Pricing the Credit Risk of Secured Debt and Financial Leasing

Journal of Business Finance & Accounting, 33(7) & (8), 1298–1320, September/October 2006, 0306-686X doi: 10.1111/j.1468-5957.2006.00619.x Pricing the Credit Risk of Secured Debt and Financial Leasing Marco Realdon∗ Abstract: This paper presents closed form solutions to price secured bank loans and financial leases subject to default risk. Secured debt fair credit spreads always increase in the debtor’s default probability, whereas financial leasing fair credit spreads may well decrease in the lessee’s default probability and even be negative. The reason is that the lessor, unlike the secured lender, can gain from the lessee’s default, especially when the leasing contract envisages initial prepayments or the lessee’s terminal options to either purchase the leased asset or to extend the lease maturity. This result, which critically depends on contractual and bankruptcy code provisions, can explain some of the empirical evidence and the use of financial leases as an alternative to secured bank lending to finance small, risky and relatively opaque firms. Keywords: default risk, secured debt, debt valuation, collateral asset, leasing valuation, leasing options. 1. INTRODUCTION AND LITERATURE Secured bank loans and financial leasing contracts are considered by the renegotiated Basel accord on the capital requirements of financial institutions. The new accord makes capital requirements depend on the credit risk of the institution’s assets. This has spurred interest in modelling the credit risk of secured loans and financial leases. This work intends to contribute to such modelling effort by proposing simple formulae to price secured bank loans and financial leases that are subject to the risk of default. According to Brealey, Myers and Allen (2005) ‘in many respects, a financial lease is equivalent to a secured loan’, but ‘. . . lessors and secured creditors have different rights when the asset user gets into trouble.’ 1 Indeed the present analysis highlights how the credit risk of financial leases is profoundly different from the credit risk of secured loans. * The author is from the University of York. He thanks an anonymous referee for suggesting substantial improvements to the previous drafts of this paper. (Paper received March 2005, revised version accepted January 2006. Online publication June 2006) Address for correspondence: Marco Realdon, Department of Economics and Related Studies, University of York, Alwin College, Heslington Y010 5DD, UK. e-mail: [email protected] 1 Brealey Myers and Allen (2005, pp. 708–09). 2006 The Author C 2006 Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK Journal compilation  and 350 Main Street, Malden, MA 02148, USA.  C 1298 PRICING THE CREDIT RISK OF SECURED DEBT 1299 Bank loans are often secured on collateral assets such as real estate properties or financial securities. Collateral enhances the recovery value of the loan in case of default and is key to the valuation of loan contracts. Financial leases resemble secured loans in that the lessor can seize the leased asset if the lessee defaults. But the structure of financial leases is more akin to a financial swap whereby use of the asset is exchanged for periodic rental payments. Terminal options to purchase the leased asset and a variety of other contract provisions complicate the valuation of financial leases. According to Leaseurope, in 2003 leasing reached 17% of business investments in fixed assets in the European Union. As for secured loans, this paper mainly contributes new closed form valuation formulae whereby default on the loan can occur at any time, since default is modelled as per ‘reduced’ models involving default intensities. Instead the past literature on secured debt valuation employed ‘structural’ models along the lines of Merton (1974), it resorted to numerical solutions and it made the severely restrictive assumption that default can only take place at debt maturity. Such is the case in Stulz and Johnson (1985) and also in Jokivuolle and Peura (2003), who proposed a structural model to value secured bank loans and estimate expected default losses. These two contributions assumed that the value of the collateral asset follows a diffusion process correlated with the value process of the debtor’s assets. Then this paper shows that credit spreads on financial leases, unlike credit spreads on secured loans, may well decrease in the lessee’s default probability. This means that the value of a financial lease to a lessor may increase as the lessee’s default probability rises. The reason is that the lessor, unlike the secured lender, can gain from the lessee’s default especially when the leasing contract envisages initial prepayments or the lessee’s terminal options to purchase the leased asset or to extend the lease maturity. Contractual and bankruptcy code provisions are shown to critically affect the lessor’s payoff upon the lessee’s default and hence the credit risk of financial leases. The present analysis may also explain stylized facts and empirical evidence which have appeared in past literature. For example, Sharpe and Nguyen (1995) found that leasing reduces the cost of borrowing of low credit quality firms and indeed the present analysis reveals how the lease fee can decrease in the lessee’s default probability. This fact can also explain why Krishnan and Moyer (1994) found that firms of lower credit quality are more likely to choose financial leases, and why Ang and Peterson (1984) found that firms with leases tend to be more levered. Then it also makes sense to conclude, as Lasfer and Levis (1998) did, that tax savings are not a major determinant of the leasing decision of small UK firms. Small and relatively opaque firms of low credit quality should often choose financial leases anyway. Schallheim et al. (1987) observed that yields on US lease contracts seem to rise with the lessee’s profitability and decrease with leverage. Their evidence suggests that the relation between yields on leases and the lessee’s default probability may be negative as well as positive. Again this is a prediction of the present analysis. DeLaurentis and Geranio (2001), Schmit and Stuyck (2002) and Schmit (2004) found that the leasing industry enjoys relatively high recovery rates, which supports the assumption of the present analysis whereby the lessor may gain from the lessee’s default. This paper is close in spirit to Grenadier (1995, 1996 and 2003), who analysed leasing contracts and their credit risk in a structural model framework in which the default barrier is constant over time. He also modelled a variety of provisions used in leasing contracts. Grenadier provided useful closed form solutions for lease valuation, 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation   C 1300 REALDON while assuming that the value of the lessee’s assets and that of the leased asset follow correlated diffusion processes. The lessor repossesses the asset net of bankruptcy costs after the lessee defaults. This paper differs from his contributions in that the lessee’s default probability is driven by a default intensity rather than by a barrier hitting time. Unlike in Grenadier, this paper shows that