Directory UMM :Data Elmu:jurnal:J-a:Journal Of Banking And Finance:Vol24.Issue1-2.2000:

Determination of the adequate capital for

default protection under the one-factor

Gaussian term structure model

Daisuke Nakazato

*

The Industrial Bank of Japan Ltd, IBJ-DL Financial Technology, 5-1, Ootemachi 1-chome, Chiyoda-ku, Tokyo 100-0004, Japan

Abstract

In practice, credit risk is measured by one of the two di€erent methodologies. One measures the prices and sensitivities of the credit linked instruments. Another measures the required collateral or capital needed to cover a potential default loss. This paper introduces a pricing methodology, which also determines the required capital.

Conventionally the value-at-risk method is used to determine collateral requirements. Artzner et al. demonstrated that the resulting adequate capital measures may fail to capture the credit diversi®cation e€ect, which is critical in credit risk management. (cf. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. 1997. De®nition of coherent measures of risk. Paper presented at the Symposium on Risk Management, European Finance Association, Vienna, August 27±30). In order to overcome this problem, this paper introduces a new approach, which combines closed form solutions and the Hull±White trinomial tree. This combined approach is computationally faster than a naively im-plemented Monte-Carlo-based VAR methodology. The pricing model is based on the rating-based Gaussian term structure model developed by Nakazato.

This approach is applicable to a wide variety of credit derivatives and their portfolios in a coherent fashion. In this paper, however, attention is concentrated on determining the collateral requirement for the counter party risk when there is the risk of credit rating change on an issue and also risk of default on that issue or on any protection already written on the issue. Ó _{2000 Elsevier Science B.V. All rights reserved.}

www.elsevier.com/locate/econbase

*_{Tel.: +813-5200-7611; fax: +813-3201-0698.}

E-mail address:d.nakazato@alum.mit.edu (D. Nakazato).

0378-4266/00/$ - see front matter Ó _{2000 Elsevier Science B.V. All rights reserved.}

JEL classi®cation:G33; G21; E43

Keywords:Capital adequacy; Credit; Term-structure; Risk-management

1. Introduction

Collateral is required to cover losses resulting from a potential default event. If collateral is not present, then any loss must be withdrawn from capital. Regulatory authorities require adequate capital to be available for holdings of credit linked contingent claims, and recommend the use of certain formulas for the calculation of this capital requirement. These methods are fairly simplistic, hence when volumes of credit transactions increase, the accumulated required capital becomes unreasonably large. Therefore a rational computational methodology for capital adequacy is essential for survival in credit related businesses.

The conventional method for determining adequate capital is the Value-at-Risk Quantile method. From the plot on the probability distribution for the present value of loss, the collateral is set at the required con®dence (quantile) level. This method is know in practice to be time consuming, especially when the distribution is generated by a Monte Carlo technique without any so-phisticated acceleration scheme. An alternative method is to use the contingent claim method. This method considers a contingent claim which covers the excess loss that is not covered by the collateral at the time of default. The collateral amount is then set so that the price of the contingent claim is su-ciently small compared to the insurance premium against the underlying de-fault. This method may also be time consuming if the contingent claim price does not have an analytic solution.

Artzner et al. (1997a,b) de®ne the term coherence for risk measurement. Essentially, a credit risk methodology will give a coherent risk measurement if the capital required to protect a portfolio of two positions is no greater than the sum of the capitals required for each position. Hence coherent risk mea-sures capture the credit diversi®cation e€ect. Artzner et al. criticize in particular the quantile method for failing this test. They also point out that replacing percentile criteria with standard deviation does not solve this problem. Other contingent claim methods are not yet known to be coherent. The new meth-odology proposed in this paper is a contingent claim method which gives an-alytic solutions and is shown to be coherent.

of ratings information. Furthermore such an approach ®ts comfortably with the way credit risk is controlled in most ®nancial institutions.

In general, solving the pricing-based capital adequacy problem requires a ¯exible model. In particular, determination of the adequate capital for default protection requires a dual-party model, where both the issuer of the bond and the protector face separate credit rating changes as well as default risks.

