Property Derivatives for Managing European Real-Estate Risk

Propery Derivaives for Managing European Real-Esae Risk Frank J. Fabozzi is Professor in he Pracice of Finance, Yale School of Managemen, New Haven, CT frank.fabozzi@yale.edu Rober J. Shiller is Arhur M. Okun Professor of Economics, Yale Universiy, New Haven, CT rober.shiller@yale.edu Radu Tunaru is Senior Lecurer in Financial Mahemaics, Cass Business School, Ciy Universiy, London r.unaru@ciy.ac.uk 1

Propery Derivaives for Managing European Real-Esae Risk Absrac Alhough propery markes represen a large proporion of oal wealh in developed counries, he real-esae derivaives markes are sill lagging behind in volume of rading and liquidiy. Over he las few years here has been increased aciviy in developing derivaive insrumens ha can be uilised by asse managers. In his paper, we discuss he problems encounered when using propery derivaives for managing European real-esae risk. We also consider a special class of srucured ineres rae swaps ha have embedded real-esae risk and propose a more efficien way o ailor hese swaps. Keywords: real-esae markes, propery derivaives, balance guaraneed swaps JEL classificaion: G15, G20 2

1. Inroducion The propery marke represens he larges marke in developed counries in Europe and he world, esimaed beween 30% and 40% of he value of all he underlying physical capial. From a heoreical perspecive, residenial real-esae asses are viewed as a combinaion of a consumpion asse and a leveraged invesmen. A decade ago Shiller and Weiss (1999) poined ou ha his very characerisic poses greaer risk for he financial sabiliy for households due o geographic flucuaions in propery prices. Clearly, his predicion was realized in he U.S. residenial real esae marke beginning in 2007. Moreover, consequences were no limied o he U.S. real-esae marke bu were fel in he marke for oher financial asses hroughou he world. Ideally, propery derivaives on some relevan real-esae indices could be used o miigae real-esae risk and poenial reduce he volailiy of propery prices. Alhough real-esae derivaives should be preferred o insurance-ype conracs because of direc selemen, he liquidiy of propery derivaives is key o heir uilisaion by individuals and professional asse managers. However, liquidiy can be esablished only afer banks decide o paricipae more acively in he real-esae index fuures and opions markes, as suggesed by Case, Shiller, and Weiss (1993). Hinkelman and Swidler (2008) invesigaed if housing price risk can be hedged wih fuures on oher commodiies and financial indices ha already rade in he markes. They could no find significan evidence of a sysemaic relaionship beween naional housing prices and prevailing raded fuures conracs. Hence, esablishing propery fuures markes would help o complee he real-esae markes. Today i is well known now ha housing prices are sicky when hey are going down, sellers being relucan o sell a a price below he psychological price a which he propery was iniially purchased, as firs noed by Case, Quigley, and Shiller (2003). This makes he 3

marke more localized, wih informaion asymmeric, and paricipans price expecaions srongly influenced by he mos recen series of prices. I is less likely, a leas for now, ha a significan fracion of individual homeowners will hedge direcly heir house prices. Neverheless, banks and building socieies holding morgage porfolios more diversified naionally have an incenive o hedge heir exposure wih propery derivaives where he underlying is a local index. The use of index-based fuures conracs and opions for hedging morgage risk, defaul risk, and real-esae price risk has long been advocaed by Case and Shiller (1996). Fisher (2005) provides an overview of oal reurn swaps aciviy in he Unied Saes while Clayon (2007) discusses several indices uilised for derivaives rading. The inroducion of derivaives in he real-esae marke is no easy because, as noed earlier, liquidiy is difficul o esablish when reurns are predicable. An exensive discussion highlighing he imporan psychological barriers ha need o be passed for he esablishmen of real-esae derivaives is provided in Shiller (2008). Carlon (1984) argues ha if changes in marke prices are predicable, hen changes in prices canno be perceived as risky. One major obsacle hindering he inroducion and developmen of real-esae derivaives was he fac ha when reurns follow rends a cerain poins in ime, he marke senimen is unidirecional, making i is difficul o find counerparies. Neverheless, we argue in he paper ha if a fuures conrac is already rading for a series of fuure mauriies, hen he shape of he forward curve on real-esae index becomes imporan. Trades may be execued on he curve, rading simulaneously say shor a fuures wih a long mauriy and long a fuures wih a shorer mauriy. Thus, a counerpary in a propery derivaives conrac does no have o be necessarily a speculaor in order o ake a posiion agains a marke rend. Wih fuures and opions on fuures, an enire specrum of rading sraegies becomes available and marke paricipans such as hedge funds, invesmen houses, and privae equiy funds may 4

provide much needed liquidiy. For example, Alpha Bea Fund Managemen (ABFM), an open-end invesmen company based in Ireland does no inves direcly in physical propery asses bu insead seeks more efficien means for accessing real-esae index reurns. The firm arges pension funds and oher invesors ha may benefi from access o Briish housing by dealing in an over-he-couner propery derivaives marke which racks he Halifax House Price Index (HHPI). The exponenial growh of he subprime morgage marke from 2002 o 2007 was driven by he use of securiizaion as a process of ring-fencing he risks of a collaeral porfolio on one side and he inroducion of he real-esae collaeralized deb obligaion (RE CDO) concep on he oher side. 1 The ever-growing demand for credi risky bonds pushed he boundaries of his new srucured credi marke ino new erriory, residenial morgagebacked securiies (RMBS) backed by home equiy loans (i.e., loans o credi impaired borrowers) and commercial morgage-backed securiies (CMBS). The cash flows of a RMBS, CMBS, and RE CDO depend fundamenally on he performance of a pool of morgage loans, which in urn depend on he behavior of individual homeowners and commercial borrowers. These real-esae srucured producs are radically differen from a corporae CDO where he collaeral consiss of corporae credis whose corporae names could be moniored and heir balance shee scruinized regularly. Hence, he real-esae risk drivers prepaymens and defauls clusering and iming as well as recovery raes, could influence he financial sabiliy of companies and insiuions no direcly relaed o he spo real-esae marke. Hedging he poenial disrupion of scheduled cash flows is no an easy ask and here are only a few insrumens available in he marke. The advances in fuures markes on real-esae indices may improve efficiency in spo markes and improve price discovery. Since ransacion coss are high and creae a barrier for 1 RE CDOs are also known as srucured finance CDOs. 5