leasing credit spreads can be negative and may decrease as the lessee’s default probability rises. Then this paper considers the lessor’s right to be compensated for the lost profit due to the lessee’s default, on top of repossessing the asset, and considers also the lessee’s right to keep honouring the leasing contract even after bankruptcy filing in violation of the par condicio creditorum. These rights are shown to significantly affect the credit risk of financial leases. Another branch of the literature on leasing valuation abstracts from default risk. Myers et al. (1976) and Franks and Hodges (1978) evaluate the tax advantages of leasing over debt financing. Instead this paper abstracts from tax considerations and focuses on credit risk pricing. McConnell and Schallheim (1983) provide a theoretical analysis to value leasing contracts under a variety of contractual options, but they do not model the lessee’s default risk, which is the thrust of this paper. Indeed the present analysis highlights important interactions between contractual lease options and default risk. This paper models the credit risk of secured loans and financial leases according to the ‘reduced form’ approach, whereas past literature employed ‘structural’ approaches. Reduced and structural models hinge on different assumptions as to the information set available to investors and are both valid. Structural models provide economic insight and imply that default is a predictable event, but they pose the implementation problem of estimating the value process of the firm’s assets. This process is difficult to estimate when the firm’s stock is not traded, and most secured lending and financial lease contracts concern firms with no traded stock. To make estimation even more problematic, Peterson (1999) highlights that small businesses are often quite ‘opaque’ to banks and that transparency is likely to be even less in the lessee-lessor relation. Reduced models are amenable to simpler implementation than structural models in that they often just require estimates of default intensities or term structures of default intensities, which can be inferred by looking at debt contracts with similar credit risk or credit rating. Estimates of default intensities can easily be made consistent with the internal rating based (IRB) calculation of capital requirements of the Basel accord and with approaches such as Credit Risk+ to model loan portfolios and leasing portfolios (see Schmit, 2002 and 2004). Hence, on the grounds of simpler estimation/implementation and of tractability, this paper proposes reduced models rather than structural models in order to price and assess the credit risk of secured bank loans and financial leases. Furthermore, reduced models seem preferable to structural models for loans and leases of maturities of less than five years, and such maturities are very common. The reason is that, in the short term, structural models notoriously tend to predict too low probabilities of default and too low credit spreads, as documented in Covitz and Downing (2002) and as reported in Sundaresan (2000, p. 1594), whereas reduced models do not suffer from this limitation. Hence in reduced models the short and medium term credit spreads on loans tend to be much higher than in structural models, and the short and medium term credit spreads on financial leases, which can be either positive or negative, tend to be much further from zero. Also structural models can predict negative credit spreads on leases, but negative spreads are much more likely when early default is envisaged as in reduced models.  C 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation  PRICING THE CREDIT RISK OF SECURED DEBT 1301 More generally reduced models also have the advantage that they do not make any assumption as to the causes of defaults on loans or leases. 2. SECURED DEBT VALUATION This section analyses secured debt, such as secured bank loans, under the assumptions of a flat and constant term structure of interest rates and of constant default intensity of the debtor. Later these restrictive assumptions are relaxed and some generalisations are introduced. We assume that the debt contract envisages maturity T , face value P and fixed coupons continuously paid at the yearly rate cP . Without loss of generality, we assume universal risk neutrality. The term structure of interest rates is flat and constant. The debt obligation is secured on a collateral asset worth S. The collateral asset value evolves according to a geometric Brownian motion such that: d S = m Sdt + σ Sdz (1) where m is the drift of S in the risk-neutral world and dz is the differential of a Wiener process; m and σ are constant. The debtor can default at any time with constant risk-neutral default intensity λ, such that the probability of default occurring in any infinitesimal time interval dt is λdt. Later we consider the case whereby λ is a function of time. As the debtor defaults at time τ ≤ T, the secured creditor receives: R(S) = [S − max(S − P , 0) + π max(P − S, 0)] = [min(S, P ) + π max(P − S, 0)] (2) where π (with 0 ≤ π ≤ 1) is the recovery rate to unsecured creditors after default. This condition states that the secured creditor can satisfy his claim by selling the collateral asset worth S. But if S < P , the secured creditor also concurs with unsecured creditors to satisfy the residual amount of his claim, i.e. max (P − S, 0), and receives a fraction π of such amount, since π is the recovery rate for unsecured creditors. The absence of arbitrage opportunities implies that before default secured debt value D(S, t) at any time t ≤ T must satisfy the following equation and conditions (to simplify notation we set D(S, t) = D): d2 D 1 2 2 dD dD + σ S + mS − (r + λ) D + c P + λR (S) = 0 dt d S2 2 dS (3) d2 D →0 S→∞ d S 2 (4) lim lim D → P S→0    c πλ  + 1 − e −(λ+r )(T−t) + P e −(λ+r )(T−t) r +λ r +λ D (S, T) = P . (5) (6) Condition 4 states that D(S, t) becomes a linear function of S as S → ∞. Condition 5 states that, when the collateral asset becomes worthless, D(S, t) tends to the value 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation   C 1302 REALDON of an unsecured debt with the same default intensity, coupon, maturity and principal and with default recovery value of P π . It can be shown that such unsecured debt would have present value equal to:    c πλ  (7) P + 1 − e −(λ+r )(T−t) + P e −(λ+r )(T−t) . r +λ r +λ λ c + rπ+λ ) is the present value of perpetual debt that is subject to Notice how P ( r +λ default with intensity λ, that pays coupons at rate Pc until default and that recovers π P in case of default. Condition 6 states that at maturity T the face value P is reimbursed if the borrower has not previously defaulted. The solution to the above equation and conditions is: D (S, t) =  Pc  1 − e −(λ+r )(T−t) + P e −(λ+r )(T−t) r +λ   λS + 1 − e −(r −m+λ)(T−t) r −m+λ      λP  λS −(λ+r )(T−t) −(λ+r −m)(T−t) +π − + O (S, t) 1−e 1−e r +λ r +λ−m − O (S, t). (8) Notice that the first line of this equation is the present value of a defaultable debt that recovers nothing in case of default, while the second line is the present value of a claim that pays S at the time of default. The third line is the present value of a claim