Interest Rate Swaps and Corporate Default

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1090 November 2013 Interest Rate Swaps and Corporate Default Urban J. Jermann Wharton School of the University of Pennsylvania and NBER Vivian Z. Yue Board of Governors of the Federal Reserve System NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

Interest Rate Swaps and Corporate Default (cid:3) Urban J. Jermann Wharton School of the University of Pennsylvania and NBER Vivian Z. Yue Federal Reserve Board November 20, 2013 Abstract This paper studies (cid:133)rms(cid:146)usage of interest rate swaps to manage risk in a model economydrivenbyaggregateproductivityshocks,in(cid:135)ationshocks,andcounter-cyclical idiosyncratic productivity risk. Consistent with empirical evidence, (cid:133)rms in the model are(cid:133)xed-ratepayers,andswappositionsarenegativelycorrelatedwiththetermspread. Inthemodel,swapsa⁄ect(cid:133)rms(cid:146)investmentdecisionsanddebtpricingverymoderately, and the availability of swaps generates only small economic gains for the typical (cid:133)rm. Key words: Interest Rate Swaps, Corporate Default, Risk Management, Swap Position, Debt Pricing. JEL Codes: E44, G1 We are grateful for comments from seminar participants at Columbia University, NYU, Ohio State, (cid:3) Rochester, CKGSB,theFederalReserveBoard, JohnsHopkinsUniversity, the2013MinnesotaMacro-Asset Pricing Conference, as well as from Adrien Verdelhan, Suresh Sundaresan, and Xiaoji Lin. We thank Michael Faulkender for sharing his data. Yue acknowledges the (cid:133)nancial support from European Central Bank Lamfalussy Research Fellowship. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as re(cid:135)ecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. 1

1 Introduction Interest rate swaps are derivative contracts through which two parties exchange (cid:133)xed and (cid:135)oating rate coupon payments. Such swaps were (cid:133)rst used in the early 1980s. By now they areamongthemost popularderivativecontracts. As showninFigure1, thenotional amount of outstanding interest rate swaps denominated in dollars, for non-(cid:133)nancial institutions, is about 10 trillion dollars. In surveys of derivative usage, a sizable fraction of the larger (cid:133)rms in the US typically indicate that they are using interest rate swaps, and that swaps are their favorite derivative contract for managing interest rate risk.1 Several empirical and theoretical studies have examined why (cid:133)rms use swaps, and how (cid:133)rm characteristics a⁄ect the use of swaps. However, to the best of our knowledge, there are no quantitative models of swap choice, and little is known about the impact of the use of interestrateswapsoncorporatedefaultrates, borrowingdecisions, realproductiondecisions, and the economy more generally. In this paper, we develop a model of (cid:133)rms(cid:146)production and (cid:133)nancing decisions with (cid:133)rms that are subject to aggregate and idiosyncratic productivity risk, as well as interest rate risk and in(cid:135)ation risk. The objective is to examine the optimal swap choice and the impact of the use of interest rate swaps on the pricing of corporate debt. This is done in an equilibrium setting, by considering production and (cid:133)nancing decisions as well as default in a consistent way. As a (cid:133)rst take on these issues, we consider a relatively simple business cycle model that, in addition to an interest rate swap, includes only the most standard ingredients. In particular, we abstract from explicit agency problems and behavioral issues that have been relatedtoswapusageinsometheoreticalstudiesandinsomenon-structuralempiricalstudies 1See Marshall and Kapner (1993) for historical and institutational background, and Bodnar et al. (2011) for current derivative usage. 2

in the literature. AwelldocumentedstylizedfactofU.S.(cid:133)rms(cid:146)interestrateswapusageisthatnon(cid:133)nancial (cid:133)rms overall tend to be (cid:133)xed rate payers. Typically, a (cid:133)rm with (cid:135)oating rate debt can enter a swap contract to convert the variable payments into payments known in advance. For instance, Li and Mao (2003) (cid:133)nd that in their sample of U.S. non-(cid:133)nancial (cid:133)rms that use swaps 44% are (cid:133)xed payers, 20% are (cid:135)oating payers, 18% are both, and for 18% this information isn(cid:146)t available. These proportions are in line with Saunders(cid:146)(1999) (cid:133)ndings that were derived from di⁄erent data. This is also consistent with Chernenko and Faulkender (2011), and with the survey of Bondnar et al (2011). In a fully speci(cid:133)ed model, where swap payouts are correlated with a number of economic risk factors, (cid:133)rms(cid:146)do not necessarily choose to be (cid:133)xed rate payers. Our analysis can help us better understand under what conditions this is the case, and what the key determinants of (cid:133)rms(cid:146)swap choices are. Another characteristic that has attracted attention is that (cid:133)rms seem to be using interest rate swaps for timing the market. That is, when the yield curve is steep, (cid:133)rms use swaps to pay a (cid:135)oating rate on their debt, that(cid:150)at least in the short run(cid:150)is relatively low; and, when the yield curve is (cid:135)at (or inverted), (cid:133)rms use swaps to lock in the relatively attractive (cid:133)xed (long-term) rate. Figure 2, based on swap usage data from Chernenko and Faulkender (2011), shows a strong negative correlation between the aggregate net (cid:133)xed swap position and the spread between 10-year and 6 month government yields. The contemporaneous correlation is -0.42, and with annual growth rates -0.21; if the swap is lagged by one year, these correlations are -0.56 and -0.57, respectively. While it is tempting to explain this type of behavior as (cid:133)rms being myopic, it is important to know under what assumptions this negative correlation can emerge with fully rational agents with no explicit con(cid:135)icts of interest. 3

We (cid:133)nd that our model can explain these stylized facts about swap usage. In the benchmark version of our model, (cid:133)rms are (cid:133)xed-rate payers and swap positions are negatively correlated with the yield spread. Counter-cyclical idiosyncratic productivity risk turns out to be a key contributor to (cid:133)rms being (cid:133)xed-rate payers. Without counter-cyclical idiosyncratic productivity risk, (cid:133)rms in our model would typically choose to be (cid:135)oating rate payers to hedge the risk of de(cid:135)ation on their nominal debt. We also (cid:133)nd that in the model swaps a⁄ect (cid:133)rms(cid:146)investment decisions and debt pricing very moderately, and the availability of swaps generates only small economic gains for the typical (cid:133)rm. Our analysis is related to a number of theoretical studies that have considered the motivations for using interest rate swaps, or derivative contracts more generally. Smith and Stulz (1985) present a number of reasons why value-maximizing (cid:133)rms use derivative contracts for hedging. As in their analysis, our model focuses on bankruptcy costs as a motive for risk management. Wall (1989) links swap usage to some classic agency problems associated with long-term debt. For instance, Myers (1977) pointed out that long term debt can lead to under-investment, because creditors share the bene(cid:133)t of new investments through a reduced probabilityof bankruptcy. Barnea, HaugenandSenbet (1980) suggestedthat longtermdebt encourages risk shifting behavior, that is, the adoption of high risk projects after debt has been issued. Both of these agency problems can be addressed by issuing short term debt instead. Wall (1989) argues that swaps are useful in this context, because they allow (cid:133)rms to hedge the interest rate risk associated with short term borrowing. Titman (1992), building on ideas of Flannery (1986), suggests that short term debt allows (cid:133)rms with positive private information to lower the expected cost of borrowing. Interest swaps allow (cid:133)rms to hedge the interest rate risk associated with short term borrowing. While not modelling any of these agency problems explicitly, in our model, (cid:133)rms are limited to short term debt as a simple 4

way of representing an inherent advantage for short term debt. Our analysis is also related to some recent studies that consider equilibrium models with defaultable debt, such as Gomes and Schmid (2010), or Bhamra, Kuehn and Strebulaev (2008). As in these studies we model (cid:133)rm default explicitly. In addition to the defaultable debt considered in these studies, our model also allows (cid:133)rms to trade derivative contracts. Morerecently, Begenau, Piazzesi, andSchneider(2012)estimatebanks(cid:146)exposurestointerest rate risk including swap positions. Section 1 presents the model. Section 2 presents a simpli(cid:133)ed model that admits a closed form solution that we can use to better understand our main model. Section 3 contains the quantitative analysis, and Section 4 concludes. 2 Model A key ingredient of our model are (cid:133)rms with long-termprojects that are (cid:133)nanced with shortterm debt. This market incompleteness highlights the role played by interest rate swaps as a tool to manage risks associated with interest rates, in(cid:135)ation and productivity. We consider a general equilibrium model where asset prices are endogenously determined. Given the huge size of the aggregate value of the outstanding swap contracts, it seems a priori reasonable to examine the impact of swaps on the aggregate economy. Another advantage of the general equilibrium analysis is that we do not need to exogenously parameterize a process for the state prices and its relation to aggregate productivity, as in a partial equilibrium analysis. Of course, this comes at the cost of possible misspeci(cid:133)cation. To minimize the e⁄ects of misspeci(cid:133)ciation, our quantitative analysis seeks to match as closely as possible important empirical properties. 5

