Multiple discount and forward curves

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1 Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of IBC 1 1 1

2 Content Dscountng and curve buldng pror to the fnancal crss The effects of credt crss Multple forward curves n a sngle currency Collateral Multple currences Out of scope Fundng curve CVA adjustment 2

3 Curve buldng For lnear nterest rate products (.e. swaps, loans, futures, FRAs) the market value (or net present value PV) s calculated by dscountng the projected cash flows on the approprate dscount curve. PV 1 CF ( t, ccy ) The dscount functon (t) s the value of a unt cash flow (zero coupon bond) n the future for any partcular currency. Alternatve we use the concept yeld curve Y(t) to descrbe the term structure of nterest rates. However the precse value of the yeld curve depends upon the day count conventon (e.g. Act/365 or Act/Act), compoundng frequency (annual, semannual, contnuously) and busness day conventon (followng, modfed followng). Most commonly used are the contnuously compoundng zero rate (z cc ) and annual zero rate (z a ): ( t) exp( zcc( t) t) ( t ) (1 1 z a ( t )) t 3

4 Valuaton and dscountng n pre-credt crss era Lbor (Eurbor) s assumed to be the rsk free rate o arbtrage condton Dscount functon can be used for dscountng and for calculatng forwards: ( t, t ) ( t, t ) ( t, t ) ( t ) 1 E[ Lbor ( t, t )] { 1} / ( t, t ) 1 1 ( t ) Valung an nstrument reduces to determnng the dscount functon at dscrete term pont and the nterpolaton between these ponts: Select a number of fnancal nstruments wth known prces (deposts, FRAs and money market futures, nterest rate swaps). For the short end of the curve we use Lbor rates. For the mddle part of the curve we use ether futures or FRAs. For the long-end we use par swaps. 4

5 Valuaton and dscountng n pre-credt crss era (cont) For deposts (Lbor rates) and FRAs/futures the dscount functon can be calculated n drectly: 1 1 1, r For swaps the fxed rate s such that the value of the fxed and floatng leg are dentcal:, 0 1 C 1 j 1 M E[ L ( j 1, j j)] j M The floatng leg becomes par, because we use Lbor both for dscountng and for calculatng forwards. The result s a closed form soluton for the dscount functon, whch can be solved teratvely. Ths process s called bootstrappng ( t ) 1 C C 1, 1, 5

6 EUR Yeld curve 18 March

7 EUR 3M forward curve 18 March

8 Development of 3M vs 6M EUR 5Y bass swap 8

9 Valuaton and dscountng n post-credt crss era The assumpton that you can always borrow and nvest at Lbor s no longer vald. In the market you can notce that: Bass swap spreads are no longer near zero The fxed rate of a par swap now depends upon the reprcng frequency of the floatng leg The prce of e.g. a 3-9 FRA s no longer close to the average of a 3-6 and 6-9 FRA The market quotes refer to collateralsed transactons. There are no unambguous quotes for non-collateralsed transactons. Ths results n dfferent curves for dfferent reprcng frequences. In constructng these curves we should consstently use nstruments wth the same reprcng frequency. We can stll assume that the value of a floatng leg equals the value of the fxed leg. However we do not assume anymore that the value of the floatng leg s par. You should use the terms yeld curve and dscount functon wth care. If we use market nstruments to construct the dscount functon, then ths can only refer to the o/n curve. The other curves are also based upon collateralsed nstruments and can n prncple only be used for calculatng forwards The smple no-arbtrage condton and relatonshp between forward rates and the dscount functon no longer holds 9

10 Curve buldng (sngle currency, post-crss) Only the o/n rate (EOIA, OIS, SOIA, etc) s regarded as the rsk-free rate. We assume that ths s the collateral rate. The bootstrappng process now requres a specfc order Frst we construct the overnght dscount curve. Ths curve s used for dscountng collateralsed cash flows and for calculatng forward o/n rates. Then we construct the curves (mostly 3M and 6M), for whch drect quotes are avalable. These curve are only used to calculate forward rates Fnally we buld the curves based upon bass swaps (mostly 1M). Also these curves are only used for calculatng forward rates. For dscountng non-collateralsed transactons, each nsttuton should choose hs natural reprcng frequency. For most nsttutons ths wll be 3M or 6M. The dscount curve s now created the other way round. From the collateralsed bootstrappng process the forwards are known and the dscount functon s calculated by usng the smple pre-crss relatonshp between the forward rates and dscount functon 10

11 Curve buldng (sngle currency, post-crss) o/n curve For the o/n curve we can stll use the tradtonal bootstrappng method. We only use the superscrpt c for the dscount functon to denote ths as the dscount functon for collateralsed transactons: c ( t ) 1 C C 1, 1, c We neglect the fact that the o/n swaps have a 1-day settlement leg. 3M curve The other curves are only used for calculatng forwards. The frst ponts on the 3M and 6M curves are straghtforward, as these are already forwards (Lbor rates or futures / FRAs). For convenence we defne the 3M dscount functon as derved from the 3M forwards as: 3 M ( t ) 1 (1 ( t 1 1, t ) fr 3 M ( t 1, t )) However ths s not the dscount functon, whch s used for dscountng. It s only a tool to express the 3M yeld curve 11