that pays π max(P − S, 0) at the time of default. This claim corresponds to the secured creditor’s right to concur with unsecured creditors to satisfy his residual unsatisfied claim after receiving S < P at default. Finally O(S, t) is the value of a claim that pays max(S − P , 0) at the time of default. It can be shown that:   O (S, t) = O(S) − C O h (S) · 1 S≥P , t − C (Ol (S) · 1 S<P , t) (9) O(S) = O h (S) · 1 S≥P + Ol (S) · 1 S<P (10) where 1 S>P is the indicator function such that 1 S>P = 1 if S >P and 1 S>P = 0 if S ≤ P . O(S) is the value of a perpetual claim that pays max (S − P , 0) at the time of default, i.e. lim T→∞ O(S, t) → O(S). O (S) will equal Oh (S) when S ≥ P , i.e. when max(P − S, 0) = 0. But O(S) will equal Ol (S) when S < P , i.e. when max(P − S, 0) > 0.C (O h (S) · 1 ST ≥P , t) can be thought of as the present values at time t ≤ T of a claim whose payoff is Oh (S) at time T provided that τ >T and ST ≥ P .C (Ol (S) · 1 ST <P , t) is the present value of a claim whose payoff is Ol (S) at time T provided that τ >T and ST < P . We recall that the condition τ >T means that default must not occur before T . The solutions for Oh (S) and Ol (S) are:  q − λP S λS (11) − + Ah O h (S) = r +λ−m r +λ P   Pλ 1 1 λP q + Ah = − − (r + λ − m) (q + − q − ) r +λ−m r + λ (q + − q − ) (12)  C 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation  1303 PRICING THE CREDIT RISK OF SECURED DEBT q± =  − m− Ol (S) = Al Al =  S P σ2 2 ±  m− σ2 2 2 + 2σ 2 (r + λ) (13) σ2 q + (14) Pλ q− 1 + Ah · . r + λ − m q+ q+ (15) The solutions for C (Oh (S) · 1 S ≥ P , t) and C (Ol (S) · 1 S<P , t) are: C (O h (S) · 1 S≥P , t) =    S Ah (−(r +λ)+n(q − ))(T−t) q− , q− ·S ·N d ·e (P )q − P    λS S + · e −(r +λ−m)(T−t) · N d r +λ−m P     √ λP S − · e −(r +λ)(T−t) · N d −σ T −t r +λ P (16) C (Ol (S) · 1 S<P , t) =    S Al (−(r +λ)+n(q + ))(T−t) q+ · S · N −d ·e , q+ (P )q + P (17) with:     q ln (z) + n (q ) + 21 σ 2 q 2 (T − t) ln (z) + m + 21 σ 2 (T − t) , d (z, q ) = , d (z) = √ √ σ T −t qσ T − t 1 1 n (q ) = mq + q (q − 1) σ 2 , N (x) = √ d 2 2 π x u2 e − 2 du. −∞ We denote with G(S, t) the value of the collateral provision in the secured debt contract. Thus G(S, t) is the difference between secured debt value and unsecured debt value. We can also think of G(S, t) as the value of a third party guarantee backing an unsecured bank loan. The guarantee would be limited to the value of a pledged asset worth S. At default the collateral provision or the guarantee grant the debt holder the total payoff of [S − max(S − P , 0) + π max(P − S, 0)] rather than π P , which would be the payoff to the unsecured debt holder. It follows that G(S, t) is the present value of the difference in the default payoff with and without the collateral provision/guarantee. Thus G(S, t) is the present value of S − max (S − P , 0) + π max (P − S, 0) − π P 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation   C 1304 REALDON received at τ . It can be shown that: G (S, t) =   λS · 1 − e −(λ+r −m)(T−t) − O (S, t) r +λ−m      λP  λS +π 1 − e −(λ+r )(T−t) − 1 − e −(λ+r −m)(T−t) + O (S, t) r +λ r +λ−m  λπ P  (18) · 1 − e −(λ+r )(T−t) . − r +λ Empirical evidence confirms the importance of collateral provisions to enhance the post-default recovery value of bank loans. Gupton et al. (2000) estimated that the postdefault fair value of US senior secured bank loans averages 69.5% of face value and the post-default fair value of senior unsecured loans averages 52.1%. A similar formula to that for G(S, t) can be derived also to value a second order guarantee, i.e. a guarantee which is subordinated to a higher order guarantee on the same asset worth S. Moreover, the value of an asset that has been pledged to guarantee a debt obligation is simply S − G(S, t). We stress that the above formulae for valuing secured loans and guarantees are in closed form, whereas Stulz and Johnson (1985) and Jokivuolle and Peura (2003) employ numerical solutions and make the restrictive assumption that default can only take place at maturity. Next we explore the predictions of these formulae. (i) Comparative Statics Figures 1 and 2 show that the secured debt valuation model predicts that credit spreads increase in the default intensity λ and in the volatility σ of the collateral asset. This follows from comparing the base case credit spreads, which assume λ = 0.02 and σ = 0.2, with the credit spreads labelled ‘λ = 0.01’ and ‘λ = 0.04’ in Figure 1 and ‘volatility = 0.1’ and ‘volatility = 0.4’ in Figure 2. Volatility depresses secured debt value since the latter is an everywhere concave function of the collateral asset value S. Such concavity is apparent in Figure 3 and becomes accentuated when the default intensity λ rises. Figures 1 and 2 show how the term structures of credit spreads on secured debt contracts are typically upward sloping. Unreported simulations show that the term structure is downward sloping when the initial collateral requirement is such that PS > 1 and m > 0. These results somewhat parallel Merton’s (1974) upward (downward) sloping term structures of credit spreads for low (high) risk firms. The case labelled ‘m = −0.2’ in Figure 2 shows that the term structure of credit spreads rises as m decreases. Typically m < 0 if S is an asset subject to economic depreciation and m = r if S is a traded asset that pays no dividend. As the recovery parameter π rises, secured debt becomes less sensitive to S, m and σ , since the potential loss given default is reduced. If the lender initially requires that the collateral asset value be significantly higher than the loan face value (e.g. PS < 0.75), then debt value becomes less sensitive to changes in volatility σ . Thus tightening collateral requirements makes errors in the estimate of collateral asset volatility less material. Figure 3 illustrates how D(S, t) varies with S and explains also the terms of the trade-off between the level of the interest rate c required by the bank and the tightness of collateral requirements. At least when the default intensity λ is reasonably low, for  C 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation  PRICING THE CREDIT RISK OF SECURED DEBT 1305 Figure 1 Term Structures of Credit Spreads on Secured Debt: The Effects of Collateral and of Default Intensity Notes: Maturity is measured in years. The base case assumes S = 100, P = 100, m = −0.1, σ = 0.2, r = 0.04, c = 0.05, λ = 0.02, π = 0. All other scenarios assume the same inputs as the base case but for the different inputs displayed in the respective lebels. example less than 5%, a slight increase in c has the same effect on D(S, t) as a dramatic increase in the initial collateral requirement. For example, in the base case secured debt is worth 102.1% of par when S = P = 100, but if the coupon rate is raised from 5% to 6% secured debt is worth 102.1% of par when S = 21. In other words a 1% increase in the coupon rate is offset by a decrease in the loan-to-value ratio PS from 1 to about 0.2. This shows that secured debt is quite insensitive to tight collateral requirements when the borrower is not risky, i.e. when λ is low. As λ rises, the concavity of D(S, t) becomes more accentuated and secured debt value becomes much more sensitive to S and σ . Now we relax some of