BA3202-L6 & L7 - BA3202 Actuarial Statistics Lecture 5...

Info icon This preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
BA3203 L6 BA3202 Actuarial Statistics Lecture 5 Summary: - Risk models
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
BA3203 L6 Short-term insurance Aggregate loss within the contract period ? ? : number of claims during the contract period ? is a discrete random variable with pdf ? ? = Pr ? = ? cdf 𝑃 ? = Pr(? ≤ ?) ? ? ? : MGF for ? ? 𝑖 : i.i.d. RV representing loss amount for each individual claim ? 𝑖 ’s are defined to be + ve cdf ? ? = Pr(? 𝑖 ≤ ?) , ? > 0 pdf ? ? is assumed to exist ? ? ? : MGF for ? 𝑖 ’s ? 𝑘 : ? ?ℎ moment of ? 𝑖 ? is independent of ? 𝑖 ’s ? = ? 𝑖 ? 𝑖=1 Actuarial Science NTU Jade Nie [email protected] 2
Image of page 2
BA3203 L6 The collective risk model Moments of ? ? ? = ? ? ? ? = ? ? ? 1 ??? ? = ? ? ? 2 + ??? ? − ? ? ? 1 2 MGF of ? ? ? ? = ? ? ? ? ? = ? ? ?∗ln ? 𝑋 ? = ? ? ln ? ? ? Examples: The compound Poisson distribution The compound Binomial distribution The compound Negative Binomial distribution Actuarial Science NTU Jade Nie [email protected] 3
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
BA3203 L6 Excess of loss reinsurance (XL) ? 𝐼 = ? 𝑖 ? 𝑖=1 : aggregate loss for direct insurer ? ? 𝐼 = ? ? ? 𝐼 ? = ? ? ? ? 𝑖 ??? ? 𝐼 = ? ? ? ? 𝑖 2 + ??? ? − ? ? ? ? 𝑖 2 ? ? = ? 𝑖 ? 𝑖=1 : aggregate loss for the reinsurer Reinsurer may not know the total number of claims ? , e.g. they may only know the number of claims that are larger than ? Let ? ? = ? 𝑖 ?? 𝑖=1 ? 𝑖 = ? 𝑖 − ? |? 𝑖 > ? , cdf of ? 𝑖 : ? ? = Pr ? 𝑖 ≤ ? = Pr(? 𝑖 − ? ≤ ?|? 𝑖 > ?) = Pr(? 𝑖 ≤?+? 𝑎?? ? 𝑖 >?) Pr ? 𝑖 >? = ? ?+? −? ? 1−? ? pdf of ? 𝑖 : ? ? = 𝜕? ? 𝜕? = ? ?+? 1−? ? ?? is the number of ? 𝑖 that is larger than ? ?? = 𝐼 1 + 𝐼 2 + ⋯ + 𝐼 ? , where 𝐼 𝑖 = 0, 𝑖? ? ≤ ? 1, 𝑖? ? > ? = 0, ?𝑖?ℎ ??????𝑖?𝑖?? ?(?) 1, ?𝑖?ℎ ??????𝑖?𝑖?? 1 − ?(?) 𝐼 𝑖 is a binomial R.V. Actuarial Science NTU Jade Nie [email protected] 4
Image of page 4
BA3203 L6 Individual risk model ? = ? 𝑖 ? 𝑖=1 ? 𝑖 has a compound Binomial distribution ? ? 𝑖 = ? 𝑖 ? 𝑖 ??? ? 𝑖 = ? 𝑖 𝜎 𝑖 2 + ? 𝑖 1 − ? 𝑖 ? 𝑖 2 ? 𝑖 = 0 , ?𝑖?ℎ ??????𝑖?𝑖?? 1 − ? 𝑖 ? 𝑖 , ?𝑖?ℎ ??????𝑖?𝑖?? ? 𝑖 ? ? = ? ? 𝑖 ? 𝑖=1 = ? 𝑖 ? 𝑖 ? 𝑖=1 ??? ? = ??? ? 𝑖 ? 𝑖=1 = [? 𝑖 𝜎 𝑖 2 + ? 𝑖 1 − ? 𝑖 ? 𝑖 2 ] ? 𝑖=1 due to independence When ? 𝑖 are also identically distributed ? 𝑖 = ? , ? 𝑖 = ? , 𝜎 𝑖 = 𝜎 for all 𝑖 ? ? = ??? ??? ? = ??𝜎 2 + ?? 1 − ? ? 2 Actuarial Science NTU Jade Nie [email protected] 5
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
BA3203 L6 Collective Vs Individual 100 risks within a portfolio: ? 1 , ? 2 , … , ? 100 Individual risk model: ? = ? 𝑖 100 𝑖=1 Claims experience from these 100 risks: Collective risk model: ? = ? 1 + ? 4 + ⋯ + ? 98 + ? 99 Let ? be the number of claims ? 𝑖 be the amount of claims for each claims arisen ? = ? 𝑖 ? 𝑖=1 Collective risk model focus on claims; individual risk model focus on individual risk profile Actuarial Science NTU Jade Nie [email protected] 6 Risks 𝒀 ? 𝒀 ? 𝒀 ? 𝒀 ? 𝒀 ?? 𝒀 ?? 𝒀 ?? 𝒀 ??? Claims ? 1 0 0 ? 4 0 ? 98 ? 99 0
Image of page 6
BA3203 L6 BA3202 Actuarial Statistics Lecture 6: - Ruin Theory
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
BA3203 L6 Objectives Explain the concept of ruin for a risk model.
Image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern