Asset financing with credit risk

Journal of Banking & Finance 37 (2013) 43–59

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Asset financing with credit risk Steven Golbeck a,⇑, Vadim Linetsky b a

Department of Applied Mathematics, University of Washington, Guggenheim Hall #414, Seattle, WA 98195-2420, United States Department of Industrial Engineering and Management Sciences, McCormick School of Engineering and Applied Sciences, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, United States b

a r t i c l e

i n f o

Article history: Received 9 January 2012 Accepted 9 August 2012 Available online 23 August 2012 JEL classification: C51 G12 G13 G18 Keywords: Credit risk Asset-backed finance Equipment finance Leasing Bankruptcy law

a b s t r a c t This paper develops a model for the unified valuation of all forms of real asset financing, such as bank loans, leases, securitization vehicles, and credit guarantees, secured by assets that generate a stochastic service flow to the operator, or a rental stream to the lessor, and depreciate over a finite economic life to their scrap value. Examples include mobile equipment, such as aircraft, railroad equipment, ships, trucks and trailers, as well as energy generation assets, heavy factory equipment and construction equipment. In the event of obligor default, after a repossession delay and incurring costs of repossession, maintenance, re-marketing and re-deployment, the lender repossesses the asset and sells it on the secondary market and is, thus, subject to the risk of decline in the market value of the asset. The model we develop in this paper treats all forms of asset financing in a unified fashion as contingent claims on the collateral asset and the credit of the borrower. As an application, we estimate the collateral asset model on historical secondary market data for aircraft values and calibrate the financing model to the Enhanced Equipment Trust Certificates (EETCs) issued in 2007 by Continental Airlines and secured by a fleet of new aircraft. We then apply the calibrated model to value private market financing, including bank loans, leases, and credit guarantees, consistently with the capital market financing, and assess the impact of repossession delays on credit spreads. This analysis leads to a policy insight suggesting that bankruptcy laws limiting asset repossession delays lead to lower costs of asset financing. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction This paper develops a stochastic model for the unified valuation of all forms of financing secured by assets that generate a stochastic service flow to the operator, or a rental stream to the lessor, and depreciate over a finite economic life to their scrap value. Examples include mobile equipment, such as aircraft, railroad equipment, ships, trucks and trailers, as well as energy generation assets, heavy factory equipment and construction equipment. The problem is of major economic importance. The paper focuses on the aviation sector as a representative example. According to the 2010 forecast by The Airline Monitor (2010), new commercial aircraft deliveries are projected to reach over 3.3 trillion US dollars over the 2010–2030 period. Assuming the orders are financed with 15% airlines’ own equity and 85% financing, the total amount of commercial aircraft financing will reach 2.8 trillion US dollars over the next 20 years. In the aviation ⇑ Corresponding author. Tel.: +1 206 543 5493. E-mail addresses: [email protected] (S. Golbeck), [email protected]. edu (V. Linetsky). URLs: http://faculty.washington.edu/golbeck/ (S. Golbeck), http://users.iems. northwestern.edu/~linetsky (V. Linetsky). 0378-4266/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jbankfin.2012.08.010

sector, forms of commercial financing include secured bank loans, operating leases, and public debt issues secured by aircraft (in particular, Enhanced Equipment Trust Certificates (EETCs)). In addition to commercial financing, governmental Export Credit Agencies (ECAs) provide export credit financing support either in the form of direct export credit loans or export credit guarantees that support commercial financing. In some cases manufacturers also provide financing. A typical bank loan financing a purchase of a new commercial aircraft might have a twelve year financing term, an initial loanto-value (LTV) of 85%, mortgage-style principal amortization, and quarterly payment schedule. In the event of default, the secured lender repossesses the aircraft and sells it on the secondary market or leases it to another operator. In either case, the lender faces the risk of declining market prices or market lease rates for used aircraft, as well as a variety of possible costs and delays associated with aircraft repossession and re-marketing. An aircraft is a depreciating asset that has a typical useful economic life of between 30 and 35 years. During its economic life it generates a stochastic revenue stream for the airline owner-operator or a rent stream for the owner-lessor, while its economic value depreciates as the aircraft ages. At the end of its useful life the aircraft is salvaged for its scrap value. When forecasting used aircraft

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values, practitioners use the residual value curve (RVC) that represents an estimate of the expected economic value of a used aircraft as a function of its age. However, market prices of used aircraft realized in secondary market transactions have substantial volatility around these expected residual values. The secondary market for used aircraft is an over-the-counter market with a reasonable amount of liquidity, at least for popular aircraft models with substantial market penetration and during non-distressed market conditions. For example, Gavazza (2010) reports that, in the twelve months between May 2002 and April 2003, of the total world stock of 12,409 commercial aircraft used for passenger transportation that were at least two years of age, 720 traded. That represents a turn over of more than 5% of the total world commercial fleet in one year. Thus, in addition to the expected economic depreciation, the lender who repossesses a used aircraft in the event of default faces a significant risk due to secondary market volatility. Furthermore, as discussed in Marray (1999), the lender also faces costs and delays in the repossession process. The expected economic depreciation, market volatility, as well as costs and delays of the repossession process, measured against the loan amortization, determine the lender’s loss-given-default (LGD). The present paper develops a model for the unified valuation of all forms of asset financing, including leases, loans, export credit guarantees, and securitization vehicles, subject to the risk of default. Our goal is to present a practically useful model that can be calibrated to the available market data. This goal requires a balance between the model complexity, tractability, and the realities of market data availability. Our model is developed within the framework of continuous-time dynamic asset pricing theory. The key part of our asset financing model is the model for market values of real assets with stochastic service flow and finite economic life. Since real assets, such as airplanes, derive their value from a (generally stochastic) service flow (revenue stream) to the owner-operator or a rent stream to the owner–lessor over a finite useful economic life, we take the instantaneous percentage lease rate (the short lease rate) to a default-free lessee as the fundamental state variable in our asset model. To gain analytical tractability, we model the short lease rate under both the physical probability measure and the risk-neutral probability measure by Ornstein– Uhlenbeck (OU) processes with time dependent long-run means (different under the physical and risk-neutral measures) and time-dependent volatility, similar to the extended Vasicek model in the interest rate modeling literature (as well as in credit risk literature). The parameter time dependence is required since the real asset dynamics has a strong dependence on the asset’s age. Under these assumptions, the market value of the real asset with finite economic life follows a process similar to the zero-coupon bond price dynamics in interest rate models. While the zero-coupon bond price appreciates toward its face value at maturity, the asset value in our model depreciates down to its final salvage value at the scrap time (‘‘maturity’’ of the asset) at the stochastic depreciation rate. This modeling approach to depreciating assets constitutes one of the innovations in the present paper, as it has never appeared in the literature before. Under the OU model specification, we are able to obtain an explicit solution for the asset value process. Under the physical probability measure, the asset value fluctuates around its expected residual value curve (RVC), with the fluctuations modeled by the exponential of an Ornstein–Uhlenbeck process. When valuing secured financing transactions under the risk-neutral probability measure, the risk-adjusted residual value curve (RA-RVC) plays the role of the forward curve and takes the place of the RVC as the asset value’s mean under the riskneutral measure. In the OU model we are able to derive an explicit expression for the risk adjustment factor in terms of the market price of asset risk (the asset’s Sharpe ratio), as well as explicitly determine prices of call and put options on the asset that are given

by Black (1976) type formulas with the risk-adjusted residual value curve in place of the forward curve and the age-dependent asset volatility in place of the constant volatility. The practical advantages of this model are its analytical tractability, as well as the ability to calibrate it to market data. In contrast to more complex modeling choices, the model does not include any unobservable parameters or hidden variables that cannot be directly estimated from data. This allows us to calibrate our asset model to historical secondary market aircraft data. We employ our depreciating real asset model as the key component of our asset financing model. We assume that default arrives at the first jump time of a time-inhomogeneous Poisson process with the arrival rate calibrated to the Credit Default Swap (CDS) spreads on senior unsecured corporate debt of the obligor. The twin advantages of calibrating the credit component of the model to CDS are that, first, the latter already incorporate default risk premium and, second, CDS are natural hedging instruments against the risk of corporate default. We then develop the valuation of all forms of asset financing, including secured loans, leases, securitization vehicles, such as EETC tranches, and export credit guarantees, as hybrid contingent claims on the credit of the borrower and the collateral asset, taking into account the realities of the asset repossession process. The recoveries in the event of default for various forms of secured financing are seen as options on the asset, with strikes and expirations adjusted by costs and delays of repossession. The result is a model that allows us to determine in a unified fashion lease rates, secured loan spreads, and export credit guarantee premiums, consistent with the CDS spreads, historical asset value statistics, the market price of asset risk, and realities of the repossession process, including costs and delays. We then imply the market price of asset risk that defines the risk adjustment from the physical to the risk-neutral measure from benchmark capital market financing transactions. Since EETCs are the only public capital markets aircraft financing transactions with publicly available data, with all of the other forms of private aircraft financing, such as bank loans and operating leases, being confidential, we take EETCs as the benchmark transactions and calibrate our model to imply the market price of asset risk from EETCs. In particular, we develop a case study of EETCs issued by Continental Airlines in 2007, imply the market price of risk, and apply our calibrated model to evaluate other forms of financing, including leases, loans, and credit guarantees. Thus, our model allows us to value all forms of financing relative to and consistent with the chosen capital markets benchmark. This is similar to how the concept of implied volatility is used in derivatives markets. The derivatives models are calibrated first to benchmark exchangetraded options, and then used to price over-the-counter derivatives consistent with these benchmarks. This approach is also supported by the fact that the benchmark capital markets securities can be used as hedging instruments. This makes our model useful to practitioners for consistent pricing and risk management across the entire portfolio of asset financing transactions. To the best our knowledge, the program of developing a unified modeling framework for all forms of financing of real assets generating a stochastic service flow over a finite economic life undertaken in this paper, from the model through to the implementation and calibration to the market data, has never been attempted in the literature. The existing literature has focused separately either on the valuation of leases or on debt financing, with little overlap. Furthermore, virtually all of the existing literature has primarily focused on real estate. Moreover, the lease valuation literature until recently has focused almost exclusively on the asset value risk and largely ignored the possibility of lessee default, going back to the classic works of Miller and Upton (1976) for a single period, McConnell and Schallheim (1983) in a multi-period discrete time framework, and through to the recent works such as the continuous-time

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setting of Stanton and Wallace (2008). Grenadier (1996) is the first to model credit risk in leasing. This paper models default within a structural approach by explicitly modeling the value of the firm leasing the asset and deriving the default time endogenously within the structural model. While providing valuable economic insights, structural models are often difficult to use in practice in the credit portfolio setting due to their complexity, as they require explicit modeling of the capital structure of each individual obligor. This has stimulated active development of reduced-form models of credit risk (see Bielecki and Rutkowski (2004), Duffie and Singleton (2003), and Lando (2004) for surveys). However, the vast reducedform credit risk literature has focused primarily on unsecured debt. Clapham and Gunnelin (2003) are the first to include some discussion of the impact of the lessee default on modeling lease rates in the context of reduced form modeling. They do not consider debt financing. On the other hand, Kijima and Miyake (2004) model secured debt financing with the risk of default in the reduced-form framework. However, they focus on secured loans only and do not consider leases. Realdon (2006) is the first paper in the literature that models both leases and secured loans in the same framework as contingent claims on the asset. However, the primary focus of these papers is on real estate, and as such their assumptions are not directly suitable for depreciating assets with finite economic life, such as capital equipment, for a variety of reasons discussed in detail in Section 2. The present paper is the first in the literature to focus specifically on financing depreciating assets, such as capital equipment, and to develop a unified modeling framework for all forms of equipment financing. It is also the first in the literature to calibrate a fully specified asset financing model to market data. Finally, the paper is the first to explicitly introduce repossession delays and costs in the model as the key variables in asset financing (all of the above referenced papers do not consider repossession delays and costs). This allows us to analyze the impact of repossession delays and costs on credit spreads of financing transactions and directly assess financial impact of bankruptcy laws limiting the repossession delay, such as Section 1110 of the US bankruptcy code and the Cape Town Treaty limiting the aircraft repossession delay to sixty days. This quantitative analysis linking law and finance leads to a policy insight suggesting that bankruptcy laws limiting asset repossession delays may lead to lower costs and greater availability of asset financing. The rest of the paper is organized as follows: Section 2 develops our model for depreciating assets that generate a stochastic service flow over a finite economic life. The key results are Theorem 2.1 which characterizes the asset value process under the risk-neutral measure and develops the risk adjustment linking the physical and risk-neutral measures, and Theorem 2.2 that develops option pricing. Section 3 develops the valuation of leases and determines the term structure of fixed-rate lease rates with the risk of default. Section 4 develops secured debt valuation and determines the term structure of credit spreads on secured loans, securitization vehicles such as EETC tranches, and establishes export credit guarantee premiums. Section 5.1 calibrates our real asset model to the historical aircraft market data. Section 5.2 analyzes a sample aircraft financing transaction, EETCs issued by Continental Airlines in 2007, calibrates the default arrival model to the Continental CDS market spreads, and implies the market price of aircraft risk. Section 5.3 applies the calibrated model to evaluate a range of other aircraft financing transactions, including aircraft loans, leases, and export credit guarantees, with special focus on the repossession delay as the key determinant of the risk and value of asset financing transactions. Section 6 concludes the paper. Supplementary material on OU processes, selected proofs and additional data related to our calibration exercise are included in the Appendices.

2. Modeling assets with stochastic service flow and finite economic life Suppose a new piece of equipment, such as an airplane, has an initial spot price of S0 dollars and is placed in service at time t = 0. We further assume that the asset has a useful economic life of T years (typically around 30–35 years for commercial aircraft) and that at the end of its useful economic life it will be retired from service and salvaged for its scrap value projected to be ST > 0. To simplify the development, we assume that the salvage value is not random. Between its known initial price S0 and its terminal scrap value ST the market price of the asset is stochastic, {St, t 2 [0, T]}. To develop our stochastic model, we start with a probability space ðX; F ; PÞ supporting a one-dimensional Brownian motion BP . Let F ¼ fF t ; t P 0g denote its (augmented) natural filtration. We model the asset price over the useful economic life of the asset [0, T] as a strictly positive Itô process adapted to the Brownian filtration F: P

dSt ¼ dt St dt þ rS ðtÞSt dBt

ð2:1Þ

with the instantaneous depreciation rate {dt, t 2 [0, T]} assumed to  RT  1 d2 dt P 2 0 t < 1 and bounded volatility {rS(t), t 2 [0, T]} satisfy E e adapted to F. We model the short lease rate, {‘t, t 2 [0, T]}, also referred to in the literature as the instantaneous percentage service flow rate or instantaneous percentage rent, accruing to the owner of the asset (either through the use of the asset or through leasing the asset) as a time-inhomogeneous Ornstein–Uhlenbeck (OU) diffusion process under the statistical probability measure P: P

d‘t ¼ jðhP ðtÞ  ‘t Þdt þ rðtÞdBt

ð2:2Þ

with the constant rate of mean-reversion j, time-dependent (deterministic) mean level hP ðtÞ, time-dependent (determinisitic) volatility r(t), and driven by the one-dimensional Brownian motion BP . The advantage of this specification is that the OU process is a Gaussian process and is, thus, analytically tractable with the Gaussian transition density. OU processes are extensively used in financial modeling in the Vasicek (1977) model of the term structure of interest rates and in the exponential OU model for commodity prices of Schwartz (1997), among others. The time dependence in hP ðtÞ and r(t) is introduced in order to capture dependence on the age of the asset to allow the parameters to change as the asset ages from brand new (t = 0) to scrap (t = T). We assume the instantaneous default-free interest rate (the short rate) rt is a given deterministic function of time. Then the value at time t of one share of the money market account is Rt r du At ¼ e 0 u , and the discount factor from time s to time t is Rs  r du Pðt; sÞ ¼ At =As ¼ e t u for any s P t P 0. The service flow or rent accruing to the owner of the asset over an infinitesimal time interval is ‘tStdt. In the context of this paper, the service flow from operating the asset or, alternatively, the rent from leasing out the asset in effect constitute stochastic dividends. Therefore, we apply the standard dynamic asset pricing theory for assets that pay dividends at a stochastic dividend rate (e.g., Duffie (2008), Section 6.L, pp. 123–125). In particular, the deflated gain process {Gt, t 2 [0, T]} measuring the total gain (capital gain plus the cumulative dividends – the cumulative service flow or rent in our case) to the owner of the asset relative to the money market account (expressed in the units of the money market account as Rt the numeraire asset) is Gt ¼ St =At þ 0 ð‘u Su =Au Þdu. By Itô’s formula, Rt Rt P Gt ¼ S0 þ 0 ð‘u  du  r u ÞðSu =Au Þdu þ 0 rS ðuÞðSu =Au ÞdBu . Define the market price of risk process:

gt :¼

‘t  d t  r t : rS ðtÞ

ð2:3Þ

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Then we can define a new probability measure Q equivalent to P on F T by:

  Z T  Z dQ 1 T 2 ¼ exp  gu dBPu  gu du :  dP F T 2 0 0

ð2:4Þ

BQ t

BPt

Rt

Then, by Girsanov’s theorem, the process :¼ þ 0 gu du is a standard Brownian motion under Q, the deflated gain process G is Rt Q a Q-martingale, Gt ¼ S0 þ 0 rS ðuÞ ASuu dBu ; and the spot price and

assumptions of the finite economic life and the known scrap value, and does not use the Gaussian OU assumption. Lemma 2.1. (a) When the asset has a finite economic life T > 0 and fixed scrap value ST > 0, the spot price St at any time 0 6 t 6 T can be written as:

   RT  ‘ du St ¼ Pðt; TÞST EQ e t u F t :

the short lease rate processes take the form under Q: Q

dSt ¼ ðr t  ‘t ÞSt dt þ rS ðtÞSt dBt ; Q

d‘t ¼ jðhQ ðtÞ  ‘t Þdt þ rðtÞdBt ; P

¼ h ðtÞ 

1

j

rðtÞgt :

ð2:5Þ

   RT  ‘ du Fðt; sÞ ¼ Pðs; TÞST EQ e s u F t :

From Eq. (2.5), under Q the total return to the owner of the asset over an infinitesimal time interval is the sum of the price change and service flow and is equal to dSt + ‘tStdt = rtStdt + rS(t)StdBt, the sum of the expected return at the risk-free rate rt and martingale uncertainty modeled by the Brownian motion. Thus, Q is the riskneutral measure equivalent to P (equivalent martingale measure (EMM)). Re-arranging Eq. (2.3) in the form

‘t ¼ r t þ dt þ rS ðtÞgt ;

ð2:7Þ

we see that the short lease rate is equal to the short interest rate plus the asset depreciation rate plus the risk premium. Thus, the lessor who purchases the asset and finances the acquisition at the short rate (by rolling over short-term financing) would charge a default-free lessee the short-term lease rate equal to the interest rate on financing the asset plus the expected asset depreciation rate plus the risk premium due to the asset’s volatility. The forward price F(t, s) for the delivery of the asset at time s negotiated at time 0 6 t 6 s that makes the present value of the forward contract equal to zero at time t is equal to the expected asset price under Q:

Fðt; sÞ ¼ EQ ½Ss jF t :

(b) The forward price for the delivery at time s P t P 0 can be written as:

where hQ ðtÞ ð2:6Þ

ð2:8Þ

Indeed, the value of the forward contract at maturity at time s is equal to Ss  F(t, s) (take delivery of the asset in exchange for the payment of the forward price F(t, s) set at the contract inception t). The present value at time t is equal to Pðt; sÞðEQ ½Ss jF t   Fðt; sÞÞ. Setting it equal to zero, so no cash changes hands at the inception of the forward contract at time t, and solving for the forward price F(t,s), we arrive at Eq. (2.8). In contrast to the securities markets, forward prices for physical assets such as airplanes may not be directly observed in the market. Instead, for depreciating physical assets with finite economic life, the data readily available to practitioners include the residual value curve that shows the expected depreciation of the asset value over time as the asset ages over time from new to scrap. The residual value curve (RVC) is the expected future asset value under the statistical measure P (given historical data on used asset values at different ages available to practitioners): RVð0; tÞ ¼ EP ½St . We will now prove the key theorem of this section that, under the assumptions of the finite useful economic life T, the known scrap value ST, the OU dynamics (2.1) for the short lease rate, and the additional assumption that the market price of risk gt is a deterministic function of time, explicitly describes the dynamics of the asset price and its forward curve under the statistical and the risk-neutral probability measures. Before presenting the theorem, we first prove the following lemma that uses only the

ð2:9Þ

ð2:10Þ

Proof. Rt nR Rt t Q ‘ du (a) The process e 0 u St =At ¼ S0 exp 0 rS ðuÞdBu  12 0 r2S ðuÞdug is a Q-martingale due to our assumption that the volatility is bounded. Then for any two times 0 6 t 6 s 6 T we can write:

   Rs  ‘ du St ¼ Pðt; sÞEQ e t u Ss F t : Setting s = T and taking the constant ST outside of the expectation, we arrive at (2.9). (b) Substituting the result (2.9) into Fðt; sÞ ¼ EQ ½Ss jF t , we arrive at (2.10). h Before proceeding with the formulation our key theorem, it is convenient to express the process ‘t in terms of the OU process X Pt with zero long-run level and zero initial condition, ‘t ¼ DPt þ X Pt ; where DPt is the deterministic function of time

DPt ¼ ejt ‘0 þ j

Z

t

ejðtuÞ hP ðuÞdu;

ð2:11Þ

0

and P

P

dX t ¼ jX Pt dt þ rðtÞdBt ;

X P0 ¼ 0:

ð2:12Þ

When the market price of risk gt is assumed to be a deterministic function of time, the short lease rate also follows the OU process Q Q (2.6) under Q and we can write ‘t ¼ DQ t þ X t where X t is the OU process with zero long-run mean and zero initial condition under Q Q

Q

dX t ¼ jX Q t dt þ rðtÞdBt ; and jt DQ ‘0 þ j t ¼ e

Z

t

XQ 0 ¼ 0;

ejðtuÞ hQ ðuÞdu:

ð2:13Þ

ð2:14Þ

0

Since the OU process is analytically tractable (see A), we are able to compute the conditional expectations in Eqs. (2.9) and (2.10) in closed form similar to the Vasicek interest rate model with timedependent parameters and, hence, obtain the risk-neutral asset price and forward curve dynamics, including an explicit expression for the asset volatility rS(t) in terms of the short lease rate volatility r(t), the rate of mean reversion j, and the useful economic life T. By changing the probability measure back to the statistical measure, we then obtain the asset price and forward curve dynamics under the statistical measure. Theorem 2.1. (a) In the Gaussian model (2.6) when the asset has a finite economic life T > 0 and fixed scrap value, the spot price St at any time 0 6 t 6 T follows the exponential OU process under Q:

S. Golbeck, V. Linetsky / Journal of Banking & Finance 37 (2013) 43–59

St ¼ Fð0; tÞ exp

( XQ t

)

1 ð1  ejðTtÞ Þ  R2 ð0; tÞ ; 2

j

ð2:15Þ

where X Q t is the OU process (2.13) with zero long-run level and zero initial condition, {F(0, t), t 2 [0, T]} is the initial forward curve at time zero, and R(u, t) is given by:

R2 ðu; tÞ ¼

1

j

2

ð1  ejðTtÞ Þ2

Z

t

r2 ðv Þe2jðtv Þ dv ;

u

0 6 u 6 t:

ð2:16Þ

(b) The spot price process (2.5) has the instantaneous volatility:

rðtÞ rS ðtÞ ¼ ð1  ejðTtÞ Þ: j

ð2:17Þ

(c) For each fixed delivery date 0 < s 6 T the forward price F(t, s) follows the process under Q: XQ t

jðTsÞ ÞejðstÞ þ1R2 ðt;sÞ1R2 ð0;sÞ 2 2

Fðt; sÞ ¼Fð0; sÞe j ð1e

;

t 2 ½0; s 

ð2:19Þ

where c(t, s) :¼ ej(st)(1  ej(Ts))/(1  ej(Tt)). (d) Suppose the market price of risk gt is the deterministic function of time. Then the initial forward curve is equal to the residual value curve adjusted by the risk premium factor:

Fð0; tÞ ¼ RVð0; tÞ   Z t 1  exp  ð1  ejðTtÞ Þ gs rðsÞejðtsÞ ds :

j

0

ð2:20Þ We thus call the forward curve the risk-adjusted residual value curve (RA-RVC). (e) Under the statistical measure P, the asset evolves according to

St ¼ RVð0; tÞ exp

 P Xt

j

 1 ð1  ejðTtÞ Þ  R2 ð0; tÞ ; 2

The further advantage of the Gaussian model (2.15) is that the call and put option pricing formulas on the asset are given in closed form by Black’s formula. h

Theorem 2.2. The prices at time t P 0 of call and put options on the asset with expiration s > t and strike K > 0 when the asset price at t is St are given by Black’s (1976) formulas with the forward price F(t, s) pffiffiffiffiffiffiffiffiffiffi and the R(t, s) given by Eq. (2.16) in place of r s  t:

Callðt; St ; s; KÞ ¼ Pðt; sÞ½Fðt; sÞNðdþ ðt; sÞÞ  KNðd ðt; sÞÞ;

ð2:22Þ

Putðt; St ; s; KÞ ¼ Pðt; sÞ½KNðd ðt; sÞÞ  Fðt; sÞNðdþ ðt; sÞÞ;

ð2:23Þ

where N() is the standard normal CDF and:

d ðt; sÞ ¼

logðFðt; sÞ=KÞ  12 R2 ðt; sÞ : Rðt; sÞ

ð2:24Þ

Proof. See B.

ð2:18Þ

cðt;sÞ St Fð0; tÞ   1 1 1  exp cðt; sÞR2 ð0; tÞ þ R2 ðt; sÞ  R2 ð0; sÞ ; 2 2 2

¼ Fð0; sÞ

47

ð2:21Þ

with the mean given by the residual value curve RV(0, t) and uncertainty around this mean modeled by the OU process (2.12) with volatility r(t) and mean-reversion rate j. Proof. See B. Eq. (2.15) expresses the risk-neutral dynamics of the spot price of the asset with short lease rate (2.6) and finite economic life T in terms of the driving OU process (2.13) and the initial forward curve F(0, t). The expression Eq. (2.17) for the spot price volatility is similar to the expression for the bond price volatility in the Vasicek model. It vanishes as the asset approaches the end of its useful economic life, t ? T, and, hence, the end of its useful economic life. The spot price then converges to the scrap value ST. For any t 2 [0, T], taking the expectation of Eq. (2.15) under Q leads to the identity EQ ½St  ¼ Fð0; tÞ. At t = 0, F(0, 0) = S0, i.e. the initial asset price when brand new. At t = T, F(0, T) = ST, the scrap value. Between t = 0 (brand new) and t = T (scrap), the asset evolves according to Eq. (2.15) under Q with the mean given by the initial forward curve F(0, t) and risk described by the OU process (2.6) with the volatility r(t) and mean-reversion rate j. The dynamics under the statistical measure P is similar with the residual value curve RVð0; tÞ ¼ EP ½St  taking place of the forward curve.

We note that these expressions enable one to price options on both new and used equipment. If the equipment is placed in service at time zero, these expressions give the price at time t P 0 of call and put options on the asset expiring at time s. At the pricing time t, the equipment is either new if t = 0 or used if t > 0 (has the current age t) and has the current spot market value St. At maturity s, the equipment is used and has the age s. h

Remark 2.1. We can introduce the term structure of depreciation rates fDP ð0; tÞ; t 2 ½0; Tg as follows: DP ð0; tÞ :¼  lnðRVð0; tÞ=S0 Þ=t; P so that RVð0; tÞ ¼ eD ð0;tÞt S0 . Here DP ð0; tÞ is the annualized depreciation rate for the period [0,t] under the physical measure P. Similarly, we can introduce the risk-adjusted term structure of depreciation rates: DQ ð0; tÞ :¼  lnðFð0; tÞ=S0 Þ=t; so that Fð0; tÞ ¼ Q eD ð0;tÞt S0 . Here DQ ð0; tÞ is the annualized risk-adjusted depreciation rate for the period [0, t] under the risk-neutral measure Q. Remark 2.2. The key features of our model are as follows. The short lease rate is taken as the fundamental state variable and is modeled as a mean-reverting OU process with time-dependent mean and volatility. Under the assumption of the finite economic lifetime T when the asset is scrapped, this choice leads to the analytical solution for the asset price process as the exponential OU process with time-dependent mean and volatility (Eq. (2.20)) and, hence, analytical solutions for options (Theorem 2.2) and the risk adjustment (Eq. (2.25)). There are three innovations in this modeling choice relative to the existing literature on leasing and financing that is primarily focused on real estate. First, we model the percentage short lease rate or instantaneous rental yield ‘t expressed in percent of the asset value per annum as the fundamental state variable following an OU process. Then the nominal lease rate expressed in dollars per annum is Rt = ‘tSt. The short lease rate ‘t = Rt/St thus serves as the stochastic rent-to-price ratio (RTP ratio). It is known from empirical data that the rental yield or RTP ratio of a real asset generally fluctuates over time with some volatility, as well as exhibits mean reversion toward some average rental yield levels (e.g., see Gallin (2008) for an empirical study of the RTP ratio in the US housing market over the period 1970–2003). The existing real estate leasing modeling literature largely models directly the nominal lease rate in dollars per annum, rather than the rental yield. In many such models the RTP ratio is either constant (c.f. Eq. (9) in Stanton and Wallace (2008) under the geometric Brownian motion assumption for the nominal lease rate) or is a complicated implicitly given function of the nominal lease rate that requires one first to solve the model for the asset

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value process in terms of the nominal lease rate and then take the ratio of the rent to price (c.f. Eqs. (9) and (35) in Clapham and Gunnelin (2003) under the exponential OU assumption for the nominal lease rate). In contrast, we explicitly model the RTP ratio as an OU process. Second, we explicitly assume that the real asset has a finite economic life, while all of the existing literature considers infinitely lived assets. This leads us to the result, Eq. (2.9), that, in turn, leads to the asset price process, Eq. (2.15), as well as the risk adjustment, Eq. (2.20). These results for assets with finite economic life are original to this paper and have never appeared in the literature. Third, we introduce explicit age dependence in the mean and volatility of the real asset, in contrast to the constant parameters usually assumed in the real estate literature. It is of importance for real assets with finite economic life, such as aircraft, since their market dynamics strongly depends on their age. We note that due to the relationship dt = ‘t  rt  rS(t)gt, deterministic assumption for the market price of risk gt and short rate rt, and the explicit expression (2.22) for the asset volatility, the instantaneous depreciation dt under the statistical measure follows a time-inhomogeneous OU process ddt ¼ jðhPd ðtÞ  dt Þdt þ rðtÞdBPt with the long-run mean hPd ðtÞ ¼ hP ðtÞ

 rS ðtÞÞ and the same rate of mean reversion þ dðgtdt rt  rS ðtÞgt  j1 dr dt

j and volatility r(t) appearing in the process for short lease rate. We remark that the possibility of negative short lease rates under the OU specification is not generally a problem in our model. While our short lease rate is specified to be an OU process, empirical long run levels for the short lease rate are high, since it can be represented as the sum of the interest rate rt plus the asset depreciation rate dt plus the risk premium rS(t)gt. Coupled with mean reversion evidenced by empirical data (as estimated in Section 5), the probability of the short lease rate becoming negative within the economically relevant time horizons is negligible. We thus do not generally see the need to switch to a positive process, such as the CIR diffusion, for the short lease rate. The cost of such a switch would be the need to replace all of the explicit analytical solutions in Theorems 2.1 and 2.2, as well as the analytical solutions for recoveries on leases and loans in Sections 3 and 4, with numerical procedures generally requiring numerical Fourier transform inversions. That would substantially complicate the model calibration in Section 5 and increase computation times by orders of magnitude, without any material benefits, given the negligible probability of negative short lease rates under our OU specification. We also remark that our asset process Eq. (2.15) arising as the solution of Eq. (2.9) under our OU specification of the short lease rate looks similar to the classical one-factor commodity model of Schwartz (1997) (see also Clewlow and Strickland, 1999) that models the market price of a consumption commodity, such as crude oil, by an exponential OU process with the mean given by its forward curve. The major difference is that in commodity markets one models the evolution of the market price of a fixed unit of a generic commodity over time, such as a barrel of crude oil. Here we model the evolution of the market price of the given real asset, from the time the brand new asset is placed in service to the time it is retired from service and scrapped. Hence, we have both the finite economic life T > 0 and the interpretation of time as the age of the given asset, rather than the time evolution of the price of a fixed quantity of a generic commodity. This results in the time dependence structure that is strikingly distinct from that of commodity models, such as Schwartz (1997) and Clewlow and Strickland (1999), as evidenced by the factor (1  ej(Tt))/j in front of the OU process in Eq. (2.9), as well as the structure of the volatility function, Eq. (2.17). As such, the dynamics of the asset is closer to that of the zero-coupon bond with maturity T > 0 in interest rate models, where the bond price appreciates toward the bond face

value at maturity, while in our case the value of the asset depreciates to its scrap value. 3. Fixed-rate leases 3.1. Default-free fixed-rate leases The short lease rate ‘t models the rate the owner of the asset would charge at time t on a hypothetical default-free instantaneous operating lease (to a default free lessee). In this section we consider default-free fixed rate operating leases and establish the term structure of default-free lease rates. Consider an operating lease starting at time t P 0 and lasting until time s > t. At the end of the lease the lessee returns the asset to the lessor, who is then free to sell the asset at the then-prevailing market price or lease it to another lessee at then-prevailing market lease rates. To simplify exposition, here we do not consider various options commonly included in lease contracts, such as the lessee’s option to purchase the asset at the end of the lease at a pre-specified residual value. These options are considered at the end of Section 3.2. Suppose the contractual lease payments are made at the contractual lease rate Lf(t, s) at times tk = t + kD, k = 1, . . . , N, s = tN (e.g., a five-year lease with monthly payments would have sixty lease payments at monthly intervals, D = 1/12, N = 60). The (annualized) lease rate is expressed as a percentage of the initial value of the asset St at the start of the lease. Thus, each lease payment at time tk is equal to DLf(t, s)St dollars. The problem is now to establish the default-free lease rate Lf(t, s). Theorem 3.1. The arbitrage-free default-free lease rate is given by:

