The Allocaion of Ineres Rae Risk and he Financial Secor Juliane Begenau Sanford Monika Piazzesi Sanford & NBER May 2012 Marin Schneider Sanford & NBER Absrac This paper sudies US banks exposure o ineres rae risk. We exploi he facor srucure in ineres raes o represen many bank posiions as porfolios in a small number of bonds. This approach makes exposures comparable across banks and across he business segmens of an individual bank. We also propose a sraegy o esimae exposure due o ineres rae derivaives from regulaory daa on noional and fair values ogeher wih he hisory of ineres raes. PRELIMINARY & INCOMPLETE. Email addresses: begenau@sanford.edu, piazzesi@sanford.edu, schneidr@sanford.edu. We hank seminar paricipans a Arizona Sae, Chicago, he Economeric Sociey Meeing in Seoul, he Federal Reserve Bank of Chicago, he Federal Reserve Bank of San Francisco, NYU, Princeon Universiy, SITE, and Wharon. 1
1 Inroducion The economic value of financial insiuions depends on heir exposure o marke risk. A radiional bank borrows shor erm via deposis and lends long erm via loans. Modern insiuions have increasingly borrowed shor erm in he money marke, for example via repurchase agreemens and len long erm via holding securiies such as morgage bonds. Modern insiuions also play a prominen role in derivaives markes. The value of posiions aken as a resul of hese aciviies changes if ineres raes change, for example because of news abou fuure moneary policy or defaul raes. Measuring financial insiuions risk exposure is clearly imporan for regulaion, bu i is also relevan for economic analysis more broadly. Insiuions are he main players in markes for fixed income insrumens. For example, many shor erm insrumens are no raded direcly by households (for example commercial paper.) Moreover, banks choose risk exposures ha are differen from each oher and herefore have differen experiences when condiions change (for example, Lehman versus JP Morgan during he 2007-2009 financial crisis.) This has moivaed a lieraure ha aims o explain asse prices from he ineracion of heerogeneous insiuions. To quanify such models, we need o know banks exposures. I is diffi cul o discern exposures from insiuions repored credi marke posiions. Indeed, common daa sources such as annual repors and regulaory filings record accouning measures on a large and diverse number of credi marke insrumens. Accouning measures are no necessarily comparable across posiions. For example, he economic value of wo loans wih he same book value bu differen mauriies will reac quie differenly o changes in ineres raes. A he same ime, many insrumens are close subsiues and hus enail essenially he same marke risk. For example, a 10 year governmen bond and a 9 year high-grade morgage bond will end o respond similarly o many changes in marke condiions. This paper consrucs comparable and parsimonious measures of insiuions exposure o marke risk by represening heir posiions as porfolios in a small number of bonds. We sar from balance shee daa from he US Repors on Bank Condiions and Income ( call repors ). We show how o consruc, for any bank and for each major class of credi marke insrumens, replicaing porfolios of bonds ha have approximaely he same condiional payoff disribuion. We hen compare porfolios across posiions as well as across banks. Our findings sugges ha he overall posiion of he major dealer banks is a porfolio which is long in long-erm bonds and shor in cash. We also find ha hese banks have large ne posiions in ineres-rae derivaives. This ne derivaive posiion comes close in magniude o he ne posiion in oher fixed income derivaives. We documen ha, 2
during much of our sample, he ne-ineres rae derivaive posiion does no hedge oher balance-shee posiions. Insead, banks increase heir ineres rae exposure hrough derivaives. Because of is large size, i is imporan o accoun for he ne posiion in ineres-rae derivaives when measuring exposure. The key diffi culy in measuring he exposure in ineres-rae derivaives is ha banks do no repor he sign of heir posiion wheher hey represen bes on ineres rae increases (e.g., pay-fixed swaps) or decreases (e.g., pay-floaing swaps.) Moreover, here is no deailed informaion abou he mauriies of hese ne (as opposed o gross) derivaives posiions or he sar day of hese derivaives (and hus heir associaed locked-in ineres raes). To deal wih he lack of repored informaion, we propose a novel approach o obain he exposure conained in he ne posiion in ineres-rae derivaives. We specify a sae space model of a bank s derivaives rading sraegy. We hen use Bayesian mehods o esimae he bank s sraegy using he join disribuion of ineres raes, bank fair and noional values as well as bid-ask spreads. Inuiively, he idenificaion of he bank s sraegy relies on wheher he ne posiion (per dollar noional) gains or loses in value over ime, ogeher wih he hisory of raes. If raes go up and he bank s derivaive posiion experiences gains, he Bayesian esimaion pus more probabiliy on a derivaive posiion wih a pay-fixed ineres rae. Our approach is moivaed by he saisical finding ha he marke value of fixed income insrumens exhibi a low-dimensional facor srucure. Indeed, a large lieraure has documened ha he prices of many ypes of bonds comove srongly, and ha hese common movemens are summarized by a small number of facors. I follows ha for any fixed income posiion, here is a porfolio in a few bonds ha approximaely replicaes how he value of he posiion changes wih innovaions o he facors. For loans and securiies, he replicaion porfolio is derived from deailed informaion on he mauriy disribuion provided by he call repors. For loans repored a book value, we follow Piazzesi and Schneider (2010) and represen loan porfolios as bundles of zero coupon bonds. For securiies repored a marke value, we use hose marke values ogeher wih he properies of zero coupon bond prices. For derivaives, he replicaion porfolio becomes an observaion equaion for a sae space sysem, which has unobservable replicaion weighs ha can be esimaed. Relaed lieraure [o be wrien] 3
2 Insiuions fixed income porfolios: an organizing framework Our goal is o undersand financial insiuions fixed income sraegies. We wan o compare sraegies across insiuions, as well as relae differen componens of an individual insiuion s sraegy, for example is loan porfolio and is derivaives rading business. We use a discree ime framework for our analysis. Fix a probabiliy space (S, S, P). Here S is he sae space: one elemen s S is realized every period. Denoe by s he hisory of sae realizaions. I summarizes all coningencies relevan o insiuions up o dae, including no only aggregae evens (such as changes in ineres raes), bu also evens specific o an individual insiuion, such as changes in he demand for loans and deposis, or he order flow for swaps. We hink of a fixed income insrumen as simply a hisory-coningen payoff sream y = {y (s )} ha is denominaed in dollars. The simples example is a safe zero coupon bond issued a some dae τ ha pays off one dollar for sure a he mauriy dae τ + m, say. More generally, payoffs could depend on ineres raes for example, an ineres rae swap or an adjusable rae morgage promise payoff sreams ha move wih a shor erm ineres rae or on oher evens, such as cusomers decisions o prepay or defaul on a morgage. We assume ha every insrumen of ineres can be assigned a fair value. If he payoff sream of he insrumen is y, we denoe is fair value π. Following GAAP accouning rules, we view he fair value as he price a which he insrumen could be sold in an orderly ransacion. For insrumens raded in a marke, fair values can be read off marke prices. For nonraded insrumens, such as loans, fair values have o be consruced from he payoffs of comparable insrumens. The fair values of fixed income insrumens exhibi a low-dimensional facor srucure. In paricular, he overwhelming majoriy of movemens in bond prices is due o he overall level of ineres raes. The laer can be summarized by any paricular ineres rae, for example a riskless nominal shor rae. Since fixed income insrumens are fairly predicable payoff sreams, i is naural ha changes in discoun raes drive heir value. Our key assumpion is ha fair values of all relevan fixed income insrumens can be wrien as funcions of a small number of facors f, as well as possibly calendar ime. Le f denoe an (N 1)-vecor-valued sochasic process of facors. Here each f is a random variable ha depends on he hisory s, bu we mosly suppress his dependence in wha follows. The fair value π (f, ) of a fixed income insrumen depends on he facors and calendar ime, which is imporan because he mauriy dae is par of he descripion of he payoff sream. 4
