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In a study of the relation between the amount of information available and use of buses in eight comparable test cities, bus route maps were given to residents of the cities at the beginning of the test period. The increase in average daily bus use during the test period was recorded. 'The numbers of maps and the increase in bus use are given in the table below (both in thousands) Number of maps (x) 80 220 140 120 180 100 200 160 Increase in hus use (v) 0.60 6.70 5.30 400 6.55 215 6.60) 5.75 For these data: Ex=1,200, Ex' = 195,800, Ey =37.65, _j' =213.4875, _ xy = 6,378 (i) Construct a scatterplot of the data and comment on the relationship between the increase in bus use and the number of maps distributed. [4] (ii) The equation of the fitted linear regression is given by y =-1.816+0.04348x. Ferform an appropriate statistical test to assess the hypothesis that the slope in this fitted model suggests no relationship between the increase in bus use and the number of maps distributed. Any assumptions made should be clearly stated. [6] (iii) The fitted responses and the residuals from the linear regression model fitted in part (ii) are given below: Fitted values (v) 1.66 7.75 4.27 3.40 6.01 2.53 6.88 5.14 Residuals (e) -1.06 -1.05 1.03 0.60 0.54 -0.38 -0.28 0.61 Plot the residuals against the values of the fitted responses and comment on the adequacy of the model. (4] (iv) A new city is added to the study, and 250,000 maps are distributed to its citizens. Calculate the prediction of the increase in bus use in this city according to the model fitted in part (ii) and comment on the validity of this prediction. [2] [Total 16](0) Derive an expression for the theta of an option under the Black-Scholes model involving delta and gamma. (4] (1i) Explain why a deep out of the money call option in the Black-Scholes world will experience a rate of return close to the risk-free rate of return [2] [Total 6] Consider a set of risky assets in a mean-variance framework where: V: = variance of the return for asse. i Cy= covariance between the returns of assets i and / where i * j Derive an expression for the variance of a portfolio of / such assets where ; is the relative weight of asset / in the portfolio. Assume that the weights sum to unity and that short selling is prohibited. [3] (ii) Show that the variance of the returns of a very large portfolio of equally weighted allocations to the assets depends mainly on the average covariance between the asset returns. [5] [Total 8] Consider a two period recombining binomial model for $,, the price of a non-dividend paying security at times r =0, 1 and 2, with real world dynamics: Sel =5, u with probability p =S, d with probability I- p and # > 1 3 0. There also exists a risk-free instrument that offers a continuously compounded rate of return of 5% per period. The state price deflator in this model after one period is: A =0.7510 when S, = Squ =1.5220 when S, = Sod The price of a derivative at time 0 that pays 1 at time 2 if Sy So using the risk-neutral probability measure derived in (ii). 121 [Total 9]