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1

Credit Spread Arbitrage in

Emerging Eurobond Markets

Caio Ibsen Rodrigues de Almeida *

Antonio Marcos Duarte, Jr. **

Cristiano Augusto Coelho Fernandes ***

Abstract

Simulating the movements of term structures of interest rates plays an important role when

optimally allocating portfolios in fixed income markets. These movements allow the

generation of scenarios which, on their turn provide the assets’ sensibility to the fluctuation of

interest rates. The problem becomes even more interesting when the portfolio is international.

In this case, there is a need to synchronize the different scenarios for the movements of the

interest rate curves in each country. An important factor to consider, in this context, is credit

risk. For instance, in the corporate Emerging Eurobond fixed income market there are two

main sources of credit risk: sovereign risk and the relative credit among the companies issuers

of the eurobonds. This paper presents a model to estimate, in a one step procedure, both the

term structure of interest rates and the credit spread function of a diversified international

portfolio of eurobonds, with different credit ratings. The estimated term structures can be used

to analyze credit spread arbitrage opportunities in Eurobond markets. Numerical examples

taken from the Argentinean, Brazilian and Mexican Eurobond markets are presented to

illustrate the practical use of the methodology.

Please address all correspondence to:

Cristiano Augusto Coelho Fernandes

Rua Marquês de São Vicente, 225 - Gávea

22453-900, Rio de Janeiro, RJ, Brazil

Electrical Engineering Department

Phones: 55-21-5299415 or 55-21-5375905

E-mail: Cris@ele.puc-rio.br

* Pontifícia Universidade Católica do Rio de Janeiro, Brazil. E-mail: caio@ele.puc-rio.br

** Unibanco S.A., Brazil. E-mail: antonio.duarte@unibanco.com.br

*** Pontifícia Universidade Católica do Rio de Janeiro, Brazil.

2

1. Introduction

The prices of fixed income assets depend on three components (Litterman

and Iben [1988]): the risk free term structure of interest rates, embedded options

values and credit risk. Optimally allocating portfolios in fixed income markets

demands a detailed analysis of each of these components.

Several authors have already considered the risk free term structure

estimation problem. For example, Vasicek and Fong [1982] suggest a statistical model

based on exponential splines. Litterman and Scheinkman [1991] verified that there are

three orthogonal factors which explain the majority of the movements of the US term

structure of interest rates. These three factors form the basis for many fixed income

pricing and hedging applications. For instance, these factors are used in Singh [1997]

to suggest optimal hedges.

Some bonds present embedded options. In general, the price of an embedded

option is a nonlinear function of its underlying bond price on all dates before the

option maturity date. An embedded option depends not only on the actual term

structure of interest rates, but also on the evolution of this term structure during the

life of the option. Several models have been proposed for the evolution of the term

structure of interest rates. These models are classified in two major groups (Duffie

[1992]): equilibrium models (Vasicek [1977], Cox et al. [1985], among others) and

arbitrage free models (Ho and Lee [1986], Black et al. [1990], Heath et al. [1992],

among others). At this point in time, the pricing of embedded options using arbitrage

free models is perceived as the most appropriate because the parameters can be

chosen to be consistent with the actual term structure of interest rates and,

consequently, to the actual prices of bonds (Heath et al. [1992]). The process modeled

can be the short-term interest rate, the whole term structure of interest rates, or the

3

forward rates curve. No matter what the process is, when it is Markovian, it is usually

implemented using binomial trees (Black et al. [1990]) or trinomial trees (Hull and

White [1993]).

Almeida et al. [1998] presented a model to decompose the credit risk of term

structures of interest rates using orthogonal factors (Sansone [1959]), such as

Legendre [1785] polynomials. In this model, the term structure of interest rates is

decomposed in two curves: a benchmark curve and a credit spread function. The last

one is modeled using a linear combination of Legendre polynomials.

In this article we present a model to estimate, in a single step, both the term

structure of interest rate and the credit spread function of an international portfolio of

bonds with different credit ratings. This model extends the approach proposed in

Almeida et al [1998]. It allows the joint estimation of the credit spread function of any

international portfolio with different credit ratings. This extension is crucial when

analyzing credit spread arbitrage opportunities in fixed income markets. For the

purpose of illustration, we concentrate on the Emerging Corporate Eurobond market,

studying the three most important in Latin America: Argentina, Brazil and Mexico.

However, the methodology is quite general, and can be applied to any fixed income

portfolio composed by bonds with different credit ratings.

This article is organized as follows. Section 2 explains the model. Section 3

presents the estimation process for its parameters. Section 4 explains the methodology

used for optimally allocating portfolios using the model. Section 5 presents three

practical examples of detection and exploitation of arbitrage opportunities in the Latin

American Eurobond market. Section 6 presents a summary of the article, and the

conclusions.

4

2. The Model

We want to analyze a portfolio in the Emerging Eurobond market. Assets with

the same cash flow and embedded option structures, but different credit ratings, ought

to have different prices. For this reason, when structuring fixed income portfolios, it is

fundamental to estimate and simulate the movements of different term structures of

interest rates, one for each credit rating in the portfolio. One possibility would be to

estimate a term structure, for each credit rating. There is a statistical problem with the

amount of data available when relying on this approach: in the emerging eurobond

market there are usually very few liquid bonds for each credit rating. A joint

estimation procedure is necessary.

