Constructing LIBOR zero and forward curves from spot market LIBOR rates and CME
eurodollar futures.
Presented below is an alternative method to bootstrapping from Treasury securities to
construct the term structure of interest rates. The term structure constructed from LIBOR,
CME futures and swap rates is of lower credit quality than the term structure constructed
from Treasury securities.
A eurodollar is a dollar deposited in a U.S. bank outside the US. The rate earned on these
deposits are benchmarked to the London Interbank Offer Rate (LIBOR). LIBOR refers to
the rate at which banks are willing to lend (LIBOR) or borrow (LIBID) these funds. The
British Banker’s Association posts a daily LIBOR value. The British Banker’s Association
daily posting is usually the rate referred to when considering LIBOR. Similar rates are
available for all major currencies. LIBOR rates from this market are referred to as spot
market rates and quoted assuming add-on interest based on a 360-day year.
Bridge-Telerate:
US@?GLUS3M, US@?GLUS6M, US@?GLUS1Y LIBOR
US@?GE3M, US@?GE6M, US@?GE1Y
EURIBOR
Futures contracts offered at the Chicago Mercantile Exchange (CME) on the 3-MO
eurodollar rate have delivery dates of near months plus March, June, September, December
cycle extending up to 10-years in the future. Settlement prices for the CME eurodollar
contract can be obtained on a daily basis from the exchange, http://www.cme.com/ . The
delivery date for these contracts is the third Wednesday of the delivery month.
A futures contract price of 96 implies a futures rate of 4%. This futures rate is expressed
with quarterly compounding and an actual/360 day count convention. Conversion of the
futures rate to a forward rate requires three sequential adjustments 360365,
quarterlycontinuous, a convexity adjustment.
Adjustment to futures contract price necessary to account for first two conversions;
Adjusted futures =(365/91)*log(1+(91/360)*0.01*(100-Settlement Price))
The futures rate, f, is biased upwards relative to the forward rate due to non-linearity of the
bond pricing relationship. CME Eurodollar futures prices (100 – f) reflect a present value
(bond price).
Convexity adjustment (Ho-Lee model) : T.S. Ho S. Lee (1986) “Term Structure
Movements and Pricing Interest Rate Contingent Claims,” Journal of Finance, vol. 41.
F  f  0.5 *  2 * t1 * t 2 where
F = forward rate
f = adjusted futures rate
1
 = volatility of annual changes in L
t1 = time in years till futures contract delivery date
t2 = time in years till maturity of the rate underlying the futures contract.
F = Adjusted futures – Convexity adjustment
Estimating volatility of annual changes in 3-MO LIBOR
http://www.federalreserve.gov/releases/
euro dollar 3m year end
1971
6.58
1987
7.07
0.0037
1972
5.41
1973
9.26
-0.0117
1988
7.85
0.0078
0.0385
1989
9.16
0.0131
1974
1975
11.04
0.0178
1990
8.16
-0.01
7.03
-0.0401
1991
5.86
-0.023
1976
5.58
-0.0145
1992
3.7
-0.0216
1977
6.03
0.0045
1993
3.18
-0.0052
1978
8.73
0.027
1994
4.63
0.0145
1979
11.96
0.0323
1995
5.93
0.013
1980
14
0.0204
1996
5.38
-0.0055
1981
16.79
0.0279
1997
5.61
0.0023
1982
13.08
-0.0371
1998
5.45
-0.0016
1983
9.57
-0.0351
1999
5.31
-0.0014
1984
10.75
0.0118
2000
6.45
0.0114
1985
8.27
-0.0248
2001
3.7
-0.0275
1986
6.7
-0.0157
2002
1.73
-0.0197
2001
3.7
0.0197
2002
1.73
-0.0197
2003
1.07
-0.0066
stdev
0.021004
The LIBOR zero curve (term structure) is constructed from observations of spot LIBOR
and forward rates calculated from CME eurodollar futures contract prices. Conversion to
continuous compounding simplifies this process (however may not be in accordance with
day count conventions).
t0---------- z1 -------------------t1
t0---------------------------------t1-------- F1,2---------t2
t0-------------------------- z2 ----------------------------t2
Definition of forward rate: The forward rate for a specified forward period is the rate
earned during the forward period making total return of a sequential investment equivalent
to the total return of a long term investment.
2






 z1 * t1t0 365   F12 * t2 t1 365   z 2 * t2 t0 365 
e
 * e

  e
 

 

Given z1 (spot market rate) and F1,2 (forward rate) derived from futures contract;
z2 
F1,2 * (t 2  t1 )  z1 (t1  t 0 )
.
(t 2  t 0 )
The set of LIBOR zero rates is calculated by extending this computation to successive
periods corresponding to the March, June, September, December eurodollar futures cycle.
