Exercise 1.17 (Investment under taxation)

An investor has a portfolio of $n$ different stocks. He has bought $s_i$ shares of stock $i$ at price $p_i$, $i=1,\dots ,n$. The current price of one share of stock $i$ is $q_i$. The investor expects that the price of one share of stock $i$ in one year will be $r_i$. If he sells shares, the investor pays $n$ 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 $0.30\times (50,000-30,000)=\$6,000$ on capital gain taxes and $0.01\times (50,000)=\$500$ 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.