An interesting possibility is to capture the difference in risk between credit

ratings using different credit spread functions. Using this approach, it is possible to

estimate in a single step the term structures for different credit ratings by modifying

the proposal in Almeida et al. [1998].

The equation that describes the term structure of an Emerging Eurobond

market (that is, fixed income instruments of one country, issued in a same currency,

with the same credit rating) can be written as

)1( ].,0[,)1

2

()()( 0

l

l∈∀−+= ∑

≥t

t

PctBtRn

nn

where

t

denotes time, )(tB is a benchmark (for example, the US term structure), n

P is

the Legendre polynomial of degree

n

, n

c is a parameter to be estimated, and l is the

largest maturity of a bond in the Emerging Eurobond market under consideration.

The price of a bond ( A

P) is related to the term structure of interest rates as

5

)2( ))(exp(

1

∑

=−= A

n

iiiiAtRtCP

where i

C denotes the th

i cash flow paid by the bond at time i

t, and A

n denotes the

total number of cash flows paid by the bond.

Setting up the notation, variables ,,...,1,Jjrj=denote different credit ratings.

For instance, a credit rating such as AAA may be associated with 1

r, a credit rating

such as AA1 may be associated with 2

r, and so on.

An extension of )1( is to consider the spread function depending on the

different credit ratings, such as

)3( ].,0[),,...,,()(),...,,(11 l∈∀+= trrtCtBrrtRJJ

The spread function ),...,,(1J

rrtC can be modeled as a linear combination of

orthogonal polynomials in order to exploit the modeling and estimation advantages

illustrated in Almeida et al. [1998].

An application of this equation that captures the difference in risk between

credit ratings using only a translation factor is given by

( )

)4( .,...2,1],,0[,)1

2

()()( 11 Jjt

t

PcStBtRn

nn

j

ii

j=∀∈∀−++= ∑∑ ≥=

l

l

where i

S is a nonnegative spread variable (that is, jiSi,...,2,1 0=∀≥) that measures

the difference in risk between the th

i)1(− and the th

i credit ratings, and

J

represents

the total number of credit ratings.

A limitation of Model )4( is that all

J

estimated term structures are parallel.

Although very limited, Model )4( captures the fact that bonds with higher ratings

ought to have smaller prices (everything else being equal). In other words: the higher

the rating, the higher the interest rates used to price bonds with that particular rating.

6

Exhibit 1 depicts a possible output for Model )4(. It is possible to exhibit more

general models than that given in )4( (that is, a model which allows the term

structures obtained for different credit ratings to differ not only by a translation

factor). Exhibit 2 presents a schematic drawing of possible term structures of interest

rates for different credit ratings, in a more general model.

3. Joint Estimation of Term Structures

Let us consider the simplest case first (that is, Model (4)). The objective is to

estimate the variables JiSi,...,1,=and the coefficients ,...,, 321 ccc . The final results

of this estimation process are J different term structures of interest rates, each related

to a different rating.

Let us define the discount function )(

)( tDjfor rating j

r to be

(

)

)5( ,...,1],,0[,)(exp)( )()( JjtttRtDjj =∈∀−= l

We assume that

m

eurobonds are available to estimate the coefficients in

Model )4(. We assume that j

m eurobonds possess a rating j

r. The residual term k

e

of the statistical fit obtained for the price of the th

k eurobond satisfies

(6) ,...,2,1,)(11 1

)( mketDuooap k

k

f

lkkl

j

kl

c

call

k

p

put

kkk =∀+=−++ ∑

=

where k

p denotes the price of the th

k eurobond, k

a denotes the accrued interest of

the th

k eurobond, put

k

1 and call

k

1 are dummy variables (Draper and Smith [1966])

indicating the existence of embedded put and call options in the eurobond,

o

p and

o

c

are unknown parameters related to the prices of the embedded put and call options, k

f

denotes the number of remaining cash flows of the th

k eurobond, kl

t the time

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remaining for payment of the

l

th cash flow kl

u of the th

k eurobond, and k

j denotes

the rating of the th

k eurobond (for instance, if the rating of the th

k eurobond is 3

r,

then 3=

k

j).

The estimation process is based in a two step procedure:

1. Identify influential observations (Rousseeuw and Leroy [1987]) using an

extension of Cook’s statistics (Atkinson [1988]). This first step is important

because in the Emerging Eurobond market there are many illiquid or “badly”

priced bonds. If these bonds are not appropriately handled during the estimation

phase, they may distort the term structures estimated (as illustrated in Almeida et.

al. [1998]).

2. Use a duration weighted estimation process after removing all the influential

observations detected in the first step. The estimation should preferably use robust

techniques, such as the Least Sum of Absolute Deviation or the Least Median of

Squares (Rousseeuw and Leroy [1987]). The use of duration weights incorporates

heterocedasticity in the nonlinear regression model by allowing the volatility of

the eurobond prices to be proportional to its duration (as suggested in Vasicek and

Fong [1982]).

