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- 1. Evolution of Interest Rate Curves Special CPT Seminar Francois Choquet, Advanced Specialist Bloomberg L.P. December 8, 2010
- 2. Amounts outstanding of over-the- counter (OTC) derivatives (in Billions of USD) Credit Equity Default Linked, 6, Swaps, 31 Commodity, 867 Breakdown by Interest Rate Instruments ,057 3,273 Foreign Total FRAsExchange, options 12% 62,933 11% Interest Rate, 478, 092 Swaps 77% Source: BIS June 2010 S/A Survey
- 3. Floating Rate Notes (Libor) Amount Outstanding in millions of US$2,000,0001,800,0001,600,0001,400,0001,200,000 1 mo1,000,000 3 mo 800,000 6 mo 600,000 400,000 200,000 - AUD EUR GBP JPY USD Source: Bloomberg
- 4. Fixed to Float Bonds (Libor) Amount Outstanding in millions of US$ 300,000.00 250,000.00 200,000.00 150,000.00 1 mo 3 mo 100,000.00 6 mo 50,000.00 - AUD EUR GBP JPY USDSource: Bloomberg
- 5. Liquidity “freeze”• Banks reluctant to lend long term in the inter-bank cash market (widening of basis spread)• Events: – Sept 7 – Fannie Mae and Freddie Mac are put into receivership – Sept 14 – Bankruptcy of Lehman; Merrill acquired buy BAC – Sept 16- AIG bailout from the treasury – GS and Morgan Stanley lose their status of broker dealer and converted into bank holding companies – Sept 19 – TARP announced by the US Treasury – Sept 28- Half of Fortis Bank capital is nationalized – Wachovia to be bought by Citi (later bought by wells Fargo) – Sept 30- Bailout money made available to Dexia Bank – Sept 30 – LIBOR rises from 4.7% to 6.88%.• These events forced participants to review the data used in building their interest curves.
- 6. LIBOR – OISUnder the normalcircumstances prior to thefinancial turmoil thatstarted in the summer of2007, OIS rates tended tomove just below thecorresponding currencyLibor in a very stablemanner. After the onsetof the financialturmoil, however, theLibor-OIS spreadswidenedsubstantially, particularlyfor the dollar LIBORspread.
- 7. FX SWAP IMPLIED USD 3MO RATE vs. USD LIBORThe EUR/USD FX swapmarket acts as asubstitute forEuropean banks toraise USD funding. Theincreased demand fordollar funding led tolarge shift in the FXforward prices withthe implied dollarfunding rate risingsharply above the 3month libor.
- 8. Curve Builder• Use most liquid benchmark instruments for different segments of the curve – Prevent abnormal spikes in the implied forward curve; – Best reflect the expected shape of the curve in the market.• Avoid overlapping between rates – Cash or deposit rates for the short end; – Futures or forwards (FRAs) for the intermediate portion; – Swaps for long end.• Data availability may vary by currency
- 9. Libor and swap rates to build curves• Data used on the next slide shows USD forward curves on 7 specific days and bootstrapped using cash and swap rates• Days used – Feb 18, June 20, Sep 1, Sep 15, Oct 20, 2008 – Jan 5, 2009• Data used – Cash rates from 1 week to 12 months – Swap rates from 2 to 30 years
- 10. Forward Curves (Cash + Swap rates)765 18-Feb-08 20-Jun-084 1-Sep-08 15-Sep-08 30-Sep-083 20-Oct-08 5-Jan-0921 15x18 mo: 1.10%0 3 6 9 12 15 18 2 3 4 5 6 10 mo Yr
- 11. Cash, IR Futures and Swap rates• The data used shows curves on 7 specific days where curves were bootstrapped using cash, IR Futures and swap rates.• The same days were used from the previous examples• Data: – Cash rates: overnight and 1 week – Futures going out to 2 years on cycle (March, June, Sept and Dec) – Swap rates used: 3 to 30 years
- 12. Forward Curve (Cash, Futures, Swaps)765 18-Feb-08 20-Jun-084 1-Sep-08 15-Sep-08 30-Sep-083 20-Oct-08 5-Jan-09210 3 6 9 12 15 18 2 3 4 5 6 10 mo Yr
- 13. Curve Comparison6543210 3 6 9 12 15 18 2 3 4 5 6 10 mo Yr 30-Sep-08 30-Sep-08 with futures 5-Jan-09 5-Jan-09 with futures
- 14. Key Facts• Use instruments that are liquid• Review the forward curves you create to ensure there are not strange “peaks and valleys”• Incorporate the use of futures or FRAs for the mid part of the curve.• Bloomberg Standard Curves use a combination of cash, FRAs or Futures and swap rates depending on the currency.