enry ino spo markes, fuures markes may also help indicae he level of spo prices for fuure and curren marke volailiy. Anoher benefi of real-esae derivaives is ha hey are useful ools ha allow invesors access o an imporan asse class ha would be hard o access oherwise. Indeed, he capial asse pricing model (CAPM) has as a fundamenal resuls ha invesors should be holding all risky asses in proporion o he amouns ousanding, which suggess ha real esae should be a major elemen in all porfolios involving risky asses. Furhermore, due o he lack of correlaion of housing prices wih equiy prices, expanding diversified porfolios o include real esae could be highly beneficial, paricularly for hose who do no now inves significanly in a diversified porfolio of real esae, including many insurance companies and pension funds. A furher of his benefi is provided by Webb, Curcio, and Rubens (1988) and Seiler, Webb, and Myer (1999). In addiion, Englund, Hwang, and Quigley (2002) poin ou ha here could be large poenial gains from insrumens ha would allow propery holders o hedge heir lumpy invesmens in housing. Obviously, he firs sep in hedging is he selecion of a suiable hedging insrumen. A primary facor in deciding which derivaive conrac will provide he bes hedge is he degree of correlaion beween he facors ha drive he price of he derivaive insrumen under consideraion as he hedging vehicle and he underlying risk which invesors seek o eliminae. Correlaion is no, however, he only consideraion when he hedging program is of significan size. If, for example, an invesor wans o hedge a very large cash posiion, liquidiy becomes an imporan consideraion and i migh be necessary o spli he hedge among wo or more differen ypes of derivaives. Real-esae derivaives are useful o several caegories of end users. The firs caegory consiss of individuals who are propery owners and privae invesors specializing in real esae. Alhough his caegory of users is very large, in pracice, no many may employ 6

propery derivaives due o knowledge and ransacion coss barriers. The second caegory consiss of asse managers hedging heir price risk exposure in boh domesic and foreign real esae. An adjacen caegory includes dealers and porfolio managers in srucured producs seeking o hedge heir posiions. Finally, real-esae derivaives can be embedded by srucurers ino newly designed srucured producs. The risks ha users in hese caegories are hedging wih propery derivaives may vary. For example, while he members of he firs caegory will hedge price risk, he users in he oher caegories may also consider propery derivaives in connecion wih ineres-rae risk and, possibly, currency risk. Propery derivaives can have many applicaions. The mos basic one is for hedging a rade in synheic asses such as forward or oal reurn swaps in order o recoup he poenial loss on primary real-esae asses. Anoher imporan applicaion is porfolio diversificaion or acical allocaion. Consider an invesor or hedge fund seeking o ge exposure o U.K. real esae wihou buying physical propery in he UK because he anicipaed appreciaion may ake a long ime o be realized and carries high ransacion coss. The invesor could simply buy fuures conracs on Eurex. An imporan applicaion ha should no be overlooked is he replicaion of marke bea. Since real-esae markes represen such a high proporion of he world marke, i is inconceivable o achieve marke bea on a porfolio ha does no include real-esae. Hence, for diversificaion purposes, a global asse manager should have a posiion in propery markes and derivaives offer he bes ool o achieve ha a a reduced cos of enry and of exi. Propery funds can adjus secor allocaions and can inves in secors of he marke where hey would no normally operae. The so-called counry swap rades, where an invesor rades in opposie direcions on real esae indices in wo differen counries, may allow he shifing in risk allocaion or aking advanage of srucural breaks of propery index series in one counry while hings remain unchanged in he oher. Invesors may also arbirage pricing inconsisencies beween he propery derivaives and real esae invesmen 7

russ (REIT) markes. Furhermore, i is also possible o have imbalances of supply/demand among propery markes hemselves ha may occur due o various facors, such as invesor senimen driven by news abou possible evens or changes in moneary policy. Pension funds may decide o aler asse allocaion o propery wihou acually rading in he underlying asse (i.e., employing an asse allocaion overlay sraegy wih derivaives). In his paper, we describe he real-esae derivaives available worldwide and discuss he issues relaed o he pricing of hese insrumens and o he managing of hedging insrumens over ime. 2. Financial insrumens relaed o propery risk In his secion, we describe he developmen of real-esae linked derivaives in Europe. The mos developed marke from a financial produc innovaion poin of view is he Unied Kingdom, wih growing aciviy noiced recenly in oher developed European counries oo. The insrumens can be classified by he ype of real-esae risk hey are hedging (1) housing price risk, (2) commercial propery price risk, and (3) morgage loan porfolio amorizing risk. 2.1 Hedging Housing Price Risk A major componen of he real-esae asse class is represened by residenial housing. Housing prices are deermined by macroeconomic and credi marke condiions and by he behaviour of he individuals buying and selling properies. We have seen in recen experience ha here is indeed a risk of sharp downurn or fall in housing prices. In addiion o he owner of he propery, his risk is a major concern o banks and oher lending insiuions and invesors in srucured producs backed by residenial morgage loans. Here we review he financial insrumens ha have been designed o be used for hedging his risk. 8

In 1991 he London Fuures and Opions Exchange (FOX) launched rading in residenial fuures (as well as commercial propery fuures). The conracs were based on a hedonic index of propery prices. However, he markes lased only hree monhs, from May o Ocober, and ended wih a wash rading scandal. 2 In he Unied Kingdom, residenial real-esae derivaives are wrien on a house price index, he mos common being he HHPI series, he longes running monhly housing price series wih daa available since January 1983. This hedonic index is based on he larges monhly sample of morgage daa, ypically covering around 15,000 house purchases per monh. In 2003, Goldman Sachs issued he firs series of a range of covered warrans based on he Halifax All-Houses All-Buyers seasonally-adjused index on he London Sock Exchange (LSE). More recenly, based on he HPPI, in Augus 2007, Morgan Sanley agreed on an exoic swap wih an undisclosed counerpary, worh more han 1 million. This is he UK s firs residenial propery derivaive rade ha included an embedded exoic opion, a knock-in pu opion. This derivaive allows he counerpary o gain if he HHPI rises subjec o a maximum payou. The invesor s capial is proeced unless he HHPI falls below an iniially specified value. 2.2 Hedging Commercial Propery Price Risk The oher major componen of he real-esae asse class is represened by commercial properies. Price risk for commercial properies is similar o housing price risk bu i is no generally influenced by he behaviour of he same paricipans as in he residenial housing marke. Commercial propery prices are deermined by supply and demand and specialized marke paricipans. Alhough some degree of correlaion is expeced beween commercial 2 We omi here discussion of spread-being markes for home prices, such as have been offered in he UK by CiyIndex.co.uk, IGIndex.co.uk and Spreadfair.co.uk, because hese did no appear o be used generally for risk managemen and all of hese markes have been shu down as of 2008 9

