1 If T, and K, are random variables measuring the complete and curtate future lifetimes, respectively,...

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1 If T, and K, 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: (i) (ii) Ax (0) (ii) xn 2 The benefit payable under a special assurance policy has present value random variable: W=max(K, +1,n) xn where K is the curtate future lifetime of a person currently aged exactly x. (i) Describe the benefit paid under this policy. Express E[W] in terms of standard actuarial notation. Express var[W] in terms of standard actuarial notation. 3 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: a 20-year pure endowment, with a benefit of £10,000 a deferred whole life assurance with a deferred period of 20 years, under which the death benefit of £20,000 is paid at the end of the year of death, as long as this occurs after the deferred period has elapsed. (i) 4 Using an interest rate of 6% pa effective and AM92 Ultimate mortality, calculate: A¹ 50:15 (ii) A50:15 5 If /40=1,000 and /40+t=140-5t for t=1,2,...,10, calculate the value of A40:10] at 6% pa interest. 1 If T, and K, 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: (i) (ii) Ax (0) (ii) xn 2 The benefit payable under a special assurance policy has present value random variable: W=max(K, +1,n) xn where K is the curtate future lifetime of a person currently aged exactly x. (i) Describe the benefit paid under this policy. Express E[W] in terms of standard actuarial notation. Express var[W] in terms of standard actuarial notation. 3 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: a 20-year pure endowment, with a benefit of £10,000 a deferred whole life assurance with a deferred period of 20 years, under which the death benefit of £20,000 is paid at the end of the year of death, as long as this occurs after the deferred period has elapsed. (i) 4 Using an interest rate of 6% pa effective and AM92 Ultimate mortality, calculate: A¹ 50:15 (ii) A50:15 5 If /40=1,000 and /40+t=140-5t for t=1,2,...,10, calculate the value of A40:10] at 6% pa interest.

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1 The symbols Ax and xl can be expressed as the expectation of a random variable as follows Ax EeTx xl EKx where Tx is the complete future lifetime random variable Kx is the curtate future lifetime ra... View the full answer