discounting curves
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7/29/2019 Discounting Curves
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1Raffaele Giura Interest rate yield curves before and after the crisis
Interest Rate Curves before and afterthe crisis
Raffaele GiuraBanca IMI - Risk Trading Fixed Income
raffaele.giura@bancaimi.com
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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 the3X6 forward rate..
..otherwise arbitrage opportunitiescan exist in the market place
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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 computeit using the 6m spot rate; or the 3m spot rate and the 3X6m forwardrate; or the 1m spot rate, the 1X2 forward rate, the 2X3 forward
rate...
Rates 0X3 3X6
Rate 0X6
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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..
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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)
+ =
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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 forwardrate
3X6 F.R.A. in the market was not actually priced from the 3mand 6m Euribors
This difference was explicitely priced in the single currencybasis swap market
Not= +
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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 and10y maturities: 2004 - 2007
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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 and10y maturities: 2006 - 2010
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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
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10Raffaele Giura Interest rate yield curves before and after the crisis
New curve building framework
Forwarding curves Used only for computing the forwardfixings implied in the market. Madefrom instruments with homogeneousunderlying fixing
Discounting curves Used only for cashflow discounting
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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
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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
FixingDate
StartDate End Date
FixingRate
PaymentDate Cash Flow Discount
Cash FlowNPV
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
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13Raffaele Giura Interest rate yield curves before and after the crisis
How can we practically build forwarding curves?
Direct solution..
Euribor
instrumentsDiscount factors Forward fixings
We start from a set of market instrument with the same underlyingeuribor fixing
We directly convert them in a discount factors
We will compute the forward fixings we need from these discountfactors
It looks similar to the pre-crisis framework but it is more complex..
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14Raffaele Giura Interest rate yield curves before and after the crisis
Lets compute the forwarding curve discount factors
3m Euribor
marketdata set
F.R.A.
0X3
3X6
6X9
9X12
........
6m Euribor
marketdata set F.R.A.
0X6
6X12
12X18
........
0X3 3X6 6X9 .......9X12
disc. 3m 6m 9m 12m .....
0X6 .......6X12
disc. 6m 12m .....
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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 computethe 1m disc. factor?
Pre-crisis: from a market quoted 1mmaturity instrument!
After-crisis: there is no market quoted
1m maturity instrument with a 6meuribor underlying...
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16Raffaele Giura Interest rate yield curves before and after the crisis
Does it help to add more F.R.A.s?
..not enough to compute3 discount factors...
F.R.A. 0X6
0m 6m
F.R.A. 1X7
1m 7m
d1m d7md6m
2 market instruments..
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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 n1
0x6
6x12
12x1818x24
.........Interpolation
1X7 7X13
1m 7m 13m
F.R.A. Set n2
1x7
7x13
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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
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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
!!
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20Raffaele Giura Interest rate yield curves before and after the crisis
What about discounting?
In examples so far we built the forwarding curve from marketsets made only of F.R.A. When you put to zero F.R.As NPV
you put to zero also their cashflows. This allowed us to ignorethe discounting curve issue
In real market conditions you have to include in the market setfrom 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 themoney market instruments
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21Raffaele Giura Interest rate yield curves before and after the crisis
Discounting and the crisis
Before the crisis discounting was based on the single Euriborcurve
The crisis implied that you have to use multiple curves. Itwouldnt make sense to discount with a forwarding curvewhose purpose is only to compute the forward fixings. Youneed something specific for discounting cashflows
The crisis increased the awareness of the counterparty risk, sothat on the interbank swap markets most of the new deals arenow centrally cleared or at least protected by a CSA
agreement.
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22Raffaele Giura Interest rate yield curves before and after the crisis
CSA discounting
The discounting rate for the cashflows of a collateralized dealhas to be computed keeping in account the CSA features. You
can have:
Standard CSA, that means: daily NPV calculations; dailymargin calls; cash only collateral, in the same currency inwhich 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 thedeal is written, margin call thresholds, asymmetrical collateraland in general any feature different from the standard CSA.Here specific discounting functions have to be built
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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 clearingtheir new deals in Swapclear. This means that the I.R.S. prices
seen in the broker pages are meant to be good forcounterparties that are clearing in Swapclear
Swapclear variation margin system works in a way comparablewith a standard collateral. This allows you to discount thecashflows of the EUR deals cleared in Swapclear at Eonia
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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 netpresent value (NPV) of trades registered with it on a daily basis() The Clearing House (...) distributes PAI on cumulativevariation margin received from an SCM. The PAI rate for Sterling isSONIA (Sterling Overnight Index Average); for Euro is EONIA(Euro Overnight Index Average); for USD is FEDFUNDS1; for YENis TONAR, for Swiss Francs is TOIS, and for Danish, NorwegianKrone and Swedish Krona, Australian, Hong Kong, New Zealandand Canadian Dollars, Polish Zloty and South African Rand, the
appropriate overnight input into our end-of-day yield curves isused.