price risk and housing price risk, here are sufficien differences beween hese wo risk o warran a separae analysis and separae hedging insrumens. The major underlying capuring he dynamics of he commercial real-esae in he Unied Kingdom and coninenal Europe is he family of Invesmen Propery Daabank (IPD) indices. The IPD UK Monhly index is based on open marke appraised valuaions compiled from 74 monhly valued funds (around 3,700 properies) worh around 56 billion. I includes commercial and oher invesmen properies represening more han 90% of he combined value of he propery asses held in UK uni russ and oher uni linked propery invesmen funds. Derivaives are issued coningen on he IPD UK Annual index which has a slighly differen calculaion mehodology. I is envisaged in he fuure ha derivaives on IPD will be issued on a quarerly IPD index. The index has been franchised on a large scale in Europe. Afer he failure in 1991 of he London Fox commercial propery fuures, aciviy moved o he over-he-couner (OTC) marke. The insrumen ha survived was he propery income cerificae. Pael (1994) suggess ha he failure of he FOX propery fuures conrac was aribuable o he problems in he consrucion of he index ha caused lag dependence, and o a lesser exen he cash-marke illiquidiy ha led o considerable ime-basis risk. Jus hree years afer he failure of he FOX conracs, Barclays began issuing is Propery Index- Cerificaes in 1994 and is Propery Index-Forwards in 1996, wih he index comprising UK commercial propery only. The conracs are currenly raded in he OTC marke and are issued on wo monhly indices published by IPD: he IPD Toal-Reurn and IPD Capial- Growh indices. These conracs have mauriies of hree o four years. The propery derivaives marke has expanded in he Unied Kingdom from 850 million in 2005, 3.9 billion in 2006 o 7.2 billion in 2007. The IPD Index Propery Derivaives volume in 2008 was 7.73 billion, no far from 8.30 billion in 2007 s record 10

year. In Europe, in 2007 he rading volume on he IPD French All Office Index was 787 million, compared o 283 million on he German All Propery Index. The firs French propery swap on he IPD France Offices Annual index was raded in December 2006 by Merrill Lynch and AXA Real Esae Invesmen Managers. The firs opion on an IPD index ouside he Unied Kingdom was referenced o he German IPD/ DIX Index and i was raded in January 2007 wih Goldman Sachs acing as a broker. Oher European markes where propery derivaives have been raded include Swizerland and Ialy. Thireen major invesmen banks acquired licenses for he IPD index family, alhough some of hese banks wen ino bankrupcy or have been acquired by oher banks. In Europe, propery derivaives wrien on IPD indices raded OTC wih a noional over 18.8 billion in rades execued by Ocober 2008. Eurex began rading propery fuures on February 9, 2009. These fuures conracs are annual conracs based on he oal reurns of he IPD UK Annual All Propery Index 3 for individual calendar years. Wih he inroducion of his fuures conrac, he Eurex seeks o eliminae counerpary risk, o improve liquidiy o he commercial propery secor of he real-esae propery marke, and o arac a complee range of poenial paricipans in his asse class. Addiional fuures conracs are going o be launched by Eurex on IPD propery indices, such as UK secor indices (Offices, Reail, Indusrial) and oher European indices (iniially France and Germany) on a demand-led basis. Proposals for improved commercial real esae indices and insrumens based on hem will find heir way ino he marke. For example, Horrigan e.al (2009) demonsrae how o consruc segmen-specific indices of propery marke reurns from REIT reurn daa, bond daa, and propery holding daa, as well as how o use hose indices o make pure, argeed 3 A he end of 2007, he IPD index covered 12,234 properies wih a oal value of 184 billion - equivalen o 49% of he UK invesmen marke. The IPD UK Annual Propery Index measures unleveraged oal reurns o direc UK propery invesmens and i is calculaed using a ime-weighed mehodology wih reurns compued monhly and hereafer compounded for he purposes of he annual index consrucion. 11

invesmens in he commercial real esae marke while reaining he liquidiy benefis of he well-developed public marke in REITs. I is possible o reconsruc he indices a daily frequency wihou significan increases in noise and a various levels of segmen granulariy. Moreover, i seems ha hese new indices suggesed by he auhors lead ransacions-based direc propery marke indices during marke urns. Thus, hese REIT-based commercial propery reurn indices have he poenial o be used o develop hedging sraegies in he realesae marke and suppor derivaives rading. In he commercial morgage-backed securiies (CMBS) marke several dealers have offered swaps on various CMBS indices and heir subsecors. Goodman and Fabozzi (2005) provide insigh ino he mechanics and he economic raionale of he oal reurn swaps issued on hese indices. They also explain he aracion o hose who wan o ge exposure o a CMBS in a synheic way. 2.3 Morgage loan porfolio amorizing risk A he porfolio level, morgage loans carry wo inrinsic risks ha require hedging ools: defaul risk and prepaymen risk. Defaul risk is he risk of loss of principal and/or ineres due o he failure of he borrower o saisfy he erms of he lending agreemen. This risk is high for RMBS ha are backed by morgage pools conaining subprime morgages (or non-conforming morgages 4 as hey are called in he U.K. marke). In he Unied Kingdom subprime morgages are loans made o borrowers who are viewed as having greaer credi risk borrowers who do no have a credi hisory and borrowers having a hisory of failing o mee heir obligaions. 4 The classificaion of loans in Europe by borrower credi is differen from ha in he Unied Saes morgage marke where loans are classified as prime and subprime loans. Prime loans are hen classified as hose ha are conforming and hose ha are non-conforming. By non-conforming i is mean ha he borrower or he propery fails o mee he underwriing sandards of Ginnie Mae, Fannie Mae, or Freddie Mac. Thus, in he Unied Saes, a non-conforming morgage can be a prime or subprime morgage. 12

