The questions below are macroeconomics questions.

If Ty and Ky are random variables measuring the complete and curtate future lifetimes, respectively, of a life aged x, write down an expression for each of the following symbols as the expectation of a random variable: Ax A - (iii) The benefit payable under a special assurance policy has present value random variable: W =max (K*+1,n) where Kx is the curtate future lifetime of a person currently aged exactly x. (i) Describe the benefit paid under this policy. (i) Express E[W] in terms of standard actuarial notation. Express var[w] in terms of standard actuarial notation. Calculate the expectation and standard deviation of the present value of the benefits from each of the following contracts issued to a life aged exactly 40, assuming that the annual effective interest rate is 4% and AM92 Ultimate mortality applies: (i) a 20-year pure endowment, with a benefit of f10,000 (i) a deferred whole life assurance with a deferred period of 20 years, under which the death benefit of E20,000 is paid at the end of the year of death, as long as this occurs after the deferred period has elapsed. Using an interest rate of 6% pa effective and AM92 Ultimate mortality, calculate: (i) 1 `50:15 (i) A50:151 If /40 = 1,000 and /40+t = /40 -5t for t=1,2,..,10 , calculate the value of Ago:10] at 6% pa interest.An enterprising scientist is modelling the population of "tribbles", an alien species renowned for its capacity to reproduce. The size of the population at time , X, is thought to follow a Markov jump process with states {1, 2, 3, ... } . The scientist assumes that the population starts with one individual at time 0, Xo = 1, which causes no problem as the tribble is asexual! The transition rates of the process depend on the current population size. The monthly rates are: 2' if j=i+1 My =3-2' if j = i 0 otherwise (i) Write down the generator matrix (transition rate matrix) of the process X, . [1] (ii) State the distribution of the random variable 7,, which measures the holding time in State i , and its expected value. [1] (iii) Calculate >E[Z, ]. [1] 1=1 (iv) Comment briefly on your answers. [1] [Total 41