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State 1 State 2 Healthy Temporarily sick or disabled State 3 Permanently sick or disabled State 4 Dead The force of transition from state / to state / (/ = /) at age x is denoted by fry , and the probability that a life, who is in state / when aged x, will be in state j at age x +t is + p, . (i) Derive from first principles a differential equation for , P," , stating all assumptions made. [5] (ii) Given that, for x = 40,41 : 1p 2 = 0.03, 1P, = 0.002, 1 PX = 0.001, 1PX = 0.4, 1023 =0.1, 1024 =0.01 and 1PX =0.3 calculate 2P40- [2] (iii) An insurance company issues a combined sickness, disability and assurance contract that provides the following benefits: an income payable while the policyholder is temporarily sick or disabled; and a lump sum payable either on becoming permanently sick or disabled, or on death. The contract terminates as soon as the lump sum has been paid. Explain how the model could be simplified for the purpose of modelling the claims process involved. State how your answer to (i) would be altered as a result of this change. (You are not required to derive this result from first principles). [2]