Question 4 (Total: 10 marks) (a) Let .L denote the future loss random variable for a whole life insurance policy issued to a life aged x. Premiums of $P are payable at the beginning of each year throughout the policy term, and the sum insured of $S is paid at the end of the year of death. (b) (i) Show that Var[L] =| S+ Var[L]=(S+)* (^, ^). (ii) (2 marks) For a life aged 50 at issue, the premiums are $1,000 per year determined by the Equivalence Principle, mortality follows the Standard Ultimate Life Table and the interest rate is 5% per year. Find S. Hence, calculate the standard deviation of L. (3 marks) A life insurance company issued a 20-year term insurance policy with a death benefit of $10,000 to a policyholder aged 50. The death benefit is payable at the end of the year of death. Level premiums and expenses are payable annually in advance as long as the policyholder is still alive, for at most 10 years. The premium expenses are 20% of gross premium in the first year and 5% in the renewal years. The policy expenses are $50 in the first year and $25 in the renewal years. The settlement expenses are $100 upon death claim. Given mortality follows the Standard Ultimate Life Table and the interest rate is 5% per year, determine the gross premium, correct to 2 decimal places. (5 marks)