A prepaymen is he amoun of principal repaymen ha is in excess of he regularly scheduled repaymen due. A prepaymen can be for he enire amoun of he remaining principal balance (i.e., complee payoff of he loan) or for only par of he ousanding morgage balance (referred o as a curailmen). Prepaymen risk is greaer for RMBS han CMBS because commercial loans have provisions o miigae prepaymen risk boh a he loan level due o lockou periods, prepaymen penalies, yield mainenance, and defeasance, and a he srucure level.. From an invesor s perspecive, prepaymen risk is he risk ha borrowers will prepay heir loan (in whole or in par) when ineres raes decline. This acion by borrowers would force invesors o reinves a lower ineres raes. Noe ha if borrowers prepay when ineres raes rise, prepaymens are beneficial o invesors because proceeds received can be reinvesed a a higher ineres rae. From he perspecive of a porfolio manager or risk manager who is seeking o hedge an RMBS posiion agains ineres rae risk, prepaymen risk exiss even if prepaymens occur when ineres raes rise. This is because in esablishing a hedge, he amoun o be hedged will vary depending on a projeced prepaymen rae. A he ouse of a hedge, an amorizaion schedule is projeced based on he projeced prepaymen rae. Acual prepaymen experience of a hedged asse or can cause a deviaion beween he projeced principal ousanding based on he amorizaion schedule designed a he ouse for he hedging insrumen and he acual principal. This can resul in over or under hedging a posiion. This sochasic naure of he amorizaion scheduled due o sochasic prepaymens ha hedgers face migh more aply be described as amorizaion risk. In Europe, dealers holding invenory for whole loans waiing o be securiized or securiized producs held in invenory, as well as asse managers, are exposed o ineres rae risk linked o he amorizaion of morgage loan porfolio dynamics. The amorizaion is sochasic and is driven mainly by prepaymen and defaul speeds. As a response o his ype 13

of hedging problem, a se of financial insrumens linked indirecly o real-esae risk drivers appeared in he marke: srucured swaps. These swaps have been used by building socieies and asse-backed securiies desks in major invesmen banks in he Unied Kingdom and Europe, o hedge ineres rae exposure on morgage loan porfolios. INSERT FIGURE 1 HERE In addiion o he propery derivaives discussed earlier, in pracice here are hree ypes of srucured swaps used by paricipans in he real-esae marke in relaion o securiisaion: (1) a balance guaraneed swap, (2) a cross-currency balance guaraneed swap, and (3) a balance guaraneed LIBOR-base rae Balance Guaraneed Swaps There are many problems relaed o he design of all hree ypes of srucured swaps. One problem is ha he collaeral coupon leg is paid in arrears since morgage paymens are colleced every day in he period. The enor may also differ from deal o deal, he monhly one being more common because his is also he frequency of morgage paymens and consequenly he calculaion of defauls and prepaymens. Quarerly is also used in he marke on occasion. Since he usual reference floaing rae is hree-monh LIBOR, here is a basis beween he reference hree-monh LIBOR colleced monhly from he swap and he same reference hree-monh LIBOR paid quarerly o he noe holders ha funded he securiisaion. Swiching o a reference rae of one-monh LIBOR does no solve he problem of he basis. Alhough he name balance guaraneed swap or LIBOR-base rae guaraneed swap may allude o some form of locking in he level of he oal balance for he referencing 14

morgage loan porfolio, he srucured swaps do no hedge prepaymen risk or defaul risk because i does no guaranee he balance: on he conrary i is exposed o i. The wrier is usually a bank ha held he loans in a warehouse before securiising hem and herefore i is well aware of he qualiy of he collaeral porfolio. Moreover, here are many insances when a bank may underwrie a balance guaraneed swap based on he informaion provided on he porfolio by a hird pary. Srucured swaps became par of he securiizaion process because heir use was mandaed by raing agencies in order o hedge he ineres rae risk mismach beween fixed coupons collecable from he pool of morgage loans and he floaing coupon paymens ha have o be paid pos securiizaion o invesors. Given he large volume of RMBS deals beween 2005 and 2007, one can imagine ha here are a good number of insiuions holding many posiions in his ype of swap. In a balance guaraneed swap he paries o he conrac exchange he colleced coupons on a collaeral porfolio of morgage loans for a reference LIBOR plus a spread. The noional is he oal noional of surviving loans for he period. I is a complex swap where he noional is sochasic, being deermined by he prepaymens, defauls, and arrears in he referencing porfolio. The oal coupon paid on an exchange dae is also sochasic for wo reasons: (1) he noional on which i is calculaed evolves in an uncerain manner and (2) even if he noional is known, because of he differen possible mixure of loans surviving in he reference porfolio, he coupon could differ for he same noional. Cross-Currency Balance Guaraneed Swaps A cross-currency balance guaraneed swap is a more complex swap produc ha deals wih cross-currency deals whereby he coupons on one leg of he swap and also he noional are deermined in a foreign currency. This swap deals wih an exra level of risk hrough is 15

foreign-exchange exposure. All characerisics described above for a balance guaranee swap apply bu he noional is sochasic and in a differen currency. One subley wih his produc is ha i also has embedded some macroeconomic risk of he counry where he obligors reside. This risk may resurface even if here are no changes in he currencies specified in he swap. For example, he ebbs and flows of he poliical and social environmen in he borrowers counry may cause job losses or price inflaion ha may, in urn, rigger high defaul raes in he collaeral porfolio. Balance Guaraneed LIBOR-Base Rae Swaps To undersand our nex swap a balance guaraneed LIBOR-base rae swap we need o review he linkages beween he official base rae and he sandard variable rae charged on morgage accouns by lending banks in he Unied Kingdom. The official base rae is he rae a which he Bank of England lends o oher financial insiuions. The Bank of England s Moneary Policy Commiee mees every monh o deermine wha needs o be done in response o economic condiions. While he official base rae plays a very imporan role in hese markes, banks have he freedom o use a differen rae on heir loans, which is he sandard variable rae. The sandard variable rae is linked direcly o he official bank rae, bu i is usually a lile higher, reflecing facors such as he real rae a which banks borrow from each oher, he business coss associaed wih lending operaions, he volume and mauriy profile of loans, and he funding arrangemens. The sandard variable rae is likely o vary from bank o bank and is used o deermine he cash flows linked o RMBS. 5 While a balance guaraneed swap is useful for convering fixed-rae coupons ino LIBOR- 5 In he Unied Saes here are variable rae morgages ha lend hemselves o hedging in he same way. 16

based coupons, here are siuaions when he collaeral pool for RMBS and CMBS, will conain loans ha pay variable coupons deermined by he variable rae esablished by he lender. The level of he variable rae is deermined by he lending bank in relaion o boh he base rae deermined by he naional or federal banks and oher funding coss driven by LIBOR-swap raes. In general, he basis beween he sandard variable rae (or is proxy he official base rae) and LIBOR is almos consan and no very large during periods of low ineres rae volailiy. However, in urbulen periods, such as he subprime crisis ha began in he summer of 2007, he basis beween he sandard variable rae and LIBOR may increase dramaically. Morgage raders herefore employ our hird swap, a balance guaraneed LIBOR-base rae swap, o hedge his basis risk. One leg of he swap pays LIBOR plus a spread while he oher leg pays he average official base rae during he period. For his ype of swap, he LIBOR leg pays he coupon fixed a he beginning of he period bu he official base rae leg paymen is fixed a he end of he period. The conrac allows he noional o be any value wihin a pre-specified band. The pary paying he average base rae has he righ o choose he size of he noional a he beginning of each period. This is a difficul economic decision o make because only he LIBOR is known a he beginning of each period. 3. Mehodological issues relaed o real esae derivaives 3.1 The Incompleeness of Propery Markes Derivaives require homogeneiy of he underlying for esablishing liquidiy in heir rading. The lack of homogeneiy in real-esae markes was one of he main obsacles o he developmen of propery derivaives. The securiisaion mechanism, in conras, has brough a wider paricipaion in hese markes and an increased demand for hedging ools. 17

