Question 1

Write down an expression for the gross future loss random variable at issue for a deferred life

annuity with a deferred period of n years. Premiums of amount G are paid annually in advance

for a maximum of n years, the annuity benefit of B is paid for life annually in advance starting in

n years' time, and no benefit is paid if the life does not survive to time n . Assume that there are

regular expenses payable annually in advance during the premium payment term of e , additional

initial expenses of I at the start of Year 1, and regular benefit payment expenses of e? payable

annually in advance while the benefit is being paid.

20.2 Show how you would modify the premium equation below to allow for the expenses indicated:

??

xn xn : : Pa SA ?

(i) initial expenses of 2% of the sum assured

(ii) renewal expenses of 2% of each premium, including the first

(iii) claim expenses of 2% of the sum assured

(iv) initial expenses of 50% of first premium plus renewal expenses of 3% of each premium

excluding the first.

20.3 A life office sells 5-year term assurance policies to lives aged 60. Each policy has a sum assured of

10,000 payable at the end of the year of death. Premiums of 200 are payable annually in

advance throughout the 5-year term or until earlier death.

Let L denote the present value of the insurer's loss on one of these policies, at policy outset,

ignoring expenses.

(i) Write down an expression for L . [2]

(ii) Assuming AM92 Ultimate mortality and 5% pa interest, calculate the expected value and

standard deviation of L . [9]

[Total 11]

20.4 Calculate the annual premium payable in advance by a life now aged exactly 32, in respect of a

deferred annuity payable from age 60 for 5 years certain and for life thereafter. The amount of

the annuity is 400 pa, payable annually in arrear, and the insurer incurs an additional

administration cost of 2 when each annuity payment is made. The premium is paid throughout

the deferred period or until the earlier death of the policyholder.

A pension scheme only allows retirement at exact age 65. An investigation of the mortality of the retired members of the scheme was carried out over the period 1 January 2001 to 31 December 2006. The following data were obtained: Date of death Member Date of retirement (if occurred during the investigation period) 1 April 1998 30 April 2005 1 August 2000 1 February 2001 1 June 2002 31 August 2004 DJOUAWN 1 August 2002 31 December 2006 1 March 2004 1 May 2004 30 November 2006 1 January 2005 All months should be assumed to be of equal length. (i) Explain the form or forms of censoring that are present in these data and in this observational plan. [2] (ii) Calculate the Kaplan-Meier (product-limit) estimate of the survival function 565 (t) from these data, stating clearly any additional assumptions that you make. [or] (iii) Estimate the force of transition from alive to dead for a two-state Markov model for age 67 last birthday, using the data given. The exposed to risk should be calculated exactly. State all assumptions required, and state the age to which the estimate would be assumed to apply. [7] [Total 19]