:Solve the following questions.

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 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]Q.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]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 (I) is negatively affected by interest rate at a given point in time (r). 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) R' = 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 R2Exercise 33 Table 3 .3 is an extract from a (hypothetical) select life table with a select period of two years. Note carefully the layout each row relates to a xed age at selection. Use this table to calculate (a) the probability that a life currently aged 75 who has just been selected will survive to age 35. (b) the probability that a life currently aged 76 who was selected one year ago will die between ages 35 and 3?, and (C) 4|24m1+1 - Table 3.3. Extrectfrom a (hypothetical) select life table. I In] low 31+: I + 2 25 15 930 15663 15 236 T? 26 15503 15224 14316 T3 '1'? 15050 14244 14310 '19 30 12 576 32 31 11923 33 32 11 250 34 33 10 542 35 34 9 312 36 35 9 064 3'? Exercise 3.4 CMI (Table A23) is based on UK data from 1999 to 2002 for female non-smokers who are term insurance policyholders. It has a select period of ve years. An extract from this table, showing values of q[x_;]+;, is given in Table 3.9. Use this survival model to calculate (3) 2pm] 1 03) sq[73]+2, Table 3.9. Mortality rates for female non-smokers who have term insurance policies. Duration ( Duration 1 Duration 2 Duration 3 Duration 4 Duration 5+ Age, X 9[x ] 4[x-1]+1 9[x-21+2 4[x-3]+3 4[x-4]+4 69 0.003974 0.004979 0.005984 0.006989 0.007994 0.009458 70 0.004285 0.005411 0.006537 0.007663 0.008790 0.010599 71 0.004704 0.005967 0.007229 0.008491 0.009754 0.011880 72 0.005236 0.006651 0.008066 0.009481 0.010896 0.013318 73 0.005870 0.007456 0.009043 0.010629 0.012216 0.014931 74 0.006582 0.008361 0.010140 0.011919 0.013698 0.016742 75 0.007381 0.009376 0.011370 0.013365 0.015360 0.018774 76 0.008277 0.010514 0.012751 0.014988 0.017225 0.021053 77 0.009281 0.011790 0.014299 0.016807 0.019316 0.023609 (c) 1/9165]+4 , and (d) 7p [70] . Exercise 3.5 CMI (Table A21) is based on UK data from 1999 to 2002 for female smokers who are term insurance policyholders. It has a select period of five years. An extract from this table, showing values of q[x-/]+1, is given in Table 3.10. Calculate (a) 7P[70] , (b) 1/29[70]+2, and (c) 3.89[70]+0.2 assuming UDD. Table 3.10. Mortality rates for female smokers who have term insurance policies. Duration ( Duration 1 Duration 2 Duration 3 Duration 4 Duration 5+ Age, x 9[x] 4[x-1]+1 9[x-21+2 4[x-3]+3 9[x-4]+4 70 0.010373 0.013099 0.015826 0.018552 0.021279 0.026019 71 0.011298 0.014330 0.017362 0.020393 0.023425 0.028932 72 0.012458 0.015825 0.019192 0.022559 0.025926 0.032133 73 0.013818 0.017553 0.021288 0.025023 0.028758 0.035643 74 0.015308 0.019446 0.023584 0.027721 0.031859 0.039486 75 0.016937 0.021514 0.026092 0.030670 0.035248 0.043686 76 0.018714 0.023772 0.028830 0.033888 0.038946 0.048270 77 0.020649 0.026230 0.031812 0.037393 0.042974 0.053262