On: 30 June 2014, At: 04:15 Publisher: Routledge
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Applied Financial Economics
Publication details, including instructions for authors and subscription information:
http://www.tandfonline.com/loi/rafe20
The valuation of special purpose vehicles by issuing
structured credit-linked notes
Chia-Chien Chang a , Chou-Wen Wang b & Szu-Lang Liao c a
Department of Finance , National Sun Yat-sen University , Taiwan b
Department of Risk Management and Insurance , National Kaohsiung First University of Science and Technology , Taiwan
c
Department of Money and Banking , National Chengchi University , Taiwan Published online: 20 Jan 2009.
To cite this article: Chia-Chien Chang , Chou-Wen Wang & Szu-Lang Liao (2009) The valuation of special purpose vehicles by issuing structured credit-linked notes, Applied Financial Economics, 19:3, 227-256, DOI: 10.1080/09603100701765190
To link to this article: http://dx.doi.org/10.1080/09603100701765190
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no
representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any
form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions
The valuation of special purpose
vehicles by issuing structured
credit-linked notes
Chia-Chien Chang
a,*, Chou-Wen Wang
band Szu-Lang Liao
c aDepartment of Finance, National Sun Yat-sen University, Taiwan b
Department of Risk Management and Insurance, National Kaohsiung First University of Science and Technology, Taiwan
c
Department of Money and Banking, National Chengchi University, Taiwan
With the intersection of market and credit risk, the first contribution is to derive the analytic formulas of the Credit Linked Notes (CLNs) and the leveraged total return CLNs issued by an Special Purpose Vehicle (SPV) or the protection buyer. The second contribution is to prove that the values of structured CLNs issued by an SPV are higher than the ones issued by the protection buyer. When the credit quality of the reference obligation and protection buyer becomes worse or the leverage effect is higher, it is a superior solution for the structured CLNs issued through an SPV. Third, the empirical results of credit spreads do not incorporate the correlation coefficient of spot rate and market index into their regression models and show that they are positively correlated with the volatilities of spot rate and return on market index; however, we find that the relationship among them depends on the sign of correlation coefficient of spot rate and equity index market. Finally, using the differences in the maturities of the note and the reference obligation as the proxy for basis risk measure, we demonstrate that the purpose of the SPV is not used to eliminate the basis risk but the credit risk of protection buyer.
I. Introduction
Special Purpose Vehicles (SPVs) are business entities formed for the purpose of conducting a clearly-specified activity, such as collecting a specific group of accounts receivable or credit risks and so on. When investing in a project or financial instruments with well-defined risk and return, many investors may prefer the isolated and uniquely identifiable nature of an SPV to a more diffusely defined corporate form. Currently, SPVs are
applied for the asset-based securities and similar financial products, such as collateralized debt obliga-tions, mortgage-backed securities and structured Credit Linked Notes (CLNs).
According to 2002 survey by the British Bankers Association (BBA), credit-linked obligations are seen to be the second hot product followed by the credit default swaps, with 22% of market share by 2001 and 26% of market share by 2004. Besides, by the end of 2006 the size of global credit derivatives market
*Corresponding author. E-mail: u7501247@yahoo.com.tw
Applied Financial EconomicsISSN 0960–3107 print/ISSN 1466–4305 onlineß 2009 Taylor & Francis 227 http://www.informaworld.com
DOI: 10.1080/09603100701765190
will be $20 trillion and BBA predicts that at the end of 2008 the global credit derivatives market will have expanded to $33 trillion.
In practice, the CLNs can be issued either by the protection buyer or by an SPV. When the issuer is an SPV, the proceeds from the noteholders are used to buy high-quality collaterals that are held by the SPV. Hence the valuation of the note is only related to the credit event of reference obligation. Otherwise, when the issuer is the protection buyer, the proceeds are held on the balance sheet of the issuer as cash and hence the valuation of CLNs depends on both the credit events of reference obligation and its buyer (the issuer), and thus is different from the ones issued by SPVs. Consequently, the motivation of this article is to price structured CLNs such as the CLNs and the leveraged total return CLNs that are issued by an SPV or the buyer himself. Meanwhile, we also provide the fair fee charged by an SPV when issuing the structured CLNs. The fee is determined by the price difference of structured CLNs issued by the two entities.
Structured CLNs, such as the CLNs and the (leveraged) total return CLNs, are one type of credit derivatives that can be used by the protection buyer to hedge against the credit risk induced by some reference entities and invested by noteholders to link to the reference obligation, which is of noninvestment grade or illiquid. The payoff of a CLN is linked to the credit event of reference entities. If a credit event occurs, no further coupon payment is paid. At the termination date, the noteholders receive par value unless a credit event occurs, in which case they will be redeemed immediately for the credit event payment, which could be the nominal amount multi-plied by the recovery rate of reference obligation.
The total return CLN could be also structured by incorporating the leverage factor. For a leveraged total return CLN, the payoff is linked to both the market price and credit event of reference obligation. For instance, if a financial institution issues $5 million nominal of a 5-year leveraged total return CLN and the reference obligation is a zero coupon bond with $25 million nominal, then the leverage factor is set to be five. The actual coupon payment also equals five times the coupon value. The coupon payment ceases immediately if a credit event occurs. Furthermore, at the termination date, the investors receive an additional payment called capital price adjustment. In this example, the payment equals five times $5 million nominal times the change in the price of reference obligation.
There have been few studies of structured CLNs. Hui and Lo (2002) use, by extending, Merton (1974) corporate bond pricing model to value CLNs by
incorporating the asset value of the reference entity as an addition variable. However, due to the unobser-vable parameters such as firm’s value, their pricing methodology is difficult to implement. Another approach is the reduced-form model in which default time is a stopping time of some given hazard rate process, and the payoff upon credit event is specified exogenously. Hence, the probability of default in the next time partition is determined by a specified hazard rate. This approach has been widely consid-ered by Lando (1994, 1998), Jarrow and Turnbull (1995, 2000), Duffie and Singleton (1999), Jarrow and Yu (2001) and so on. Jarrow and Turnbull (1995) uses the analogous cross-currency framework to evaluate the financial derivatives with credit risk by assuming a hazard function that is independent of spot interest rate. However, Kao (2000) documents that the interest rate level and the Russell 2000 index return have significant explanatory power for change in the credit spread index level. Campbell and Taksler (2003) demonstrate that the idiosyncratic equity volatility can explain about one-third of the variation in yield spreads. Janosi et al. (2002) find that the default intensity depends on the spot interest rate. Huang and Kong (2003) indicates that high interest rates and steep yield curves are usually associated with an expanding economy and low credit spread, whereas the higher interest rate volatility is usually associated with wider credit spread, especially for high-yield bond indexes. They also discover that a higher equity market index return will reduce credit spreads, and higher equity volatility will signif-icantly widen credit spread. Athanassakos and Carayannopoulos (2001), Batten et al. (2005, 2006) and Batten and In (2006) indicate that relative credit spreads returns are negatively related to both changes in interest rate and changes in equity return. Therefore, the hazard process is dependent on spot interest rate and market index; and hence CreditMatrics, CreditRiskþ
and KMV methodolo-gies cannot be consistent with those empirical results, given their assumption of constant interest rate.