Lf ðt; sÞ ¼

  1  Pðt; sÞ Fðt; sÞ Pðt; sÞ þ 1 ; Aðt; sÞ St Aðt; sÞ

ð3:1Þ

where P(t, s) is the default-free discount factor for the period [t, s], A(t, s) is the annuity factor for the period [t, s] (present value at time t of the annuity that pays D dollars at each of the lease payment dates tk), P Q Aðt; sÞ ¼ D Nk¼1 Pðt; tk Þ, and Fðt; sÞ=St ¼ eD ðt;sÞðtsÞ is the ratio of the forward price for delivery at the end of the lease to the spot price of the asset at the start of the lease. Proof. The present value at t of the cash flow stream from the lease is DL(t, s)A(t, s)St. The present value at time t of receiving the asset at time s is P(t, s)F(t, s). Thus, the present value of the lessor’s position is DLf(t, s)A(t, s)St + P(t, s)F(t, s). To prevent arbitrage, this should be equal to the spot price of the asset at time t:

St ¼ DLf ðt; sÞAðt; sÞSt þ Pðt; sÞFðt; sÞ:

ð3:2Þ

Solving for the lease rate yields Eq. (3.1). h From Eq. (3.1), we see that the default-free lease rate is equal to the (s  t)-maturity swap rate (or par bond rate) (1  P(t, s))/A(t, s) plus the spread accounting for the risk-adjusted depreciation of the asset during the term of the lease ð1  Fðt; sÞ=St ÞPðt; sÞ= Q Aðt; sÞ ¼ ð1  eD ðt;sÞðtsÞ ÞPðt; sÞ=Aðt; sÞ. 3.2. Defaultable fixed-rate leases We now turn to modeling fixed-rate operating leases to a risky lessee. Under the risk-neutral measure Q, we model the time of default s as the first jump time of a time-inhomogeneous Poisson process with given time-dependent deterministic intensity (arrival rate) k(t). Then, given no default up until time t, the risk-neutral probability of noR sdefault (survival probability) from time t to time  kðuÞdu s is given by e t . Then, given no default up until time t, the price at time t of a defaultable zero-coupon bond with unit face value, zero recovery in the event of default prior to maturity, and

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maturity at time s is then given by P d ðt; sÞ ¼ Pðt; sÞe

Rs t

kðuÞdu

, where

P(t, s) is the price of the corresponding default-free zero-coupon bond. We assume that if the lessee defaults at time s, the lessor will repossess the asset at time s + d, where d P 0 is the repossession delay, upon which the asset is sold for its then current market value. Note that the present value of a lease’s payments plus the residual claim on the asset is equal to the current asset value as in Eq. (3.2). Thus, the recovery assumption of selling the asset at its market value is equivalent to re-leasing it at the market lease rate. We also assume that in the process of repossession the lessor may incur fixed costs Cs+d P 0, as well as proportional costs cs+dSs+d at the rate cs+d P 0 of the asset value. The costs may generally be time dependent as they may increase with the age of the asset and with the repossession delay. Then the recovery in the event of default is:

Rsþd ¼ maxðð1  csþd ÞSsþd  C sþd ; 0Þ:

ð3:3Þ

The repossession delay and costs are important variables in the valuation of defaultable leases, as well as secured loans. The costs the lessor or secured lender may incur in the repossession process may include legal costs of the repossession process, costs of the physical repossession (e.g., flying the aircraft to the desired destination), costs of repair and maintenance of the asset (can be substantial if the bankrupt lessee or borrower fails to properly maintain the asset, and no separate maintenance reserves have been provisioned in the lease or loan contract), costs of downtime (e.g., storage fees and insurance premiums), costs of re-marketing the asset to find a new lessee or buyer, as well as possible super-priority liens, such as mechanics’ liens or airport fees attached to the aircraft in the event the bankrupt lessee or borrower does not pay those obligations. Re-marketing costs may have fixed, as well as proportional components (e.g., broker commissions). All other costs are generally fixed. To simplify our analysis, we do not explicitly consider downtime from the time of repossession to the time of sale or lease of the asset, but roll the costs due to downtime into the fixed costs parameter C. We thus work with the repossession delay d and the total costs C. The costs of downtime could have also been separated from the other costs and added onto the repossession delay (for the study of idle time in oil rig leases see Kenyon and Tompaidis, 2001). The repossession delay d depends on the type of asset and the legal jurisdiction of the lessee or borrower. If the bankruptcy results in a stay of repossession, the repossession delay can take many months or even years. During that period the asset continues to depreciate, lease or loan payments are not made, and the various costs continue to mount. In the US, Section 1110 of the US bankruptcy code grants aircraft lessors and secured lenders exemption from the repossession stay and grants them the right to repossess aircraft sixty days after payment default on a lease or secured loan. Internationally, the Cape Town Treaty grants aircraft lessors and secured lenders the right to repossess an aircraft sixty days after default in jurisdictions that have ratified it. Section 1110 in the US and the Cape Town Treaty provide valuable protections to aircraft lessors and secured lenders by limiting the repossession delay. In addition to the repossession of the asset, the lessor may claim certain losses and costs as a senior unsecured claim in the bankruptcy process. What is actually recoverable depends on the details of the lease contract. We assume that the costs Cs+d + cs+dSs+d and unpaid and accrued lease payments AI from the last completed payment at time tk prior to default s up to the time s + d when the lessor repossesses the asset are recoverable as senior unsecured claims. Assuming a recovery rate u 2 [0, 1] on senior unsecured debt, the unsecured part of recovery is u(Cs+d + cs+dSs+d + AI). To simplify exposition, in this section we set u = 0, thus

neglecting the unsecured part of the recovery. It can be easily added to the analysis and is considered in Section 4. We now turn to the problem of establishing the defaultable lease rate Ld(t, s). Under our assumption that the default intensity is a given deterministic function of time k(t), we have the following result for the defaultable lease rate. Theorem 3.2. In equilibrium, the defaultable lease rate is given by:

 Pd ðt;sÞ d ðt;sÞ Ld ðt; sÞ ¼ 1P þ 1  Fðt;sÞ Ad ðt;sÞ St Ad ðt;sÞ Ru Rs  kðv Þdv  St A 1ðt;sÞ t kðuÞe t ð1  cuþd ÞCall

ð3:4Þ

d

ðt; St ; u þ d; C uþd =ð1  cuþd ÞÞdu; where Pd(t, s) is the defaultable discount factor for the period [t, s] (the price at time t of a defaultable zero-coupon bond with unit face value, maturity at time s and no recovery in default), P d ðt; sÞ ¼ Rs  kðuÞdu Pðt; sÞe t ; Ad(t, s) is the defaultable annuity factor for the period [t, s] (present value at time t of the defaultable annuity that pays D dollars at each of the payment dates tk until either maturity s = tN or default, whichever comes first, and has no recovery in the event of P default), Ad ðt; sÞ ¼ D Nk¼1 Pd ðt; t k Þ, Call (t, St; u + d, Cu+d/(1  cu+d)) denotes the price at time t of a call option on the asset with expiration at time u + d, strike price Cu+d/(1  cu+d), and asset value St at time t (the pricing formula is given in Theorem 2.2), and other notation as in Theorem 3.1. Proof. The sum of the present values of (1) the promised lease payments until either the end of the lease or default, whichever comes first, (2) return of the asset at the end of the lease if no default occurs, and (3) the recovery via repossession of the asset in the event of default is equal to:

DLd ðt; sÞSt

Z s Ru N X  kðv Þdv Pd ðt; t k Þ þ Fðt; sÞPd ðt; sÞ þ kðuÞe t Pðt; u k¼1

t

Q

þ dÞE ½Ruþd jF t du: Recalling Eq. (3.3), the recovery term can be expressed in terms of the call option on the asset:

Pðt; u þ dÞEQ ½Ruþd jF t  ¼ ð1  cuþd ÞCallðt; St ; u þ d; C uþd =ð1  cuþd ÞÞ: To prevent arbitrage, the sum of the three terms should be equal to the spot price, St, of the asset at time t. Solving for the lease rate yields Eq. (3.4). h From Eq. (3.4), we see that the defaultable lease rate is equal to the (s  t)-maturity defaultable par bond rate (1  Pd(t, s))/Ad(t, s) assuming zero recovery plus the spread that accounts for the risk-adjusted depreciation of the asset during the term of the lease (1  F(t, s)/St)Pd(t, s)/Ad(t, s) when the asset is returned at the end of the lease if no default occurs minus the present value of the repossession recovery if default occurs. The difference Ld(t, s)  Lf(t, s) between the defaultable lease rate and the default-free lease rate on an otherwise identical lease is the lease credit spread compensating the lessor for the risk of default. To reduce the loss-given-default, lessors often require a down payment at the inception of the lease. Assuming a down payment aSt (a fraction a of the initial asset value St), the present value in the proof of Theorem (3.2) is set equal to the initial value of the asset minus the down payment (1  a)St. Solving for the lease rate, the resulting lease rate is given by the expression (3.4) minus the annuitized down payment rate, a/Ad(t, s). Real asset leases often include options. One of the most common options is the option to purchase the asset at a fixed price at lease maturity. Grenadier (1996) considers this option for credit risky real estate leases in a structural credit risk model where the

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asset follows a Geometric Brownian Motion process. Here we consider a reduced-form credit risk model with the asset model as proposed in Eq. (2.15). Let the lease be initiated at time t with maturity s > t. Without the option, the defaultable lease rate is established by the no-arbitrage argument of Eq. (3.4). If the lessee has the option to purchase the asset at lease maturity at the fixed price K, the lessee is effectively granted a European call on the asset with the same maturity as the lease. Assuming the option price is rolled into the lease rate, the expression for the arbitrage-free lease rate is modified by adding to the right-hand side of Eq. (3.4) the annuitized option term: 

e

Rs t

ku du

Callðt; St ; s; KÞ : St Ad ðt; sÞ

ð3:5Þ

Remark 3.1. We stress that the default intensity k entering Eq. (3.4) is the risk-neutral (or risk-adjusted) intensity. It may differ from the default intensity (arrival rate) under the physical probability measure P. When calibrating our model to market data in Section 5.2, we imply the intensity from market quotes of credit default swap (CDS) spreads, rather than calibrate it to historical default frequencies corresponding to the credit rating of the borrower. A number of empirical studies have documented that risk-adjusted default intensities implied by market CDS spreads (as well as market prices of corporate bonds) are generally higher than historical default frequencies. The ratio of the risk-neutral to the historical default intensity represents the risk premium investors demand in compensation for the exposure to the risk of default (more precisely, to the risk of experiencing a jump in prices of debt instruments in response to a default event). In the context of our model, by calibrating it to CDS spreads, the default risk component of the secured financing transaction is priced consistently with the unsecured debt.

Remark 3.2. Our choice of the time-inhomogeneous Poisson process model is motivated by, on one hand, our desire to calibrate to the term structure of CDS spreads and, on the other hand, to keep the combined asset-credit model single-factor to facilitate analytical tractability, calibration to the available data, and to avoid introducing any unobservable model parameters that we cannot directly estimate from the available data.

