modeling euro area sovereign bonds - an application

8/2/2019 Modeling Euro Area Sovereign Bonds - An application

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Mathematical Finance Practitioners Seminar

Final Project

Modeling Euro Area Sovereign Bonds

An application

Bas Ummels (su2171)

Ezequiel Zambaglione (eaz2109)


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Introduction

The idea of this Project is to explore the European Bond Market and to perform a trading strategy

to exploit the current uncertainty around its default probabilities. To do that, we select Euro Area

Government bonds. The selected bonds are bonds in euro currency, non-callable, non-putable,

senior unsecured straight bonds with fixed coupons and no sinking fund provisions, havingremaining maturities no longer than 5 years.

Secondly, we built a bond valuation model using the risk free rate as a discount factor and default

probabilities to weigh the cash-flows. We take the default probabilities implied in the bond prices

and use them as an input of the model. After that, we test the model in several Government

Bonds of different Euro Area countries in order to be sure that we can replicate market prices.

Once we are convinced that our model is correct, we analyze the past behavior of some variables

of these countries in order find some patterns that can make us have a better understanding

about the possible future scenarios.

After make the analysis, we present several scenarios based on changes in the parameters of the

model (basically risk free rate and default probabilities) and show how the prices of the bonds

would be if these scenarios actually occurs. Finally we present the conclusions.


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The Model

The model we select for this project uses the risk free rate as a discount factor and the default

probabilities to calculate the cash-flows. For the estimation of the risk free rate, we use a

stochastic process for the short rate.

The default probabilities are obtained from the bond prices providing us with a default probability

for every bond maturity, and then we use a second stochastic process to simulate the event of the

default.

To estimate the risk free rate we use the Ho-Lee (1986) model,

drf= dt + dWt (1)

where rf is the risk free rate, is the drift of the process, is the volatility of the risk free rate and

Wt is a standard Brownian Motion.

Because the bonds we are going to valuate are expressed in euro currency, we choose as a risk free

rate the Euribor rate1. To calibrate the model to market data we use as a drift of the process the

changes in the implicit forward rate of the euro swap curve. By doing this we make the process

arbitrage free. Finally, to estimate the volatility of rf we use historical volatility. As a result, we

obtain the Term Structure of the risk free interest rates in Euros.

Secondly, we extract the value of the parameter lambda, the default probability, using the z-

spread. The z-spread (Sz) is the spread you have to add to the risk free rate structure in order to get

the market price of the bond.

where P is the market price of the bond.

However, the z-spread is not a spread related straight to default probabilities, this spread holds all

the risks that the risky bond has over the risk free rate, for example liquidity, tax benefits, etc. That

is why to get the default spread (d-spread) we have to apply this transformation,

Once the d-spread is obtain, is easy to find the default probabilities implied in every bond and

construct the Term Structure of default probabilities for every country. It is important to remark

that we are working with bonds in Euro, and the CDS in Euro dont have the same liquidity of the

ones in USD, moreover, you are not able to get prices of them every day or for every maturity.

1European Interbank Offered Rate


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The implied survive probabilities for the different countries are the following,

Cumulative Survive Probabilities(Units vs Maturity)

On the other hand, we didnt use the CDS implied default probabilities because with them we

arrived to lower bond price. One explanation we found of it, is that after the Greece event, when

the bond restructuration wasnt trigger an default event, maybe the market put some p remium

risk on it. Another explanation could be counterparty risk, because CDS are OTC (over of the

counter) assets.

Finally, to simulate the time to default, we use a uniform random variable, so in every time period

the value of this function is

td=min{t|u(0,1)>tt}

where u(0,1) is a uniform random variable between 0 and 1 and is the probability of default.

After calibrating and finding all the parameters, we are ready to value the bonds with the model.

The bond price on every path is given by,

[]

[]

where tdis the time to default, tmq is the time to maturity of the bond q, Cq are the coupon

payments of the bond q, []is an indicating function.Because we are using a Monte Carlo method, the price of the bond is the average of all the paths,


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and the error of the model is

Testing the model

The purpose of this section is to test the model to be sure that the bonds prices it calculates are

exactly the market prices. To do that, we use market data to set the parameters2

(bond structure,

rf, volatility of rf,z-spread, we assume a Recovery of 40%3), and with this information we obtain the

model price.

We test the model against 59 bonds, which is our bond sample and includes bonds of Italy (25),

Spain (18), Netherlands (10) and France (6) with maturities lesser than five years. We select these

countries because all of them have a considerable amount of bonds (more than five) and wewanted to have in the sample countries with uncertainty about the future payments of its debt

(Italy and Spain) and others with almost certainty about it (Netherlands and France).

