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Interest Rate Futures and TED Spread Trading

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This workshop presented to staff in Sales and Analytics is an overview of the eurodollar futures contract. It focuses on the trading mechanics including some arbitrage and hedging samples and an …

This workshop presented to staff in Sales and Analytics is an overview of the eurodollar futures contract. It focuses on the trading mechanics including some arbitrage and hedging samples and an in-depth study case on trading the TED spreads with details on the calculation and presentation of Bloomberg analytics.

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  • 1. Eurodollar Futures and TED spread Trading Training Workshop François Choquet Advanced Application Specialist July 2011
  • 2. Motivations and Applications• Speculation on views on interest rates.• Hedge against fluctuations in Interest Rates. 1. Convert fixed rate loans into floating rate loans. 2. Convert Floating rate loans into fixed rate loans. 3. Hedging using a stack of Eurodollar Contracts 4. Hedging using a strip of Eurodollar Contracts
  • 3. Motivations and Applications• Speculation on views on interest rates.• Hedge against fluctuations in Interest Rates. 1. Convert fixed rate loans into floating rate loans. 2. Convert Floating rate loans into fixed rate loans. 3. Hedging using a stack of Eurodollar Contracts 4. Hedging using a strip of Eurodollar Contracts
  • 4. Speculating with IR Futures• Trading by holding an outright positions i.e. long or short or trading a spread• The long trader bets that interest rate will fall so the price of the futures will rise• The short trader bets that interest rate will rise so the price of the futures will fall.• The spread trader bets that interest rate curve will steepen or flatten.
  • 5. Outright Position• (1) The trader believes that short term rates will rise and execute the following trade: Date Futures Market 5/16/2011 Sell one SEP 11 ED Futures at 99.685 8/14/2011 Buy one SEP 11 ED Futures at 99.505 Profit=18 basis points Total Gain=18 x 25 x 1 =450• To profit from rising rates, the trader must be short IR futures. Accordingly the trader sells one SEP11 contracts at 99.30. Five days later IR have risen and the futures contract trades at 99.12.• The trader gains 18 basis points. As each basis point is worth $25, the total profit is $450.
  • 6. Spreads• Intracommodity spread: Speculation on the changing shape of the IR curve. E.g. spread between a nearby and more distant futures contract.• Intercommodity spread: Shifting risk from two different instruments: Libor-OIS spread.• Today, a trader considers the following Libor rates and futures yields: Time to mty Libor Spot Futures Contract Ticker Futures Futures Rates Yield Price 3m 0.264 SEP 11 - 4.4 months EDU1 comdty 0.32 99.68 6m 0.27485 DEC 11 - 7.4 months EDZ1 comdty 0.415 99.585 9m 0.30915 MAR 12 - 10.4 months EDH2 comdty 0.575 99.425 1y 0.35741• The yield curve is upward sloping with a spread between 12 month and 3 month showing 9 basis points.• The futures yields are consistent with the forward rates implied from the Eurodollar curve.
  • 7. Spread Curve Trade• (2) The trader speculates that the curve will flatten within the next 6 months and decides to execute the following trade: Date Futures 11-May-11 Buy the MAR12 ED contract at 99.425 Sell the DEC11 ED contract at 99.585 30-Jul-11 Buy the DEC11 ED contract at 99.635 Sell the MAR12 ED contract at 99.525 Profit=5 basis points Total Gain=$125• By buying the more distant MAR12 contract and selling the DEC11 contract today , the trader bets that the yield differential of 16 bps of will narrow.• On July 30th, the yield spread diff. between MAR12 and DEC11 is 11 bps.• No matter whether rates rise or falls, this spread strategy will produce a profit.
  • 8. FRA/OIS Spread• Speculating on changing risk structure of interest rates.• E.g. risk of widespread default triggers widening of spread between OIS and Libor reflecting the changing perception of the risk involved in holding Eurodollar deposits in the face of potentially very large loan losses.• Assume the spread between the 3 month IMM OIS and FRA is 17bps.• The banks’ riskiness is perceived to increase, we might expect the spread to widen. (This would be the case whether interest rates are rising or falling).