As an example, le he payoff sream correspond o a riskfree zero coupon bond wih mauriy dae + m ha was issued a dae or earlier. Le i (m) denoe he yield o mauriy on an m-period zero coupon bond quoed in he marke a dae. The price of he payoff sream y a dae is exp( i (m) m). A any laer dae + j before mauriy dae (so j < m), he price is exp( i (m j) +j (m j)). The payoff sream hus saisfies our assumpion as long as he ineres rae depends on he facors. We assume furher ha he disribuion of he facors is given by a saionary Gaussian AR(1) process. We hus represen he disribuion of f under P by a saionary process ha saisfies f +1 = φf + σε +1, ε +1 i.i.n (0, I N ). (1) We assume ha he riskless one period ineres rae is a linear funcion of he facors. i = δ 0 + δ 1 f. The linear Gaussian dynamics are no necessary for he approach o work, bu hey simplify he analysis. They also provide a reasonable descripion of ineres-rae dynamics for quarerly daa. More generally, i would be possible o exend he analysis o allow for changes in he condiional volailiy of he facors or nonlineariies in heir condiional mean. We approximae he change in he fair value of he insrumen as a linear funcion in he shocks σε +1. If ime were coninuous, Io s lemma would deliver his resul exacly, given normaliy and he smoohness of π. Here we use a second-order Taylor expansion and he properies of normal disribuions. We wrie π (f +1, + 1) π (f, ) π f (f, ) (f +1 f ) + π (f, ) + 1 2 σπ ff (f, ) σ = π f (f, ) (E f +1 f + σε +1 ) + π (f, ) + 1 2 σπ ff (f, ) σ =: a π + b π ε +1, (2) where he firs (approximae) equaliy uses he fac ha he hird momens of a normal disribuion are zero and higher momens are an order of magniude smaller han he firs and second momens. The coeffi cien a π is he condiional expeced change in fair value. If we divide a π by he curren fair value, π (f, ), we ge he expeced reurn. The 1 N slope coeffi ciens b π is he exposure of he fair value o he facor risks, ε +1. We are now ready o replicae he payoff sream of any insrumen by N + 1 simple securiies. Tha is, we define, for each dae, a porfolio of N + 1 securiies ha has he same value as he insrumen in every sae of he world a dae + 1. We always 5
ake one of he securiies o be he riskless one period bond; le θ 1 denoe he number of shor bonds in he porfolio a dae. Since θ 1 is also he face value of he one-period riskless bonds, we will refer o θ 1 as cash. For he payoff sream corresponding o a shor bond, he coeffi ciens in (2) are given by a π = i e i and b π = 0. Consider N addiional spanning" securiies ha saisfy ˆP +1 ˆP = â + ˆb ε +1 (3) The N 1 vecor ˆθ denoes he holdings of spanning securiies a dae. In our one facor implemenaion below he only spanning securiy will be a long bond (so ha ˆθ will be a scalar). For each period, we equae he change in he values of he payoff sream y and is replicaing porfolio. This means ha for every realizaion of he shocks ε +1, he holdings of cash θ 1 and spanning securiies ˆθ solve ( a π b π ) ( 1 ε +1 ) = ( θ 1 ) ( i e i 0 ˆθ â ˆb These are N + 1 equaions in N + 1 unknowns, he holdings ) ( 1 ε +1 ( θ 1, ˆθ ). (4) ) of cash and longer spanning bonds. If he marix on he lef hand side is nonsingular hen we can find porfolio holdings (θ 1, ˆθ ) ha saisfies his equaion. If he marke prevens riskless arbirage, hen he value of he replicaing porfolio a dae should be he same as he value of he payoff sream π (f, ). Suppose o he conrary ha he value of he replicaing porfolio, π (f, ) say, was lower han π (f, ). Then one could sell shor one uni of he payoff sream y, buy one uni of he replicaing porfolio and inves he difference π (f, ) π (f, ) in he riskless asse. Since he change in value for y and he replicaing porfolio is idenical, his sraegy delivers a riskfree profi ha consiss of he ineres earned on π (f, ) π (f, ). I follows ha one period ahead a posiion in he payoff can be equivalenly viewed as a posiion in he replicaing porfolio: i has he same value a dae as well as in each sae of he world a dae + 1. Once posiions are represened as porfolios, we can measure risk by considering how he value of he posiion changes wih he prices of he long erm spanning securiies ˆP, or equivalenly wih he facor innovaions ε +1. If he shor ineres rae is he only facor, hen he exposure of he posiion is closely relaed o duraion, which is defined as (minus) he derivaive of a bond s value wih respec o is yield. In his case, he holdings of he spanning bonds ˆθ are he dela of he posiion, and he change in value (4) can be used for VaR compuaions ha deermine he hreshold loss ha occurs wih a cerain probabiliy. For example, we migh deermine ha a given bond has a 6
one-quarer 5% VaR of 90 cens. This would correspond o a 5% probabiliy ha he bond s price will fall by more han 90 cens over he quarer. The advanage of he porfolio represenaion (4) over VaR is ha i fully describes he condiional disribuion of risk in he insrumen, no jus he probabiliy of a cerain ail even. Anoher advanage is ha he replicaing porfolios of various fixed-income posiions are addiive, making hese posiions easy o compare. The same is no rue for VaR compuaions of complex posiions. Moreover, our approach can easily incorporae facors in addiion o he shor rae, such as liquidiy facors. 3 Daa Our daa source for bank porfolios are he Bank Repors of Condiions and Income, or "call repors, filed quarerly by US commercial banks and bank holding companies (BHCs). The call repors conain deailed breakdowns of he key iems on an insiuion s balance shee and income saemen. The breakdowns are for mos iems more deailed han wha is conained in corporaions SEC filings for banks. A he same ime, he call repors conain all banks, no simply hose ha are publicly raded. They also conain addiional informaion ha helps regulaors assess bank risk. Of paricular ineres o us are daa on he mauriy disribuion of balance shee iems such as loans and borrowed money, as well as on he noional value and mauriy of ineres rae derivaive conracs. Table 1 shows a bank balance shee which is based on he Consolidaed Financial Saemens for Bank Holding Companies (FR-Y-9C) from December 31, 2011. These financial saemens are required by law and are filed by Bank holding companies o he Board of Governors of he Federal Reserve Sysem. The abbreviaions in he balance shee are domesic offi ce (DO), foreign offi ce (FO), ineres bearing (IB), nonineres bearing (NIB), Federal Funds sold (FFS), and Federal Funds purchased (FFB). In his paper we are ineresed in represening banks ne exposures due o differen ypes of business. To provide a firs impression, Figure (1) shows various ne posiions as a percenage of asses for he larges bank in recen years, JP Morgan Chase. In paricular, he doed dark blue line shows he ne fair value of ineres rae derivaives. The solid dark blue line describes a ne fixed income posiion wihou ineres rae derivaives: i comprises loans plus securiies plus ne rading asses less deposis and oher deb. To pu hese numbers in perspecive he red line is (book) equiy over asses. Finally, he ligh blue line labeled "ne oher" is a residual defined so ha all hree blue lines ogeher add up o equiy. The remainder of his paper is abou undersanding he risk exposure inheren in he leveraged fixed income posiions represened by he dark blue lines. 7
