discounting curves

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

[email protected]


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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 the3X6 forward rate..

..otherwise arbitrage opportunitiescan exist in the market place


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..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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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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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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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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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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..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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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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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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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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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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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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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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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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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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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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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


19/40


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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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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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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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


23/40


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


24/40


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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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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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


27/40


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)


28/40


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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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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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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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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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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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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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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If needed slides


36/40


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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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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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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Why we should discount the cashflows

of the collateralized deals with the CSArate?

Test explanation


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What is a future cashflow worth?

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

NPV

discounting curves

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