the restrictive assumptions of the present setting. (ii) Introducing Time Dependence Closed form solutions are possible even when the default intensity λ, the instantaneous interest rate r and the face value of outstanding debt P change over time according n−1 n−1 to step functions, i.e. when λ(t) = i=0 1ti <t<i+1 · λi , r (t) = i=0 1ti <t<i+1 · r i , P (t) = n−1 i=0 1ti <t<i+1 · Pi , where ti are the dates when the time dependent functions λ(t), r (t) and P (t) change in value. In this setting secured debt is worth: n−1 e i=0 i−1 j =0 −(Tj +1 −Tj )(λ j −λi +r j −r i ) [D (S, t; λi , Pi , Ti+1 ) − D (S, t; λi , Pi , Ti )] 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation   C (19) 1306 REALDON Figure 2 Term Structures of Credit Spreads on Secured Debt: The Effect of Volatility and Drift of the Collateral Asset Notes: Maturity is measured in years. The base case assumes S = 100, P = 100, m = −0.1, σ = 0.2, r = 0.04, c = 0.05, λ = 0.02, π = 0. All other scenarios assume the same inputs as the base case but for the different inputs displayed in the respective labels. σ measures ‘volatility’. where D (S, t; λ i , Pi , Ti ) is just a more thorough notation to indicate D(S, t) when assuming that λ = λ i , P = Pi , T = Ti . This setting encompasses a non-flat term structure of interest rates, time dependent default intensity and amortisation of secured debt as Pi decreases over time. Amortisation reduces exposure to default risk. The merit of debt amortisation is greater when S is subject to economic depreciation, i.e. when m < 0, so that the decrease in Pi over time somehow matches the expected decline in the value of the collateral asset. In passing we note how, when π = 0, equation 19 can be re-interpreted to also value unsecured bonds if min(S, P ) is taken to be the amount of assets available to satisfy bondholders in case of default, i.e. if min(S, P ) is the recovery value of defaulted bonds. Having analyzed secured debt, now we analyse the credit risk of financial leasing and contrast it with that of secured debt. 3. FINANCIAL LEASING VALUATION This section analyses financial lease contracts and compares the credit risk of financial leasing with that of secured debt. Although financial leasing and secured loans are often assimilated, this section shows the different impact of credit risk on the valuation of these two contract types. While retaining the assumption of universal risk neutrality, we can also re-interpret the above model for secured debt in order to value financial leasing. In particular now S is the value of the leased asset (e.g. real estate property or a piece of equipment). In  C 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation  PRICING THE CREDIT RISK OF SECURED DEBT 1307 Figure 3 The Value of a Secured loan: Comparative Statics Notes: The base case again assumes S = 100, P = 100, m = −0.1, σ = 0.2, r = 0.04, c = 0.05, λ = 0.02, π = 0. Scenarios other than the base case assume the same inputs as the base case but for the different inputs displayed in the respective labels. a world of certainty d S = mSdt and often m < 0 to reflect the loss of leased asset value due usage, obsolescence, etc.. . Now λ is the constant instantaneous default intensity of the lessee. T is the lease termination date. The lessee continuously pays a constant fee at the yearly rate f . Following Grenadier (1996) we assume that uSdt is the market value of use of the asset (i.e. of the service flow) for the infinitesimal period dt. u is constant. U (S, t) is the present value of the expected service flow captured by using the asset during the whole life of the lease or until the lessee’s default. In the absence of arbitrage U (S, t) satisfies the following equation and conditions: dU (S, t) d 2 U (S, t) 1 2 2 dU (S, t) + σ S + mS − (r + λ)U (S, t) + uS = 0 dt d S2 2 dS s.t. : lim U (S, t) → ∞, lim U (S, t) → 0, U (S, T) = 0. S→∞ S→0 (20) uS The solution to this equation is U (S, t) = r +λ−m (1 − e −(λ+r −m)(T−t) ) provided r > m. Notice that, if λ = 0 and T → ∞, then U (S, t) → S, i.e. the value of a default-free perpetual lease must coincide with the value of the leased asset itself. This implies that u = r − m. We denote with τ (with t ≤ τ ≤ T) the time of the lessee’s default (e.g. the time when the lessee misses a contractual payment of the leasing fee). Following Grenadier (1996), we assume that at default the lessor repossesses the asset and leases it again at the then prevailing market conditions to a default free lessee until time T . In this process the lessor typically incurs legal costs or opportunity costs due to the impossibility 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation   C 1308 REALDON of immediately leasing the asset again. Hence at the lessee’s default the lessor incurs bankruptcy costs equal to wU (S, τ , λ = 0) = wS(1 − e −(r −m)(T−τ ) ) where w (with 0 ≤ w ≤ 1) is the parameter that captures the costs due to the lessee’s default and U (S, τ , λ = 0) is equal to U (S, τ ) when λ = 0. Thus the default payoff to the lessor is:   Rl (S, τ ) = −L (S, τ ) − wS 1 − e −(r −m)(T−τ ) (21) where L(S, τ ) denotes the value of the leasing contract to the lessor at time τ . Notice that the default payoff to the lessor Rl (S, τ ) is equal to the lost value of the leasing contract −L(S, τ ) minus the default related costs. Next we need to determine L(S, t) for a typical contract. Typically at maturity T the lessee, provided he has not previously defaulted, has the right to acquire the asset by paying K to the lessor. In this setting the absence of arbitrage opportunities implies that the value of the leasing contract to the lessor L(S, t) must satisfy the following equation and conditions: d L (S, t) d L (S, t) d 2 L (S, t) 1 2 2 + σ S + mS − r L (S, t) − uS + f + λRl (S, t) = 0 2 dt dS 2 dS   f 1 − e −(λ+r )(T−t) (22) d 2 L (S, t) s.t. : lim → 0, lim L (S, t) → , 2 S→∞ S→0 dS r +λ L (S, T) = − max (S − K , 0) . The lower boundary condition for S → 0 implies that the lessee keeps paying the leasing fees even as the leased asset becomes worthless. But as S → 0 the lessee may be tempted to default strategically, if default could liberate the lessee from its fee payment obligations toward the lessor. As in Grenadier (1996) we abstract from this possibility. Later we will also consider the case of those jurisdictions whereby default does not liberate the lessee from its obligations to the lessor, since the lessee is required to compensate the lessor for any losses due to its default. In such case the lessee’s strategic default would not make sense. The solution to (22) can be written as: L(S, t) = L s (S, t) − F (S, t, T) − e −λ(T−t) L o (S, t) L s (S, t) =  f  1 − e −(λ+r )(T−t) − U (S, t) r +λ F (S, t, T) =   λ + (r − m) e −(r +λ−m)(T−t) − (λ + r − m) e (m−r )(T−t) wS λ+r −m (23) (24) (25) L o (S, t) =        √ S S − e −r (T−t) K N d −σ T −t . Se −(r −m)(T−t) N d K K (26) L s (S, t) is the value of a claim that pays the fixed fee f in exchange for the service flow uS until the lessee’s default. F (S, t, T ) is the present value of the expected bankruptcy  C 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation  PRICING THE CREDIT RISK OF SECURED DEBT 1309 costs incurred by the lessor as the lessee defaults. It can