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In the Poisson model, if the average number of events occurring to each member of a population in a given period of time is 2, then the probability of observing exactly a events occurring to any one individual in the same period of time is: Pr[ D = /]=- exp( 2)A d! (i) Derive the maximum likelihood estimator under the Poisson model of the average rate at which events occur. , in a population where the exposed to risk for each person i is E. [4] A university runs a bus service between its teaching campus and its student halls of residence. Traffic conditions mean that the arrival of buses at the bus stop on the teaching campus can be considered to follow a Poisson process. The university decided to commission a study of how long students typically have to wait at the bus stop for a bus to arrive. Students were asked to record the time they arrive at the stop, and the time the next bus arrived. Students who became tired of waiting at the stop and left before the next bus arrived were asked to record the time they left. Below are given data from 10 students. Student Time arrived Time left our Time next left before bus arrived next bus arrived 4.00 p.m. 4.05 p.m. 4. 10 p.m. 4.35 p.m. 4.20 p.m. 4.30 p.m. 4.30 p.m. 4.35 p.m. 4.40 p.I. 4.50 p.I. 4.45 p.m. 4.50 p.m. 4.55 p.m. 5.05 p.m. 5.00 p.m. 5.20 p.m. 5.10 p.m. 5.40 p.m. 5.10 p.m. 6.10 p.m. (ii) Calculate the maximum likelihood estimate of the hourly rate at which buses arrive at the bus stop, using the Poisson estimator, and assuming that only one bus arrived at any given time. [3] (iii) Comment on the use of the Poisson model for this investigation. [3] [Total 10]Let R; denote the return on security i given by the following multifactor model R; = a;+ bill + bialy+ ... + billy + c; a; and c are the constant and random parts respectively of the component of the return unique to security i. I .. It are the changes in a set of L indices. bak is the sensitivity of security i to factor k. (i) State the category of the above model where: (a) index 1 is a price index index 2 is the yield on government bonds index 3 is the annual rate of economic growth (b) index 1 is the level of Research and Development expenditure index 2 is the price earnings ratio [2] (ii) Determine the number of parameters to be estimated in a single index model and in a multifactor model. [4] [Total 6] The following unusual model has been proposed for the (real-world) stochastic behaviour of the short term interest rate: dr, = ur, di + dZ,, where u > 0 and o are fixed parameters and Z is a standard Brownian motion under the proposed real-world measure P. Under the same measure P, a (zero coupon) bond with maturity T has price at time f B(t, T) = exp(-(T-D)r, + 6 (T-);6). (a) Derive the SDE satisfied by B(t. 7). (b) Determine the market price of risk and deduce the corresponding SDE for r, under the risk neutral measure @. [7]