4. Secured debt 4.1. Secured loans We now turn to the valuation of secured loans. Consider a mortgage-style fixed-rate secured loan that originates at time t, matures at time s > t, and has initial principal P0. Let m = m(t, s) denote the fixed mortgage interest rate and C the corresponding periodic payment in dollars. The payments are made at times tk = t + kD, k = 1, 2, . . . , N, with t0 = t and tN = s. The total number of payments is equal to N = (s  t)/D. Then, according to the mortgage amortization schedule, the remaining principal Pk after the payment is made at time tk and the periodic payment amount p are equal to:

recovery is: min{Pb(st)/Dc + AIs+d, (1  cs+d)Ss+d  Cs+d}, where Pb(st)/Dc denotes the principal outstanding at the time of default (bxc denotes the integer part of x, so that b(s  t)/Dc is the number of the last completed payment before default) and AIs+d is the interest accrued from the last completed payment before default up to the time of repossession s + d (we assume the interest accrues at the contractual rate m), and the costs incurred in the repossession process are as discussed in Section 3.2. The recovery is equal to either the outstanding principal plus accrued interest or the asset value net of all the costs, whichever is less. In addition to the secured recovery, the secured lender may claim the shortfall

maxfPbðstÞ=Dc þ AIsþd þ C sþd  ð1  csþd ÞSsþd ; 0g as a senior unsecured claim in the bankruptcy process. Assuming the senior unsecured recovery rate u 2 [0, 1], the total recovery can be written in the form:

Rsþd ¼ P bðstÞ=Dc þ AIsþd  ð1  uÞ maxfPbðstÞ=Dc þ AIsþd þ C sþd  ð1  csþd ÞSsþd ; 0g: Then, using the previous results in Sections 2 and 3, we can write the present value of the mortgage loan at origination as the sum of the present value of the promised periodic payments p discounted with the defaultable discount factors Pd(t, tk), and the recovery in the event of default:

Z s Ru N X  kðv Þdv Mt ¼ p Pd ðt; t k Þ þ kðuÞe t Pðt; u k¼1

t

þ dÞEQ ½Ruþd jF t du;

ð4:2Þ

with the present value of recovery given by:

Z

Ru  kðv Þdv kðuÞe t Pðt; u þ dÞEQ ½Ruþd jF t du t Z s Ru  kðv Þdv kðuÞe t Pðt; u þ dÞK uþd du  ð1  uÞ ¼ t   Z s Ru K uþd þ C uþd  kðv Þdv du; kðuÞe t ð1  cuþd ÞPut t; St ; u þ d;  1  cuþd t s

 K þC uþd is the where Ks+d = Pb(st)/Dc + AIs+d and Put t; St ; u þ d; uþd 1c uþd

price of the put at time t with expiration at time u + d, strike price (Ku+d + Cu+d)/(1  cu+d), and asset price St. The pricing formula is given in Theorem 2.2. This result yields the present value of the loan, given the contractual loan rate m. To determine the loan rate at origination, one needs to solve the equation setting the loan’s present value equal to the initial loan principal:

Mt ðmÞ ¼ P0 :

ð4:3Þ

ð4:1Þ

Since the loan’s present value is a non-linear function of the loan rate m, this equation has to be solved numerically by an iterative root finding algorithm such as bisection. The solution exists and is unique since the loan value is a monotonically increasing function of m. In this section we have considered a loan with mortgage-style amortization. Other types of loans, such as balloon loans or loans with level principal amortization, can easily be considered in the same framework.

The present value of the defaultable mortgage consists of two parts: the present value of the promised periodic payments up until the time of default or maturity, whichever comes first, and the present value of recovery in the event of default. Assuming the lender repossesses the asset after a delay d, the secured part of

Remark 4.1. We note that the costs and delays in the event of default and asset repossession on secured debt may generally be different than those on leases. In fact, Habib and Johnsen (1999) and Eisfeldt and Rampini (2009) argue that leasing and lending are not perfect substitutes because lessors generally enjoy a

Pk ¼ P0

1  ð1 þ DmÞðNkÞ 1  ð1 þ DmÞN

;

p¼

DmP 0 1  ð1 þ DmÞN

:

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repossession advantage in Chapter 11 bankruptcy. From the standpoint of our model, while both leases and secured debt are treated in a unified fashion as contingent claims on the credit of the obligor and on the asset, different cost and delay structures for leases and secured debt can easily be considered, thus providing a vehicle for analyzing differences between various forms of financing.

at the time of default, while it takes time for the ECA to repossess the aircraft. Assuming that the asset value dynamics are given by Eq. (2.15) and the default time s is the first jump time of a timeinhomogeneous Poisson process with intensity k(t), we therefore have that the guarantee value is:

CG ¼

Z

s



Ru

kðv Þdv

Pð0; uÞðP bu=Dc þ AIu  Pðu; u þ dÞ Z s Ru  kðv Þdv kðuÞe 0 ð1  ðPbu=Dc þ AIuþd ÞÞdu þ kðuÞe

0

0

4.2. Export credit guarantees

0

To help support exports of equipment, such as aircraft, railroad equipment, ships, and certain other categories, Export Credit Agencies (ECAs), such as Export–Import Bank of the United States, under certain conditions provide export credit guarantees to lenders for loans conforming to the export credit eligibility requirements (see, e.g., Murphy (1998) for general background). In the aircraft sector, the Organization for Economic Cooperation and Development (OECD) Sector Understanding on Export Credits for Civil Aircraft (the ‘‘OECD Aircraft Sector Understanding (ASU)) governs the terms and conditions of ECA support for aircraft. Some of the current terms and conditions under the new ASU that came into force in 2011 are as follows. The maximum term for the guaranteed loan is twelve years, the maximum LTV of the loan is 85% of the net purchase price of the aircraft, and the principal amortization schedule is mortgage-style with quarterly payments. Furthermore, according to the credit rating of the borrower, certain risk mitigants are applied to the basic terms and conditions, such as the reduction in LTV, shortening loan maturity to 10 years, and/or replacing the mortgage-style principal amortization profile with the level principal amortization. The purpose of risk mitigants is to reduce the risk of the loan to reduce the risk exposure of the ECA guaranteeing it. An export credit guarantee is essentially a pure cover insurance contract. In the event of default, the ECA takes over the defaulted loan from the commercial financier and pays off the outstanding loan balance and the accrued interest to the financier. The financier thus is relieved from the cost and delay of having to repossess the aircraft and the risk of any shortfall in the event the recovered value of the repossessed aircraft net of all costs is less than the outstanding loan balance plus accrued interest. The ECA charges an upfront exposure fee expressed as percentage of the loan principal amount. The exposure fee is paid by the borrower (e.g., the airline acquiring the aircraft financed by a third party financier). We now show how to value export credit guarantees in our modeling framework. If default occurs at time s, the guarantee payoff at default to the commercial lender is equal to the outstanding loan balance plus accrued interest from the last completed quarterly payment, Pbs/Dc + AIs. In exchange, the ECA takes over the defaulted loan and, after a repossession delay d, repossesses the aircraft. Thus, the ECA’s recovery at time s + d via repossession of the aircraft, net of costs, is determined similar to the loan recovery:

minfPbs=Dc þ AIsþd ; ð1  csþd ÞSsþd  C sþd g ¼ Pbs=Dc þ AIsþd  maxfPbs=Dc þ AIsþd þ C sþd  ð1  csþd ÞSsþd ; 0g: Then the present value of the credit guarantee at time zero (loan origination) to the commercial lender is: 

 CG ¼ EQ 1fs
þ i : þ EQ 1fs
ð4:4Þ

If the repossession delay is equal to zero, d = 0, then the value of the guarantee collapses to the value of the put option on the asset exercisable in the event of borrower default. With repossession delay, there is an additional term in the value of the guarantee accounting for the fact that the guarantee itself pays off the commercial lender

 cuþd ÞPutð0; S0 ; u þ d; Kðu; dÞÞdu;

ð4:5Þ

where the strike price of the put option is K(u, d) = (Pbu/Dc + AIu+d + Cu+d)/(1  cu+d). 4.3. Securitization: enhanced equipment trust certificates Enhanced Equipment Trust Certificates (EETCs) are debt securities issued by airlines to finance acquisitions of new aircraft or refinance existing fleet. Typically there are several aircraft in the pool, with the corresponding aircraft mortgages placed in the trust. To appeal to different types of investors, several tranches of different seniority are issued. Each tranche has its own amortization schedule, coupon rate, maturity, and priority in the payment of interest and principal relative to the other tranches. A subordination agent distributes scheduled interest payments and principal repayments to the investors in each tranche. In the event of the airline’s default, the trustee has the right to repossess the aircraft collateral, sell the aircraft, and distribute the proceeds net of costs incurred in the repossession and sale process to the tranche investors according to the subordination structure. In the US, EETCs enjoy the protection of Section 1110 of the US bankruptcy code that allows the trustee to automatically repossess the aircraft sixty days after the airline default. Any shortfall after the repossession and sale of the aircraft becomes a senior unsecured claim in the bankruptcy proceedings and, as such, generally receives the same recovery rate as senior unsecured corporate bonds. Further discussions of the EETC market can be found in Benmelech and Bergman (2009), Littlejohns and McGairl (1998), and Morrell (2001). The EETC structure is enhanced with a stand-by liquidity facility provided by a third party financial institution. In the event of default, the liquidity facility covers up to eighteen months of interest payments on enhanced tranches, until the collateral aircraft are repossessed and sold. The liquidity facility provider then has the first priority claim on the recovery to reimburse these payments. The liquidity facility payments are, in effect, an advance payment toward the recovery received by the enhanced tranches. To simplify exposition, we do not explicitly model the liquidity facility in what follows. The impact of this simplification on the tranche values and interest rates is negligible due to the first priority claim of the liquidity provider on the recovery. Consider the typical case with three tranches: senior (A), mezzanine (B), and junior/subordinate certificates (C). Suppose that the airline defaults at time t, and the collateral aircraft is subsequently repossessed at time t + d. Then a typical EETC issue consisting of three tranches has the following priority structure for receipt of recovery payments: (1) Accrued interest of A, K ð1Þ :¼ AIðAÞ t ; (2) t ðBÞ ð2Þ Accrued interest of B, K t :¼ AIt ; (3) Accrued interest of C, ðAÞ ð4Þ K ð3Þ :¼ AIðCÞ :¼ Pbt=Dc ; (5) Outt ; (4) Outstanding principal of A, K t t ðBÞ standing principal of B: K ð5Þ :¼ P ; (6) Outstanding principal of t bt=Dc ðCÞ C: K ð6Þ :¼ P bt=Dc . Here bt/Dc is equal to the number of the last cout ðAÞ pon payment completed prior to default, so that P ½t=D is the tranche A principal outstanding at the time of default, etc. Our modeling framework allows us to explicitly value each of these recoveries in the EETC priority structure as options on the collateral asset. Complete details are presented in C, where explicit expressions

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for the present values of each tranche’s recovery, as well as the present value of the entire EETC issue are given.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u 1 X k

2 Rð0; tÞ ¼ t xt  xt : N  1 k¼1

5. Applications to aircraft financing 5.1. Estimation of the aircraft value process The key inputs into our valuation formulas are the risk-adjusted residual value curve F(0, t) (the forward asset price under the riskneutral probability measure equal to the mean of the risk-neutralized asset price distribution) and the volatility R(0, t) of the asset value distribution around that mean. In this section we estimate the residual value curve RV(0, t) and volatility R(0, t) from historical data on used aircraft values. Since F(0, t) is related to RV(0, t) by the risk-adjustment factor of Eq. (2.20), the market price of risk g is required to fully specify the risk-adjusted residual value curve (the forward curve). In the next section we imply the market price of risk g from capital market pricing of EETCs. We then use the calibrated model to value aircraft leases, aircraft-backed loans, and export credit guarantees. The Aircraft Value Analysis Company (AVAC) provided the authors access to a data set of annual current market value (CMV) aircraft appraisals starting as far back as 1967 and up until 2008. Our dataset contained 36 different aircraft models, each with multiple vintages (year of production), for a total of 450 time series of annual CMV appraisals. Each time series was inflation-adjusted using the consumer price index (CPI). All the prices were normalized relative to the initial appraisal so all brand new aircraft are assumed to be valued at 100. In order to estimate the residual value curve and volatility, all inflation-adjusted and normalized CMV appraisals for one year old aircraft, two year old aircraft and so on, were separately pooled together. The mean and volatility were estimated from each age pool in the following way. Let Skt be the kth CMV in the pool for a t-year old aircraft. According to our model Eq. (2.21), under P the asset value has a lognormal distribution:

St ¼ RVð0; tÞe

Rð0;tÞZ12R2 ð0;tÞ

;

 Defining xkt ¼ log Skt =RVð0; tÞ , we then estimate the volatility as the age-dependent standard deviation:

ð5:6Þ

where Z is a standard normal random variable. Thus, we treat historical values Skt as sample observations drawn from this distribuP tion. The mean is estimated by computing RVð0; tÞ ¼ Nk¼1 Skt =N, where the sum is taken over all observations in the pool of t-year old aircraft. Note that when using the estimates of the mean in pricing, it is necessary to include back the effect of expected inflation over the term of interest. Fig. 1a displays the 450 time series in our data set, each normalized and inflation-adjusted. Each time series represents a particular aircraft model/vintage after normalization and inflation adjustment. The mean estimates RVð0; tÞ are also displayed on the graph. Fig. 1b displays our estimates RVð0; tÞ for the mean residual values, with the corresponding standard errors marked on the graph (see also Table D.4 for numerical values). We also fit the exponential RVC of the form RV(0, t) = exp (D t)S0 for the first 12 years. The depreciation rate of D = 8.08% gives the best fit. The corresponding exponential RVC is plotted on the graph, suggesting that the inflation-adjusted depreciation is close to exponential for the first 12 years of the aircraft’s life (typical term of aircraft mortgages). Referring to Fig. 1b, we see that there is little residual value remaining for aircraft 30 years and older. From our estimated residual value curve, the mean value of the 35 year old aircraft is only 0.65% of the new value (on the inflation adjusted basis). Commercial aircraft are typically retired from service and salvaged for their scrap value around 30–35 years of age. We take the useful economic life of the aircraft to be 35 years (T = 35).

ð5:7Þ

Fig. 2a displays the estimate of R(0, t) for the first 15 years of the aircraft life, along with the standard errors (see also Table D.4 for numerical values). It is clear from the graph that there is significant volatility in aircraft values, and furthermore that as the aircraft ages, the volatility grows larger. For the purpose of valuing the recovery terms in aircraft backed loans and EETCs, we need to imply the volatility function r(t) entering the OU SDE for the short lease rate ‘t from R(0,t). For practical purposes we specify the volatility function {r(t), t 2 [0, T]} as a piece-wise constant function:

rðtÞ ¼

bT= Dc X

rk 1½tk ;tkþ1 Þ ðtÞ;

ð5:8Þ

k¼0

where the volatility rk:¼r(tk) is in effect for the time interval t 2 [tk, tk+1), tk = kD, and b T/Dc is the integer part of the ratio T/D. As the AVAC appraisal time series is annual, we make the assumption that r(t) is piece-wise constant also on an annual basis (D = 1). For this specification, the squared asset volatility, Eq. (2.17), is given by:

R2 ð0; tÞ ¼

1 ð1  ejðTtÞ Þ2 2j3

½t= D1 X

r2k ðe2jðttkþ1 Þ  e2jðttk Þ Þ

k¼0

 þr2½t=D ð1  ejðt½t=DDÞ Þ : We can then back out the volatility r(t) of the short lease rate from the asset volatility R(0, t) via the recursion:

r20 ¼

1 2j3 R2 ð0; t 1 Þ; 2 j D ð1  e Þ ð1  ejðTt1 Þ Þ2

and for j > 1: 2 j

r

" 1 2j3 ¼ R2 ð0; t jþ1 Þ 2 j D ð1  e Þ ð1  ejðTtjþ1 Þ Þ2 # j1 X 2 2jðt jþ1 t kþ1 Þ 2jðt jþ1 t k Þ  rk ðe e Þ : k¼0

The rate of mean reversion j is estimated by maximum likelihood estimation (MLE) carried out on the CMV appraisal time series. Time series of at least 8 years in length were used, for a total of 6969 data points, with the result that j = 0.187 with standard error 0.003. Fixing j = 0.187, T = 35 and using the asset volatility estimates RðtÞ of Eq. (5.7), the piece-wise constant short lease rate volatility rðtÞ as a function of the aircraft age is then fully determined  ðtÞ and r  S ðtÞ as defined by the above recursion. Fig. 2b plots both r in Eq. (2.17). The coupon rate, amortization schedule, principal amount, and loan-to-value for each tranche of the EETC issue, as well as appraisals of the underlying pool of aircraft, can be obtained from the prospectus of the initial public offering. The only remaining quantity necessary to evaluating Eq. (C.4), the total EETC value, is the market price of risk g entering into the risk-adjustment factor of Eq. (2.20) linking the historical RVC and the risk-adjusted RVC of the pricing formulas (risk-neutral forward curve). In the following section we imply the market price of risk and, hence, the risk-adjusted residual value curve, from the EETC coupon rates.

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A

B Residual Value Curve (Normalized & Inflation−Adjusted)

100% 100*exp(−D*t), D=8.08% RVC

90% 80%

% of Initial Value

70% 60% 50% 40% 30% 20% 10% 0% 0

5

10

15

20

25

30

35

Aircraft Age (years) Fig. 1. (A) Displays a total of 450 Current Market Value (CMV) appraisal time series, normalized and inflation-adjusted, as well as the estimated Residual Value Curve. (B) Displays the Estimated Residual Value Curve (RVC), normalized and inflation-adjusted. Standard errors are also displayed.

A

B

Fig. 2. (A) Displays the estimated aircraft volatility R(0, t) as a function of aircraft age. Standard errors are also displayed. (B) Displays the estimated r(t) and rS(t) as functions of aircraft age. Note that the estimated r(t), which we assume to be piece-wise constant in between annual nodes, is increasing in the aircraft age.

5.2. Case study: Continental 2007 EETC issue We now apply our methodology to the EETCs issued by Continental Airlines on April 10th, 2007. This issue financed the acquisition of 30 new aircraft by Continental Airlines, including 12 Boeing 737-824 and 18 Boeing 737-924ER. Appraisals of the aircraft were made by three independent appraisers and were given in the initial public offering prospectus for the EETC issue. For each aircraft in the pool, we chose the lowest appraisal among the three independent appraisals as the initial asset value in the model. The total initial value of the collateral was $1,554,200,000. The issue consisted of three tranches, with the A tranche being the most senior, the B tranche subordinate to A, and the C tranche subordinate to both A and B. The subordination structure of this issue is as described in Section 4.3. The characteristics of the issue are summarized in Table 1, with the principal repayment schedule presented in Table D.5. The ‘‘Swap Rate’’ given for each tranche is the par rate

Table 1 Summary of the CAL 2007 EETC issue.

Principal Swap rate Prospectus rate Spread (bps) Maturity Cumulative LTV Asset value Total principal

Tranche A

Tranche B

Tranche C

$756,762,00 5.29% 5.98% 69 15 48.7% $1,554,200,000 $1,146,810,000

$221,850,000 5.26% 6.90% 164 15 63%

$168,198,000 5.14% 7.34% 220 7 73.8%

obtained from the LIBOR and swap curve for the cash flows of each EETC tranche. The zero-coupon curve P(0, t) is stripped from the LIBOR and swap curve, with the rates given in Table D.3. The par rates on each of the tranches are determined by the tranche amortization profile and the zero-coupon curve.

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S. Golbeck, V. Linetsky / Journal of Banking & Finance 37 (2013) 43–59

The risk-neutral default intensity k(t) is recovered from Continental Airlines senior unsecured corporate debt CDS spreads on the date of the EETC issue as reported by Bloomberg by assuming a piece-wise constant functional form of the intensity (constant between available CDS maturities of 1, 2, 3, 4, 5, 7, and 10 years). See Table D.3 for CDS spreads and Fig. D.6a for the market-implied risk-neutral probability of survival over 10 years. To imply the risk-neutral default intensity from the market CDS spreads, we assume a recovery rate of 19%, corresponding to the loss-given-default (LGD) of 81%. This recovery rate is estimated from a dataset of senior unsecured US airline corporate bond recoveries provided to us by Professor Edward Altman of the Stern School of Business, New York University. The definition of recovery in this dataset is the market value of the corporate bond after the default announcement. It is thus the market’s estimate of the ultimate recovery through the bankruptcy proceedings. To account for inflation over the 15 year term of the EETC, we use the Federal Reserve Bank of Cleveland’s estimate1 of the expected rate of inflation over 15 years as of April, 2007. The Cleveland Fed’s inflation forecasting model relies heavily on the spreads between market yields on Treasury Inflation Protected Securities (TIPSs) and straight Treasury notes (see Haubrich et al. (2008)). Assuming the annual inflation rate of i = 2.4857% per the Fed’s estimate, we used the corresponding continuously compounded inflation rate of ic = log (1 + i) = 2.4553% to inflate our inflation-adjusted aircraft Residual Value Curve for the use in EETC valuation. To value the recovery component of the EETCs, a repossession delay of 60 days (d = 1/6) is assumed in accordance with Section 1110 of the US Bankruptcy Code. Fixed costs of the repossession process are assumed to be increasing with the aircraft’s age. According to industry sources, these costs may range from around 5% for new aircraft to substantially greater than 10% for older aircraft. In our implementation we take the costs to be linearly increasing in the aircraft age according to Ct = (0.05 + 0.0075t)S0. That is, the fixed costs are assumed to start at five percent of the initial value of the aircraft and increase by 0.75% per year thereafter. No proportional costs are assumed (ct = 0). We assume that the market price of risk g is constant, and imply it from the market value of the EETC by setting the model value of the EETC issue with the prospectus quoted coupons as given by Eq. (C.4), to its total principal and uniquely solving the resulting nonlinear equation for g:

PVðEETCÞðgÞ ¼ Par: The solution is unique since the EETC value is a monotonically decreasing function of g. The resulting implied value of g was determined to be g = 0.49. We note that from the standpoint of the classical Capital Asset Pricing Model (CAPM), the market price of risk g could be alternatively estimated as the product of the correlation between changes in aircraft market values and the stock market index and the historical Sharpe ratio of the stock market index (ratio of the excess index return relative to the index return volatility). The value of the implied market price of risk we obtain is higher than would be expected from the CAPM. The historical S& P 500 Sharpe ratio estimated over the period 1928–2009 from S& P 500 total returns and one-month T-bill returns (as the risk-free return) using CRSP data is 0.368. Since the correlation between changes in aircraft market values and the stock index is generally less than one, the CAPM would predict g < 0.368. It is not surprising that the market price of risk implied by our model from the EETC market pricing is greater than predicted by the classical CAPM, as our model does not explicitly account for liquidity and market friction effects in 1