In this paper, the model employed is a one-factor Gaussian term structure model with credit rating classes, as developed in Nakazato (1997a). Lando (1998) independently developed a similar model, however this model was not addressed as a generalized Heath, Jarrow and Morton term structure model see e.g., Heath et al. (1992). Hence its one-factor implementation is not completely designed to ®t to the observed individual credit spread term structures for each credit rating class while simultaneously maintaining the arbitrage-free condi-tions. The Nakazato model extends to the multiple party case with ane Gaussian multi-factors while keeping the integrity of the single party HJM equilibrium as a special case of the multiple party extension.

Because of the complex nature of this problem, it is necessary to use tensor notation. This notation, although straightforward to use and common in physics, is rarely met in ®nance, hence an explanation is given in Appendix A. In this paper, it is shown that, in the one-factor case especially, algorithmic implementation is straightforward. The method is essentially this: a Hull± White (1990) tree is prepared, however, at each node of the tree, instead of a single value, a two-dimensional array representing a matrix of corresponding credit ratings is determined before discounting backwards. When accumulating stepwise discounted values, occasional matrix multiplication is necessary. To speed up the calculation, coding of the closed form expression may be used, which is typical in the Hull±White approach. This numerical evaluation takes a fraction of a second on a standard PC.

methodology to evaluate the contingent claim analytically and algorithmically. In the case of multi-factor evaluation, an ecient high-dimensional lattice generation technique must be used. For details refer to Nakazato (1998).

The approach in this paper is general, however, the focus is on determining the adequate capital for default protection when both a bond and also a credit default option on that bond have been purchased. In this case there is not only the risk of credit rating changes or default of the bond, but also the risk that the writer of the option (known as the protector) may default on their obli-gation. Hence it is necessary to determine the price of a contingent claim which protects against excess loss due to credit protection default.

2. Coherent risk measures

Artzner et al. de®ne a risk measure K(X), for a future net loss X from a

portfolio, to be coherent when the following four conditions are satis®ed:

‰Sub-additivityŠK…X‡Y†6K…X† ‡K…Y†:

‰HomogeneityŠ K…cX† ˆcK…X†:

‰MonotonicityŠK…X†6K…Y†; if X6Y:

‰Risk-free conditionŠ K…XÿRc† ˆK…X† ÿc;

R:risk-free money market discount bond

f g:

Among these conditions, Sub-additivity is the most important. This constraint is the mathematical restatement of the diversi®cation e€ect, which is at the heart of credit risk management. The failure of the VAR method to satisfy these criteria was illustrated in the article by Artzner et al. (1997b). For ex-ample, suppose a company has a policy to protect against any risk that has more than 5% chance of occurring, and that the same criterion is applied during approval of any individual position with such risks. A problem could occur if the company individually approved two short positions in out-of-the-money options. A short position in an out-of-out-of-the-money call option, with a strike in the upper 4% tail of the distribution, would be approved with no required protection or collateral. Similarly, an out-of-money put option with a strike in the lower 4% tail of the distribution would also be approved with no required protection. However, together the two positions represent an 8% risk, breaching the policy as no collateral protection has been allocated.

claim is suciently small, when compared to an insurance premium that would cover the total loss against the underlying default. The insurance premium represents the expected loss amount. When expressed mathematically, the adequate capitalK(X) for the loss distributionXis de®ned as follows, for some given suciently smalle>0:

K…X† ˆ infK: E…X ÿK†‡6eEX:

This functionK(X) satis®es the most essential part of Artzner et al.'s de®nition

of the coherent risk measurement, which is sub-additivity, i.e. K…X‡Y†6

K…X† ‡K…Y†. The proof is as follows: Sub-additivity is clearly satis®ed if

E…X ‡YÿK…X† ÿK…Y††‡6eE…X‡Y†;

since

K…X ‡Y† ˆ infK :E…X ‡YÿK†‡6eE…X‡Y†:

By convexity

E…X ‡YÿK…X† ÿK…Y††‡6E…X ÿK…X††‡E… ÿY K…Y††‡:

Then by the de®nition

E…X ‡YÿK…X† ÿK…Y††‡6eEX ‡eEY ˆeE…X‡Y†:

In this proof, it was assumed that EX> 0. Otherwise, replacing X with the

positive partX‡_{gives the general result, provided that the negative part (pro®t)}

of this particular loss is not used for netting purposes.

