Evolution of Interest Rate Curves since the Financial Crisis

Evolution of Interest Rate Curves

Special CPT Seminar
Francois Choquet, Advanced Specialist
Bloomberg L.P.
December 8, 2010

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 FRAs
Exchange,
options 12%
62,933
11%
Interest
Rate, 478,
092 Swaps
77%

Source: BIS June 2010 S/A Survey

Floating Rate Notes (Libor)
Amount Outstanding in millions of US$
2,000,000

1,800,000

1,600,000

1,400,000

1,200,000
1 mo
1,000,000
3 mo
800,000
6 mo
600,000

400,000

200,000

-
AUD EUR GBP JPY USD

Source: Bloomberg

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 USD

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)
7

6

5
18-Feb-08
20-Jun-08
4
1-Sep-08
15-Sep-08
30-Sep-08
3
20-Oct-08
5-Jan-09
2

1 15x18 mo: 1.10%

0
3 6 9 12 15 18 2 3 4 5 6 10
mo Yr

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)
7

6

5
18-Feb-08
20-Jun-08
4
1-Sep-08
15-Sep-08
30-Sep-08
3
20-Oct-08
5-Jan-09
2

1

0

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

Curve Comparison
6

5

4

3

2

1

0
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

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 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
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.

Pricing a Callable Step Floater

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
$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)

6 month Curve 3 month Curve Difference
(no basis)
Principal $ -380,262.44 $ -414,247.25 $ 33,984.81
Par Coupon 1.17% 1.06% 11 bps
DV01 $3,508.36 $3,071.18 $437.18

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.

Pricing a Eurodollar Strip

PV FV * [1 r /(t / 360)] 1
A eurodollarstrip is composedof n deposit periods- each witha unique
interestrate(ri ) and term(ni ). So, we can write:
PVi FVi * [1 ri (ti / 360)] 1
PVi present va at thestart of theith deposit period
lue
FVi future value at theend of theith deposit
ri interestratefor theith deposit period
i number of thedeposit period,i 1,2,3...,
n

Solving for the PV of a sequence of
investments starting from n to n-1
T hestrip is a sequence of investment : T heproceedsat theterminati of one deposit are
s on
fully and immediatel reinvestedin thenext deposit periodas a sequence.So, thepresent
y
value for a given periodis thefuture value of theprecedingperiod.FVi 1 PVi . Applying
thisequation t say, the thirddeposit period:
o,
PV3 FV3 *[1 r3 * (t3 / 360)] 1
to find thepresent va of thisdeposit,we must discount it over the
lue secondperiod:
PV2 FV2 * [1 r2 * (t 2 / 360)] 1
PV2 PV3 *[1 r2 * (t 2 / 360)] 1
or
PV2 FV3 *[1 r3 * (t3 / 360)] 1
*[1 r2 * (t 2 / 360)] 1

Solving for the PV of a sequence of
investments from n to today
We arriveat thepresent va of thecash flow at thesart of the
lue
deposit 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)] 1
T hequantity[1 ri * (ti / 360)] 1 is thediscount factor,dfi , for periodi
over 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.
, lue
di [1 ri * (ti / 360)] 1

Discount Factors

We can thenexpressthePV as :
PV FVn * (df1 * df2 * df3 ...* dfn )
T heright most termbetween th parenthese is theproduct of then discount fact ors
e s
thatcomposethest rip.It is called thediscount funct ionand is writ tenas :
DFn (df1 * df2 * df3 ...* dfn )
where dfi discount fact orfor periodi
DFn discount funct ioncomposedof theproduct of then - perioddiscount fact ors.
It gives PV FV * DFn .

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)
+ρ(S,r)
Futures: Daily Settlement

+ρ(S,r)
Futures Contract Exchange Traded Contract

OTC agreement between two
Forward Contract counterparties

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 9/17/2009
Quote 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)
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
, on
Eurodollar Future March 20102 (EDM2) as of Sep 17, 2010
Last trade 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) 0.9892% (0.010025-0.000132381)
see next slide for calc prove out

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 0.000132381
2.0333 1.7833 * 0.03
0.03 ( 2.0333 1.7833 )
1 e
B (t1 , t 2 ) 0.248767
0.03
0.03*1.78333
1 e
B (0, t1 ) 1.736437
0.03
a 0.03
0.88%

USD FRA
Settle discount spot
/Term 9/21/2010 ASK BID Term Period expiry days factor rates
3m 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

Futures Discount Factors (no cnvx. adj.)
contract Expiry Term Period Rate The front stub is the
BBA LIBOR USD Overnight 9/23/2010 1 D 0.22788 rate that spans the
USD DEPOSIT T/N 9/24/2010 2 D 0.25 period from settlement
BBA LIBOR USD 1 Week 9/29/2010 1 W 0.2515 (Sep 22) to the expiry
BBA LIBOR USD 2 Week 10/6/2010 2 W 0.25181 of the front contract
BBA 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 linearly
BBA 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 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)*0.999329
EDH1 0.470 3/16/2011 6/15/2011 91 a360 0.997123 =1/(1+0.470*91/36000)*0.998307
EDM1 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=1.00942819
2 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)

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.
N

100 C N dfn 100 df N
n 1
100 C N AN 100 df N
N

AN dfn AN 1 df N
n 1
1 C N AN 1
df N
1 CN
For 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)
*

Bootstrapped IRS Curve w/
Cash, Future Strip and Swap Rates
settle date 9/22/2010
stub 84
contract 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%

Additional references

• DOC 2055462 : Complete curve builder methodology.

Evolution of Interest Rate Curves since the Financial Crisis

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