Question

Problem 4 EXERCISE 7.15 (BF 409)      (Formulation only – no computer work is needed)

Exercise 7.15 (Investment Management under taxation) Last year, Ellen Grunbaum purchased Si Shares of stock I at price qv i = 1, … , n. The current price per share one year from now will be rv i = 1, … n. If she sells any shares, she must pay a transaction cost of 1% of the amount transacted. In addition, she must pay a capital-gains tax at the rate of 30% on any capital gains at the time of the sale. For example, suppose that Ellen Grunbaum sells 1,000 shares of a stock today at $50 per share, which she had originally purchased at $30 per share. She would receive $50,000. However, she would have to pay capital gains taxes of 0.30 x (50,000-30,000) = $6,000, fore, by selling 1,0000 shares of this stock, she would have a net cash flow of enough shares of the stock in her portfolio today to generate an amount of cash C today for use as a down payment on a home. Formulate the problem of selecting how many shares of which stocks she needs to sell in order to generate the cash amount C, net of capital gains and transaction costs, while maximizing the expected value of her portfolio for next year.

This requires the following steps:

Formulate the problem

Define the variables

Set up the constraints

Define the objective function

List and describe any assumptions that you made.

Answer 1

Variables :

Si = number of shares purchased

qvi = purchase price per share (i =1,....,n)

rvi = current price per share (i =1,....,n)

Let Xi be the number of shares sold (i =1,....,n)

Constraints:

• Si>=0 (number of shares purchased is greater than or equal to 0)
• Si>=Xi (number of shares purchased is greater than or equal to number of shares sold)
• Xi>=0 (number of shares sold is greater than or equal to 0)

Objective:

Maximize:

Subject to the above constraints.

Assumption: Current market price(rvi) is greater than the purchase price(qvi)