The Homogeneity condition is trivially satis®ed by this measure. But there are some pathological cases when the Monotonicity condition is violated. See Appendix C for an example. In these rare occasions, this violation may cause minor discrepancies in setting collateral levels too high for individual low-risk positions, but when the credit business grows and diversi®cation is in e€ect, then these discrepancies can be ignored.

The Risk-free condition states that risk-free assets can go in and out of the portfolio freely, causing a change in risk measure proportional to the change in risk-free asset. This measure does not satisfy the Risk-free condition as stated above, however it does satisfy the criteria when equality is replaced with in-equality. Unlike the violation of sub-additivity of VAR, this relaxation does not have serious economic consequences. Hence the following modi®ed Risk-free condition is satis®ed:

‰Risk-free conditionŠ K…XÿRc†6K…X† ÿc;

R: risk free money market discount bond:0

f 6cg:

allowed. In practice, if reinvestment is allowed, then liquidity can become an issue. In the event of default, the collateral must be easily liquidated, but this has not always proved to be possible in practice. For this reason most prudent risk managers prefer to inhibit reinvestment of collateral. However capital managers prefer to reinvest, thus avoiding the depreciation of dormant capital over time, and also reducing their capital adequacy requirement. Hence the capital adequacy problem should be formulated di€erently, depending on the company policy on capital reinvestment treatment.

In essence, the pricing-based capital adequacy determination should be considered as a coherent risk measure for credit derivative risk management, despite minor violations of the technical conditions, because it does capture the credit diversi®cation e€ect.

3. The pricing model

In order to determine the adequate capital for default protection, a ¯exible pricing model is required. In practice, a rating-based model is preferable as, conventionally, credit risk is monitored by rating-based categorizations. The term structure of yield spreads of each credit class must be re¯ected in the model as well as the credit-risk-free interest rate term structure. Also the dual-party risk must be considered simultaneously. The two parties are the issuer of

the bond and the provider of protection in the event of the issuer_Õs default.

These are the minimum requirements for a model to approach this particular problem. The model should be further required to ®t the observed volatility term structure of the yield curve and spreads. Each term structure of every rating class must evolve consistently with the speci®cations in a no arbitrage

manner as in Heath et al. (1992). To the best of the authorÕs knowledge, the

Nakazato (1997a) is the only model at present that meets all these require-ments. In this paper, for ease of algorithmic implementation, the requirement for the volatility term structure is dropped and the term structures are modeled by a simple one-factor extended Vasicek type model.

Now, the results from the Gaussian term structure model with credit rating classes developed by Nakazato (1997a) are summarized. For this particular problem, the dual-party one-factor ane Gaussian model is assumed. It is also assumed that both parties come from the same credit rating category; this means that when they are in the same credit class, they become statistically indistinguishable. This assumption is imposed because of computational sim-plicity, for a more general treatment of individual credit transition, refer to

Nakazato (1997a). The credit rating classes are 1;. . .;n and 0, the last one

dis-count bond price from an initial state (a,b) at tto a terminal state (a, b) at maturity Tis denoted byDa;ab;b(t,T). The bond pays the principal value of 1

only when the terminal state is reached at maturity. The discount bond price function is assumed to be of the form

Da;ba;b…t;T† ˆUa;bn1;n2exp

Here, Einstein_Õs convention is used, i.e. the summation signsPn

n1ˆ0

Pn

n2ˆ0 are omitted. An explanation for the tensor notation is given in Appendix A. From this formula it can be seen that, for a ®xed maturity, each tensor discount bond

can be expressed as a linear combination of the commonn2_{hypothetical assets}

(spectrum discount bonds) exp…ÿRT

t F…n1;n2†…t;s†ds†. The basic tensorsUare

constant. These are computed from the eigenvectors of the single-party tran-sition probability matrix. This is explained in more detail below. The bene®ts of tensor notation become apparent when evaluating complex credit contingent claims, which are contingent on the credit states of multiple parties at various cash ¯ow timings, in which case keeping track of indices can easily get out of hand. Each spectrum then satis®es the equilibrium condition:

Et d exp

Each spectrum discount bond behaves as if it has its own closed economy. In

other words, for ®xed indices (n1;n2), the spectrum discount bonds have a

unique yield curve and volatility term structure.

