(c) to (f)

Question 2 (14 marks) Inspired by all you have learned about survival models this semester, you have decided to build a model for the mortality of domestic pet rats. You have been able to collect some data and have determined that the survival function is estimated as Pr(To >t)= exp(-0.2t) Ost15 where To is the random variable for the complete future lifetime of a new-born rat and t is in years. You will note that no pet rat survives to age 15. Where relevant, round your answers to 4 decimal places or the nearest dollar. (a) Under this survival function, what is the value of o? (1 mark) (6) Calculate the probability that a new-born rat will survive 10 years. (1 mark) c) Calculate the probability that a rat aged 6 months will die aged 10 last birthday. (3 marks) (d) What is the complete expectation of life for a new-born rat? (3 marks) (e) Suppose you also want to offer a rat food annuity. That is an annuity that would pay $1 a week (assume this is frequent enough to be considered continuously) for the life of the rat in order to cover the costs of feeding the rat. What would be the expected present value of that benefit for a new-born rat? Assume a force of interest of 5% per annum. (3 marks) (f) Suppose you wanted to offer rat life insurance. That is, you would pay $50 to the owner of a rat as soon as the rat dies so that they can buy a new one. Using what you know about the relationship between assurances and annuities, what would be the expected present value of this benefit to a new-born rat? Continue to assume a force of interest of 5% per annum. (3 marks) Question 2 (14 marks) Inspired by all you have learned about survival models this semester, you have decided to build a model for the mortality of domestic pet rats. You have been able to collect some data and have determined that the survival function is estimated as Pr(To >t)= exp(-0.2t) Ost15 where To is the random variable for the complete future lifetime of a new-born rat and t is in years. You will note that no pet rat survives to age 15. Where relevant, round your answers to 4 decimal places or the nearest dollar. (a) Under this survival function, what is the value of o? (1 mark) (6) Calculate the probability that a new-born rat will survive 10 years. (1 mark) c) Calculate the probability that a rat aged 6 months will die aged 10 last birthday. (3 marks) (d) What is the complete expectation of life for a new-born rat? (3 marks) (e) Suppose you also want to offer a rat food annuity. That is an annuity that would pay $1 a week (assume this is frequent enough to be considered continuously) for the life of the rat in order to cover the costs of feeding the rat. What would be the expected present value of that benefit for a new-born rat? Assume a force of interest of 5% per annum. (3 marks) (f) Suppose you wanted to offer rat life insurance. That is, you would pay $50 to the owner of a rat as soon as the rat dies so that they can buy a new one. Using what you know about the relationship between assurances and annuities, what would be the expected present value of this benefit to a new-born rat? Continue to assume a force of interest of 5% per annum