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A motor insurance portfolio produces claim incidence data for 100,000 policies over one year. The table below shows the observed number of policyholders making 0, 1, 2, 3, 4, 5, and 6 or more claims in a year. No. of claims No. of policies 87,889 11,000 1,000 100 4 10 un + 26 Total 100,000 (i) Using the method of moments, estimate the parameter of a Poisson distribution to fit the above data and hence calculate the expected number of policies giving rise to the different numbers of claims assuming a Poisson model. [3] (ii) Show that the estimate of the Poisson parameter calculated from the above data using the method of moments is also the maximum likelihood estimate of this parameter. [4] (iii) Using the method of moments, estimate the two parameters of a Type 2 negative binomial distribution to fit the above data and hence calculate the expected number of policies giving rise to the different numbers of claims assuming a negative binomial model. [6] (iv) Explain briefly why you would expect a negative binomial distribution to fit the above data better than a Poisson distribution. [2] [Total 15]1. For which violation of the classical linear regression assumption does Durbin-Watson test is used for detection? And outline the procedure for the test merely formulating the hypotheses and stating the decision rule using the measure of autocorrelation, p, and the Durbin-Watson statistics, d-statistics. 2. The economic theory of investment states that investment expenditure at a given point in time (It) is negatively affected by interest rate at a given point in time (rt). Suppose that the functional relationship in the stated economic theory is linear and estimated values of the intercept and slope are 82 and 0.6 respectively. Based on the stated economic theory and the associated information, derive the estimated econometric model and predicate the value of the dependent variable provided that 50 is given as value of the independent variable. 3. The following estimated equation was obtained by OLS regression using quarterly data for 1978 to 1996 inclusive. Y = 2.20+0.104X, (3.4) (0.005) Standard errors are in parentheses, the explained sum of squares and the residual sum of squares were 109.6 and 18.48 respectively. Thus, a) Test a 5% level of significance for the statistical significance of the parameter estimates using t-test technique b) Test the overall test of significance ant 1% c) Calculate the coefficient of determination 4. Consider the following estimated regression model of monthly earnings of 25 employees selected at random from the pool of employees in a given organization. Assuming the monthly earning (ME) is affected by two explanatory variables-SEX and AGE of employees, the regression output is given as ME = -1.65 + 0.33SEX + 0.4 AGE (-0.60) (0.18) (0.03) R2 = 0.825 Given the standard errors of each estimate in brackets answer the following questions A. Test the statistical significance of the intercept using standard error test B. Test the statistical significance of SEX at 5% level of significance C. Test the overall significance of the model at 1% level of significance D. Interpret R2Q.2) (i) What are the different types of loans? Describe in brief. (3) (ii) A loan is being repaid with 25 annual payments of Rs.300/ each. With the 10" payment, the borrower pays an extra Rs. 1000/-, and then repays the balance over 10 years with a revised annual payment. The effective rate of interest is 8%. Calculate the amount of the revised annual payment. (3) (iii) An investor borrows an amount at an annual effective interest rate of 5% and will repay all interest and principal in a lump sum at the end of 10 years. She uses the amount borrowed to purchase a Rs.1000/- par value 10-year bond with 8% semiannual coupons bought to yield 6% convertible semiannually. All coupon payments are reinvested at a nominal rate of 4% convertible semiannually Calculate the net gain to the investor at the end of 10 years after the loan is repaid. (4) (iv) A loan is repaid with level annual payments based on an annual effective interest rate of 7%. The 8th payment consists of Rs.789/- of interest and Rs.211/- of principal. Calculate the amount of interest paid in the 18" payment. (5) (v) Define the characteristics of government index linked bonds. Explain in practice why most index linked securities carry some inflation risk in practice. (3) [18]().6) A company is adopting a particular investment strategy such that the expected annual effective rate of return from investments is 7% and the standard deviation of annual returns is 9%. Annual returns are independent and (1 + i) is lognormally distributed where it is the return in the (" year. The company has received a premium of Rs.1,000/- and will pay the policyholder Rs.1,400/- after 10 years. Page 4 of 5 ASI CT1 1006 (1) Calculate the expected value and standard deviation of an investment of Rs.1,000/- over 10 years, deriving all formulae that you use. (9) (ii) Calculate the probability that the accumulation of the investment will be less than 50% of its expected value in ten years time. (8)