  • 9. • (3) To take advantage of this view, the trader could sell the SEP IMM FRA buy the SEP IMM OIS contract.Date FuturesToday Sell $1mm SEP IMM 3MO FRA at a rate of 0.32% Buy $1mm SEP IMM OIS at a rate of 0.16%August Sell $1mm SEP IMM OIS at a rate of 0.17%15th Buy one SEP IMM 3MO FRA at a rate of 0.40%Profit = 7 basis pointsTotal Gain = 7 x 25 = $175• On August 15 the spread between the two contracts has widened by 16 basis points, in line with the trader’s expectations which produces a profit of $400.• The futures prices already embed the expectation of higher rates and spread between Eurodollar and OIS. Thus, by engaging into this strategy, the trader speculates AGAINST the rest of the market.• It is not enough to expect yield spreads to widen, but the trader must expect them to widen MORE than the market EXPECTS.
  • 10. Motivations and Applications• Speculation on views on interest rates.• Hedge against fluctuations in Interest Rates. 1. Convert fixed rate loans into floating rate loans. 2. Convert Floating rate loans into fixed rate loans. 3. Hedging using a stack of Eurodollar Contracts 4. Hedging using a strip of Eurodollar Contracts
  • 11. Creating a Synthetic Fixed Rate Loan• A construction firm plan a project that will take six months to complete. It is worth $100 million. The bank provides funds for 6 months at a single rate, that is 200 bps above the 90- day Libor.• The rate for the second quarter is 200 bps above the 90 day Libor rate that prevails at that date.• The company must pay interest at the end of 3 month and interest plus principal at the end of the 6 month period.
  • 12. Schedule Cash Market Futures MarketJune 20th, 2011 Borrow $100 m at 2.316% Sell 100 Sept. Eurodollar for three months from the Futures Contracts at 99.66 bank who commit to extend which corresponds to a the loan for 3 additional 0.34% yield. months at 200 bps above 3 month Libor.September 20th, 2011 The company pays interest Offset 100 Sept. contracts of $591,886.67. The 3 Mo at 99.06 reflecting a 0.84% Libor is now at 0.84% so the yield. The trade produces company borrows for a profit of $125,000. another 3 months at 2.84%. (50*25*100)December 20th, 2011 Pay interest of $717,888.89 Futures Profit: $125,000. and repay principal of $100m. Total interest expense $1,309,755.56 Net Interest Expense after Hedging: $1,184,755.56
  • 13. Synthetic Floating Rate Loan• The bank decides to let the company borrowing at a fixed rate.• The bank’s cost of funds is 90 day Libor and expect to pay 0.316% this quarter and 0.34% next quarter, so an average of 0.328% over 6 months.• Therefore the bank decides to make a fixed rate 6 month loan to the construction company at 2.328%.• The bank’s expected profit is the 200 basis points between the lending rate and the bank’s Libor based cost of funds.• If Libor rises by 50 bps to .% for the second quarter, the bank will have to pay an additional $125,000 in interest. To avoid that the bank will transact as follows:
  • 14. Schedule Cash Market Futures MarketJune 20th, 2011 Borrow principal of $100m Sell 100 September at 0.316% and lend it for 6 Eurodollar contracts at months at 2.316% to the 99.66 (.34% yield) construction company.September 20th, 2011 Pay Interest of $80,755.56 Offset the 100 Sept. Libor is now at .84% so the contracts at 99.16 bank borrows $100m @ reflecting the .84% yield. It .84%. produces a profit of $125,000.00March 20th, 2011 Pay interest of $212,333.33 and repay principal of $100m. Total Expense=$293,088.89 Profit=$125,000Net interest expense after hedging: $168,088.89
  • 15. Multi-Period Funding• In the previous example, the interest risk focuses on a single date. Often the period of the loans comes at a number of different dates at which the rate might be reset.• The company makes a more realistic assessment of the completion date of the project: 1 year.• The bank insists on making a floating rate loan for 3 months at a rate of 200 basis points above the 90 day Libor rate prevailing at the time. – 3 month Libor: 0.316% – SEP Eurodollar: 0.34% – DEC Eurodollar: 0.416% – MAR Eurodollar: 0.595%• The cost of funds is then 2.316%, 2.34%, 2.416% and 2.595% or @100m @ an average rate of 2.41675%.• In a stack hedge, all of the futures contracts are concentrated or stacked in a single futures expiration date.
