This problem set is to be completed separately by each student. You can discuss the portfolios and the methodology with other students.
1(a). You have been given $1,000,000 to invest in a combination of futures contracts and a money market fund using closing prices of November 9. Hence, your positions are proposed trades (i.e. you don't know precisely the prices you will get.) If you go slightly over the $1,000,000 then you can borrow at 2% (annualized) over the 30-day Treasury rate. If you are under, then invest at the bill rate. If there are big differences, you will have to downsize/upsize your trades. You must liquidate your position no later than the close of trading on trading day before the assignment is due. Your $1,000,000 should be used either as margin for futures positions (earning no interest), or set aside in a money market fund (earning the one month bill rate).
You may go long, short, or spread as many contracts as you wish on any futures market quoted in the WSJ. Fractional positions are not allowed, but you may hold contracts in a number of different markets. Initial and maintenance margins are those for small speculators quoted on the attached margin sheets, which are attached.
Along with your position, you must submit an estimate of your leverage employed in each futures market, based upon your allocation of your $1,000,000 to the various positions that you take. You should also show the ratio of your allocation of funds to each position to the margin required for the position. Finally, you should obtain an estimate of the percentage (annualized) standard deviation of an unlevered position in each of your contracts, and compute an estimate of your standard deviation as a fraction of your equity investment in each contract.
The best estimate of volatility is the implied (annualized) volatility for the corresponding option. A crude measure of the annualized standard deviation is 4 times the margin on the contract. I will hand out hardcopies of the Goldman Sachs volatility review.
I have also collected margin information from a top brokerage house. This is available in Margins.
The table that you should use to present this information is the following:
Sample Position Table
Position 1 Position 2 Totals
Futures market Treasury bonds Live cattle Exchange CBT CME Contract month June 1992 June 1993 Long or short Long Long Number of contracts 5 5 Value per contract (approx.) $ 66,000 $ 27,000 Total value of assets 330,000 135,000 $465,000 Margin per contract 2,000 1,200 Total margin of pos'n 10,000 6,000 16,000 Money market reserve 60,000 24,000 84,000 Total equity committed 70,000 30,000 100,000 Ratio: Equity/margin 7.0 5.0 6.2 Leverage: Assets/equity 4.7 4.5 4.6 Annual $ std dev'n/contract 8,000 4,800 Total $ std dev pos'n 40,000 24,000 64,000 Ratio: Total $ std dev/assets 12.1% 17.8% 13.8% Ratio: Total $ std dev/equity 57.7% 80.0% 64.0%
With both your options assignment and the futures assignment, you may change your portfolio by making trades during the time you hold your portfolio. Similar to the options assignment, trading is allowed by e-mail to ceb4@mail.duke.edu. Remember that the trade takes effect at the close of trading on the day the trade is initiated. The price that you buy at or sell at is uncertain. Bear in mind that each of these trades increases your record keeping requirements in Part B.
1(b). You must also compute an estimate of the dollar standard deviations of your individual portfolio positions for holding period of the assignment, rather than for a 1-year holding period. Assuming returns are independent and have the same variance, just use the formula we employed with options: [sigma x sqrt{T}]. Roughly approximate the standard deviation of your whole portfolio by taking the average of the volatilities. (You can adjust this downwards if you have hedged positions -- use judgement.)
1(c). For comparison purposes, each student must also compute the returns to a position that invests the $1,000,000 in the Dow Jones Industrial Average. For example, if the DJIA is approximately 5000 at the initiation of your portfolio, you buy ($1,000,000/5000) = 200 "shares" of the DJIA. You may assume that the annualized standard deviation on the DJIA is 15%. Fill in the above table for this investment in the DJIA, too.
1(d). Should your futures position be more or less risky than the DJIA? What are the approximate daily standard deviations of the stock and futures positions, respectively? Show your computations.
1(e). Note: if you attach $5 to the Options assignment, then you are in the running for a cash prize.
2(a). On the above date, you should hand in a day-by-day table of your gains and losses for each different futures position and for the DJIA, using a form similar to the sample spreadsheet and in the lecture note. For simplicity, you do not have to compute the daily interest earned on the money market fund.
If your funds for one position are insufficient to meet its margin calls and you have excess funds in another position, then you may shift funds across accounts. However, if your total equity balance ever becomes insufficient to cover total margin requirements, then you must liquidate some of your position and inform my secretary (treat this as a trade and she must initial the trade which will become effective at the close of trading).
2(b). With this day-by-day worksheet on your futures positions, you should supply a paragraph of comments on whether or not you took too large or too small a position for your $1,000,000 capital. Compare the volatility of your profits and losses with that of a $1,000,000 investment in the Dow Jones Industrials for the same period. Compare it also to the return you would have had if your bought the DJIA on margin with just 50% down, i.e., buying $2,000,000 of stocks by borrowing $1,000,000 at 2% above the one month bill rate from your broker (as you may do in stocks).
2(c). Compute the actual standard deviation of daily price changes for each position and for the futures and stock portfolios for the holding period, comparing actual results with your Part A estimates.
2 (d). What, if anything, would you do differently the next time? What would have happened if you had put no money aside in a money market fund for margin calls (taking a proportionately larger position). What would you have held if you only had $100,000 to invest? How about $20,000? Finally, how would you adjust your portfolio if you planned to hold your position for a year, rather than for 2 weeks?