For the reminder of this section, we (cid:133)rst describe (cid:133)rms and their decisions, and then close the model with investors and consumers. 2.1 Firms with long-term projects The focus of our analysis is on (cid:133)rms with projects that have a longer maturity than the available debt (cid:133)nancing. It is in this case that an interest rate swap is potentially useful. In this subsection, we describe their technology and available (cid:133)nancial instruments, and we characterize optimal (cid:133)nancing and production decisions. 2.1.1 Technology A (cid:133)rm(cid:146)s life extends over three periods, 0; 1 and 2. Production requires an initial capital input Kl R+ in period 0, and total output is given by the (concave) production function 2 F Kl;" in period 2. The logarithm of the productivity level " is determined by aggre- 00 00 ga(cid:0)te and(cid:1)idiosyncratic shocks that are realized in period 1 and 2. We label the aggregate productivity shock by A A = (A ;::A ), and assume it follows a Markov chain, and the 1 n 2 idiosyncratic shock by z Z R. The idiosyncratic shock is independently drawn from a 2 (cid:26) continuous distribution (cid:8)(z A), conditional on A, with the expectation of exp(z) equal to j 1. Productivity depends on the history of aggregate and idiosyncratic shocks such that " = A +A +z +z : 00 0 00 0 00 A and A are the aggregate productivity shocks in period 1 and 2, respectively. The idio- 0 00 syncratic shocks z and z are assumed to be known in period 1 and 2, respectively. 0 00 In each time period, a continuum of measure one of new (cid:133)rms enter the market. Firms 6

are ex-ante identical, but di⁄er ex-post due to the realization of the idiosyncratic shocks z. Therefore, in each period, there are three vintages of (cid:133)rms active in production. There is a (cid:133)rst group of identical (cid:133)rms that are investing in new projects. In the second group, (cid:133)rms have active projects in place, and their (cid:133)rst period productivity shocks are revealed. In the third group, the second period shocks are revealed, production is completed, and (cid:133)rms exit. 2.1.2 Financial contracts A (cid:133)rm can borrow from investors in the form of one-period nominal debt. A (cid:133)rm initially borrows Bl 2 R+ to (cid:133)nance its investment. R 1 c is the (gross) interest rate on the (cid:133)rst period debt, where the superscript c stands for corporate debt, which is subject to default. After one period, the aggregate shock A and the idiosyncratic shock z are realized. At this point, 0 0 the (cid:133)rm needs to roll over its debt because the project has not produced any output yet. That is, the (cid:133)rm needs to borrow BRc for another period. No additional investment in the 1 project is needed. If the (cid:133)rm can roll over the debt, the new interest rate is Rc. Because the 2 (cid:133)rst period idiosyncratic shock z is realized, and (cid:133)rms at this stage are heterogeneous, the 0 second period debt interest rates Rc depends on the (cid:133)rm speci(cid:133)c realization of z . At the 2 0 end of the project(cid:146)s second stage, the (cid:133)rm gets output from the project, repays the second period debt, and pays out pro(cid:133)ts. Default on the debt is a possibility. In the second period, the (cid:133)rm defaults if its output is insu¢ cient to repay the second period debt. In the (cid:133)rst period, the (cid:133)rm defaults, if it cannot roll over its initial debt. The (cid:133)rm cannot roll over its debt, if the (cid:133)rm(cid:146)s assets are valued less than its debt. In addition to one-period debt, (cid:133)rms have access to an interest rate swap that is chosen in period 0. According to a swap contract, one party agrees to pay a (cid:133)xed coupon with a 7

gross rate RS to the other party in period 1 and period 2; RS is called the swap rate. In exchange, payments that are indexed to the one-period risk free rates R and R are received 1 2 in period 1 and 2, respectively; this is the (cid:135)oating rate. We assume that the swap rate RS is determined so that the present value of the contract (abstracting from default costs) is zero under the pricing measure used in the economy. The amount of the swap is denoted by s, which is measured as a fraction of (cid:133)rst-period debt Bl. Due to the (cid:133)rm(cid:146)s default risk on the interest rate swap, the (cid:133)rm also needs to pay a default spread in each period. In particular, we assume that in period 1, the (cid:133)rm needs to pay s Bl RS +ss Bl;s R ; 1 (cid:1) (cid:0) (cid:2) (cid:0) (cid:1) (cid:3) and in period 2 s Bl RS +ss Bl;s;X ;z R (X ) ; 1 0 0 2 0 (cid:1) (cid:0) (cid:2) (cid:0) (cid:1) (cid:3) where ss(:) and ss (:) denote the default spreads, and X is the vector of next period(cid:146)s 1 0 realized aggregate state variables which include productivity A and in(cid:135)ation (to be de(cid:133)ned 0 below). To save on notation, for now, our notation does not explicitly acknowledge the dependence on endogenous state variables, which will be described later. We assume that ss Bl;s;X ;z isdeterminedinperiod1, andthatitrepresentsatthattimethefairmarket 1 0 0 valu(cid:0)e of the loss(cid:1)es expected for period 2. Similarly, ss Bl;s is determined at time 0 and represents the compensation for possible losses due to d(cid:0)efaul(cid:1)t that might have occurred at time 1. We assume absolute priority for the swap, that is, the swap will (cid:133)rst get fully paid before lenders get anything.2 We assume that the (cid:133)rms(cid:146)swap counterparties will always 2To the extent that default risk is priced in interest rate swaps, it is typically build into the (cid:133)xed rate. Thus, our assumption of a variable default premium deviates somewhat from the most immediate speci(cid:133)cation. We make this assumption for numerical tractability, as becomes clear below. Given that defaults on swaps are extremely rare in the model, this assumption does not a⁄ect any of our results. 8

fully honor their commitments. Clearly, at the time the swap contract is signed, the future one-period interest rate R is 2 not known; it will be known at time 1. Therefore, swap payouts are a function of the future one-period interest rate R . Because future interest rates are correlated with a number of 2 risks the (cid:133)rm is exposed to, swaps allow the (cid:133)rm to manage these risks.3 2.1.3 Firms(cid:146)problem An individual (cid:133)rm takes as given the interest rate schedule for corporate debt, the schedule for the swap spreads, as well as the risk-free interest rates that determine the swap payouts. In period 0, a new (cid:133)rm chooses the amount of investment (equivalently debt), and the amount of the swap to maximize its expected value under the owners(cid:146)pricing measure. In addition to exogenous aggregate and idiosyncratic productivity shocks, the (cid:133)rms and investors are subject to shocks in the rate of in(cid:135)ation, (cid:25). In(cid:135)ation is assumed to follow a Markov chain. To keep the model as simple as possible, the in(cid:135)ation process is assumed to be exogenous. However, money is not neutral in this economy, because in(cid:135)ation matters for default, andthus, (cid:133)rmstakeitintoaccountwhenmakinginvestmentand(cid:133)nancingdecisions. For compactness the aggregate shock is denoted by X (A;(cid:25)).4 (cid:17) We denote the joint distribution of (X ;z ) conditional on the aggregate shocks X as 0 0 P , X ;z X 0 0j P = (cid:8)(z X )(cid:0)(X X); X ;z X 0 0 0 0 0j j j 3Typically, (cid:133)xed-for-(cid:135)oating interest rates swaps have maturities of several years, and coupon exchanges take place at quarterly, bi-annual or annual frequencies. We choose a two-period swap contract as the most tractable speci(cid:133)cation that captures the swap(cid:146)s risk management properties. 4Assuming an exogenous in(cid:135)ation process is not necessarily that restrictive. In particular, in(cid:135)ation will be allowed to be correlated with aggregate productivity. 9

with transition matrix (cid:0)(X X), and the corresponding pricing measure is denoted by 0 j P = P m(X X); (1) X(cid:3) 0 ;z 0j X X 0 ;z 0j X 0 j where m(:) is the (real) stochastic discount factor; m(:) also depends on the endogenous state variables to be de(cid:133)ned later. This representation expresses the fact that idiosyncratic risk does not require a risk premium. In period 2, conditional on the (cid:133)rmnot having defaulted on its obligations before, default occurs if output is not su¢ cient to cover the (cid:133)rm(cid:146)s (cid:133)nancial obligations. In this case, the counterparties for the swap and the debt obtain the available output according to the established priority rules. We assume that the counterparties do not incur any additional cost for default in period 2. This assumption helps making the problem more tractable numerically. Due to this assumption, the value of the (cid:133)rm at time 1 is not a⁄ected by the possibility of defaultattime2. Thisisbecausethedefaultspreadsontheswapandontheone-perioddebt determined in period 1 are such as to exactly compensate the investors for expected default losses in period 2. Without a cost of default, the (cid:133)rm values the ability to default as being equal to the default spread determined by the investors. Of course, this result also depends crucially on the assumption that the (cid:133)rm values pro(cid:133)ts with the same stochastic discount factor as the investors. This is a version of Modigliani-Miller(cid:146)s theorem; see Appendix A for details. Under the made assumptions, in period 1 the (equity) value of the (cid:133)rm is given by bl R R (X ) F bl;" dP Rc bl;s +s R +ss bl;s R + S (cid:0) 2 0 : (2) Z 00 X(cid:3) 00 ;z 00j X 0 (cid:0) (cid:25) 0 (cid:26) 1 (cid:20) S (cid:0) 1 R 2 (X 0 ) (cid:21)(cid:27) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) The present value of the (cid:133)rm(cid:146)s operational revenues is given by F bl;" dP , and 00 X(cid:3) ;z X 00 00j 0 R (cid:0) (cid:1) 10