A numerical example illustrating the practical use of this methodology is

presented next.

4. A Numerical Example of the Estimation Process

As an example we consider the joint estimation of Brazilian and Mexican

eurobonds term structures. Fifty-two eurobonds are used: twenty-five Brazilian;

8

twenty-seven Mexican. The eurobonds are classified in seven different credit ratings1:

BB1, BB2, BB3, B1, B2, B3 and NR. Exhibit 3 presents the main characteristics of

the fifty-two eurobonds.

Three leverage points were detected in the first step of the estimation

process: one Brazilian (Iochpe); two Mexican (Bufete and Grupo Minero). Exhibit

4 and Exhibit 5 provide the results obtained.

Note that for the Brazilian term structures, the translation factor varies just a

few basis points when different ratings are compared: for instance, the difference

between the B1 and the B3 translation factors is only 34 basis points. On the other

hand there is a difference of 130 basis points between the B1 and B3 Mexican

translation factors. The next sections illustrate how the term structures in Exhibit 5

can be used to exploit arbitrage in the Emerging Eurobond market.

5. Detection and Exploitation of Arbitrage Opportunities

The following five steps are proposed to detect and exploit arbitrage in Latin

American Eurobond markets:

1. Choose a set of eurobonds with a common rating.

2. Estimate the term structures of interest rates for each country.

3. Based on the estimated term structures, consider possible future scenarios for their

relative movement.

4. Analyze the sensibility of different eurobond portfolios to these scenarios

generated.

5. Suggest a portfolio that better adjusts to the scenarios generated.

Three numerical examples are presented:

9

1. Brazil and Mexico: B1 Eurobonds.

2. Brazil and Mexico: B3 Eurobonds.

3. Argentina and Mexico: BB2 Eurobonds.

All the data were collected in June the third of 1998, on J.P. Morgan’s web

site4.

5.1 Brazil and Mexico: B1 Eurobonds

Exhibit 5 depicts the Brazilian and Mexican B1 term structures. The Mexican

term structure lies below the Brazilian term structure, indicating that the Brazilian B1

eurobonds are cheap when compared to Mexican B1 eurobonds. The large difference

between the translation, rotation and torsion factors of the two term structures

suggests as the most probable future scenarios the ones where the curves become

closer. That is, if there are no economic conditions leading these countries to behave

radically different, we could expect a convergence between term structures of assets

of the same rating. Exhibit 6 depicts a scenario representing the convergence of the

term structures. The arrows indicate the direction of the movements that would be

realized by each term structure in this situation. However, in spite of the suggested

convergence, we generate for the analysis of portfolio’s sensibility, unbiased future

scenarios. There are 63 possibilities of generating future scenarios changing a subset

of the term structures factors: Brazilian translation factor, Brazilian rotation factor,

Brazilian torsion factor, Mexican translation factor, Mexican rotation factor and

Mexican torsion factor. In what follows, without loosing generality, we fix as possible

future scenarios the ones where just the Brazilian translation factor and the Mexican

rotation factor change their values. A set of twelve scenarios for each term structure

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is generated and we calculate the prices of the eurobonds on each of these scenarios.

Exhibit 7 depicts the future scenarios for the Brazilian and the Mexican term

structures. Exhibit 8 presents the prices of the Mexican eurobonds on each future

scenario for the Mexican term structure and Exhibit 9 presents the prices of the

Brazilian eurobonds on each future scenario for the Brazilian term structure.

For instance, suppose the occurrence of the scenario depicted in Exhibit 6, that

is, a diminution in both the Brazilian translation factor and the Mexican rotation

factor, with all the other factors remaining the same. This could mean a decrease in

the external long term lend rate for emerging markets. We note in Exhibit 5 that the 9

years maturity is the largest one in both B1 eurobonds markets. The Legendre

polynomials symmetry suggests an analysis of the term structures based in two

regions: region I, for maturities less than 4.5 years, and region II, for maturities

greater than 4.5 years. In the occurrence of such a scenario both term structures attain

a decrease of interest rates in region II. Exhibit 8 and Exhibit 9 show that in this

situation, all the Brazilian bonds would increase their values, short-term Mexican

bonds would decrease their values and long-term Mexican bonds would increase their

values. A good strategy would be buy Brazilian bonds and Mexican long-term bonds,

and sell Mexican short-term bonds.

Exhibit 10 presents the percent variation of a proposed portfolio long US$ 1.5

million in the Mexican eurobond Azteca (2007), short US$ 1 million in the Mexican

eurobond Vicap (2002), and long US$ 1 million in the Brazilian eurobond Votorantin

(2005). Note that in the worst case scenario the investor of such a portfolio would

loose 9% of its initial wealth. On the other hand, he would profit only 5.6% in the

best-proposed situation, which would occur when the Mexican rotation factor goes

down 94 basis points (sixty percent of its value) and simultaneously the Brazilian

