ST227 Review Exercises to be handed in 26-02-08

ST227 Review Exercises to be handed in 26-02-081. [40 marks](a)[15 marks] Let T be a random lifetime with survival function F (t) and let k > 0.Prove thatE[T k ] = k∫ ∞0t k−1 F (t)dtprovided that E[T k ] < ∞.(b)[15 marks] Assume now that the hazard rate is given in Weibull form, that iswe have for t ≥ 0 and parameters α, β > 0 thatµ (t) = βα −β t β−1 .Find the survival function F (t). Show that the conditional hazard rate µ x+t isagain of Weibull form only when β = 1.(c)[10 marks] Prove that( )E[T k ] = kαk kβ Γ .βHere Γ denotes the gamma function defined byfor z > 0.Γ(z) =∫ ∞0u z−1 e −u du2.[10 marks]A man receives the Rolls Royce of his uncle aged x years, with remaining life lengthT x . In return he agrees to pay his uncle a whole-life annuity with continuous rateof £ 8000 (per year), plus a term insurance of £ 50000 in case his uncle dies withinthe next 5 years. Derive a formula for the expected present value of this contract,in terms of r, τ p x , and µ x+τ . Here r is the interest rate (assumed to be constant),τp x the survival functions and and µ x+τ the mortality intensity given the personhas already survived x years.1

3. [25 marks]The contract of James Bond specifies that the secret agent (resp. some of hisrelatives) will receive the benefits of a term insurance depending on his cause ofdeath. If 007 dies in immediate action (state d 1 ), the benefit is £ 100000. Onthe other hand, a natural death not related to his job (state d 2 ) will yield onlya benefit of £ 50000. Her majesty’s secret service estimates that the transitionfrom active (state a) to d 1 happens with a certain rate µ whereas the transitionfrom a to d 2 happens with rate ν. Moreover, James receives each time he spendsin hospital (transition from a to hospital state i with rate σ, and from i to a withrate ρ) a continuous rate payment with benefit rate b. Transition from i to d 2happens with rate ν ′ whereas it is assumed that it is not possible to have a directtransition from i to d 1 .(a)[5 marks] Draw a flowchart specifying the possible states and the transitionrates.(b)[10 marks] Derive the forward differential equation for the transition probabilityp ad1 and give the initial condition.(c)[10 marks] Find a differential equation for the reserve in states a and i. Youmay use the general Thiele’s equations.4. [25 marks]Denote by X t a Poisson process with rate λ > 0. (Recall that a Poisson processhas state space Z = {0, 1, 2, 3, ...} and the only possible transitions are withintensity λ from state i to state i + 1 for any i ∈ Z). As the Poisson processis a homogeneous Markov chain (the intensities do not depend on time) we canwrite p i,j (t) := p i,j (0, t).(a)[5 marks] Proof that for any i ∈ Zp i,i (t) = e −λt .(b) [10 marks] Prove that the forward equation is given byddt p i,j (t) = λp i,j−1 (t) − λp i,j (t) (1)for j > i ≥ 0. What is p i,j (0)?(c)[10 marks] Check that equation (1) is fulfilled byfor t > 0 and j > i ≥ 0.−λt (λt)j−ip i,j (t) = e(j − i)!2