bl is Bl de(cid:135)ated by the price level, that is, the debt (and the capital stock) in units of the numeraire good.5 Interest due on debt is represented by Rc bl;s . The term in square 1 brackets displays the per unit value of the swap including the(cid:0)pay(cid:1)ments due in period 1. Given our assumption that default in period 2 is not subject to a deadweight cost, and that a fairly valued default spread is added to the swap payment, the present value of the swap payment due in period 2, (R R (X ))=R (X ), can be obtained by discounting at the S 2 0 2 0 (cid:0) risk free rate, R (X ). As is clear in the equation above, realized in(cid:135)ation (cid:25) a⁄ects the real 2 0 0 value of the debt and the swap. If the value of the (cid:133)rm at time 1 is positive, then the one-period debt due Rc bl;s bl=(cid:25) 1 0 is rolled over. We can assume that the payment received, or due, from the swap(cid:0) (cid:1) R +ss bl;s R sbl=(cid:25) is used to decrease/increase the debt, although the exact timing S 1 0 (cid:0) (cid:2)of the set(cid:0)tleme(cid:1)nt of t(cid:3)he swap is unimportant here. If the (cid:133)rm value given in equation 2 is negative, the (cid:133)rm is in default at time 1. In particular, the (cid:133)rm will be unable to (cid:133)nd new investors to roll over its existing debt because the value of the remaining assets is below the value of the existing debt. We assume that this will generate a default cost modelled as a reduction in the output available for the creditors to F bl;" with 1. In default, we assume that the swap is honored fully 00 (cid:20) before debt holde(cid:0)rs get(cid:1)paid. This is intended to capture that quite often swaps involve some form of collateral and can thus be thought as representing a more senior claim than debt. Counterparties are given shares in the remaining (cid:133)rm assets. Speci(cid:133)cally, if bl R R (X ) F bl;" dP s R +ss bl;s R + S (cid:0) 2 0 ; (3) (cid:1) Z 00 X(cid:3) 00 ;z 00j X 0 (cid:21) (cid:25) 0 (cid:26) S (cid:0) 1 R 2 (X 0 ) (cid:27) (cid:0) (cid:1) (cid:0) (cid:1) 5There is some abuse of notation for the sake of brievity, the pricing schedules, such as Rc bl;s , are 1 henceforth denoted as function of the debt in real terms bl. (cid:0) (cid:1) 11

swap counterparties get fully paid, and the reminder is distributed to the debt holders. In case the swap has a positive value for the (cid:133)rm, that is, the RHS of Equation (3) is negative, debt holders, as the new owners, take over as creditors from the (cid:133)rm. If bl R R (X ) F bl;" dP < s R +ss bl;s R + S (cid:0) 2 0 ; (4) (cid:1) Z 00 X(cid:3) 00 ;z 00j X 0 (cid:25) 0 (cid:26) S (cid:0) 1 R 2 (X 0 ) (cid:27) (cid:0) (cid:1) (cid:0) (cid:1) then the swap counterparties get all the remaining assets from the (cid:133)rm, that is, they get F bl;" dP , the debt holders get nothing. Depending on the realization of the (cid:1) 00 X(cid:3) 00 ;z 00j X 0 idioRsync(cid:0)ratic(cid:1)shock z we have either one of the three outcomes. Typically, for values of 0 z above a given threshold there is no default. Below that threshold, default reduces the 0 debt holders(cid:146)claim. And for very low values of z , swap counterparties do not get fully paid 0 (assuming they are owed anything). The aggregate shock realization X and the endogenous 0 state variables a⁄ect these cuto⁄levels. We can now state the (cid:133)rm(cid:146)s objective as maximizing the expected value as of period 0 of its expected value as of period 1 over the no-default set for (X ;z ) denoted by ND bl;s , 0 0 1 (cid:0) (cid:1) F bl;" dP 8 00 X(cid:3) 00 ;z 00j X0 9 max R (cid:0) (cid:1) dP : bl;s ZND1 (bl;s) > > > > < > > > > = X(cid:3) 0 ;z 0j X (cid:0)(cid:25) bl 0 R 1 c bl;s +s (cid:1) R S +ss bl;s (cid:0) R 1 + RS R (cid:0) 2 R (X 2( 0 X ) 0 ) > > > > > (cid:16) (cid:0) (cid:1) h (cid:0) (cid:1) i(cid:17) > > > : ; In our model, as a function of the (cid:133)rms(cid:146)decisions, creditors and counterparties are compensated for default losses through additional payments in the no-default states. In order to solve for the (cid:133)rms(cid:146)problem, we need to derive the default spreads built into the corporate interest rate Rc bl;s and the default spread for the swap in the (cid:133)rst period 1 ss bl;s . To solve for opti(cid:0)mal d(cid:1)ecisions, we can aggregate the default spread into a total (cid:0) (cid:1) 12

default spread ts bl;s Rc bl;s R +s ss bl;s : (cid:17) 1 (cid:0) 1 (cid:2) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) By no-arbitrage, for a given choice bl;s , the default spread ts is given by (cid:0) (cid:1) E m ts Bl = E(m default loss to creditors); (cid:1) (cid:1) (cid:1) (cid:0) (cid:1) where m is the stochastic discount factor introduced in eq. (1). We assume that bond investors and swap counterparties use the same pricing measure as the (cid:133)rms(cid:146)equity holders. Substituting in the creditors(cid:146)default loss and rearranging yields ts bl;s = D1(X 0 ) R (cid:25) 1 0 + (cid:25) s 0 R S (cid:0) R 1 + RS R (cid:0) 2 R (X 2( 0 X ) 0 ) (cid:0) =bl F bl;" 00 dP X(cid:3) 00 ;z 00j X 0 dP X(cid:3) 0 ;z 0j X : (cid:0) (cid:1) R n h ND1(X 0 i ) (cid:25) 1 0 d(cid:0)P X(cid:3) 0 ;z 0 (cid:1) j X R (cid:0) (cid:1) o R The numerator represents the value lost at default relative to default free contracts, the denominator is the risk-adjusted no-default probability. Oncethemodelissolved, includingts;itisstraightforwardtocomputethedefaultspread for debt and the swap separately, following the seniority rule outlined above. Similarly, default spreads for period 2 can be computed after the model is solved. Using default spreads de(cid:133)ned in this way, after some algebra, the (cid:133)rm(cid:146)s objective can be rewritten as F bl;" dP dP bl max 00 X(cid:3) 00 ;z 00j X 0 X(cid:3) 0 ;z 0j X (cid:0) ; (5) 8 9 bl;s > < (cid:0) D1 ( R bl; h s R ) (cid:0)(1 (cid:0) (cid:1))F bl;" 00 i dP X(cid:3) 00 ;z 00j X 0 dP X(cid:3) 0 ;z 0j X > = h i R R (cid:0) (cid:1) > > : ; where D bl;s stands for the default set. This representation makes it clear that the (cid:133)rm 1 (cid:0) (cid:1) 13

value is given by the following three components PV(output) bl debt bl PV( default cost) bl;s (cid:0) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) where PV(.) stands for present value. This decomposition shows that the swap only a⁄ects the (cid:133)rm value through the default cost. This implies that without default cost, that is, if = 1 the swap choice is indeterminate. This is because the swap then simply represents a contract that has a zero market value. With default costs < 1 the (cid:133)rm optimally chooses the swap to reduce default costs. This property is related to the analysis of Smith and Stulz (1985). While in(cid:135)ation does not seem to appear in Equation (5), in(cid:135)ation (cid:25) does enter 0 through its e⁄ect on the default set D bl;s , as shown in Equations (3) and (4). 1 (cid:0) (cid:1) 2.2 Firms with short-term projects In addition to (cid:133)rms with long-term projects that require two periods to produce output, there are (cid:133)rms with short-term projects that produce output in one period. At a conceptual level, the short-term (cid:133)rms add very little to our analysis. They are however needed to produce reasonable quantitative implications. In particular, our model without these (cid:133)rms would typically produce strongly oscillating movements in short-term interest rates. This is due to the fact that the consumers(cid:146)/investors(cid:146)desire to smooth consumption between two adjacent periods will have to be frustrated with such interest rate movements because there would be no technology to transfer resources from one period to the one immediately next. This problem is reminiscent of the oscillating interest rates in models with time-to-build in investment (see for instance, Rouwenhorst (1991) or Kuehn (2009)). With short-term (cid:133)rms, the economy has the ability to physically transfer resources from one period to the next, and 14

this essentially eliminates oscillating interest rates. We keep this sector as simple as possible. FirmsinvestKs attime0, andproduceoutputoneperiodlateraccordingtoaproduction function F (Ks;A +z ) 0 s0 that depends on aggregate and idiosyncratic productivity A and z . Capital is (cid:133)nanced 0 s0 with one-period defaultable debt. For computational simplicity, we assume no default cost. Given our presentation of the long-term (cid:133)rms problem, the short-term (cid:133)rms(cid:146)problem is straightforward and thus omitted from the presentation. 2.3 Consumers/investors and equilibrium We assume a continuum of measure one of identical, in(cid:133)nitely-lived, consumers/investors. We can thus focus on a representative whose goal is to maximize lifetime utility given by 1 E (cid:12)tu(C ); 0 t t=0 X with (cid:12) the time discount factor, C consumption in period t, and u(:) a concave momentary t utility. In every period, the consumers earn pro(cid:133)ts from owning (cid:133)rms, receive interest, and make or receive payments associated with swaps. To save on notation, we do not write out the budget constraint. We assume that consumers/investors take all prices as given and choose consumption and investment strategies for debt and swaps to maximize lifetime utility. In equilibrium, bond markets and swap markets clear, and the economy-wide resource constraint holds. The later is given by Yl +Ys = C +Il +Is; 15

