Provide precise solutions to the questions below. Address all questions.

An actuary is fitting the following linear regression model through the origin: Y = Bx te, e, ~ N(0,0-) 1=1,2....n (i) Show that the least squares estimator of / is given by: B =LLY [3] (ii) Derive the bias and mean square error of & under this model. [4] [Total 7]A man takes out a loan over 25 years. He makes monthly repayments in arrears. The first payment is $50 and the remaining payments increase by $5 each time until the payments reach 1770. The remaining payments are then all $770. The annual effective rate of interest is 7.5%. (i) Show that the amount of money that the man borrowed is $66,767. [6] (ii) Calculate the amount of capital outstanding after 6 years. [4] (iii) Calculate the amount of capital repaid in the 6th year. [3] (iv) Calculate the interest paid in the 73rd payment. Comment on your answers to parts (i), (ii), (iii) and (iv). [3] (v) At the end of the sixth year (after the payment then due) the man decides to reschedule the loan over a term of ten years with level monthly payments. He negotiates a deal with the loan company to reduce the interest he pays by 0.1% pa. Calculate how much he saves (compared to his original loan) under this arrangement. [7] [Total 23]