- 15. Eurodollar rates as forward rates• Eurodollar futures rates are considered forward three- month rates whose values reflect market expectations for future three-month Libor. – Each contract represents a deposit for a future, or forward, period, the contract rate is thought of as a forward rate.• You can think of buyers of a particular contract as agreeing to receive that forward rate—the rate at which they are willing to lend money in the future.• Conversely, contract sellers agree to pay the forward rate, meaning, to lock in now a finance rate for future borrowing.
- 16. Eurodollar Contract CME Eurodollar Futures (ED) : EDA <Cmdty> CT <go>Trade Unit Eurodollar Time Deposit have a principal value of $1,000,000 with a three month maturityPoint Description 1 point=.005=$12.50Contract Listing Mar (H), Jun (M), Sep (U), Dec (Z)Deposit Rate 100-QuoteBloomberg Ticker EDZ0, EDH1, EDM1, EDU1 Cmdty <Go>Contract Value 10,000*[100-.25*(100-Quote)] Libor (%) Quote Contract Price Sep 19, 2010 0.41 99.59 998,975 Dec 2010 0.405 99.595 998,987.5 Gain/Loss 0.005bps 12.5bps
- 17. Eurodollar Strip• Investors can create longer forward periods by trading a sequence of two or more contiguous contracts, effectively fusing adjacent deposit periods into an extended single period.• Such a sequence of contracts is called a Eurodollar strip.• The individual forward rate of each component contract in the strip is known, so, it is possible to compute an equivalent single rate—called a Eurodollar strip rate—for the strip as a whole. Then we can use the strip rates to present-value, or discount cash flows.
- 18. Bloomberg Curve Builder ICVSICVS allows you tofully customize a swapcurve with your choiceof instruments anduse it to derive eitherthe current value orthe historical mark tomarket value of aswap on SWPM. It canalso be used todetermine the assetswap spread and z-spread on ASW, theprice of floaters andstructured notes onYASN. See IDOC2054526 to set thecustom curve.
- 19. Forward Curve
- 20. ICVS Curve on SWPM
- 21. Pricing a Callable Step Floater
- 22. Valuation on YASN
- 23. Standard vs. Non-Standard Curves• Contracts that are used to build an interest rate curve refer to the same tenor of the underlying benchmark i.e. 3 month libor. – A curve can be used to price swaps that reference to the same tenor (standard). – Cannot be used to price instruments that reference to a different tenor (non-standard) – Spread adjustment required to get the correct curve for calculating implied forwards.• Basis swap: A tenor of the index that is swapped for a different tenor periodically.
- 24. Non Standard Curves on ICVSICVS allows you togenerate forwardcurves adjusted tothe basis i.e. 3month vs. 6 monthLibor. In turn, it canbe used to calculatethe market value ofswaps referencedagainst the nonstandard benchmarke.g. 6 month Libor.
- 25. Pricing a Non Standard Swap$10MM 5 year pay swap @ 2.42% effective 1/5/2009 against 6 mo US Liborpriced on December 6th 2010 (pays and resets semi-annually on both fixed andfloating sides) 6 month Curve 3 month Curve Difference (no basis)Principal $ -380,262.44 $ -414,247.25 $ 33,984.81Par Coupon 1.17% 1.06% 11 bpsDV01 $3,508.36 $3,071.18 $437.18
- 26. Non Standard Swap on SWPM
- 27. Curve BuilderAPPENDIX
- 28. 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 occurirng at the final maturity of the strip.
