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Homework #1: BASIC RETURNS CALCULATIONS Questions 7 - 10 are to be done in Excel. You are invited to use Excel for all of these problems, but it is not required for Q.4 and 6. 4. A bank account pays interest at an annual rate expressed as 3.50% (APR). They compound monthly. a) How much money would you actually have at the end of 6 months if you deposited $500? b) Using continuous compounding with a different APR rate would give the same total amount of dollars in the account after 6 months. What is the continuous compounding APR that corresponds to this 3.50% APR with monthly compounding? c) Starting in 1933 and continuing until 1986, the maximum interest rate a commercial bank was allowed to pay on a savings deposit was set by law at 5%. It was 5 1/4% for a Savings and Loan bank. But this was the annual percentage rate and there was no limit on how frequently the bank could compound interest. What was the maximum EAR that a Savings and Loan could achieve (by compounding continuously) 6. A gold futures contract as traded on the NYMEX calls for delivery of 100 ounces of gold. On September 15, you sell 5 DEC gold futures contracts at a futures price of $1360 per ounce. The margin requirements per contract are: initial margin $2000, maintenance margin $1200. Assume your position is marked to market today at $1360 a) What is the minimum amount you have to deposit to open the position? b) Suppose tomorrow's settlement price for the DEC contract is $1358. What would your variation margin cash flow tomorrow be? c) Suppose tomorrow's settlement is different from $1358. At what futures price would you get a margin call? d) Suppose that tomorrow the futures price jumps from 1360 to 1372. Will you get a margin call? If so, how much will it be for? e) Suppose you carry this position all the way to expiration, at which point the futures expire at a final settlement price of $1325. Do you have to make a delivery of gold, or do you accept delivery from someone else? How much cash will change hands on the day when the gold is delivered? Do problems 7 - 10 in Excel. 7. Consider the US 5 3/8 of Feb 2031. This is a US Treasury bond with a 5.375% coupon, maturing on Feb. 15, 2031 and paying half its annual coupon each year on Feb 15 and half on Aug 15 (or the first business day after that, if the date is Saturday, Sunday, or a holiday such as Presidents Day, Feb. 16, 2015). Assume today is the first Wednesday in September of the current year and right now the bond is quoted in the market at a price of 98-14 (i.e., 98 and 14 / 32). 1 The first thing you need to do is figure out how many days are in the current coupon period and the amount of interest that accrues each day. a) If you buy this bond at the quoted price for delivery tomorrow, what will the invoice price be per 100 face value (the invoice price is the quoted price plus accrued interest)? b) Use the built-in Excel spreadsheet function YIELD to calculate the yield to maturity on the bond at this price. c) Suppose the yield to maturity were 1 basis point higher than what you just calculated. How much different would the price be? (Use the built-in Excel function PRICE to do this.) This difference is called the "price equivalent of a basis point" or the "DV01". 8. The closing prices for stocks ABC and XYZ last week were: Day ABC XYZ Mon Tues Weds Thurs Fri 51 53 52.50 51 53 31 32.50 33 31.50 32 a) Calculate the mean logarithmic returns for each stock, the variances, and standard deviations. (Note: The logarithmic, or continuously compounded return, is defined as Rt = ln(St / St-1). ) b) Annualize the means and variances of the log returns (assume 255 trading days per year). c) Compute the volatilities. (Volatility is the annualized standard deviation of log returns.) d) What are the covariance and the correlation coefficient between ABC's and XYZ's returns? 9. Do problem 8 again, but use simple returns instead of log returns. (Note: The simple return is calculated as Rt = (St / St-1) - 1. For "volatility" in part c), just calculate the standard deviation of the annualized percent return. How similar or different are the answers from the two ways of doing the calculations? 10. Futures mature every few months, so in estimating optimal hedge ratios, it is common to create a historical series of returns on the "nearby" contract (i.e., the one closest to expiration) by stringing together past returns drawn from a sequence of futures contracts. This must be done carefully to avoid having a spurious jump in the price (and therefore, a weird return) on the day when you roll over to the next maturity. You must first compute the returns for the SEP and DEC contract months and then connect the returns into a single returns series. Suppose these are the prices for the Sep and Dec futures contracts in the week when you want to roll over from the Sep to the Dec future. Rollover day is Thursday, meaning that you use the returns from the SEP contract through Weds, and from the DEC contract for Thursday and Friday. 2 Day SEP DEC Fri Mon Tues Weds Thurs Fri 1311.00 1298.00 1274.50 1262.20 1243.20 1271.70 1322.70 1310.30 1286.30 1275.70 1251.80 1283.00 Construct one returns series for this futures contract for the week, with a rollover from the Sep to the Dec contract on Thursday. (Note: for the "return" on a future, just use the percentage price change.) 3