futures

# futures - Forwards and Futures 1 Forward Rates no-arbitrage...

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) Emma Rasiel, 2009 1 Forwards and Futures 2 Compare two possible investments with \$1 mil Forward Rates: no-arbitrage approach 1. Lend \$1,000,000 at 7.5% per annum for two years 2. Lend \$1,000,000 at 7.2% for one year, and when it matures, reinvest all proceeds for a further year at rate f 1,1, which is guaranteed today Strategy 1 Time Dollar Value Annual rate t = 0 1,000,000 7.5% t = 1 1,075,000 7.5% t = 2 1,155,625 Strategy 2 Time Dollar Value Annual Rate t = 0 1,000,000 7.2% t = 1 1,072,000 ???% t = 2 1,072,000*(1+?) Suppose we can 'lock in' today the one-year rate f 11 that we will get one year from now For no-arbitrage: 1.075 2 = 1.072*(1+ f 11 ) f 11 = 0.078 General Formula for f mn : the n-period rate beginning m periods from now (given today's spot m+n and m-period rates y m+n and y m ) (1+ y m+n ) m+n = (1+ y m ) m (1 + f mn ) n

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) Emma Rasiel, 2009 3 Forward Rates: riskless arbitrage example Suppose r 1 = 7% r 3 = 7.4% f 1,2 = 7.5%. (1 + r 3 ) 3 = (1 + 7.4%) 3 = 1.2388 (1 + 7%)(1 + 7.5%) 2 = 1.2365 Invest at 7.4% annually for 3 years. Total return is 23.88% Invest at 7% for 1 year, and 7.5% per annum the next 2 years. Total return is 23.65% Is there an arbitrage opportunity? If so, how do we take advantage of it? Step 1 : Is the forward rate f 1,2 too high or too low? HINT: do we get a better overall return using the 1 year “spot” rate and the forward rate, or the 3 year “spot” rate? NOTE: how do we know that it is the forward rate (rather than one of the spot rates) that is “wrong”? 4 Forward Rates: riskless arbitrage Step 1 : Is the forward rate f 1,2 too high or too low? TOO LOW! (why?) Step 2 : Create an arbitrage using all three rates… Time 0 1 3 Lend \$100 for 3 yrs at r 3 –100 +100 (1.074) 3 = 123.88 Borrow \$100 for 1yr at r 1 +100 –100 (1.07) = –107.00 Borrow \$107 for 2yrs at f 1,2 +107 –107(1.075) 2 = –123.65 0 0 +0.23 Rates: r 1 = 7% r 3 = 7.4% f 1,2 = 7.5%.
) Emma Rasiel, 2009 5 Applications of Forwards and Futures A gold-mining company has projections for gold extraction quantity, fixed and variable mining costs, etc., but is exposed to fluctuations in the price of

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