Interest Rate Curves before and after
the crisis
Raffaele Giura
Banca IMI - Risk Trading Fixed Income
[email protected]
Raffaele Giura – Interest rate yield curves before and after the crisis
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One of the first notions in finance textbooks..
..is that investing at the 6m spot rate
=
should be equivalent to
investing at the 3m spot rate
+
and reinvesting the proceeds at the
3X6 forward rate..
..otherwise arbitrage opportunities
can exist in the market place
Raffaele Giura – Interest rate yield curves before and after the crisis
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..which implies that..
..if you have to compute the 6m discount factor you can compute
it using the 6m spot rate; or the 3m spot rate and the 3X6m forward
rate; or the 1m spot rate, the 1X2 forward rate, the 2X3 forward
rate...
Rates 0X1 1X2 2x3 3x4 4x5 5x6
Rates
0X3
Rate
Raffaele Giura – Interest rate yield curves before and after the crisis
3X6
0X6
3
A typical pre-crisis Euribor curve building framework
Market
Tenors
Market Instruments:
Deposits
Euribor futures
Bootstrapping
I.R.S.
Pricing:
I.R.S.
F.R.A
Interpolation
..Others..
Raffaele Giura – Interest rate yield curves before and after the crisis
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Disc.
Factors
Example: 3X6 F.R.A. pricing pre-crisis
3m d.factor
3m depo (euribor)
6m d.factor
6m depo (euribor)
Pricing F.R.A. 3X6
+
=
Raffaele Giura – Interest rate yield curves before and after the crisis
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The single currency basis swap:
ƒ Even before the crises in the market investing at the 6m euribor
spot rate was not equivalent to investing at the 3m euribor spot
rate and reinvesting the proceeds at the 3X6 euribor forward
rate
Not=
+
ƒ 3X6 F.R.A. in the market was not actually priced from the 3m
and 6m Euribors
ƒ This difference was explicitely priced in the single currency
basis swap market
Raffaele Giura – Interest rate yield curves before and after the crisis
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If before the crisis the 3m-6m basis was small..
Market prices of the 3m vs 6m euribor basis swap for the 1y and
10y maturities: 2004 - 2007
Raffaele Giura – Interest rate yield curves before and after the crisis
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..after the crises you can not ignore it anymore
Market prices of the 3m vs 6m euribor basis swap for the 1y and
10y maturities: 2006 - 2010
Raffaele Giura – Interest rate yield curves before and after the crisis
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The old framework performance after the crisis
Euribor 6m fwd
4
3.5
3
2.5
2
1.5
7-Jan-11 26-Jul-11 11-Feb-12 29-Aug-12 17-Mar-13 3-Oct-13 21-Apr-14
Pre-crisis framework
Raffaele Giura – Interest rate yield curves before and after the crisis
Correct mkt data
9
New curve building framework
Forwarding curves
Used only for computing the forward
fixings implied in the market. Made
from instruments with homogeneous
underlying fixing
Discounting curves
Used only for cashflow discounting
Raffaele Giura – Interest rate yield curves before and after the crisis
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Forward fixing curve
Euribor 3M Fwd
3.3000
2.8000
2.3000
1.8000
1.3000
0.8000
17-May- 25-Aug- 3-Dec- 13-Mar- 21-Jun- 29-Sep- 7-Jan- 17-Apr- 26-Jul2009
2009
2009
2010
2010
2010
2011
2011
2011
Raffaele Giura – Interest rate yield curves before and after the crisis
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3-Nov- 11-Feb2011
2012
Swap floating leg pricing
Euribor 3M Fwd
3.3000
2.8000
2.3000
1.8000
1.3000
0.8000
17-May2009
3-Dec-2009
21-Jun2010
7-Jan-2011 26-Jul-2011
11-Feb2012
Fixing
Date
Start
Date
End Date
Fixing
Rate
Payment
Date
Cash Flow
Discount
Cash Flow
NPV
3-Jul-09
7-Jul-09
7-Oct-09
1.0636
7-Oct-09
271,809
0.9986
271,430
5-Oct-09
7-Oct-09
7-Jan-10
1.0408
7-Jan-10
265,982
0.9969
265,154
5-Jan-10
7-Jan-10
7-Apr-10
1.0315
7-Apr-10
257,875
0.9950
256,582
1-Apr-10
7-Apr-10
7-Jul-10
1.2127
7-Jul-10
306,544
0.9926
304,270
Float Leg NPV =
Raffaele Giura – Interest rate yield curves before and after the crisis
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1,097,436
How can we practically build forwarding curves?
Direct solution..
