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

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]An investor buys, for a premium of 187.06, a call option on a non-dividend paying stock whose current price is 5,000. The strike price of the call is 5,250 and the time to expiry is 6 months. The risk free rate of return over this period is 5% p.a. The Black-Scholes price for a call option on a non-dividend paying share is c = S,Q(d ) - Kerr-