Post on 11-Nov-2014
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1Raffaele Giura – Interest rate yield curves before and after the crisis
Interest Rate Curves before and after the crisis
Raffaele GiuraBanca IMI - Risk Trading Fixed Income raffaele.giura@bancaimi.com
2Raffaele Giura – Interest rate yield curves before and after the crisis
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
3Raffaele Giura – Interest rate yield curves before and after the crisis
..which implies that..
Rates 0X1 1X2 2x3 3x4 4x5 5x6
..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 forwardrate...
Rates 0X3 3X6
Rate 0X6
4Raffaele Giura – Interest rate yield curves before and after the crisis
A typical pre-crisis Euribor curve building framework
Market Instruments:
Deposits
Euribor futures
I.R.S.
Bootstrapping
Market Disc. Tenors Factors
Interpolation
Pricing:
I.R.S.
F.R.A
..Others..
5Raffaele Giura – Interest rate yield curves before and after the crisis
Example: 3X6 F.R.A. pricing pre-crisis
3m d.factor
6m d.factorPricing F.R.A. 3X6
3m depo (euribor)
6m depo (euribor)
+ =
6Raffaele Giura – Interest rate yield curves before and after the crisis
The single currency basis swap:
Even before the crises in the market investing at the 6m euriborspot rate was not equivalent to investing at the 3m euribor spotrate and reinvesting the proceeds at the 3X6 euribor forward rate
3X6 F.R.A. in the market was not actually priced from the 3m and 6m Euribors
This difference was explicitely priced in the single currency basis swap market
Not= +
7Raffaele Giura – Interest rate yield curves before and after the crisis
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
8Raffaele Giura – Interest rate yield curves before and after the crisis
..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
9Raffaele Giura – Interest rate yield curves before and after the crisis
The old framework performance after the crisis
Euribor 6m fwd
1.5
2
2.5
3
3.5
4
7-Jan-11 26-Jul-11 11-Feb-12 29-Aug-12 17-Mar-13 3-Oct-13 21-Apr-14
Pre-crisis framework Correct mkt data
10Raffaele Giura – Interest rate yield curves before and after the crisis
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
11Raffaele Giura – Interest rate yield curves before and after the crisis
Euribor 3M Fwd
0.8000
1.3000
1.8000
2.3000
2.8000
3.3000
17-May-2009
25-Aug-2009
3-Dec-2009
13-Mar-2010
21-Jun-2010
29-Sep-2010
7-Jan-2011
17-Apr-2011
26-Jul-2011
3-Nov-2011
11-Feb-2012
Forward fixing curve
12Raffaele Giura – Interest rate yield curves before and after the crisis
Euribor 3M Fwd
0.8000
1.3000
1.8000
2.3000
2.8000
3.3000
17-May-2009
3-Dec-2009 21-Jun-2010
7-Jan-2011 26-Jul-2011 11-Feb-2012
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 = 1,097,436
Swap floating leg pricing
13Raffaele Giura – Interest rate yield curves before and after the crisis
How can we practically build forwarding curves? Direct solution..
Euribor instruments
Discount factors Forward fixings
We start from a set of market instrument with the same underlying euribor fixing
We directly convert them in a discount factors
We will compute the forward fixings we need from these discount factors
It looks similar to the pre-crisis framework but it is more complex..
14Raffaele Giura – Interest rate yield curves before and after the crisis
Let’s compute the forwarding curve discount factors
3m Euribor market data set
F.R.A.
0X3
3X6
6X9
9X12
........
6m Euribor market data set
F.R.A.
0X6
6X12
12X18
........
0X3 3X6 6X9 .......9X12
disc. 3m 6m 9m 12m .....
0X6 .......6X12
disc. 6m 12m .....
15Raffaele Giura – Interest rate yield curves before and after the crisis
Problem: how do we compute the “in-between”discount factors now?
0X6 .......6X12
1m 6m 12m 18m ......
12X18F.R.A.
Disc. factors
How do we compute the 1m disc. factor?
Pre-crisis: from a market quoted 1m maturity instrument!
After-crisis: there is no market quoted 1m maturity instrument with a 6m euribor underlying...
16Raffaele Giura – Interest rate yield curves before and after the crisis
Does it help to add more F.R.A.s?
..not enough to compute 3 discount factors...
F.R.A. 0X60m 6m
F.R.A. 1X71m 7m
d1m d7m d6m
2 market instruments..
17Raffaele Giura – Interest rate yield curves before and after the crisis
Interpolation could be an answer..
0X6 6X12
d.factors 6m 12m 18m 24m ......
12X18 18X24 ..........F.R.A. Set n°10x66x1212x1818x24.........
Interpolation
1X7 7X13
1m 7m 13m
F.R.A. Set n°21x77x13
18Raffaele Giura – Interest rate yield curves before and after the crisis
Interpolation: important to run stress tests on the results. Constrained cubic spline on rate*time..
1.5
2
2.5
3
3.5
4
4.5
06-07-09 18-11-10 01-04-12 14-08-13 27-12-14 10-05-16
19Raffaele Giura – Interest rate yield curves before and after the crisis
Interpolation: important to run stress tests on the results. Quartic spline on fwd-fwd rates..
1.5
2
2.5
3
3.5
4
4.5
06-07-09 18-11-10 01-04-12 14-08-13 27-12-14 10-05-16
!!
20Raffaele Giura – Interest rate yield curves before and after the crisis
What about discounting?