be shown that: T F (S, t, T) = t T = t e −(λ+r )(τ −t) λwE t [U (S, τ, λ = 0)] dτ   e −(λ+r )(τ −t) λwSe m(τ −t) 1 − e −(r −m)(T−τ ) dτ (27) where E t [..] denotes the expectation conditional on information at time t. The term e −λ(T−t) L o (S, t) is the present value of the lessee’s terminal option to purchase the leased asset provided he has not defaulted. Notice that the lessee loses such option in case of default and e −λ(T−t) is the probability of the lessee not defaulting in the time interval [t, T ]. We could generalise these results to allow the default intensity λ(t) and the default free interest rate r (t) to be step-wise functions of time as we did for secured debt in the previous section. Closed form solutions would still be available, but for simplicity the following analysis assumes that λ and r are constant. We can already notice that, if the leasing contract does not envisage the lessee’s terminal purchase option, then L o (S, t) = 0 and the value of the leasing contract L(S, t) does not depend on the leased asset volatility σ . Now we can determine fair leasing fees in this setting. But we have to consider that the lessee is often required to make a prepayment at contract initiation time t 0 . Assuming that at time t 0 the market value of the leased asset is S 0 , such prepayment is typically a fraction a (with 0 ≤ a ≤ 1) of S 0 . If a = 0 the fair leasing fee f ∗ is such that the initial present value of the leasing at t 0 is zero to both the lessee and the lessor, i.e. L(S 0 , t 0 ) = 0. If a > 0, then the fair leasing fee f ∗ is such that the leasing is initially worth L(S 0 , t 0 ) = −aS 0 to the lessor, which entails that: f ∗ = (r + λ) −aS0 + F (S, t, T) + e −λ(T−t) L o (S, t) + U (S, t) . 1 − e −(λ+r )(T−t) (28) Now we can compute term structures of leasing fees and of credit spreads on the leasing fees. Following Grenadier (1996), we define the credit spread cs on a financial leasing as the difference between the fair leasing fee f ∗ when λ > 0 and the fee f ∗ when λ = 0. Term structures of fees and of credit spreads are shown in Table 1 and Figures 4 and 5, which highlight a number of points. When a = 0 and K → ∞, i.e. when no prepayment or terminal purchase option is envisaged, credit spreads on financial leasing are nonnegative. This is shown in the three columns under the ‘base case’ heading in Panel A of Table 1. Then credit spreads increase as m decreases, i.e. as the leased asset depreciates more quickly. The reason is that the expected value of L s (S, τ ) for any possible default date t 0 ≤ τ ≤ T is: E t [L s (S, τ )] =  (r − m) Se m(τ −t0 )   f  1 − e −(λ+r )(T−τ ) − 1 − e −(λ+r −m)(T−τ ) > 0 r +λ r +λ−m (29) since the present value of the remaining leasing fees is higher than the expected value of the remaining service flow of the leased asset. In other words, since the leased asset 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation   C 1310 REALDON Figure 4 Term Structures of Credit Spreads on Financial Leasing: The Effects of Prepayment and of the Purchase Option Exercise Price 60 40 20 Credit spreads 1 2 3 4 5 6 7 8 9 10 -20 -40 Base case -60 K = S exp[-(r-m)(T-t)] K=0 -80 a = 25% K = 0, a = 25% -100 K = 40 -120 Time to maturity (in years) Notes: This is a plot of the credit spreads in Table 1 shown in the columns headed cs for the various scenarios. depreciates exponentially, the lessee’s default causes the lessor to lose the lease contract when such contract is a net asset for the lessor. It follows that as m decreases E t [L s (S, τ )] rises as well as credit spreads. This can be seen by comparing the columns under the heading ‘base case’ in Panel A of Table 1, which assumes m = −0.2, and the columns under the headings ‘m = 0’ and ‘m = −0.04’ in Panel C. These considerations explain why spreads are positive even when the bankruptcy costs parameter is w = 0, as in the base case. Of course credit spreads rise as w rises. When m is positive, which may often be the case if the leased asset is a real estate property, then E t [L s (S, τ )] < 0 and we tend to observe even negative credit spreads. When the lease contract envisages prepayment or the lessee’s terminal purchase option, spreads may decrease in the lessee’s default intensity λ and even negative credit spreads are often possible especially for shorter lease maturities. In other words, when a > 0 or K → 0, it can happen that the fair lease fee decreases as the lessee’s default probability rises and that the fair lease fee to a default-free lessee can be higher than the fair fee to a default-risky lessee, ceteris paribus. This can be seen in Panel A of Table 1 by comparing the columns under the heading ‘base case’, which assumes a = 0 and K → ∞, and the columns under the headings ‘a = 25%’, ‘K = 0’, ‘K = 0 and a = 25%’. These results are confirmed in Panel B. Credit spreads decrease as K decreases and negative spreads are possible. The reason is that the lessee’s default may be beneficial to the lessor when the lease contract envisages the lessee’s terminal purchase option or prepayment at contract initiation. We can explain this as follows. First upon default the lessee loses the valuable  C 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation  Panel A Maturity T-t T-t T-t T-t T-t T-t T-t T-t T-t T-t =1 =2 =3 =4 =5 =6 =7 =8 =9 = 10 Panel B Maturity T-t T-t T-t T-t T-t T-t T-t T-t T-t T-t =1 =2 =3 =4 =5 =6 =7 =8 =9 = 10 Base Case ∗ ∗ a = 25% f (λ = 2%) f (λ = 0%) cs 21.77 19.86 18.21 16.78 15.55 14.47 13.53 12.71 12.00 11.36 21.77 19.83 18.16 16.69 15.42 14.31 13.33 12.47 11.71 11.03 1 3 5 9 13 17 21 25 29 33 a = 10% ∗ ∗ K=0 f (λ = 2%) f (λ = 0%) cs 6.59 9.10 9.75 9.76 9.51 9.16 8.78 8.40 8.04 6.83 9.31 9.93 9.90 9.62 9.23 8.81 8.40 8.00 - 23 - 21 - 12 - 15 - 11 -7 -3 0 4 ∗ K = 0, a = 25% f (λ = 2%) f (λ = 0%) cs f (λ = 2%) f ∗ (λ = 0%) cs 101.22 51.41 34.91 26.72 21.85 18.64 16.37 14.68 13.38 12.35 102.01 52.03 35.37 27.05 22.07 18.75 16.38 14.61 13.23 12.13 - 20 - 02 - 47 - 33 - 21 - 11 -1 7 15 22 75.46 38.14 25.80 19.69 16.07 13.68 11.99 10.74 9.79 9.03 76.51 39.02 26.53 20.29 16.55 14.06 12.28 10.95 9.92 9.10 - 105 - 23 - 73 - 00 - 42 - 32 - 29 - 21 - 14 -7 K = S exp[-(r-m)(T-t)] ∗ ∗ K = 20 K = 40 λ = 2% λ = 0% cs λ = 2% λ = 0% cs λ = 2% λ = 0% cs λ = 2% λ = 0% cs 11.47 14.55 14.57 13.97 13.23 12.49 11.78 11.14 10.56 10.03 11.57 14.63 14.62 13.99 13.21 12.43 11.69 11.00 10.38 9.82 -9 -2 -5 -2 2 6 9 14 12 21 92.41 47.14 32.15 24.71 20.30 17.39 15.33 13.82 12.67 11.77 93.12 47.67 32.53 24.96 20.43 17.41 15.27 13.67 12.45 11.48 - 71 - 53 - 32 - 25 - 13 -2 7 15 22 22 81.81 41.99 28.82 22.32 18.51 16.06 14.38 13.16 12.23 11.48 82.41 42.42 29.10 22.46 18.54 15.99 14.23 12.95 11.96 11.17 - 60 - 43 - 22 - 15 -2 6 14 21 22 22 62.40 32.67 23.22 18.88 16.45 14.87 13.72 12.80 12.03 11.38 82.81 32.91 23.32 18.88 16.38 14.74 13.52 12.56 11.75 11.05 - 40 - 24 - 10 -0 2 14 19 24 29 33 PRICING THE CREDIT RISK OF SECURED DEBT 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation   C Table 1 Term Structures of Fair Leasing Fees and Credit Spreads 1311 1312 Table 1 (Continued) Panel C Maturity T-t T-t T-t T-t T-t T-t T-t T-t T-t T-t =1 =2 =3 =4 =5 =6 =7 =8 =9 = 10  C 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation  T-t T-t T-t T-t T-t T-t