I has been emphasized by Case and Shiller (1989, 1990) ha he housing marke in general is inefficien due o serial correlaion and ineria in housing prices, as well as in he excess reurns. A possible explanaion has been offered by Case, Quigley, and Shiller (2004) who argue ha he price expecaion of he majoriy of marke paricipans is backward looking. The real-esae lieraure suggess ha repea-sales indices have hree main characerisics: (1) hey are no subjec o he noise caused by a change in he mix of sales, (2) hey are highly auocorrelaed, and (3) hey are predicable wih a forecas R-squared roughly 50% a a one-year horizon. The hedonic indices are subjec o model risk semming from he mulivariae regressions used o consruc hose indices. The facors employed in hedonic regressions are hose characerisics of properies ha have been hisorically found o explain housing prices. The mehods associaed wih hedonic indices inheri all he common problems known for regression analysis, namely spurious regression, muli-collineariy, and model selecion. When pricing or hedging various insrumens, i should be remembered ha shor sales of real-esae properies is impossible and rading may no be feasible for fracional unis. These unique characerisics of real-esae properies have profound implicaions regarding valuaion principles. They srongly violae he assumpions of he Black-Scholes framework, ha insananeous rades in and ou are no only feasible bu also cosless.. Moreover, he predicabiliy in housing prices makes i difficul o esablish hedging procedures. However, one may argue ha a well-esablished fuures marke may feed informaion ino curren prices and some balance can be achieved wih a dual informaion ransmission mechanism. The finance lieraure dealing wih real-esae pricing models can be classified ino wo main frameworks: equilibrium models and no-arbirage marke models. Gelner and Fisher (2007) is he main reference for he former caegory. For he laer, he earlies 18

reference is Timan and Torous (1989) who considered a credi-risk ype of model for pricing commercial morgages. Buimer, Kau, and Slawson (1997) were he firs o propose pricing real-esae derivaives such as oal reurn swaps under a Black-Scholes framework. Their modelling approach was revisied by Bjork and Clapham (2002) who focus on some imporan heoreical issues relaed o no-arbirage models. They proved ha he spread of a oal-reurn swap should be zero and poin o some approximaion errors inroduced by Buimer e.al (1997). Pael and Pereira (2006) exended he no-arbirage models o include counerpary risk. The value of a real-esae oal reurn swap is no longer zero when counerpary risk is aken ino accoun. The spread over LIBOR is highly dependen on he volailiy of index reurns and on counerpary defaul risk. The Black-Scholes risk-neural mehodology was applied by Ciurlia and Gheno (2008) for pricing real-esae derivaives such as European and American opions on real-esae asses wih a wo-facor model. Syz (2008) gives an overview of he markes and discusses pricing of forwards in he Black- Scholes framework idenifying he deficiencies bu no providing a soluion. All he no-arbirage models jus menioned were formulaed wihin he conex of complee markes. However, real-esae derivaives are very much incomplee markes. Oaka and Kawaguchi (2002) developed a model for pricing commercial real-esae under incomplee markes ha consiss of (1) a securiy marke where socks, bonds, currencies, and derivaive securiies are raded wihou fricion, (2) a space marke wih he rens of buildings, and (3) a propery marke where he prices of real properies are deermined. More recenly, Baran e al. (2008) used he Schwarz and Smih (2000) model for pricing real-esae derivaives. Pricing is considered under a maringale measure and he main innovaion of heir paper is relaed o he consrained maximum likelihood mehod combined wih Kalman filer applied for parameer esimaion. 19

The no-arbirage argumen is difficul o apply because of he impossibiliy of shor selling and he non-homogeneiy of he propery as an asse. Moreover, he underlying index is only an observable variable and no an asse ha could be raded freely wihou any fricions, alhough real esae porfolios mimicking exacly he index are a possibiliy for a disinc class of invesors. The new generaion of producs dependen on real-esae risks canno be priced in a risk-neural framework eiher. This is due o he relaionship beween ineres rae risk along he erm srucure of ineres raes and he risk riggers in RMBS space such as defauls, prepaymens, and arrears. Hence, a real-world measure approach based on saisical calibraion of hisorical daa is required for pricing real-esae derivaives. A new mehod focused on he marke price of risk as a modelling ool is described by Fabozzi and Tunaru (2009). Their models ake ino consideraion he mean-reversion o a nonlinear long-run rend of real-esae indices. Propery derivaives markes are incomplee because he primary asse underpinning his marke suffers from lack of homogeneiy. Black (1986) emphasizes ha for he smooh funcionaliy of derivaives, a homogeneous underlying asse is helpful. However, similar properies in close geographical proximiy should have similar prices. This high correlaion is helpful for cross-hedging spo real-esae porfolios wih fuures on local indices. Neverheless, he lack of homogeneiy implies ha hedging wih real-esae derivaives is always going o be less han perfec. Baum (1991) remarks ha managers of real-esae porfolios wih asses worh individually a large amoun of money have exra exposure o idiosyncraic risk ha canno be hedged away easily wih propery derivaives. An imporan characerisic o accoun for when modelling an underlying real-esae index is reversion o a long-run rend. Similar o commodiy markes where he underlying asse is a consumpion asse, he supply and demand forces on real-esae markes drive realesae prices in a differen way han a sock index or bond index. 20