% asses 8 6 4 2 0 2 ne fixed income ne i.r. derivs equiy ne oher 1995 2000 2005 2010 Figure 1: Balance shee posiions of JP Morgan Chase The risk exposures in hese posiions are no eviden from he variaion in he fair values in Figure 1 over ime. The reason is ha hese fair values only represen he overall value of heir replicaing porfolio π (f, ) = θ 1 e i + ˆθ ˆP in he noaion of Secion 2. To learn abou he risk exposure of he porfolio, we would need o know he porfolio weighs ˆP (n) /π (f, ) on each of he n = 1,..., N risky ˆθ (n) spanning securiies and how hese weighs change over ime. The spanning securiies depend on he risk in he facors ε +1 hrough he loadings ˆb in equaion (3). Therefore, once we know he porfolio weighs for each period, we know how he overall porfolio depends on he risk facors. In he res of he paper, we will compue he replicaing porfolio ( θ 1, ˆθ ) for each of he fair values in Figure 1 and for each of he U.S. banks. 8
Sample selecion We are ineresed in he risk exposure of domesic BHCs. We hus work wih daa series ha are consolidaed a he BHC level. We consider only BHCs ha are he op ier company in heir BHC, and hus eliminae BHC ha are subsidiaries of anoher BHC. We also eliminae all BHC ha have a foreign paren. The risk exposure of a US subsidiary of a foreign bank is likely o depend on very differen consideraions han ha of a US op ier bank. Mos daa series are direcly available from he consolidaed BHC files in he call repors. However, he mauriy disribuion of loans, securiies and borrowed money is more deailed in he bank daa. For hese iems, we hus sum up he bank-level holdings over all banks in he same BHC o obain he BHC level mauriy disribuion. We verify ha his procedure comes up wih he correc aggregae holdings. Our sample is 1995:Q1-2011:Q4. We choose his period because accouning rules allow consisen definiions of he main fair value and noional value series. In paricular, he fair value of ineres rae derivaives posiions is available over his whole period and we have hree mauriy buckes for noionals. Our sample period also conains he years 2009-2011, during which he call repors also conain he major surviving invesmen banks, Goldman Sachs and Morgan Sanley. This fac ogeher wih new regulaory requiremen on he reporing of credi exposure in derivaives markes makes his laes par of he sample paricularly ineresing for sudying swap posiions. Holding companies wih less han $500 million asses repor semiannually o he Federal Reserve. For iered bank holdings companies, only he op-ier holding company mus file a repor. We use informaion on merger and acquisiion aciviies of our sample from he Federal Reserve Bank of Chicago. This daa has dae of merger, he ideniy number of he non-surviving and he acquiring bank and heir respecive bank holding company ideniy number. We conver he merger dae o he quarer dae. Daa on loans and securiies Under radiional accouning rules, deposis and loans are recorded in balance shees a face value. The face value of a deposi posiion is he amoun of money deposied in he accoun. The face value of a loan is usually he amoun of money disbursed when he loan is aken ou (alhough here can be small difference, for example, when a morgage borrower buys poins.) The balance shee herefore does no conain a proper measure of economic value, and i canno answer quesions on how he loan porfolio is exposed o ineres rae risk. Under he radiional rules, flucuaions in ineres raes show up only in he income saemen. Indeed, ineres paid on deposis or earned on loans is recorded as par of ineres income and expense, respecively. 9
T 1: B B S Asses Liabiliies 1. Cash 13. Deposis NIB balances, currency and coin BHCK0081 a. In DO: (1) NIB BHDM6631 IB balances in US offi ces BHCK0395 (2) IB BHDM6636 IB balances in FO BHCK0397 b. In FO, Edges, IBFs: (1) NIB BHFN6631 (2) IB BHFN6636 2. Securiies 14. FFP a. Held-o-mauriy securiies BHCK1754 a. FFP in DO BHDMB993 b. Available-for-sale securiies BHCK1773 b. Securiies Sold o Repurchase BHCKB995 3. FFS 15. Trading Liabiliies a. FFS in DO BHDMB987 BHCK3548 b. Securiies Purchased BHCKB989 4. Loans & Leases 16. Oher Borrowed Money a. Loans & leases held for sale BHCK5369 Includes morgage, indebness, BHCK3548 d. Loans & leases, ne of unearned BHCKB529 and obligaions under capialized leases income and allowance for (iems 17., 18. are no applicable) loan & lease losses 5. Trading Asses 19. Subordinaed Noes BHCK3545 Subordinaed noes and debenures BHCK4062 Subordinaed noes payable o russ BHCKC699 6. Premises and fixed Asses 20. Oher Liabiliies BHCK2145 BHCK2750 Oher Invesmen 21. Toal Liabiliies 7. Oher real esae owned BHCK2150 BHCK2948 8. Invesmens in uncons. subsidiaries BHCK2130 9. Direc & indirec invesmens BHCK3656 in real esae venures 10. Inangible asses Equiy a. Goodwill BHCK3163 Toal Equiy Capial BHCKG105 b. Oher inangible asses BHCK0426 11. Oher Asses BHCK2160 12. Toal Asses BHCK2170 Abbreviaions: domesic offi ce (DO), foreign offi ce (FO), ineres baring (IB), nonineres baring (NIB), Federal Funds sold (FFS), and Federal Funds purchased (FFB). 10
Recen saemens by he Financial Accouning Sandard Board (FASB) have moved US GAAP rules increasingly owards marked-o-marke (MTM) accouning. Saemen FAS 115, issued in 1993, inroduced a hree way spli of posiions ino "held o mauriy" (HTM), "available for sale" (AFS), and "held for rading" insrumens. The laer wo caegories are recorded a fair value on he balance shee, while HTM insrumens are recorded a face value. The difference beween AFS and rading asses is how changes in fair values affec earnings: rading gains and losses direcly affec ne income, whereas gains and losses on AFS asses ener oher comprehensive income (OCI), a componen of equiy. The call repors show how many loans and securiies are designaed as "available for sale" and recorded a fair value versus "held o mauriy" and recorded a face value. Over our sample, he majoriy of posiions in loans, deposis and "oher borrowed money" is recorded a face value, while he majoriy of posiions in securiies is recorded a fair value. We hus work wih face value numbers for loans and deposis and compue fair values, as described furher below. We work wih fair value numbers for securiies. Loans or securiies held for rading mus be held wih he purpose of resale in he near fuure. The call repor show hese rading asses separaely. Ineres rae swaps: erminology and marke srucure In erms of boh noionals and gross fair values, ineres rae swaps are by far he mos imporan derivaives used by banks. A plain vanilla single currency ineres rae swap is an agreemen by wo paries o exchange ineres paymens a regular inervals. The ineres paymens are proporional o a noional amoun. One pary pays a fixed ineres rae, he swap rae, while he oher pary pays a floaing rae. The paymens are made a a cerain frequency up o a given mauriy. The sream of fixed ineres rae paymens ogeher wih he noional value paid a mauriy, is referred o as he fixed leg of he swap. Similarly, he sream of floaing paymens ogeher wih he noional value a mauriy is called he floaing leg. Alhough he noional values cancel exacly, including hem in he sreams is helpful in calculaions. Consider a fricionless marke wihou bid ask spreads. The swap rae is hen chosen a he incepion dae (when he swap agreemen is wrien) o equae he presen values of he fixed and floaing legs. In oher words, he fair value of he swap a incepion is zero. Afer he incepion dae, he fair value of he swap moves wih marke ineres raes. In paricular, he fair value of a pay fixed (receive floaing) swap becomes posiive if ineres raes rise above wha hey were a he incepion dae. This is because higher floaing raes are received. Similarly, he fair value of a pay floaing (receive fixed) swap increases when raes fall, as lower floaing raes are paid. I is helpful o resae hese effecs by comparing swaps wih bonds. Consider he value of he wo paymen sreams. On he one hand, he presen value of he fixed leg 11