Before explaining the notation used, it is necessary to review the single party case, which gives a basis for the construction of the dual-party case. Suppose that in practice, from historical observation, a state transition rate matrix is

estimated as a constant Q. The problem of calibrating parameters from the

market and historical data is explained in detail in Nakazato (1997a). ThenQ

should be of the form

Each element of~kis the default rate multiplied by one minus the recovery rate. A diagonal matrix is denoted by~xwhen its diagonal elements form the vector~x. All vectors are row vectors, unless stated otherwise; the transpose~xis explicitly

used to indicate a column vector. The matrix U of row eigenvectors of the

minor matrix qÿ~k may now be computed, normalized toU~10_ˆ~₁0_{. This}

as-sumes that all eigenvalues are distinct and non-zero, however if this is not the

case then perturbQslightly. The basic tensors are now constructed as follows:

U0

In tensor notation, the order of subscripts and superscripts plays an important role. Elements in the inverse matrix are expressed by reversing the order of subscripts and superscripts. In the dual-party case, the basic tensors are ob-tained by multiplication of the two basic tensors for the single-party case:

Ua1;a2

These are the linear weights for the hypothetical assets when constructing the tensor discount bond, and are simply the product of two elements from the eigenvector matrix of the transition rate matrix of the single-party credit rating transition.

Next, the model is ®tted to the initial yield and spread term structure. For

each credit class 1 through n as well as the ``credit'' risk-free case, smooth

discount yield curves are determined from the market data. Prices at timetof

pure discount bonds maturing at time T are denoted D~1…t;T†;. . .;D~n…t;T†,

where the subscripts denote credit rating class. In the risk-free case, the price is denotedD…t;T†.

The instantaneous forward spectra are de®ned as

The dual-party instantaneous forward spectra are simply linear combinations of the single-party instantaneous forward rates.

In the Hull and White model, i.e. the extended Vasicek model, where a one-factor ane Gaussian model is assumed, it is natural to use the risk-free spot

ratert as the unique common factor. However, for the one-factor Nakazato

model, when coecients of the spot rate are expressed in terms of the current term structure of the yield and spreads, the calculation of these coecients is non-trivial. The use of a one-factor ane Gaussian model implies that

Z T

t

F…n1;n2†…t;s†dsˆa…n1;n2†…t;T† ‡b…n1;n2†…t;T†rt:

The single-party case is described in the same way except with one less index. It

is feasible but dicult to ®nd the coecient functionsa,bfor both single- and

dual-party cases which satisfy all the constraints discussed above. In this paper, an alternative formulation of the Vasicek model is given which yields simpler expressions for the coecient functions. The formulation uses the martingale parto_{Y…0;t†of the spot rate as the common factor, rather than the spot ratert} itself.

The extended Vasicek model is now brie¯y reviewed. Under this model, the process for the risk-free spot rate is given by

drtˆ…h…t†‡Q…t†rt†dt‡n…0†dwt;

dwtN 0;r2…t†dt

ÿ

;

where the mean reversion coecient is denotedQ…t†. The following terms are

now de®ned:

P…t;T† ˆ exp

Z T

t

Q…t†dt

;

D…t;T† ˆ

Z T

t

P…t;s†ds;

also

rYY…t;T† ˆ

Z T

t

P…s;T†2r2…s†ds;

rYy…t;T† ˆ

Z T

t

D…s;T†P…s;T†r2…s†ds;

ryy…t;T† ˆ

Z T

t

D…s;T†2r2_s†ds:

o_{Y…t;T† ˆ}

From the equilibrium condition:

Et dexp

The processes for the accumulative instantaneous forward rates and the ac-cumulative spot rate are now obtained by integration:

Z T

These results are now extended to the spot spectra.