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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 indexcomputed as a weighted average of all overnight unsecured lending
transactions in the Euro interbank market. By comparison theEuribor calculation only takes into account offered rates and notrates at which transactions have been made. The bankscontributing 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 atthe end of each interest rate period.
Daily compounding at Eonia rateFloating leg cashflow
Fixed leg cashflow
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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 correctlycompute 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
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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 fillin the in between discount factors in an arbitrary way, as we
did on the Euribor curve. We can compute them directly frommarket quoted instruments (the 1m discount from the 1m Eoniarate)
Lastly if we are also discounting at Eonia (which Ill assume we
are doing from now on) we will have the same curve used bothfor forwarding and discounting: in other worlds the Eonia swapscan still be priced correctly with the old single curve framework(even if the eonia curve bootstrapping has its own specific
features)
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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 tobuild an Euribor-Eonia forward spread term structure
Any Euribor fixing can be seen as the sum of the equivalentEonia swap rate and a spread
So if we had, for example, to compute the 1X4 F.R.A. wecould think to compute the 1X4 Eonia swap rate first andafter add the appropriate spread in order to get the correctprice
1X4m F.R.A. 1X4m Eonia + Spread
0m 1m 4m 0m 1m 4m
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29Raffaele Giura Interest rate yield curves before and after the crisis
What does the Euribor3m-Eonia spread tell us
(a traders point of view)
Intuitively this represents the difference between lending directlyfor 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, theEonia is a traded (mid) rate. This factor should amount to6.25bp = (Euribor-Euribid)/2 and should remain constant
Counterparty and liquidity risk, wich together give a measureof the banking sector stress, as perceived by the interbankmoney markets (Morini 2009)
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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 havea term structure and can be bootstrapped from the Euribor andEonia market data
Euribor3m-Eon ia 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
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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 termstructure we can accordingly compute any implied forwardfixing:
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- Ju l- 09 1 8- No v- 10 01 -A pr -12 1 4- Aug -13 27- Dec -14 10- May -16 2 2- Se p- 17 04- Feb- 19 18- Ju n- 20 31 -Oc t- 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
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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 anEonia swap plus a changing spread on the floating leg. Thesespreads are taken from the forward spread term structure we havejust seen
Euribor3m-Eonia Spd
0
0.05
0 .1
0.15
0 .2
0.25
0 .3
0 6- Ju l- 09 1 8- No v- 10 0 1- Ap r- 12 1 4- Au g- 13 2 7- De c- 14 1 0- Ma y- 16 2 2- Se p- 17 0 4- Fe b- 19 1 8- Ju n- 20 3 1- Oc t- 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
Euribor fixings Eonia fixings + spreads
=
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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 marketextended to long and extra long maturitites: you can not apply
this kind of solution to all currencies. However, if you discount atEonia equivalent rates, you will need an Eonia curve anyway
In real life curve calibration can be more difficult, mainly inmarket stress situations
Eonia + spread solution is likely to require more work toimplement it in the pricing, risk and revaluation systems
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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 nothave 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 informationsfrom 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 ratesare expected to evolve in the future; from the Euribor-Eoniaterm structure you can extract informations on the way thebanking sector stress is expected to evolve in the future
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35Raffaele Giura Interest rate yield curves before and after the crisis
If needed slides
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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 Euribor6m 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
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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 onthe 9 mn we received yesterday; the rate of interest we usefor this calculation is the CSA rate; if we agreed to use theovernight as CSA rate, and the overnight was 1.25% wehave to pay 312.5 eur (9mn * 1.25% / 360)
-net payment: we rec 999.687,5 eur
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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=
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39Raffaele Giura Interest rate yield curves before and after the crisis
Why we should discount the cashflows
of the collateralized deals with the CSArate?
Test explanation
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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
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