where Yi and Ii are aggregate output and investment in the two sectors. As is clear from the resource constraint, it is assumed that default costs are not deadweight costs at the level of the economy. Given that we typically think of default costs as due to legal costs or asset (cid:133)re sales, losses to (cid:133)rms and investors correspond to gains for lawyers and vulture investors. In this model, the asset pricing measure is given by (cid:12)u (C ) 0 0 P = P : X(cid:3) 0 ;z 0j X X 0 ;z 0j X u 0 (C) 2.4 Functional forms and aggregate state variables We use the standard utility function u(C) = C1 (cid:13)=(1 (cid:13)), with risk aversion coe¢ cient (cid:13). (cid:0) (cid:0) The production function for long-term and short-term (cid:133)rms is represented by F (K;") = exp(")[ZK(cid:11) +(1 (cid:14))K]; (cid:0) with (cid:11) < 1. Z > 0 is a scale parameter. Parameter values can potentially be di⁄erent for the long-term and short-term (cid:133)rms. In particular, we set Zl = 1 for the long-term sector as a normalization, Zd then determines the relative size of the short-term sector. The fact that (1 (cid:14))K is also multiplied by the productivity shock exp(") is somewhat nonstandard, and (cid:0) e⁄ectively introduces stochastic depreciation. This speci(cid:133)cation reduces the number of state variables, and thus improves numerical tractability. Includedintheaggregatestatevector(cid:23) arethecurrent aggregateshocks X = (A;(cid:25)) plus two variables that summarize the amount of resources in the economy that are at di⁄erent stages of production. In particular, this includes bl the amount of debt/capital chosen by 1 (cid:0) long-term(cid:133)rmsthatwerestartedinthepreviousperiod,aswellasexp(A ) bl (cid:11) +(1 (cid:14) )bl + (cid:0) 1 (cid:0) 2 (cid:0) l (cid:0) 2 (cid:0)(cid:2) (cid:3) (cid:1) 16

Zd bs (cid:11) +(1 (cid:14) )bs which summarizes the accumulated aggregate resources due to long- 1 (cid:0) s 1 (cid:0) (cid:0) term(cid:2) (cid:133)r(cid:3)ms started two periods ago and the short-term (cid:133)rms started in the previous period. The model is solved numerically by guessing policy functions for bl((cid:23)) and bs((cid:23)). Based on these, aggregate consumption and the state-prices can be computed for future periods 1 and 2. Then, the (cid:133)rms(cid:146)problems are solved, including default cost schedules. The policy functions bl((cid:23)) and bs((cid:23)) are updated until convergence. 3 Simpli(cid:133)ed model of swap choice To help understand model mechanisms, this section describes a simpli(cid:133)ed model that focuses on the swap choice only, and that admits a closed-form solution. Under a set of simplifying assumptions, the (cid:133)rms(cid:146)problem for choosing the swap is equivalent to minimizing the variance of the total position at time 1 that is given in equation 2. Most important is the assumption that default losses are not state contingent. While the assumptions needed are not exactly satis(cid:133)ed in our main model, the simple model here illustrates some key quantitative channels. See Appendix B for an explicit derivation that spells out all the required assumptions. Consider the problem of a (cid:133)rm choosing the fraction of debt that is swapped s so as to minimize the variance of the value of equity at time 1, b RS R minvar E [m F ] R +ts+s RS R + (cid:0) 2 ; 1 1;2 00 1 1 s (cid:18) (cid:0) (cid:25) 0 (cid:20) (cid:18) (cid:0) R 2 (cid:19)(cid:21)(cid:19) where E [m F ] represents the expected value of output conditional on productivity real- 1 1;2 00 17

ized at time 1. Slightly rewriting yields b[R +ts] b RS minvar E [m F A] 1 s RS R + 1 (6) 1 1;2 0 1 s (cid:18) j (cid:0) (cid:25) 0 (cid:0) (cid:25) 0 (cid:20)(cid:18) (cid:0) R 2 (cid:0) (cid:19)(cid:21)(cid:19) with the stochastic terms in bold (m ;A;(cid:25) ;R ). In the same order, these are: the 1;2 0 0 2 stochastic discount factor, productivity realized at time 1, in(cid:135)ation realized at time 1, and theone-periodinterestattime1. The(cid:133)rsttwotermsinthevarianceinequation(6)represent the (cid:133)rm(cid:146)s exposure to risk, that is, the initial position that is subject to interest rate, productivity, and in(cid:135)ation risk. The third (and last) term represents the e⁄ect of the swap. Minimizing the variance of the hedged position is typically the way (cid:133)nance practitioners(cid:146) text books present the choice of an optimal hedge; see, for instance, Hull (2000). Linearizing the terms in the variance gives F b[R +ts] b RS ^ ^ 1 ^ A R +E(cid:25) + (cid:25) +s R (7) R =(cid:25) 0 (cid:0) 2 00 (cid:25) 0 (cid:25) R 2 2 00 (cid:18) 0 (cid:19) (cid:18) 0 2(cid:19) (cid:16) (cid:17) c b ^ wherex^ dx=x. NotethatR = E(m^ )+E(cid:25) , thatis, the(log-linearized)nominalinter- 2 1;2 00 (cid:17) (cid:0) est rate equals the real interest rate plus expected in(cid:135)ation. Also note that realized in(cid:135)ation c doesn(cid:146)tmultiplytheswapbecausetheexpectedvalueasoftime0of RS R + RS R2 = 0, (cid:0) 1 R(cid:0) 2 (cid:16) (cid:17) and this is the point of linearization. Thus, to a (cid:133)rst-order approximation, the swap doesn(cid:146)t create any exposure to realized in(cid:135)ation. Taking (cid:133)rst-order conditions with respect to s and rearranging terms gives the optimal level of the swap as F R +ts s (cid:3) = R2=(cid:25) 00 (cid:12)R^ 2 (cid:0) E(cid:25) 00 (cid:12)A 0 1 (cid:12)(cid:25) 0 ; (8) b RS 2 (cid:0) 3(cid:0) RS=R (cid:25) 0 R2 ! Realinterest rcate risk Producticvity risk (cid:18) 2 (cid:19) In(cid:135)atiobn risk 4 5 | {z } |{z} |{z} 18

where (cid:12)x^ cov x^;R ^ =var R ^ , the coe¢ cient in a regression of the nominal interest 2 2 (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) rate on x^.6 These interest rate betas measure the contribution to the optimal hedge due to each of the three sources of risk. 1. The (cid:133)rst term in brackets represents the hedge against real interest rate risk for the (cid:133)rm(cid:146)s assets that comes from the (real) stochastic discount factor. The more volatile the real interest rate relative to the nominal interest rate, the more risk there is, and the larger is the swap position. Unless expected in(cid:135)ation is very volatile, this term will be positive. If expected in(cid:135)ation is non-stochastic (cid:12)R^ 2 (cid:0) E(cid:25) 00 = 1. c 2. Thesecondterminbrackets, (cid:12)A 0, representsthehedgeagainstproductivityrisk. Given c that output and nominal interest rates are typically negatively correlated (empirically, at least, over the long run), this term contributes positively to the swap position. 3. Thethirdtermrepresentsthehedgeagainstin(cid:135)ation(orde(cid:135)ation)riskassociatedwith the nominal debt. Typically, realized in(cid:135)ation is positively correlated with nominal interest rates, thus we expect the overall contribution of this term to the swap position to be negative. Intuitively, in(cid:135)ation risk on the nominal debt is hedged by getting a swap that requires the (cid:133)rm to pay the (cid:135)oating rate. For instance, when realized in(cid:135)ation is low, there is a capital loss due to the increase in the real value of the debt, and this can be compensated by lower (cid:135)oating rate payments on the swap. Thus, this term contributes negatively to the swap position s (which we have de(cid:133)ned as a swap that requires the (cid:133)rm to pay (cid:133)xed and receive (cid:135)oating). 6For instance, for min var(x~ hy~), with random variables x~ and y~, setting the derivative with respect h (cid:0) to h to 0 yields h var(y~)=cov(x~;y~). (cid:1) 19

The two terms in round brackets capture the exposures due to (cid:133)rm assets and debt divided by the sensitivity of the swap to changes in interest rates. Both of these terms are positive. To a (cid:133)rst approximation F = b RS equals the total value of the (cid:133)rm divided R2=(cid:25) 00 (cid:25) 0 R2 (cid:16) (cid:17) by debt, that is, the inverse of the leverage ratio. In the aggregate, we would expect this term to be slightly larger than 2, while R1+ts is slightly larger than 1. Thus, the (cid:133)rst two RS=R2 (cid:16) (cid:17) hedging motives are magni(cid:133)ed compared to in(cid:135)ation risk. Finally, we can see that persistent expected in(cid:135)ation, everything else equal, scales up the swap position. If the swap position is positive, higher (persistent) in(cid:135)ation requires a bigger swap position. To see this, assume R = Rr(cid:25), and RS = RS;r(cid:25), which holds to a (cid:133)rst-order 2 2 approximation, and (cid:25) = (cid:25) = (cid:25), then 00 0 F s = (cid:25) R 2 r (cid:12)R^ 2 (cid:0) E(cid:25) 00 (cid:12)A 0 R 1 r +trs (cid:12)(cid:25) 0 : ( bR R S 2 r ;r ! h c (cid:0) ci (cid:0) (cid:18) RS;r=R 2 r (cid:19) b ) Intuitively, in(cid:135)ationcontributestoshrinkingtherealsizeoftheswappositionovertimegiven that the notional is in nominal terms. The optimal hedge counters this e⁄ect by scaling up the position. We will return to this model when discussing the implications of the fully speci(cid:133)ed model. 4 Quantitative analysis In this section the model is parameterized and quantitative properties are examined. 20