12 Curve buldng (sngle currency, post-crss) For the long end of these curves the value of the fxed leg equals the value of the floatng leg, f dscounted on the o/n curve: 3M curve M j 1 c fr C j, j 1 j, j 1 j, 1 1 c However ths requres a dfferent knd of bootstrappng: The dscount functon on all ponts s known and only the last forward rate(s) s (are) unknown. In case there s only 1 unknown forward rate, ths equaton can be solved drectly In case there s more than 1 unknown forward rate (whch s mostly the case) an assumpton about the nterpolaton of these unknown forward rates have to be made. Unless you choose a lnear nterpolaton of these forward rates, ths equaton has to be solved teratvely. 12

13 Curve buldng usng bass swaps 1M curve Bass swaps are quoted as a spread rate on the shortest reprcng frequency for a specfc tenor, e.g. 1M Eurbor + 15 bps versus 3M Eurbor for a 5Y perod. To compute the 1M forward curve, frst the 3M forward curve and o/n dscountng curve have to be calculated. The bootstrappng process for the 1M forward curve then s smlar to the bootstrappng process for the 3M forward curve: M j 1 1M 1M b fr ) j, j 1 j, j 1 c j ( K 1 3 M, 1 fr 3 M, 1 c It s noted that: For the short end of the curve we use standard fxed-floatng swaps, dscounted on the o/n curve. The soluton for the forward rate s straghtforward In general the coupon payments concde wth the reprcng frequency, so M=3K and the accrual factors are dfferent on both sdes Quotes are avalable for annual bass swaps 13

14 EUR Yeld curves 17 October

15 Curve buldng (non-collateralsed, post-crss) C curve For dscountng non-collateralsed transactons, each nsttuton should choose hs natural reprcng frequency. For most nsttutons ths wll be 3M or 6M. The dscount curve s now created the other way round. From the collateralsed bootstrappng process the forwards are known and the dscount functon s calculated by usng the smple pre-crss relatonshp between the forward rates and dscount functon C 1 C 1 1, fr 3 M 1,, C 0 1 As ths curve s also used for dscountng credt nstruments (loans,bonds, deposts) t s common practse to add a credt spread to ths curve. Be aware that: Most systems add the credt spread to the zero curve, whereas t should be added to the forward curve Calculatng the nterest senstvty by shftng the zero curve can create sgnfcant non-zero senstvtes for a portfolo of floatng loans and lead to ncorrect hedges 15

16 Development of EUR/USD 5Y bass swap 16

17 Cross currency bass spreads We also notce that Cross currency bass swap spreads n the major markets (EUR, USD, GBP, JPY) are non-zero. e.g, You pay 3M USD Lbor and receve 3M Eurbor mnus 50 bps Under some CSA agreements USD collateral s posted for EUR swaps and vce versa These spreads reflect a market, whch s not n equlbrum Assumng the same cost of fundng n ther home market and local currency a USD based bank has a compettve advantage over a EUR based bank. It s no longer possble to use both EOIA and OIS as rsk-free rate Identcal cash flows have dfferent values for dfferent market partcpants 17

18 Constructon of consstent mult currency curves A possble approach Step 1: construct EUR curves usng the sngle currency approach as descrbed before. The EUR o/n curve wll be used to dscount all EUR cash flows wth EUR collateral The EUR o/n, 1M, 3M and 6M forward curves Step 2: construct USD curves usng the sngle currency approach as descrbed before. The USD o/n curve wll be used to dscount all USD cash flows wth USD collateral The USD o/n, 1M, 3M and 6M forward curves Step 3: use 3M EUR vs 3M USD cross currency bass swaps for: The USD dscount functon wth EUR as collateral currency The EUR dscount functon wth USD as collateral currency 18

19 USD dscount functon wth EUR collateral Assume the XCCY quotes wll get EUR collateral Ths yelds (spread b (currently negatve) on the EUR leg): EUR [ 1 M j 1 { EUR,3 M j, j 1 ( b fr EUR,3 M j, j 1 ) c j ( EUR ) ( EUR )} c ( EUR ) ( EUR )] fx USD [ 1 M j 1 { USD,3 M j, j 1 fr USD,3 M j, j 1 c ( EUR j ) ( USD )} c ( EUR ) ( USD )] The only unknown s the USD dscount functon under EUR collateral, thus ths can be solved by the tradtonal bootstrappng methods 19

20 EUR dscount functon wth USD collateral Assume the XCCY quotes wll get USD collateral Same EUR/USD bass swap wth spread b * : (whch s not necessarly the same as the prevous bass spread) EUR [ 1 M j 1 { EUR,3 M j, j 1 ( b * fr EUR,3 M j, j 1 ) c j ( USD ) ( EUR )} c ( USD ) ( EUR )] fx USD [ 1 M j 1 { USD,3 M j, j 1 fr USD,3 M j, j 1 c ( USD j ) ( USD )} c ( USD ) ( USD )] The only unknown s the EUR dscount functon under USD collateral, thus ths can be solved by the tradtonal bootstrappng methods Usng other XCCY spreads ths results n for e.g. all EUR cash flows: one dscountng curve per collateral currency. 20

21 Cross currency bass spreads Ths approach: - values all par swaps at par - neglects the cross currency spread n the cost of fundng n foregn currences: e.g. the XCCY spread has no nfluence on the dscountng of a standard USD swap. Alternatve: Choose EUR as base currency: Include the cross currency bass spread n the valuaton of all non-eur based fnancal nstruments. However a the value of non-eur par swap wll not be par anymore. How? Would ths be preferred? 21

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