To incorporate the state variables such as spot interest rate and market index into the default intensity function, Lando (1998) uses the doubly stochastic Poisson processes of default that allow the hazard function to depend on state variables. Jarrow and Turnbull (2000) assume that the hazard function is dependent on the interest rate and index return volatility. Recently, the default model is extended to the multivariate underlying asset case to value a credit swap of the basket type. Duffie (1998) provides the valuation of a first-to-default type claim that depends on the first default time of a given list of credit events. Kijima (2000) and Kijima and
Muromachi (2000) consider the joint survival prob-ability of occurrence times of a credit event under the assumption of conditional independence to value the default swaps of basket type.
We assume that the protection buyer can comple-tely transfer the credit risk of reference obligation to the noteholders via the structured CLNs. Conditional independence between the default times of protection buyer and reference obligation with respect to the filtration generated by the spot interest rate and return of the market index is also assumed. In this situation, the first contribution of this article is to price the structured CLNs, such as CLNs and leveraged total return CLNs, which are issued by an SPV or the protection buyer, respectively. Further, in practice most of the protection buyers issue the structured CLNs through the SPVs. As a result, the second contribution of this article is to demonstrate the best timing for the structured CLNs issued by SPVs instead of protection buyers. In addition, we also derive the suitable fee that the protection buyer must pay to an SPV for the purpose of issuing the structured CLNs. From the numerical analyses, we investigate the characteristics for the credit spreads of structured CLNs directly issued by the protection buyers. Meantime, the empirical results, such as Kao (2000), Campbell and Taksler (2003) and Huang and Kong (2003), neglect the effect of the correlation coefficient of spot rate and market index, and they show the positive relationship among the credit spread, the volatilities of spot rate and return on market index. However, we demonstrate that the relationship among the values and credit spreads of structured CLNs, spot rate volatility and market index volatility depend on the sign of correlation coefficient of spot rate and equity index. Finally, the important sources of basis risk in credit derivatives discussed in the literature are the differences in the maturities of the hedging instrument and the refer-ence obligation. In addition, Azarchs (2003) indicates that issuers could not eliminate basis risk by issuing credit derivatives, thus by defining the differences in the maturities of the note and the reference obligation as the proxy for the basis risk measure, we find that the value of an SPV is not an increasing function of the basis risk measure, but is positively related with the default intensity of the reference obligation and protection buyer. Therefore, we demonstrate that the purpose for issuing the structured CLNs through SPVs is to hedge the credit risk induced by the protection buyer but not the basis risk.
An outline of the article is as follows. In Section II, we introduce the trading economy. In Sections III and IV, we derive the analytic formula of CLNs and leveraged total return CLNs,
respectively. We present the characteristics of the structured CLNs in Section V. Summary and conclusions is presented in Section VI.
II. Setup of Economy
Let the uncertainty in the economy be described by the filtered probability space ð, F, P, ðFtÞT
t¼0Þ.
We assume the existence and uniqueness of P, so that bond markets are complete. Let A and R stand for the default times of the protection buyer and reference entity and given as
i¼inf t : ZT t iðXsÞds Ei , for i ¼ A or R where we assume that the construction of the doubly stochastic Poisson processes of default (also called a Cox process) with an intensity function i(Xt), ðXtÞT
t¼0is a right continuous with left limits R
d -valued process and represents d state variables underling the evolution of the economy, such as the spot rate, market index, credit ratings or other variables deemed relevant for predicting the likelihood of default. Ei is a unit exponential random variable which is independent of state variables and i. The default time can be thought of as the first jump time of a Cox process with stochastic intensity process i(Xt) and are conditional independent with respect to the filtration generated by X under P.
Some empirical studies, such as Kao (2000), Campbell and Taksler (2003), Huang and Kong (2003), Batten et al. (2005, 2006), indicate that credit spreads are negatively related to both in interest rate and equity return. To describe the dependence of the default process on the state of the economy and incorporate the empirical results into our reduced-form model, we introduce the enlarged filtration F by setting
Ft ¼Frt_FtI_HAt _HRt
where Frt ¼ðrðsÞ, 0 s tÞ, FIt ¼ðIðsÞ, 0 s tÞ and Hit¼ð1fisg, s tÞ, i ¼ A or R. 1{} is the indicator function. r(t) is the spot rate at time t. I(t) denotes a time-t market index such as the Standard and Poor 500 stock index. As a result, FrT_FIT
contains complete information on the spot rate and the market index. In an economy that evolves according to the filtration Fr
T_FIT, it is possible
to select a nonnegative, Fr
T_FIT-measurable
pro-cess i, and satisfies R0ti½rðsÞ, IðsÞds P–a.s. for all t 2 ½0, T. An inhomogeneous point process can be
defined, using the realized history of the process ias
its intensity function. Consequently, the conditional distributions of iare given as
P i4 t FrT_FIT ¼exp Zt 0 i½rðsÞ, IðsÞds t 2 ½0, T
Hence, by the law of iterated expectation, we have Pði4 tÞ ¼ E exp Zt 0 i½rðsÞ, IðsÞds , t 2 ½0, T
Let p(t, T) be the time-t price of a zero coupon bond paying one dollar at time T. Y(t, T) is the yield-to-maturity of p(t, T). B(t) corresponds to the wealth accumulated by an initial one-dollar investment at short-term interest rate r(t) in each subsequent period. Therefore, pðt, TÞ ¼ exp yðt, TÞ ðT tÞ½ ¼E BðtÞ BðTÞ Ft and BðtÞ ¼exp Z t 0 rðsÞds ð1Þ We assume that the point processes governing default for the spot rate and market index is
iðuÞ ¼ f½rðuÞ, ZðuÞ where
zðuÞ log IðuÞ Ið0Þ log BðuÞ Bð0Þ
Jarrow and Turnbull (1995, 2000), Lando (1998) and Jarrow and Yu (2001) assume the hazard rate function is linear when modelling their hazard rates. Thus this article follows their framework and our linear hazard rate function admits the following representation:
iðuÞ i0þi1rðuÞ þ izðuÞ
¼ i0ilog Ið0Þ Bð0Þ þi1rðuÞ þ ilog IðuÞ BðuÞ ¼i0þi1rðuÞ þ ilog IðuÞ BðuÞ , for i ¼ A or R ð2Þ where i
0is the spontaneous default intensity of entity
i, i
1measures the sensitivity of entity i to the level of
spot rate and irepresents the sensitivity of entity i to the excess return on market index.