4.1 Parameterization Parameter values are chosen according to two types of criteria. A (cid:133)rst set of parameters are given values based on direct evidence on the parameters or to keep the model simple. A second set of parameters values are chosen so that selected moments of endogenous model quantities match empirical counterparts. Table 1 shows the (cid:133)rst set of parameter values. Table 1 Parameter values Depreciation (annual) (cid:14) 0:1 Risk aversion (cid:13) 15 Capital curvature (cid:11) 0:5 Short term sector size Z 0:75 d The depreciation rate (cid:14) at 0.1 annually is a standard value. We assume a model period of (cid:133)ve years. Based on this, the depreciation rates in the model are 1 (1 (cid:14))T with T (cid:0) (cid:0) equal to 5 or 10 for short-term and long-term (cid:133)rms, respectively. Zd is set to 0:75 to have the short term sector roughly produce half of the economy(cid:146)s output, the other half being producedbythelong-termsector. Thecapitalcurvatureparameter(cid:11)issetto0:5. Plantlevel estimates of this parameter from Cooper and Haltiwanger (2006) and Cooper and Ejarque (2003) are around 0:6 0:7, the typical values used in business cycle studies for aggregate (cid:0) production functions are around 0:3 0:4. We set risk aversion to 15, as a way of generating (cid:0) nonnegligeable risk premiums. Alternatively, we could introduce a richer utility speci(cid:133)cation as typically used in asset pricing studies. Given the computational cost, and given that the focus of the paper is not on the ability to match the overall size of risk premiums, we prefer this more parsimonious approach. Table2presentsalistofparameterswithvalueschosensothatthemodelcloselyreplicates 21

empirical counterparts. Model statistics are based on a very long simulated sample. The joint process for aggregate productivity, A, and in(cid:135)ation, (cid:25), is built from two two-state Markov chains. For both, the diagonal elements are set to determine the serial correlation coe¢ cients. In the combined four-state Markov process, elements that are associated with relativelyhigh(orlow)realizationsforboth,productivityandin(cid:135)ation,areincreasedbycr=4, the others are decreased by cr=4. This is a convenient way to introduce non-zero correlation between productivity and in(cid:135)ation. For instance, as is easy to see, if both productivity and in(cid:135)ation are IID, then cr equals the correlation coe¢ cient between productivity and in(cid:135)ation. Given that in(cid:135)ation is exogenous, it is easy to perfectly match mean, standard deviation and (cid:133)rst-order serial correlation of the data. All other target moments are endogenous, and the simulated model comes very close to matching the calibration targets. The empirical counterparts for consumption and output are based on constructing aggregates for 5 year periods, the interest rate is taken to be a (cid:133)ve year rate. Logarithms are taken of consumption, output, interest rates and in(cid:135)ation. The Appendixcontainsamoredetaileddescriptionofhowempiricalcounterpartsareconstructed. Following Bloom et al (2009), idiosyncratic productivity risk varies countercyclically with aggregate productivity, so that with low aggregate productivity it is (cid:27) +(cid:1) and with high z aggregate productivity (cid:27) (cid:1). The level of (cid:27) is set to match default rates in the data, and z z (cid:0) the variation, (cid:1), is in the range of the empirical evidence presented in Bloom et al (2009). In particular, they report the % increase of various dispersion measures in recessions relative to expansions to be between 0:23 and 0:67; our model target is set to 0:5. The correlation between in(cid:135)ation and productivity, corr((cid:25);A), is set so that the model is reasonably close to the data for the correlations of output growth with both in(cid:135)ation and the nominal interest rate. The model cannot exactly match both of these correlations. As we show below, this 22

parameter matters mainly for the correlation of the swap and the term spread. Table 2 Parameters set to match moments Parameter Target Statistic Data Model (cid:12) 0:78 E(r) 0:302 0:314 0:5 E(Recovery rate) 0:40 0:43 (cid:27) 0:37 E(Default rate) 0:0709 0:0722 z (cid:1), in (cid:27) (cid:1) 0:075 Dispersion (Rec.)/Disp(Exp.) 1 0:23 0:67 0:5 z (cid:6) (cid:0) (cid:0) Std(A) 0:0625 Std((cid:1)c) 0:032 0:035 (cid:26)(A) 0 (cid:26)((cid:1)c) 0:06 0:03 (cid:0) (cid:0) E((cid:25)) 0:184 E((cid:25)) 0:184 0:184 Std((cid:25)) 0:11 Std((cid:25)) 0:11 0:11 (cid:26)((cid:25)) 0:45 (cid:26)((cid:25)) 0:45 0:45 corr((cid:25);A) 0:3 Corr((cid:25);(cid:1)y) 0:2 0:23 (cid:0) (cid:0) (cid:0) Corr(r;(cid:1)y) 0:31 0:22 (cid:0) (cid:0) 4.2 Model implications for swaps Twofactsabout(cid:133)rms(cid:146)swapusageemergefromempiricalstudies: swappositionsarepositive ((cid:133)rms arenet (cid:133)xedpayers), andswappositionscommovenegativelywiththetermspread, so that (cid:133)rms increase net (cid:133)xed rate obligations when long-term interest rates are relatively low. As shown in Table 3, the model(cid:146)s benchmark calibration can produce these two properties, with the expected swap level at 0:33 and the correlation between the swap and the term spread at 0:75. As is clear from Table 3, the model(cid:146)s ability to produce these facts hinges (cid:0) critically on the types of risks included in the model. To better understand the determinants of the level and the cyclical behavior of the swap, it is useful to consider the solution of the simpli(cid:133)ed model, which we reproduce here for 23

convenience F R +ts s (cid:3) = R2=(cid:25) 00 (cid:12)R^ 2 (cid:0) E(cid:25) 00 (cid:12)A 0 1 (cid:12)(cid:25) 0 : (9) b RS 2 (cid:0) 3(cid:0) RS=R (cid:25) 0 R2 ! Realinterest rcate risk Producticvity risk (cid:18) 2 (cid:19) In(cid:135)atiobn risk 4 5 | {z } |{z} |{z} 4.2.1 Swap level We compare di⁄erent model versions, starting with version A in Table 3 that has only aggregate productivity shocks. Idiosyncratic productivity shocks with constant variance are included in versions A, B and C; the benchmark version features idiosyncratic productivity shocks with countercyclical variance. For case A, the level of the swap is positive and large, with an expected value of 5:98. The high level is easy to understand with the solution of the simpli(cid:133)ed model in equation (9). Indeed, as shown in Table 3, the real interest rate beta, (cid:12)R^ 2 (cid:0) E(cid:25) 00, is equal to one, because there is no expected in(cid:135)ation risk. Productivity risk also c makes a large positive contribution. Interest rates and productivity are negatively related, and (conditional) interest rate volatility is low, thus the large value for (cid:12)A 0. The in(cid:135)ation (cid:0) c beta, (cid:12)(cid:25) 0, is obviously equal to zero in this case. Note, Table 3 reports the average of the b conditional betas, computed from the model solution, as it is the conditional relation that determines the swap choice. Also, betas are computed under the risk neutral probabilities (that is, physical probabilities are scaled by the stochastic discount factor), as this is what (cid:133)rms care about (not the physical probabilities). Column B in Table 3 displays the case with productivity and in(cid:135)ation shocks, assuming that the two are uncorrelated. Unlike the case A with only productivity risk, this case produces a negative swap position, with an expected value of 1:96. Clearly, in(cid:135)ation risk (cid:0) has a (cid:133)rst-order impact on swap choice. As shown in Table 3, the change in the betas drives the strongly lower, and now negative, swap position. Most importantly, the in(cid:135)ation 24