- 29. Pricing a Eurodollar StripPV FV * [1 r /(t / 360)] 1A eurodollarstrip is composedof n deposit periods- each witha uniqueinterestrate(ri ) and term(ni ). So, we can write:PVi FVi * [1 ri (ti / 360)] 1PVi present va at thestart of theith deposit period lueFVi future value at theend of theith depositri interestratefor theith deposit periodi number of thedeposit period,i 1,2,3..., n
- 30. Solving for the PV of a sequence of investments starting from n to n-1T hestrip is a sequence of investment : T heproceedsat theterminati of one deposit are s onfully and immediatel reinvestedin thenext deposit periodas a sequence.So, thepresent yvalue for a given periodis thefuture value of theprecedingperiod.FVi 1 PVi . Applyingthisequation t say, the thirddeposit period: o,PV3 FV3 *[1 r3 * (t3 / 360)] 1to find thepresent va of thisdeposit,we must discount it over the lue secondperiod:PV2 FV2 * [1 r2 * (t 2 / 360)] 1PV2 PV3 *[1 r2 * (t 2 / 360)] 1orPV2 FV3 *[1 r3 * (t3 / 360)] 1 *[1 r2 * (t 2 / 360)] 1
- 31. Solving for the PV of a sequence of investments from n to todayWe arriveat thepresent va of thecash flow at thesart of the luedeposit period- thatis, today- by discountin it over the g first period,PV1 FV3 *[1 r3 * (t3 / 360)] 1 *[1 r2 * (t 2 / 360)] 1 *[1 r1 * (t3 / 360)] 1T hequantity[1 ri * (ti / 360)] 1 is thediscount factor,dfi , for periodiover any deposit periodsn over whichFVn is discounted T hediscount factor .determines in present va - at thestart of period,i of a sum paid at theend of periodi. , luedi [1 ri * (ti / 360)] 1
- 32. Discount FactorsWe can thenexpressthePV as :PV FVn * (df1 * df2 * df3 ...* dfn )T heright most termbetween th parenthese is theproduct of then discount fact ors e sthatcomposethest rip.It is called thediscount funct ionand is writ tenas :DFn (df1 * df2 * df3 ...* dfn )where dfi discount fact orfor periodiDFn discount funct ioncomposedof theproduct of then - perioddiscount fact ors.It gives PV FV * DFn .
- 33. Futures Vs. Forwards• Assumption is often that 100-F = forward rate• Not exact for several reasons: – Interest differentials on margin surplus & funding. – Futures are marked to market(p&l settled daily =PV gain/loss). – “Convexity” - stochastic interest rates give rise to differences
- 34. Eurodollar vs. Forward Rates (FRAs) +ρ(S,r)Futures: Daily Settlement +ρ(S,r) Futures Contract Exchange Traded Contract OTC agreement between two Forward Contract counterparties
- 35. Exercise (Libor FRA convexity)• Sell $100mm 3x9 IMM dated FRA today• Hedge by selling futures• Assume that the yield curve is flat• Work out:• Equivalent futures position• Gain or loss on FRA and equivalent Futures position for parallel shifts +/- 2%
- 36. Pricing convexity• If not priced – Short futures buys convexity for free• If priced – Forward rates implied by FRA’s differ from forward rates implied by futures.
