This project involves the creation of a seven year investment strategy for a client of yours. The client has identified ten investments of interest, and assigned a risk factor to each. The particulars of these investments are shown in Table 1. The different investments are labeled with the letters A through J. Your client can commit to any non-negative number of units of an investment, even fractional units. (There may be an upper bound on the number of units in which a single person or entity can participate. Read on for details.) Each unit of an investment will involve a two way flow of money. Table 1 shows this cash flow, with negative entries indicating cash flowing from

your client into the investment and positive entries indicating a return from the investment to your client. As an example, suppose your client chooses to participate in one unit of investment G. Your client needs to send $1.00 to the investment at the beginning of years 1, 3 and 5. In return for these payments, you client would receive $0.50 at the beginning of year 2, $0.60 at the beginning of year 4 and $2.00 at the beginning of year 6. These investment opportunities are realistic. Investment A is like a bond, in which you give money to a company now and they return your money, with interest, at a later point in time. Investment I is a bit like a loan or mortgage, where you get money from someone else up front and have to pay them back later. You and your client need to decide in how many units of each investment your client should participate. Your client has one million dollars at the beginning of year 0 and is willing to tie up all of this money over the next seven years. The overall goal for your client is to maximize the amount of cash on hand after the transactions at the beginning of year 6, subject to some constraints outlined below. The first constraint has to do with risk. The integrity and financial stability of the investment groups vary from project to project. Your client is willing to take on some risk, but also wants control over the riskiness of the investment portfolio. Your client has established the riskiness of all the investments, as shown in Table 1. Higher numerical values indicate higher risk. The specific measure of risk that interests your client is the unit-weighted average risk, the average risk of the investments weighted by the number of units in each investment. As an example, suppose your client has decided to participate in 5 units of investment A

(risk 11), 12 units of investment B (risk 8) and 50 units of investment C (risk 14). The unit- weighted average risk of this collection of investments would be

The second major constraint has to do with the maximum number of units available of any in-

vestment. Due to various governmental and stock exchange regulations, your client may participate in no more than 750,000 units of any single investment. The third, and last, constraint has to do with cash balances. It may be to your clients advantage to leave some money uninvested in some year. Any such money is called cash-on-hand, which is available for use in subsequent years and earns no interest. Your client must maintain a non-negative balance of cash-on-hand at all times during this investment cycle. All transactions, money in and money out, happen simultaneously at the beginning of each year. After each years transactions, the amount of cash-on-hand must be nonnegative. The million dollars your client has to invest may be considered the cash-on-hand before the Year 0 transactions.

The most basic information your client seeks is an optimal investment strategy (maximize cash-on- hand at the end of the seven years) given the constraint that the unit-weighted average risk must

be at most 10. Responding to your clients desire for this information involves providing a list of the number of units of each investment your client should participate in, the amount of cash that will be on hand at the end of the seven years, and the cash-on-hand balance for each of the seven years of the investment period. Your client is also curious about a few other matters. For one thing, your client wonders how much potential gain is being lost due to risk aversion. To answer this question, provide your client with a list of investment outcomes giving varying levels of risk acceptance. For every integer from 7 to 15, inclusive, indicate the amount of cash your client could have on hand at the end of the investment period if this number were the upper bound on acceptible average risk. You do NOT need to specify the investment schedule that would provide this return. A table and/or graph would be a nice way to present this information. Your client is thinking about developing some political pull to change some of the regulations concerning the number of investment units in which a single person can participate. Given an average risk tolerance of 10, indicate the optimal amount of cash that your client could have at the end of the investment period if your client were allowed to purchase up to 1,000,000 units, or 2,000,000 units, of any investment. If your client was successful at having the unit ownership caps removed entirely, what would be the maximum potential cash-on-hand at the end of the seven year investment period? Next, suppose your client decides to go for broke, specifically arranging for the political clout

needed to remove all investment limits while agreeing to accept unlimited risk. Under these cir- cumstances, what is the maximum amount of money your client can have on hand at the end of

the seven years? The last thing you should address is not a question from your client. You are a very clever financial analyst, not one to miss an opportunity. You find yourself wondering if it is possible to make money for yourself by personally participating in this investment scheme. The only problem is that all your cash resources are tied up and, due to bad luck and prior financial misadventures, no one will give you a loan. Even with zero dollars of capital available, you think you might be able to make some money for yourself by cleverly participating in investment units under the same regulations as apply to your client. Comment on whether your intuition is correct, carefully and fully justifying your conclusion. Your discussion of this should appear AFTER the third section of your paper, in the form of a note in your personal financial diary.

Table 1: Investment Payoff Schedule and Risk Analysis Year Risk 0 1 2 3 4. 5 6 A 11 -1.00 0.00 1.07 0.00 0.00 0.00 0.00 B 8 -1.00 0.03 0.03 0.40 0.30 0.20 0.10 14 -1.00 -1.00 0.00 0.00 0.40 0.88 0.80 D 17 0.00 -1.00 -1.00 -1.00 1.03 1.03 1.03 E 11 0.00 -1.00 0.00 1.07 0.00 0.00 0.00 F 26 0.00 0.00 -1.00 -1.00 -1.00 -1.00 1.56 1.56 G 20 0.00 -1.00 0.50 -1.00 0.60 -1.00 2.00 H 8 0.00 0.00 0.00 0.00 -1.00 0.00 1.06 I 19 1.00 1.00 1.00 1.00 -1.30 -1.30 -1.50 J 14 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 6.08 (5 x 11) + (12 x 8) + (50 x 14) 5 + 12 + 50 12.701 Table 1: Investment Payoff Schedule and Risk Analysis Year Risk 0 1 2 3 4. 5 6 A 11 -1.00 0.00 1.07 0.00 0.00 0.00 0.00 B 8 -1.00 0.03 0.03 0.40 0.30 0.20 0.10 14 -1.00 -1.00 0.00 0.00 0.40 0.88 0.80 D 17 0.00 -1.00 -1.00 -1.00 1.03 1.03 1.03 E 11 0.00 -1.00 0.00 1.07 0.00 0.00 0.00 F 26 0.00 0.00 -1.00 -1.00 -1.00 -1.00 1.56 1.56 G 20 0.00 -1.00 0.50 -1.00 0.60 -1.00 2.00 H 8 0.00 0.00 0.00 0.00 -1.00 0.00 1.06 I 19 1.00 1.00 1.00 1.00 -1.30 -1.30 -1.50 J 14 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 6.08 (5 x 11) + (12 x 8) + (50 x 14) 5 + 12 + 50 12.701