beta makes a large negative contribution. This is because realized in(cid:135)ation and expected in(cid:135)ation (and thus interest rates) are positively correlated. With interest rates now being driven to a large extent by expected in(cid:135)ation (see the conditional interest rate volatility), the real interest rate beta and the productivity beta are now making a considerably weaker contribution. Table 3 Model implications: Swap choice depending on risks in the model (A) (B) (C) Benchmark Statistic (cid:27) (cid:27) ;(cid:27) (cid:27) ;(cid:27) ;(cid:27) (cid:27) ;(cid:27) ;(cid:27) ;(cid:1) A A I A I A;I A I A;I E(swap) 5:98 1:96 1:37 0:32 (cid:0) (cid:0) Corr(swap, term spread) 0:24 0:95 0:92 0:76 (cid:0) (cid:0) Std(swap) 0:05 0:16 0:13 0:12 Corr(swap, interest rate) 0:27 0:81 0:43 0:96 (cid:0) (cid:0) Corr(swap, in(cid:135)ation) 0:94 0:94 0:72 (cid:0) (cid:0) Std(r) 0:0626 0:0791 0:0872 0:0873 E StdQ(r t 1) 0:0180 0:0476 0:0528 0:0528 j (cid:0) Std StdQ(r t 1) 0:0012 0:0004 0:0030 0:0030 (cid:0) j (cid:0) (cid:1) (cid:12)R^ 2 (cid:0)(cid:0) E(cid:25) 00 (cid:1) 1 0:14 0:23 0:23 (cid:12)A 0 c 3:08 0:44 0:63 0:63 (cid:0) (cid:12)(cid:25)c0 0 1:91 1:73 1:73 (cid:0) (cid:0) (cid:0) (cid:0) ((cid:27) ,baggregate productivity shocks, (cid:27) , in(cid:135)ation shocks, (cid:27) negatively A I A;I correlated productivity and in(cid:135)ation, (cid:1) time-varying idiosyncratic productity) Case C introduces negative correlation between exogenous aggregate productivity and in(cid:135)ation. Relative to case B with uncorrelated productivity and in(cid:135)ation, this has a modest e⁄ect on the level of the swap. Among the betas, the contribution of the productivity beta changes the most. This is now larger, because with negatively correlated productivity and in(cid:135)ation, the negative correlation between productivity and the interest rate is stronger. The Benchmark case shown in Table 3, in addition to the shocks in case C, also features 25

idiosyncratic productivity shocks with countercyclical variance. As shown in Table 3, this has a big e⁄ect on the level, making it positive, with a mean of 0:32. While the simpli- (cid:133)ed model doesn(cid:146)t explicitly include time-varying idiosyncratic risk, the model nevertheless suggests how this can increase the level of the swap. With idiosyncratic risk higher when productivity is lower, idiosyncratic productivity risk ampli(cid:133)es aggregate productivity risk, and this ampli(cid:133)cation can be seen as requiring an increase in the swap position for hedging against aggregate productivity risk. Overall, time-varying idiosyncratic productivity risk is a necessary ingredient to generate positive swap positions and to overturn the strong negative e⁄ect from in(cid:135)ation risk. 4.2.2 Swap (cid:135)uctuations Table 3 presents correlations of the swap with the term spread, and several other moments that characterize the (cid:135)uctuations of the swap. In(cid:135)ation shocks again play a crucial role. First, in(cid:135)ation shocks make the swap more volatile, as seen in case B. Second, when in(cid:135)ation is negatively correlated with productivity as in case C, the swap moves negatively with the term spread, as does its empirical counterpart. Cyclical properties are only moderately a⁄ected when adding idiosyncratic productivity risk in the benchmark case. AsshowninTable3, incaseB,theintroductionofin(cid:135)ationshockssubstantiallyincreases the volatility of the swap compared to A, and this produces a strong positive correlation between the swap and the term spread. Figure 3 illustrates how (cid:135)uctuations in the swap are related to the components of the swap choice from the simple model presented in equation (9). The upper row shows (cid:135)uctuations due to each of the three sources of risk. The lower panels show the corresponding betas. For instance, "total productivity risk" refers to the e⁄ect of (cid:135)uctuations in F = b RS (cid:12)A 0 on the approximate swap from the simple model. R2=(cid:25) 00 (cid:25) 0 R2 (cid:16) (cid:17) c 26

In this case, only this term is allowed to change, while the other terms in equation (9) are held constant at their average level. The (cid:133)gure displays a random sample of 100 periods. To produce this (cid:133)gure, the swap of the simple model is scaled to match the mean and standard deviation of the swap from the full model, because the level of the swap in the simple model does not match exactly the level in the full model. In the third panel on the top row of Figure 3 we can see that total in(cid:135)ation risk, R1+ts (cid:12)(cid:25) 0, closely tracks the movements of the swap. Given that the in(cid:135)ation beta (cid:0) RS=R2 do (cid:16) es not (cid:17) trac b k the swap that closely, it is the "exposure" term R1+ts that makes an im- RS=R2 (cid:16) (cid:17) portant contribution to the variation in the swap. As is clear, this term is strongly positively correlated with the current one-period rate, R , and because of the negative sign, this pro- 1 duces the negative correlation of 0:81 of the swap and the current interest rate, shown in (cid:0) Table 3. Given that the term spread ln RS=R moves negatively with the short rate R , 1 1 this explains the positive correlation betw(cid:0)een the(cid:1)swap and the term spread in case B. As shown in Table 3, introducing a negative correlation between productivity and in- (cid:135)ation (going from case B to C) dramatically changes the cyclical behavior of the swap. The correlation between the swap and the term spread is now negative, at 0:92. Figure 4 (cid:0) provides some indication about why the correlation of the swap and the term spread changes so much. Indeed, total in(cid:135)ation risk now plays a minor role, but the betas, and the productivity beta in particular, are dominant drivers. The movement in the in(cid:135)ation beta mostly o⁄sets the movement in the "exposure" term R1+ts , this is why in(cid:135)ation risk now makes RS=R2 (cid:16) (cid:17) a marginal contribution to the time-variation. Clearly, from the movements in the betas we can see that there is some heteroscedasticity in the model. Table 3 shows that the standard deviation of the short rate, R , under the risk free measure, is about 8 times larger with 1 correlated in(cid:135)ation than with uncorrelated in(cid:135)ation. While quantitatively small in itself, 27

this e⁄ect has a signi(cid:133)cant impact on the cyclical (cid:135)uctuations of the swap. The main results highlighted here are not very sensitive to changes in most of the calibrated parameter values. As suggested by the discussion in the preceding paragraph, one parameter that can have some impact on some of the results is the correlation between in- (cid:135)ation and productivity. For the benchmark calibration this is set to corr((cid:25);A) = 0:3, (cid:0) as shown in Table 3. With this value, the model correlation of output growth and in(cid:135)ation corr((cid:1)y;(cid:25)) is 0:23, in the data this is 0:2, and the model correlation of output growth (cid:0) (cid:0) and the nominal interest rates is 0:22, in the data this is 0:31. Setting corr((cid:25);A) to (cid:0) (cid:0) either 0:25 or 0:4, so that the model either exactly matches corr((cid:1)y;(cid:25)) or corr((cid:1)y;r) (cid:0) (cid:0) respecetively, makes the correlation of the swap with the term spread to be either 0:22 or (cid:0) 0:93 respectively, in the benchmark case it is 0:76. In sum, while varying the value of (cid:0) (cid:0) corr((cid:25);A) can have a signi(cid:133)cant quantitative impact, within the range of reasonable values our main conclusions remain unchanged. 4.2.3 Other model implications Table 4 presents a set of model implications and their empirical counterparts that demonstrate that the model captures reasonably well a number of additional facts it was not calibrated to. Importantly, the model roughly matches the interest rate volatility and the volatility of output. The model falls somewhat short on the investment volatility. However, the empirical volatility of investment is particularly high due to the selected sample period that the includes the Great Depression. 28

Table 4 Model implications Statistic Data Model Std(r) 0:114 0:087 Std((cid:1)y) 0:097 0:071 Std((cid:1)inv) 0:231 0:153 Std (default rate) 0:037 0:066 Std(recovery rate) 0:047 0:021 4.3 Consequences and value of swap usage So far we have focused on the optimal swap choice, we consider now how the availability of interest rate swaps changes other equilibrium outcomes in the model, and how much value is created by (cid:133)rms(cid:146)access to interest rate swaps. For this purpose, we introduce a swap transaction cost that is proportional to the swap position. In period 1, with a swap, the (cid:133)rm now needs to pay s Bl RS +ss Bl;s +sgn(s)(cid:17) R ; 1 (cid:1) (cid:0) (cid:2) (cid:0) (cid:1) (cid:3) with (cid:17) 0, and in period 2 (cid:21) s Bl RS +ss Bl;s;X ;z +sgn(s)(cid:17) R (X ) : 1 0 0 2 0 (cid:1) (cid:0) (cid:2) (cid:0) (cid:1) (cid:3) The transaction cost (cid:17) adds to the bid-ask spread of the swap. If the (cid:133)rm is a (cid:133)xed rate payer, s > 0, then (cid:17) is added to the (cid:133)xed rate that is paid by the (cid:133)rm and received by the swap counterparty, and vice versa with s < 0. Empirically, (cid:133)rms pay for swaps through a bid-ask spread of this type. So far, the swap default spreads ss(:) and ss (:) allowed for 1 some bid-ask spread. However, given the vanishingly small default probabilities on the swap in the cases considered, the spread has been essentially zero. Clearly, making (cid:17) large enough will lead (cid:133)rms to stop using interest rate swaps. The sensitivity of the swap choice to (cid:17) is 29

also informative about the value created by the swap. Consider (cid:133)rst the issue how swap usage a⁄ects (cid:133)rm behavior and debt pricing. For this we can compare column 2 and column 4, in Table 5. Both consider the benchmark calibration with all the risks included. Column 2 is the benchmark case with no transaction cost (and is the same case as reported in Table 3). Column 4 has a transaction cost of 10 basis points, (cid:17) = :001, and with this level of cost (cid:133)rms stop using swaps. No real quantities are included in Table 5, as there are no signi(cid:133)cant real di⁄erences in the two cases in the model. There are some di⁄erences in default behavior and the debt risk premiums, though these are also relatively minor. As shown in equation (5), the swap choice is driven by the objective to minimize default costs. Table 5 shows that despite this objective default rates are not necessarily lower when (cid:133)rms have access to swaps. But, default becomes slightly less frequent in states with low aggregate productivity(cid:150)for which state prices are relatively high(cid:150)and defaults become more common in high productivity state that have lower state prices. The outcome of this shift of defaults can be seen in the default spread and the debt risk premium, both are moderately reduced by the presence of the swap.7 Tofurtherillustratethee⁄ectsofswapsondefaultanddebtpricing,considerthecasewith only productivity shocks, displayed in the last two columns of Table 5. Without the swap, that is the case where (cid:17) = 0:001, the default rate is 7:86% with low aggregate productivity and 3:10% with high productivity. Allowing for the swap without costs equalizes average default rates conditioned on low and high productivity. In this model version, there are is only one type of aggregate risk, the productivity shock, and conditionally, productivity and interest rates are perfectly (negatively) correlated. Therefore, the swap is very e⁄ective in managing default risk induced by aggregate shocks. 7See the Appendix for a formal de(cid:133)nition of the debt risk premium. 30