  • 16. Scenario 1: Parallel Shift• Shortly after the company enters the hedge, Libor rates jump by 50 basis points. The borrowing rate for the next 3 quarters are then: – September 11 – December 11 : 0.84% – December 11– March 12: 0.916% – March 12 – June 12: 1.095%• Hedge $100 m with 300 September Eurodollar Futures contracts.
  • 17. Eurodollar Stack Hedge Cash Market Futures MarketJun 20th, 2011 Borrow $100 m at 2.316% for 3 months and Sell 300 Dec Eurodollar futures contracts commit to roll over the loan for 3 quarters @ 99.66 which corresponds to a yield of at 200 basis points over the prevailing Libor 0.34%. rate.Sep 20th, 2011 Co pays interest of $591,866.67. Libor is Offset 300 Dec Eurodollar contracts @ now 0.84% so the co borrows $100m @ 99.16 which reflects the yield of 0.84%. 2.84%. The trade produces a profit of 50*25*300=$375,000.Dec 20th, 2011 Co pays interest of $717,888.89. and borrows $100 m for 3 months @ 2.916%.Mar 20th, 2012 Co pays interest of $737,100.00 and borrows $100m for 3 months @ 3.095%.June 20th, 2012 Co pays interest of $790,044.44 and repays principal of $100m. Total interest expense: @$2,837,800.00 Futures profit : $375,000Total interest expense net of hedging: $2,462,800.00Initial cost without 50 basis point increase: $2,457,029.17 (2.41675%*100m*366/360)
  • 18. Scenario 2: Steepening Curve• Shortly after the company enters the hedge, Libor rates jump unevenly across the Libor curve. The borrowing rate for the next 3 quarters are then: – September 11 – December 11 : 0.43% (+9bps) – December 11– March 12: 0.93% (+51bps) – March 12 – June 12: 1.5% (+55 bps)• Hedge $100 m with 300 September Eurodollar Futures contracts.• With this changes the company will suffer an increase in borrowing costs as follows: New rate Days in Cost for the period period June – September: 2.32% 92 591,866.67 September-December: 2.43% 91 614,250.00 December-March 2.93% 91 740,638.89 March-June 3.50% 92 894,444.44 $ 2,841,200.00• This change in rates produces an increase in costs of $348,171.00 from the initially expected level of $2,457,029.17 to $2,841,200.00• Here the DEC contract produces only a gain of which is equal to: 0.09/0.005*12.5=$67,500. It isn’t sufficient to cover the increase in cost.
  • 19. Interest Rate Curve Scenarios1.60%1.40%1.20%1.00% expected cost of funding today0.80% Cost of funding (+50 bps parallel shift) Cost of funding (steepening)0.60%0.40%0.20%0.00% today (June 20th 2011) Sep 11-Dec 11 Dec 11-Mar 12 Mar 12-June 12
  • 20. A Strip Hedge• Unlike a stack hedge which concentrates the position on a single expiration date, a strip hedge uses an EQUAL number of contracts for each futures expiration over the hedging horizon.• For a $100 mln financing requirements at risk for three quarters, the co sells 100 ED contracts each of the SEP, DEC and MAR futures instead of the 300 contracts on SEP futures.• With the hedge in place, each quarter of the coming year is hedged against shifts in IR for that quarter.• (see next table) Timing of the futures hedge to that of the market risk exposure: The performance of the strip hedge results from the alignment of the futures market hedges with the actual risk exposure of the firm.• Performance depends on the horizon and the liquidity of the most distant contracts.
