Exercise 1.17 (Investment under taxation) An investor has a portfolio of different stocks. He has bought si shares of stock i at price Pi; i = 1, ...,N. n The current price of one share of stock i is qi. The investor expects that the price of one share of stock i in one year will be ri. If he sells shares, the investor pays transaction costs at the rate of 1% of the amount transacted. In addition, the investor pays taxes at the rate of 30% on capital gains. For example, suppose that the investor sells 1,000 shares of a stock at $50 per share. He has bought these shares at $30 per share. He receives $50,000. However, he owes 0.30x(50,000 30,000) = $6,000 on capital gain taxes and 0.01x(50,000) = $500 on transaction costs. So, by selling 1,000 shares of this stock he nets 50,000 6,000 500 = $43,500. Formulate the problem of selecting how many shares the investor needs to sell in order to raise an amount of money K, net of capital gains and transaction costs, while maximizing the expected value of his portfolio next year. = =