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## Three Problems In Quantitative Risk Management

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**Author**

Mitra, Abhimanyu

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**Abstract**

This thesis deals with the approximation of the probability of remote risk regions. The simplest example is to compute P[X > x] for a one-dimensional random variable X and a large threshold x. Such probabilities give useful measures of risk. We consider three problems related to the approximation of the probability of a risk region. The first, an important problem in finance and insurance, is to approximate the probability that a sum of losses, X + Y, exceeds a large threshold. We investigate a common case where the distribution of (X, Y ) belongs to the maximal domain of attraction of a bivariate Gumbel distribution with X and Y being asymptotically independent [18, pages 18, 229] so that both X and Y are in the maximal domain of attraction of the Gumbel distribution. We obtain sufficient conditions to guarantee tail equivalence of X + Y and X , that is lim x[RIGHTWARDS ARROW][INFINITY] P(X + Y > x)/P(X > x) ∈ (0, [INFINITY]). Under the further assumption of nonnegativity of losses, the result is extended to aggregation of any finite number of losses. We explore the asymptotics of finite linear combinations of losses n i =1 ai Xi with ai [GREATER-THAN OR EQUAL TO] 0, i = 1, 2, [MIDDLE DOT] [MIDDLE DOT] [MIDDLE DOT] , n, which we then use to suggest an approxi- mate solution for an optimization problem applicable to portfolio design. As opposed to aggregation of a fixed number of losses dealt with in the first problem, in the second problem we deal with aggregation of a random number of losses. This problem arises from warranty claims modeling. Consider a retail company, for example a car company, that sells items each of which is covered by a warranty for a period W. To decide on a reserve for the next quarter, the company has to estimate the quantiles of the distribution of the total warranty cost for the next quarter, based on historical data. Here, each warranty claim arriving in the next quarter is a loss to the retail company and the total cost is the aggregation of such losses. However, the number of claims that will arrive in the next quarter is random. We approximate the distribution of total warranty cost using minimal assumptions on the sales process and the nature of arrival of claims thus making the approximation robust against model error. We suggest a method of computing quantiles of the distribution of the total warranty cost in the next quarter using historical data, which is applied to warranty claims data from a car manufacturer for a single car model and model year. The third problem deals with joint tail probability estimation, for example P[Z 1 > x, Z 2 > y] for two large thresholds x and y. The joint tail probability P[Z 1 > x, Z 2 > y] is a useful measure of risk which helps us understand the tail-dependence of Z 1 and Z 2 . Under the standard model for heavy-tailed losses, multivariate regular variation (abbreviated MRV) [47, page 172] often estimates P[Z 1 > x, Z 2 > y] as zero but hidden regular variation (HRV) [46] offers a refinement of MRV which provides a non-zero and more accurate estimate of P[Z 1 > x, Z 2 > y]. In prior work, HRV was defined only on the cone E(2) = {x ∈ [0, [INFINITY]]d : x(2) > 0}, where x(2) is the second largest component of x. We extend HRV on other sub-cones E(l) = {x ∈ [0, [INFINITY]]d : x(l) > 0} of E(2) as well, 3 [LESS-THAN OR EQUAL TO] l [LESS-THAN OR EQUAL TO] d, where x(l) is the l-th largest component of x. For d > 2, this extended model of HRV significantly improves the accuracy of the estimates of joint tail probabilities compared to the earlier model of HRV. We suggest some exploratory methods of detecting the presence of HRV on E(l) , 2 [LESS-THAN OR EQUAL TO] l [LESS-THAN OR EQUAL TO] d. Using HRV, we devise a method of estimating joint tail prob- abilities P[Z i1 > xi1 , Z i2 > xi2 , [MIDDLE DOT] [MIDDLE DOT] [MIDDLE DOT] , Z il > xil ] for 2 [LESS-THAN OR EQUAL TO] l [LESS-THAN OR EQUAL TO] d, 1 [LESS-THAN OR EQUAL TO] i1 < i2 < [MIDDLE DOT] [MIDDLE DOT] [MIDDLE DOT] < il [LESS-THAN OR EQUAL TO] d from data. We apply our method to Internet traffic data to compute a measure of burstiness.

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**Date Issued**

2011-05-31#####
**Subject**

Quantitative risk mamangement; Hidden regular variation; Aggregation of risks

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**Committee Chair**

Resnick, Sidney Ira

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**Committee Member**

Jarrow, Robert A.; Todd, Michael Jeremy

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**Degree Discipline**

Operations Research

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**Degree Name**

Ph.D. of Operations Research

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**Degree Level**

Doctor of Philosophy

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**Type**

dissertation or thesis