Le 0 X be a sochasic process represening a real-esae index. Then consider he process on he log-scale Y ln( X ). There is a clear linear ime rend for he IPD UK Monhly index 6 since 1986 as illusraed in Figure 2 using a log-scale and for he Halifax House Price Index in Figure 3 based on a quarerly hisorical series. One could argue ha he flucuaions of he observed values of boh indices, on he log-scale, are wrapped around a posiive linear ime rend. A ime-rended mean-revering Ornsein Uhlenbeck process could be appropriae o describe his phenomenon and any model for pricing real-esae derivaives on hese indices should accoun for his saisical propery, hough care should be aken abou he possibiliy of spurious rend esimaion. The failure of an Ornsein Uhlenbeck process o capure shor-erm momenum may also be a problem.. Bu as a firs approximaion i may do, and is in line wih he framework proposed by Lo and Wang (2005) for pricing derivaives on asses ha are predicable. INSERT FIGURE 2 HERE INSERT FIGURE 3 HERE Therefore, we can assume ha 0 Y is a process ha is mean revering owards a deerminisic linear rend. In oher words, he log-index is he sum of a zero-mean saionary auoregressive Gaussian process and a deerminisic linear rend. One could also fi nonlinear rends if necessary. I is possible hen o calculae he soluion of his equaion in closed form which can be useful for pricing derivaives. Hence, a real-esae index can be modelled wih a geomeric Ornsein-Uhlenbeck process ha always ensures posiive levels. Because he realesae marke is incomplee, for pricing purposes one needs o specify exogenously he marke price of risk.. 6 The IPD Monhly Index is based on an open marke appraised valuaions of real buildings and i covered 3,700 properies worh around 47 billion in Sepember 2006. I includes commercial and oher invesmen properies represening more han 90% of he combined value of he propery asses held in UK uni russ and oher uni linked propery invesmen funds. 21

I is criical o be able o model real-esae indices as close as possible o real-world marke condiions since many morgage-relaed securiies are marked o model in he absence of a liquid marke. A large bias in forecasing fuure levels of a real-esae index will be refleced, say, in marking he profi and loss posiion of a real-esae posiion, and his could be exremely derimenal o banks holding posiions in hese securiies. 3.2 Risk Managemen Problems of Srucured Swaps Since i is impossible o deermine a priori he fuure noional and coupons ha will be paid, i is very difficul o price a balance guaranee swap wih risk-neural or maringale pricing echniques. Moreover, as he subprime-liquidiy crisis has shown, i may be impossible o use he marke o rerieve informaion abou key facors affecing valuaion. In oher words, i may be difficul o mark real-esae posiions o marke and, as a resul, banks and asse managers may need o resor o in-house valuaions or marking o model. The only soluion is o look a fundamenals, employ hisorical daa, and price cash flows wihin a riskadjused framework by aggregaing across a large se of simulaed scenarios. This approach is rooed in acuarial science and i is appropriae in his conex because he noional in hese conracs is driven by defaul risk and prepaymen risk. These risks canno be deermined wihin his marke or by real-esae marke paricipans. The hree ypes of swap described in Secion 2.3.,_depending on a sochasic noional, canno be priced wih echniques applied o oher ineres and credi derivaives. A saisical modelling approach ha calibraes daa available and ha would allow a Mone Carlo exercise for he deerminaion of he probabiliy disribuion of defauls and prepaymens is necessary. I is far more imporan, as discussed in Pavlov (2001), o capure he sensiiviies of defaul and prepaymen raes o he inrinsic characerisics of he collaeral porfolio han being worried abou riskless arbirage siuaions ha are difficul, alhough no impossible, o 22

appear in his conex. Advances in ineres rae derivaives modelling can sill play a role since he fuure levels of ineres raes can be used as an inpu in hese saisical regression models for defaul and prepaymen raes. The sandard mehod for pricing a balance guaranee swap is o consider a region ha will cover he realised amorising balance of he porfolio of loans. Figure 4 describes his echnique. The prepaymen rae used in he consrucion of he figure is he condiional prepaymen rae (CPR), he measure mos commonly used for RMBS. The upper curve is associaed wih a low CPR while he lower curve is deermined from a high CPR. While he upper curve canno be aken beyond he 0% CPR, he lower curve choice is deermined by he risk appeie of he underwrier of he balance guaranee swap. Hence, he shaded region in Figure 4 represens he risk aken by he seller of his produc. If he realised curve moves below he lower curve, hen he mechanism of he pricing ha is described nex does no funcion. Inser Figure 4 Here The seller will price he balance guaraneed swap wih a porfolio of an amorising swap wih noional following he upper CPR curve and a series of Bermudan receiver swapions wih noionals deermined by he difference beween he upper curve and lower curve. The amorising swap will pay a fixed coupon pre-specified by he seller, ideally as close as possible o he oal fixed coupons colleced from he porfolio of morgage loans under he low CPR scenario. The swapions will have an exercise price equal o he difference beween he fixed leg raes of he amorising swap and he price of he amorising swap. The frequency and mauriies of he swapions are in correspondence wih he amorising swap. The role of he swapions is o allow he compression of he noional of 23

his hedging porfolio when he balance of he collaeral porfolio moves below he upper curve and LIBOR decreases. In order o calculae he price of he balance guaranee swap, hree elemens are considered: (1) he price of he amorising swap (i.e., he spread added o LIBOR o bring he swap on par), (2) he premia of he series of receiver swapions, and (3) he reserve amoun o compensae for unhedged risk. The hird elemen is he shaded region in Figure 4. If he buyer does no wan o face any upfron coss, hen he firs wo elemens should be ransformed ino an equivalen running rae. Neverheless, here is sill a difficul problem he seller of he balance guaraneed swap faces, namely he economic decision when o exercise he swapions. Exercising any swapions leads o auomaic enry ino swaps receiving fixed and paying LIBOR and his process is irreversible. If a a laer dae he balance of he collaeral morgage loans jumps back again, hen here is a danger of possible losses due o periods of low levels of LIBOR. The decision o exercise he swapions is no easy because LIBOR can easily flucuae from one period o anoher. This replicaing sraegy requires acive managemen and excellen economic forecasing skills of ineres rae markes. The srucure of paymens over one period, usually one monh, is described in Table 1. Inser Table 1 here. The major problem in pricing and hedging a balance guaranee swap is he difference beween N, c he sochasic noional (balance) for he balance guaraneed swap and he fixed, and unknown a ime zero, coupons ha are colleced over he period on one side and U,, he amorising noional given by he upper curve of low CPR scenario and corresponding fixed coupons on he oher side. The price of his swap is given by he spread rae y in his srucure and i may no coincide wih he spread from a par calculaion on 24