- 37. Convexity Adjustment (Ho-Lee)Eurodollar Future March 20102 (EDM2) asof 9/17/2009Quote 99.9901Rate 0.99%Continuously compounded rate 1.0025% (LN(1+0.99%/4)*365/90Volatility of change in short rate 0.88%Delivery 1.783 yearsDelivery + 90 days 2.033 yearsForward rate (after convexity adjustment) 0.9866% (1.0025-0.5*0.88%^2*1.783*2.03) Forward rate = Futures Rate – 0.5σ2T1T2
- 38. Convexity Adjustment (Hull White) B (t1,t 2 ) 2 at1 B (t1 , t 2 )(1 e ) 2aB(0, t1 ) 2 t 2 t1 a a (T t ) 1 e B (t , T ) a a mean reversionspeed volat ilit ycaplet vol forward rat e(t1 , t 2 ) t hatexpriesat t1 , onEurodollar Future March 20102 (EDM2) as of Sep 17, 2010Last trade 99.9901Rate 0.99%Continuously compounded rate 1.0025% (LN(1+0.99%/4)*365/90Volatility of change in short rate 0.88%Delivery 1.783 yearsDelivery + 90 days 2.033 yearsForward rate (after convexity adjustment) 0.9892% (0.010025-0.000132381) see next slide for calc prove out
- 39. Convexity Adjustment (Hull White)B (t1,t 2 ) 2 at1 B (t1 , t 2 )(1 e ) 2aB(0, t1 ) 2 t 2 t1 a 0.248767 2*0.03*1.7833 2 0.88% 0.2487671 e ( ) 2 * 0.03*1.736437 0.0001323812.0333 1.7833 * 0.03 0.03 ( 2.0333 1.7833 ) 1 eB (t1 , t 2 ) 0.248767 0.03 0.03*1.78333 1 eB (0, t1 ) 1.736437 0.03a 0.03 0.88%
- 40. USD FRA Settle discount spot /Term 9/21/2010 ASK BID Term Period expiry days factor rates3m LIBOR 0.29156 3 m 12/21/2010 91 0.999263544 0.292%6m 3X6 0.422 0.402 6 m 3/21/2011 91 0.998198743 0.357%12m 6X9 0.4837 0.4637 9 m 6/21/2011 92 0.996966371 0.400%18m 9X12 0.57 0.555 12 m 9/21/2011 92 0.995516235 0.443% D3m=1/(1+0.29156*91/36000)=0.99263544 D3-6=1/(1+0.422*91/360000)=0.999834414 D6m=D3m*D3-6=0.99263544*0.999834414=0.998198743
- 41. Futures Discount Factors (no cnvx. adj.)contract Expiry Term Period Rate The front stub is theBBA LIBOR USD Overnight 9/23/2010 1 D 0.22788 rate that spans theUSD DEPOSIT T/N 9/24/2010 2 D 0.25 period from settlementBBA LIBOR USD 1 Week 9/29/2010 1 W 0.2515 (Sep 22) to the expiryBBA LIBOR USD 2 Week 10/6/2010 2 W 0.25181 of the front contractBBA LIBOR USD 1 Month 10/22/2010 1 M 0.2575 (12/15/10- ED Dec 10).BBA LIBOR USD 2 Month 11/22/2010 2 M 0.27438 Here, it is linearlyBBA LIBOR USD 3 Month 12/22/2010 3 M 0.29156 interpolated between 2 and 3 mo Libor (23 0.27438+23/30*(0.29156-0.27438)=0.28755 days) Days in Day- Discount contract yield Start Date End Date period count factorsLibor* 0.28755 9/22/2010 12/15/2010 84 a360 0.999329 =1/(1+.28755*84/36000)EDZ0 0.405 12/15/2010 3/16/2011 91 a360 0.998307 =1/(1+0.405*91/36000)*0.999329EDH1 0.470 3/16/2011 6/15/2011 91 a360 0.997123 =1/(1+0.470*91/36000)*0.998307EDM1 0.555 6/15/2011 9/21/2011 98 a360 0.995619 =1/(1+0.555*98/36000)*0.997123 9/22/2010 9/22/2011 365 a360 0.995600 =0.995619+1/90*(0.99396-0.995619)EDU1 0.660 9/21/2011 12/21/2011 91 a360 0.993960 Future strip=0.995600*365/360=1.009428192 year swap 0.682 9/22/2010 9/24/2012 722 30360 0.986389 =(1-0.682/100*0.995600*365/360)/(1+0.682/100)