Lando (1998) uses the doubly stochastic Poisson processes of default that allow the linear hazard function to depend on state variables. Jarrow and Turnbull (2000) assume that the linear hazard
function is dependent on the interest rate and index return volatility. Jarrow and Yu (2001) model the linear hazard function is only affected by the interest rate. Hence, we extend the model of Jarrow and Yu (2001) to incorporate the market index into our hazard rate function. Note that, Jarrow and Turnbull (2000) uses index return volatility to capture the effect on credit spread, but we consider the return of the market index which can describe the level of market index return and index return volatility. We also assume that the stochastic processes of r(t) and I(t) are given as follows:
drðtÞ ¼ ½ðtÞ ðtÞrðtÞdt þ rdWrt ð3Þ
dIðtÞ
IðtÞ ¼rðtÞdt þ IdW
I
t ð4Þ
where (t) represents the long-term equilibrium value of the process; (t) is a nonnegative mean reversion speed; and r and I are the volatilities of spot rate and return on market index, respectively. Wr
t and WIt are Brownian motions with respect to Ft
and satisfy EðWr
tWItÞ ¼rIdt, where rI is the
correlation coefficient of spot rate and market index. The assumption of normality for hazard rate function allows the derivation of closed form solutions, such as expression (5) below. One of the disadvantages of this assumption is that the intensity function can be negative. However, in lattice-based models, this difficulty can be avoided via the use of nonlinear transformations, as in Jarrow and Turnbull (1997).
Let A(t, T) and R(t, T) be the prices of risky zero coupon for the protection buyer and reference obligation, respectively. A and R are correspond-ingly the recovery rates of the protection buyer and reference entity. Therefore, it is noteworthy that under the setup of equality (2), the valuations of risky zero coupon bonds A(t,T) and R(t, T) are presented in the following theorem.
Theorem 1: The prices of risky zero coupon bond for entity i, i ¼ A or R, admits the following representation iðt, TÞ ¼ pðt, TÞ iþ1fi4 tgð1 iÞexp i0ðT tÞ i1Yðt, TÞðT tÞ þ i 1ð1 þ i1Þ 2 2ðt, TÞ þðT tÞ 2 4 i 2 I ðT tÞ3 6 2 iI2 ð1 þ i1ÞiIrI ZT t ðT uÞbðu, TÞdu , i ¼ A or R ð5Þ
where 2ðt,sÞ¼Vt Zs t rðuÞdu ¼ Zs t bðu,sÞ2du bðt,TÞ¼rDðt,TÞ, Dðt,TÞ¼ ZT t expf½ðsÞðtÞgds ðtÞ¼ Zt 0
aðuÞdu, aðuÞ¼exp ðtuÞ½
b(t,T) is denoted as the volatility of default-free zero coupon bond, and Vt() is the variance conditional on Ft.
We prove Theorem 1 in Appendix 1.
If the recovery rate is zero, we can rewrite equality (5) as
0 iðt, TÞ
pðt, TÞ 1fi4 tgexp cs½ iðt, TÞ ðT tÞ 1, i ¼ Aor R
where csi(t, T) is the credit spread function of risky zero coupon bond of the entity i and is defined as follows: csiðt, TÞ ¼ i0þ i 1Yðt, TÞ ðT tÞi2I 4 ði 1Þ 2þi 1 2 2ðt, TÞ T t þ ðT tÞ22iI2 6 þð1 þ i 1Þ T t iIrI Z T t ðT uÞbðu, TÞdu
If i¼0, the credit spread function reduces to the result of Jarrow and Yu (2001), in which the hazard rate function is only affected by the level of interest rate. Thus csi(t,T) is the generalization of Jarrow and Yu (2001).
Next, we discuss the properties of the credit spread function. Using the chain rule of differentiation, we have @csiðt, TÞ @Yðt, TÞ ¼ i 1, i ¼ A or R @csiðt, TÞ @rI ¼ð1 þ i 1Þir ðT tÞ ZT t ðT uÞbðu, TÞdu, i ¼ Aor R @csiðt, TÞ @2 r ¼ ð1 þ i 1Þ 2ðT tÞ2 r i12ðt, TÞ þ iIrI Z T t ðT uÞbðu, TÞdu , i ¼ Aor R ð6Þ @csiðt, TÞ @2 r ¼ ðT tÞi 4 þ ðT tÞ22 i 6 þ ð1 þ i 1ÞirI 2IðT tÞ ZT t ðT uÞbðu, TÞdu, i ¼ Aor R ð7Þ Under the restrictions that
i04 0, i5 0, 1 5 i15 0 we have @csiðt, TÞ @Yðt, TÞ 5 0, @csiðt, TÞ @rI 4 0
As a result, a higher correlation coefficient of spot rate and market index or lower yields of default-free zero coupon bonds are associated with the wider credit spreads of risky zero coupon bonds. Furthermore, if rI0, we have
@csiðt, TÞ @2 r 4 0 and @csiðt, TÞ @2 I 4 0 ð8Þ
Equality (8) coincides with the empirical results such as Kao (2000), Campbell and Taksler (2003) and Huang and Kong (2003). Or equivalently, under the case that rIis nonnegative (usually corresponding to an expanding economy or recession), since the hazard rate function is an increasing function of the spontaneous default intensity and negatively sensitive to the level of spot rate as well as to the excess equity return, we can obtain that the credit spread of risky zero coupon bonds increases as the volatilities of spot rate or the volatility of return on market index increase. Hence, the characteristics of credit spread of risky zero coupon bonds match the empirical results for credit spreads.
However, the equalities (6) and (7) may be negative if rI is negative. As a result, the correlation coefficient of spot rate and market index plays an important role in the relationship among the credit spread, spot rate volatility and market index vola-tility. Nevertheless, the empirical results, such as Kao (2000), Campbell and Taksler (2003), Huang and Kong (2003), etc., do not incorporate the correlation coefficient of spot rate and market index into their regression models and then they may ignore the possibilities of negatively relationship among the credit spread, spot rate volatility and market index volatility. Consequently, for further research, we suggest that the dependent variable rI should be included in the empirical regressions to more exactly capture their relationship.
III. Pricing Model for CLNs
In this section, we first derive the analytic formula of CLNs when the issuers are respectively an SPV and the protection buyer. Then we provide the fair fee that the buyer should pay to the SPV for the purpose of issuing the CLNs. Let T be the maturity date of CLNs. Csis the coupon payment1at time s, s 2 ½t, T and M is the nominal principle. A and R are the recovery rate of reference obligation and the issuer, respectively.