Euribor
instruments
Discount factors
Forward fixings
ƒ We start from a set of market instrument with the same underlying
euribor fixing
ƒ We directly convert them in a discount factors
ƒ We will compute the forward fixings we need from these discount
factors
ƒ It looks similar to the pre-crisis framework but it is more complex..
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Let’s compute the forwarding curve discount factors
0X3
3m Euribor
market
data set
0X3 3X6 6X9 9X12 .......
3X6
F.R.A.
6X9
disc. 3m
6m 9m
12m .....
9X12
........
0X6
6m Euribor
market
data set
0X6
6X12
.......
6X12
F.R.A.
12X18
disc.
........
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6m
12m
.....
Problem: how do we compute the “in-between”
discount factors now?
F.R.A.
Disc. factors
0X6
1m
6X12
6m
12X18
12m
.......
18m
......
Pre-crisis: from a market quoted 1m
maturity instrument!
How do we compute
the 1m disc. factor?
Raffaele Giura – Interest rate yield curves before and after the crisis
After-crisis: there is no market quoted
1m maturity instrument with a 6m
euribor underlying...
15
Does it help to add more F.R.A.s?
0m
6m
F.R.A. 0X6
1m
7m
2 market instruments..
F.R.A. 1X7
d1m
Raffaele Giura – Interest rate yield curves before and after the crisis
d6m d7m
16
..not enough to compute
3 discount factors...
Interpolation could be an answer..
F.R.A. Set n°1
0X6
6X12
12X18
18X24
..........
0x6
6x12
12x18
18x24
d.factors 6m
12m
18m
24m
......
.........
Interpolation
F.R.A. Set n°2
1x7
7x13
1X7
1m
7X13
7m
13m
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Interpolation: important to run stress tests on the
results. Constrained cubic spline on rate*time..
4.5
4
3.5
3
2.5
2
1.5
06-07-09
18-11-10
01-04-12
Raffaele Giura – Interest rate yield curves before and after the crisis
14-08-13
18
27-12-14
10-05-16
Interpolation: important to run stress tests on the
results. Quartic spline on fwd-fwd rates..
4.5
4
3.5
3
2.5
!!
2
1.5
06-07-09
18-11-10
01-04-12
Raffaele Giura – Interest rate yield curves before and after the crisis
14-08-13
19
27-12-14
10-05-16
What about discounting?
ƒ In examples so far we built the forwarding curve from market
sets made only of F.R.A. When you put to zero F.R.A’s NPV
you put to zero also their cashflows. This allowed us to ignore
the discounting curve issue
ƒ In real market conditions you have to include in the market set
from which you build the curve also the I.R.S. Once you start
using multicashflow I.R.S. discounting matters
ƒ In practice you build the discounting curve first; then you find
the forwarding curve that zeroes the NPVs of your set of “at the
money” market instruments
Raffaele Giura – Interest rate yield curves before and after the crisis
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Discounting and the crisis
ƒ Before the crisis discounting was based on the single Euribor
curve
ƒ The crisis implied that you have to use multiple curves. It
wouldn’t make sense to discount with a forwarding curve
whose purpose is only to compute the forward fixings. You
need something specific for discounting cashflows
ƒ The crisis increased the awareness of the counterparty risk, so
that on the interbank swap markets most of the new deals are
now centrally cleared or at least protected by a CSA
agreement.
Raffaele Giura – Interest rate yield curves before and after the crisis
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CSA discounting
ƒ The discounting rate for the cashflows of a collateralized deal
has to be computed keeping in account the CSA features. You
can have:
ƒ Standard CSA, that means: daily NPV calculations; daily
margin calls; cash only collateral, in the same currency in
which the deal is denominated, yielding the overnight rate.
Market consensus is Eonia discounting here for EUR deals.
ƒ Non standard CSA, that means: collateral delivery options,
collateral paid in currencies different from the one in which the
deal is written, margin call thresholds, asymmetrical collateral
and in general any feature different from the standard CSA.