- 42. Bootstrapping Discount Factors and Zero Rates from Swap RatesA swap Rate is the coupon rate which the fixed side is going to pay for the par swap. The procedure to solvethe discount factor from a quoted swap rate is called bootstrapping. As shown above, To solve the 2-yeardiscount factor, we need 1 year discount factor. To solve 6-year discount factor, we need 1 year, 2 year, 3year, 4 year, 5 year discount factors. Thus we have to go step by step to solve the discount factors. N100 C N dfn 100 df N n 1100 C N AN 100 df N NAN dfn AN 1 df N n 1 1 C N AN 1df N 1 CNFor example, we solvethe two year discountfactor from the 2 year swap rate :df2 * 100 coupon df1 * coupon 100 df2 1 coupon df1 /( 1 coupon) *Similarly,we solvefor the three year discountfactor from the 3 year swap rate :df3 * ( 100 coupon) df2 * coupon df1 * coupon df * ( 100 coupon) coupon df2 * df1 * 100 df3 1 coupon ( df2 * df1 ) /( 1 coupon)So, we can solvefor any discountrate using:dfn ( 1 coupon previousannuity) /( 1 coupon) *
- 43. Bootstrapped IRS Curve w/ Cash, Future Strip and Swap Rates settle date 9/22/2010 stub 84contract term freq Start expiry ask ask (dec) days to Time between Discount Future Strip spot rates expiry contract Factor (S/A cmpd) expiry dates (years)LIBOR USD O/N 1 D 9/22/2010 9/23/2010 0.22788 0.002279 0.002778 0.0027 0.999994 0.2279%LIBOR USD 1W 1 W 9/22/2010 9/29/2010 0.2515 0.002515 0.019444 0.0167 0.999951 0.2515%LIBOR USD 2W 2 W 9/22/2010 10/6/2010 0.25181 0.002518 0.038889 0.0194 0.999902 0.2518%LIBOR USD 1M 1 M 9/22/2010 10/22/2010 0.2575 0.002575 0.083333 0.0444 0.999785 0.2575%LIBOR USD 2M 2 M 9/22/2010 11/22/2010 0.27438 0.002744 0.169444 0.0861 0.999535 0.2744%LIBOR USD 3M 3 M 9/22/2010 12/22/2010 0.29156 0.002916 0.252778 0.0833 0.999264 0.2916%90DAY EURO$ FUTR Dec10 3 M 12/15/2010 3/16/2011 0.405 0.00405 0.479452 0.2528 0.998307 0.3527%90DAY EURO$ FUTR Mar11 3 M 3/16/2011 6/15/2011 0.47 0.0047 0.728767 0.2528 0.997123 0.3946%90DAY EURO$ FUTR Jun11 3 M 6/15/2011 9/21/2011 0.555 0.00555 0.99726 0.2722 0.995619 0.4393%USD SWAP SEMI 30/360 2YR 2 Y 9/22/2010 9/24/2012 0.682 0.00682 2.008219 1.0139 0.986389 1.00942819 0.6813%USD SWAP SEMI 30/360 3YR 3 Y 9/22/2010 9/23/2013 1.015 0.01015 3.005479 0.9972 0.969925 1.0134%USD SWAP SEMI 30/360 4YR 4 Y 9/22/2010 9/22/2014 1.361 0.01361 4.00274 0.9972 0.946639 1.3653%USD SWAP SEMI 30/360 5YR 5 Y 9/22/2010 9/22/2015 1.703 0.01703 5.00274 1.0000 0.917603 1.7115%USD SWAP SEMI 30/360 6YR 6 Y 9/22/2010 9/22/2016 1.992 0.01992 6.005479 1.0000 0.885971 2.0059%USD SWAP SEMI 30/360 7YR 7 Y 9/22/2010 9/22/2017 2.262 0.02262 7.005479 1.0000 0.85126 2.2856%USD SWAP SEMI 30/360 8YR 8 Y 9/22/2010 9/24/2018 2.458 0.02458 8.010959 1.0056 0.81815 2.4898%USD SWAP SEMI 30/360 9YR 9 Y 9/22/2010 9/23/2019 2.633 0.02633 9.008219 0.9972 0.784602 2.6748%USD SWAP SEMI 30/360 10Y 10 Y 9/22/2010 9/22/2020 2.777 0.02777 10.00822 0.9972 0.751997 2.8277%USD SWAP SEMI 30/360 11Y 11 Y 9/22/2010 9/22/2021 2.872 0.02872 11.00822 1.0000 0.722755 2.9278%USD SWAP SEMI 30/360 12Y 12 Y 9/22/2010 9/22/2022 3.003 0.03003 12.00822 1.0000 0.689406 3.0734%
- 44. Additional references• DOC 2055462 : Complete curve builder methodology.

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