CLNs issued by an SPV
If the note issuer is an SPV, then the proceeds from the noteholders are used to buy some high-quality collateral that is held by the SPV, thus the valuation of the note is uncorrelated with the credit event of the protection buyer who owns the reference obligation. The payoff structures of CLN is as follows: (1) For coupon payment, if the reference entity does not default at the coupon payment date s, the payments are Cs. Otherwise, the payment is zero; (2) For the principal, if there is no default prior to time T, the payments are M. If there is a default event prior to time T, the credit event payment RM is paid immediately at default time R. Therefore, the valuation of the CLN equals
CSPVðtÞ ¼ E "ZT t 1fR4 sgCs BðtÞ BðsÞds þ 1fR4 Tg BðtÞ BðTÞM þ1ft5 RTg BðtÞ BðRÞRM Ft #
The analytic formula of the CLN issued by an SPV is provided in the following theorem.
Theorem 2: The analytic formula of the CLN is
CSPVðtÞ ¼ ZT t Cspðt, sÞG1ðt, sÞds þ M pðt, TÞG1ðt, TÞ þR 0RM ZT t pðt, sÞG1ðt, sÞds þ R1RM ZT t pðt, sÞG2ðt, sÞds þRRM RT t pðt, sÞG3ðt, sÞds if R4 t RM if R¼t 0 if R5 t 8 > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > : ð9Þ where G1ðt, sÞ ¼ exp R0ðT tÞ R1Yðt, sÞðs tÞ þ R 1ð1 þ R1Þ 2 2ðt, sÞ þðs tÞ2 4 R 2 1 þðs tÞ 3 6 2 RI2 ð1 þ R1ÞRIrI Zs t ðs yÞbðy, sÞdy ¼exp cs½ Rðt, sÞ ðs tÞ ð10Þ is a credit risk adjusted discount factor induced by spontaneous default intensity of reference obligation if the issuer is an SPV and
G2ðt, sÞ ¼ G1ðt, sÞ fðt, sÞ R1 2 bðt, sÞ 2 RIrI Zs t ðu, sÞðs yÞdy
is a credit risk adjusted discount factor induced by the sensitivity of reference obligation to the level of spot rate if the issuer is an SPV and 0ðt, sÞ Et½rðsÞ ¼
fðt, sÞ Rtsðu, sÞbðu, sÞdu G3ðt, sÞ represents the
credit risk adjusted discount factor induced by the sensitivity of reference obligation to the excess return on market index if the issuer is an SPV and is defined as follows: G3ðt, sÞ ¼ G1ðt, sÞ I2ðs tÞ 2 ½1 Rðs tÞ þ ð1 þ R1ÞIrI Zs t bðy, sÞdy
A detailed proof is sketched in Appendix 2. Using the chain rule of differentiation for G1(t, s), we have @G1ðt, sÞ @R 0 ¼ ðs tÞG1ðt, sÞ 5 0, @G1ðt, sÞ @rI ¼ ðs tÞG1ðt, sÞ @csRðt, sÞ @rI 5 0 @G1ðt, sÞ @2 r ¼ ðs tÞG1ðt, sÞ @csRðt, sÞ @2 r , @G1ðt, sÞ @2 I ¼ ðs tÞG1ðt, sÞ @csRðt, sÞ @I2 ð11Þ 1
We use continuous time to calculate the accrued interest payment instead of discrete time, since the credit event may happen prior to the coupon payment date.
Hence, the discounted factor is decreasing functions of spontaneous default intensity of reference entity and rI. In addition, if rI40 in view of the equality (8), we have @G1ðt, TÞ @2 r 5 0 and @G1ðt, TÞ @I2 5 0
This means that under the case for rI40, higher volatilities of spot rate and market index return and default probability of reference obligation are usually associated with wider credit spread and lower credit risk discounted factor as well as the values of CLNs. However, particularly, if rI< 0, the equalities (6) and (7) may be negative, which is different from the empirical results for credit spreads, and then the equality (11) may be positive. Meanwhile, it is also apparent that
@pðt, sÞG1ðt, sÞ
@Yðt, sÞ ¼ ð1 þ
R
1Þðs tÞG1ðt, sÞ 5 0
As a result, we conclude that higher yield of default-free zero coupon bond is likely to be accompanied by the lower values of CLNs.
Issued by the protection buyer
If the issuer is the protection buyer, the note value
is related to the default intensities of both the protection buyer and reference obligation. To show its payoff structure, for the coupon payment, if both the reference entity and the protection buyer do not default at the coupon payment date s, the payments are Cs. If the protection buyer defaults prior to the coupon payment date s but the reference entity does not, the payment are ACs. Otherwise, the payment is 0. For the principal of the CLN, if default does not occur prior to time T, the payments is M. If the buyer defaults prior to time T but the reference entity does not, similar to the case of the common bonds that are already in default, the default payment AMis paid at
time T. Besides, when the reference entity defaults prior to time T, two additional scenarios should be discussed. First, if the default time of reference entity is prior to the one of the buyer, the credit event payment RMis paid immediately at default time R whether or not the default time of the buyer is earlier than the date. If the first to default is the buyer, the payment ARMis paid immediately at time R, then the value of the CLN is as follows:
CBPðtÞ ¼ E ZT t 1fA4 sg1fR4 sgCs BðtÞ BðsÞds þ ZT t 1fAsg1fR4 sgACs BðtÞ BðsÞds þ 1fA4 Tg1fR4 Tg BðtÞ BðTÞM þ1ft<ATg1fR4 Tg BðtÞ BðTÞAM þ1fA4 Rg1ft<RTg BðtÞ BðRÞRM þ1fARg1ft<RTg BðtÞ BðRÞARM Ft
The analytic formula of the CLN issued by the protection buyer is provided in Theorem 3.