11

translation factor goes down 132 basis points (thirty percent of its value). The

question here is why invest in this portfolio if its profit and loss distribution is

asymmetric with higher extreme loss values than extreme profit values? The answer

depends on the probabilities assigned to each scenario. In the uniform case, where all

the scenarios receive the same probability, this portfolio would not be a good choice

because in average the investor would loose money. However, if the investor

perceives greater chances for occurrence of scenarios favorable to the portfolio, that

is, the ones where the Mexican rotation factor and the Brazilian translation factor fall,

than it would worth investing in such a portfolio. Exhibit 11 presents the portfolio

sensibility to the Brazilian translation factor and the Mexican rotation factor. The

portfolio value is more sensible to changes in the Brazilian translation factor than to

changes in the Mexican rotation factor. For instance, take a variation of 10 percent in

the Brazilian translation factor (44 bps). We observe, in Exhibit 10, that this scenario

would generate around 1.3 percent variation in the portfolio value. Nevertheless, if we

had taken the same percent variation in the Mexican rotation factor (16 bps) it would

generate around 0.6 percent variation in the portfolio value. We also observe that the

portfolio value varies almost linearly with respect to the Brazilian translation factor

but non-linearly with respect to the Mexican rotation factor.

5.2 Brazil and Mexico: B3 Eurobonds

Exhibit 5 also depicts the Brazilian and Mexican B3 term structures of interest

rates. These curves cross each other at the 3.2 years maturity. Due to the curves

disposition and to the Legendre polynomials symmetry, we split the analysis of the

term structures in three regions: region I, for maturities less than 3.2 years, region II,

12

for maturities between 3.2 and 4.5 years, and, region III, for maturities greater than

4.5 years. On region I, the Brazilian term structure of interest rates presents values

greater than the Mexican ones, with the difference approaching 150 basis points, for

short maturities. On regions II and III, there is an inversion of the situation: the

Mexican curve presents values greater than the Brazilian curve values, with the

difference approaching 100 basis points, for maturities around 9 years. The major

cause for this difference between the term structures relies in the fact that the Mexican

term structure rotation factor (156 basis points) is much superior to the Brazilian

one (57 basis points).

Just looking at the term structures picture, we could suggest, as a possible

future scenario, an approximation of the curves slopes, that is, the convergence of the

rotation factors.

The B3 eurobonds characteristics are listed in Exhibit 3. Note that the three

eurobonds have maturities less than or equal to two years, meaning that their pricing

is based on region I values of the term structures. According to the supposed future

scenario, the curves approximate each other, on region I, implying a depreciation of

the Mexican eurobond, and an appreciation of the Brazilian eurobonds. In this

context, a good strategy would be buy the Brazilian bonds and sell the Mexican one.

5.3 Argentina and Mexico: BB2 Eurobonds

In this example we consider the joint estimation of Argentinean and Mexican

eurobonds term structures. Exhibit 12 depicts the Argentinean and Mexican BB2 term

structures of interest rates. Exhibit 13 presents the factor values for both curves. We

observe that the Mexican rotation factor is much higher than the Argentinean rotation

13

factor. Suppose a fixed income manager is positioned in these markets on a portfolio

composed by the eurobonds listed in Exhibit 14. The manager is long US$ 1 million

in the Argentinean eurobond Multicanal (2007), long US$ 1 million in the

Argentinean eurobond Perez (2007), long US$ 1 million in the Mexican eurobond

Televisa (2006) and short US$ 2 million in the Mexican eurobond Cemex (2006).

Suppose also the economic committee is expecting a scenario of reduction on long-

term Mexican interest rates combined to a reduction on Argentinean interest rates.

How could the fixed income manager generate scenarios based on the described

expectations to analyze the portfolio exposure? He could begin decreasing the

Argentinean translation factor to get the effect of reducing Argentinean interest rates.

On the other hand, a combination of changes in the Mexican rotation and torsion

factors would provide the desired results for the Mexican term structure. As a matter

of fact, if he reduced the Mexican rotation factor he would decrease long-term interest

rates but also increase short-term interest rates (an undesired effect). Next, increasing

the absolute value of the Mexican torsion factor he would decrease both the short-

term rates and long-term rates, compensating the increase in short-term rates

generated by the rotation factor movement. Exhibit 15 depicts different scenarios

incorporating the economic committee expectations, that is, scenarios of reduction of

the Argentinean translation factor, reduction of the Mexican rotation factor and

reduction of the Mexican torsion factor. We note that the most extreme scenario for

the Mexican term structure really reflects a reduction of long-term Mexican interest

rates. In this scenario it would happen a reduction around a hundred basis points in the

Mexican long-term interest rate, an increase around thirty basis points in the medium-

term interest rate and an increase around ten basis points in the short-term interest

rates. Exhibit 16 presents the prices for the Argentinean eurobonds in the manager’s

14

portfolio for the different scenarios for the Argentinean Term Structure. Exhibit 17

presents the prices for the Mexican eurobonds in the manager’s portfolio for the

different scenarios for the Mexican Term Structure. Exhibit 18 presents, for each

scenario observed in Exhibit 15, the percent variation of the portfolio. We note, in

Exhibit 16 that the reduction of Argentinean interest rates contributes positively to the

portfolio value due to the long position in the BB2 Argentinean market. However,

without the help of Exhibit 17 and Exhibit 18, it would not be a simple task to identify

the exact effects in the portfolio value caused by the expected changes in the Mexican

interest rates. In Exhibit 18, we note that the reduction on Mexican long-term interest

rates decreases the portfolio value (in 1.79%) in the worst case scenario, where the

Argentinean Interest rates do not change. This behavior is attributed to the short

position in the Mexican eurobond Cemex which pursues a very high coupon (12.5%)

responsible for increasing its sensibility to interest rates movements. We also note in

Exhibit 18 that the portfolio would profit in the major portion of the scenarios

proposed. We conclude that translating abstract interest rates expectations to possible

factor movements is a very good analysis tool, because it allows an easy interpretation

of the portfolio sensibility to interest rates movements.

