Evolution of Interest Rate Curves

Special CPT SeminarFrancois Choquet, Advanced Specialist

Bloomberg L.P.December 8, 2010

Amounts outstanding of over-the-counter (OTC) derivatives (in Billions of USD)

In-terest Rate; 478,0

92

For-eign Ex-

change;

62,933

Credit Default Swaps; 31,057

Equity Linked; 6,867

Commodity; 3,273 FRAs12%

Swaps77%

Total options11%

Breakdown by Interest Rate Instruments

Source: BIS June 2010 S/A Survey

Floating Rate Notes (Libor)

AUD EUR GBP JPY USD -

200,000

400,000

600,000

800,000

1,000,000

1,200,000

1,400,000

1,600,000

1,800,000

2,000,000

1 mo3 mo6 mo

Amount Outstanding in millions of US$

Source: Bloomberg

Fixed to Float Bonds (Libor)

AUD EUR GBP JPY USD -

50,000.00

100,000.00

150,000.00

200,000.00

250,000.00

300,000.00

1 mo3 mo6 mo

Amount Outstanding in millions of US$

Source: Bloomberg

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.

LIBOR – OIS

Under the normal circumstances prior to the financial turmoil that started in the summer of 2007, OIS rates tended to move just below the corresponding currency Libor in a very stable manner. After the onset of the financial turmoil, however, the Libor-OIS spreads widened substantially, particularly for the dollar LIBOR spread.

FX SWAP IMPLIED USD 3MO RATE vs. USD LIBOR

The EUR/USD FX swap market acts as a substitute for European banks to raise USD funding. The increased demand for dollar funding led to large shift in the FX forward prices with the implied dollar funding rate rising sharply above the 3 month libor.

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

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

Forward Curves (Cash + Swap rates)

0

1

2

3

4

5

6

7

18-Feb-0820-Jun-081-Sep-0815-Sep-0830-Sep-0820-Oct-085-Jan-09

3 6 9 12 15 18 2 3 4 5 6 10mo Yr

15x18 mo: 1.10%

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

Forward Curve (Cash, Futures, Swaps)

0

1

2

3

4

5

6

7

18-Feb-0820-Jun-081-Sep-0815-Sep-0830-Sep-0820-Oct-085-Jan-09

3 6 9 12 15 18 2 3 4 5 6 10mo Yr

Curve Comparison

0

1

2

3

4

5

6

30-Sep-08 30-Sep-08 with futures 5-Jan-093 6 9 12 15 18 2 3 4 5 6 10

mo Yr

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.

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.

Eurodollar ContractCME Eurodollar Futures (ED) : EDA <Cmdty> CT <go>

Trade Unit Eurodollar Time Deposit have a principal value of $1,000,000 with a three month maturity

Point Description 1 point=.005=$12.50

Contract Listing Mar (H), Jun (M), Sep (U), Dec (Z)

Deposit Rate 100-Quote

Bloomberg 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

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.

Bloomberg Curve Builder ICVS

ICVS allows you to fully customize a swap curve with your choice of instruments and use it to derive either the current value or the historical mark to market value of a swap on SWPM. It can also be used to determine the asset swap spread and z-spread on ASW, the price of floaters and structured notes on YASN. See IDOC 2054526 to set the custom curve.

Forward Curve

ICVS Curve on SWPM

Pricing a Callable Step Floater

Valuation on YASN

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.

Non Standard Curves on ICVS

ICVS allows you to generate forward curves adjusted to the basis i.e. 3 month vs. 6 month Libor. In turn, it can be used to calculate the market value of swaps referenced against the non standard benchmark e.g. 6 month Libor.

Pricing a Non Standard Swap

6 month Curve 3 month Curve (no basis)

Difference

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

$10MM 5 year pay swap @ 2.42% effective 1/5/2009 against 6 mo US Libor priced on December 6th 2010 (pays and resets semi-annually on both fixed and floating sides)

Non Standard Swap on SWPM

APPENDIXCurve Builder

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.

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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

Eurodollar vs. Forward Rates (FRAs)

Forward Contract

Futures Contract

OTC agreement between two counterparties

Exchange Traded Contract

Spot Price (S) of underlying+ρ(S,r)

Futures: Daily Settlement

Excess

Margin Margin

Call

Spot Price (S) of underlying +ρ(S,r)

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%

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.

Convexity Adjustment (Ho-Lee)

Eurodollar Future March 20102 (EDM2) as of 9/17/2009Quote 99.9901

Rate 0.99%

Continuously compounded rate 1.0025% (LN(1+0.99%/4)*365/90

Volatility of change in short rate 0.88%

Delivery 1.783 years

Delivery + 90 days 2.033 years

Forward rate (after convexity adjustment) 0.9866% (1.0025-0.5*0.88%^2*1.783*2.03)

Forward rate = Futures Rate – 0.5σ2T1T2

Convexity Adjustment (Hull White)

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Volatility of change in short rate 0.88%

Delivery 1.783 years

Delivery + 90 days 2.033 years

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USD FRASettle /

Term 9/21/2010 ASK BID Term Period expiry daysdiscount

factorspot rates

3 m LIBOR 0.29156 3 m 12/21/2010 91 0.999263544 0.292%6 m 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

Futures Discount Factors (no cnvx. adj.)

contract yield Start Date End Date Days in period

Day-count

Discount factors

Libor* 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)

contract Expiry Term Period RateBBA LIBOR USD Overnight 9/23/2010 1 D 0.22788USD DEPOSIT T/N 9/24/2010 2 D 0.25BBA LIBOR USD 1 Week 9/29/2010 1 W 0.2515BBA LIBOR USD 2 Week 10/6/2010 2 W 0.25181BBA LIBOR USD 1 Month 10/22/2010 1 M 0.2575BBA LIBOR USD 2 Month 11/22/2010 2 M 0.27438BBA LIBOR USD 3 Month 12/22/2010 3 M 0.29156

0.27438+23/30*(0.29156-0.27438)=0.28755

The front stub is the rate that spans the period from settlement (Sep 22) to the expiry of the front contract (12/15/10- ED Dec 10). Here, it is linearly interpolated between 2 and 3 mo Libor (23 days)

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Bootstrapping Discount Factors and Zero Rates from Swap Rates

A swap Rate is the coupon rate which the fixed side is going to pay for the par swap. The procedure to solve the discount factor from a quoted swap rate is called bootstrapping. As shown above, To solve the 2-year discount factor, we need 1 year discount factor. To solve 6-year discount factor, we need 1 year, 2 year, 3 year, 4 year, 5 year discount factors. Thus we have to go step by step to solve the discount factors.

Bootstrapped IRS Curve w/ Cash, Future Strip and Swap Ratessettle date 9/22/2010stub 84

contract term freq Start expiry ask ask (dec) days to expiry

Time between contract

expiry dates (years)

Discount Factor

Future Strip spot rates (S/A cmpd)

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%

Additional references

• DOC 2055462 : Complete curve builder methodology.