  • 21. Eurodollar Strip Hedge Cash Market Futures MarketJun 20th, 2011 Borrow $100 m at 2.316% for 3 Sell 100 for each of Sept, Dec and months and commit to roll over Mar @ 99.66, 99.584, 99.405 the loan for 3 quarters at 200 basis respectively. points over the prevailing Libor rate.Sep 20th, 2011 Co pays interest of $591,866.67. Offset 100 Sep contracts @ 99.57. Libor is now 0.43% so the co Profit=$22,500.00 borrows $100m @ 2.43%.Dec 20th, 2011 Co pays interest of $614,250.00 Offset 100 Dec contracts @ 99.07. and borrows $100 m for 3 months Profit=$128,500.00 @ 2.93%.Mar 20th, 2012 Co pays interest of $740,638.89 Offset 100 Mar contracts @ 98.5. and borrows $100m for 3 months Profit=$226,250.00 @ 3.5%.June 20th, 2012 Co pays interest of $894,444.44 and repays principal of $100m. Total interest expense: Total Profit = $377,250.00 $2,841,200.00Total interest expense net of hedging: $2,463,950.00
  • 22. Arbitraging and Hedging Treasuries against Eurodollars.TED SPREAD
  • 23. Speculating with IR Futures• Trading by holding an outright positions i.e. long or short or trading a spread• The long trader bets that interest rate will fall so the price of the futures will rise• The short trader bets that interest rate will rise so the price of the futures will fall.• The spread trader bets that: – Interest rate curve will steepen or flatten. – The correlation between the ED futures rates and yield on Treasuries will change over time (TED).
  • 24. G7 Macro Situation Today Events with Significant Impact• Strong recovery of the global economy in 2010 to 1st quarter 2011 but outlook for growth tilted on the downside amid weaker consumer sentiment.• Price risk is rising but expectations remain anchored to central banks’ objective of keeping inflation close to 2%.• Expectations for higher policy rates from ECB & BOE.• Severe stress in the bond markets reflecting the on-going sovereign crisis in the Euro-zone. Downgrades of Greece and Portugal.• Large exposure of G7 banks to Greece, Ireland, Portugal and Spain.• Geopolitical tensions and North African and the middle east.
  • 25. -40 -30 -20 -10 -12 -10 -8 -6 -4 -2 10 20 30 40 0 0 2 4 8 6Mar-04 Dec-98 Jul-04 Aug-99Nov-04 Apr-00Mar-05 Dec-00 Jul-05 U.S. Aug-01 U.S.Nov-05Mar-06 Apr-02 Jul-06 Dec-02Nov-06 Aug-03 EurozoneMar-07 Apr-04 Eurozone Jul-07 Dec-04Nov-07 Aug-05Mar-08 U.K. Jul-08 Apr-06 U.K.Nov-08 Dec-06Mar-09 Aug-07 Japan Jul-09 Apr-08 Industrial Production YoY % JapanNov-09 Dec-08Mar-10 Aug-09 Real Output Growth YoY changes % Jul-10Nov-10 Apr-10Mar-11 Dec-10 -60 -40 -20 40 60 0 20 10 20 30 40 50 60 70 0Dec-96 Mar-04 Jul-97Feb-98 Aug-04Sep-98 Jan-05Apr-99 Jun-05 U.S.Nov-99Jun-00 Nov-05Jan-01 Apr-06Aug-01Mar-02 Sep-06Oct-02 Feb-07 EurozoneMay-03Dec-03 Jul-07 Jul-04 Dec-07Feb-05 May-08 U.K.Sep-05 Exports YoY %Apr-06 Oct-08Nov-06 Mar-09Jun-07Jan-08 Aug-09 Japan PMI CompositeAug-08 Jan-10Mar-09 Global Purchasing Manager Index Jun-10Oct-09May-10 Nov-10Dec-10 Apr-11