Since the assumption is that the each discount function has a common Gaussian single factoro_{Y…0;t†, which is independent of the maturityTand the} state (credit rating class), the processes of the spot spectra become

dR…n†…t† ˆ h…n†…t†

All spot spectra move instantaneously parallel with di€erent ampli®cationsn.

Et dexp

Ampli®cation constants n of the spectrum volatility are determined by the

relationship

Similarly, the processes for the accumulative instantaneous forward spectrum and the accumulative spot spectrum are given by

Z T

Note that the accumulative forward spectra for the discount bonds have the

common Gaussian factor o_{Y…0;t†. The tensor discount functions are now}

completely determined. The money market discount process in tensor form is obtained by taking the exponential, then multiplying the basic tensors.

This is a tensor extension of the money market discount bond in a sense

Now all the necessary pricing tools are de®ned.

3.1. Use of the Hull and White tree

An innovation in option pricing came when Cox, Ross and Rubinstein in-troduced the binomial tree as a discretization of the Black and Scholes con-tinuous model. This innovation was extended to interest rate term structure modeling by Hull and White, (1990), who discretized the extended Vasicek model. The major contribution of the Hull±White approach is the combining of trinomial trees and analytical solutions. This allows the ecient evaluation of complex interest rate derivatives. In this subsection, the Hull±White ap-proach is extended to include credit rating changes and default.

In this one-factor setting, it is possible to evaluate the whole structure by the Hull±White tree. Given a Hull±White tree, that is a trinomial lattice of the risk-free spot rater…no_t†, then for each lattice point, there is a simple conversion formula to the state variable:

o_Y…0;no_{t† ˆ} 1

n…0† r…no_t†

… ÿf…0;no_{t†† ÿn}

…0†rYy…0;not†:

For interest rate derivative evaluation, it is sucient to store the price of the contingent claim at each node of the tree. The price is then discounted back by the risk-free spot rate r…no_{t†, multiplied by the appropriate branching} equi-librium probability. With credit classes, the prices become a tensor. Thus the tensor, i.e. the two-dimensional array for the dual-party case, must be stored at each node. The branching probability stays the same, however, the prices are now discounted by the tensors.

For tensor discounting of each time step, if the time stepo_{tis chosen to be}

small enough, then the following approximation holds:

Hence it is not necessary to compute the supplementary factoro_y…0;no_{t†. In} general, Nakazato (1997a) demonstrated that this model is extendable to the multi-factor Gaussian model so that individual movements of the risky yields corresponding to the di€erent credit classes can be re¯ected. The multi-factor model can be evaluated eciently using a high-dimensional lattice. For more details, refer to Nakazato (1998) for algorithmic implementation of the multi-factor ane Gaussian models.

4. Evaluation of the excess loss protection by the Hull±White tree

In the case of the protectorÕs default it is necessary to purchase protection from the ``credit'' risk-free party. Thus the loss due to the protectorÕs default is equivalent to the insurance fee for secondary protection. Suppose that default takes place at timetand the credit rating of the issuer isn, then the loss is given by

where the hazard rateg_a…t†can be considered as the instantaneous insurance

fee to cover the principal of the bond. Here the credit rating class of the issuer of the bond prior to default isa, and the maturity of the underlying bond isT. Note that the tensor indices run through non-defaulting states only, because it is only necessary to consider the case when neither party defaults. Hence the loss can be interpreted as a ``tensor discounted'' total of the instantaneous insurance fee.

The loss can be expressed in a closed form:

Ln…t† ˆUnm

The proof is given in the appendix. The excess loss over the collateral K is

· _{when reinvestment is prohibited:}

Ln…t†

… ÿK†‡ˆmax…Ln…t† ÿK; 0†:

This restriction is a conservative one and it comes from the limitation of the Hull±White tree, because the tree spans ino_{Y…0;t†direction only.}

· _{when reinvestment is allowed:}

Ln…t†

In the latter case, when building the Hull±White tree it is necessary to keep

track of the accumulated money market value exp Rt

0rtdt

in each node.

