1. 1 Math 479 / 568 Casualty Actuarial Mathematics Fall 2014 University of Illinois at Urbana-Champaign Professor Rick Gorvett Session 14: Credibility October 21, 2014

2. 2 Agenda Credibility –Basic concept –Uses of credibility theory –Types of credibility calculations Classical / limited fluctuation Bühlmann Bayesian

3. 3 The Basic Concept “ The concept of credibility has been the casualty actuaries ’ most important and enduring contribution to casualty actuarial science. ” –Matthew Rodermund, Preface, Foundations of Casualty Actuarial Science, 1 st edition

4. 4 The Basic Concept (cont.) Credibility: –Ultimately, we want to know what the “ optimal ” combination of data is New versus old data Individual versus group data –A function of variability Target-shooting example

5. 5 Uses of Credibility Theory Example 1 – New versus old (current) Premium rate in effect during 2014, based on 2013 data –For 2015, we could just “ update ” the 2014 rate (e.g., for loss trend) –But we now also have 2014 data –How should we “ combine ” the current rate with the new data?

6. 6 Uses of Credibility Theory (cont.) Example 2 – Individual versus group Insurance applicant has $0 of actual historical losses –Applicant is classified in a group with an annual rate of $1,000 –How shall we “ combine ” actual ($0) individual loss experience with the group indication?

7. 7 Uses of Credibility Theory (cont.) Mathematically, the credibility-weighted premium is: where Z = credibility factor, 0 ≤ Z ≤ 1. = new data, or individual experience. M = old data, or class rate

8. 8 Types of Credibility (1) Classical (limited fluctuation) –Quantity (number of exposures, claims, or dollars of loss) needed for data to achieve “ full credibility ” (Z = 1) –Formula for cases where Z < 1. (2) Bühlmann (least squares) –Motivation: target-shooting example (3) Bayesian –Prior and posterior probabilities

9. 9 Classical Credibility Two “ parameters ” : within +/- k of the expected value at least 100p% of the time Volume needed for full credibility can be expressed in terms of –Number of exposures –Number of expected claims –Aggregate dollars of loss

10. 10 Classical Credibility (cont.) Formula for full-credibility standard (in terms of number of claims) for aggregate losses (or pure premium):

11. 11 Classical Credibility (cont.) Full-credibility formula parameters: –  (y) = (1+P) / 2 –P = probability that losses (or pure premium) is within +/- k of true mean value –f indicates frequency distribution; S indicates severity distribution

12. 12 Classical Credibility (cont.) Partial credibility –If volume is greater than or equal to the standard for full credibility, then Z = 1.00 –If volume is less, then

13. 13 Bühlmann Credibility Z = n / (n + k), where k = v / a –n = “ volume ” of data or experience –v = expected value of the process variance –a = variance of the hypothetical means Relationships: –n increases  Z increases –v increases  Z decreases –a increases  Z increases

14. 14 Bayesian Credibility Adjust the probabilities associated with possible distributions or classifications, in light of emerging experience –A priori –A posteriori

15. 15 Example # 1 What is the full credibility standard if the desired precision is a 90% chance of being within +/- 5% of the true value, when the frequency is assumed to be Poisson, and all claims are assumed to have the same size?

16. 16 Example # 2 What is the full credibility standard if the desired precision is a 95% chance of being within +/- 10% of the true value, when the frequency is assumed to be Poisson, and the claim size distribution has a coefficient of variation of 2.0?

17. 17 Example # 3 If the full-credibility standard is 2,000 claims, what is the credibility assigned to data with 800 claims?

18. 18 Example # 4 You are given the following information: (i) Claim counts follow a Poisson distribution. (ii) Claim sizes follow a lognormal distribution with a mean of 5,000 and a variance of 64,000,000. (iii) Claim sizes and claim counts are independent. (iv) The number of claims in the first year was 431. (v) The aggregate loss in the first year was 2.25 million. (vi) The manual pure premium for the first year was 1.80 million. (vii) The exposure in the second year is identical to the exposure in the first year. (viii) The full credibility standard is to be within 4% of the expected aggregate loss 90% of the time. (a)Using classical (or “ limited fluctuation ” ) credibility, determine the full credibility standard based on the number of claims. (b)Using your result from (a), determine the credibility associated with the observed number of claims in the first year. (c)Use your result from (b) to estimate the classical credibility pure premium estimate, in millions, for the second year.