Theorem 3: The analytic formula of the CLN issued by the protection buyer is as follows:
where G4ðt, sÞ ¼ exp ðR0þA0ðs tÞ ðR1þA1ÞYðt, sÞðs tÞ þð R 1þA1Þð1 þ R1þA1Þ 2 2ðt, sÞ þðs tÞ 2 4 ðRþAÞ 2 1þ ðs tÞ3 6 ðRþAÞ 2 I2 ð1 þ R1þA1ÞðRþAÞIrI Zs t ðs uÞbðu, sÞdu CPBðtÞ ¼ ZT t Cspðt, sÞ ½ AG1ðt, sÞ þ ð1 AÞG4ðt, sÞds þMpðt, TÞ ½ AG1ðt, TÞ þ ð1 AÞG4ðt, TÞ þR 0RM ZT t pðt, sÞ ½ AG1ðt, sÞ þ ð1 AÞG4ðt, sÞds þR1RM ZT t pðt, sÞ ½ AG2ðt, sÞ þ ð1 AÞG5ðt, sÞds þRRM ZT t pðt, sÞ ½ AG3ðt, sÞ þ ð1 AÞG6ðt, sÞds if A4 t, R4 t ACSPVðtÞ if At, R4 t 0 otherwise 8 > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > : ð12Þ
is a discount factor induced by spontaneous default intensity of reference obligation if the issuer is a protective buyer, and
G5ðt, sÞ ¼ G3ðt, sÞ fðt, sÞ ðR 1 þA1Þ 2 bðt, sÞ 2 ðRþAÞIrI Zs t ðu, sÞðs uÞdu
is a credit risk adjusted discount factor induced by the sensitivity of reference obligation to the level of spot rate if the issuer is a protective buyer. G6(t, s) expresses the credit risk adjusted discount factor induced by the sensitivity of reference obligation to the excess return on market index if the issuer is a protective buyer and is displayed as follows: G6ðt, sÞ ¼ G3ðt, sÞ ð2 Iðs tÞ 2 ½1 ðRþAÞðs tÞ þ ð1 þ R1þA1ÞIrI Z s t bðu, sÞdu
We prove Theorem 3 in Appendix 3.
It is apparent that equality (9) is a special case of equality (11) by substituting A¼A0 ¼A1 ¼0. Or
equivalently, a CLN is free of credit exposure for the protection buyer if the issuer is an SPV. Moreover, we can rewrite G4(t, s) as follows:
G4ðt, sÞ ¼ exp ðH0Þðs tÞ H1Yðt, sÞðs tÞ þ H 1ð1 þ H1Þ 2 2ðt, sÞ þðs tÞ 2 4 H 2 I þðs tÞ 3 6 2 H2I ð1 þ H1ÞHIrI Zs t ðs uÞbðu, sÞdu exp½ðcsHðt, sÞðs tÞ ð13Þ where H 0 A0 þR0, H1 A1 þR1 and H¼Aþ
R. In view of (2), the default hazard rate function2
H(u) of entity H defined in this way is indeed the sum of A(u) and R(u), where u 2 ½t, T.
Hence, we have @G4ðt, sÞ @i 0 ¼ ðs tÞG4ðt, sÞ 5 0, i ¼ Aor R @G4ðt, sÞ @rI ¼ ðs tÞG4ðt, sÞ @csHðt, sÞ @rI 5 0 @G4ðt, sÞ @2 r ¼ ðs tÞG4ðt, sÞ @csHðt, sÞ @2 r , @G4ðt, sÞ @2 I ¼ ðs tÞG4ðt, sÞ @csHðt, sÞ @2 I ð14Þ
Similar to the case for a CLN issued by an SPV, the discounted factor is increasing with spontaneous default intensity of the reference entity, spontaneous default intensity of the protective buyer and rI. Additionally, if rI40, we can see that G4(t, s) is decreasing functions of spontaneous default intensi-ties for both the protection buyer and reference entity and the volatilities of spot rate and return on market index. However, it is possible to obtain the opposite outcomes when rI< 0. Moreover, it is also clear that @pðt, sÞG4ðt, sÞ
@Yðt, sÞ ¼ ð1 þ
H
1Þðs tÞG4ðt, sÞ 5 0, i ¼ A or R
Consequently, this in turn shows that higher yields of default-free zero coupon bonds are also related to lower values of CLNs no matter who the issuers are. Fair fee charged by an SPV
In practice, the CLNs can be issued either by the protection buyer or by an SPV. When the issuer is an SPV, the proceeds from the noteholder are used to buy high-quality collateral that is held by the SPV. Otherwise, the proceeds are held on the balance sheet of the protection buyer as cash. Thus, when it comes to investing in CLNs, many noteholders may prefer the isolated and uniquely identifiable nature of an SPV to a more diffusely defined corporate form such as the protection buyer. Since the SPVs play an important role in practice, when financial institutions who own the reference obligation issue CLNs through SPVs, it is imperative to determine the values or fair fees charged by SPVs with issuing the CLNs, especially for accounting purposes. Given the pricing formulas in equalities (9) and (11), the fair
2
The default time of entity H can be explained as the minimum of A and R and hence entity H is the first-to-default contingent claim. The readers can refer to Bielecki and Rutkowski (2002) for a full treatment.
fee charged by an SPV with issuing a CLN is equal to equality (9) minus equality (12) and is presented at the following corollary.
Corollary 1: From Theorems 2 and 3, the fair fee charged by an SPV issuing a CLN, defined as SPVfee, is given as follows:
From Corollary 1, if G1(t, s)4G4(t, s), G2(t, s)4 G5(t, s) and G3(t, s)4G6(t, s), we can see that the fee charged by an SPV is always greater than zero. In view of (10) and (13), G4(t, s) consider both default risks of reference obligation and protection buyer, but G1(t, s) only consider default risk of reference obligation. Hence, it is obvious to obtain that G1(t, s)4G4(t, s). Alone line as spot rate and index are positive, it can be shown that G2(t, s)4G5(t, s) and G3(t, s)4G6(t, s). Therefore, we conclude that the appropriate fee charged by an SPV with issuing the CLNs is a positive amount.