After constructing the LIBOR zero curve from spot rates and forward rates, linear
interpolation between zero rates can be used to find zero rates corresponding to the
payment dates of financial contracts (interest rate swap).
Utilizing zero rates for a swap contract’s payment dates the relationship above can be
inverted to determine the relevant forward rates for determining the equilibrium swap rate.
For example if t1 is a reset date and t2 the corresponding payment date, the forward rate F1,2
, determines the variable rate payment for purposes of calculating the equilibrium swap rate
is
z * (t 2  t 0 )  z1 * (t1  t 0 )
F1,2  2
(t 2  t1 )
.
3
3/08/04
91
182
3MO-Spot 1.04% 6MO-Spot 1.08%
eurodollar futures settlement prices as of 03/08/04 07:00 pm (cst)
Term Structure
Delivery
DDLV
Settle
Adj. Futures Convexity Forward Maturity DTM Zero Forward
4-Sep
9/15/2004 191
282
98.7
1.3159%
0.0089% 1.3070% 6/7/2004
91
1.04%
4-Dec
12/15/2004 282
373
98.465
1.5533%
0.0174% 1.5359% 9/6/2004
182
1.08% 1.12%
5-Mar
3/16/2005 373
464
98.175
1.8461%
0.0286% 1.8174% 12/15/2004 282
1.16% 1.31%
5-Jun
6/15/2005 464
555
97.825
2.1992%
0.0426% 2.1565% 3/16/2005 373
1.25% 1.54%
5-Sep
9/21/2005 562
653
97.47
2.5570%
0.0607% 2.4962% 6/15/2005 464
1.36% 1.82%
5-Dec
12/21/2005 653
744
97.15
2.8792%
0.0804% 2.7988% 9/14/2005 555
1.49% 2.16%
6-Mar
3/22/2006 744
835
96.9
3.1308%
0.1028% 3.0280% 12/21/2005 653
1.62% 2.32%
6-Jun
6/21/2006 835
926
96.665
3.3671%
0.1280% 3.2392% 3/22/2006 744
1.76% 2.80%
6-Sep
9/20/2006 926
1017 96.445
3.5883%
0.1559% 3.4324% 6/21/2006 835
1.90% 3.03%
6-Dec
12/20/2006 1017
1108 96.235
3.7992%
0.1865% 3.6127% 9/20/2006 926
2.03% 3.24%
7-Mar
3/21/2007 1108
1199 96.055
3.9800%
0.2199% 3.7601% 12/20/2006 1017 2.16% 3.43%
7-Jun
6/20/2007 1199
1290
95.88
4.1556%
0.2560% 3.8996% 3/21/2007 1108 2.28% 3.61%
7-Sep
9/19/2007 1290
1381 95.715
4.3212%
0.2949% 4.0263% 6/20/2007 1199 2.39% 3.76%
7-Dec
12/19/2007 1381
1472 95.555
4.4816%
0.3365% 4.1452% 9/19/2007 1290 2.50% 3.90%
8-Mar
3/19/2008 1472
1563
95.42
4.6169%
0.3808% 4.2361% 12/19/2007 1381 2.60% 4.03%
8-Jun
6/18/2008 1563
1654
95.29
4.7472%
0.4279% 4.3193% 3/19/2008 1472 2.69% 4.15%
8-Sep
9/17/2008 1654
1745 95.165
4.8724%
0.4777% 4.3947% 6/18/2008 1563 2.78% 4.24%
8-Dec
12/17/2008 1745
1836
95.04
4.9976%
0.5303% 4.4674% 9/17/2008 1654 2.87% 4.32%
9-Mar
3/18/2009 1836
1927
94.94
5.0977%
0.5856% 4.5122% 12/17/2008 1745 2.95% 4.39%
9-Jun
6/17/2009 1927
2018 94.845
5.1928%
0.6436% 4.5492% 3/18/2009 1836 3.02% 4.47%
9-Sep
9/16/2009 2018
2109 94.755
5.2829%
0.7044% 4.5785% 6/17/2009 1927 3.09% 4.51%
9-Dec
12/16/2009 2109
2200 94.665
5.3729%
0.7679% 4.6050% 9/16/2009 2018 3.16% 4.55%
10-Mar
3/17/2010 2200
2291
94.58
5.4580%
0.8342% 4.6238% 12/16/2009 2109 3.22% 4.58%
10-Jun
6/16/2010 2291
2382
94.5
5.5380%
0.9032% 4.6348% 3/17/2010 2200 3.28% 4.61%
10-Sep
9/15/2010 2382
2473
94.42
5.6180%
0.9750% 4.6430% 6/16/2010 2291 3.33% 4.62%
10-Dec
12/15/2010 2473
2564
94.35
5.6880%
1.0495% 4.6385% 9/15/2010 2382 3.38% 4.63%
11-Mar
3/16/2011 2564
2655 94.285
5.7529%
1.1267% 4.6262% 12/15/2010 2473 3.43% 4.64%
11-Jun
6/15/2011 2655
2746 94.215
5.8229%
1.2067% 4.6162% 3/16/2011 2564 3.47% 4.64%
11-Sep
9/21/2011 2753
2844 94.155
5.8828%
1.2959% 4.5870% 6/15/2011 2655 3.51% 4.63%