T-t T-t T-t T-t =1 =2 =3 =4 =5 =6 =7 =8 =9 = 10 ∗ m=0 ∗ ∗ λ = 4% ∗ ∗ ω = 10% f (λ = 2%) f (λ = 0%) cs f (λ = 2%) f (λ = 0%) cs f (λ = 4%) f (λ = 0%) cs λ = 2% λ = 0% cs 36.34 30.53 26.07 22.63 19.92 17.78 16.07 14.68 13.54 12.59 36.31 30.45 25.92 22.40 19.62 17.41 15.63 14.17 12.98 11.98 2 2 15 23 30 32 44 50 56 61 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 - 21.78 19.89 18.26 16.87 15.67 14.64 13.74 12.96 12.29 11.70 21.77 19.83 18.16 16.67 15.42 14.31 13.33 12.47 11.71 11.03 1 5 11 17 25 33 41 50 52 66 21.80 19.90 18.26 16.84 15.61 14.54 13.60 12.79 12.07 11.44 21.77 19.83 18.16 16.69 15.42 14.31 13.33 12.47 11.71 11.03 3 6 10 14 19 23 22 32 36 40 Base Case Grenadier f ∗ 21.77 19.83 18.16 16.70 15.42 14.31 13.34 12.48 11.73 11.07 ∗ a = 25% Grenadier f (λ = 0%) cs 21.77 19.83 18.16 16.69 15.42 14.31 13.33 12.47 11.71 11.03 0 0 0 0 0 1 1 2 3 4 f ∗ 6.83 9.31 9.93 9.90 9.62 9.23 8.82 8.41 8.02 ∗ K = 0 Grenadier f (λ = 0%) cs 6.83 9.31 9.93 9.90 9.62 9.23 8.81 8.40 8.00 -0 -0 -0 -0 0 0 1 1 2 f ∗ 102.01 52.03 35.37 27.04 22.00 18.61 16.19 14.41 13.05 11.98 ∗ K = 0, a = 25% Grenadier f (λ = 0%) cs f∗ f ∗ (λ = 0%) cs 102.01 52.03 35.37 27.05 22.07 18.75 16.38 14.61 13.23 12.13 0 0 -0 -2 -7 - 14 - 19 - 10 - 19 - 16 76.51 39.02 26.53 20.27 16.48 13.92 12.09 10.75 9.72 8.92 76.51 39.02 26.53 20.29 16.55 14.06 12.28 10.95 9.10 9.10 0 0 -0 -2 -7 - 14 - 19 - 21 - 20 - 18 Notes: Maturity is measured in years. The base case assumes S = 100, m = −0.2, σ = 0.2, r = 0.04, a = 0, K = ∞, w = 0. All other scenarios assume the same inputs as the base case but for the different inputs displayed in the respective headings. The columns headed cs display fair leasing credit spreads. Credit spreads are equal to the difference between the defaultable leasing fee and the corresponding default-free leasing fee (multiplied by S = 100). Thus credit spreads are expressed in basis points. The column headed ‘K = Se (r −m)(T−t) ’ in Panel B is of interest since E t [ST ] = Se (r −m)(T−t) . Panel D repeats the simulations in Panel A using Grenadier’s 1996 model and the calculations are explained in the Appendix. REALDON Panel D Maturity m = −0.4 ∗ 1313 PRICING THE CREDIT RISK OF SECURED DEBT Figure 5 Term Structures of Credit Spreads on Financial Leasing: The Effects of Default Intensity, of Drift of the Leased Asset Value and of the Costs of Default 70 m = - 0.4 60 m=0 Credit spreads 50 w = 10% Base case 40 30 20 10 1 2 3 4 5 6 7 8 9 10 Time to maturity (in years) Notes: This is a plot of the credit spreads in Figure 4 shown in the columns headed cs for the various scenarios. terminal option to purchase the leased asset. Thus the lessee’s default liberates the lessor from the corresponding obligation to relinquish the asset at maturity. Secondly, if a > 0 because an initial prepayment is required of the lessee, default may liberate the lessor from the lease contract when the contract is a net liability for the lessor, in other words it may happen that: E t [L s (S, τ )] =  (r − m) Se m(τ −t0 )   f  1 − e −(λ+r )(T−τ ) − 1 − e −(λ+r −m)(T−τ ) < 0. r +λ r +λ−m (30) This is more likely when default takes place soon after the initiation date t 0 , i.e. when τ is close to t 0 . Notice that, as default time τ gets closer to contract maturity T , the value of the remaining service flow approaches the value of the remaining lease fees, i.e. lim τ →T E t [L s (S, τ )] → 0, and default tends to cause no gain or loss if w = 0 and K → ∞. These considerations make the credit risk of a lease contract somewhat similar to the credit risk of a financial swap, whose ‘marked-to-market’ value may be either positive or negative when one party defaults. These results differ from those of Grenadier (1996), who found that leasing credit spreads, defined in the same way as above, always increase in the lessee’s default probability. The reasons are that his simulations of credit spreads assume no terminal option for the lessee to purchase the leased asset and no prepayment. Panel D of Table 1 repeats the simulations of Panel A using the model in Grenadier (1996). The parameters in Grenadier’s model are chosen so that the ten year cumulative 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation   C 1314 REALDON default probability predicted by Grenadier’s model is the same as that predicted by the model of this section. The relevant calculations are explained in the Appendix. The different credit spreads in Panels A and D are mainly due to the fact that in Grenadier’s structural model framework there is a very low probability of the lessee’s default soon after contract initiation, whereas in the ‘reduced form’ model of this section early default is quite likely. The effects of this are apparent if we compare the columns under the heading ‘a = 25%’ in Panel A and in Panel D: in Panel A spreads are negative and lower than in Panel D since early default is beneficial rather than detrimental to the lessor, if the lessee has made a prepayment. Grenadier’s model somehow overlooks the possibility that the lessor may gain from the lessee’s default in that it understates the probability of ‘early’ default. And yet, as Schmit (2004) reports, negative losses-givendefault for lessors are a real possibility. When the lease envisages a terminal purchase option, ‘late’ default can also be beneficial to the lessor. That is why the longer term credit spreads under the headings ‘K = 0’ and ‘K = 0 and a = 25%’ are higher in Panel D than in Panel A. Grenadier’s structural model predicts that ‘late’ default is more likely than the model of this section predicts. These results also show how the credit risk of financial lease contracts markedly differs from the credit risk of secured debt contracts when prepayments and terminal purchase options are considered and why such contractual provisions are so popular. In fact the lessor may gain upon the lessee’s default, whereas secured lenders usually do not gain from the borrower’s default. 2 As a consequence negative fair credit spreads are possible for financial leases, but usually not for secured loans. 3 This may explain why leasing companies typically serve small risky businesses of low credit quality. Schmit (2004) finds that default probabilities are higher for leasing than for similar private debt contracts. Moreover, whereas the value of secured debt becomes relatively insensitive to the borrower’s default probability when the value of the collateral asset exceeds face value, the value of financial leases is quite sensitive to the lessee’s default probability even when the leased asset value greatly exceeds the present value of the remaining lease fees. Then, unlike in the case of secured debt whose value always decreases in the volatility of the collateral asset, the value of a lease contract will hardly depend on the volatility of the leased asset. When the lessee has no terminal purchase option (i.e. when L o (S, t) = 0), the lease is independent of asset volatility. When the lessee does have a terminal purchase option, usually the exercise price K is so low (e.g. K ⋍ 0.01 · S 0 ) that the −λ(T−t) ‘vega’ of the purchase option (i.e. d(e dσLo (S,t)) ) usually approaches zero. Again this entails that the lease contract value L(S, t) and the fair lease fee f ∗ will hardly change with asset volatility, whereas secured debt value becomes sensitive to the collateral asset volatility σ as the borrower’s credit quality declines. Thus leasing may alleviate the costs of asymmetric information about the collateral asset volatility that are associated with borrowing through secured loans. 