If the economy goes into a recession and incomes fall, what happens in the markets for inferior goods?

The price of a one year zero coupon bond is currently G.94. Dn maturity it will pay 1. An equity also has a current price of 0.94. In one year the price of the equity will depend on the state of the economy. lithe economy is in a good state the equity will have a value of 1. 15 and if the economy is in a bad state the equity will have a value of 1190. It is estimated that the economy will be in a good state with probability [LE and in a bad state with probability [1.4. A call option is sold on the equity which will expire in 1 year and has a strike price of G.95. {i} Show that the option price can be derived using the binomial model. [5] {ii} A pricing kernel K: , is a stochastic process such that the total return index for any asset P'f obeys the relationship. P, K, = em K; for all s :r r The value of the pricing kernel at time 1 depends on the state of the economy at that time. (a) Using the binomial model for equities and bonds. derive the values of the pricing kernel at the end of one Year in the good and bad states of the economy. (b) Show that the option price calculated in (i) can also be derived using the pricing kernels from (iia). [T] 8 (i) Consider a fund where the surplus at time t, S(t), is measured as the excess of the value of a portfolio of assets at time t, A(), over the value of a portfolio of liabilities at time t, L((). (a) Write down a differential equation for S(t) in terms of A(t), L(t), the deterministic rate of return earned on the assets, r, and the deterministic rate of increase in the liabilities, ry. (b) State the circumstances under which the surplus diminishes. [3] (ii) Consider a charitable fund that is invested in a portfolio of assets and disburses continuously a constant cash flow at a rate of L per annum. (a) Write down and solve a differential equation for F(t), the fund size at time t, in terms of the deterministic annual rate of return earned on the assets, r , and L. (b) State the circumstances under which the fund is constant through time. (c) Describe the alternatives to ensure that the fund grows. [7] (iii) Outline how asset liability modelling may be done in practice to analyse the problems in (i) and (ii) in the cases where r is stochastic. [3]