IV. Pricing the Leveraged Total Return CLNs
Issued by an SPV
For pricing the leveraged total return CLNs, we assume that the maturity date of a leveraged total return CLN is T, L is the leverage factor, LCsis the coupon payment at time s, s 2 ½t, T and M is the principal amount. In addition, vR(t0, U) is the price at time t of a risky zero coupon bond that pays one dollar at time U and issues by the reference obligation, where t0t T U T*. t0is the launch-ing date of the leveraged total return notes. Similarly, if the issuer is an SPV, the note value is uncorre-lated with the credit event of the protection buyer. For their payoff structure, if the reference entity does not default at the coupon payment date s, the coupon payment is LCs. Otherwise, the coupon payment is zero. In view of the principal amount, if there
is no default prior to time T, the noteholders receive the leveraged principal, which is defined as the sum of the principal plus the principal multiplied by both the leverage factor and the return on the price of risky zero coupon bond of reference obligation. Otherwise, the noteholders receive the leveraged
principal immediately at time R. Hence, the valua-tion of the leveraged total return CLN is as follows:
TCSPVðtÞ ¼ E Z T t 1fR4 sgLCs BðtÞ BðsÞds þ 1fR4 Tg BðtÞ BðTÞ M 1 þ Lv RðT, UÞ vRðt0, UÞ vRðt0, UÞ þ1ft5 RTg BðtÞ BðRÞ M 1 þ LRpð R, UÞ vRðt0, UÞ vRðt0, UÞ Ft
The pricing formula for the leveraged total return CLN issued by an SPV is shown in Theorem 4. Theorem 4: The analytic formula of the leveraged total return CLN is TCSPVðtÞ ¼ Z T t LCspðt, sÞG1ðt, sÞds þ M ð1 LÞ pðt, TÞG1ðt, TÞ þ Z T t pðt, sÞ½R 0G1ðt, sÞ þR1G2ðt, sÞ þ RG3ðt, sÞds þ L vRðt0, UÞ Rpðt, TÞG7ðt, T, UÞ þ ð1 RÞpðt, UÞG1ðt, UÞ þ ZT t Rpðt, sÞ½R0G7ðt, s, UÞ þ R1G8ðt, s, UÞ þRG9ðt, s, UÞds if R4 t M 1 þ LRpðt, UÞ v Rðt0, UÞ vRðt0, UÞ if R¼t 0 if R5 t 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : ð15Þ SPVfeeðtÞ ¼ ð1 AÞ ZT t Cspðt, sÞ G½ 1ðt, sÞ G4ðt, sÞds þ ð1 AÞM pðt, TÞ G½ 1ðt, TÞ G4ðt, TÞ þ ð1 AÞR0RM Z T t pðt, sÞ G½ 1ðt, sÞ G4ðt, sÞds þ ð1 AÞR1RM Z T t pðt, sÞ G½ 2ðt, sÞ G5ðt, sÞds þ ð1 AÞRRM ZT t pðt, sÞ G½ 3ðt, sÞ G6ðt, sÞds if A4 t, R4 t ð1 AÞCSPVðtÞ if At, R4 t 0 otherwise 8 > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > :
where G7ðt, T, UÞ ¼ exp " R0ðT tÞ Yðt, UÞðU tÞ ðR1 1ÞYðt, TÞðT tÞ þðT tÞ 2 4 R 2 1 þð R 1Þ 2 R 1 2 2ðt, TÞ þðT tÞ 3 6 2R12þR11ðt, T, UÞ þR12ðt, T, UÞ þ R1R13ðt, TÞ #
is a credit risk adjusted discount factor induced by spontaneous default intensity of reference obligation if the leveraged total return CLN is issued by an SPV and
G8ðt, s, UÞ ¼ G7ðt, s, UÞ
0ðt, sÞ þ
Zs t
ðy, sÞbðy, UÞdy R 1 2 bðt, sÞ 2 RIrI Zs t ðy, sÞdy
is a credit risk adjusted discount factor induced by the sensitivity of reference obligation to the level of spot rate if the leveraged total return CLN is issued by an SPV. G9(t, T, U) expresses the credit risk adjusted discount factor induced by the sensitivity of reference obligation to the excess return on market index if the leveraged total return CLN is issued by an SPV and is defined as follows: G9ðt, s, UÞ ¼ G7ðt, s, UÞ 2 1ðs tÞ 2 ½1 Rðs tÞ þIrI Zs t bðy, UÞdy þ R1IrI Zs t bðy, sÞdy 1ðt, T, UÞ ¼ ZT t
bðy, UÞbðy, TÞdy 2ðt, T, UÞ ¼ rI ZT t ðT yÞbðy, UÞdy, 3ðt, TÞ ¼ rl ZT t ðT yÞbðy, TÞdy
The detailed proof for Theorem 4 is given by Appendix 4.
By virtue of (13), if U equals T, we obtain G7(t,T,U) ¼ G1(t,T) and thereby G7(t,T,U) is com-posed of decreasing functions of spontaneous default intensity, the interest rate volatility and the volatility of return on market index when prI40. Conversely, the relationship among G7(t,T,U), spot rate volatility and market index volatility may not be consistent
with the empirical results when prI< 0. Meanwhile, it is also apparent that
@pðt, TÞG5ðt, T, UÞ
@Yðt, TÞ ¼
@pðt, TÞG1ðt, TÞ
@Yðt, TÞ
¼ ð1 þ R1ÞðT tÞG1ðt, TÞ 5 0
Therefore, we can document that higher level of yield of default-free zero coupon bond is associated with lower values of the leveraged total return CLNs under the condition of U ¼ T.
Issued by the protection buyer
Similarly, if the issuer is the protection buyer, the note value is related with the default intensities of the protection buyer and the reference obligation. For the coupon payment, if neither the reference entity nor the protection buyer default at the coupon payment date s, the payments are LCs. If the protection buyer defaults at the coupon payment date s but the reference entity does not, the payments are ALCs. For the leveraged principal, the following four cases should be discussed. First, if there is no default prior to maturity date T, the noteholders receive the leveraged principal at the maturity. Second, if the reference entity does not default prior to maturity T but the protection buyer does default, similar to the case that risky bond defaults, the noteholders receive the amount equal to leveraged principal multiplied by the recovery rate of the buyer at the maturity. Third, if both the entities default prior to maturity T but the first-to-default is the reference entity, the noteholders receive the leveraged principal immediately at time R. Finally, if both the entities default prior to maturity T but the first-to-default is the protection buyer, the noteholders receive the amount equal to leveraged principal multiplied by the recovery rate of the protection buyer immediately at time R. In brief, the payoff of a leveraged total return CLN is as follows:
TCPBðtÞ ¼ E ZT t 1fA4 sg1fR4 sgLCs BðtÞ BðsÞds þ ZT t 1fAsg1fR4 sgLACs BðtÞ BðsÞds þ1fA4 Tg1fR4 Tg BðtÞ BðTÞ M 1 þ Lv RðT, UÞ vRðt 0, UÞ vRðt 0, UÞ þ1ft5 ATg1fR4 Tg BðtÞ BðTÞA M 1 þ Lv RðT, UÞ vRðt 0, UÞ vRðt 0, UÞ
þ1fA4 Rg1ft5 RTg BðtÞ BðRÞ M 1 þ LRPð R, UÞ vRðt 0, UÞ vRðt 0, UÞ þ1fARg1ft5 RTg BðtÞ BðRÞA M 1 þ LRpð R, UÞ vRðt 0, UÞ vRðt 0, UÞ Ft
We provide the analytic formula of a leveraged total return CLN issued by the protection buyer in Theorem 5.
Theorem 5: The analytic formula of the leveraged total return CLN is as follows:
where G10ðt, T, UÞ ¼ exp h H0ðT tÞ Yðt, UÞðU tÞ ðH1 1ÞYðt, TÞðT tÞ þðT tÞ 2 4 H 2 Iþ ðH 1Þ2H1 2 2ðt, TÞ þðT tÞ 3 6 2 H2IþH11ðt, T, UÞ þHI2ðt, T, UÞ þ H1HI3ðt, TÞ i is a discount factor induced by spontaneous default intensity of reference obligation is
leveraged total return CLN is issued by a protection buyer.