is he sum of a coupon bond ha pays he swap rae every period unil mauriy plus a zero coupon bond ha pays he noional value a mauriy. The presen value of he fixed leg hus works like a long bond ha falls as ineres raes increase. On he oher hand, he presen value of he floaing leg is simply equal o he noional value and does no respond o ineres raes. This is because owning he floaing leg is equivalen o owning he noional in cash and rolling i over a he shor ineres rae unil mauriy boh sraegies give rise o a floaing sream of ineres paymens plus he noional a mauriy. Anoher way o undersand he effecs of rae changes on fair value is hus o view a pay fixed (pay floaing) swap as a leveraged posiion in long (shor) bonds which loses (gains) as ineres raes rise. In pracice, mos swaps are raded over he couner. As for many classes of bonds, a few large dealers make he marke and frequenly rerade swaps among each oher. The concenraion of he marke is illusraed in Figure 2. I shows he oal noionals of ineres rae derivaives held for rading, for all BHCs as well as for he op hree BHCs in erms of ineres rae derivaives hgeld for rading. Here we exclude he Goldman-Sachs and Morgan Sanley, firms ha became BHCs only afer he financial crisis. There is an imporan difference in how swap dealing and bond dealing affec a dealer s posiion. A bond dealer makes he marke by buying and selling bonds. He makes money because he buys a a lower bid price and sells a a higher ask price. The invenory of bonds currenly held is recorded on he dealer s books as rading asses (or rading liabiliies if he dealer allows a shor sale). Once he dealer sells a bond, i is no longer on he dealer s balance shee. The bidask spread eners as income once i is earned. In conras, a swap dealer makes he marke by iniiaing a swap wih one clien a and hen iniiaing an offseing swap wih anoher clien. The dealer makes money by adjusing he swap raes o incorporae a spread. In paricular, he swap rae on a pay-fixed (pay-floaing) swap is ypically lower (higher) han he rae ha makes he fair value zero. Moreover, he boh swaps remain in he accouns of he dealer and conribue o he repored numbers for noional and fair values. The income on he swap is earned only period by period as he swap paymens are made and are recorded as income when hey received. Ineres rae derivaives: accouning rules & daa Banks hold a variey of derivaives for example, opions, fuures or swaps wih payoffs ha depend on credi evens, exchange raes, sock prices or ineres raes. FAS 133 requires ha all derivaives are carried on he balance shee a fair value. Banks hus compue for every derivaive posiion wheher he fair value is posiive or negaive. Posiions wih posiive (negaive) fair value are included on he asse (liabiliy) 12
Trillions $US 160 140 for rading no for rading op 3 dealers 120 100 80 60 40 20 1995 2000 2005 2010 Figure 2: Toal noionals in ineres-rae derivaives of US banks. The noionals are for rading, no for rading, and he op hree dealer banks. side of he balance shee. In he call repors, schedule HC-L provides boh fair values and noional values for derivaives by ype of exposure. For ineres rae derivaives, here is also informaion on he mauriy disribuion: i is known how many noionals have mauriy less han one year, beween one and five years or more han five years. Unforunaely, here is no informaion abou he direcion of rades. Thus, we do no know wheher, for example, swaps are pay-fixed or pay-floaing. The call repors disinguish beween derivaives "held for rading purposes" or "no held for rading. The difference lies in how changes in fair value affecs income, as for nonderivaive asses. However, he meaning of "held for rading" is broader for derivaives han for loans and securiies and does no only cover shor erm holdings. The broad scope of he erm "held for rading" is clarified in he Federal Reserve Board s Guide o he BHC performance repor: "Besides derivaive insrumens used in dealing and oher rading aciviies, his line iem [namely, derivaives held for rading purposes] covers aciviies in which he BHC acquires or akes derivaives posiions for sale in he 13
near erm or wih he inen o resell (or repurchase) in order o profi from shorerm price movemens, accommodae cusomers needs, or hedge rading aciviies. In conras, derivaives "no held for rading" comprise all oher posiions. Independenly of wheher a derivaive is designaed as "for rading", FAS 133 provides rules for so-called hedge accouning. The idea is o allow businesses o sheler earnings from changes in he fair value of a derivaive ha is used o hedge an exising posiion (a "fair value hedge") or an anicipaed fuure cash flow (a "cash flow hedge"). In boh cases, here are sringen requiremen for demonsraing he correlaion beween he hedging insrumen and he risk o be hedged. If he derivaive qualifies as a fair value hedge, hen he fair value on he hedged posiion may be adjused o offse he change in fair value of he derivaive. This is useful if he hedged posiion is no iself marked o marke, for example if i is fixed rae deb and he derivaives is a pay floaing swap. If he derivaive qualifies as a cash flow hedge, hen a change in is fair value can iniially be recorded in OCI, wih a laer adjusmen o earnings when he hedged cash flow maerializes. An unforunae implicaion of curren accouning rules is he call repors canno be used o easily disinguish hedging, speculaion and inermediaion. In paricular, here is no clean mapping beween "held for rading" and shor erm holdings due o inermediaion or shor erm speculaion, and here is no clean mapping beween "no held for rading" and hedging. On he one hand, "held for rading" derivaives could conain long erm speculaive holdings, bu also hedges, in principle even qualifying accouning hedges. On he oher hand, derivaives no held for rading could conain speculaive holdings, as long as hey are no shor erm. A he same ime, we ake away hree observaions ha help us inerpre our findings below. Firs, shor erm holdings, due o inermediaion or shor-erm speculaion, mus be "held for rading". Second, hedging of posiions in (nonderivaive) rading asses or securiies are likely o be held for rading. If he posiion o be hedged is in he balance shee a fair value wih changes going o direcly o income, hen i makes sense o accoun for he derivaive he same way. Finally, derivaives ha hedge posiions ha are no marked o marke are more likely o be "no held for rading", unless hey saisfy he requiremens for fair value hedges. We obain informaion on bid ask spreads in he swap marke by mauriy from Bloomberg. 14
4 A Porfolio View of Bank Call Repors In his secion we replicae major bank posiions in he call repors by porfolios in wo spanning zero coupon bonds a one quarer bond (which we ofen refer o as "cash") as well as a five year bond. Zero coupon bonds are useful because mos insrumens can be viewed as collecions of such bonds, perhaps wih adjusmens for defaul risk. For example, a loan or a swap can be viewed as collecion of zero coupon bond posiions of many differen mauriies one for every paymen. We now describe a pricing model ha gives rise o a linear represenaion of fair values as in (??) as well as he pricing of zero coupon bonds for ha model. 4.1 Summarizing ineres rae dynamics We consider an exponenial affi ne pricing model ha describes he join disribuion of riskfree nominal governmen bonds and risky nominal privae secor bonds. The nominal pricing kernel process M +1 represens one sep ahead dollar sae prices (normalized by condiional probabiliies) for dollar payoffs coningen on he facor innovaion ε +1. In paricular, for any payoffy (s, ε +1 (s )) he dae price is E [M +1 (s +1 ) y (s, ε +1 (s )) s ]. We choose he funcional form M +1 = exp ( i 12 ) λ λ λ ε +1 λ = l 0 + l 1 f Since ε +1 is sandard normal, he price of a cerain payoff of one is simply he one period zero coupon bond price P (1) = exp ( i ). The price of he payoff exp (ε +1,n 1/2) is given by exp ( i λ,n ). In his sense λ,n is he marke price of he risk inroduced by he nh facor innovaion. Marke prices of risk can in general vary over ime wih he facors. Riskfree governmen bonds The price of an n-period riskless zero coupon bond is given recursively by P (n) ( ) ( s = E [M ) +1 s +1 P (n 1) ( ( +1 s, ε )) ] +1 s s. This recursion sars wih he bond s payoff a mauriy, P (0) assumpions ensure ha is can be wrien as P (n) = exp ( ) A n + Bn f (5) = 1. Our funcional form (6) 15