6. Conclusion

This article presents a methodology for jointly estimating term structures of

interest rates for clusters of bonds with different credit ratings. The model is based on

an optimization procedure, which assumes that the term structures movements are

driven by orthogonal factors. The estimated curves are useful for risk analysis,

derivatives pricing and portfolio selection. The methodology is efficient from the

15

computational point of view and is particularly useful when analyzing markets with

few liquid bonds such as Emerging Eurobond markets.

The methodology is completely compatible with scenario analysis models

which have become a very interesting alternative to single point forecasting

methodologies such as the mean-variance model (Markovitz [1959]). The

decomposition of the term structures movements in polynomial orthogonal factors

allows easy generation of interest rates scenarios, even turning it possible to apply the

methodology to dynamic asset-liability models (Cariño et al. [1994]). However, an

important problem involving scenario analysis concerns the choice of the probability

distribution that generates the future scenarios. Although this is not a simple problem,

it could be addressed by applying historical simulation (Jorion [1997]), using the

historical distribution of the term structure factors, or by applying Monte Carlo

simulation (Jorion [1997]) by adjusting a model for the dynamic of the term

structures.

Latin America Eurobond markets were chosen to illustrate the practical use of

the methodology. We explore some simple examples of arbitrage between

international term structures for the same rating, using scenario analysis to select

portfolios. Although the joint estimations realized in the article involve just pairs of

countries (such as Mexico, Brazil and Mexico, Argentina) the joint estimation process

could involve many countries, not necessarily the same number of countries used in

each arbitrage analysis.

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Endnotes

1 The ratings were made public by Bloomberg Agency. They were created as a

combination of Moody’s Investors Service ratings and Standard and Poors ratings.

The rating “NR” denotes eurobonds not rated by both rating agencies. For more

details see the web page www.bloomberg.com.

2 The eurobonds in Exhibit 3 whose names are marked by a star are step-up bonds,

that is, coupon increasing bonds. For each of these bonds, the coupon shown in

Exhibit 3 is the current coupon on June the third of 1998.

3 We constrain the translation factor for the rating NR to be the greatest among all the

translation factors for the different ratings.

4 www.jpmorgan.com.

5 The prices shown in bold on Exhibit 8 were obtained by discounting the Mexican

eurobonds cash flows using the Mexican B1 estimated term structure.

6 The prices shown in bold on Exhibit 9 were obtained by discounting the Brazilian

eurobonds cash flows using the Brazilian B1 estimated term structure.

7 The prices shown in bold on Exhibit 16 were obtained by discounting the

Argentinean eurobonds cash flows using the Argentinean BB2 estimated term

structure.

8 The prices shown in bold on Exhibit 17 were obtained by discounting the Mexican

eurobonds cash flows using the Mexican BB2 estimated term structure.

9 In this context, we consider as a scenario for the BB2 Mexican term structure, a

combination of simultaneous changes in the Mexican rotation and torsion factors.

17

References

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the Movements of Term Structures of Interest Rates in Emerging Eurobonds

Markets”, Journal of Fixed Income, 1 (1998), pp. 21-31.

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Black, F., E.Derman, and W.Toy. “A One-Factor Model of Interest Rates and its

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Cox, J.C., J.E.Ingersoll, and S.A.Ross. “A Theory of the Term Structure of Interest

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Draper,N., and H.Smith. Applied Regression Analysis. New York: Wiley, 1966.

Duffie, D. Dynamic Asset Pricing Theory. Princeton: Princeton University Press,

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Heath, D., R.Jarrow, and A.Morton, “Bond Pricing and the Term Structure of Interest

Rates”, Econometrica, 60 (1992), pp. 77-105.

Ho, T.S.Y., and S.B.Lee, “Term Structure Movements and the Pricing of Interest Rate

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1011-1029.

18

Hull, J., and A.White, “Numerical Procedures for Implementing Term Structure

Models I: Single Factor Models”, Journal of Derivatives, 2 (1994), pp. 7-16.

Jorion, P. Value at Risk. New York: McGraw-Hill, 1997.

Legendre, A.M. Sur l'Attraction des Sphéroides. Mémoires Mathematiques et

Physiques Présentés à l'Acadamie Royal Des Sciences, X, 1785.

Litterman, R. and T.Iben “Corporate Bond Valuation and the Term Structure of Credit

Spreads”, Technical Report, Financial Strategies Series, Goldman Sacks, November

1988.