  • 26. Inflation Rates YoY changes % Global Commodity Indices U.S. Eurozone U.K. Japan Agriculture Metal Energy 6 450 5 400 4 350 3 300 2 250 1 200 0 150-1 100-2 50-3 0 May-09 May-10 May-11 Nov-09 Nov-10 Jul-09 Sep-09 Jul-10 Sep-10 Jan-09 Mar-09 Jan-10 Mar-10 Jan-11 Mar-11 Aug-99 Apr-00 Aug-01 Apr-02 Aug-03 Apr-04 Aug-05 Apr-06 Aug-07 Apr-08 Aug-09 Apr-10 Dec-98 Dec-00 Dec-02 Dec-04 Dec-06 Dec-08 Dec-10 Public Debt / GDP % Budget Deficit (-) %160 10140 Greece, 144 5120 0 Italy, 118.1 Italy, -4.6 -5100 Spain, -9.2 Ireland, 94.2 -10 80 Portugal, 83.2 Portugal, -9.1 -15 Greece, -10.5 60 Spain, 63.4 -20 40 -25 20 -30 Ireland, -32.4 0 -35 2003 2004 2005 2006 2007 2008 2009 2010 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
  • 27. Tensions in the Government Bond Markets Government Bond Spreads in 2010 and 2011 1400 1200 Greece, 1209.4 1000Spreads in bps 800 Ireland, 709.7 600 Portugal, 547.6 400 200 Spain, 193.1 Italy, 121.7 0
  • 28. Deterioration in perceived debt sustainability of “PIGS” Five Year CDS spreads16001400 greece cds usd sr1200 5y, 1248.3971000 800 portug cds usd sr 5y, 606.5 600 400 200 spain cds usd sr 5y, 232.248 0
  • 29. Banks Exposure to “PIGS” End of Q3 2010; in billion of US dollars – Source BIS Germany France Italy Other Euro Area UK Japan U.S. R.O.W Total exposure $2.512 trillion Germany, 685.6 France, 632.5 UK, 609.3 Germany, 570.7 523.7 UK, 421.2 U.S., 426.7 287.5 France, 247.3 193 Germany, 137.1Germany, 179.2 France, 200.8 151.7 France, 128.5 112.3 111.5 97.3 93.2 UK, 92.8 98.7 82.8 UK, 55.8 64.1 66 63.8 17.7 26.1 20.5 15.7 5.9 8 Greece Ireland Portugal Spain
  • 30. Counterparty Risk 3 Month Libor OIS Spreads 50 45 40 35 OIS GBP, 27.94Spreads in bps 30 OIS EUR, 26 25 20 15 OIS USD, 15.95 10 5 0
  • 31. AXE• Less accommodative monetary policy resulting in an increase in interbank rates.• Growing concerns about PIGS’ sustainability of public finances and fiscal outlook. Talks amongst EU leaders about debt restructuring for Greece.• Large exposure of banks to “PIGS”.• Flight to safety resulting in a decrease in AAA rated government bond yields. – > BUY TREASURY, SELL EURODOLLARS/EURO FUTURES
  • 32. TED Spread Speculative trades on TED are executed in anticipation of a change in the spread between Treasury and Eurodollar deposits based on the assumption that the correlation between returns of the two instruments will change overtime.• Long position in TSY and a short position in a strip of euro-dollar contracts with similar maturity.• Position is established when the spread is narrow. The spread between the two yields is constantly changing as it is affected by the turmoil or uncertainty in the international markets and banks’ overall liquidity position.• A manager takes a position on the on-the-run 2 year TSY when the spread is at 16 basis points.• The manager anticipates that the spread will widen to 26 basis points allowing him to exit the trade at a profit…(see next slide)
