# BA3202-L8 - BA3202 Actuarial Statistics Lecture 6&7 Summary...

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BA3203 L8 BA3202 Actuarial Statistics Lecture 6 &7 Summary: - Ruin Theory

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BA3203 L8 At time ? = 0 , an insurer set aside certain amount of money for an insured risk portfolio called the initial surplus ? , ? ≥ 0 For this portfolio, the insurer has a continuous premium income at a rate of 𝑐 per unit of time total premium income for period 0, ? is 𝑐? ? ? : aggregate claim for period 0, ? Surplus process ? ? ?≥0 : ? ? = ? + 𝑐? − ? ? The surplus process Actuarial Science NTU Jade Nie 2 ?′ ? ? = ? + 𝑐? ? 𝑖 𝑁 ? 𝑖=1 where ? ? = 2 Surplus ?(?) Time ? Initial surplus= ? Premium income at rate of 𝑐 ? 1 ? 2 ? 3 ? 4 0
BA3203 L8 Probability of ruin When the surplus falls below zero the insurer has run out of money and it is said that ruin had occurred Probability of ruin: Ultimate probability of ruin given initial surplus ? : 𝜓 ? = Pr[? ? < 0, ??? ??𝑚? ?, 0 < ? < ∞] Probability of ruin before time ? , given initial surplus ? : 𝜓 ?, ? = Pr ? 𝜏 < 0, ??? ??𝑚? 𝜏, 0 < 𝜏 < ? Actuarial Science NTU Jade Nie 3 ?(?) ? ? ? 1 ? 2 ? 3 ? 4 ? 1 ? 2 ? 3 ? 4 0 Ruin occurred: Time of ruin ? 4 ? ? 4 = 4

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BA3203 L8 The Poisson Process Key properties: ? ? ~𝑃?𝑖 𝜆? for any ? > 0 Denote ? 𝑖 to be the time of the 𝑖 ?ℎ claim ? 𝑖 = ? 𝑖 − ? 𝑖−1 : the waiting time between 𝑖 − 1 ?ℎ and 𝑖 ?ℎ claim Then ? 𝑖 ~?𝑥? 𝜆 Actuarial Science NTU Jade Nie 4 ? ? ? 1 ? 2 0 ? ? 1 ~𝑃?𝑖 𝜆? 1 ? ? 2 ~𝑃?𝑖 𝜆? 2 ? 1 ? 2 ? 3 ? 1 ? 2 ? 3 ? 4 ? 4
BA3203 L8 Lundberg’s inequality Lundberg’s inequality states that 𝝍 𝑼 ≤ 𝒆 −𝑹𝑼 ? : the initial surplus 𝜓 ? : the Ultimate probability of ruin given initial surplus ? ? : the adjustment coefficient The larger ? smaller ? −?𝑈 smaller upper bound for 𝜓 ? Derivation out of syllabus ? −?𝑈 is often used as an approximation to 𝜓 ? Actuarial Science NTU Jade Nie 5

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BA3203 L8 The adjustment coefficient ? : the adjustment coefficient A measure of risk for a surplus process When ? ? ?≥0 is a compound Poisson process, ? is defined to be the unique +ve root of equation ? ? = 𝜆? 𝑋 ? − 𝜆 − 𝑐? = 0 So that ? is given by 𝜆 + 𝑐? = 𝜆? 𝑋 ? If we write 𝑐 = 1 + 𝜃 𝜆𝑚 1 , we have ? 𝑋 ? = 1 + 1 + 𝜃 𝑚 1 ? The equation is independent of 𝜆 , but only depend on 𝜃 : the premium loading factor 𝑚 1 : the expected individual claim amount It can be proved that function ? ? has only one +ve root (self- read) A minor note: for those who remembers martingale, ? is found such that ? ? −𝑟𝑈 ? = ? −𝑟𝑈 or ? ? 𝑟 ?−𝑐 = 1 Actuarial Science NTU Jade Nie 6
BA3203 L8 BA3202 Actuarial Statistics Lecture 8: - Run-off Triangles

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BA3203 L8 Introduction Run-off triangles are techniques used to estimate claims Usually arise in types of non-life insurance where it may take some time after a loss until the full extent of the claims which have to be paid are known.
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