Litterman, R. and J.A. Scheinkman. “Common Factors Affecting Bond Returns.”,

Journal of Fixed Income, 1 (1991), pp. 54-61.

Markowitz, H.M. Portfolio Selection: Efficient Diversification of Investments. New

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Rousseeuw,P.J., and A.M.Leroy. Robust Regression and Outlier Detection. New

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Singh, M.K. “Value-at-Risk Using Principal Components Analysis.” Journal of

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19

Exhibit 1. An Example of Term Structures for Different Ratings in

Model (4)

BB3

BB2

AA2

interest rate

AA1

term

20

Exhibit 2. An Example of Term Structures for Different Ratings in a

General Model

BB3

BB2

AA2

interest rate

AA1

term

21

Exhibit 3. Eurobonds Used to Illustrate the Estimation Process

Eurobond

2Rating Country Coupon (%) Maturity

Bco Bradeco

B2

Brazil

8.000

28-Jan-2000

Bco Excel B2 Brazil 10.750 08-Nov-2004

Bco Itau B2 Brazil 7.500 11-Jul-2000

Bco Safra B2 Brazil 8.125 10-Nov-2000

Bco Safra B2 Brazil 8.750 28-Oct-2002

Bco Safra B2 Brazil 10.375 28-Oct-2002

CEMIG NR Brazil 9.125 18-Nov-2004

CESP*NR Brazil 9.125 28-Jun-2007

Copel NR Brazil 9.750 02-May-2005

CSN Iron B2 Brazil 9.125 01-Jun-2007

CVRD NR Brazil 10.000 02-Apr-2004

Ford B2 Brazil 9.250 22-Jan-2007

Ford Ltd B2 Brazil 9.125 08-Nov-2004

Gerdau NR Brazil 11.125 24-May-2004

Iochpe NR Brazil 12.375 08-Nov-2002

Ipiranga

*NR Brazil 10.625 25-Feb-2002

Klabin NR Brazil 10.000 20-Dec-2001

Klabin

*NR Brazil 12.750 28-Dec-2002

Lojas NR Brazil 11.000 04-Jun-2004

Minas X WR-A B3 Brazil 7.875 10-Feb-1999

Minas X WR-B B3 Brazil 8.250 10-Feb-2002

Parmalat

*NR Brazil 9.125 02-Jan-2005

RBS B1 Brazil 11.000 01-Apr-2007

Unibanco B2 Brazil 8.000 06-Mar-2000

Votorantim B1 Brazil 8.500 27-Jun-2005

Altos Hornos B2 Mexico 11.375 30-Apr-2002

Altos Hornos B2 Mexico 11.875 30-Apr-2004

Azteca B1 Mexico 10.125 15-Feb-2004

Azteca B1 Mexico 10.500 15-Feb-2007

Banamex BB2 Mexico 9.125 06-Apr-2000

Bufete B3 Mexico 11.375 15-Jul-1999

Cemex BB2 Mexico 8.500 31-Aug-2000

Cemex BB2 Mexico 9.500 20-Sep-2001

Cemex BB2 Mexico 10.000 05-Nov-1999

Cemex BB2 Mexico 10.750 15-Jul-2000

Cemex BB2 Mexico 12.750 15-Jul-2006

Coke FEMSA BB2 Mexico 8.950 01-Nov-2006

Cydsa NR Mexico 9.375 25-Jun-2002

DESC BB3 Mexico 8.750 15-Oct-2007

ELM NR Mexico 11.375 25-Jan-1999

Empresas ICA B1 Mexico 11.875 30-May-2001

Gruma BB1 Mexico 7.625 15-Oct-2007

Grupo IMSA BB2 Mexico 8.930 30-Sep-2004

Grupo Minero BB1 Mexico 8.250 01-Apr-2008

Hylsa BB3 Mexico 9.250 15-Sep-2007

Pepsi-Gemex BB3 Mexico 9.750 30-Mar-2004

Televisa BB2 Mexico 015-May-2008

Televisa BB2 Mexico 11.375 15-May-2003

Televisa BB2 Mexico 11.875 15-May-2006

Tolmex BB2 Mexico 8.375 01-Nov-2003

Vicap B1 Mexico 10.250 15-May-2002

Vicap B1 Mexico 11.375 15-May-2007

22

Exhibit 4. Values of Factors for the Brazilian and Mexican Term

Structures for Different Ratings

Factor3Value (bps)

Brazilian B1 Translation 441

Brazilian B2 Translation 451

Brazilian B3 Translation 475

Brazilian NR Translation 485

Brazilian Rotation 57

Brazilian Torsion -56

Mexican BB1 Translation 249

Mexican BB2 Translation 322

Mexican BB3 Translation 322

Mexican B1 Translation 379

Mexican B2 Translation 509

Mexican B3 Translation 509

Mexican NR Translation 532

Mexican Rotation 156

Mexican Torsion -114

23

Exhibit 5. A Comparison of Mexican and Brazilian Term Structures of

Interest Rates for Different Credit Ratings

0 1 2 3 4 5 6 7 8 9 10

500

600

700

800

900

1000

1100

1200

Years to Maturity

Interest Rate (bps)