Table 5 Model implications as a function of swap transaction cost, (cid:17) (2) (3) (4) (5) (6) Benchmark Productivity shocks only (cid:17) = 0 (cid:17) = (cid:17) = (cid:17) = 0 (cid:17) = Statistic 0:00025 0:001 0:001 E(swap) 0:32 0:06 0 5:98 0 Corr(swap, term spread) 0:76 0:76 0:24 (cid:0) (cid:0) (cid:0) (cid:0) E(default rate) 0:0722 0:0719 0:0718 0:0624 0:0548 E(default rate A ) 0:1274 0:1285 0:1288 0:0624 0:0786 low j E(default rate A ) 0:0169 0:0152 0:0147 0:0623 0:0310 high Default spread j , Eln Rc 0:0607 0:0608 0:0608 0:0362 0:0384 R1 Debt risk premium 0:0177 0:0180 0:0180 0:002 0:007 (cid:0) Based on the fact that with a transaction cost of 10 basis points, (cid:133)rms stop using swaps, this can be taken as the upper bound of the value (per period) created by the swap. It is probably uncontroversial that this is a rather small number. It might be interesting then to compare this to the empirical level of costs for interest rate swaps. According to current Reuter(cid:146)s quotes, bid-ask spreads for 10-year interest rate swaps in USD are about 1 basis point (at an annualized basis). That is, from the end-users(cid:146)perspective, the cost of a swap is about 1/2 of a basis point on an annualized basis, and 2.5 basis point for a 5-year period. Introducingsuchatransactionspreadintothemodelleads(cid:133)rmstosubstantiallyreduceswap usage. As shown in Table 5, with a 2.5 basis points cost, swap usage is reduced to an average of 0.06 compared to 0.33 in the case with (cid:17) = 0. There is no change on the correlation of the swap and the term spread. The conclusion from this analysis is that swaps do not create a lot of value for the typical (cid:133)rms in this model. But, because swaps are cheap, it nevertheless makes sense to use them. The simplifying assumption that all (cid:133)rms are initially identical seems to be one reason why swaps do not create a lot of value in our model. To the extent thatsome(cid:133)rmshaverelativelyhighprobabilitiesof default, theywouldattachahighervalue 31

to the insurance provided by the swap. 5 Conclusion We have built a simple equilibrium model to study swap usage. Based on available empirical evidence, the focus is on whether this model can produce positive positions for endusers of swapsthatrequire(cid:133)xedratepayments,andanegativeco-movementbetweenswapsandterm spreads. The calibrated model with uncertainty about aggregate productivity and in(cid:135)ation, andwithcountercyclicalidiosyncraticproductivityshocks, canproducethesetwoproperties. Speci(cid:133)cally, (cid:133)rms optimally reduce the share of debt that is swapped into (cid:133)xed when long termratesarehighrelativetoshortrates. Inthemodel, all(cid:133)rmsbehaverationally, andthere are no explicit agency con(cid:135)icts between managers and shareholders. Considering the impact ofswapusage,we(cid:133)ndthatswapshaveessentiallynoe⁄ecton(cid:133)rms(cid:146)realinvestmentbehavior. The (cid:133)rms(cid:146)objective to use swaps to minimize default costs does not result in a reduction of the overall default frequency. However, (cid:133)rms endupwithmoderatelydecreaseddefault rates in recessions(cid:150)for which stochastic discount factors are high(cid:150)and with moderately increased default rates in expansions(cid:150)when stochastic discount factors are low. Thus, the cost of debt (cid:133)nancing is somewhat reduced by the swap. Overall, we (cid:133)nd only small economic gains from swap usage. Given that the conclusions of the analysis depend on the modelling assumptions, one can speculate about how general these (cid:133)ndings are. The model was designed to be as simple as possible and to only include (cid:133)rst-order macroeconomic risks. Additional risks and richer processes have the potential to change some of the conclusions about the level and cyclical behavior of interest rate swaps. For swaps to have a larger economic impact, 32

more fundamental changes in the model are needed. In typical macroeconomic models, risk does not have (cid:133)rst-order e⁄ects on production choices, therefore, our conclusions on this dimension are likely to be robust for a large class of models. 33

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[9] Flannery, Mark J., 1986, Asymmetric Information and Risky Debt Maturity Choice, Journal of Finance, vol. 41(1), pages 19-37. [10] Gomes, Joao, and Lukas Schmid, 2010, Equilibrium Credit Spreads and the Macroeconomy, unpublished manuscript. [11] Hull, John C., 2000, Options, Futures & Other Derivatives, Prentice Hall. [12] Kuehn, Lars-Alexander, 2009, Disentangling Investment Returns and Stock Returns: TheImportanceofTime-to-Build, unpublishedmanuscript, CarnegieMellonUniversity. [13] Li, Haitao and Connie X. Mao, 2003, Corporate Use of Interest Rate Swaps: Theory and Evidence, Journal of Banking and Finance, Volume 27, Issue 8, August, Pages 1511-1538. [14] Marshall,JohnFrancisandKennethR.Kapner,1993,UnderstandingSwaps,JohnWiley & Sons. [15] Moody(cid:146)sInvestorsService,2006,DefaultandRecoveryRatesofCorporateBondIssuers, 1920-2005. [16] Myers, Stewart, 1977, Determinants of Corporate Borrowing, Journal of Financial Economics, Volume 5, Issue 2, November, Pages 147-175 [17] Rouwenhorst, K. Geert, 1991, Time to Build and Aggregate Fluctuations: A Reconsideration, Journal of Monetary Economics, vol. 27(2), April, pages 241-254. [18] Smith, Cli⁄ord, W, and Rene M, Stulz, 1985, The Determinants of Firms(cid:146)Hedging Policies, Journal of Financial and Quantitative Analysis, Vol. 20, no. 4, December, pp. 391-405. 35

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A Default spread and deadweight cost This section shows how default spreads are determined, and how deadweight costs introduce a wedge between the spread determined by the investor and the valuation of the option to default for the (cid:133)rm. This section is added for completeness, the derived results are standard. Securities are priced by the investors (the market) through the stochastic discount factor M so that the price of the security V is given by V = E(M Payout); (cid:1) where Payout is the state contingent payment to be received by the owner. Consider a defaultable version of this security and consider splitting the set of possible states into two: no-default states and default states. In a no-default state the security pays the promised Payout and in a default state the security pays (cid:13)(s) <Payout(s). Clearly, this security will have a lower price than the non-defaultable claim. To justify the same price as the non-defaultable security, the defaultable security needs to o⁄er a default spread (cid:26) which we take to be a constant, so that in no-default states the security pays Payout(s)+(cid:26) while in default states it pays (cid:13)(s). The default spread is found as the solution to E(M Payout) = E [M (Payout +(cid:26))]+E [M (cid:13)(s)]; ND D (cid:1) (cid:1) (cid:1) e e where E denotes the partial expectation over the set j. This implies j e E (M [Payout-(cid:13)(s)]) = E [M (cid:26)], or D ND (cid:1) (cid:1) E[M (cid:26)] = E (M [Payout+(cid:26)-(cid:13)(s)]) e eD (cid:1) (cid:1) e e 37

which implies E (M [Payout-(cid:13)(s)]) D (cid:26) = (cid:1) : (10) E [M] ND e Thus, there is an easy way to keep the price of the defaultable security identical to the none defaultable one. It involves adding a default spread that equals the value of the loss given default E (M [Payout-(cid:13)(s)]) scaled by the value of the no-default states. D (cid:1) From the (cid:133)rm(cid:146)s ( the borrower(cid:146)s) perspective, the default spread is the price that is paid e for the option to default in some states. It is easy to see that if there are no deadweight costs for default, then a borrower that uses the same M values this option to default also as (cid:26). However, with a deadweight cost this is no longer the case, and the borrower(cid:146)s value of the default option will be lower than (cid:26), because the borrower e⁄ectively pays the deadweight cost. Assuming as in the main text that (cid:13)(s) = g(s), with < 1 capturing the deadweight cost, that is, while at default the borrower pays g(s) the lender gets only a fraction < 1 of it. In this case, the value that the borrower attaches to the option to default is given by E [M (1 g(s))] E [M (1 g(s))] (cid:26)Bor = D (cid:1) (cid:0) < D (cid:1) (cid:0) = (cid:26); E [M] E [M] ND ND e e e e that is, it is smaller than the spread actually paid. The wedge between the cost and the value of the option to default for the borrower is given by E [M (g(s))] (cid:26)Bor (cid:26) = D (cid:1) (1 ): (cid:0) E [M] (cid:0) ND e e Clearly, this wedge is larger, the smaller the recovery parameter , the more likely default is, and the more expensive default states are. Without deadweight costs, = 1, there is no 38