Therefore, for more general applications, one additional dimension o_y…0;t†

should be added when the evaluation lattice is generated.

In Appendix B, the closed form solution is given when the collateral can be invested in the money market. The tree approach is recommended because of the complexities in the closed form solution. Therefore in the rest of the paper, it is assumed that no reinvestment is allowed.

Now suppose the issuer is in credit classn1, and the protector is inn2, then

the instantaneous insurance fee to cover the excess loss is given by

gn2…t†…Ln1…t†ÿK† ‡_;

where the hazard rate is given as

gn…t† ˆUnmgm…t†; surance fee is then analytically determined. The prices of the contingent claims which cover the excess loss are:

Va1;a2…0;K† ˆE0

This can be interpreted as a ``tensor discounted'' total of the instantaneous insurance fee to cover the excess loss. Thus at each node of the tree, this for-mula can be recursively evaluated as

Va1;a2…t;K† ˆga2…t†…La1…t†ÿK†

term is associated with subsequent branch nodes. These prices are at time t

when the issuer is in the credit class a1, and the protector is in a2. Further

reduction of the computation time is possible by eliminating tensor multipli-cation at each node; discounting the tree backward in the spectra, then com-puting the price at the root of the tree from the accumulated spectrum:

Vm1;m2_{t;K† ˆg}m2_t†Un

m1…Ln…t†ÿK† ‡_o

t‡EtV

m1;m2_t‡ot;K†

‰ Š

1‡R…m1;m2†…t†ot ;

Va1;a2…0;K† ˆUa1;a2

m1;m2_Vm1;m2_0;K†:

Using this price, the collateralKis set by the criteriaVa1;a2…0;K† ˆeVa1;a2…0;0†

for suciently smalle. The valueKcan be found numerically using the secant

method or the bisection method.

4.1. Numerical example

The pricing model must be ¯exible enough to allow the pricing-based capital adequacy determination to be applied to a wide variety of instruments. This subsection illustrates the pricing model ¯exibility and shows that the adequate capital required can be numerically solved, given the credit rating classes, current yield and spreads. The adequate capital required with and without reinvestment is also compared.

The plots in Fig. 1 were calculated using a Hull±White model with a stant price volatility of 1% and mean reversion coecient of 2%. For con-ciseness of the illustration, the transition probabilities and the credit risk-free yield curve data are omitted. The numbers used are hypothetical. See Fig. 1 for the spread curves used.

For di€erent credit rating classes of the protector and the issuer with a risk tolerance level of 0.1% and maturity 5 years, the adequate collateral was computed as given in Fig. 2.

With these 3-D plots, it is dicult to show the re-investment e€ect of the collateral. For di€erent risk tolerance levels with the 5 year ``A'' bond pro-tected by the ``Aa'' protector, Fig. 3 illustrates the re-investment e€ect.

And for varying maturities with a 0.1% tolerance level, the same comparison is made (see Fig. 4).

5. Conclusion

The purpose of this paper is to provide a practical algorithmic solution to the problem of determining the adequate level of capital for complex credit

Fig. 2. Collateral vs. credit ratings.

derivatives. The example given in this paper concerns the most common problem in credit derivatives, namely default protection. However, this con-tingent claim is suciently complex to demonstrate the novelty of the ap-proach, since the price depends on the credit ratings and default risk of both the protector and the issuer of the protected bond.

As far as the algorithmic construction is concerned, the Hull and White tree is one of the simplest term structure models available. The additional practical implementation is also straightforward, because of the credit evaluation model. The model naturally combines transition data between di€erent credit rating classes with current yield curves observed in the market for di€erent credit classes.