http://www.clevelandfed.org/research/data/inflation_expectations/index.cfm

the aircraft market, as well as the possible dependence between default arrival and collateral asset values (collateral channel effects c.f. Benmelech and Bergman, 2011) and contagion effects (c.f. Giesecke and Weber (2006) and references therein) that may be in play in the aircraft financing market. The market price of risk value implied from the market pricing of EETCs captures any possible market pricing of these effects even though we do not explicitly model them in our model. This situation is similar to implied vs. historical volatility in equity options prices. On the surface of things, the Black–Scholes–Merton (BSM) model is inadequate as the pricing model to establish absolute levels of options prices. Followed literally, the BSM model requires one to estimate historical volatility from historical price data, and then compute options prices using this volatility. This is generally inadequate, as the BSM model does not capture many real-world features of asset returns, such as stochastic volatility, jumps and various market frictions. However, the BSM model is still very useful as a relative pricing device. In the implied approach, one calculates implied volatilities from benchmark options, such as highly liquid exchange-traded options, and then uses the implied values to establish relative pricing of less liquid derivatives, such as various over-the-counter option contracts, relative to the benchmark securities. It is in this spirit we calibrate our model to a benchmark EETC capital market aircraft financing transaction, and then use it to establish relative pricing of other private market financing transactions, such as bank loans, leases, and export credit guarantees. In the aircraft financing market, EETCs are the only publicly traded securities with transparent pricing, all other transactions being highly confidential private market transactions. In the subsequent sections, we develop this application and establish relative pricing of aircraft leases, loans, and export credit guarantees relative to the benchmark EETC issue. We also stress that we value the Continental 2007 EETC issue relative to the Continental Credit Default Swaps on the same date. Since CDS are hedges for the senior unsecured debt, we have thus isolated the market price of collateral value risk and separated it from the risk of corporate default. Fig. 3a displays the corresponding risk-adjusted residual value curve along with the historical RVC. The risk adjustment shifts the residual value curve downward so that the asset depreciates at the faster rate under the risk-neutral probability measure than under the historical probability measure. Fig. 3b also presents the plots of the risk-neutral depreciation rate DQ ð0; tÞ relative to the historical depreciation rate DP ð0; tÞ. 5.3. Application: the impact of repossession delay on asset financing We now apply our model calibrated to the historical aircraft data in Section 5.1 and with the market price of risk implied from the Continental 2007 EETC issue to evaluate various forms of private market financing, including leases, loans, and credit guarantees. As a particularly interesting application, we focus on the impact of the repossession delay on valuation of these transactions. Section 1110 of the US Bankruptcy Code provides a secured lender the ability to repossess the aircraft within sixty days after a bankruptcy filing unless the airline cures all defaults. The right of the lender to take possession of the aircraft is not hampered by the automatic stay provisions of the US Bankruptcy Code. Section 1110 has payed a pivotal role in allowing US airlines to access capital markets via the issuance of EETCs. Senior EETC tranches are rated investment grade by credit ratings agencies. The A tranche of the Continental 2007 issue we analyzed in Section 5.2 was rated A by Standard & Poor’s and Baa1 by Moody’s, while Continental’s senior unsecured debt was rated B by Standard & Poor’s and B3 by Moody’s at the time of the EETC issue. The investment grade rating of the EETC was predicated on the combination of the high

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S. Golbeck, V. Linetsky / Journal of Banking & Finance 37 (2013) 43–59

A

B

Fig. 3. (A) Displays the Residual Value Curve (RVC) and risk-adjusted RVC. (B) The depreciation rates under the physical and risk-neutral probability measures, respectively DP ð0; tÞ and DQ ð0; tÞ. The risk-adjustment factor and risk-neutral depreciation rate are implied from the 2007 EETC issued by Continental Airlines.

value of the aircraft collateral and the assurance of Section 1110 that in the event of airline bankruptcy the EETC trustee would be able to repossess the collateral after a delay limited to sixty days. Section 1110 protection is a crucial element of the EETC structure, due to the fact that the collateral is only as good as the ability of the creditor to access it in the event of default. However, Section 1110 of the US bankruptcy code is unique among national insolvency laws around the world in giving secured creditors with interests in the aircraft the automatic ability to repossess after a fixed sixty day delay, thus removing any potential uncertainty in the handling of the creditor’s rights by the courts. As a result, while the US airlines have enjoyed access to capital markets due to Section 1110, non-US airlines have so far been largely unable to issue any public debt securities collateralized by aircraft on competitive terms. The primary reason has been

A

the reluctance of bond investors to accept the depreciating aircraft collateral in the face of potentially lengthy repossessions delays. While for many of the larger US airlines EETC issues have been the primary source of aircraft financing, the primary sources of financing for non-US airlines have been bank loans, leases, and export credit financing. The reasons why private market financiers, such as banks and operating lessors, are more willing to lend against aircraft collateral in the face of possible repossession delays is their specialized expertise in the aircraft markets and their ongoing relationships with the airlines that may give them higher confidence in their ability to repossess swiftly, while EETC trusts are stand alone financing vehicles managed by the trustee on behalf of bond holders that may lack the specialized expertise of a large bank or lessor extensively involved in aircraft finance on an on-going basis. Campello and Hackbarth (2012) support this line

B

Loan credit spread as a function of Maturity and Repossession Delay 200

Guarantee Premium (% of loan principal)

180 Repossession Delay=2m Repossession Delay=4m

160

Repossession Delay=6m Repossession Delay=12m Repossession Delay=18m

Credit Spread (bps)

Guarantee Premium as a function of Maturity and Repossession Delay

12%

140 120 100 80 60

Repossession Delay=2m

10%

Repossession Delay=4m Repossession Delay=6m Repossession Delay=12m Repossession Delay=18m

8%

6%

4%

40 2% 20 0 5

6

7

8

9

10

11

Maturity (years)

12

13

14

15

0% 5

6

7

8

9

10

11

12

13

14

15

Maturity (years)

Fig. 4. (A) Displays the loan credit spread term structure for selected repossession delays, d, with LTV = 85%. (B) Diplays the ECA guarantee premium term structure for selected repossession delays, d, again with LTV = 85%.

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S. Golbeck, V. Linetsky / Journal of Banking & Finance 37 (2013) 43–59

A

B

Fig. 5. (A) Displays the default-free lease rate term structure, with and without the option to purchase at maturity, as compared to the swap rate curve prevailing on 4/10/ 2007. (B) Displays the lease credit spread term structure for selected repossession delays, d.

of reasoning, arguing that firms facing financing frictions gain greater access to credit, often at a lower cost, as contracts become more enforceable, partly due to increased tangibility of assets. The positive impact of Section 1110 on US airlines’ aircraft financing, including the airlines’ access to capital markets via EETC issues, lead to the desire by many aircraft market participants to replicate the success of Section 1110 globally. The Cape Town Convention on International Interests in Mobile Equipment and the Protocol to the Convention on International Interests in Mobile Equipment on Matters Specific to Aircraft Equipment, together usually referred to as the Cape Town Treaty (CTT), is a recent international treaty (entry into force in 2006) intended to standardize aircraft financing transactions and harmonize diverse national insolvency laws. In particular, the OECD qualifying declarations with the Protocol Article XI, Alternative A with a maximum repossession delay period of sixty days essentially replicate Section 1110 internationally among the jurisdictions ratifying the CTT. A number of jurisdictions around the world have already ratified the CTT, with a significant number currently considering ratification. In this section we study the impact on theoretical loan and lease credit spreads and export credit guarantee premiums of reducing the repossession delay from various levels to sixty days, corresponding to the CTT ratification by jurisdictions with various expected delay levels. According to industry sources, the actual repossession experiences in various non-CTT jurisdictions around the world have varied greatly, from sixty days or less to as long

as two years, depending on the jurisdiction’s legal system, national insolvency laws, court expediency, and the circumstances of each case. In the examples in this section we consider a range of repossession delays from 2 months to 18 months and compute the term structures of lease credit spreads, loan credit spreads, and export credit guarantee premiums. For these examples we take the term structures of Continental CDS spreads and USD LIBOR and interest rate swap rates on the date of the CAL 2007 EETC issue as in Section 5.2. The results for credit spreads for loans with 85% LTV, quarterly payments and maturities from five to fifteen years are presented in Fig. 4a. The corresponding export credit guarantee premiums insuring these loans against the loss in the event of default are presenting in Fig. 4b. We assume that the loan rate in the guaranteed loan is equal to the corresponding interest rate bootstrapped from the LIBOR and swap curve. The results for credit spreads on leases with monthly payments and no down payment with maturities ranging from 2 to 7 years and with and without the embedded option to purchase the aircraft at the end of the lease at its expected residual value are presented in Fig. 5b. Fig. 5a also includes the base default-free lease rates with and without the purchase option, as well as the corresponding base interest rates derived from the interest rate swap curve. The value of the option is rolled into the lease rate as suggested in the discussion surrounding Eq. (3.5). In the case of a lease with the option to purchase, the credit spread is computed by comparing the rate of a default-free lease with the option to the rate of a defaultable lease, also with the

Table 2 Benchmark loans, guarantees and leases. Upfront guarantee premiums and lease credit spreads are considered for a range of repossession delays. The percentage change in the guarantee premium and credit spread corresponding to the shift from d > 2 m to d = 2 m is also shown, illustrating the impact of adopting a statutory maximum two month delay. d=2m

d=4m

d=6m

d = 12 m

d = 18 m

12y Loan

Credit spread (bps) % Reduction if d = 2 m

38 –

47 18.7%

57 32.6%

90 57.5%

126 69.6%

12y Guarantee

Premium (% of P0) % Reduction if d = 2 m

2.12 –

2.58 17.7%

3.08 31.2%

4.79 55.8%

6.65 68.1%

5y Lease w/o Option

Credit Spread (bps) % Reduction if d = 2 m

56 –

65 13.9%

74 24.3%

100 43.7%

124 54.6%

5y Lease w/ Option

Credit Spread (bps) % Reduction if d = 2 m

52 –

62 14.7%

70 25.5%

96 45.3%

120 56.2%

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option. It is observed that credit spreads are larger for leases without the option to purchase compared to leases with the option. This is due to the option being a liability to the lessor as it is only exercised if the strike price is at or below market value at maturity. In the event of default, the lessor is freed of this liability, and this is reflected by a reduction in the credit spread relative to a lease without the option. Table 2 presents credit spread values for the 12 year loan, the corresponding guarantee premium, and the 5 year lease for repossession delays of 2, 4, 6, 12, and 18 months, as well as the percentage reductions in the value of the credit spread and the guarantee premium achieved by reducing the repossession delay from 4, 6, 12, and 18 months to the sixty day period under the CTT. The corresponding reductions can be viewed as the potential economic benefits of ratifying the CTT for jurisdictions with the corresponding expected repossession delays. We observe that even jurisdictions with generally creditor friendly national insolvency laws and expedient courts may substantially benefit, as the reduction from 4 months to 60 days already garners an 18% reduction in theoretical credit spreads. The policy implication of our analysis of asset financing is that insolvency laws that give the creditor swift access to the collateral, such as the Section 1110 and the CTT in the aircraft sector, lead to substantially lower financing costs.

with the initial state X0, rate of mean-reversion j > 0, and (generally time-dependent) volatility r(t) > 0. A unique strong solution to this SDE is known explicitly (e.g., Karatzas and Shreve (1991)):

6. Conclusion

C XX ðs; tÞ ¼

This paper develops an option-theoretic model for the valuation of defaultable leases and debt financing of capital equipment assets that generate a stochastic service flow and depreciate over a finite economic life. The model treats leases and secured debt in a unified fashion as contingent claims on the collateral asset market value. Term structures of lease rates and secured debt credit spreads are determined based on the term structure of default-free interest rates, the term structure of unsecured credit spreads, the parameters of the collateral asset’s value process, including the residual value curve and volatility, characteristics of the repossession process, including delays and costs, and the market price of asset risk. As an application, we calibrate our real asset model to the historical secondary market data for aircraft and calibrated our financing model to the Continental Airlines 2007 Enhanced Equipment Trust Certificates (EETCs) issue secured by a fleet of new Boeing aircraft. We apply the calibrated model to value private market financing, including loans, leases, and credit guarantees, consistently with the capital market financing, and analyze the impact of repossession delays on credit spreads of financing transactions. Explicit inclusion of the repossession delay as the key determinant of the asset financing transaction’s value allow us to explicitly assess and quantify the financial impact of bankruptcy laws limiting the repossession delay, such as Section 1110 of the US bankruptcy code and the Cape Town Treaty limiting the aircraft repossession delay to 60 days. This analysis leads us to a policy insight suggesting that bankruptcy laws limiting asset repossession delays lead to substantially lower costs of asset financing.