2 Secured debt holders cannot gain from the borrower’s default if the coupon rate is not less than the risk free interest rate and if the term structure of interest rates is assumed flat and constant. Moreover secured lenders rarely realise a significant gain from the borrower’s default even when the term structure of interest rates is stochastic. 3 So, although we can also assimilate increasing the size of the required leasing pre-payment to the tightening of collateral requirements on secured loans (since both policies reduce the credit risk exposure of the lender or lessor), the former may bring about negative credit spreads unlike the latter.  C 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation  PRICING THE CREDIT RISK OF SECURED DEBT 1315 We notice that the empirical evidence provided by Sharpe and Nguyen (1995) supports the conclusions of this section. They found that leasing reduces contracting costs due to asymmetric information about the firm’s credit quality and more generally the cost of financing of low credit quality firms. Indeed the above analysis has shown how the lease fee may decrease in the lessee’s default intensity. Further support is provided by the evidence in Krishnan and Moyer (1994), in that firms of lower credit quality are more likely to choose financial leasing, in Ang and Peterson (1984), in that firms with leases tend to be more levered, in Lasfer and Levis (1998), in that tax savings are not a major determinant of the leasing decision of small UK firms, and in Schallheim et al. (1987), in that the observed yields on US leasing contracts seem to rise with the lessee’s profitability and decrease with leverage. The above analysis can help make sense of such evidence, since we have seen that the credit spreads on (secured) loans increase with the borrower’s default risk, while the credit spreads on leases often decrease with default risk. Then it should not be surprising to observe that financial leasing is used more by small and relatively opaque firms of low credit quality. Furthermore, the above analysis of leasing hinges on the possibility that the lessor may gain from the lessee’s default. This is supported by DeLaurentis and Geranio (2001), Schmit and Stuyck (2002) and Schmit (2004). These researchers find that the leasing industry enjoys relatively high recovery rates. (i) Valuation of the Option to Extend the Lease Maturity What has been said about the lessee’s terminal option L o (S, t) to buy the leased asset at the maturity of the lease is applicable also to the lessee’s terminal option to extend the life of the lease contract, which is another popular contractual feature. In fact on the lease maturity date T the lessee often has the option to continue the lease until Te >T at a reduced fee of f e < f . The payoff to this option is again similar to that of a call option, i.e.   1 − e −(λ+r )(T−T) · max ( f T − f e , 0) r +λ (31) where f T is the fair leasing fee at time T , i.e.   u·ST 1 − e −(λ+r −m)(Te −T) F (ST , T, Te ) + r +λ−m   . f T = (r + λ) 1 − e −(λ+r )(Te −T) (32) The present value at t < T of such ‘extension’ option is e −λ(T−t) L e (S, t), where: λ + (r − m)e −(r +λ−m)(T−t) − (λ + r − m)e (m−r )(T−t) L e (S, t) = w λ+r −m         √ S S · Se −(r −m)(T−t) N d − e −r (T−t) Se N d −σ T −t . Se Se (33) 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation   C 1316 REALDON and where Se is such that:   u·Se F (Se , T, Te ) + r +λ−m · 1 − e −(λ+r −m)(Te −T)   . f e = (r + λ) 1 − e −(λ+r )(Te −T) The presence of this extension option in the contract, like the presence of the terminal purchase option, decreases leasing credit spreads, since the lessee’s default would liberate the lessor from the obligation associated with the extension option. (ii) Alternative Payoffs at Default There are two main possible variations to the lessor’s payoff as the lessee defaults. First, the lessor may, on top of repossessing the leased asset, require compensation for the gain he loses due to the lessee’s default. Secondly, the debtor may keep honouring the leasing contract even after default on other debt obligations of his. Next we examine these two cases. (a) Lessor’s Compensation for Lost Gain The lessor may have the right to require compensation for the lost gain and bankruptcy costs brought about by the lessee’s default. At default time τ this right of the lessor’s would be worth max (L(S, τ ) + wS(1 − e −(r −m)(T−τ ) ), 0), it would typically depend on legal jurisdiction and on contractual provisions, and it would clearly be intended to avert a sort of strategic default on the part of the lessee should the market value of the leased asset fall below the present value of the lease fees yet to be paid. Thus upon default in the present setting the lessor’s complete payoff becomes:   Rl (S, τ ) = −L (S, τ ) − wS 1 − e −(r −m)(T−τ )      + π max L (S, τ ) + wS 1 − e −(r −m)(T−τ ) , 0 (34) where π (with 0 ≤ π ≤ 1) is the recovery rate for unsecured creditors since the lessor would be an unsecured creditor as he claims to recover lost gain and bankruptcy costs. Notice that this default payoff is the same as before when π = 0. This default payoff entails that now equation (22) needs solving numerically through a finite differences scheme. Unreported simulations show that, as expected, as π rises lease credit spreads decrease. In particular credit spreads decrease even further as the lessee’s default probability rises. The reasons are the same as above and additionally the expected default payoff to the lessor is even higher in the present setting. (b) The Option to Continue Honouring the Lease Contract Even After Default Some bankruptcy codes (e.g. in Europe and the US) let the liquidator or receiver of a bankrupt lessee choose whether or not to continue the lease contract, even during bankruptcy. If continuation is chosen, the lessee firm must keep making lease fee payments regularly and the lessor would receive preferential treatment over other creditors in violation of the par condicio creditorum. But this possible violation to the par  C 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation  PRICING THE CREDIT RISK OF SECURED DEBT 1317 condicio may be beneficial to the lessee’s other creditors, since honouring the leasing contract in order to eventually exercise the option to purchase the leased asset may be in the creditors’ collective interest, at least when S is sufficiently high. Otherwise the lessee’s default could entail loss of the possibly very valuable terminal purchase option to the detriment of all creditors. This option is often valuable since the exercise price K , to purchase the leased asset at the end of the lease, typically is a small fraction of S 0 . Indeed the present analysis supports the bankruptcy code provisions that allow the mentioned violations to the par condicio creditorum. Then if after the lessee’s default the liquidator acts so as to minimise the