STAT 479: Short Term Actuarial Models

Transcription

1 STAT 479: Short Term Actuarial Models Jianxi Su, FSA, ACIA Purdue University, Department of Statistics Week 1.

2 Some important things about this course you may want to know... The course covers a large portion of the learning objectives for the Exam C. Two mid-term exams and four quizzes. Three major components: Frequency, severity, and aggregation models (Exam 1); Construction of empirical and parametric models (Exam 2); Credibility. Part 1. Models. Final exam is cumulative. Part 2. Estimations. Part 3. Applications. The course will cover a lot of theoretical contents learning the theories is very valuable so that you can apply and modify them to solve non-traditional problems. About 150 application questions are posted.

3 Exam C vs. Exam STAM... Removes basic material on (life table) estimation, and adds pricing and reserving. The last Exam C will take place in June, 2018 (registration deadline: May 8th, 2018). Exam C is credited from both the SOA and the CAS. Exam STAM is only credited from the SOA. No instant result of the STAM exam (at least for the first few sittings). Study materials for Exam C are more mature Society of Actuaries Student Research Case Study Challenge:

4 Review of probability (Chapter 2 of Loss Models). Real valued random variables (rv s) are functions from sample space Ω to real R := {, }. We use capital letters to denote rv s (e.g, X,Y,Z) and lower case letters to denote numerical values (e.g., x,y,z). The support of a rv is the set of all possible values. The cumulative distribution function (cdf) for a rv X is F X (x) := P(X x), for x R. Any cdf must satisfy the following conditions: 0 F X (x) 1, x; F X (x) is non-decreasing (flat or increasing, eg., P(X 1) P(X 2)); F X (x) is right continuous; lim x F X (x) = 0 and lim x F X (x) = 1.

5 A rv is discrete if the support contains at most a countable number of values (can be finite or infinite). Eg., the number of insurance claims (aka. frequency). A rv is continuous if the cdf is continuous, and it is differentiable everywhere except at a countable number of values. Eg., the dollar amount of an individual insurance claim (aka. severity). A rv is mixed if it is not discrete, but the cdf continuous everywhere except at at least one but almost a countable number of values. Think a little bit differently... Notice P(X = x) = P(X x) lim δ 0 P(X x δ) = F X (x) F X (x ). Let D = {x : F X (x) F X (x ) > 0}. X is discrete if F X (x) F X (x ) = 1. x D X is continuous if F X (x) F X (x ) = 0. x D X is mixed if 0 < F X (x) F X (x ) < 1. x D

6 The survival function of rv X is given by S X (x) := P(X > x) = 1 P(X x) = 1 F X (x), for all x R. Any survival function must satisfy the following conditions: 0 S X (x) 1, x; S X (x) is non-increasing (eg., P(X > 1) P(X > 2)); F X (x) is left continuous; lim x S X (x) = 1 and lim x S X (x) = 0. For any a b, P(a < X b) = P(X b) P(X a) = F X (b) F X (a). Alternatively, P(a < X b) = P(X > a) P(X > b) = S X (a) S X (b).

7 The probability mass function (pmf) of a (discrete) rv X is for all x R. For discrete rv, p X (x) := P(X = x), F X (x) = P(X = k), k x and S X (x) = k>x P(X = k). For a continuous rv X, P(X = x) = 0 for all x R. The probability density function (pdf) of a (continuous) rv is given by f X (x) = d dx F X (x) = d dx S X (x). Note: pdf may not always exist since cdf may not differentiable everywhere. For a continuous rv X, F X (x) = and S X (x) = x x f X (x)dx, f X (x)dx.

8 The hazard rate function of a (continuous) rv X is formulated as for x R, given the pdf exists. h X (x) := f X (x) S X (x), In the literature of survival analysis, the hazard rate function is also termed the force of mortality and failure rate. Properties: and assume X > 0, h X (x) = d dx ln(s X (x)), x S X (x) = exp{ h X (x)dx}. 0 The mode of a rv is the most likely value.