In the figures below one can see the results for Italy and Spain,

Model Bond prices vs Market prices

As we can see, the results of the model are pretty accurate with a statistical error of 0.1759% to

Italy and 0.1898% to Spain.

We reach same results for Netherlands and France. It is important to remark that the implieddefault probabilities for these countries are nearly zero, but we include them because in the next

section we perform an analysis of the impact in their bonds of an increasing in the default

probabilities.

2In this Project all the market data was taking from Bloomberg and the prices are from 03/30/2012.

3This assumption does not change the results, because changes in the recovery rate are going to change the

probabilities of default and the results are nearly the same.


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Model Bond prices vs Market prices

The average error between the model price and the market price for every country was below 1%.

To end this section, we show that the recovery rate dont affect the final prices of the bonds. The

default probabilities under two different recovery values are related through a simple analytical

relationship . Thats why if you change the recovery value, the implied

probabilities are going to change too, but the effect on the bond prices is going to be negligible.

Looking back

The idea of this section is to show the past behavior of the Credit Default Swaps (CDS) spreads for

different countries of the Euro Area. The focus is on the years 2010 (specially the second and third

quarter) and 2011 (third quarter), when the European Sovereign Bond crisis reached its worst

levels.


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To perform the analysis, we decided to pick seven countries in order to see the evolution of the

CDS spread of each of them over time. These countries are Greece, Portugal, Ireland, Spain, Italy,

France and Netherlands. Furthermore, we decided to classify them in four different categories,

During the first quarter of 2010, Portugal, Ireland, Spain and Italy used to live in the second

category (between 100bp and 250bp). However, after a second quarter characterized by high

volatility in all the countries, during the third quarter the first two jump into the third category and

end up the year with a more pronounced positive slope than the second ones, showing the first

differentiation between these countries spread.

On the other hand, Netherlands and France showed a more stable behavior with a small positive

trend at the end of the year, especially France which finished the year in the second category. In

this graph, we can see how the different countries start to differentiate of each other, with

Portugal and Ireland lying in the third category, Spain and Italy in the second one and France and

Netherland in the first one.

Selected European Countries - CDS Spread 2010(Basis Points)

Categories

I 0 100II 100 250

III 250 500

IV 500 >500

CDS spread


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If we move ahead to 2011, the next graph shows some interesting things. First of all, Portugals

and Irelands CDS spreads jump to the fourth category, following the previous year Greeces CDS

spread trend.

Secondly, the CDS spread levels of Spain and Italy lived in the third category most of the second

half of the year, finishing it on that category. Thereby, they practically replicated the behavior ofthe previous year of Portugal and Ireland.

Selected European Countries - CDS Spread 2010/11(Basis Points)

Lastly and maybe more important, the CDS spread of Netherlands and France, definitely broke the

level of the first category (100bp) with a marked positive slope. Furthermore, France ended the

year 2011 close to the upper bound of the second level, with a lot of similarities to Italys 2010

performance.

It is interesting how the countries seem to move in pairs, replicating year after year the patterns of

its neighborhood pair, for example Spain and Italy seem to replicate in 2011 the trajectory of

Portugal and Ireland, and the same for France with Italy.

In summary, we followed the evolution of the CDS spread of most of the principal countries of the

Euro Area. We showed that in the last 2 years the Sovereign Bond Crisis seemed to be transformed

from a crisis in the peripheral economies (Greece, Portugal and Ireland) to important economies,

like Spain and Italy (third and fourth nominal GDP contributors of the Area). Moreover, France (the

second biggest economy) started to show at the end of 2011 the same pattern than Spain and Italy

in 2010.

On the other hand, if we take a look to the interest rates, we can see that the Term Structure

didnt show parallel shift in those years. Instead of that, it showed decrease in the long term rates

IV Category

IIICategory

II Category

I Category


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or flattening on the entire curve. Moreover, especially in 2011, the short rates increased during

the stressing periods (II and III quarters).

Eurpean Yield Curve

2010 2011

Looking forward

In this section, we show the power of the model developed in the first section. Based on the

patterns analyzed in the previous section, we present several possible future scenarios for the

parameters of the model (default probabilities, interest rates, and volatility on interest rates) in

order to take a look at what would be the price changes of the bonds if these scenarios actually

occur in the future.

The idea is to quantify the actual effect that the new set of parameters have on the bond prices.