11-Dec
12/21/2011 2844
2935
94.09
5.9478%
1.3815% 4.5662% 9/14/2011 2746 3.55% 4.62%
12-Mar
3/21/2012 2935
3026 94.035
6.0027%
1.4699% 4.5328% 12/21/2011 2844 3.57% 4.26%
12-Jun
6/20/2012 3026
3117
93.97
6.0676%
1.5611% 4.5065% 3/21/2012 2935 3.60% 4.57%
12-Sep
9/19/2012 3117
3208
93.91
6.1275%
1.6550% 4.4726% 6/20/2012 3026 3.63% 4.53%
12-Dec
12/19/2012 3208
3299
93.85
6.1874%
1.7516% 4.4358% 9/19/2012 3117 3.65% 4.51%
13-Mar
3/20/2013 3299
3390 93.805
6.2324%
1.8510% 4.3814% 12/19/2012 3208 3.68% 4.47%
4
13-Jun
6/19/2013 3390
3481 93.745
6.2923%
1.9531% 4.3392% 3/20/2013 3299 3.70% 4.44%
13-Sep
9/18/2013 3481
3572
6.3372%
2.0580% 4.2792% 6/19/2013 3390 3.72% 4.38%
13-Dec
12/18/2013 3572
6.4020%
2.1656% 4.2365% 9/18/2013 3481 3.73% 4.34%
999
999
93.7
3663 93.635
999
999
999
999
12/18/2013 3572 3.75% 4.28%
3/19/2014 3663 3.76% 4.24%
Maturity DTM
Zero Forward
Fixed for floating interest rate swap valuation
The timing of cash flows for a plain-vanilla swap: the realization at time ti (reset time) of
the spot rate Li spanning the period [ti ti+1] determines a floating payment per unit of
notional principal at time ti+1 (payment date) of magnitude Li*i per dollar of notional
principal. The distance i is given by the number of days in the period [ti ti+1] divided by
360 or 365, as dictated by the appropriate conventions. For a plain-vanilla swap, the fixed
payment per unit of notional principal, X*i ,also occurs at time ti+1.
Li
i
x -----------------------------------|---------------|
t0
ti
ti+1
Li is the spot 3-MO or 6-MO LIBOR (reference rate for swap contract) rate prevailing at
time ti .
Absence of arbitrage swap pricing:
Given both LIBOR forward and spot rate curves, the unknown cash flows in the floating
leg of the swap must be set equal to the forward rates to prevent arbitrage opportunities.
Define P(0,t) = the price of a discount bond maturing at time t. Using continuous
compounding the P(0,t) are functions of the LIBOR zero rates, zt .
P(0, t )  exp(  zt * (t / 365))
Define Ft = the LIBOR forward rate for the period [t t +1]
The equilibrium swap rate (coupon rate) is defined as the fixed rate X such that today’s
present value of the fixed and floating rate payments over the swap’s tenor (legs) are equal.
5
n
n
 NP * X *  i * P(0, t i 1 )   NP * Fi *  i * P(0, t i 1 )
i 1
i 1
n
 NP * Fi *  i * P(0, t i 1 )
X  i 1
n
 NP *  i * P(0, t i 1 )
i 1
n
 Fi * P(0, t i 1 )
 i 1
n
 P (0, t i 1 )
i 1
X is the equilibrium swap rate that makes today’s value of the swap equal for both
counterparties. X is a weighted average of the projected forward rates analogous to the
calculation of Macaulay’s duration.
After an equilibrium swap has been entered, the swap will in general no longer have zero
value, since interest rates will in general not have followed the implied forward curve. The
swap will maintain zero value only if the realized values Li are the projected forward rates
on the date the equilibrium (zero value) swap was struck.
Deviations of realized Li from time 0 projected forward rates will cause the swap value to
deviate from zero.
Replacement value of a swap after the initial date can be found from the initial equilibrium
swap rate and the forward rate curve on the valuation date.
