1. * A f100 bond is redeemable at par in 27 years. It pays semiannual coupons at a rate of 6% per annum payable semiannually. This means that the bondholder receives $3 (half of 6% of [100) every half a year for 27 years, and a further payment of $100 after 27 years. (a) Compute the price of the bond which achieves a yield of 5% per annum effective by adding the value of the coupons to the value of the redemption payment. (b) Repeat the same with the premium / discount formula. (c) What is the price (still assuming a yield of 5%) if the maturity term is 17 years? 7 years? (d) What is the price if the maturity term is 27 years and the yield is 4% p.a. ef- fective? If the yield is 3% p.a. effective? (e) Compute the price of a 27-year bond with annual coupons at a rate of 6% p.a. effective bought to yield 5% p.a. effective.Q2. (a) Show that N(u, (?), is not a member of the Regular Exponential Class. [Marks 4] (b) Consider a random sample of size n from a Normal distribution, N ((1, g?) where u is known. (i) Find a sufficient statistic for of using the Factorization Criterion. [Marks 4] (ii) Show that the statistic found in (i) is also sufficient using the conditional joint density method. [Marks 4] (iii) Show that the statistic found in (i) is also complete from definition and without using membership in Regular Exponential Class. [Marks 6] (iv) Construct a Uniformly minimum variance unbiased estimator for or. [Marks 8] (v) Derive the Cramer Rao Lower Bound for Unbiased Estimators of or. [Marks 4] (vi) Derive the asymptotic distribution of the Maximum Likelihood Estimator for (7. [Marks 4]Consider a model of contract design under asymmetric information, A regulator (or policy maker) wants to provide a subsidy, by designing a voluntary incentive contract, to a firm (who is a regulated monopolist) for exerting effort to reduce pollution from its production process. The regulator is the principal and the firm is an agent. A firm's utility is: U = 1 - w(be). where "(De) is the cost function of taking efforts with ;"(.) > 0 and v"(.) > 0 t is the monetary transfer from the regulator to the firm assume the firm has social preferences toward environmental protection and o captures such social preferences ($ 3 0) define & as the firm's intrinsic valuation of money (Le. how much a firm loves money over environment) a firm with low e means that the firm's cost of per unit of e exerted is less than a firm with high e 9 is a firm's private information A firm's effort has social value $ and social cost C = 8 - De, where f is the efficiency parameter of the firm and e is effort. The efficiency parameter, S, is fixed and common knowledge. The regulator observes the cost ex-ante. The regulator cannot observe 9. She knows the firm is one of two types depending on two levels of 8. Assume a firm can be one of two types depending on their social preferences toward environmental protection. D e (8.9). and g 0 is the social value of public funds used by the regulator to compensate the firm. Please Turn Over 5 of 5 ECON20022/30022 A firm suffers a monetary loss when it exerts efforts to reduce pollution, It will voluntarily participate if it gets monetary transfer from the regulator to compensate Its loss. The regulator is going to offer up a monetary transfer, t, to the firm in exchange for effort by the firm, ie. she designs a contract to maximize social welfare from pollution reduction subject to the firm's participation constraint. Social welfare is the utility of private firms and consumers, and cost of monetary compensation scaled by social value of public funds. a) Set up the objective function of the regulator and the constraint under complete information about # and e. Show and explain the optimal conditions. [Word limit: 100 words] [10 marks] b) Under asymmetric information, however, the regulator only knows that a firm could be one of two types but does not know which type it is. Construct and explain the constraints that the regulator must consider under asymmetric information. Compare the constraints with the complete information case and explain intuitively. Using the constraints, identify the volume of 'information rent', if any. Give a roadmap how to solve the problem (no need to solve, you need to explain the steps) [Word limit: 250 words] [206. Two different companies (ACME and BuildIT) are working on construction projects across the country. The companies agreed to be monitored and provide data to the government about their performance as part of a competition for a large contract. ACME provided data on a sample of 83 of its projects and 32 of those projects were successful in meeting all deadlines. BuildIT met all deadlines of 39 of 111 sampled projects. Use Stat Key to generate a bootstrap distribution for the difference in proportions of projects meeting all deadlines and estimate the 95% confidence interval using the 2 standard error approach. BE sure to tell me what direction you set up (BuildIT - ACME or ACME - BuildIT). For full credit include all of the following: a) The image or very detailed sketch of the bootstrap distribution from stat key. If the values are blurry give me the values for the mean and the standard error. b) Show the calculation of the interval using the 2 SE approach 7. Based on your answer to 6, does the interval provide us a reason to conclude that the population proportions are significantly different from each other? Answer yes or no and supply the reason