Here specific discounting functions have to be built
Raffaele Giura – Interest rate yield curves before and after the crisis
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LCH-Swapclear discounting
ƒ Nowadays most players in the EUR I.R.S. market are clearing
their new deals in Swapclear. This means that the I.R.S. prices
seen in the broker pages are meant to be good for
counterparties that are clearing in Swapclear
ƒ Swapclear variation margin system works in a way comparable
with a standard collateral. This allows you to discount the
cashflows of the EUR deals cleared in Swapclear at Eonia
Raffaele Giura – Interest rate yield curves before and after the crisis
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LCH-Swapclear regulation
11.7 Price Alignment Interest
“The Clearing House collects and distributes changes in the net
present value (NPV) of trades registered with it on a daily basis
(…) The Clearing House (...) distributes PAI on cumulative
variation margin received from an SCM. The PAI rate for Sterling is
SONIA (Sterling Overnight Index Average); for Euro is EONIA
(Euro Overnight Index Average); for USD is FEDFUNDS1; for YEN
is TONAR, for Swiss Francs is TOIS, and for Danish, Norwegian
Krone and Swedish Krona, Australian, Hong Kong, New Zealand
and Canadian Dollars, Polish Zloty and South African Rand, the
appropriate overnight input into our end-of-day yield curves is
used.”
Raffaele Giura – Interest rate yield curves before and after the crisis
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A second solution for forwarding curves:
Eonia + spread
ƒ Eonia (Euro OverNight Index Average) is an interest rate index
computed as a weighted average of all overnight unsecured lending
transactions in the Euro interbank market. By comparison the
Euribor calculation only takes into account offered rates and not
rates at which transactions have been made. The banks
contributing to Eonia are the same as the Panel Banks quoting for
Euribor.
ƒ Eonia can be used as reference rate for Euro denominated I.R.S.
where variable interests are reset daily, compounded, and paid at
the end of each interest rate period.
Fixed leg cashflow
Floating leg cashflow
Daily compounding at Eonia rate
Raffaele Giura – Interest rate yield curves before and after the crisis
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The Eonia curve building: practical advantages
vs Euribor curve building after the crisis (1)
ƒ When you consider the Eonia swap market you can still correctly
compute the 3X6m forward rate from the 3m and 6m spot
rates.There is no 3m vs 6m basis as in the Euribor swap market.
3X6m Eonia swap
0X3m Eonia swap
The fixings are the same
0X6 Eonia swap
Raffaele Giura – Interest rate yield curves before and after the crisis
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The Eonia curve building: practical advantages
vs Euribor curve building after the crisis (2)
ƒ When we build the Eonia forwarding curve we do not have to fill
in the “in between” discount factors in an arbitrary way, as we
did on the Euribor curve. We can compute them directly from
market quoted instruments (the 1m discount from the 1m Eonia
rate)
ƒ Lastly if we are also discounting at Eonia (which I’ll assume we
are doing from now on) we will have the same curve used both
for forwarding and discounting: in other worlds the Eonia swaps
can still be priced correctly with the old single curve framework
(even if the eonia curve bootstrapping has it’s own specific
features)
Raffaele Giura – Interest rate yield curves before and after the crisis
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So far so good with Eonia: but I need Euribor
curves...
ƒ To build an Euribor curve from an Eonia curve we need to
build an Euribor-Eonia forward spread term structure
ƒ Any Euribor fixing can be seen as the sum of the equivalent
Eonia swap rate and a spread
ƒ So if we had, for example, to compute the 1X4 F.R.A. we
could think to compute the 1X4 Eonia swap rate first and
after add the appropriate spread in order to get the correct
price
0m
1m
4m
0m 1m
1X4m F.R.A.
Raffaele Giura – Interest rate yield curves before and after the crisis
4m
1X4m Eonia + Spread
28
What does the Euribor3m-Eonia spread tell us
(a trader’s point of view)
ƒ Intuitively this represents the difference between lending directly
for 3 months and rolling cash o/n for the same 3 months. On the
market the Euribor-Eonia spread is supposed to depend on:
ƒ The offer vs mid spread. The Euribor is an offer rate, the
Eonia is a traded (mid) rate. This factor should amount to
6.25bp = (Euribor-Euribid)/2 and should remain constant
ƒ Counterparty and liquidity risk, wich together give a measure
of the banking sector “stress”, as perceived by the interbank
money markets (Morini 2009)
Raffaele Giura – Interest rate yield curves before and after the crisis
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Euribor-Eonia forward spread term structure
ƒ The Euribor-Eonia forward spreads are not constant. They have
a term structure and can be bootstrapped from the Euribor and
Eonia market data
E ur ibor 3m-E onia S pd
0 .3
0 .2 5
0 .2
0 .1 5
0 .1
0 .0 5
0
06- Ju l- 0 9
18- Nov - 1 0
0 1- A pr - 1 2
14 - A ug - 13
2 7- Dec - 14
Raffaele Giura – Interest rate yield curves before and after the crisis
10 - May - 1 6
30
22- Se p- 1 7
04 - Feb- 1 9
1 8- Jun- 20
31 - Oc t- 21
Generic 3m Euribor forward fixing
ƒ Once we built an Eonia curve and an Euribor-Eonia spread term
structure we can accordingly compute any implied forward
fixing:
Euribor3m-Eonia Spd
Eonia implied daily fwds
0.3
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0.25
0.2
0.15
0.1
0.05
0
06-Jul-09
18-Nov-10
01-Apr-12
14-Aug-13
27-Dec-14
10-May-16
22-Sep-17
04-Feb-19
18-Jun-20
31-Oct-21
18-Nov- 01-Apr- 14-Aug- 27-Dec- 10-May- 22-Sep- 04-Feb- 18-Jun10
12
13
14
16
17
19
20
E.g. 27m Euribor3m-eonia spread + 27x30m Eonia swap
=
27x30m F.R.A.