G11ðt, s, UÞ ¼ G10ðt, s, UÞ
0ðt, sÞ þ
Zs t
bðy, UÞðy, sÞdy H 1 2 bðt, sÞ 2 HIrI Zs t ðy, sÞdy
is a credit risk adjusted discount factor induced by the sensitivity of reference obligation to the level of spot rate if the leveraged total return CLN is issued by a protection buyer.
G12(t,T,U) means the credit risk adjusted discount factor induced by the sensitivity of reference obligation to the excess return on market index if the leveraged total return CLN is issued by a protection buyer and is
shown as follows: G12ðt, s, UÞ ¼ G10ðt, s, UÞ 2 Iðs tÞ 2 ½1 Hðs tÞ þIrI Zs t bðy, sÞdy þ H1IrI Zs t bðy, sÞdy G13ðt, T, UÞ ¼ exp " R0ðU tÞ A0ðT tÞ R1Yðt, UÞðU tÞ A1Yðt, TÞðT tÞ
þðT tÞ 2 4 A 2 Iþ ðU tÞ2 4 R 2 I TCPBðtÞ ¼ ZT t LCspðt, sÞ½ð1 AÞG4ðt, sÞ þ AG1ðt, sÞds þ M ð1 LÞ pðt, TÞ½ð1 AÞGAðt, TÞ þ AG1ðt, TÞ þ ð1 AÞ Z T t pðt, sÞ R0G4ðt, sÞ þ R1G5ðt, sÞ þ RG6ðt, sÞ ds þ A ZT t pðt, sÞ R0G1ðt, sÞ þ R1G2ðt, sÞ þ RG3ðt, sÞ ds þ L vRðt 0, UÞ Rpðt, TÞ ð1 ½ AÞG10ðt, T, UÞ:
þAG7ðt, T, UÞ þ ð1 RÞpðt, UÞ ½ AG1ðt, UÞ þ ð1 aAÞG13ðt, T, UÞ
þ R A: ZT t pðt, sÞ R0G7ðt, s, UÞ þ R1G8ðt, s, UÞ þRG9ðt, s, UÞds þ ð1 AÞ ZT t pðt, sÞ½R0G10ðt, s, UÞ þ R1G11ðt, s, UÞ þRG12ðt, s, UÞds if A4 t, R4 t ATCSPVðtÞ if At, R4 t 0 otherwise 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : ð16Þ
þð A 1Þ 2A 1 2 2ðt, TÞ þðR1Þ 2þR 1 2 2ðt, UÞ þðU tÞ 3 6 2 R 2 Iþ ðT tÞ3 6 2 A 2 I þ ð1 þ R1ÞA11ðt, T, UÞ þ ð1 þ R1Þ AI2ðt, T, UÞ þ A1AI3ðt, TÞ þ ð1 þ R1ÞRI4ðt, UÞ þA1RI5ðt, T, UÞ þ ARI26ðt, T, UÞ is a discount factor induced by spontaneous default intensity of reference obligation if the leveraged total return CLN is issued by a protection buyer. Being different with G10(t,T,U), G10(t,T,U) reflects that the reference obligation still survive after T, yet G13(t,T,U) represents that reference obligation is still alive after U.
4ðt, UÞ ¼ rI
ZU t
ðU yÞbðy, UÞdy 5ðt, T, UÞ ¼ rI
ZT t
ðU yÞbðy, TÞdy, 6ðt, T, UÞ
¼ ZT
t
ðU yÞðT yÞdy
The detailed proof for Theorem 5 is given by Appendix 5.
By virtue of equalities (15) and (16), if we assume that A¼A0 ¼A1 ¼A¼0, it can be seen that
equality (15) is a special case of equality (16), i.e., the leveraged total return CLN is free of credit exposure for the protection buyer since the issuer
is an SPV. Meanwhile, without loss of generality, if we assume that U ¼ T, we obtain G10(t,T,U) ¼ G4(t,T) ¼ G13(t,T,U), G11(t,T,U) ¼ G5(t,T) and G12(t,T,U) ¼ G6(t,T). In addition, the relationship among the spot rate volatility, the market index volatility and G10(t,T,U) or G13(t,T,U) is similar to the case of G4(t,T). In addition, it is also clear that
@pðt, TÞG10ðt, T, UÞ @Yðt, TÞ ¼ @pðt, TÞG4ðt, TÞ @Yðt, TÞ ¼@pðt, TÞG13ðt, T, UÞ @Yðt, TÞ ¼ ð1 þ H1ÞðT tÞG4ðt, TÞ 5 0
Hence, the higher yields of risk-free zero coupon bonds are also correlative with lower values of the leveraged total return CLNs no matter who the issuers are.
Fair fee charged by an SPV
In practice, the leveraged total return CLNs can also be issued either by the protection buyer or by an SPV, and it is important to determine the values or fair fees charged by SPVs with issuing the leveraged total CLNs. Given the pricing formulas of equalities (15) and (16), the fair fee that the SPV charges with issuing the leveraged total return CLN is equal to equality (15) minus equality (16), as presented in the following corollary.
Corollary 2: The fair fee charged by an SPV with issuing the leveraged total return CLN, defined as TCSPVfee, is given as follows:
TCSPVfeeðtÞ ¼ ZT t LCspðt, sÞð1 AÞ½G1ðt, sÞ G4ðt, sÞds þ M ð1 LÞð1 AÞ ½G1ðt, TÞ G4ðt, TÞ þ R0ð1 AÞ Z T t pðt, sÞ½G1ðt, sÞ G4ðt, sÞds þR1ð1 AÞ Z T t pðt, sÞ½G2ðt, sÞ G5ðt, sÞds þ Rð1 AÞ Z T t pðt, sÞ½G3ðt, sÞ G6ðt, sÞds þ L vRðt 0, UÞ
Rð1 AÞpðt, TÞ½G7ðt, T, UÞ G10ðt, T, UÞ þ ð1 RÞð1 AÞpðt, UÞ
½G1ðt, UÞ G13ðt, T, UÞ þ ð1 AÞRR0 ZT t pðt, sÞ½G7ðt, s, UÞ G10ðt, s, UÞds þ ð1 AÞRR1 ZT t pðt, sÞ½G8ðt, s, UÞ G11ðt, s, UÞds þ ð1 AÞRR ZT t pðt, sÞ½G9ðt, s, UÞ G12ðt, s, UÞds if A4 t, R4 t ð1 AÞTCSPVðtÞ if At, R4 t 0 otherwise 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > :
From Corollary 2, similar to the case for the CLNs, if G7(t,s,U)4G10(t,s,U), G1(t,U)4G13(t,s,U), G8(t,s,U)4G11(t,s,U) and G9(t,s,U)4G10(t,s,U), we can see that the fee charged by an SPV is always greater than zero. Since G1(t,U), G7(t,s,U), G8(t,s,U) and G9(t,s,U) are the values that consider only the default event of reference obligation, but G10(t,s,U), G13(t,s,U), G11(t,s,U) and G12(t,s,U) are the values that consider both the default events of reference obligation and protection buyer, it is reasonable that the fair fee charged by an SPV for issuing the leveraged total return CLNs is positive.
