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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, _y? =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 . Fe forn 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 Flot 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](1) Explain the difference between an efficient market and an arbitrage-free market. 141 Empirical investigations of stock market returns have revealed a fractal dimension of 1.4. (ii) Explain what this means about the distribution of returns. [2] (iii) Explain how mean-reversion in the stock market can be consistent with an efficient market. (2] (iv) Outline the claim and test of excessive volatility in stock markets made by Shiller, along with four criticisms made of the test. 171 [Total 15] (i) Describe three types of credit risk model. [6] Assume that a company has fixed debt of $40m with term 10 years, the value of the equity in the company is E20m and the Merton model for credit risk holds true. 'The risk free rate of return is $% p.a. and there are no other dividends or interest payments. (ii) Explain how to calculate the (risk neutral) probability of default. You do not have to calculate the probability, but should state how each value would be calculated [6] In a particular two state model for credit rating with deterministic transition intensity, the risk free rate is a constant, r, the recovery rate is & and the zero coupon bond price is given by: B(t, m) =e-(7-1 1-(1-8) 1-e 4 (iii) (a) State the general formula for the zero coupon bond prices in a two state model for credit ratings. (b) Deduce the risk-neutral default intensity for the particular two state model above. [4] [Total 16]