A rational computational methodology alternative to the VAR quantile method is introduced in this paper. The method is based on coherent ana-lytical evaluation of the protection required against the excess default loss over and above the coverage provided by the collateral. The advantage of a co-herent approach is that the risk measurement captures the diversi®cation ef-fect. This is the essence of credit business and credit risk management. In addition, the computational speed is increased over the conventional Monte Carlo method, using the recombining Hull±White tree and the bene®t of the tensor-spectrum decomposition. It is, in fact, possible to solve the problem in closed form without using the tree. The computation time then reduces fur-ther.

Unlike the adequate capital criteria recommended by the regulators, the methodology developed in this paper does not unreasonably increase the capital requirement when it is applied to a large portfolio of the credit linked contingent claims.

Acknowledgements

The author is grateful for Richard Bateson, Michael Dempster, Toshifumi Ikemori, Patricia Jackson, Farshid Jamshidian, Masaaki Kijima, Hiroshi Shirakawa, Akihiko Takahashi and Domingo Tavella for comments, and most importantly Alan Ambrose of the Quantz Ltd. for patiently coding the com-plex formulae. The author owes to Mark Davis of the Tokyo Mitsubishi In-ternational for showing the counter example. The author especially would like to thank Stephen Hancock and Alex McGuire of IBJ International for careful reading and clari®cation of the paper. Of course any remaining errors are those of the author alone.

Appendix A. Introduction to matrix term structure model

In this appendix, the general theory behind the rating-based model devel-oped by Nakazato (1997a) is introduced. The initial breakthrough in this area came from Due and Singleton (1997) when they formulated the price of a defaultable discount bond as

D…t;T† ˆEt exp

ÿ

Z T

t

rt

f ‡ktgdt

;

wheretis the evaluation time,Tthe maturity,rtthe risk-free spot rate andktis

the mean loss rate (e€ective hazard rate), which is de®ned as the product of the hazard rate and one minus recovery rate. This formulation is powerful because not only does this give a direct economic interpretation of the yield spread as the mean loss rate, but also the mathematical elegance of the result allows extensions and applications to more complicated problems. The hazard rate is the continuous state Markov chain analogy to the transition rate matrix of a discrete state Markov chain. Thus it is a natural extension to substitute the e€ective hazard rate with the e€ective transition rate matrix. This allows the inclusion of credit rating classes. In this technique the risky discount bond becomes a matrix discount bond. The economic justi®cation for this matrix extension is rather lengthy, therefore the interested reader should refer to Nakazato (1997a). The meaning of the matrix discount bond is clear since each element is de®ned as a tensor in the pricing model section of this paper.

Caution is required when substituting matrices in such a formula since commutativity does not hold:

exp…A‡B† 6ˆ exp…A†exp…B† 6ˆ exp…B†exp…A†:

D…t;T† ˆEt

aT

t

expf

"

ÿ frtÿQ…t†gdtg

# :

As usual the product integralY…o;t† ˆ`t

oexpfAtdtgis the unique solution of

the ordinary di€erential equation d=dtY…o;t† ˆY…o;t†At,Y…o;o† ˆI.

Extension to the multiple party case is straightforward. Using the decom-position result of Markov chains, the transition rate matrix is decomposed into the sum of the transition rate matrices of each party. For example, in the dual-party case:

D…t;T† ˆEt

aT

t

expn

"

ÿ rt

n

ÿQ1…t† IÿIQ2…t†

o

dto #

;

where the symbolis the direct product in the tensor algebra.

In tensor notation, the discount bond price from an initial state (a,b) attto a terminal state (a;b) at maturityTis denoted byDa;ba;b…t;T†. The bond pays

the principal value of 1 only when the terminal state is reached at maturity. The discount bond price function is assumed to be of the form

Da;ba;b…t;T† ˆUa;bn1;n2exp

ÿ

Z T

t

F…n1;n2†…t;s†ds

Ua;b n1;n2:

Here, EinsteinÕs convention is used, i.e. the summation signsPn

n1ˆ0

Pn

n2ˆ0 are omitted.

In general, a certain handling rule for subscripts and superscripts needs to be agreed. The state space ofD…t;T†is indexed by a vector coordinate (n1;. . .;nm),

this means that partyiis in credit classniforiˆ1;. . .;m. Ifniˆ0 then partyi

is in the default state. For notational simplicity, an elementf_i;ni…t;T†is written

f_ni†…t;T†, and the 0th elementf_0†…t;T† is writtenf(t,T). Under this conven-tion, tensor notations are now reviewed.