where he coeffi ciens A n and B n saisfy a sysem of difference equaions wih boundary condiions A n = δ 0 and B n = δ 1. The difference equaions are A n+1 = A n B n σl 0 + 1 2 B n σσ B n δ 0 B n+1 = B n (φ σl 1 ) δ 1 The recursion of he coeffi ciens B n shows how he difference equaion reflecs he expecaions hypohesis of he erm srucure. Indeed, wih risk neural pricing (λ = 0), he log price is minus he sum of expeced fuure shor raes (plus a Jensen s inqualiy erm.). Wih risk adjusmen, he mechanics are he same, bu expecaions are aken under a risk-adjused probabiliy. Afer risk adjusmen, expecaions are formed using differen AR(1) coeffi ciens φ σl 1 for he facors and a differen long-run mean ( σl 0 raher han 0.) The expeced excess reurns on a riskfree n-period bond held over one period is E log P (n 1) +1 log P (n) + 1 ( ) 2 var log P (n 1) +1 i (7) = A n 1 + B n 1E f +1 A n B n f δ 0 δ 1 f = B n 1σλ The amoun of risk in he excess reurn on a long bond is B n 1σ, which is a vecor describing he amoun of risk due o each of he shocks ε +1. The vecor λ of marke prices of risk capures a conribuion o expeced excess reurns ha is earned as a compensaion for a uni exposure o each shock. Suppose ha here is a single facor (N = 1) which is posiively relaed o he riskless shor rae, ha is δ 1 > 0. A large posiive shock ε +1 means an increase in he shor rae, which lowers he one-period bond price P +1. (1) If λ < 0, a higher shor rae represens a bad sae of he world. Wih a negaive λ, he pricing kernel M +1 depends posiively on ε +1, which means ha payoffs in bad saes are valued highly. Since bond prices are exponenial-affi ne (6) and he coeffi cien B n 1 is negaive, he condiional sandard deviaion of he log reurn on he long-erm bond is B n 1 σ. This suggess an alernaive inerpreaion of λ in equaion (7) as he (posiive) Sharpe raio of he bond, is expeced excess reurn divided by he reurn volailiy. The expeced excess reurn on long bonds and heir Sharpe raio is posiive if long bonds have low payoffs in bad saes in which case hey are unaracive asses ha need o compensae invesors wih a posiive premium. Risky privae secor bonds Privae secor bonds are subjec o credi risk. For each dollar invesed in risky bonds beween and + 1, here is some loss from defaul. We rea a risky bond is a claim on 16
many independen borrowers, such as a morgage bond or an index of corporae bonds. For every dollar invesed in he risky bond a dae, here will be some loss from defaul beween daes and + 1 due o he law of large numbers. This loss can be larger or smaller depending on he sae of he economy a dae, capured by he facors f, as well as he sae of he economy a dae + 1, capured by f +1 (or equivalenly, given knowledge of f, by he innovaion ε +1. To reain he racabiliy of he affi ne model, we follow Duffi e and Singleon (1999) and assume ha he recovery value on a bond in defaul is proporional o he value of he bond. In paricular, suppose is he value of an n-period zero coupon bond rading a dae. As of dae, invesors anicipae he (n 1) value of he bond a + 1 o be +1 P +1, where he loss facor +1 = exp ( d 0 d 1 f 12 d 2 d 2 d 2 ε +1 ). capures joinly he probabiliy of defaul and he recovery value. The prices of risky bonds are deermined recursively as risk adjused presen values: P (n) ( ) [ ( s = E M ) +1 s +1 exp ( d 0 d 1 f 12 ) ] d 2 d 2 d 2 ε +1 P (n 1) ( ) +1 s, s +1 s As for riskless bonds, here is an exponenial affi ne soluion soluion P (n) = exp (Ãn + B ) n f, where Ãn and B n saisfy a sysem of difference equaions P (n) Ã n+1 = Ãn B n σ (l 0 + d 2 ) + 1 2 B n σσ Bn δ 0 d 0 + d 2 l 0 B n+1 = B n (φ σl 1 ) δ 1 d 1 + d 2 l 1 (8) wih boundary condiions Ã1 = δ 0 d 0 + d 2 l 0 and B 1 = δ 1 d 1 + d 2 l 1. The privae secor shor rae is given by ĩ = log P (1) = i + d 0 + d 1 f d 2 λ and incorporae a spread over he riskless rae ha depends on he parameer of. Wih risk neural pricing, he spread ĩ i reflecs only he expeced loss per dollar invesed d 0 + d 1 f which can vary over ime wih f. More generally, he spread can be higher or lower han he risk-neural spread because of risk premia. In paricular, λ < 0 means ha high ineres raes are a bad sae of he world. If d 2 > 0 means less payoff afer aking ino accoun when raes are high (since is lower when ε +1 is large.) 17
Togeher we have a posiive expeced excess reurn on he one-period risky bond over he shor rae E log +1 + 1 2 var (log +1 ) log P (1) i = d 2 λ So we can hink of d 2 as giving he expeced excess reurn on risky bonds over riskless bonds. Replicaion of risky zero coupon bonds The affi ne model leads o simple formulas for he coeffi ciens a π and b π in (2) if he payoff sream is a risky zero coupon bond. Taking defaul ino accoun, he change in he porfolio value beween and + 1 is +1 P (n 1) +1 P (n) = = P (n) P (n) + + P (n) (n) P P (n) ( ( ) (n 1) (n) log +1 P +1 log P + 1 ( 2 var ( d 0 d 1 f + Ãn 1 + B ) n 1 (φ 1) f Ãn ( ( Bn 1 B ) n f + 1 ) 2 B n 1σσ Bn 1 ( ) B n 1σ d 2 ε +1 ( ( ) ) ( ) i + B n 1σ d 2 λ + B n 1σ d 2 ε +1, where he second equaliy uses he coeffi cien difference equaions. Using hese coeffi ciens ã n +1 P (n 1) +1 P (n) log +1 P (n 1) +1 ) ) and b n, he percenage change in value can be wrien as P (n) = i + ( ) B n 1σ d 2 (λ + ε +1 ) Noe ha he parameers d 0 and d 1 do affec he replicaion of he change in value only hrough he coeffi ciens B n 1. This is because hey represen predicable losses from defaul which affec he value of he risky posiion, bu no is change over ime. The P (n) coeffi cien ã (n) / is he expeced log reurn ( on he risky ) bond, which is equal o he riskless shor rae plus he risk premium B n 1σ d 2 λ. The risk premium has wo erms as i compensaes invesors for boh ime variaion in he bond price a + 1 as well as he defaul loss beween and + 1. For he riskless ( bond, his) risk premium is jus equal o Bn 1σλ as compued above. The coeffi cien B n 1σ d 2 is he volailiy of he reurn on he bond beween and + 1. In he one facor case, he marke prices of risk λ is again he Sharpe raio. 18
Replicaion wih a single facor Suppose we have a single facor, so ha we can replicae any insrumen using cash θ 1 and a public bond ˆθ wih spanning mauriy m. To replicae a privae bonds wih mauriy n, we equae he changes in value ( ( ) ) θ 1 i + P (m) (n) ˆθ (i B m 1 σ (λ + ε +1 )) = P i + Bn 1 σ d 2 (λ + ε +1 ). P (1) The replicaing porfolio does no depend on ime and is given by P (m) ˆθ P (n) = B n 1 σ d 2 B m 1 σ on he m-period public bond, which is consan over ime. To ranslae his porfolio (n) weigh ino holdings θ, we also mach he value P If he bond we are replicaing is riskless, he porfolio weigh has he simpler formula P (m) ˆθ P (n). = B n 1 B m 1 Inuiively, if m = n, he porfolio weigh is equal o 1. Moreover, he Bs are negaive and heir absolue value increases in mauriy, so we will find a larger porfolio weigh if n > m and smaller oherwise. If n = 1, hen B 0 = 0, and he porfolio weigh on he long riskless bond is zero, because he replicaing porfolio consiss only of cash. A risky, privae secor bond is like a riskless bond wih a differen duraion. Wheher i is shorer or longer depends on he parameers of. There are wo effecs. Firs, he replicaing porfolio capures exposure o he ineres rae induced by losses from defaul beween and + 1. The direcion of his effec depends on he sign of d 2. If d 2 > 0, hen here is more defaul (or a lower payoff in defaul) when ineres raes are high. As a resul, a riskier bond will have more exposure o changes in ineres raes and is hus more similar o a longer riskless bond. In conras, if d 2 < 0 hen he loss beween and + 1 induces less exposure o ineres rae risk and he risky bond will be more similar o a shorer riskless bond. The second effec comes from he difference beween he coeffi ciens B and B. From (8), his effec depends no only on d 2, bu also on d 1. If d 1 > 0, hen here are larger expeced losses from defaul if ineres are high. This means ha risky bond prices are more sensiive o ineres raes han riskless bond prices of he same mauriy, so ha again risky bonds work like longer riskless bonds. The opposie resul obains if d 1 < 0. In addiion o he effecs of d 1, he coeffi ciens B also depend on he produc d 2 l 1. However, in our esimaions his par of he risk premium urns ou o be an order of magniude smaller han d 1. 19
