ACTUARIAL STATISTICS — DU B.Sc (hons) Statistics Previous Year Question Papers

that the policy pays 10 units. For all the parts assume that the insurer will use an annual effective interest rate of i = 0.06. (b) Express the variance of the loss L associated with n-year endowment insurance in terms of Actuarial present values for fully continuous premiums. 2. Attempt six questions in all selecting three from each section. 1. (a) (i) Explain what you mean by 'Proportional Reinsurance'. Suppose that an insurer sells this type of reinsurance for a claim amount X. Clearly explain the insurer's and reinsurer's position. What happens in case if X~Gamma(a, 2)? Instead, if X~Lognormal(u, 02) how does your answer change? (ii) Suppose an insurer enters into Excess of Loss Reinsurance Arrangement with retention limit M. Obtain the c.d.f. of the random variable W denoting the amount of non-zero payments made by the reinsurer. For the continuous case, obtain the corresponding p.d.f. Suppose • that the insurer incurs claims that follow an Exponential distribution with parameter 2. Obtain and identify the p.d.f. of the non-zero payments made by the reinsurer. Comment The policy will pay 1 unit at the end of the year of death in exchange for the payment of a premium P at the beginning of each year, provided the life survives. Find the annual premium, P, as determined by: (i) Principle I: P will be the least annual premium such that the insurer has a probability of positive financial loss of at most 0.25. (ii) Principle II : P will be the annual premium such that the insurer, using a utility of wealth function, u(x)=x-0.01?, x<50, with initial wealth W=10 will be able to maximize his expected utility. (iii) Principle III: P will be the annual premium such that the insurer, using a utility of wealth function, u(x) = -e-0lx, will be indifferent between accepting and not accepting the risk. For this case assume (a) A unit is used to purchase a combination benefit consisting of a life income of I per year payable continuously while (x) survives and an insurance of J payable immediately on death of (x). Write the present value random variable for this combination and give its mean and variance. (b) Prove and interpret the given relations: where notations have their usual meaning. 8. (a) An insurer is planning to issue a policy to a life age 0 whose curtate future lifetime, K, is governed by the probability function on your answer. (b) Let the random variable X have distribution function F given by forx < 20 2. (a) What is a Utility Function? Which are the properties that it must satisfy? How is it linked to the coefficient of risk aversion? Explain in detail the quadratic utility function and obtain the coefficient of risk aversion for this function. An insurer whose wealth is W has a choice between investments 1 and 2, which will result in wealth of W + X, and W + X, respectively, offering complete insurance cover against a random loss X such that