Exercise 1. A mortgage lender holds a portfolio of 100 mortgages to private home-owners that belong...

 

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Exercise 1. A mortgage lender holds a portfolio of 100 mortgages to private home-owners that belong in one category with regards to their income characteristics. It is known that the borrower's income affects the probability of default in mortgages. Assume that the income of each borrower changes once a year. Let X₁, i = 1,..., 100 be the income of borrower i for the next year. Assume that the X, 's are independently and identically distributed with a normal distribution with mean £27,000 and variance 90,000. The lender has estimated that a mortgage of this type will default in the next year if the income of the borrower falls below £26,400. (i) Calculate the probability that 10 out of the 100 mortgages default within the next year and the probability that all mortgages default. (ii) Calculate the probability that no more than 2 defaults are observed in the portfolio within the next year. Now assume that the incomes are not independent but their dependence is described by a Clayton copula given by C(u₁,..., u100) = (₁++100-100+1)- with 0 = 2. (iii) Calculate the probability that all mortgages default and compare it with the probability you calculated in (a). Explain the difference. Exercise 1. A mortgage lender holds a portfolio of 100 mortgages to private home-owners that belong in one category with regards to their income characteristics. It is known that the borrower's income affects the probability of default in mortgages. Assume that the income of each borrower changes once a year. Let X₁, i = 1,..., 100 be the income of borrower i for the next year. Assume that the X, 's are independently and identically distributed with a normal distribution with mean £27,000 and variance 90,000. The lender has estimated that a mortgage of this type will default in the next year if the income of the borrower falls below £26,400. (i) Calculate the probability that 10 out of the 100 mortgages default within the next year and the probability that all mortgages default. (ii) Calculate the probability that no more than 2 defaults are observed in the portfolio within the next year. Now assume that the incomes are not independent but their dependence is described by a Clayton copula given by C(u₁,..., u100) = (₁++100-100+1)- with 0 = 2. (iii) Calculate the probability that all mortgages default and compare it with the probability you calculated in (a). Explain the difference.