Answer all of them

1. Consider a zero-coupon bond Bt

with three years to maturity. The bond pays $100at maturity if it has not defaulted, or $30 if it has defaulted. Thecontinuouslycompounded risk-free rate is r. In a two-state model the default intensity is l underthe probability measure P, and the bond price is:

(i) Show that P is an equivalent martingale measure. [4]

A derivative pays $35 at time 3 if the bond has defaulted and $0 otherwise.

(ii) (a) Determine a constant portfolio containing the bond and cash which

replicates this derivative.

(b) Derive an expression for the arbitrage-free price for the derivative attime 0 in terms of r and l. [5]

(iii) Explain how your answers to parts (i) and (ii) are related through the value ofthe portfolio in part (ii) also being a martingale. [3]

2.

A random variable X follows a Pareto distribution with density function: 5 X20 (1 +x) For a given estimate d of x, the loss function is defined as: x4 - 4d-x2 + d' (a) Show that the expected loss is given by: E(L(x,d)) =1- 2d2 3 (b) Determine the optimal estimate for d under the Bayes rule.The trustees of a defined benefit pension scheme with a large deficit have agreed to close the scheme to both new members and future accrual with immediate effect. A funding valuation is due shortly. (i) Outline how the closure might affect the funding valuation. [4] (ii) Discuss the actions that the trustees might take to address the key risks now that the scheme is closed to future accrual. [8] The trustees have commissioned an Asset Liability Model (ALM) as part of their analysis. (iii) Describe the two most commonly used types of economic model in Asset Liability Modelling. [4] (iv) List four key inputs for an ALM in order to set assumptions. [2] A new scheme has been suggested where the level of benefits are linked to the share price of the sponsoring employer. (v) Outline the advantages and disadvantages of such a scheme. [2]