he hedging amorising swap. A lower y may be jusified o cover he premia on ineres rae derivaives upfron coss and he unhedged risk area highlighed in Figure 4. If L is he LIBOR fixed a ime and w represens he weigh ha will make he noional of he receiver swapions equal o he difference beween he upper and lower curve hen he ne payoff o he seller of a balance guaraneed swap a he end of period [, +1], is N ( c L y) U (1 w )( L ) (1) There is no guaranee ha his payoff will be posiive all he ime in all marke scenarios. Here we propose a differen pricing mechanism ha will circumven he swapions exercising problem. We propose hedging he risk posed by selling a balance guaraneed swap wih a porfolio of an amorising swap off he upper curve as before and a series of amorising floors wih noional deermined again by he difference beween he upper and lower curve. The exercise price of he floors will be he same as he exercise price of he swapions so boh ses of insrumens have he same payoff when in he money. However, he floors do no have any downside so he seller will no have any ouflow associaed wih he floors. For his reason, he floors will always be more expensive han he swapions bu he seller also has wo major advanages. Firs, here is no economic decision o be made over he life of he swap. If an opion on he floor (i.e., a floorle) is in he money, he opions are exercised and he same economic oucomes relaed o he managing of he balance guaranee swap are realised for ha period. Second, even if hey are no needed for noional compression, he floors will always be exercised if hey are in he money. The addiional cash flow will offse some of he increased premia paid for he floors when he swap is sold. Anoher possibiliy would be o replicae a balance guaraneed swap wih an amorising swap off he lower curve and a porfolio of caps wih exercise price equal o he difference beween he fixed leg raes of he amorising swap and he price of he amorising swap. The caps will help o bridge he gap beween LIBOR and fixed coupon paymens. 25

There are similar advanages in erms of managing he hedging porfolio. Table 2 illusraes he flow of paymens under his hedging sraegy. The ne paymen o he seller of a balance guaraneed swap is N ( c L y) U ( L ) wu max( L,0) (2) This shows ha he possible negaive paymens faced by he underwrier of a balance guaraneed swap are confined o he region where he balance of he collaeral porfolio is ouside he chosen CPR region. Inser Table 2 Here The LIBOR-base swaps are difficul o price mainly because here is no model available for an ineres rae variable such as he Bank of England base rae (BBR). The hisorical daa for he BBR indicae ha for long periods his rae is consan and he dynamics consiss only for up or down jumps given by he decisions of he economic moneary commiee. While he LIBOR leg of he swap could be priced as usual off he forward curve, he BBR leg of he swap is difficul o price. For his insrumen here is an even more sringen need for a good macroeconomic forecasing model ha will help generae projecions of he BBR under various fuure economic scenarios. The sub-prime crisis clearly demonsraed ha i is difficul o forecas he iming and size of he reducion in BBR. While a LIBOR-based swap is wihou any doub very useful, he pricing needs once again needs o be done looking a fundamenal macroeconomic analysis, no-arbirage principle being of lile help in his case. 4. Conclusions 26

Real esae markes represen a very large proporion of he oal wealh in developed counries. Ye, he risk managemen ools available for hedging real-esae risk are very much in heir infancy and have problems ranging from illiquidiy of rading o lack of heoreical developmen in erms of modelling. In his paper, we surveyed he se of financial insrumens available for he managemen of risks arising in European propery markes. Given he incomplee characer of real-esae markes, we advocae a suiable pricing and hedging framework. A special place is dedicaed o morgage-backed securiies which require srucured swaps capable of handling he embedded prepaymen risk and defaul risk. We discuss some of he risk managemen problems faced by asse managers of RMBS deals and we sugges some soluions. References Baran, L.C., Buimer, R.J., and Clark, S.P., Calibraion of a Commodiy Price Model wih Unobserved Facors: The Case of Real Esae Index Fuures, Review of Fuures Markes, Vol. 16, 2008, pp. 455-469. Baum, A., Propery Fuures, Journal of Propery Valuaion and Invesmen, Vol. 1, 1991, pp. 235-240. Black, D., Success and Failure of Fuures Conracs: Theory and Empirical Evidence, Monograph Series in Finance and Economics, Monograph 1986-1. New York Universiy, 1986. Bjork, T., and Clapham, E., On he Pricing of Real Esae Index Linked Swaps, Journal of Housing Economics, Vol. 11, 2002, pp. 418-432. Buimer, R.J. Jr., Kau, J.B., and Slawson, C.V., A Model for Pricing Securiies Dependen upon a Real Esae Index, Journal of Housing Economics, Vol. 6, 1997, pp. 16-30. 27

Carlon, D. W., Fuures Markes: Their Purpose, Their Hisory, Their Growh, Their Success and Failures Journal of Fuures Markes, Vol. 4, 1984, pp. 237-271. Case, K. E., Quigley, J.M. and Shiller, R.J., Home-Buyers, Housing, and he Macroeconomy. In Anhony Richards and Tim Robinson (Eds.), Asse Prices and Moneary Policy, (Reserve Bank of Ausralia, 2004), pp. 149-188. Case, K. E., and Shiller, R.J., The Efficiency of he Marke for Single Family Homes, American Economic Review, Vol. 79, 1989, pp. 125-37. Case K. E. and Shiller, R.J., Forecasing Prices and Excess Reurns in he Housing Marke, AREUEA Journal, Vol. 18, 1990, pp. 253-273. Case, K. E. and Shiller, R.J., Morgage Defaul Risk and Real Esae Prices: The Use of Index Based Fuures and Opions in Real Esae, Journal of Housing Research, Vol. 7, 1996, pp. 243-58. Case, K. E., Shiller, R.J. and Weiss, A.N., Index-Based Fuures and Opions Trading in Real Esae, Journal of Porfolio Managemen, Vol. 19, 1993, pp. 83-92. Ciurlia, P. and Gheno, A., A Model for Pricing Real Esae Derivaives wih Sochasic Ineres Raes, Working Paper, (Munich Personal RePEc Archive, No. 9924, 2008). Clayon, J. Commercial Real Esae Derivaives: They re Here Well, Almos. PREA Quarerly, Winer 2007), pp. 68-71. Englund, P., Hwang, M. and Quigley, J. Hedging Housing Risk, Journal of Real Esae Finance and Economics, Vol. 24, 2002, pp. 167-200. Fabozzi, F. and Tunaru, R., Pricing Models for Real-Esae Derivaives, Working Paper, (Yale School of Managemen, Inernaional Cener for Finance, 2009). Fisher, J. D., New Sraegies for Commercial Real Esae Invesmen and Risk Managemen, Journal of Porfolio Managemen, Vol. 32, 2005, pp. 154-161. Hinkelman, C. and Swidler, S., Trading House Price Risk wih Exising Fuures Conracs. Journal of Real Esae Finance and Economics, Vol. 36, 2008, 37-52. Horrigan, H. T., Case, B., Gelner, D., and Pollakowski, H.O., REIT-Based Commercial Propery Reurn Indices: A Model o Suppor and Improve Segmen-Specific Invesmen in he Real Esae Markes., Working Paper, (MIT Working Paper, 2009). Gelner, D. and Fisher, J., Pricing and Index Consideraions in Commercial Real Esae Derivaives, Journal of Porfolio Managemen, Special Real Esae Issue, 2007, pp. 99-117. Goodman, L. and Fabozzi, F., CMBS Toal Reurn Swaps, Journal of Porfolio Managemen, Special Issue, 2005, pp. 162-167 Lo, A. and Wang, J., Implemening Opion Pricing Models When Asse Reurns Are Predicable, Journal of Finance, Vol. 50, 2005, pp. 87-129. 28

