Assume a1 " 02 " 0.2, in " 0.2, #2 " 0.1, gm " 0.5,p11 " 0.9, p21 " 0.01. Find the optimal policy to this problem. You can use dynamic programming (backward induction) and / or any methods including numerical fsimulation methods. (e) Compare your answer in (d) to that in (a), what conclusions can you draw? 4 Models for market frictions 4.1 Asset allocation under costly trading Suppose someone has initial endowment X5 " (so, .51, 52, - -- ,5a') (in dollar amount) in cash and d risky assets. r, R1, - -- ,Rd are the gross return of each asset, where R ~ N(p., )3). Suppose trading each risky asset is costly, with cost function Odo-3) " $3,511,\") + k|m| + 1352532. Withdrawing or saving money in the risk-free asset is costfree. 00 (ac) " 0. (Refer to L10 for more detailed explanation) Consider the one-period asset allocation problem in this market. (a) Suppose one wants to maximize the expectation of total book value (before liquidation) and to minimize variance of total book value. How to formulate this problem as an optimization problem? (b) Suppose one wants to maximize the expectation of total value after liquidation and to min- imize variance of total liquidated value. How to formulate this problem as an Optimization problem? (c) This is a counterpart to Markowitz's model in this market. Describe a way to solve this problem and get the efcient frontier. (You can explain your approach using mathematical formulas and plain words)