4.2 The esimaed one facor model We esimae he governmen and privae secor yield curves using quarerly daa on Treasury bonds and swap raes from 1995:Q1-2011:Q4. As our single facor, we choose he wo-year swap rae, which will capure boh ineres rae risk as well as credi risk. The esimaion of he governmen yield curve is in several seps. Firs, we esimae he parameers φ and σ wih OLS on he (demeaned) facor dynamics (1). Then we esimae he parameer δ 0 as he mean of he riskless shor rae and δ 1 wih an OLS regression of he shor rae on he facor. Finally, we esimae he parameers l 0 and l 1 by minimizes he squared errors from he model where min l 0,l 1,n î (n) ( i (n) = A n n B n n f. ) 2 î (n) (9) The esimaion of he privae secor yield curve ges he parameers d 0 and d 1 from minimizing he squared errors ( ) min ĩ (n) ĩ (n) 2 (10) d 0,d 1,d 2,n where he model-implied privae yields are ĩ (n) = Ãn n B n n f. Panel A in Table 2 conains he esimaion resuls ogeher wih Mone Carlo sandard errors. The parameer δ 0 imes four is he average shor rae, 3.07%. The riskless shor rae has a loading of almos one on he facor, δ 1 = 0.999. The facor is highly persisen wih a quarerly auoregressive coeffi cien of 0.97. The marke prices of risk are on average negaive, l 0 = 0.25 (since he facor has a mean of zero), implying ha high nominal ineres raes represen bad saes of he world. Invesors wan o be compensaed for holding asses such as privae or public nominal bonds ha have low payoffs (low prices) in hose saes. These prices of risk are, however, imprecisely esimaed in small samples. The spreads of risky over riskless bonds is posiive and covaries posiively (d 1 > 0) wih he level of ineres raes. Moreover, d 2 < 0, indicaing ha he payoff in defaul is higher when raes are surprisingly high. Panel B in Table 2 shows average absolue fiing errors around 30 basis poins (per year), wih larger fiing errors for 30-year Treasuries. The spreads beween risky and riskless bonds are fied wih an error of roughly 20 basis poins. 20
T 2: Y C E P A: P δ 0 0.0077 (0.4215) φ 0.9702 (0.0079) δ 1 0.9990 (0.1249) σ 0.0012 (0.0001) l 0 0.2523 (39.773) d 0 0.0010 (0.0002) l 1 0.0018 (40.305) d 1 0.0814 (5.9903) d 2 0.0022 (0.0124) P B: M (% ) mauriy n (in qrs) 1 4 8 12 20 40 120 public yields i (n) 0.35 0.31 0.27 0.29 0.38 0.51 0.68 spreads ĩ (n) i (n) 0.35 0.20 0.24 0.28 0.25 0.15 0.29 Noe: Panel A repors parameer esimaes and small sample sandard errors. The daa are quarerly zero coupon yields from Treasuries and swaps, 1995:Q1-2011:Q4. The single facor is he wo-year zero-coupon yield from swaps. The sequenial esimaion procedure is described in he ex. The small sample sandard errors are compued from 10,000 Mone Carlo simulaions wih he same sample lengh as he daa. Panel B repors mean absolue fiing errors for public ineres raes i (n) and spreads ĩ (n) i (n) beween privae and public ineres raes in annualized percenage poins. 4.3 Replicaion: loans, securiies, deposis and borrowed money For shor erm asses and liabiliies, book value and fair value are ypically very similar. Here shor erm refers no o he mauriy dae, bu raher o he nex repricing dae. For example, a 30 year adjusable rae morgage ha reses every quarer will also have a fair value close o is book value. We rea all asses and liabiliies wih repricing dae less han one quarer as a one quarer bond, applying a privae secor or governmen shor rae depending on he issuer. In conras, long erm securiies are revalued as news abou fuure ineres raes and paymens arrive. For hose long erms posiions ha are recorded a book value in paricular mos loans and long erm deb i is hus necessary o consruc measures of marke value as well as replicaing porfolios from book value daa. For long erm securiies where we have fair value daa, he consrucion of replicaing porfolios is sraighforward. 21
Loans and long erm deb We view loans as insallmen loans ha are amorized following sandard formulas. We derive a measure of marke value for loans by firs consrucing a paymen sream corresponding o a loan porfolio, and hen discouning he paymen sream using he yield curve. The resuling measure is no necessarily he marke price a which he bank could sell he loan. Indeed, banks migh hold loans on heir porfolios precisely because he presence of ransacion coss or asymmeric informaion make all or pars of he porfolio hard o sell. A leas par of he loan porfolio should hus bes be viewed as a nonradable endowmen held by he bank. Neverheless, our presen value calculaion will show how he economic value of he endowmen moves wih ineres raes. The firs sep is o find, for each dae, he sequence of loan paymens by mauriy expeced by he bank. Le x m denoe he loan paymen ha he expeced as of dae by he bank in +m, (x m ). To consruc paymen sreams, we use daa on he mauriy disribuion of loan face values (N m ) ogeher wih he yield o mauriy on new loans by mauriy (r m ). For he firs period in he sample, we assume ha all loans are new. We hus deermine he paymens (x m ) by a sandard annuiy formula: N m mus equal he presen value of an annuiy of mauriy m wih paymen x m and ineres rae r m. We can also deermine how much face value from he iniial vinage of loans remains in each following period, assuming ha loans are amorized according o he sandard schedule. We hen calculae recursively for each period he amoun of new loans issued, as well as he expeced paymens and evoluion of face value associaed wih ha period s vinage. In paricular, for period we compue new loans as he difference beween oal loan face values observed in he daa and he parially amorized old loans remaining from earlier periods. This procedure produces a complee se of paymen sreams for each dae and mauriy. The marke value can hen be calculaed by applying he appropriae privae or public secor prices o he paymen sreams. For long erm deb, we follow a similar procedure for consrucing vinages. The difference is ha long erm deb is reaed as coupon bonds issued a a par value equal o he face value. As a resul, he paymen sream consiss of a sequence of coupon paymens ogeher wih a principal paymen a mauriy, and he face value is no amorized. Mauriy daa in he call repors are in he form of mauriy (or repricing) buckes. The buckes conain mauriies less han one quarer, 1-4 quarers, 1-3 years, 3-5 years, 5-15 years and more han 15 years. We assume ha mauriies are uniformly disribued wihin buckes and ha he op coded bucke has a maximal lengh of 20 years. Securiies & rading asses Suppose here is a pool of securiies for which we observe fair values by mauriy 22
