Interest rate curves after the crisis- Raffaele Giura

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 [email protected]


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


..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


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..


Example: 3X6 F.R.A. pricing pre-crisis

3m d.factor

6m d.factorPricing F.R.A. 3X6

3m depo (euribor)

6m depo (euribor)

+ =


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= +


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


..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


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


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


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


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


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..


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 .....


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...


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..


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


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


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

!!


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


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.


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


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


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.”


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


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


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)


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


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)


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


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


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

=


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


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


“If needed” slides


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


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


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=


Why we should discount the cashflows of the collateralized deals with the CSA rate?

Test explanation


What is a future cashflow worth?

days 0X1 1X2 2x3 3x4 4x5 5x6 6x7 7x8 n x n+1.. Disc. rates

NPV

Interest rate curves after the crisis- Raffaele Giura

Economy & Finance

Transcript of Interest rate curves after the crisis- Raffaele Giura