre; the mean is dragged to the right due to high outliners; the mode is the peak of the distribution Negative Distribution the peak lies to the right of centre; the mean has been dragged to the left due to very low outliners; the mode is the peak of the distribution Pearson’s Coefficient measures the degree of skew – I.e. how far of Skewness the mean is from the mode; the lower the value calculated, the nearer the data is to a normal distribution; formula automatically gives the direction of skew mean mode

  37. (4) Deriving probability

  38. Deriving probability PROBABILITY The contributions of the many will be used to pay for the losses of the few To know how much each insured must contribute to the pool, we must know what the likely losses are going to be Probability theory is a formal mechanism for measuring likelihood

  39. Deriving probability Deriving probabilities A priori probability is one that applies when all the possible outcomes of the event are known before the event occurs Relative Frequency based upon empirical (historical) information of what has happened in the past Subjective probability deal with occurrences that have not happened at all before or very infrequently

  40. Deriving probability Probability rules Alternative Events P(A or B)– “additions” rule (1) Events mutually exclusive (2) Events not mutually exclusive - need to remove the aspect of double counting (best visualised by Venn Diagrams) Joint Events P(A and B) – “multiplication” rule (1) Independent events (2) Dependent events

  41. Deriving probability Probability rules Alternative events: Mutually exclusive P(A or B) = P(A) + P(B) Not Mutually exclusive P(A or B) = P(A) + P(B) - P(A and B) Joint Events: Independent P(A and B) = P(A) x P(B) Dependent P(A and B) = P(A) x P(B/A)

  42. (5) Probability distributions

  43. Probability distributions Expected value When tossing a coin 100 times you would expect to get 50 heads because ½ times 100 = 50 When rolling a die 60 times you would expect 10 sixes because 1/6 times 60 = 10  Expected value E = P(x).x In frequency distributions the expected values of the various outcomes have to be added to find the total: E = P(x).x

  44. Probability distributions Expected Frequency Number Number Probability Expected number of of thefts of shops distribution thefts per shop x P(x) P(x).x 0 700 0.7 0 1 200 0.2 0.2 2 60 0.06 0.12 3 30 0.03 0.09 4 100.010.04 1,000 1.00 0.45

  45. Probability distributions Law of large numbers The actual number of events occurring will tend towards the expected number where there are a large number of similar situations

  46. Probability distributions Expected severity Cost Frequency Probability Midpoint Expected per theft (£) distribution cost loss per theft P(x) x P(x).x 0<300 270 0.6 150 90.0 300<600 135 0.3 450 135.0 600<900 27 0.06 750 45.0 900<1,200 14 0.03 1050 31.5 1,200<1,500 40.01 1350 13.5 450 1.00 315.0

  47. Probability distributions Premium calculation: Expected number of thefts per shop: 0.45 Expected loss per theft: £315.00 Pure premium per shop: £315 x 0.45 = £141.75

  48. Probability distributions NORMAL DISTRIBUTION Symmetrical bell shaped curve Mean under the apex, coincides with mode and median Tails never touch the horizontal Specific areas around the mean can be measured

  49. Probability distributions The width of the bell depends upon the actual spread of the distribution of data measured by the Standard Deviation. Data with the same mean but different Standard Deviations will produce different curves Knowing the mean and standard distribution for different groups of data enable comparisons to be made between them and a normal distribution curve

  50. 34.13% 34.13% 13.59% 13.59% 2.15% 2.15% 0.13% 0.13% -3 -2 -1  +1 +2 +3 68.26% 95.44% 99.74% Probability distributions Percentages under the normal curve 34.13% 34.13% 13.59% 13.59% 2.15% 2.15% 0.13% 0.13% -3 -2 -1  +1 +2 +3 68.26% 95.44% 99.74%

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