(F V m ). Wihou informaion on face values, i is diffi cul o consruc direcly he paymen sream promised by he securiies. As a resul, he consrucion of he replicaing porfolio from payoff sreams is no feasible. However, we can use he mauriy informaion o view securiies as zero coupon bonds ha can be direcly replicaed. Consider he case of riskless bonds here we coun boh governmen bonds and GSE-insured morgage bonds. We assume ha he fair value F V m is he marke value of θ (m) = F V m /P (m) riskless zero coupon bonds. We hen replicae hese bonds according o (4). Similarly, for privae secor bonds all privae secor bonds ha are no GSE-insured we can find θ (m) = F V m (m) / P and hen replicae he resuling porfolio of privae secor zero coupon bonds. As for loans, he call repors provide mauriy buckes for differen ypes of securiies. We again proceed under he assumpion ha he mauriy is uniform condiional on he bucke and ha he maximal mauriy is 20 years. For securiies held for rading, deailed daa on mauriies is no available. This iem consiss of bonds held in he shor erm as invenory of marke making banks. We proceed under he assumpion ha he average mauriy is similar o ha of securiies no held for rading. From he breakdown of bonds held for rading ino differen ypes we again form privae and public bond groups and replicae wih he respecive weighs. 4.4 Ineres rae derivaives The daa siuaion for ineres rae derivaives is differen han ha for loans and securiies. In paricular, we do no observe he direcion of rades, ha is, wheher a bank wins or loses from an increase in ineres raes. For his reason, we infer he direcion of rade from he join disribuion of he ne fair value in ineres rae derivaives ogeher wih he hisory of ineres raes. Inuiively, if he bank has a negaive ne fair value and ineres raes have recenly increased, we would expec ha he bank has posiion ha pays off when ineres raes fall, for example i has enered in pay-fixed swaps or i has purchased bonds forward. The srengh of his effec should depend on he bid and ask prices ha he bank deals a. Our goal is o approximae he ne posiion in ineres rae derivaives by a replicaing porfolio. We work under he assumpion ha all ineres rae derivaives are swaps. In fac, swaps make up he majoriy of ineres rae exposures, followed by fuures which behave similarly as hey also have linear payoffs in ineres raes. A more deailed reamen of opions, which have nonlinear payoffs, is likely o be no of primary imporance and in any case is no feasible given our daa. To value swaps wih our pricing model, he following noaion is helpful. Define as he dae price of a privaely issued annuiy ha promises one dollar every period up o dae + m. Consider now a pay fixed swap of mauriy m ha promises fixed C (m) 23
(1) paymens a he swap rae s and receives floaing paymens a he shor rae log P. As explained in Secion 3, he fixed leg is he sum of a zero coupon bond and an annuiy, (m) and he fair value of he floaing leg is equal o he noional value. Using P o again denoe a privae secor zero coupon bond of mauriy m, he fair value of a pay fixed swap can be wrien as he difference beween he floaing and fixed legs ( ) F V = N sc (m) (m) + P N =: F (s, m) N Here F (s, m) is he fair value of a pay fixed swaps wih a noional value of one dollar. A he same ime, he fair value of a pay floaing swap wih noional value of one dollar is equal o F (s, m) We now develop he relaionship beween ne and oal noionals and heir effecs on he fair value. Le N m+ denoe he amoun of noionals in pay fixed swaps of mauriy m held a dae and le N m denoe he mauriy m pay floaing noionals. Assume furher ha all pay-fixed swaps are of mauriy m have he same locked in swap rae s m + z m. Here s m is he midmarke swap rae (ha is, he rae a he midpoin of he bidask spread) and z m is one half he bidask spread for mauriy m. Moreover, all pay floaing swaps of mauriy m have he same locked in rae s m z m. Wih his noaion, he fair value of he ne posiion in pay fixed swaps is F V = m N m+ F (s m z m, m) m N m F (s m + z m, m) = m ( N m+ ) N m F (s m, m) + m ( N m+ ) + N m z m C m = : m N m ω m F (s m, m) + m N m z m C m (11) where N m = N m+ + N m is he oal amoun of noionals of mauriy m and ω is he ne posiion in pay fixed swaps expressed as a share of oal noionals. For every mauriy, he fair value hus naurally decomposes ino wo pars. The firs sum, F V a say, is he ne fair value due o he bank rading on is accoun, valued a he midmarke rae. Is sign depends on he direcion and size of he bank s rade (capured by he sign and size of ω m ) as well as on he hisory of ineres raes since he swap rae s m was locked in. The second erm F V b say, consiss of he presen value of bidask spreads, which scales direcly wih oal noionals. Our esimaion sraegy reas he wo erms separaely. The reason is ha we have daa on he mauriy disribuion for oal noionals, bu no for ne noionals. Since oal noionals are poenially much larger han ne noionals, especially for large 24
dealer banks, we canno know a wha mauriies he banks rade on heir own accoun. We herefore use daa on bidask spreads and mauriies o obain an esimae of F V a. We hen subrac ha esimae from he oal ne fair value o obain an esimae of F V a. We hen specify a sae space model ha replicaes F V a by a porfolio of 5 year swaps and cash, up o measuremen error. This esimaion sep allows replicaion in he absence of mauriy informaion on F V a. Rens from marke making For every mauriy m, he spread facor z m in (11) reflecs (one half) he average bidask spread for all he swaps currenly on he bank s books. To he exen ha bidask spreads change over ime, is magniude depends on how many curren swaps were iniiaed in he pas when bidask spreads were, say, higher. To capure his effec, we consruc a vinage disribuion of swap noionals analogously o he vinage disribuions of loans and long erm deb discussed above. We use daa on bidask spreads on new swaps o find, for each mauriy and period, he oal bidask spread paymen earned by swaps of ha mauriy in ha period. More specifically, suppose we know he disribuion (N m ) of oal noional values by mauriy as well as he disribuion of bidask spreads on new swaps by mauriy, ha is, he sequence (2z m ). We assume ha in he firs sample period, all swaps are new, and we record he sream of bidask spread paymens (z1 m ) on hose swaps. We hen proceed recursively: for each period and mauriy, new swaps are defined as he difference beween oal noionals for ha period and old noionals ha remain from he previous period, aking ino accoun ha he old swaps have aged by one period. We hen use he curren bidask spreads o add o he sream of paymens for all fuure periods. Trading on own accoun A any poin in ime he ne posiion in pay fixed swaps is replicable by a porfolio in a long erm spanning bond and cash. Alernaively, we can hink of a posiion in cash and a long erm swap, say of mauriy ˆm. Suppose ha, a dae 1, he bank s ne posiion in pay fixed swaps per dollar of noional value can be wrien as a posiion ˆω 1 in he ˆm period pay fixed swap wih swap rae s 1 as well as K 1 dollars in cash. From (11), he fair value of his posiion a dae is (ˆω 1 F ( s 1, ˆm 1) + K 1 ) N 1. Here he fair value is he presen value of fuure paymens on he swap; curren ineres paymens are no included since i is booked as income in he curren period. Our goal is o describe he rading sraegy of he bank over ime in erms of he 25