The fixed rate payor’s replacement value at date t, where n- is the number of remaining
cash flows at (t);
n
Vt   Fi * P(t , t i 1 )  X 0 * P(t , t i 1 )* NP * i
i 1
n
Vt  ( X t  X 0 ) *  P(t , t i 1 ) * NP * i
i 1
6
Swap pricing date 3/08/04
Tenor
10-year, 39 reset 40 cash flow
NP
1000000
Swap Pricing fixed for 3MO LIBOR
Reset
DTRorDTC
Zero
6/7/2004
91
1.04%
9/6/2004
182
1.08%
12/6/2004
273
3/7/2005
Expected Cash Flow
Forward
PV
PV*F
FR-payor
PV*CF VR-payor PV*CF
0.9974
1.0373%
-6518
-6501
6518
6501
1.12%
0.9946
1.1140%
-6319
-6285
6319
6285
1.15%
1.30%
0.9914
1.2886%
-5870
-5820
5870
5820
364
1.24%
1.51%
0.9877
1.4937%
-5340
-5275
5340
5275
6/6/2005
455
1.35%
1.79%
0.9833
1.7580%
-4653
-4576
4653
4576
9/5/2005
546
1.48%
2.12%
0.9781
2.0748%
-3822
-3738
3822
3738
12/5/2005
637
1.60%
2.30%
0.9725
2.2323%
-3388
-3295
3388
3295
3/6/2006
728
1.74%
2.71%
0.9660
2.6192%
-2351
-2271
2351
2271
6/5/2006
819
1.88%
2.99%
0.9588
2.8655%
-1660
-1591
1660
1591
9/4/2006
910
2.01%
3.20%
0.9512
3.0466%
-1125
-1070
1125
1070
12/4/2006
1001
2.13%
3.40%
0.9431
3.2061%
-636
-600
636
600
3/5/2007
1092
2.26%
3.58%
0.9348
3.3482%
-181
-169
181
169
6/4/2007
1183
2.37%
3.74%
0.9261
3.4591%
202
187
-202
-187
9/3/2007
1274
2.48%
3.88%
0.9172
3.5550%
553
507
-553
-507
12/3/2007
1365
2.58%
4.00%
0.9081
3.6367%
874
794
-874
-794
3/3/2008
1456
2.68%
4.12%
0.8988
3.7075%
1173
1055
-1173
-1055
6/2/2008
1547
2.77%
4.22%
0.8894
3.7540%
1413
1256
-1413
-1256
9/1/2008
1638
2.85%
4.31%
0.8799
3.7883%
1623
1428
-1623
-1428
12/1/2008
1729
2.93%
4.38%
0.8703
3.8139%
1815
1579
-1815
-1579
3/2/2009
1820
3.01%
4.46%
0.8607
3.8346%
1997
1719
-1997
-1719
6/1/2009
1911
3.08%
4.51%
0.8511
3.8342%
2121
1805
-2121
-1805
8/31/2009
2002
3.15%
4.54%
0.8415
3.8233%
2216
1865
-2216
-1865
11/30/2009
2093
3.21%
4.57%
0.8320
3.8054%
2293
1908
-2293
-1908
3/1/2010
2184
3.27%
4.60%
0.8225
3.7842%
2360
1941
-2360
-1941
5/31/2010
2275
3.32%
4.62%
0.8131
3.7571%
2410
1960
-2410
-1960
8/30/2010
2366
3.37%
4.63%
0.8037
3.7239%
2441
1962
-2441
-1962
11/29/2010
2457
3.42%
4.64%
0.7945
3.6880%
2462
1956
-2462
-1956
2/28/2011
2548
3.46%
4.64%
0.7853
3.6438%
2457
1929
-2457
-1929
5/30/2011
2639
3.50%
4.63%
0.7763
3.5935%
2430
1886
-2430
-1886
8/29/2011
2730
3.54%
4.62%
0.7674
3.5443%
2404
1845
-2404
-1845
11/28/2011
2821
3.56%
4.32%
0.7592
3.2818%
1666
1265
-1666
-1265
2/27/2012
2912
3.59%
4.49%
0.7508
3.3691%
2077
1560
-2077
-1560
5/28/2012
3003
3.62%
4.54%
0.7423
3.3714%
2212
1642
-2212
-1642
8/27/2012
3094
3.65%
4.51%
0.7340
3.3130%
2142
1572
-2142
-1572
11/26/2012
3185
3.67%
4.48%
0.7258
3.2529%
2062
1497
-2062
-1497
2/25/2013
3276
3.69%
4.45%
0.7178
3.1912%
1973
1416
-1973
-1416
7
5/27/2013
3367
3.71%
4.40%
0.7100
3.1210%
1848
1312
-1848
-1312
8/26/2013
3458
3.73%
4.35%
0.7024
3.0554%
1735
1219
-1735
-1219
last reset
11/25/2013
3549
3.74%
4.29%
0.6949
2.9844%
1597
1110
-1597
-1110
last payment
2/24/2014
3640
3.76%
4.25%
0.6876
2.9205%
1479
1017
-1479
-1017
DTRorDTC
Zero
Forward
PV
PV*F
Sum(PV*F) 124.69%
Sum(PV)
34.1214
X
8
3.65%
FR-payor
Sum(PV*CF)
PV*CF VR-payor PV*CF
0
0
9