wedge. B Simpli(cid:133)ed model This appendix derives the simpli(cid:133)ed model of swap choice presented in the main text. As shown in Eq. (5), for a given amount of debt, the swap is chosen to minimize the expected value of default costs. For the simpli(cid:133)ed model, it is assumed that default losses are constant acrossstatesofnature, whichisapproximatelytrueifthelossimposedondefaultisrelatively large. The objective is then to minimize the probability of default (under the risk-neutral distribution), which for convenience we rewrite as EQ(cid:8)(z ) dP ; (cid:3) / ZD1(s) X(cid:3) 0 ;z 0j X whereEQ istheexpectationundertherisk-neutraldistribution, z thecuto⁄levelfordefault (cid:3) of the idiosyncratic shock, z , and (cid:8)(:) the cdf of z . 0 0 To solve min EQ(cid:8)(z ), the (cid:133)rst-order necessary condition is s (cid:3) dz EQ(cid:30)(z ) (cid:3) = 0; (cid:3) ds with (cid:30)(:) the pdf. The linear approximation of the equity value given in Eq. (7) can be written as F b[R +ts] b RS ^ ^ 1 ^ E + z^ +A R +E(cid:25) + (cid:25) +s R q R =(cid:25) 0 0 (cid:0) 2 00 (cid:25) 0 (cid:25) R 2 2 00 (cid:18) 0 (cid:19) (cid:18) 0 2(cid:19) (cid:16) (cid:17) c b where variables with (cid:145)^(cid:146)are the random variables, and x^ dx=x, and E the level of the q (cid:17) 39

equity value around which we approximate. The cuto⁄value for z^, z , is implicitly de(cid:133)ned by setting this to 0, so that 0 (cid:3) b[R1+ts] b RS z = (cid:25) 0 (cid:25) s (cid:25) 0 R2 R ^ E A ^ + R ^ +E(cid:25) (cid:3) (cid:0)(cid:16) F (cid:17) 0 (cid:0) (cid:16) F (cid:17) 2 (cid:0) q (cid:0) 0 2 00 R2=(cid:25) 00 R2=(cid:25) 00 (cid:16) (cid:17) = b (cid:25) sbb R ^ E A ^ + R ^ +E(cid:25) : c d 0 s 2 q 0 2 00 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:0) (cid:16) (cid:17) b c The (cid:133)rst-order condition can then be rewritten as dz EQ(cid:30)(z ) (cid:3) = EQ(cid:30)(z )R ^ = 0; (cid:3) (cid:3) 2 ds implying covQ (cid:30)(z );R ^ = 0; (cid:3) 2 (cid:16) (cid:17) because to a (cid:133)rst-order approximation EQ R ^ = 0. 2 (cid:16) (cid:17) Consider a (cid:133)rst-order approximation of the pdf, so that we can write (cid:30)(z ) = (cid:30) +(cid:30) z . (cid:3) 0 1 (cid:3) The (cid:133)rst-order condition can be then solved for s covQ b (cid:25) sb R ^ E A ^ + R ^ +E(cid:25) ;R ^ = 0 d 0 s 2 q 0 2 00 2 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:0) (cid:16) (cid:16) (cid:17) (cid:17) c c covQ R ^ +E(cid:25) A ^ ;R ^ covQ (cid:25) ;R ^ 1 2 00 0 2 b 0 2 (cid:0) d s = ; b s (cid:16)(cid:16) varQ R ^(cid:17) (cid:17) (cid:0) b s varQ (cid:16) R ^ (cid:17) c2 b 2 (cid:16) (cid:17) (cid:16) (cid:17) which corresponds to Eq. (8) in the main text. 40

C Data Consumption, output and investment are from NIPA. A model period being 5 years, annual data, 1930-2006, is summed over 5-year periods. Interest rates are 5-year Treasury constant maturity rate rates, 1953-2007, from Board of Governors of the Federal Reserve System, averagedover5-yearperiods. In(cid:135)ationistheCIP-U,1930-2009,averagedover5-yearperiods. Default and recovery rates are from Moody(cid:146)s (2006) for all issuers. Default rates cover 1920- 2005fortheaverageof 5-yearcumulativedefaultrates, 1970-2005forthestandarddeviation. Recovery rates cover 1982-2005. D De(cid:133)nition of default risk premium Wecanthinkofcorporatedefaultspreadsasbeingdeterminedbytwocomponents. First, the investor is compensated for the possibility of receiving less than the promised debt payment. Second, there is a risk premium due to the covariance between the realized return and the impliedstochasticdiscountfactor. Thissectionisaddedforcompleteness, thederivedresults are standard. De(cid:133)ne the realized return on a corporate bond as (cid:13)Rc, where (cid:13) 1 is the recovery rate. (cid:20) The spread can be decomposed as follows 1 ln(Rc=R ) = ln Rc=R + ln1=E((cid:13)) : 1 1 E((cid:13)) (cid:20) (cid:21) Expected default loss Risk premium | {z } | {z } The second term on the right hand side is the expected default loss, and the (cid:133)rst term, the residual, is called the risk premium. To link this (cid:133)rst term explicitly to the risk premium, 41

consider the no-arbitrage relationship u (c) 1 = E (cid:12) 0 0 (cid:13)Rc u (c) (cid:18) 0 (cid:19) that implies 1 Rc = ; E((cid:13))=R +cov (cid:12)u 0 (c 0 );(cid:13) 1 u(c) 0 (cid:16) (cid:17) withthecovariancetermcapturingtheriskpremium. Asiseasilyseen,ifcov (cid:12)u 0 (c 0 );(cid:13) = 0, u(c) 0 (cid:16) (cid:17) then ln(Rc=R ) = ln1=E((cid:13)), which explicitly justi(cid:133)es our labelling of the residual term as 1 the risk premium in the (cid:133)rst equation of this section. Alternatively, by de(cid:133)nition, 1 = EQ((cid:13))Rc=R , where EQ is the expectation under risk-neutral probabilities. Thus, the risk 1 premium equals E((cid:13))=EQ((cid:13)). 42

16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 1998-H1 1998-H2 1999-H1 1999-H2 2000-H1 2000-H2 2001-H1 2001-H2 2002-H1 2002-H2 2003-H1 2003-H2 2004-H1 2004-H2 2005-H1 2005-H2 2006-H1 2006-H2 2007-H1 2007-H2 2008-H1 2008-H2 2009-H1 2009-H2 2010-H1 2010-H2 2011-H1 2011-H2 2012-H1 US dollar Interest Rate Swaps by Non-financial Institutions (Notional Amounts in trn USD, source BIS) 43

20 3.5% 3.0% 15 2.5% 10 2.0% 1.5% 5 1.0% 0 0.5% 0.0% -5 -0.5% -10 -1.0% dexif otni deppawS % 1993-12-01 1994-12-01 1995-12-01 1996-12-01 1997-12-01 1998-12-01 1999-12-01 2000-12-01 2001-12-01 2002-12-01 )M6 - Y01( daerps dleiY US Corporate Swap usage compared to Yield Spread ( swap % of total debt, source Chernenko and Faulkender (2011) ) 44

total productivity risk -1.8 -1.9 -2 -2.1 -2.2 -2.3 -2.4 20 40 60 80 100 paws total real interest rate risk -1.8 -1.9 -2 -2.1 -2.2 -2.3 -2.4 20 40 60 80 100 paws total inflation risk -1.8 -1.9 -2 -2.1 -2.2 -2.3 -2.4 20 40 60 80 100 paws productivity beta -1.8 -1.9 -2 -2.1 -2.2 -2.3 -2.4 20 40 60 80 100 paws real interest rate beta -1.8 -1.9 -2 -2.1 -2.2 -2.3 -2.4 20 40 60 80 100 paws inflation beta -1.8 -1.9 -2 -2.1 -2.2 -2.3 -2.4 20 40 60 80 100 paws 45

total productivity risk -1.2 -1.25 -1.3 -1.35 -1.4 -1.45 -1.5 -1.55 -1.6 -1.65 20 40 60 80 100 paws total real interest rate risk -1.2 -1.25 -1.3 -1.35 -1.4 -1.45 -1.5 -1.55 -1.6 -1.65 20 40 60 80 100 paws total inflation risk -1.2 -1.25 -1.3 -1.35 -1.4 -1.45 -1.5 -1.55 -1.6 -1.65 20 40 60 80 100 paws productivity beta -1.2 -1.25 -1.3 -1.35 -1.4 -1.45 -1.5 -1.55 -1.6 -1.65 20 40 60 80 100 paws real interest rate beta -1.2 -1.25 -1.3 -1.35 -1.4 -1.45 -1.5 -1.55 -1.6 -1.65 20 40 60 80 100 paws inflation beta -1.2 -1.25 -1.3 -1.35 -1.4 -1.45 -1.5 -1.55 -1.6 -1.65 20 40 60 80 100 paws 46