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Euribor swaps as Eonia swaps + spread
ƒ When you are pricing an Euribor swap you can think to it as an
Eonia swap plus a changing spread on the floating leg. These
spreads are taken from the forward spread term structure we have
just seen
Eon ia im p lied daily fw ds
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
=
18-Nov10
01-Ap r12
14-Au g - 27-Dec- 10-May- 22-Sep - 04-F eb - 18-Ju n 13
14
16
17
19
20
E u r ib o r 3 m - E o n ia S p d
0 .3
Euribor fixings
0 .2 5
Eonia fixings + spreads
0 .2
0 .1 5
0 .1
0 .0 5
0
0 6 - Ju l- 0 9
Raffaele Giura – Interest rate yield curves before and after the crisis
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1 8 - No v - 1 0
01-A pr-12
14-A ug-13
2 7 - De c - 1 4
1 0 - Ma y - 1 6
22-Sep-17
0 4 - Fe b - 1 9
1 8 - Ju n - 2 0
3 1 - O c t- 2 1
Comparing the two forwarding curve solutions (1)
ƒ Eonia + spread needs a developed Eonia swap market
extended to long and extra long maturitites: you can not apply
this kind of solution to all currencies. However, if you discount at
Eonia equivalent rates, you will need an Eonia curve anyway
ƒ In real life curve calibration can be more difficult, mainly in
market stress situations
ƒ Eonia + spread solution is likely to require more work to
implement it in the pricing, risk and revaluation systems
Raffaele Giura – Interest rate yield curves before and after the crisis
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Comparing the two forwarding curve solutions (2)
ƒ On the other side if you use Eonia + spread you do not
have to rely on values of the “in between” discount factors
which are at the end filled in by an arbitrary way
ƒ Eonia + spread could allow you to extract more informations
from the yield curve. You have two different term structures,
each one doing a different job. From the Eonia curve term
structure you can extract informations on how interest rates
are expected to evolve in the future; from the Euribor-Eonia
term structure you can extract informations on the way the
banking sector stress is expected to evolve in the future
Raffaele Giura – Interest rate yield curves before and after the crisis
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“If needed” slides
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CSA at work: day 1
ƒ Day 1, at 12.00: IRS trader pays 1 bn 10y swap vs Euribor
6m at 3.32 to counterparty
ƒ Day 1, at 17.15: close of business:
-10y vs 6m swap rate went up
-swap NPV now = 9 mn euro
-counterparty will “post” (give us) 9mn eur as collateral
Raffaele Giura – Interest rate yield curves before and after the crisis
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CSA at work: day 2
ƒ Day 2, at 17.15: close of business
-no new deal has been made
-market moved: swap NPV is now = 10mn
-counterparty should pay us 1mn ( 10mn – 9mn =
NPV(day2)-NPV(day1)
-we should pay to the counterparty the 1 day interests on
the 9 mn we received yesterday; the rate of interest we use
for this calculation is the CSA rate; if we agreed to use the
overnight as CSA rate, and the overnight was 1.25% we
have to pay 312.5 eur (9mn * 1.25% / 360)
-net payment: we rec 999.687,5 eur
Raffaele Giura – Interest rate yield curves before and after the crisis
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Rate for discounting when deals are inside a CSA
ƒ For a discussion on this matter see Willmott forum at:
http://www.wilmott.com/messageview.cfm?catid=4&threadid
=68959&FTVAR_MSGDBTABLE=
Raffaele Giura – Interest rate yield curves before and after the crisis
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Why we should discount the cashflows
of the collateralized deals with the CSA
rate?
Test explanation
Raffaele Giura – Interest rate yield curves before and after the crisis
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What is a future cashflow worth?
NPV
days 0X1 1X2 2x3 3x4 4x5 5x6 6x7 7x8 n x n+1.. Disc. rates
Raffaele Giura – Interest rate yield curves before and after the crisis
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