value of the lease contract to the lessor, 4 the lessor’s payoff at default becomes:   Rl (S, τ ) = −L(S, τ ) − 1 L(S,τ )>0 · wS 1 − e −(r −m)(T−τ ) − max(−L (S, τ ) , 0) (35) where 1 L(S,τ )>0 is the indicator function of the event L(S, τ ) > 0. For simplicity, this payoff assumes that the default intensity describing the probability of default on the lease contract stays constant even after time τ , which is the time when the lessee defaults on his other debt obligations and the liquidator decides whether or not to continue the lease. The term max(−L(S, τ ), 0) is the payoff of the liquidator’s option to keep honouring the lease contract even after the lessee’s default. If L(S, τ ) < 0 the liquidator pays the remaining lease fees and he retains the final purchase option plus the right to use the asset until T . Of course the liquidator may first commit to honouring the lease contract even in bankruptcy and later decide to default strategically if S drops sufficiently. Here we abstract from this refinement. Notice that the lessor’s default payoff becomes Rl (S, τ ) = min(− L(S, τ ), 0) if w = 0. Equation (35) entails that L(S, t) and f ∗ now need to be found numerically by solving equation (22) through finite differences. In this setting the lessor no longer can gain from the lessee’s default much in the same way as a secured creditor cannot. Thus, unreported simulations confirm that the lease contract now behaves more like a secured debt contact: its value always decreases in the lessee’s default probability and in the volatility σ of the leased asset. Finally, we note that, in the presence of both the liquidator’s option to continue honouring the lease and of the lessor’s right to claim compensation for lost gain after the lessee’s default, the default payoff and the value of the lease contract tend to become insensitive to the probability of default and to approach their default free counterparts. Indeed, if w = 0 and π = 1, the default payoff to the lessor would become: Rl (S, τ ) = −L(S, τ ) + [max(L(S, τ ), 0)] − max(−L(S, τ ), 0) = 0. (36) 4. CONCLUSION This paper has provided simple closed form solutions for the valuation and credit risk assessment of secured bank loans and financial leases. Similar closed forms are 4 Here we also assume that the lessor simply repossesses the leased asset and cannot claim compensation for lost gain, should the lessee default on the payments of the leasing fees. 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation   C 1318 REALDON applicable also to value and assess the credit risk mitigation of a guarantee such as a pledged asset backing a debt obligation. These results are particularly relevant in the light of the new Basel agreement on the capital requirements of financial institutions. The analysis has highlighted how secured debt value is quite insensitive to strict collateral requirements when the borrower’s credit quality is high. The credit risk of secured debt differs from that of financial leases, because secured creditors do not usually gain from the borrower’s default, while the lessor can gain from the lessee’s default. As a consequence, whereas the value of secured debt always decreases in the borrower’s default probability, the value of financial leases may well increase in the lessee’s default probability. Indeed fair credit spreads on financial leases may even become negative. This is especially the case when the lessee is required to make a substantial prepayment at the start of the lease and/or when the lessee has the terminal option to purchase the leased asset or to extend the lease duration. This result suggests that such prepayments and options can significantly reduce the credit risk of financial leases and explains why financial leasing competes with (secured) bank lending to finance small, risky and relatively ‘opaque’ firms, whose default probability may be particularly high and difficult to estimate. The analysis also points outs that, while the value of secured loans always decreases in the volatility of the collateral asset value, the value of financial leases is often hardly sensitive to the volatility of the leased asset. These conclusions help us to interpret previous empirical evidence such as in Sharpe and Nguyen (1995), who suggest that leasing provides ‘a means of alleviating financial contracting costs’ associated with asymmetric information about a firm’s credit quality and more generally a means of reducing the cost of financing of low credit quality firms. The difference in the credit risk of financial leases and secured loans also depends on other contractual and bankruptcy code provisions determining the lessor’s payoff upon the lessee’s default. In particular, if the lessee’s default gives the lessor the right to claim compensation for lost gain as well as the right to repossess the leased asset, leasing credit spreads decrease and are often negative. If the bankruptcy code gives the lessee or its liquidator the option to keep honouring the lease contract even during bankruptcy in violation of the par condicio creditorum, the lessor may no longer gain from the lessee’s default and the lease value decreases in the lessee’s default probability and in the volatility of the leased asset, much as in the case of secured debt. Finally, the reduced models proposed here seem preferable to structural models for the valuation of bank loans and leases of maturities of less than five years, since reduced models, unlike structural models, predict realistic default probabilities even in the short and medium term. APPENDIX This Appendix explains how Grenadier’s (1996) leasing model is used in Panel D of Table 1. In Grenadier’s model it is assumed that the firm’s assets value V follows the risk neutral process dV = V α v dt + V σ v dz v with dzv dz = ρdt (where dz is the differential of the Wiener process driving dS). In Panel D we assume that ρ = 0, since our purpose is to compare Grenadier’s model with the leasing model of this paper. Under this assumption and adapting Grenadier’s model to the analysis of this paper, the fair leasing fee and  C 2006 The Author C Blackwell Publishing Ltd. 2006 Journal compilation  1319 PRICING THE CREDIT RISK OF SECURED DEBT fair credit spread of Grenadier’s model become: f ∗ = Sr − α − f 3 (1 − w) 1 − e −(r −m)(T−t) + S P (V , T − t) Lo (S,t) S (T−t) −r 1−e − f2 s p = f ∗ − Sr 1 − e −(r −m)(T−t) + Lo (S,t) −α  S (T−t) −r 1−e (37) (38) where α is again the prepayment fraction, where the terminal purchase option is worth SP(V , T − t) ·L o (S, t), where L o (S, t) is given in the text, where SP(V , T − t) is the survival probability from t to T so that: SP(V , T − t) = N (d1 ) −  V Vd 1−2 αv2 σv N (d2 ) y 1 u2 √ e − 2 du, 2π −∞  V   ln Vd + αv − 21 σv2 (T − t) d1 = √ σv T − t     ln VVd + αv − 12 σv2 (T − t) , d2 = √ σv T − t N (y ) = (39) where: f2 = f3 =   G (V , τ, y ) = N and where c1 = a2 −a1 ,c σv2  2 c 1 V Vd V Vd  c 2 ln = G (V , τ, a2 ) − e −r τ G (V , τ, a1 ) G (V , τ, a4 ) − e −(r −m)τ G (V , τ, a3 )  Vd  − yτ √ σv τ V a4 −a3 ,τ σv2  +  Vd V  2y2 σv N  ln  Vd  + yτ √ σv τ V = T − t, a1 = αv − 21 σv2 , a2 =  (40)  a12 + 2r σv2 , a3 = αv + ρσv σ − 21 σv2 , a4 = a32 + 2(r − m)σv2 . 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