Provide sufficient explanations to the following attachments.
An insurance company with substantial domestic pensions business is reviewing its policy with respect to derivatives across various asset classes. The actuarial risk function is concerned about the amount of longevity and Limited Price Indexation (LPI) risk inherent in the pensions business. It has noted that most of the OTC contracts available to manage these risks tend to be arranged with counterparties in the banking sector, which has been under stress recently. (i) Describe the main types of derivative instruments that can be used to hedge longevity risk. [5] (ii) Explain how counterparty risk originates in OTC contracts and how the insurance company can mitigate it. [4] (111) (a) Discuss the nature of the insurance company's LPI risk in a low inflation environment, commenting on potential ways of reducing it. (b ) Identify institutions that might be natural hedgers of inflation risk in the financial markets, commenting on how likely it is that they would want to use inflation derivatives.A financial institution has written f1 billion of five-year equity linked bonds with maturity guarantees. Rather than buying a five-year Put option as a hedge, it has been delta hedging the implicit option dynamically using equity index futures. Eighteen months from the outset, you have been asked to compare retrospectively the effect of the futures based strategy against what the institution would have experienced had it purchased a five-year option. You have broken down the difference between the two strategies into the main option sensitivities (Greeks) and separated the figures into three time periods: . Months 1-6 ("Market fall"), during which equity prices fell steadily. . Months 7-12 ("Volatile period"), during which equity prices were extremely volatile but finished at levels similar to those at the start of the period. . Months 13-18 ("Market recovery"), during which equity prices rose steadily. Your analysis assumes that market implied volatilities remained at 30% throughout the eighteen month period, this being the same volatility as was assumed throughout in the delta hedging calculations.1. Consider an economy with two time periods (labelled 0 and 1) in which a typical consumer has preferences given by: log(co) + Blog(c,), where ct is consumption in period f. Each consumer is endowed with ko units of capital at the beginning of period 0 and with one unit of time in each period. In each period, there is a price-taking, profit-maximizing firm that produces goods using capital and labor. These goods can be either consumed or saved in the form of capital that can be used in production in the next period. Let the firm's production function be: y = zknl-", where k is the amount of capital rented by the firm and n is the amount of labor rented by the firm in a given period. Because leisure is not valued (leisure does not appear in the utility function), each consumer supplies labor inelastically, i.c., he supplies one unit of labor in each time period. The only interesting decision that a consumer makes, then, is how much to save in period 0. Let k, be the amount of capital that a typical consumer saves in period 0. Then each consumer faces a pair of budget constraints: Co = Toko + Wo - ki ho and where r, is the rental price of capital in period / (expressed in terms of period- consumption goods) and w, is the wage rate in period / (again expressed in terms of period- consumption goods). Each consumer takes these prices as given when deciding how much to save. Because period 1 is the last period of his life, each consumer consumes all of his resources in period 1. In equilibrium, the markets for goods, labor, and capital must clear in both time periods. (a) Find an explicit expression (in terms of primitives) for the competitive equilibrium capital stock in period 1. Use your answer to determine the equilibrium rate of return on savings between periods 0 and 1. In addition, determine the equilibrium allocation of consumption across the two time periods. How do changes in z affect the equilibrium allocation? Explain. (10 points) (b) Formulate a social planning problem for this economy and show that the allocation chosen by the social planner is identical to the competitive equilibrium allocation that you determined in part (a). (5 points)Question 3: Two-period model with a borrowing constraint 1. Without the borrowing constraint. Ci = .()) + .). Under the assumed values of r, Y. and 8, it wants to consume 400/3, which is more than what it earns in period 1. Under the borrowing constraint, It is not allowed. so Ci -100 and C. = 100 2. The optimal value of Ci without the constraint is 200/2.2, which is smaller than Vi. Hence it is still allowed even under a borrowing constraint. The optimal value of C, is 1200. 3. For the first problem, it changes Of and 6% as the household wanted to borrow if it was allowed. In the second problem, it does not have any effect since the household did not want to borrow