  • 33. Trade Example Bond position 5/12/2011 6/13/2011 Principal : 100,142,000.00 99,953,125 Position Established on 5/12/2011 (T+1) Accrued: 22,078.80 76,426.63 Bought 100mm of 0 5/8 13@100.142 (YTM 0.552%) Total: 100,164,078 100,029,551.63 Sold 2 year Eurodollar bundle i.e. first 8 quarterly CME Profit (Loss): ($134,526.37) Eurodollar contracts. Futures Strip Position: Last Price Rate # Contracts Profit: 803*20*25=$401,500 Front Stub 99.80097 0.19903125 36 EDM1 Comdty 99.735 0.265 101 EDU1 Comdty 99.69 0.31 101 Total Gain: $266,973.63 EDZ1 Comdty 99.595 0.405 101 EDH2 Comdty 99.455 0.545 101 Margin per Contract ($650) EDM2 Comdty 99.22 0.78 101 EDU2 Comdty 98.925 1.075 100 Capital employed (803 contracts x $650 – 0% haircut)=$521,950 EDZ2 Comdty 98.615 1.385 99 EDH3 Comdty 98.34 1.66 99 Total Return on Capital for 32 days: 51.15% Position reviewed on 6/13/2011 Sell 100mm of 0 5/8 13 @ 99.951 (YTM 0.651% up 10 bps) Buy 2 year Eurodollar bundle at following prices (implying a 20 basis point increase in rates): # Contracts: Price Rate P&L Front Stub 99.60097 0.399031 0 EDM1 Comdty 99.535 0.465 50500 Face value x (days in contract/360) x discount factor strip EDU1 Comdty 99.49 0.51 50500 -------------------------------------------------------------------------- EDZ1 Comdty 99.395 0.605 50500 Risk of ED Futures x 10,000 EDH2 Comdty 99.255 0.745 50500 EDM2 Comdty 99.02 0.98 50500 EDU2 Comdty 98.725 1.275 50000 The rate used in calculating the discount factor is the ED rate. EDZ2 Comdty 98.415 1.585 49500 (the TED spread can be subtracted from it). EDH3 Comdty 98.14 1.86 49500 $401,500
  • 34. Futures TableEurodollar Contract Table Period Ticker Last Price Rate Exp. Month Exp. Date Deposit Deposit No. days in Period Start Period ends period 1 Front Stub* 99.8024 0.1976 5/12/2011 5/16/2011 6/15/2011 30 2 EDM1 Comdty 99.7350 0.2650 May-11 6/15/2011 6/17/2011 9/16/2011 91 3 EDU1 Comdty 99.6900 0.3100 Jun-11 9/21/2011 9/23/2011 12/23/2011 91 4 EDZ1 Comdty 99.6000 0.4000 Jul-11 12/21/2011 12/23/2011 3/23/2012 91 5 EDH2 Comdty 99.4550 0.5450 Aug-11 3/21/2012 3/23/2012 6/22/2012 91 6 EDM2 Comdty 99.2200 0.7800 Sep-11 6/20/2012 6/22/2012 9/21/2012 91 7 EDU2 Comdty 98.9300 1.0700 Oct-11 9/19/2012 9/21/2012 12/21/2012 91 8 EDZ2 Comdty 98.6300 1.3700 Dec-11 12/19/2012 12/21/2012 3/21/2013 90 9 EDH3 Comdty 98.3600 1.6400 Mar-12 3/20/2013 3/22/2013 6/21/2013 91 10 EDM3 Comdty 98.0950 1.9050 Jun-12 6/19/2013 6/21/2013 9/20/2013 91 11 EDU3 Comdty 97.8450 2.1550 Sep-12 9/18/2013 9/20/2013 12/20/2013 91* Libor RatesLibor Tenor Periodicity Expiration RateUS0002W index 2 W 5/30/2011 0.17975US0001M index 1 M 6/16/2011 0.19875US0002M index 2 M 7/18/2011 0.232
  • 35. How is the TED spreadcalculated? 3 methods.1. Implied Yield:The stub Libor and ED rates are usedto find the par coupon of a swapwhose cash flows correspond to thatof the treasury note. The TSY yield issubtracted from this par coupon toproduced the spread.2. SpreadIt represents the bps that must besubtracted from the stub Libor and 1 – Implied Yield TED: Par Coupon on aEurodollar futures contract rates to set swap: Not tradablethe PV of the TSY notes cash flows to 2 – Spread: Subtracting basis pointsits full market price (dirty). Act/360money market basis points. from Futures 3 – Implied Price: PV of cash flows3. Implied Price. (best).Method used in the next slide. The TSYnotes cash flows are discounted at thestub Libor and Eurodollar futuresrates. The implied yield resulting fromthe PV is subtracted from the TSYnotes yield (S/A bond equivalent basispoints).