In tensor notation, an element of the column vector is written as a covariant

vectorxiindexed by subscript; examples are price vectors, or terminal cash ¯ow

vectors. An element of the row vector is written as a contravariant vectorxi

indexed by superscript; examples are probability vectors, or Arrow±Debreu price vectors. In order to distinguish originally vector elements, all non-tensor subscripts are parenthesized; e.g.f_n

i†…t;T†. It is also assumed that any

sub-subscript is a non-tensor index. Using EinsteinÕs convention, namely

omitting the summation symbol; the right-hand side of an expression is sum-med over all sub-(super-)scripts not appearing in the left-hand side for ap-propriate ranges.

F…t;T† ˆ ÿD…t;T†ÿ1 o

Then the matrix form of the HJM model is

dF…t;T† ˆEt

Unfortunately, this result is too general. Although these expressions are compact, in order to obtain analytical solutions for the contingent claim in question, further assumptions are necessary, so it follows that a simpler version of the matrix term structure model is obtained as in the main body of this paper. This model, called the ane Gaussian model, can be viewed as an ex-tension of the Vasicek±Jamshidian model.

Appendix B. The closed form solution

Using the de®nition of the hazard rate, the expression for the protection fee can be simpli®ed to

Ln…t† ˆUnmlim

Suppose the initial collateralKis invested in the money market, then the excess

Ln…t†

The loss occurs in the event of default by the protector, thus the price of the contingent claim which covers the excess loss is

Va1;a2…0;K† ˆE0

The insurance premium for the excess loss is

V_a

Using these prices, the collateral K is set by the coherent criteria

Va1;a2…0;K† ˆeV

a1;a2…0†for suciently smalle. The valueKis then numerically

determined by the Newton±Raphson method:

KˆK1;

The closed form solution ofV

A proof of this formula is omitted. For interested readers, the proof for the more complicated casesVa1;a2…0;K†andoVa1;a2=oK…0;K†is given in the original

working paper Nakazato (1997b).

Appendix C. Counter example of monotonicity in the coherent measure

This ingenious counter example is due to Mark Davis.

Let ([0,1],du) be the unit interval probability space, andY(u)ˆu.

Then EY ˆ1

2:

And E…Yÿk†‡ˆ1 2…1‡k

2_{† ÿk:}

For example; whenkˆkY ˆ0:8; then E…Y ÿk†‡ˆ0:02:

Hence E…YÿkY†‡ˆ0:02ˆ0:04EY:

ThuskY ˆK…Y†wheneˆ0:04:

Now let

X…u† ˆ 1

2u; u6k0

u u>k0

…soX6Y†:

Then

E…XÿkY†‡ˆE…Xÿk0†‡ f*k0ˆkYgg ˆ0:04EY

>0:04EX:

For this exampleX6Y but K…X†>K…Y†.

References

Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., 1997a. De®nition of coherent measures of risk. Paper presented at the Symposium on Risk Management, European Finance Association, Vienna, August 27±30.

Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., 1997b. Thinking coherently. Risk 10 (11), 68±71. Due, D., Singleton, K. 1997. Modeling term structures of defaultable bonds. Working paper,

Graduate School of Business, Stanford University, Stanford CA.

Hull, J., White, A., 1990. Numerical procedures for implementing term structure models I: Single factor models. Journal of Derivatives 2 (1), 7±16.

Lando, D., 1998. On Cox processes and credit risky securities. Working paper, Department of Operations Research, University of Copenhagen. Forthcoming in the Review of Derivatives. Nakazato, D., 1997a. Gaussian term structure model with credit rating classes. Working paper,

Financial Engineering Department, The Industrial Bank of Japan, Tokyo.

Nakazato, D., 1997b. Determination of the adequate capital for default protection under the one-factor Gaussian term structure model. Working paper, Financial Engineering Department, The Industrial Bank of Japan, Tokyo.