This quantification let us make the best rational selection of which assets and strategy is better to

use to invest on a particular view about the future values of the parameters.

First of all, we are going to present different scenarios for the default probabilities and its

corresponding bond prices.

Scenario I: Increase in the default probabilities

In this scenario we are going to analyze the effect of an increase in the d-spread on the bond

prices using three different targets, 50bp, 100bp and 250bp.

It is interesting to see how if the movements of the d-spread are small (50bp), the percentualchanges in the bond prices of the different countries show different behavior. For example, the

maximum changes in Netherland bonds prices is around 1% while for Spain this value reaches

more than 2%.

However, if the changes in the d-spread are big (e.g. 250bp), the percentual changes in the bond

prices for all the countries are nearly the same, between 9% and 10%. Apparently, for small

changes in the d-spread, the initial values of the default probabilities play an important roll in the


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bond prices changes, but when the movements in the d-spread are big, the effect of the initial

values become relatively meaningless.

Impact of increase in the d-spread in the bond prices

As a conclusion of this scenario, if you think that there is going to be big movements in the CDS

spreads, the bonds of Italy and Spain are going to converge to similar prices. However, if the

changes are small, the initial conditions are going to play a roll and the bonds of the country withworst initial conditions are going to have worst performance.

So in a scenario of small movements in the same direction, it is possible to perform a strategy

based on the relative initial conditions. The same analysis could be made for Netherlands and

France.

Scenario II: Decrease in the default probabilities

In this scenario we are going to analyze the effect of the decrease in the d-spread on the bond

prices using the same targets as in the previous scenario. However, in this scenario we dont take

Netherlands and France bonds because the default probabilities of these countries are nearly zero,so a decrease on it is not possible.

As we can see in the following graph, the results achieve on the behavior of the bond prices when

the spreads go down are similar to the ones in the previous section. So, we arrive to the same

conclusions.

Impact of increase in the d-spread in the bond prices

Scenario III: Changes in the risk free rate

50 basis point 100 basis point 250 basis point

50 basis point 100 basis point 250 basis point


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In this scenario we are going to analyze the effect of changes in the risk free rate, and present the

effects on the bond prices. We are going to taking into account two possible changes in the Term

Structure of the risk free rate, parallel shifts and changes in the slope.

External shocks to the risk free rate

In the following graph we can see two interest things, the first one is the different effects on the

bond prices of a parallel changes and a flattening (decrease in the slope) in the Yield Curve. While

the effect of a parallel shifts impacts linearly on the different maturities, the effect of a flattening

looks like a quadratic function.

Impact of changes in the risk free rate

The second interest thing that we can observe in this graph is that the impacts of the changes in

the risk free rate on the bond prices, either parallel or flattening, are almost the same for the

different countries, no matter the initial value of their default probabilities.

This is an interesting observation, because this implies that if you want to perform a strategy

based on the relative initial default probabilities of two different countries (small changes in

Parallel shifts Flattening

Parallel shifts Flattening


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default probabilities), this means go long one country bond and short the other one, this strategy

is going to be neutral to changes in risk free rate.

Conclusions

In this paper we introduce a model to price risky bonds using the risk free rate and the defaultprobabilities. Also, using the z-spread, we show a way to find the cumulative default probabilities

implied in the bond prices. The advantage of this methodology is that you can get the default

probability of different countries and currencies. In particular, we did it for Sovereign European

Bonds in Euros.

After that, we present an analysis of the behavior of the CDS spread for the principal European

countries in the last two years, classified them into four groups, according to the level of the

spread in their CDS.

Then we showed that in the beginning, the debt problems were a particular characteristic of

peripheral countries like Greece, Portugal and Ireland, but in the last year (2011), important

economies like Spain and Italy started following similar behavior as the peripheral in 2010.

Moreover, key economies of the Euro Area like France, end up 2011 in the Spain and Italy

beginning of 2011 spread levels, showing a kind of transformation from a peripheral countries

debt crisis to a Euro Area debt Crisis.

In the last section, based on the patterns observed in the previous years, we use the model to

forecast the bond prices in different scenarios of risk free rates and default probabilities. We saw

that big changes in default probabilities reached similar percentual changes in bond price of same

maturities of the different countries, no matter the initial values of default probabilities. However,

if the changes are small, different initial default probabilities follows to different percentual

changes in the bond prices.

Furthermore, changes in the risk free rate, parallel shift as well changes in the slope, leads to

similar changes in the bond prices, suggesting that long/short strategies with bonds of different

countries are neutral to movements in risk free rate.

modeling euro area sovereign bonds - an application

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