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

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BA3203 L6 BA3202 Actuarial Statistics Lecture 5 Summary: - Risk models
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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
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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
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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
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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
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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
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BA3203 L6 BA3202 Actuarial Statistics Lecture 6: - Ruin Theory
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BA3203 L6 Objectives Explain the concept of ruin for a risk model.
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