riple (ˆω, K, s ). We define he sae space model F V a /N 1 = ˆω 1 F ( s 1, ˆm 1) + K 1 + u (ω, s, K ) = T (ω 1, s 1, K 1 ), where u is an iid sequence of measuremen errors. The ransiion equaion capures he evoluion of he sae variables which has wo pars. Firs, since ω describes he posiion in a fixed mauriy insrumen ha ages beween periods 1 and, he ransiion equaion mus adjus he ω posiion for aging. Second, he ransiion equaion mus describe how he bank s rades in long erm swaps affec is swap rae and cash posiion. We now describe hese pars in urn. Consider firs he updaing of mauriies. I is useful o view ω 1 as he long swap posiion of mauriy ˆm a he end of period 1. The bank hen eners dae wih a long swap posiion ω 1 of mauriy ˆm 1 as well as cash K 1. We wan o ransform his posiion ino a beginning of period posiion in mauriy ˆm swaps and cash, denoed ) old (ˆω, K old. Here we use he same replicaion argumen as in (4). For he fixed leg of a swap of mauriy ˆm 1, here exis coeffi ciens a s and b s such ha he fixed leg is replicaed by b s unis of he fixed leg of a swap of mauriy ˆm ogeher wih a s dollars in cash. We hus updae he posiion in long swaps by ˆω old = b s ω 1. I remains o updae he cash posiion. Replicaion of he fixed leg involves ω 1 a s dollars in cash which mus be subraced from he cash posiion. Consider now he floaing legs, which are equivalen o posiions in cash since he fair value of a floaing leg is equal o is noional value. The floaing leg of he original swap (of mauriy ˆm 1) can be viewed as a posiion of ω 1 dollars in cash. The floaing leg of he mauriy m swap is a cash posiion of only b s ω 1 dollars. We mus herefore add he difference ω 1 (1 b s ) dollars o he cash posiion. The updaing rule is herefore K old = K 1 + ω 1 (1 a s b s ). For large ˆm, such as ˆm = 20 quarers, swaps of mauriies ˆm and ˆm 1 end o be very similar. The replicaing porfolio mus capure he fac ha he mauriy ˆm 1 swap is less (bu almos as) responsive o ineres raes as he mauriy ˆm swap. As a resul, b s will be close o bu less han one and a s will be close o bu greaer han zero, and he sum will generally be close o one. This explains why he cash posiions we find end o be small in size. Consider now he rades he bank can make, ha is, how i moves from he beginning of period posiion ( ) ω old, K old o he end of period posiion (ω, K ). Since he only long swaps are of mauriy ˆm, here are wo possibiliies. On he one hand, he bank can eiher increase or decrease is exposure o hose swaps. If he bank increases is exposure, i combines ω old N 1 swaps wih he old locked in rae s 1 wih ω new N new swaps ha 26
are issued a he curren marke rae s m. The paymen sream of he combined swaps is equivalen o holding ω old N 1 + ω new N swaps a he adjused swap rae N 1 s = ωold ω N s 1 + ωnew s m. ω On he oher hand, he bank can decrease is exposure o long swaps by canceling some of he old swaps. In pracice, cancellaion is ofen accomplished by iniiaing an offseing swap in he opposie direcion. If he curren swap rae for he relevan mauriy is differen from he original locked in rae, he cancellaion will also involve a sure gain or loss. We assume ha his gain or loss is direcly booked o income and does no appear as par of he fair value afer cancellaion. The remaining long swaps hen reain he same locked in swap rae, ha is s = s 1. We assume ha, in any given period, he bank makes moves beween posiions ω old and ω in he simples possible way. In paricular, if he sign of ω remains he same, hen i makes only one of he above rades i eiher increases or decreases is exposure. he only excepion o his rule is he case where he bank changes he sign of ω : in his case we assume ha i cancels all exising long swaps and issues all new swaps in he opposie direcion. Le γ denoe he fracion of old long swaps ha is canceled in period. The ransiion for ω can be summarized by ω N = (1 γ ) ω old N 1 + ω new N. Given hese assumpion, a sequence ˆω ogeher wih iniial condiions for s 1 and K 1 implies a unique hisory of all hree sae variables. We ake a Bayesian approach o infer he sequence of ˆω. Our prior is ha changes in he sraegy are hard o predic and are broadly similar in magniude over ime. We hus assume ha ˆω follows a random walk wihou drif, wih iid innovaion ha have variance σ 2 ω. Under he prior, he variance σ 2 ω as well as he variance of he measuremen error σ 2 u follow noninformaive gamma priors and are muually independen as well as independen of he ˆω s. We fix he iniial swaprae o he swap rae a he beginning of he sample and se he iniial cash posiion o zero. We joinly esimae he sequence (ˆω ) and he variances σ 2 ω and σ 2 u using Markov chain Mone Carlo mehods. The condiional poserior of eiher one of he variances given he oher variance, he daa and he ˆω s is available in closed form. However, he condiional disribuion of he sequence (ω ) given he variances and he daa is no simple. This is because he value of ˆω affecs he swap rae and he cash posiion in a nonlinear fashion. Moreover, since we need he enire sequence of ω o infer he swap raes, he problem does no allow he applicaion of sequenial Mone Carlo mehods. 27
We hus follow a Meropolis-wihin-Gibbs approach. We draw variances in Gibbs seps. We draw sequences (ω ) in a Meropolis sep. To une he proposal densiy, we use he log adapive proposal algorihm developed by Shaby and Wells (2010). Esimaion resuls To illusrae how he esimaion works, Figure 3 shows he rading posiions for wo major dealer banks, JPMorganChase (blue/dark lines) and Bank of America (green/ligh lines.) The op panels display he daa. The op lef panel shows he evoluion of noional values. These numbers are large because of he lack of neing of inerdealer posiions in he call repors: he noionals of each bank by iself amouns o several imes US GDP. While JPMorgan Chase was for he mos par larger han BofA, he noionals held by he laer jumps wih he akeover of Merrill Lynch in 2008. The op righ panel shows he ne fair value as a share of noionals. Here we show he raio F V a /N 1 defined above he fair value is already ne of he presen value of bidask spreads F V b The boom panels display he esimaion resuls, wih poserior medians as hick solid lines and he 25h and 75h perceniles as hin dashed lines. The boom lef panel shows he esimaed sequence of posiions in long ( ˆm = 5 years) swaps ω. The boom righ panel shows he locked in swap rae s on hose swaps. In addiion o he blue and green lines ha show he poserior medians for he locked in raes for boh banks, he gray line shows he curren midmarke swap rae.. 5 Replicaion resuls Figure 4 illusraes he resuls of he replicaion exercise for JP Morgan Chase. The solid lines represen he replicaing porfolio for he bank s radiional ne fixed income posiion, defined as loans plus securiies less deposis and oher borrowings. The solid green line shows he face values of 5 year zero coupon bonds, and he solid red line shows he face value of shor bonds. The doed line shows he replicaing porfolio for he oal ne posiion in ineres rae derivaives. Finally, he dashed line presens he replicaing porfolio for bonds in he bank s rading porfolio. This posiion is broken ou separaely in par because he replicaion resuls are more uncerain for his iem due o he lack of informaion on mauriies. Figure 5 shows he replicaing porfolio for four op dealer banks. The op lef panel replicaes Figure 4, and he oher panels show Bank of America, Wells Fargo and Ciibank. 28
$ rillions 60 noionals ne fair value / noional (%) 0.1 0.05 40 20 0 1995 2000 2005 2010 omega (%) 0 0.05 0.1 1995 2000 2005 2010 5 year swap rae (% p.a.) 6 4 2 0 2 1995 2000 2005 2010 6 4 2 curren locked in 1995 2000 2005 2010 Figure 3: Trading posiions for JP MorganChase (blue/dark) and Bank of America (green/ligh.) 29
Trillions $US 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1995 2000 2005 2010 Figure 4: Replicaion porfolios for JP Morgan Chase. The porfolios are holdings of cash (in red) and a 5-year riskless zero coupon bond (in green). Solid lines are replicaing porfolios for he radiional fixed income posiion, while doed lines are for derivaives and dashed lines are for bonds held for rading. 30
Trillions $US Trillions $US Trillions $US Trillions $US JPMORGAN CHASE & CO. BANK OF AMERICA CORPORATION 1 0.5 cash 5 year 1 0.5 0 0.5 1995 2000 2005 2010 WELLS FARGO & COMPANY 0.5 0 0.5 1995 2000 2005 2010 CITIGROUP INC. 0.5 0 0 0.5 1995 2000 2005 2010 0.5 2000 2005 2010 Figure 5: Replicaion porfolios of four op dealer banks. The porfolios are holdings of cash (in red) and a 5-year riskless zero coupon bond (in green). Solid lines are replicaing porfolios for he radiional fixed income posiion, while doed lines are for derivaives and dashed lines are for bonds held for rading. 31
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