  • 36. Calculate the TED spread Step 1 – Match the cash flows of the treasury note with the Eurodollar deposit periods. Step 2 – Find the interpolated Eurodollar discount function. On-the-run Treasury 2 year note Df 9/16/2011 = [1+0.00197632*30/360]-1 Coupon 0.625 percent Maturity 4/30/2013 *[1+0.00265*91/360]-1 Settlement 5/16/2011 =0.99166031 Accrued Interest 0.0220788 percent Clean Price 100.1523438 Df 12/23/2011 = [1+0.00197632*30/360]-1 Full price 100.1744226 *[1+0.00265*91/360]-1 Yield 0.5465946 percent *[1+0.0031*91/360] -1 Face Amount $1,000,000.00 *[1+0.004*91/360]-1Cash Flows =0.998383686 Present We interpolate the discount factors for 10/31/2011,the payment date of the note. Date Interest Principal Df Value Rather than using the actual values, we use the natural log of these values (which10/31/2011 3125 0 0.9987791 3121.19 flattens or smoothen the curvature of the ED forward curve).4/30/2012 3125 0 0.9967968 3114.9910/31/2012 3125 0 0.9928456 3102.64 LN(Df9/16/2011)=LN(0.99166031)=-0.000834/30/2013 3125 1000000 0.9861227 989204.3 Total PV= 998,543.1 LN(Df12/23/2011)=LN(0.998383686)=-0.00162 Dirty Price 99.85431 As 10/31/2011 is 45 days into the Sep – Dec 11 period, the discount factor should reflect 45/91 day change for the period. Clean px 99.83223 Df 10/31/2011=-0.00083+(45/91)*(-0.00162-(-0.00083)=-0.00122 Yield 0.711475 Using e ln(x) =x, where e is the base of the natural logarithm, we have e- 0.00122=0.998779081 TED 16.48806 (0.711475-0.5465946) The discount factors for the 2nd, 3rd and 4th terms are solved similarly. All the values are show in the cash flow table.
  • 37. Appendix: How to create an ED strip• The first step is to construct a forward strip that begins with the soonest- to-expire, front futures.• It ends with the contract whose deposit contains the maturity of the contiguous swap.• A cash libor deposit that spans the period from settlement to the front contract’s expiration is added to the front of the strip: The ‘front stub’.• The resulting structure is a synthetic, long term, Libor quality deposit that begins at settlement and terminates at the end of the final contract’s deposit period.• The rates in the chain determine the future value to which a present value would grow if invested during the sequence of deposits that makes up the strip.• In other words, the chain also determines the PV of a future payment occurring at the final maturity of the strip.
  • 38. Appendix:Pricing a Eurodollar Strip PV FV * [ 1 r /( t / 360)] 1 A eurodollar strip is composedof n depositperiods - each with a uniqueinterest rate (ri ) and term (ni ). So, we can write : PVi FVi * [ 1 ri ( t i / 360)] 1 ; PVi present value at the start of the ith depositperiod. FVi future value at the end of the ith deposit;ri interest rate for the ith depositperiod i number of the depositperiod, i 1,2,3...,n Solving for the PV of a sequenceof investments starting from n to n-1 : The strip is a sequenceof investments : The proceeds at the terminatio n of one depositare fully and immediatel y reinvested in the next depositperiod as a sequence. So, the present value for a given period is the future value of the preceding period. FVi 1 PVi . Applying this equation to, say, the third depositperiod : 1 PV3 FV3 * [ 1 r3 * ( t 3 / 360)] to find the present value of this deposit,we must discountit over the secondperiod : 1 PV2 FV2 * [ 1 r2 * ( t 2 / 360)] 1 PV2 PV3 * [ 1 r2 * ( t 2 / 360)] or PV2 FV3 * [ 1 r3 * ( t 3 / 360)] 1 * [ 1 r2 * ( t 2 / 360)] 1
  • 39. Solving for the PV of a sequence of investments from n to today andDiscount FunctionWe arrive at the present value of the cash flow at the sart of thedepositperiod - that is, today - by discountin it over the firstperiod, g 1PV1 FV3 * [ 1 r3 * ( t 3 / 360)] 1 * [ 1 r2 * ( t 2 / 360)] 1 * [ 1 r1 * ( t 3 / 360)]The quantity [ 1 ri * ( t i / 360)] 1 is the discountfactor, dfi , for period iover any depositperiods n over which FVn is discounted. The discountfactordetermines , in present value - at the start of period, i of a sumpaid at the end of period i . 1di [ 1 ri * ( t i / 360)]We can then express the PV as :PV FVn * ( df1 * df2 * df3 ... * dfn )The right most term between the parentheses is the productof the n discountfactorsthat composethe strip.It is called the discountfunctionand is written as :DFn ( df1 * df2 * df3 ... * dfn )where dfi discountfactor for period iDFn discountfunctioncomposedof the productof the n - period discountfactors.It gives PV FV * DFn .