C II ðs; tÞ ¼

X t ¼ X 0 ejt þ

Z

t

rðsÞejðtsÞ dBs ; t P 0:

ðA:2Þ

0

Rt Define the integral process by It :¼ 0 X s ds, t P 0. The 2-dimensional process (X, I) is also Gaussian. Its conditional mean vector and covariance matrix can be explicitly computed from (A.2) using properties of Itô integrals and Brownian motion (here Es ½ :¼ E½jF s  and Covs ½A; B :¼ Es ½AB  Es ½AEs ½B):

Xs Es ½X t  ¼ X s ejðtsÞ ; Es ½It  ¼ Is þ ð1  ejðtsÞ Þ; j Z t 2 C XX ðs; tÞ :¼ Vars ½X t  ¼ r ðuÞe2jðtuÞ du; ðA:3Þ s  Z t Z u C XI ðs; tÞ :¼ Covs ½X t ; It  ¼ ejðtuÞ r2 ðv Þe2jðuv Þ dv du; ðA:4Þ s

s

C II ðs; tÞ :¼ Vars ½It  Z Z t 1 ð1  ejðtuÞ Þ ¼2 s

j

u



r2 ðv Þe2jðuv Þ dv du:

ðA:5Þ

s

In the time-homogeneous case of constant volatility r, the variances reduce to:

r2 2j

ð1  e2jðtsÞ Þ;

C XI ðs; tÞ ¼

r2 2j2

ð1  ejðtsÞ Þ2 ;

 r2  2jðt  sÞ þ 4ejðtsÞ  e2jðtsÞ  3 : 2j3

Due to the Gaussian nature of the process, we can readily compute the following conditional expectation for any real a and b and 0 6 s 6 t:    b E½expfaX t þ bIt gjF s  ¼ exp X s aejðtsÞ þ ð1  ejðtsÞ Þ þ bIs j  1 2 2 þ ða C XX ðs; tÞ þ b C II ðs; tÞÞ þ abC XI ðs; tÞ : 2

ðA:6Þ

To prove this result, observe that Zt :¼ a Xt + b It is Gaussian and, 1 hence, Es ½eZt  ¼ eEs ½Zt þ2Vars ½Zt  . Then compute the mean and variance of Zt using the results in this appendix. Appendix B. Proofs of Theorems 2.1 and 2.2 Proof of Theorems 2.1. We first prove part (c) of Theorem 2.1, and use the result to prove part (a). Applying the law of iterated expectations to Eq. (2.10) and using the results of A, we obtain:

     RT   ‘ du Fðt; sÞ ¼ Pðs; TÞST EQ EQ e s u F s F t    RT XQ  DQ duþ js ð1ejðTsÞ Þþ12C II ðs;TÞ  ¼ Pðs; TÞST EQ e s u F t

R T Q XQt D duþ j ð1ejðTsÞ ÞejðstÞ þ 12 ð1ejðTsÞ Þ2 C XX ðt;sÞþ12C II ðs;TÞ 2j : ¼ Pðs; TÞST e s u ðB:1Þ

Eq. (2.18) follows by setting t = 0, Acknowledgement

Fð0; sÞ ¼ Pðs; TÞST e This research was supported by the National Science Foundation under Grant CMMI-0654043. Appendix A. Ornstein–Uhlenbeck processes Let X be a time-inhomogeneous Gaussian diffusion process solving the following linear SDE

dX t ¼ jX t dt þ rðtÞdBt

ðA:1Þ

RT

1 ð1ejðTsÞ Þ2 C ð0;sÞþ1C ðs;TÞ XX 2 II 2j 2

DQ u duþ

s

;

ðB:2Þ

substituting the result back into Eq. (B.1), and letting R2 ðt; sÞ ¼ j12 ð1  ejðTsÞ Þ2 C XX ðt; sÞ. Eq. (2.15) follows by setting s = t in Eq. (2.18) and recalling that F(t, t) = St. h Proof of part 2.1(b): Observe that setting s = t in Eq. (B.1), and noting that F(t, t) = St, we obtain:

St ¼ Pðt; TÞST exp

(Z

t

T

DQ u du þ

XQ t

j

) 1 ð1  ejðTtÞ Þ þ C II ðt; TÞ : 2

58

S. Golbeck, V. Linetsky / Journal of Banking & Finance 37 (2013) 43–59

are funds remaining after the recovery of accrued interest to A, but the amount is less than the outstanding accrued interest of B, this construction yields the amount received to be equal to ð1Þ ð2Þ Stþd  C tþd  K tþd < K tþd . In general, for any subordinated recovery we have that:

Applying Itô’s formula,

 2 Q jðTtÞ jðTtÞ 2 dSt ¼ St r t  DQ  r2jðtÞ Þ dt 2 ð1  e t  Xt e þ Sjt ð1  ejðTtÞ ÞdX t þ jSt2 ð1  ejðTtÞ Þ2 d < X Q >t Q

¼ ðr t  ‘t ÞSt dt þ rjðtÞ ð1  ejðTtÞ ÞSt dBt : Q

The expression for the instantaneous asset volatility, Eq. (2.17), then follows. h Proof of part 2.1(d-e): By Eq. (2.4), the OU processes X Q and X P are related by:

XQ t

¼

X Pt

þ

Z

t

jðtsÞ

gðsÞrðsÞe

ds:

ðB:3Þ

0

Substituting this in Eq. (2.15), we obtain the asset price dynamics under P:

St ¼Fð0; tÞ exp  exp



 P Xt

j

1

j

ð1  ejðTtÞ Þ

Z

t

0

 gs rðsÞejðtsÞ ds

From the results in Appendix A, Xt conditional on F s is normally disjðtsÞ tributed with mean X Q and variance CXX(s, t). Thus, s e Q jðtsÞ XQ  X e is normally distributed with zero mean and variance t s CXX(s, t). Hence, St conditional on F s has a lognormal distribution:



 1 St ¼ Fðs; tÞ exp Rðs; tÞZ  R2 ðs; tÞ ; 2

Here we present EETC valuation details. We index the tranches A, B, C by i, j = 1, 2, 3. From the theoretical standpoint, any number of tranches can be considered in a similar way. The recovery of the accrued interest of A at time t + d is given by:

n o n o ð1Þ ð1Þ ð1Þ Rtþd ¼ min K tþd ; Stþd  C tþd þ u max K tþd  ðStþd  C tþd Þ; 0 n o ð1Þ ð1Þ ¼ K tþd  ð1  uÞ max K tþd þ C tþd  Stþd ; 0 : The recovery of the accrued interest of B at time t + d is given by,

i¼1

( ð2Þ

þu max K tþd  min ( ð2Þ

¼ K tþd  ð1  uÞ max

) ðiÞ

K tþd ; Stþd  C tþd

( 2 X i¼1

n o ð2Þ  min K tþd ; Stþd  C tþd )

ðiÞ

K tþd ; Stþd  C tþd

As each EETC tranche consists of both principal and interest payments, there are two such recovery terms for each tranche. For the sake of brevity, define:

"

R sþd

r s ds

0

i X ðjÞ K sþd þ C sþd  Ssþd

1fs2½tk1 ;tk Þg

!þ # :

ðC:1Þ

Here the k index denotes the payment period while i denotes the priority of the recovery which is reflected in the level of the strike P ðjÞ price, ij¼1 K sþd þ C sþd . The value of any EETC tranche at time 0 (contract inception) is then given by:

PVðiÞ ¼

N X

 Rt   k r ds EQ e 0 s 1fs>tk g C ik

k¼1

þ

N X

 R sþd

  r ds ðiÞ ðiþ3Þ EQ e 0 s 1fs2½tk1 ;tk Þg Rsþd þ Rsþd ;

k¼1

or in the more explicit form:

PVðiÞ ¼

N X

 Rt   k r ds EQ e 0 s 1fs>tk g C ik

k¼1

þ

N X

 R sþd   r ds EQ e 0 s 1fs2½tk1 ;tk Þg ð1 þ ðs þ d  tk1 ÞLi ÞPik1

k¼1

k¼1

Appendix C. EETC valuation details

( 2 X

i¼1

N X

Rk;i  1fi>1g Rk;i1 þ Rk;iþ3  Rk;iþ2  ð1  uÞ

where Z is the standard normal random variable, and we use the relationship R2 ðs; tÞ ¼ j12 ð1  ejðTtÞ Þ2 C XX ðs; tÞ. Thus, the option prices are given bypBlack’s ffiffiffiffiffiffiffiffiffiffi formulas with the forward F(s, t) and R(s, t) in place of r t  s. The expression for the forward F(s, t) in terms of spot Ss follows from combining Eqs. (2.15) and (2.18). h

ð2Þ

i¼1

j¼1

  jðTtÞ Þ 1 2 Q jðtsÞ ð1  e St ¼ Fðs; tÞ exp ðX Q  X e Þ  R ðs; tÞ : t s 2 j

(

( ) j X ðiÞ K tþd þ C tþd  Stþd ; 0 þ ð1  uÞ

( ) j1 X ðiÞ K tþd þ C tþd  Stþd ; 0 :  max



Taking the expectation of both sides and using the definition of the residual value curve RVð0; tÞ ¼ EP ½St , we arrive at Eq. (2.20). Eq. (2.21) follows by applying the Girsanov transformation of Eq. (B.3) to Eq. (2.15), and using the definition of the RA-RVC in Eq. (2.20). h Proof of Theorem 2.2 From Eqs. (2.15) and (2.18), we can write for any 0 6 s 6 t 6 T:

ð2Þ

ðjÞ

Rk;i  EQ e

 1 ð1  ejðTtÞ Þ  R2 ðtÞ : 2

Rtþd ¼ min K tþd ; min

ðjÞ

Rtþd ¼ K tþd  ð1  uÞ max

n

ð1Þ

)

ðC:2Þ The indicator function 1{i>1} is present in the recovery term on the right side of the equality, as this term only appears for the recovery of accrued interest for the A tranche, denoted by the index i = 1. Adding the value of the tranches together, we obtain the value of the entire EETC issue (observe that there are a number of cancellations of recovery terms when valuing the entire EETC issue): PVðEETCÞ ¼

k¼1 i¼1

)

þ min K tþd ; Stþd  C tþd ; 0 )

2 n o X ðiÞ ð1Þ K tþd þ C tþd  Stþd ; 0 þ ð1  uÞ max K tþd þ C tþd  Stþd ; 0 : i¼1

k¼1

  R sþd N X 3 X  r ds þ EQ 1fs2½tk1 ;tk Þg e 0 s ð1 þ ðs þ d  tk1 ÞLi ÞP ik1 : k¼1 i¼1

ðC:3Þ

As in Section 3.2, we assume that default occurs at the first arrival time of a time-inhomogeneous Poisson process with intensity k(t). The EETC valuation formula is then: PVðEETCÞ ¼

o

 Rt  N X 3 N X X  k r ds EQ e 0 s 1fs>tk g C ik  ð1  uÞ Rk;6

N X C EETC P d ð0;tk Þ k k¼1

þ

N Z X k¼1

tk

Rv 0

kðyÞdy

t k1

N Z X  ð1  uÞ k¼1

The first term is constructed such that the recovery of tranche B in h i ð2Þ the event of default is restricted to be in the range 0; K tþd . If there



kðv Þe tk tk1



kðv Þe

Pð0; v þ dÞK EETC v þd dv

Rv 0

kðyÞdy

Putðv þ d;K EETC v þd þ C v þd Þdv ; ðC:4Þ

P P6 ðiÞ ðiÞ EETC where C EETC ¼ 3i¼1 C k , K EETC is the total i¼1 K v þd . Here C k v þd ¼ k coupon payment on the EETC issue at time tk, and K EETC is the total t

S. Golbeck, V. Linetsky / Journal of Banking & Finance 37 (2013) 43–59

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