B1 Eurobonds Term Structures

US Strips

Brazil

Mexico

0 1 2 3 4 5 6 7 8 9 10

500

600

700

800

900

1000

1100

1200

1300

Years to Maturity

Interest Rate (bps)

B3 Eurobonds Term Structures

US Strips

Mexico

Brazil

24

Exhibit 6. Possible Convergence Scenario

0 1 2 3 4 5 6 7 8 9 10

500

600

700

800

900

1000

1100

Years to Maturity

Interest Rate (bps)

B1 Eurobonds Term Structures

Brazil

Mexico

US Strips

25

Exhibit 7. Different Scenarios for the B1 Term Structures

0 1 2 3 4 5 6 7 8 9 10

500

600

700

800

900

1000

1100

1200

1300

Years to Maturity

Interest Rate (bps)

Brazilian B1 Term Structure

US Strips

Brazil

0 1 2 3 4 5 6 7 8 9 10

500

600

700

800

900

1000

1100

1200

Years to Maturity

Interest Rate (bps)

Mexican B1 Term Structure

US Strips

Mexico

26

Exhibit 8. Price Variation5 of Mexican B1 Eurobonds for Different

Scenarios for the Mexican Term Structure

Rotation

Factor

(variation in bps)

ICA

2001

(US$)

Vicap

2002

(US$)

Azteca

2004

(US$)

Azteca

2007

(US$)

Vicap

2007

(US$)

94 108.6660 110.7541 102.5180 98.4830 79.0877

78 108.4775 108.6085 102.7050 99.7355 84.5002

62 108.2892 106.8840 102.8769 100.9473 90.3572

47 108.1013 105.5128 103.0322 102.1112 95.9951

31 107.9138 104.4372 103.1698 103.2205 100.9502

16 107.7265 103.6072 103.2897 104.2685 104.9842

0

107.5396

102.9799

103.3919

105.2489

108.0515

-16 107.3530 102.5176 103.4772 106.1557 110.2376

-31 107.1668 102.1876 103.5465 106.9832 111.6975

-47 106.9809 101.9613 103.6011 107.7264 112.6066

-62 106.7953 101.8140 103.6425 108.3804 113.1286

-78 106.6101 101.7247 103.6724 108.9411 113.3996

-94 106.4252 101.6756 103.6925 109.4048 113.5225

27

Exhibit 9. Price Variation6 of Brazilian B1 Eurobonds for Different

Scenarios for the Brazilian Term Structure

Translation

Factor

(variation in bps)

Votorantin

2005

(US$)

RBS

2007

(US$)

132 80.8641 98.4083

110 83.1444 99.6497

88 85.4123 100.9116

66 87.6610 102.1943

44 89.8838 103.4982

22 92.0737 104.8238

094.2236 106.1714

-22 96.3265 107.5414

-44 98.3751 108.9342

-66 100.3624 110.3502

-88 102.2811 111.7899

-110 104.1245 113.2537

-132 105.8856 114.7419

28

Exhibit 10. Percent Variation on the B1 Portfolio Value for Different

Scenarios for the Mexican and Brazilian Term Structures

94 78 62 47 31 16 0 -16 -31 -47 -62 -78 -94

132 -8.96 -7.86 -6.89 -6.03 -5.28 -4.62 -4.05 -3.55 -3.12 -2.76 -2.45 -2.20 -2.00

110 -8.27 -7.17 -6.19 -5.34 -4.59 -3.93 -3.36 -2.86 -2.43 -2.07 -1.76 -1.51 -1.31

88 -7.58 -6.48 -5.51 -4.65 -3.90 -3.25 -2.67 -2.17 -1.75 -1.38 -1.07 -0.82 -0.62

66 -6.90 -5.80 -4.82 -3.97 -3.22 -2.56 -1.99 -1.49 -1.06 -0.70 -0.39 -0.14 0.06

44 -6.23 -5.12 -4.15 -3.30 -2.55 -1.89 -1.32 -0.82 -0.39 -0.02 0.28 0.54 0.74

22 -5.56 -4.46 -3.49 -2.63 -1.88 -1.23 -0.65 -0.15 0.27 0.64 0.95 1.20 1.40

0-4.91 -3.81 -2.83 -1.98 -1.23 -0.57 0.00 0.50 0.93 1.29 1.60 1.85 2.05

-22 -4.27 -3.17 -2.20 -1.34 -0.59 0.06 0.64 1.14 1.56 1.93 2.24 2.49 2.69

-44 -3.65 -2.55 -1.58 -0.72 0.03 0.69 1.26 1.76 2.18 2.55 2.86 3.11 3.31

-66 -3.05 -1.95 -0.97 -0.12 0.63 1.29 1.86 2.36 2.79 3.15 3.46 3.71 3.92

-88 -2.47 -1.36 -0.39 0.46 1.21 1.87 2.44 2.94 3.37 3.73 4.04 4.29 4.50

-110

-1.91 -0.80 0.17 1.02 1.77 2.43 3.00 3.50 3.93 4.29 4.60 4.85 5.06

-132

-1.38 -0.27 0.70 1.56 2.31 2.96 3.54 4.03 4.46 4.83 5.13 5.39 5.59

Horizontal - Changes in the Mexican Rotation Factor in Basis Points

Vertical - Changes in the Brazilian Translation Factor in Basis Points

29

Exhibit 11. B1 Portfolio Sensibility for Different Scenarios

-100 -80 -60 -40 -20 0 20 40 60 80 100

-2

-1.5

-1

-0.5

0

0.5

1x 105Different Scenarios for the Mexican B1 Term Structure

Variation in Mexican Rotation Factor (bps)

