RT F70RT2 TUT8 2016 - dsfdsf PDF

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F70RT2  RISK THEORY  TUTORIAL 8 RUIN THEORY 1. There are six claims made on a portfolio of an insurance company within a year. The days on which the claims are lodged and the amounts of the claims are shown in the table below. Day 67 105 Amount 97 104

199 157

286 56

306 23

358 99

The company calculated the premium for this risk at the start of the year using a 25% security loading ( = 0.25)and an estimate of the annual claim amount based on the following information: the numbers of claims over the previous 5 years were 4, 3, 6, 2, and 5 and the mean of the individual claim amounts was £120. Construct a surplus process diagram similar to that in Figure 8.1.1 in the workbook. For how many days during the year was the surplus actually negative? What minimum initial surplus would have been required to ensure that the process never went into negative territory? Comment on the choice of . 2. Give brief verbal justifications of the following relationships satisfied by ruin probabilities for 0 < t1  t2 <  and 0  U1  U2 :

 U ,t   U  . (U2,t)   (U1,t) ;  (U2)   (U1) ;  (U,t1) (U,t2)   (U) ; lim t 

3. Consider the case in which claim amounts are exponentially distributed with mean  = 1/. (a) State the adjustment coefficient R in the cases (i)  = 1 , (ii)  = 100 in terms of the security loading  and comment in the context of working in monetary units of pounds or pence. (b) Construct a diagram similar to Figure 8.3.1 in the case  = 20 and  = 0.1. What value of U gives U) = 0.05? What is the minimum value of U such that exp( RU)   U) < 0.01 ? 4. Suppose that all claims are for a constant amount k. (a) Show that the adjustment coefficient R satisfies ekR = 1 + (1 + )kR , with R < 2/k. (b) Consider the case k = 1 and  = 0.1. Noting that R < 0.2, find R to 3dp. (You can use a formal Newton-Raphson iterative scheme or computer evaluation and search.) 5. Suppose that individual claim amounts X (in units of £1000) have the following discrete distribution: P(X = 1) = ¼, P(X = 2) = ½, P(X = 3) = ¼. The insurer uses a 10% premium loading (i.e.  = 0.1). Show that the insurer’s adjustment coefficient for this risk is between 0.08 and 0.085 and state an upper bound on the probability of ultimate ruin in the case that the insurer has initial reserves of £5000. 6. Claims occur on a portfolio of insurance policies according to a Poisson process at rate. The insurer’s initial surplus is U and the premium rate is calculated using a premium loading factor . All claims are for a fixed amount d where d > U.

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1  1 

Show that the probability that ruin occurs at the first claim is 1  exp  

 U  1    . d  

7. Suppose that claims can be for one of two amounts, 2/3 and 4/3, where the two values occur with equal probability. Assume that  = 0.1. (a) Find the equation satisfied by the adjustment coefficient R and obtain an upper bound for R. (b) Evaluate R to 3dp. (c) Compare the Lundberg bound for the three cases: exponential claims with mean 1, constant claims equal to 1, the claims distribution in this exercise. How do these bounds compare with the exact probability of ruin in the exponential case?...