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4 Describe briefly the Vasicek one-factor model of interest rates and its key statistical properties. [4] (ii) According to a particular parameterisation of this model, the instantaneous forward rate applicable at a fixed time 7>t implied by the market prices at time t is found to be: f (t,T)= ne " +(1-e ")+k(1-e-dje-or where ?=7-t and a >0. Show that, if a humped curve is required for f(t, 7), the parameter values must satisfy the condition > - - [5] Here a "humped curve" means one where the value of the function for some intermediate values of + exceeds the values for both : =0 and r= . In other words, there will be a maximum value for some positive value of T . (ii) Describe briefly the main advantages and limitations of the Vasicek model. [4] [Total 13] 5 Claims occur according to a compound Poisson process at a rate of % claims per year, Individual claim amounts, X , have probability function: P(X =50) =0.8 P(X =100) =0.2 The insurer charges a premium at the beginning of each year using a 20% loading factor. The insurer's surplus at time t is U(t) . Find P[U(2)