Price Variation

-150 -100 -50 0 50 100 150

-1.5

-1

-0.5

0

0.5

1

1.5 x 105

Price Variation

Variation in Brazilian Translation Factor (bps)

Different Scenarios for the Brazilian B1 Term Structure

30

Exhibit 12. Detecting Arbitrage Opportunities in the BB2 Argentinean

and Mexican Eurobond Markets

0 1 2 3 4 5 6 7 8 9 10

500

550

600

650

700

750

800

850

900

950

1000

1050

Years to Maturity

Interest Rate (bps)

BB2 Eurobonds Term Structures

US Strips

Argentina

Mexico

31

Exhibit 13. Values of Factors for the Argentinean and Mexican BB2

Term Structures

Factor Value (bps)

Argentinean Translation 291

Argentinean Rotation 98

Argentinean Torsion -57

Mexican Translation 322

Mexican Rotation 153

Mexican Torsion -74

32

Exhibit 14. BB2 Eurobond Portfolio Used in Example 5.3

Eurobond Country Coupon (%) Maturity

Multicanal Argentina 10.500 01-Feb-2007

Perez Argentina 8.125 15-Jul-2007

Cemex Mexico 12.750 15-Jul-2006

Televisa Mexico 11.875 15-May-2006

33

Exhibit 15. Different Scenarios for the BB2 Term Structures

0 1 2 3 4 5 6 7 8 9 10

500

550

600

650

700

750

800

850

900

950

1000

Years to Maturity

Interest Rate (bps)

Argentinean BB2 Term Structure

Argentina

US Strips

0 1 2 3 4 5 6 7 8 9 10

500

600

700

800

900

1000

1100

Years to Maturity

Interest Rate (bps)

Mexican BB2 Term Structure

US Strips

Mexico

34

Exhibit 16. Price Variation7 of the Argentinean Eurobonds in the BB2

Portfolio, for Different Scenarios for the Argentinean Term Structure

Translation

Factor

(variation in bps)

Multicanal

2007

(US$)

Perez

2007

(US$)

0112.5599 97.9793

-7 112.9957 98.3912

-13 113.4337 98.8054

-20 113.8739 99.2217

-27 114.3163 99.6403

-33 114.7608 100.0612

-40 115.2076 100.4842

-47 115.6566 100.9096

-53 116.1079 101.3372

-60 116.5614 101.7670

35

Exhibit 17. Price Variation8 of the Mexican Eurobonds in the BB2

Portfolio, for Different Scenarios9 for the Mexican Term Structure

Rotation Factor

(variation in bps)

Torsion Factor

(variation in bps)

Televisa

2006

(US$)

Cemex

2006

(US$)

0 0 113.4530 122.8172

-6 -4 113.5656 122.9501

-11 -9 113.6788 123.0836

-17 -13 113.7924 123.2177

-22 -18 113.9065 123.3524

-28 -22 114.0210 123.4877

-33 -27 114.1360 123.6236

-39 -31 114.2515 123.7602

-44 -36 114.3675 123.8974

-50 -40 114.4840 124.0352

36

Exhibit 18. Percent Variation on the BB2 Portfolio Value for Different

Scenarios for the Argentinean and Mexican Term Structures

Rotation

Torsion

0-7 -13 -20 -27 -33 -40 -47 -53 -60

0 0 0.00 1.08 2.17 3.26 4.36 5.47 6.58 7.69 8.81 9.94

-6 -4 -0.20 0.89 1.97 3.07 4.17 5.27 6.38 7.50 8.62 9.75

-11 -9 -0.39 0.69 1.78 2.87 3.97 5.07 6.18 7.30 8.42 9.55

-17 -13 -0.59 0.49 1.58 2.67 3.77 4.88 5.99 7.10 8.22 9.35

-22 -18 -0.79 0.29 1.38 2.48 3.57 4.68 5.79 6.90 8.03 9.15

-28 -22 -0.99 0.10 1.18 2.28 3.37 4.48 5.59 6.71 7.83 8.95

-33 -27 -1.19 -0.10 0.98 2.08 3.17 4.28 5.39 6.51 7.63 8.75

-39 -31 -1.39 -0.31 0.78 1.87 2.97 4.08 5.19 6.30 7.43 8.55

-44 -36 -1.59 -0.51 0.58 1.67 2.77 3.88 4.99 6.10 7.22 8.35

-50 -40 -1.79 -0.71 0.38 1.47 2.57 3.67 4.78 5.90 7.02 8.15

Horizontal - Changes in the Argentinean Translation Factor in Basis Points

Vertical – Changes in the Mexican Rotation and Torsion Factors in Basis Points