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
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
In practice you build the discounting curve first; then you findthe forwarding curve that zeroes the NPVs of your set of “at the money” market instruments
21Raffaele Giura – Interest rate yield curves before and after the crisis
Discounting and the crisis
Before the crisis discounting was based on the single Euribor curve
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
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.
22Raffaele Giura – Interest rate yield curves before and after the crisis
CSA discounting
The discounting rate for the cashflows of a collateralized deal has to be computed keeping in account the CSA features. You can have:
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.
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 collateraland in general any feature different from the standard CSA. Here specific discounting functions have to be built
23Raffaele Giura – Interest rate yield curves before and after the crisis
LCH-Swapclear discounting
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
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
24Raffaele Giura – Interest rate yield curves before and after the crisis
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.”
25Raffaele Giura – Interest rate yield curves before and after the crisis
A second solution for forwarding curves:Eonia + spread
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.
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.
Daily compounding at Eonia rateFloating leg cashflow
Fixed leg cashflow
26Raffaele Giura – Interest rate yield curves before and after the crisis
The Eonia curve building: practical advantages vs Euribor curve building after the crisis (1)
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.
0X6 Eonia swap
0X3m Eonia swap 3X6m Eonia swap
The fixings are the same
27Raffaele Giura – Interest rate yield curves before and after the crisis
The Eonia curve building: practical advantages vs Euribor curve building after the crisis (2)
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)
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)
28Raffaele Giura – Interest rate yield curves before and after the crisis
So far so good with Eonia: but I need Euribor curves...
To build an Euribor curve from an Eonia curve we need to build an Euribor-Eonia forward spread term structure
Any Euribor fixing can be seen as the sum of the equivalent Eonia swap rate and a spread
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
1X4m F.R.A. 1X4m Eonia + Spread
0m 1m 4m 0m 1m 4m
29Raffaele Giura – Interest rate yield curves before and after the crisis
What does the Euribor3m-Eonia spread tell us (a trader’s point of view)
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:
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
Counterparty and liquidity risk, wich together give a measure of the banking sector “stress”, as perceived by the interbank money markets (Morini 2009)
30Raffaele Giura – Interest rate yield curves before and after the crisis
Euribor-Eonia forward spread term structure
The Euribor-Eonia forward spreads are not constant. They have a term structure and can be bootstrapped from the Euribor and Eonia market data
Euribor3m-Eonia Spd
0
0 .05
0 .1
0 .15
0 .2
0 .25
0 .3
06-Jul-09 18-Nov -10 01-A pr-12 14-A ug-13 27-Dec -14 10-May -16 22-Sep-17 04-Feb-19 18-Jun-20 31-Oc t-21
31Raffaele Giura – Interest rate yield curves before and after the crisis
Generic 3m Euribor forward fixing
Once we built an Eonia curve and an Euribor-Eonia spread term structure we can accordingly compute any implied forward fixing:
E.g. 27m Euribor3m-eonia spread + 27x30m Eonia swap
=
27x30m F.R.A.
Euribor3m-Eonia Spd
0
0.05
0.1
0.15
0.2
0.25
0.3
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
Eonia implied daily fwds
00.5
11.5
22.5
33.5
44.5
5
18-Nov-10
01-Apr-12
14-Aug-13
27-Dec-14
10-May-16
22-Sep-17
04-Feb-19
18-Jun-20
32Raffaele Giura – Interest rate yield curves before and after the crisis
Euribor swaps as Eonia swaps + spread
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 havejust seen
Eur ibor3m-Eonia Spd
0
0 .0 5
0 .1
0 .1 5
0 .2
0 .2 5
0 .3
06-Ju l-09 18-Nov -10 01-A pr -12 14-A ug-13 27-Dec -14 10-May -16 22-Sep-17 04-Feb-19 18-Jun-20 31-Oc t-21
Eonia im plied daily fw ds
00.5
11.5
22.5
33.5
44.5
5
18-Nov-10
01-Apr-12
14-Aug-13
27-Dec-14
10-May-16
22-Sep-17
04-Feb-19
18-Jun-20
Euribor fixings Eonia fixings + spreads
=
33Raffaele Giura – Interest rate yield curves before and after the crisis
Comparing the two forwarding curve solutions (1)
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
In real life curve calibration can be more difficult, mainly in market stress situations
Eonia + spread solution is likely to require more work to implement it in the pricing, risk and revaluation systems
34Raffaele Giura – Interest rate yield curves before and after the crisis
Comparing the two forwarding curve solutions (2)
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
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
35Raffaele Giura – Interest rate yield curves before and after the crisis
“If needed” slides
36Raffaele Giura – Interest rate yield curves before and after the crisis
CSA at work: day 1
Day 1, at 12.00: IRS trader pays 1 bn 10y swap vs Euribor 6m at 3.32 to counterparty
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
37Raffaele Giura – Interest rate yield curves before and after the crisis
CSA at work: day 2
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
38Raffaele Giura – Interest rate yield curves before and after the crisis
Rate for discounting when deals are inside a CSA
For a discussion on this matter see Willmott forum at:
http://www.wilmott.com/messageview.cfm?catid=4&threadid=68959&FTVAR_MSGDBTABLE=
39Raffaele Giura – Interest rate yield curves before and after the crisis
Why we should discount the cashflows of the collateralized deals with the CSA rate?
Test explanation
40Raffaele Giura – Interest rate yield curves before and after the crisis
What is a future cashflow worth?
days 0X1 1X2 2x3 3x4 4x5 5x6 6x7 7x8 n x n+1.. Disc. rates
NPV