5. James, who joins the market at the beginning of investment with an initial wealth o and initial liability lo, plans to invest his wealth within two-period. He can reallocates his portfolio at the beginning of each period. The capital market consists of one risk- free asset, one risky asset and one liability. At time period t, the given deterministic return of the risk-free asset, the random return of the risky asset, and the random return of the liability are denoted by st (> 1), et and qt, respectively. Assume that the only information known about t and qt are their first unconditional moments, E[et], E[qt] and 2 x 2 positive definite covariance Cov et - E E (000) E (()) = [() < Let It and I be the wealth and liability of the investor at the beginning of period t respectively, then It - It is the net wealth. At time period t, if t is the amount invested in the risky asset, then xt-t is the amount invested in the risk-free asset. Assume that the liability is exogenous. This means that the liability is uncontrollable and cannot be affected by James' strategies. Define the excess return of risky asset as Pt = et - St. The two-period mean-variance model of asset-liability management is to seek the best strategy, , t = 0, 1, which is the solution of the following dynamic stochastic optimiza- tion problem, min Var(2-2) = E[(x2 - 12-d)], s.t. Ex2 2]=d, = 8txt + Ptt, lt+1=9tlt, t 0,1. [10 marks] [5 marks] (i) What is the optimal strategy of every period? (ii) What is the optimal Lagrange multiplier? (iii) What is the minimum value of Var(2-12)? [10 marks]