Oaka, M. and Kawaguchi, Y., Hedging and Pricing of Real Esae Securiies under Marke Incompleeness, Working Paper, MTB Invesmen Technology Insiue Co., Ld., Tokyo 2002. Pael, K. and Pereira, R., Pricing Propery Index Linked Swaps wih Counerpary Defaul Risk, Journal of Real Esae Finance and Economics,Vol. 36, 2008, pp. 5-21. Pavlov, A. D., Compeing Risks of Morgage Terminaion: Who Refinances, Who Moves, and Who Defauls?, Journal of Real Esae Finance and Economics, Vol. 23, 2001, pp. 185-211. Schwarz, E. and Smih, J.E., Shor-Term Variaions and Long-Term Dynamics in Commodiy Prices, Managemen Science, Vol. 46, 2000, pp. 893-911. Seiler, M.J., Webb, J.R., and Myer, N.F.C. Diversificaion Issues in Real Esae Invesmen, Journal of Real Esae Lieraure, Vol. 7, 1999, pp. 163-179. Shiller, R. J. and Weiss. A.N., Home Equiy Insurance, Journal of Real Esae Finance and Economics, Vol. 19, 1999, pp. 21-47. Shiller, R.J., Derivaives Markes for Home Prices, Yale Economics Deparmen Working Paper No. 46, (Cowles Foundaion Discussion Paper No. 1648, 2008). Shiller, R.J., Measuring Asse Value for Cash Selemen in Derivaive Markes: Hedonic Repeaed Measures Indices and Perpeual Fuures, Journal of Finance, Vol. 68, 1993, pp. 911-931. Syz. Juerg M. Propery Derivaives. (Wiley: Chicheser, 2008). Timan, S. and Torous, W. Valuing Commercial Morgages: An Empirical Invesigaion of he Coningen-Claim Approach o Pricing Risky Deb, Journal of Finance, Vol. 44, 1989, pp. 345 373. Webb, J.R., Curcio, R.J., and Rubens, J.H., Diversificaion Gains from Including Real Esae in Mixed-Asse Porfolios, Decision Sciences, Vol. 19, 1988, pp. 434-452. 29

Pool of morgages SVR coupons Bank owning loans Fixed coupons Fixed or SVR coupons LIBOR + spread Ineres rae swap desk Libor + spread Porfolio of ineres rae derivaives Ineres rae derivaives desk (swapions, caps, floors) Invesors Figure 1. The main componens affecing a balance guaraneed swap linked o a porfolio of morgage loans. The SVR is he sandard variable rae ha banks charge heir morgage borrowers when heir morgage swiches from fixed he floaing. I is driven by he official base rae bu is exac value is a he discreion of he bank. 30

Dec-86 Dec-87 Dec-88 Dec-89 Dec-90 Dec-91 Dec-92 Dec-93 Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Index IPD UK All Propery Index on Log scale 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 Figure 2. Hisorical monhly daa for he IPD UK Monhly index on he log scale. The series displays a linear ime rend. 31

1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 Index Halifax House Price Index on Log scale 7 6.5 6 5.5 5 4.5 4 3.5 3 Figure 3. Hisorical quarerly daa of he Halifax House Price Index (HHPI) on he log scale. 32

Balance Realised Curve High CPR Curve Underwriing Risk Low CPR Curve 0 1 2 3 n N Time in monhs Figure 4. The curves aken ino consideraion for pricing and managing a balance guaraneed swap. All hree curves (high CPR curve, realised curve, and low CPR curve) refer o oal balance of a porfolio of morgage loans ha are he collaeral in a ransacion. 33

Table 1. Cash flows when srucuring a balance guaraneed swap wih amorising swapions The srucure of payoffs faced by he seller of a balance guaraneed swap and is counerparies. For all paymens, is he day ime accrual for he period [,+1]; N, c are he sochasic noional (balance) for he balance guaraneed swap and he fixed, and unknown a ime zero, coupons ha are colleced over he period; L is he LIBOR fixed a ime ; U, are he amorising noional given by he upper curve of he low CPR scenario and corresponding fixed coupons; is he amorising swap price; w represens he weighs ha will make he noional of he receiver swapions equal o he difference beween he upper and lower curve. The price of he balance guaranee swap is given by spread rae y and his is he spread ha is offered by he seller and acceped by he buyer, i is no a spread resuling from a par swap calculaion. Payoff a ime +1 Produc Leg Buyer BGS Seller BGS Hedges Ouside Balance guaraneed Fixed Nc Nc - swap Libor N L y) N ( L y) - ( Amorising swap Fixed - U U Libor - U ( L ) U ( L ) Swap from swapion Fixed - wu wu Libor - wu ( L ) wu ( L ) 34

Table 2. Cash flows when srucuring a balance guaraneed swap wih amorising floors The srucure of payoffs faced by he seller of a balance guaraneed swap and is counerparies when hedging wih floors. For all paymens. is he day ime accrual for he period [,+1]; N, are he sochasic noional (balance) for he balance guaraneed swap and he fixed, and c unknown a ime zero, coupons ha are colleced over he period; L is he LIBOR fixed a ime ;, are he amorising noional given by he upper curve of low CPR scenario and U corresponding fixed coupons; is he amorising swap price; w represens he weighs ha will make he noional of he floors equal o he difference beween he upper and lower curve. The price of he balance guaranee swap is given by spread rae y and his is he spread ha is offered by he seller and acceped by he buyer, i is no a spread resuling from a par swap calculaion. Payoff a ime +1 Produc Leg Buyer BGS Seller BGS Hedges Ouside Balance Fixed Nc Nc - guaraneed Libor N ( L y) N ( L y) - swap Amorising Fixed - U U swap Libor - U ( L ) U ( L ) Floorle - wu max( L,0) wu max( L,0) payoff 35