V. Numerical Analyses of Structured CLNs In this section, we investigate the properties of the CLNs and the leveraged total return CLNs, which are correspondingly issued by an SPV or the protection buyer. Besides, we also demonstrate the appropriate fee that a protection buyer should pay to an SPV for issuing the CLNs or the leveraged total return CLNs, and then show the appropriate timing for the CLNs and the leveraged total return CLNs issued through an SPV. Finally, we examine the properties of their required yields (or credit spreads).
CLNs
We assume that the principal amount is AUD 40 000 000 and the maturity date is 2 years. The coupon rate is 5%. If a credit event occurs, no further interest will be paid at the coupon payment date. At the maturity, the noteholders receive AUD 40 000 000 unless a credit event occurs, in which case they receive the amount equal to the recovery rate Rmultiplied by the principal. We assume that ¼ 0.0254, R¼0.3, A¼0.4, A
1 ¼R1 ¼ 0:01, A¼R0.05 and the
initial term structure is flat and satisfies p(t,T) ¼ exp[ 0.05 (T t)].
Characteristics of the CLNs
We report some numerical values of the CLN by varying the different levels of spontaneous default intensities, spot rate volatility, market index volatility and correlation coefficient of spot rate and market index in Exhibit 1. From Table 1, the numerical results demonstrate that declining the spontaneous default intensity of reference entity R
0 and rI may
rise up the values of the CLN, which is issued by an SPV. Since the higher level of spontaneous default intensity of reference entity is associated with wider credit spread as well as higher default probabilities,
which make intuitive sense, it results in lower values of CLNs. It is noteworthy that when rI40, the values of the CLN are decreasing functions of rand I. Consequently, higher spot rate volatility and market index volatility widen the credit spread and increase the default probability. This result is consistent with the empirical results, such as Kao (2000), Campbell and Taksler (2003), Huang and Kong (2003), etc. Nevertheless, if rI< 0, we can see that the values of the CLN are not definitely decreasing functions of r and I. The numerical results indicate that rI is an important factor that determines the relationship among the values of CLNs, interest volatility and market index volatility. In Table 2, when the issuer is the protection buyer, the properties are similar to the ones issued by an SPV, as mentioned above. In addition, since higher levels of A
0 increase the default probability of the
protection buyer, it is conceivable that the values of the CLNs are also negatively correlated with the spontaneous default intensity of the protection buyer. Appropriate fees charged by an SPV
for issuing CLNs
When the issuer is an SPV, the proceeds from the noteholders are used to buy high-quality collaterals held by the SPV, and hence the CLNs are free of the credit exposure of the protection buyer. As a result,
Table 1. The values of the CLN issued by an SPV R 0 rI I r 0.01 0.03 0.5 0.2 0.02 98.5024 96.6495 0.05 98.5101 96.657 0.5 0.02 98.7328 96.9003 0.05 98.7729 96.9226 0 0.2 0.02 98.7328 96.8825 0.05 98.7317 96.8819 0.5 0.02 98.4955 96.6422 0.05 98.4932 96.6403 0.5 0.2 0.02 98.716 96.8648 0.05 98.6908 96.8414 0.5 0.02 98.4886 96.635 0.05 98.4763 96.6236 Notes: This table reports the price of the CLN as functions of spontaneous default intensity, interest volatility, market index volatility and correlation coefficient between interest rate and return on market index. The numerical results show that the value of the CLN, which issued by an SPV, is a decreasing function of R
0 and rI. In addition, the
values of the CLN is a decreasing function of the volatility Iand the volatility rwhen rI40. However, if rI< 0, we
find that the relationship among them may be negative and is different from the empirical results.
Appendix 5 Proof of Theorem 5 TCPBðtÞ ¼ E Z T t 1fA4 sg1fR4 sgLCs BðtÞ BðsÞ þ ZT t 1fAsg1fR4 sgALCs BðtÞ BðsÞ þ1fA4 Tg1fR4 Tg BðtÞ BðTÞ M 1 þ Lv RðT, UÞ vRðt 0, UÞ vRðt 0, UÞ þ1ft5 A5 Tg1fR4 Tg BðtÞ BðTÞA M 1 þ L vRðT, UÞ vRðt 0, UÞ vRðt 0, UÞ þ1fA4 Rg1ft5 RTg BðtÞ BðRÞ M 1 þ LRPð R, UÞ vRðt 0, UÞ vRðt 0, UÞ þ1fARg1ft5 RTg BðtÞ BðRÞ A M 1 þ L RPðR, UÞ vRðt0, UÞ vRðt 0, UÞ Ft: # ¼IE1þIE2þIE3þIE4þIE5þIE6
Using the same pricing procedure of Appendices 3 and 4, we have IE1¼1fA4 tg1fR4 tg ZT t LCspðt, sÞG4ðt, sÞds IE2¼1fA4 tg1fR4 tg ZT t ALCspðt, sÞG1ðt, sÞds 1fA4 tg1fR4 tg ZT t ALCspðt, sÞG4ðt, sÞds IE3¼E " 1fA4 Tg1fR4 Tg BðtÞ BðTÞ: M þ ð1 LÞv RðT, UÞ vRðt 0, UÞ vRðt 0, UÞ Ft # ¼JE1þJE2 where JE1¼E 1fA4 Tg1fR4 Tgð1 LÞM BðtÞ BðTÞ Ft: " # ¼1fA4 tg1fR4 tgð1 LÞMpðt, TÞG4ðt, TÞ JE2¼E 1fA4 Tg BðtÞ BðTÞ LM vRðt 0, UÞ E BðTÞ BðUÞ Rþ ð1 RÞ1fR4 Tgexp ZU T RðsÞds FT: ! 1fR4 TgjFt
We divide JE2into two parts and obtain
KE1¼E 1fA4 Tg BðtÞ BðTÞ LM vRðt 0, UÞ E BðTÞ BðUÞR1fR4 Tg Ft: " # ( ) ¼1fA4 tg1fR4 tg LMR vRðt 0, UÞ pðt, TÞG10ðt, T, UÞ KE2¼E 1fA4 Tg BðtÞ BðTÞ LM vRðt 0, UÞ E BðTÞ BðUÞ ð1 RÞ " ( 1fR4 Tgexp ZU T RðsÞds FT: ! 1fR4 Tg Ft: #) ¼1fA4 tg1fR4 tg LM vRðt 0, UÞ ð1 RÞpðt, UÞ G13ðt, T, UÞ
Finally, the proofs of IE4, IE5, IE6 also follow the same idea as ID3and is omitted.